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

Percentage Accurate: 69.1% → 99.8%

Time: 7.0s

Alternatives: 19

Speedup: 1.6×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 69.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 0.8× speedup

Math

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

FPCore

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

C

assert(x < y);
double code(double x, double y) {
	return (((y / (x + y)) / ((x + y) - -1.0)) * x) / (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 = (((y / (x + y)) / ((x + y) - (-1.0d0))) * x) / (x + y)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 66.9%

   \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
2. Add Preprocessing
3. 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 91.4

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

4. Applied rewrites 91.4%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-*r/ N/A

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

   4. lift-*.f64 N/A

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

   5. lift-+.f64 N/A

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

   6. +-commutative N/A

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

   7. *-commutative N/A

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

   8. times-frac N/A

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

   9. lower-*.f64 N/A

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

   10. lower-/.f64 N/A

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

   11. lift-+.f64 N/A

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

   12. +-commutative N/A

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

   13. lift-+.f64 N/A

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

   14. lift-+.f64 N/A

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

   15. lift--.f64 N/A

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

   16. lift-+.f64 N/A

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

   17. +-commutative N/A

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

   18. metadata-eval N/A

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

   19. metadata-eval N/A

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

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

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

   21. lift-+.f64 N/A

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

   22. metadata-eval N/A

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

   23. lift-+.f64 N/A

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

6. Applied rewrites 99.7%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. frac-times N/A

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

   5. lift-+.f64 N/A

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

   6. +-commutative N/A

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

   7. lift-+.f64 N/A

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

   8. lift--.f64 N/A

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

   9. lift-+.f64 N/A

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

   10. +-commutative N/A

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

   11. *-commutative N/A

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

   12. lift-+.f64 N/A

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

   13. +-commutative N/A

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

   14. lower-/.f64 N/A

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

   15. lower-*.f64 N/A

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

   16. lift-+.f64 N/A

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

   17. +-commutative N/A

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

   18. lift-+.f64 N/A

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

   19. +-commutative N/A

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

   20. lift-+.f64 N/A

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

8. Applied rewrites 91.4%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-+.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. times-frac N/A

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

   6. associate-*r/ N/A

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

   7. lower-/.f64 N/A

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

   8. lower-*.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. lift-+.f64 99.7

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

10. Applied rewrites 99.7%

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

Alternative 2: 97.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -1.4 \cdot 10^{+154}:\\ \;\;\;\;\frac{y}{y + x} \cdot \frac{1}{x}\\ \mathbf{elif}\;x \leq 2.4 \cdot 10^{+107}:\\ \;\;\;\;\frac{\frac{y}{x + y} \cdot x}{\left(\left(x + y\right) - -1\right) \cdot \left(x + y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x - x \cdot \frac{\mathsf{fma}\left(3, x, 1\right)}{y}}{y}}{y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -1.4e+154)
   (* (/ y (+ y x)) (/ 1.0 x))
   (if (<= x 2.4e+107)
     (/ (* (/ y (+ x y)) x) (* (- (+ x y) -1.0) (+ x y)))
     (/ (/ (- x (* x (/ (fma 3.0 x 1.0) y))) y) y))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -1.4e+154) {
		tmp = (y / (y + x)) * (1.0 / x);
	} else if (x <= 2.4e+107) {
		tmp = ((y / (x + y)) * x) / (((x + y) - -1.0) * (x + y));
	} else {
		tmp = ((x - (x * (fma(3.0, x, 1.0) / y))) / y) / y;
	}
	return tmp;
}

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -1.4e+154)
		tmp = Float64(Float64(y / Float64(y + x)) * Float64(1.0 / x));
	elseif (x <= 2.4e+107)
		tmp = Float64(Float64(Float64(y / Float64(x + y)) * x) / Float64(Float64(Float64(x + y) - -1.0) * Float64(x + y)));
	else
		tmp = Float64(Float64(Float64(x - Float64(x * Float64(fma(3.0, x, 1.0) / y))) / y) / y);
	end
	return tmp
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[x, -1.4e+154], N[(N[(y / N[(y + x), $MachinePrecision]), $MachinePrecision] * N[(1.0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 2.4e+107], N[(N[(N[(y / N[(x + y), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision] / N[(N[(N[(x + y), $MachinePrecision] - -1.0), $MachinePrecision] * N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x - N[(x * N[(N[(3.0 * x + 1.0), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision] / y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.4e154

   1. Initial program 52.2%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing
   3. 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 71.7

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

   4. Applied rewrites 71.7%

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

   5. Taylor expanded in x around inf

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

      1. lower-/.f64 75.8

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

   7. Applied rewrites 75.8%

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

   if -1.4e154 < x < 2.4000000000000001e107

   1. Initial program 71.2%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing
   3. 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 97.7

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

   4. Applied rewrites 97.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lift-*.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. times-frac N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lift-+.f64 N/A

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

      14. lift-+.f64 N/A

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

      15. lift--.f64 N/A

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

      16. lift-+.f64 N/A

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

      17. +-commutative N/A

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

      18. metadata-eval N/A

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

      19. metadata-eval N/A

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

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

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

      21. lift-+.f64 N/A

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

      22. metadata-eval N/A

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

      23. lift-+.f64 N/A

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

   6. Applied rewrites 99.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. frac-times N/A

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-+.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. lift-+.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-/.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lift-+.f64 N/A

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

      17. +-commutative N/A

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

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

      20. lift-+.f64 N/A

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

   8. Applied rewrites 97.7%

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

   if 2.4000000000000001e107 < x

   1. Initial program 57.8%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

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

      1. unpow2 N/A

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

      2. associate-/r* N/A

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

      3. lower-/.f64 N/A

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

   5. Applied rewrites 25.8%

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

Alternative 3: 97.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_0 := \left(x + y\right) - -1\\ \mathbf{if}\;x \leq -1.4 \cdot 10^{+154}:\\ \;\;\;\;\frac{y}{y + x} \cdot \frac{1}{x}\\ \mathbf{elif}\;x \leq 4.7 \cdot 10^{+20}:\\ \;\;\;\;\frac{\frac{y}{x + y} \cdot x}{t\_0 \cdot \left(x + y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x + y} \cdot \frac{x}{t\_0}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (+ x y) -1.0)))
   (if (<= x -1.4e+154)
     (* (/ y (+ y x)) (/ 1.0 x))
     (if (<= x 4.7e+20)
       (/ (* (/ y (+ x y)) x) (* t_0 (+ x y)))
       (* (/ 1.0 (+ x y)) (/ x t_0))))))

C

assert(x < y);
double code(double x, double y) {
	double t_0 = (x + y) - -1.0;
	double tmp;
	if (x <= -1.4e+154) {
		tmp = (y / (y + x)) * (1.0 / x);
	} else if (x <= 4.7e+20) {
		tmp = ((y / (x + y)) * x) / (t_0 * (x + y));
	} else {
		tmp = (1.0 / (x + y)) * (x / t_0);
	}
	return tmp;
}

Fortran

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

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

real(8) function code(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (x + y) - (-1.0d0)
    if (x <= (-1.4d+154)) then
        tmp = (y / (y + x)) * (1.0d0 / x)
    else if (x <= 4.7d+20) then
        tmp = ((y / (x + y)) * x) / (t_0 * (x + y))
    else
        tmp = (1.0d0 / (x + y)) * (x / t_0)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double t_0 = (x + y) - -1.0;
	double tmp;
	if (x <= -1.4e+154) {
		tmp = (y / (y + x)) * (1.0 / x);
	} else if (x <= 4.7e+20) {
		tmp = ((y / (x + y)) * x) / (t_0 * (x + y));
	} else {
		tmp = (1.0 / (x + y)) * (x / t_0);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	t_0 = (x + y) - -1.0
	tmp = 0
	if x <= -1.4e+154:
		tmp = (y / (y + x)) * (1.0 / x)
	elif x <= 4.7e+20:
		tmp = ((y / (x + y)) * x) / (t_0 * (x + y))
	else:
		tmp = (1.0 / (x + y)) * (x / t_0)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	t_0 = Float64(Float64(x + y) - -1.0)
	tmp = 0.0
	if (x <= -1.4e+154)
		tmp = Float64(Float64(y / Float64(y + x)) * Float64(1.0 / x));
	elseif (x <= 4.7e+20)
		tmp = Float64(Float64(Float64(y / Float64(x + y)) * x) / Float64(t_0 * Float64(x + y)));
	else
		tmp = Float64(Float64(1.0 / Float64(x + y)) * Float64(x / t_0));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	t_0 = (x + y) - -1.0;
	tmp = 0.0;
	if (x <= -1.4e+154)
		tmp = (y / (y + x)) * (1.0 / x);
	elseif (x <= 4.7e+20)
		tmp = ((y / (x + y)) * x) / (t_0 * (x + y));
	else
		tmp = (1.0 / (x + y)) * (x / t_0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := Block[{t$95$0 = N[(N[(x + y), $MachinePrecision] - -1.0), $MachinePrecision]}, If[LessEqual[x, -1.4e+154], N[(N[(y / N[(y + x), $MachinePrecision]), $MachinePrecision] * N[(1.0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 4.7e+20], N[(N[(N[(y / N[(x + y), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision] / N[(t$95$0 * N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[(x + y), $MachinePrecision]), $MachinePrecision] * N[(x / t$95$0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.4e154

   1. Initial program 52.2%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing
   3. 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 71.7

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

   4. Applied rewrites 71.7%

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

   5. Taylor expanded in x around inf

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

      1. lower-/.f64 75.8

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

   7. Applied rewrites 75.8%

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

   if -1.4e154 < x < 4.7e20

   1. Initial program 70.9%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing
   3. 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 99.2

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

   4. Applied rewrites 99.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lift-*.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. times-frac N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lift-+.f64 N/A

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

      14. lift-+.f64 N/A

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

      15. lift--.f64 N/A

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

      16. lift-+.f64 N/A

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

      17. +-commutative N/A

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

      18. metadata-eval N/A

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

      19. metadata-eval N/A

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

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

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

      21. lift-+.f64 N/A

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

      22. metadata-eval N/A

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

      23. lift-+.f64 N/A

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

   6. Applied rewrites 99.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. frac-times N/A

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-+.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. lift-+.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-/.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lift-+.f64 N/A

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

      17. +-commutative N/A

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

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

      20. lift-+.f64 N/A

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

   8. Applied rewrites 99.2%

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

   if 4.7e20 < x

   1. Initial program 61.7%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing
   3. 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 77.9

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

   4. Applied rewrites 77.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lift-*.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. times-frac N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lift-+.f64 N/A

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

      14. lift-+.f64 N/A

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

      15. lift--.f64 N/A

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

      16. lift-+.f64 N/A

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

      17. +-commutative N/A

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

      18. metadata-eval N/A

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

      19. metadata-eval N/A

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

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

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

      21. lift-+.f64 N/A

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

      22. metadata-eval N/A

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

      23. lift-+.f64 N/A

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

   6. Applied rewrites 99.7%

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

   7. Taylor expanded in x around 0

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

   9. Applied rewrites 33.8%

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

4. Final simplification 80.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.4 \cdot 10^{+154}:\\ \;\;\;\;\frac{y}{y + x} \cdot \frac{1}{x}\\ \mathbf{elif}\;x \leq 4.7 \cdot 10^{+20}:\\ \;\;\;\;\frac{\frac{y}{x + y} \cdot x}{\left(\left(x + y\right) - -1\right) \cdot \left(x + y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x + y} \cdot \frac{x}{\left(x + y\right) - -1}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 97.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_0 := \frac{y}{y + x}\\ \mathbf{if}\;x \leq -1.4 \cdot 10^{+154}:\\ \;\;\;\;t\_0 \cdot \frac{1}{x}\\ \mathbf{elif}\;x \leq 4.7 \cdot 10^{+20}:\\ \;\;\;\;t\_0 \cdot \frac{x}{\left(\left(y + x\right) - -1\right) \cdot \left(y + x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x + y} \cdot \frac{x}{\left(x + y\right) - -1}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ y (+ y x))))
   (if (<= x -1.4e+154)
     (* t_0 (/ 1.0 x))
     (if (<= x 4.7e+20)
       (* t_0 (/ x (* (- (+ y x) -1.0) (+ y x))))
       (* (/ 1.0 (+ x y)) (/ x (- (+ x y) -1.0)))))))

C

assert(x < y);
double code(double x, double y) {
	double t_0 = y / (y + x);
	double tmp;
	if (x <= -1.4e+154) {
		tmp = t_0 * (1.0 / x);
	} else if (x <= 4.7e+20) {
		tmp = t_0 * (x / (((y + x) - -1.0) * (y + x)));
	} else {
		tmp = (1.0 / (x + y)) * (x / ((x + y) - -1.0));
	}
	return tmp;
}

Fortran

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

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

real(8) function code(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = y / (y + x)
    if (x <= (-1.4d+154)) then
        tmp = t_0 * (1.0d0 / x)
    else if (x <= 4.7d+20) then
        tmp = t_0 * (x / (((y + x) - (-1.0d0)) * (y + x)))
    else
        tmp = (1.0d0 / (x + y)) * (x / ((x + y) - (-1.0d0)))
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double t_0 = y / (y + x);
	double tmp;
	if (x <= -1.4e+154) {
		tmp = t_0 * (1.0 / x);
	} else if (x <= 4.7e+20) {
		tmp = t_0 * (x / (((y + x) - -1.0) * (y + x)));
	} else {
		tmp = (1.0 / (x + y)) * (x / ((x + y) - -1.0));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	t_0 = y / (y + x)
	tmp = 0
	if x <= -1.4e+154:
		tmp = t_0 * (1.0 / x)
	elif x <= 4.7e+20:
		tmp = t_0 * (x / (((y + x) - -1.0) * (y + x)))
	else:
		tmp = (1.0 / (x + y)) * (x / ((x + y) - -1.0))
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	t_0 = Float64(y / Float64(y + x))
	tmp = 0.0
	if (x <= -1.4e+154)
		tmp = Float64(t_0 * Float64(1.0 / x));
	elseif (x <= 4.7e+20)
		tmp = Float64(t_0 * Float64(x / Float64(Float64(Float64(y + x) - -1.0) * Float64(y + x))));
	else
		tmp = Float64(Float64(1.0 / Float64(x + y)) * Float64(x / Float64(Float64(x + y) - -1.0)));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	t_0 = y / (y + x);
	tmp = 0.0;
	if (x <= -1.4e+154)
		tmp = t_0 * (1.0 / x);
	elseif (x <= 4.7e+20)
		tmp = t_0 * (x / (((y + x) - -1.0) * (y + x)));
	else
		tmp = (1.0 / (x + y)) * (x / ((x + y) - -1.0));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := Block[{t$95$0 = N[(y / N[(y + x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.4e+154], N[(t$95$0 * N[(1.0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 4.7e+20], N[(t$95$0 * N[(x / N[(N[(N[(y + x), $MachinePrecision] - -1.0), $MachinePrecision] * N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[(x + y), $MachinePrecision]), $MachinePrecision] * N[(x / N[(N[(x + y), $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.4e154

   1. Initial program 52.2%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing
   3. 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 71.7

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

   4. Applied rewrites 71.7%

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

   5. Taylor expanded in x around inf

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

      1. lower-/.f64 75.8

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

   7. Applied rewrites 75.8%

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

   if -1.4e154 < x < 4.7e20

   1. Initial program 70.9%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing
   3. 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 99.2

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

   4. Applied rewrites 99.2%

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

   if 4.7e20 < x

   1. Initial program 61.7%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing
   3. 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 77.9

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

   4. Applied rewrites 77.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lift-*.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. times-frac N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lift-+.f64 N/A

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

      14. lift-+.f64 N/A

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

      15. lift--.f64 N/A

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

      16. lift-+.f64 N/A

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

      17. +-commutative N/A

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

      18. metadata-eval N/A

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

      19. metadata-eval N/A

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

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

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

      21. lift-+.f64 N/A

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

      22. metadata-eval N/A

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

      23. lift-+.f64 N/A

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

   6. Applied rewrites 99.7%

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

   7. Taylor expanded in x around 0

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

   9. Applied rewrites 33.8%

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

4. Final simplification 80.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.4 \cdot 10^{+154}:\\ \;\;\;\;\frac{y}{y + x} \cdot \frac{1}{x}\\ \mathbf{elif}\;x \leq 4.7 \cdot 10^{+20}:\\ \;\;\;\;\frac{y}{y + x} \cdot \frac{x}{\left(\left(y + x\right) - -1\right) \cdot \left(y + x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x + y} \cdot \frac{x}{\left(x + y\right) - -1}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 96.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_0 := \frac{y}{x + y}\\ \mathbf{if}\;y \leq 6.8 \cdot 10^{+113}:\\ \;\;\;\;\frac{t\_0 \cdot x}{\left(\left(x + y\right) - -1\right) \cdot \left(x + y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0}{x + y} \cdot \frac{x}{1 + y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ y (+ x y))))
   (if (<= y 6.8e+113)
     (/ (* t_0 x) (* (- (+ x y) -1.0) (+ x y)))
     (* (/ t_0 (+ x y)) (/ x (+ 1.0 y))))))

C

assert(x < y);
double code(double x, double y) {
	double t_0 = y / (x + y);
	double tmp;
	if (y <= 6.8e+113) {
		tmp = (t_0 * x) / (((x + y) - -1.0) * (x + y));
	} else {
		tmp = (t_0 / (x + y)) * (x / (1.0 + y));
	}
	return tmp;
}

Fortran

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

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

real(8) function code(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = y / (x + y)
    if (y <= 6.8d+113) then
        tmp = (t_0 * x) / (((x + y) - (-1.0d0)) * (x + y))
    else
        tmp = (t_0 / (x + y)) * (x / (1.0d0 + y))
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double t_0 = y / (x + y);
	double tmp;
	if (y <= 6.8e+113) {
		tmp = (t_0 * x) / (((x + y) - -1.0) * (x + y));
	} else {
		tmp = (t_0 / (x + y)) * (x / (1.0 + y));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	t_0 = y / (x + y)
	tmp = 0
	if y <= 6.8e+113:
		tmp = (t_0 * x) / (((x + y) - -1.0) * (x + y))
	else:
		tmp = (t_0 / (x + y)) * (x / (1.0 + y))
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	t_0 = Float64(y / Float64(x + y))
	tmp = 0.0
	if (y <= 6.8e+113)
		tmp = Float64(Float64(t_0 * x) / Float64(Float64(Float64(x + y) - -1.0) * Float64(x + y)));
	else
		tmp = Float64(Float64(t_0 / Float64(x + y)) * Float64(x / Float64(1.0 + y)));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	t_0 = y / (x + y);
	tmp = 0.0;
	if (y <= 6.8e+113)
		tmp = (t_0 * x) / (((x + y) - -1.0) * (x + y));
	else
		tmp = (t_0 / (x + y)) * (x / (1.0 + y));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := Block[{t$95$0 = N[(y / N[(x + y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, 6.8e+113], N[(N[(t$95$0 * x), $MachinePrecision] / N[(N[(N[(x + y), $MachinePrecision] - -1.0), $MachinePrecision] * N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$0 / N[(x + y), $MachinePrecision]), $MachinePrecision] * N[(x / N[(1.0 + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < 6.80000000000000038e113

   1. Initial program 71.1%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing
   3. 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 95.8

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

   4. Applied rewrites 95.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lift-*.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. times-frac N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lift-+.f64 N/A

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

      14. lift-+.f64 N/A

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

      15. lift--.f64 N/A

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

      16. lift-+.f64 N/A

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

      17. +-commutative N/A

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

      18. metadata-eval N/A

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

      19. metadata-eval N/A

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

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

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

      21. lift-+.f64 N/A

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

      22. metadata-eval N/A

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

      23. lift-+.f64 N/A

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

   6. Applied rewrites 99.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. frac-times N/A

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-+.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. lift-+.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-/.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lift-+.f64 N/A

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

      17. +-commutative N/A

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

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

      20. lift-+.f64 N/A

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

   8. Applied rewrites 95.9%

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

   if 6.80000000000000038e113 < y

   1. Initial program 45.7%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing
   3. 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 68.6

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

   4. Applied rewrites 68.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lift-*.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. times-frac N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lift-+.f64 N/A

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

      14. lift-+.f64 N/A

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

      15. lift--.f64 N/A

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

      16. lift-+.f64 N/A

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

      17. +-commutative N/A

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

      18. metadata-eval N/A

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

      19. metadata-eval N/A

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

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

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

      21. lift-+.f64 N/A

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

      22. metadata-eval N/A

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

      23. lift-+.f64 N/A

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

   6. Applied rewrites 99.6%

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

   7. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 88.5

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

   9. Applied rewrites 88.5%

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

Alternative 6: 88.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -6 \cdot 10^{+98}:\\ \;\;\;\;\frac{y}{y + x} \cdot \frac{1}{x}\\ \mathbf{elif}\;x \leq -1.1 \cdot 10^{-146}:\\ \;\;\;\;\frac{x \cdot y}{\left(y + x\right) \cdot \left(\left(\left(y + x\right) - -1\right) \cdot \left(y + x\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y} \cdot \frac{x}{\left(x + y\right) - -1}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -6e+98)
   (* (/ y (+ y x)) (/ 1.0 x))
   (if (<= x -1.1e-146)
     (/ (* x y) (* (+ y x) (* (- (+ y x) -1.0) (+ y x))))
     (* (/ 1.0 y) (/ x (- (+ x y) -1.0))))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -6e+98) {
		tmp = (y / (y + x)) * (1.0 / x);
	} else if (x <= -1.1e-146) {
		tmp = (x * y) / ((y + x) * (((y + x) - -1.0) * (y + x)));
	} else {
		tmp = (1.0 / y) * (x / ((x + y) - -1.0));
	}
	return tmp;
}

Fortran

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

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

real(8) function code(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-6d+98)) then
        tmp = (y / (y + x)) * (1.0d0 / x)
    else if (x <= (-1.1d-146)) then
        tmp = (x * y) / ((y + x) * (((y + x) - (-1.0d0)) * (y + x)))
    else
        tmp = (1.0d0 / y) * (x / ((x + y) - (-1.0d0)))
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (x <= -6e+98) {
		tmp = (y / (y + x)) * (1.0 / x);
	} else if (x <= -1.1e-146) {
		tmp = (x * y) / ((y + x) * (((y + x) - -1.0) * (y + x)));
	} else {
		tmp = (1.0 / y) * (x / ((x + y) - -1.0));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -6e+98:
		tmp = (y / (y + x)) * (1.0 / x)
	elif x <= -1.1e-146:
		tmp = (x * y) / ((y + x) * (((y + x) - -1.0) * (y + x)))
	else:
		tmp = (1.0 / y) * (x / ((x + y) - -1.0))
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -6e+98)
		tmp = Float64(Float64(y / Float64(y + x)) * Float64(1.0 / x));
	elseif (x <= -1.1e-146)
		tmp = Float64(Float64(x * y) / Float64(Float64(y + x) * Float64(Float64(Float64(y + x) - -1.0) * Float64(y + x))));
	else
		tmp = Float64(Float64(1.0 / y) * Float64(x / Float64(Float64(x + y) - -1.0)));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -6e+98)
		tmp = (y / (y + x)) * (1.0 / x);
	elseif (x <= -1.1e-146)
		tmp = (x * y) / ((y + x) * (((y + x) - -1.0) * (y + x)));
	else
		tmp = (1.0 / y) * (x / ((x + y) - -1.0));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[x, -6e+98], N[(N[(y / N[(y + x), $MachinePrecision]), $MachinePrecision] * N[(1.0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -1.1e-146], N[(N[(x * y), $MachinePrecision] / N[(N[(y + x), $MachinePrecision] * N[(N[(N[(y + x), $MachinePrecision] - -1.0), $MachinePrecision] * N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / y), $MachinePrecision] * N[(x / N[(N[(x + y), $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -6.0000000000000003e98

   1. Initial program 38.4%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing
   3. 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 80.2

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

   4. Applied rewrites 80.2%

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

   5. Taylor expanded in x around inf

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

      1. lower-/.f64 72.2

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

   7. Applied rewrites 72.2%

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

   if -6.0000000000000003e98 < x < -1.1e-146

   1. Initial program 89.6%

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

      1. lift-*.f64 N/A

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

      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. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 89.6

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

      10. lift-+.f64 N/A

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

      11. metadata-eval N/A

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

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

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

      13. metadata-eval N/A

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

      14. metadata-eval N/A

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

      15. metadata-eval N/A

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

      16. lower--.f64 N/A

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

      17. lift-+.f64 N/A

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

      18. +-commutative N/A

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

      19. lower-+.f64 N/A

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

      20. metadata-eval 89.6

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

      21. lift-+.f64 N/A

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

      22. +-commutative N/A

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

      23. lower-+.f64 89.6

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

   4. Applied rewrites 89.6%

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

   if -1.1e-146 < x

   1. Initial program 67.1%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing
   3. 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 91.6

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

   4. Applied rewrites 91.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lift-*.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. times-frac N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lift-+.f64 N/A

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

      14. lift-+.f64 N/A

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

      15. lift--.f64 N/A

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

      16. lift-+.f64 N/A

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

      17. +-commutative N/A

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

      18. metadata-eval N/A

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

      19. metadata-eval N/A

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

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

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

      21. lift-+.f64 N/A

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

      22. metadata-eval N/A

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

      23. lift-+.f64 N/A

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

   6. Applied rewrites 99.7%

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

   7. Taylor expanded in x around 0

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

      1. lower-/.f64 59.1

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

   9. Applied rewrites 59.1%

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

Alternative 7: 88.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -6 \cdot 10^{+98}:\\ \;\;\;\;\frac{y}{y + x} \cdot \frac{1}{x}\\ \mathbf{elif}\;x \leq -1.1 \cdot 10^{-146}:\\ \;\;\;\;\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y} \cdot \frac{x}{\left(x + y\right) - -1}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -6e+98)
   (* (/ y (+ y x)) (/ 1.0 x))
   (if (<= x -1.1e-146)
     (/ (* x y) (* (* (+ x y) (+ x y)) (+ (+ x y) 1.0)))
     (* (/ 1.0 y) (/ x (- (+ x y) -1.0))))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -6e+98) {
		tmp = (y / (y + x)) * (1.0 / x);
	} else if (x <= -1.1e-146) {
		tmp = (x * y) / (((x + y) * (x + y)) * ((x + y) + 1.0));
	} else {
		tmp = (1.0 / y) * (x / ((x + y) - -1.0));
	}
	return tmp;
}

Fortran

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

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

real(8) function code(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-6d+98)) then
        tmp = (y / (y + x)) * (1.0d0 / x)
    else if (x <= (-1.1d-146)) then
        tmp = (x * y) / (((x + y) * (x + y)) * ((x + y) + 1.0d0))
    else
        tmp = (1.0d0 / y) * (x / ((x + y) - (-1.0d0)))
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (x <= -6e+98) {
		tmp = (y / (y + x)) * (1.0 / x);
	} else if (x <= -1.1e-146) {
		tmp = (x * y) / (((x + y) * (x + y)) * ((x + y) + 1.0));
	} else {
		tmp = (1.0 / y) * (x / ((x + y) - -1.0));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -6e+98:
		tmp = (y / (y + x)) * (1.0 / x)
	elif x <= -1.1e-146:
		tmp = (x * y) / (((x + y) * (x + y)) * ((x + y) + 1.0))
	else:
		tmp = (1.0 / y) * (x / ((x + y) - -1.0))
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -6e+98)
		tmp = Float64(Float64(y / Float64(y + x)) * Float64(1.0 / x));
	elseif (x <= -1.1e-146)
		tmp = Float64(Float64(x * y) / Float64(Float64(Float64(x + y) * Float64(x + y)) * Float64(Float64(x + y) + 1.0)));
	else
		tmp = Float64(Float64(1.0 / y) * Float64(x / Float64(Float64(x + y) - -1.0)));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -6e+98)
		tmp = (y / (y + x)) * (1.0 / x);
	elseif (x <= -1.1e-146)
		tmp = (x * y) / (((x + y) * (x + y)) * ((x + y) + 1.0));
	else
		tmp = (1.0 / y) * (x / ((x + y) - -1.0));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[x, -6e+98], N[(N[(y / N[(y + x), $MachinePrecision]), $MachinePrecision] * N[(1.0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -1.1e-146], N[(N[(x * y), $MachinePrecision] / N[(N[(N[(x + y), $MachinePrecision] * N[(x + y), $MachinePrecision]), $MachinePrecision] * N[(N[(x + y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / y), $MachinePrecision] * N[(x / N[(N[(x + y), $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -6.0000000000000003e98

   1. Initial program 38.4%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing
   3. 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 80.2

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

   4. Applied rewrites 80.2%

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

   5. Taylor expanded in x around inf

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

      1. lower-/.f64 72.2

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

   7. Applied rewrites 72.2%

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

   if -6.0000000000000003e98 < x < -1.1e-146

   1. Initial program 89.6%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing

   if -1.1e-146 < x

   1. Initial program 67.1%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing
   3. 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 91.6

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

   4. Applied rewrites 91.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lift-*.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. times-frac N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lift-+.f64 N/A

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

      14. lift-+.f64 N/A

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

      15. lift--.f64 N/A

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

      16. lift-+.f64 N/A

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

      17. +-commutative N/A

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

      18. metadata-eval N/A

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

      19. metadata-eval N/A

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

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

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

      21. lift-+.f64 N/A

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

      22. metadata-eval N/A

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

      23. lift-+.f64 N/A

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

   6. Applied rewrites 99.7%

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

   7. Taylor expanded in x around 0

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

      1. lower-/.f64 59.1

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

   9. Applied rewrites 59.1%

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

Alternative 8: 86.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_0 := \left(x + y\right) - -1\\ \mathbf{if}\;x \leq -5.5 \cdot 10^{+102}:\\ \;\;\;\;\frac{y}{y + x} \cdot \frac{1}{x}\\ \mathbf{elif}\;x \leq -1.6 \cdot 10^{-159}:\\ \;\;\;\;y \cdot \frac{x}{t\_0 \cdot \left(\mathsf{fma}\left(2, y, x\right) \cdot x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y} \cdot \frac{x}{t\_0}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (+ x y) -1.0)))
   (if (<= x -5.5e+102)
     (* (/ y (+ y x)) (/ 1.0 x))
     (if (<= x -1.6e-159)
       (* y (/ x (* t_0 (* (fma 2.0 y x) x))))
       (* (/ 1.0 y) (/ x t_0))))))

C

assert(x < y);
double code(double x, double y) {
	double t_0 = (x + y) - -1.0;
	double tmp;
	if (x <= -5.5e+102) {
		tmp = (y / (y + x)) * (1.0 / x);
	} else if (x <= -1.6e-159) {
		tmp = y * (x / (t_0 * (fma(2.0, y, x) * x)));
	} else {
		tmp = (1.0 / y) * (x / t_0);
	}
	return tmp;
}

Julia

x, y = sort([x, y])
function code(x, y)
	t_0 = Float64(Float64(x + y) - -1.0)
	tmp = 0.0
	if (x <= -5.5e+102)
		tmp = Float64(Float64(y / Float64(y + x)) * Float64(1.0 / x));
	elseif (x <= -1.6e-159)
		tmp = Float64(y * Float64(x / Float64(t_0 * Float64(fma(2.0, y, x) * x))));
	else
		tmp = Float64(Float64(1.0 / y) * Float64(x / t_0));
	end
	return tmp
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := Block[{t$95$0 = N[(N[(x + y), $MachinePrecision] - -1.0), $MachinePrecision]}, If[LessEqual[x, -5.5e+102], N[(N[(y / N[(y + x), $MachinePrecision]), $MachinePrecision] * N[(1.0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -1.6e-159], N[(y * N[(x / N[(t$95$0 * N[(N[(2.0 * y + x), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / y), $MachinePrecision] * N[(x / t$95$0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -5.49999999999999981e102

   1. Initial program 37.6%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing
   3. 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 79.2

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

   4. Applied rewrites 79.2%

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

   5. Taylor expanded in x around inf

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

      1. lower-/.f64 73.3

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

   7. Applied rewrites 73.3%

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

   if -5.49999999999999981e102 < x < -1.6e-159

   1. Initial program 86.4%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. distribute-rgt-in N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-fma.f64 60.6

         \[\leadsto \frac{x \cdot y}{\left(\color{blue}{\mathsf{fma}\left(2, y, x\right)} \cdot x\right) \cdot \left(\left(x + y\right) + 1\right)} \]

   5. Applied rewrites 60.6%

      \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\mathsf{fma}\left(2, y, x\right) \cdot x\right)} \cdot \left(\left(x + y\right) + 1\right)} \]
   6. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x \cdot y}{\left(\mathsf{fma}\left(2, y, x\right) \cdot x\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot y}}{\left(\mathsf{fma}\left(2, y, x\right) \cdot x\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      3. *-commutative N/A

         \[\leadsto \frac{\color{blue}{y \cdot x}}{\left(\mathsf{fma}\left(2, y, x\right) \cdot x\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      4. associate-/l* N/A

         \[\leadsto \color{blue}{y \cdot \frac{x}{\left(\mathsf{fma}\left(2, y, x\right) \cdot x\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{y \cdot \frac{x}{\left(\mathsf{fma}\left(2, y, x\right) \cdot x\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      6. lower-/.f64 69.8

         \[\leadsto y \cdot \color{blue}{\frac{x}{\left(\mathsf{fma}\left(2, y, x\right) \cdot x\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      7. lift-*.f64 N/A

         \[\leadsto y \cdot \frac{x}{\color{blue}{\left(\mathsf{fma}\left(2, y, x\right) \cdot x\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      8. *-commutative N/A

         \[\leadsto y \cdot \frac{x}{\color{blue}{\left(\left(x + y\right) + 1\right) \cdot \left(\mathsf{fma}\left(2, y, x\right) \cdot x\right)}} \]

      9. lower-*.f64 69.8

         \[\leadsto y \cdot \frac{x}{\color{blue}{\left(\left(x + y\right) + 1\right) \cdot \left(\mathsf{fma}\left(2, y, x\right) \cdot x\right)}} \]

      10. lift-+.f64 N/A

          \[\leadsto y \cdot \frac{x}{\color{blue}{\left(\left(x + y\right) + 1\right)} \cdot \left(\mathsf{fma}\left(2, y, x\right) \cdot x\right)} \]

      11. lift-+.f64 N/A

          \[\leadsto y \cdot \frac{x}{\left(\color{blue}{\left(x + y\right)} + 1\right) \cdot \left(\mathsf{fma}\left(2, y, x\right) \cdot x\right)} \]

      12. metadata-eval N/A

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

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

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

      14. metadata-eval N/A

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

      15. metadata-eval N/A

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

      16. lower--.f64 N/A

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

      17. lift-+.f64 69.8

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

   7. Applied rewrites 69.8%

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

   if -1.6e-159 < x

   1. Initial program 67.5%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing
   3. 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 91.5

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

   4. Applied rewrites 91.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lift-*.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. times-frac N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lift-+.f64 N/A

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

      14. lift-+.f64 N/A

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

      15. lift--.f64 N/A

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

      16. lift-+.f64 N/A

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

      17. +-commutative N/A

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

      18. metadata-eval N/A

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

      19. metadata-eval N/A

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

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

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

      21. lift-+.f64 N/A

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

      22. metadata-eval N/A

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

      23. lift-+.f64 N/A

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

   6. Applied rewrites 99.7%

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

   7. Taylor expanded in x around 0

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

      1. lower-/.f64 59.4

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

   9. Applied rewrites 59.4%

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

Alternative 9: 99.8% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 66.9%

   \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
2. Add Preprocessing
3. 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 91.4

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

4. Applied rewrites 91.4%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-*r/ N/A

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

   4. lift-*.f64 N/A

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

   5. lift-+.f64 N/A

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

   6. +-commutative N/A

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

   7. *-commutative N/A

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

   8. times-frac N/A

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

   9. lower-*.f64 N/A

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

   10. lower-/.f64 N/A

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

   11. lift-+.f64 N/A

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

   12. +-commutative N/A

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

   13. lift-+.f64 N/A

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

   14. lift-+.f64 N/A

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

   15. lift--.f64 N/A

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

   16. lift-+.f64 N/A

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

   17. +-commutative N/A

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

   18. metadata-eval N/A

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

   19. metadata-eval N/A

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

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

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

   21. lift-+.f64 N/A

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

   22. metadata-eval N/A

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

   23. lift-+.f64 N/A

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

6. Applied rewrites 99.7%

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

Alternative 10: 87.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 9 \cdot 10^{-149}:\\ \;\;\;\;\frac{y}{y + x} \cdot \frac{1}{1 + x}\\ \mathbf{elif}\;y \leq 1.85 \cdot 10^{+131}:\\ \;\;\;\;1 \cdot \frac{x}{\left(\left(y + x\right) - -1\right) \cdot \left(y + x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x + y} \cdot \frac{x}{\left(x + y\right) - -1}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 9e-149)
   (* (/ y (+ y x)) (/ 1.0 (+ 1.0 x)))
   (if (<= y 1.85e+131)
     (* 1.0 (/ x (* (- (+ y x) -1.0) (+ y x))))
     (* (/ 1.0 (+ x y)) (/ x (- (+ x y) -1.0))))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 9e-149) {
		tmp = (y / (y + x)) * (1.0 / (1.0 + x));
	} else if (y <= 1.85e+131) {
		tmp = 1.0 * (x / (((y + x) - -1.0) * (y + x)));
	} else {
		tmp = (1.0 / (x + y)) * (x / ((x + y) - -1.0));
	}
	return tmp;
}

Fortran

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

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

real(8) function code(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 9d-149) then
        tmp = (y / (y + x)) * (1.0d0 / (1.0d0 + x))
    else if (y <= 1.85d+131) then
        tmp = 1.0d0 * (x / (((y + x) - (-1.0d0)) * (y + x)))
    else
        tmp = (1.0d0 / (x + y)) * (x / ((x + y) - (-1.0d0)))
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (y <= 9e-149) {
		tmp = (y / (y + x)) * (1.0 / (1.0 + x));
	} else if (y <= 1.85e+131) {
		tmp = 1.0 * (x / (((y + x) - -1.0) * (y + x)));
	} else {
		tmp = (1.0 / (x + y)) * (x / ((x + y) - -1.0));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if y <= 9e-149:
		tmp = (y / (y + x)) * (1.0 / (1.0 + x))
	elif y <= 1.85e+131:
		tmp = 1.0 * (x / (((y + x) - -1.0) * (y + x)))
	else:
		tmp = (1.0 / (x + y)) * (x / ((x + y) - -1.0))
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= 9e-149)
		tmp = Float64(Float64(y / Float64(y + x)) * Float64(1.0 / Float64(1.0 + x)));
	elseif (y <= 1.85e+131)
		tmp = Float64(1.0 * Float64(x / Float64(Float64(Float64(y + x) - -1.0) * Float64(y + x))));
	else
		tmp = Float64(Float64(1.0 / Float64(x + y)) * Float64(x / Float64(Float64(x + y) - -1.0)));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 9e-149)
		tmp = (y / (y + x)) * (1.0 / (1.0 + x));
	elseif (y <= 1.85e+131)
		tmp = 1.0 * (x / (((y + x) - -1.0) * (y + x)));
	else
		tmp = (1.0 / (x + y)) * (x / ((x + y) - -1.0));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[y, 9e-149], N[(N[(y / N[(y + x), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.85e+131], N[(1.0 * N[(x / N[(N[(N[(y + x), $MachinePrecision] - -1.0), $MachinePrecision] * N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[(x + y), $MachinePrecision]), $MachinePrecision] * N[(x / N[(N[(x + y), $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 8.9999999999999996e-149

   1. Initial program 68.6%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing
   3. 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 94.4

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

   4. Applied rewrites 94.4%

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

   5. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 57.2

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

   7. Applied rewrites 57.2%

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

   if 8.9999999999999996e-149 < y < 1.84999999999999998e131

   1. Initial program 76.8%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing
   3. 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 99.8

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 65.4%

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

   if 1.84999999999999998e131 < y

   1. Initial program 46.8%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing
   3. 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 67.8

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

   4. Applied rewrites 67.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lift-*.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. times-frac N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lift-+.f64 N/A

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

      14. lift-+.f64 N/A

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

      15. lift--.f64 N/A

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

      16. lift-+.f64 N/A

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

      17. +-commutative N/A

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

      18. metadata-eval N/A

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

      19. metadata-eval N/A

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

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

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

      21. lift-+.f64 N/A

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

      22. metadata-eval N/A

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

      23. lift-+.f64 N/A

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

   6. Applied rewrites 99.6%

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

   7. Taylor expanded in x around 0

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

   9. Applied rewrites 86.0%

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

4. Final simplification 63.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 9 \cdot 10^{-149}:\\ \;\;\;\;\frac{y}{y + x} \cdot \frac{1}{1 + x}\\ \mathbf{elif}\;y \leq 1.85 \cdot 10^{+131}:\\ \;\;\;\;1 \cdot \frac{x}{\left(\left(y + x\right) - -1\right) \cdot \left(y + x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x + y} \cdot \frac{x}{\left(x + y\right) - -1}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 87.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 9 \cdot 10^{-149}:\\ \;\;\;\;\frac{y}{y + x} \cdot \frac{1}{1 + x}\\ \mathbf{elif}\;y \leq 1.85 \cdot 10^{+131}:\\ \;\;\;\;1 \cdot \frac{x}{\left(\left(y + x\right) - -1\right) \cdot \left(y + x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y} \cdot \frac{x}{\left(x + y\right) - -1}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 9e-149)
   (* (/ y (+ y x)) (/ 1.0 (+ 1.0 x)))
   (if (<= y 1.85e+131)
     (* 1.0 (/ x (* (- (+ y x) -1.0) (+ y x))))
     (* (/ 1.0 y) (/ x (- (+ x y) -1.0))))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 9e-149) {
		tmp = (y / (y + x)) * (1.0 / (1.0 + x));
	} else if (y <= 1.85e+131) {
		tmp = 1.0 * (x / (((y + x) - -1.0) * (y + x)));
	} else {
		tmp = (1.0 / y) * (x / ((x + y) - -1.0));
	}
	return tmp;
}

Fortran

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

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

real(8) function code(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 9d-149) then
        tmp = (y / (y + x)) * (1.0d0 / (1.0d0 + x))
    else if (y <= 1.85d+131) then
        tmp = 1.0d0 * (x / (((y + x) - (-1.0d0)) * (y + x)))
    else
        tmp = (1.0d0 / y) * (x / ((x + y) - (-1.0d0)))
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (y <= 9e-149) {
		tmp = (y / (y + x)) * (1.0 / (1.0 + x));
	} else if (y <= 1.85e+131) {
		tmp = 1.0 * (x / (((y + x) - -1.0) * (y + x)));
	} else {
		tmp = (1.0 / y) * (x / ((x + y) - -1.0));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if y <= 9e-149:
		tmp = (y / (y + x)) * (1.0 / (1.0 + x))
	elif y <= 1.85e+131:
		tmp = 1.0 * (x / (((y + x) - -1.0) * (y + x)))
	else:
		tmp = (1.0 / y) * (x / ((x + y) - -1.0))
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= 9e-149)
		tmp = Float64(Float64(y / Float64(y + x)) * Float64(1.0 / Float64(1.0 + x)));
	elseif (y <= 1.85e+131)
		tmp = Float64(1.0 * Float64(x / Float64(Float64(Float64(y + x) - -1.0) * Float64(y + x))));
	else
		tmp = Float64(Float64(1.0 / y) * Float64(x / Float64(Float64(x + y) - -1.0)));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 9e-149)
		tmp = (y / (y + x)) * (1.0 / (1.0 + x));
	elseif (y <= 1.85e+131)
		tmp = 1.0 * (x / (((y + x) - -1.0) * (y + x)));
	else
		tmp = (1.0 / y) * (x / ((x + y) - -1.0));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[y, 9e-149], N[(N[(y / N[(y + x), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.85e+131], N[(1.0 * N[(x / N[(N[(N[(y + x), $MachinePrecision] - -1.0), $MachinePrecision] * N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / y), $MachinePrecision] * N[(x / N[(N[(x + y), $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 8.9999999999999996e-149

   1. Initial program 68.6%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing
   3. 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 94.4

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

   4. Applied rewrites 94.4%

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

   5. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 57.2

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

   7. Applied rewrites 57.2%

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

   if 8.9999999999999996e-149 < y < 1.84999999999999998e131

   1. Initial program 76.8%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing
   3. 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 99.8

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 65.4%

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

   if 1.84999999999999998e131 < y

   1. Initial program 46.8%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing
   3. 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 67.8

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

   4. Applied rewrites 67.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lift-*.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. times-frac N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lift-+.f64 N/A

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

      14. lift-+.f64 N/A

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

      15. lift--.f64 N/A

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

      16. lift-+.f64 N/A

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

      17. +-commutative N/A

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

      18. metadata-eval N/A

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

      19. metadata-eval N/A

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

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

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

      21. lift-+.f64 N/A

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

      22. metadata-eval N/A

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

      23. lift-+.f64 N/A

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

   6. Applied rewrites 99.6%

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

   7. Taylor expanded in x around 0

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

      1. lower-/.f64 85.6

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

   9. Applied rewrites 85.6%

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

Alternative 12: 85.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 9 \cdot 10^{-149}:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(x, x, x\right)}\\ \mathbf{elif}\;y \leq 1.85 \cdot 10^{+131}:\\ \;\;\;\;1 \cdot \frac{x}{\left(\left(y + x\right) - -1\right) \cdot \left(y + x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y} \cdot \frac{x}{\left(x + y\right) - -1}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 9e-149)
   (/ y (fma x x x))
   (if (<= y 1.85e+131)
     (* 1.0 (/ x (* (- (+ y x) -1.0) (+ y x))))
     (* (/ 1.0 y) (/ x (- (+ x y) -1.0))))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 9e-149) {
		tmp = y / fma(x, x, x);
	} else if (y <= 1.85e+131) {
		tmp = 1.0 * (x / (((y + x) - -1.0) * (y + x)));
	} else {
		tmp = (1.0 / y) * (x / ((x + y) - -1.0));
	}
	return tmp;
}

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= 9e-149)
		tmp = Float64(y / fma(x, x, x));
	elseif (y <= 1.85e+131)
		tmp = Float64(1.0 * Float64(x / Float64(Float64(Float64(y + x) - -1.0) * Float64(y + x))));
	else
		tmp = Float64(Float64(1.0 / y) * Float64(x / Float64(Float64(x + y) - -1.0)));
	end
	return tmp
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[y, 9e-149], N[(y / N[(x * x + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.85e+131], N[(1.0 * N[(x / N[(N[(N[(y + x), $MachinePrecision] - -1.0), $MachinePrecision] * N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / y), $MachinePrecision] * N[(x / N[(N[(x + y), $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 8.9999999999999996e-149

   1. Initial program 68.6%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. distribute-lft-in N/A

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

      4. *-rgt-identity N/A

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

      5. lower-fma.f64 56.5

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

   5. Applied rewrites 56.5%

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

   if 8.9999999999999996e-149 < y < 1.84999999999999998e131

   1. Initial program 76.8%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing
   3. 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 99.8

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 65.4%

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

   if 1.84999999999999998e131 < y

   1. Initial program 46.8%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing
   3. 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 67.8

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

   4. Applied rewrites 67.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lift-*.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. times-frac N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lift-+.f64 N/A

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

      14. lift-+.f64 N/A

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

      15. lift--.f64 N/A

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

      16. lift-+.f64 N/A

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

      17. +-commutative N/A

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

      18. metadata-eval N/A

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

      19. metadata-eval N/A

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

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

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

      21. lift-+.f64 N/A

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

      22. metadata-eval N/A

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

      23. lift-+.f64 N/A

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

   6. Applied rewrites 99.6%

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

   7. Taylor expanded in x around 0

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

      1. lower-/.f64 85.6

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

   9. Applied rewrites 85.6%

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

Alternative 13: 85.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 9 \cdot 10^{-149}:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(x, x, x\right)}\\ \mathbf{elif}\;y \leq 1.85 \cdot 10^{+131}:\\ \;\;\;\;1 \cdot \frac{x}{\left(\left(y + x\right) - -1\right) \cdot \left(y + x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y} \cdot \frac{x}{y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 9e-149)
   (/ y (fma x x x))
   (if (<= y 1.85e+131)
     (* 1.0 (/ x (* (- (+ y x) -1.0) (+ y x))))
     (* (/ 1.0 y) (/ x y)))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 9e-149) {
		tmp = y / fma(x, x, x);
	} else if (y <= 1.85e+131) {
		tmp = 1.0 * (x / (((y + x) - -1.0) * (y + x)));
	} else {
		tmp = (1.0 / y) * (x / y);
	}
	return tmp;
}

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= 9e-149)
		tmp = Float64(y / fma(x, x, x));
	elseif (y <= 1.85e+131)
		tmp = Float64(1.0 * Float64(x / Float64(Float64(Float64(y + x) - -1.0) * Float64(y + x))));
	else
		tmp = Float64(Float64(1.0 / y) * Float64(x / 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, 9e-149], N[(y / N[(x * x + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.85e+131], N[(1.0 * N[(x / N[(N[(N[(y + x), $MachinePrecision] - -1.0), $MachinePrecision] * N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / y), $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 8.9999999999999996e-149

   1. Initial program 68.6%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. distribute-lft-in N/A

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

      4. *-rgt-identity N/A

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

      5. lower-fma.f64 56.5

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

   5. Applied rewrites 56.5%

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

   if 8.9999999999999996e-149 < y < 1.84999999999999998e131

   1. Initial program 76.8%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing
   3. 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 99.8

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 65.4%

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

   if 1.84999999999999998e131 < y

   1. Initial program 46.8%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing
   3. 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 67.8

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

   4. Applied rewrites 67.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lift-*.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. times-frac N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lift-+.f64 N/A

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

      14. lift-+.f64 N/A

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

      15. lift--.f64 N/A

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

      16. lift-+.f64 N/A

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

      17. +-commutative N/A

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

      18. metadata-eval N/A

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

      19. metadata-eval N/A

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

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

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

      21. lift-+.f64 N/A

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

      22. metadata-eval N/A

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

      23. lift-+.f64 N/A

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

   6. Applied rewrites 99.6%

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

   7. Taylor expanded in x around 0

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

      1. lower-/.f64 85.6

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

   9. Applied rewrites 85.6%

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

   10. Taylor expanded in y around inf

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

       1. lower-/.f64 85.4

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

   12. Applied rewrites 85.4%

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

Alternative 14: 81.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{+150}:\\ \;\;\;\;\frac{\frac{y}{x}}{x}\\ \mathbf{elif}\;x \leq -3 \cdot 10^{-74}:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(x, x, x\right)}\\ \mathbf{elif}\;x \leq 2.25 \cdot 10^{+86}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(y, y, y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -2e+150)
   (/ (/ y x) x)
   (if (<= x -3e-74)
     (/ y (fma x x x))
     (if (<= x 2.25e+86) (/ x (fma y y y)) (/ (/ x y) y)))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -2e+150) {
		tmp = (y / x) / x;
	} else if (x <= -3e-74) {
		tmp = y / fma(x, x, x);
	} else if (x <= 2.25e+86) {
		tmp = x / fma(y, y, y);
	} else {
		tmp = (x / y) / y;
	}
	return tmp;
}

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -2e+150)
		tmp = Float64(Float64(y / x) / x);
	elseif (x <= -3e-74)
		tmp = Float64(y / fma(x, x, x));
	elseif (x <= 2.25e+86)
		tmp = Float64(x / fma(y, y, y));
	else
		tmp = Float64(Float64(x / y) / y);
	end
	return tmp
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[x, -2e+150], N[(N[(y / x), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[x, -3e-74], N[(y / N[(x * x + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 2.25e+86], N[(x / N[(y * y + y), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / y), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < -1.99999999999999996e150

   1. Initial program 54.2%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

      1. unpow2 N/A

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

      2. associate-/r* N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 75.9

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

   5. Applied rewrites 75.9%

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

   if -1.99999999999999996e150 < x < -3.00000000000000007e-74

   1. Initial program 69.1%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. distribute-lft-in N/A

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

      4. *-rgt-identity N/A

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

      5. lower-fma.f64 64.1

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

   5. Applied rewrites 64.1%

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

   if -3.00000000000000007e-74 < x < 2.24999999999999996e86

   1. Initial program 72.5%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. distribute-lft-in N/A

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

      4. *-rgt-identity N/A

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

      5. lower-fma.f64 71.8

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

   5. Applied rewrites 71.8%

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

   if 2.24999999999999996e86 < x

   1. Initial program 57.4%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

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

      1. unpow2 N/A

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

      2. associate-/r* N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 27.6

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

   5. Applied rewrites 27.6%

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

Alternative 15: 79.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -3 \cdot 10^{-74}:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(x, x, x\right)}\\ \mathbf{elif}\;x \leq 2.25 \cdot 10^{+86}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(y, y, y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -3e-74)
   (/ y (fma x x x))
   (if (<= x 2.25e+86) (/ x (fma y y y)) (/ (/ x y) y))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -3e-74) {
		tmp = y / fma(x, x, x);
	} else if (x <= 2.25e+86) {
		tmp = x / fma(y, y, y);
	} else {
		tmp = (x / y) / y;
	}
	return tmp;
}

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -3e-74)
		tmp = Float64(y / fma(x, x, x));
	elseif (x <= 2.25e+86)
		tmp = Float64(x / fma(y, y, y));
	else
		tmp = Float64(Float64(x / y) / y);
	end
	return tmp
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[x, -3e-74], N[(y / N[(x * x + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 2.25e+86], N[(x / N[(y * y + y), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -3.00000000000000007e-74

   1. Initial program 64.5%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. distribute-lft-in N/A

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

      4. *-rgt-identity N/A

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

      5. lower-fma.f64 66.9

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

   5. Applied rewrites 66.9%

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

   if -3.00000000000000007e-74 < x < 2.24999999999999996e86

   1. Initial program 72.5%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. distribute-lft-in N/A

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

      4. *-rgt-identity N/A

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

      5. lower-fma.f64 71.8

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

   5. Applied rewrites 71.8%

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

   if 2.24999999999999996e86 < x

   1. Initial program 57.4%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

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

      1. unpow2 N/A

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

      2. associate-/r* N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 27.6

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

   5. Applied rewrites 27.6%

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

Alternative 16: 78.3% accurate, 1.6× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -3 \cdot 10^{-74}:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(x, x, x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(y, y, y\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -3e-74) (/ y (fma x x x)) (/ x (fma y y y))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -3e-74) {
		tmp = y / fma(x, x, x);
	} else {
		tmp = x / fma(y, y, y);
	}
	return tmp;
}

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -3e-74)
		tmp = Float64(y / fma(x, x, x));
	else
		tmp = Float64(x / fma(y, y, y));
	end
	return tmp
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[x, -3e-74], N[(y / N[(x * x + x), $MachinePrecision]), $MachinePrecision], N[(x / N[(y * y + y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -3.00000000000000007e-74

   1. Initial program 64.5%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. distribute-lft-in N/A

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

      4. *-rgt-identity N/A

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

      5. lower-fma.f64 66.9

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

   5. Applied rewrites 66.9%

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

   if -3.00000000000000007e-74 < x

   1. Initial program 68.0%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. distribute-lft-in N/A

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

      4. *-rgt-identity N/A

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

      5. lower-fma.f64 54.1

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

   5. Applied rewrites 54.1%

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

Alternative 17: 76.0% accurate, 1.6× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -2.7 \cdot 10^{-6}:\\ \;\;\;\;\frac{y}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(y, y, y\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -2.7e-6) (/ y (* x x)) (/ x (fma y y y))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -2.7e-6) {
		tmp = y / (x * x);
	} else {
		tmp = x / fma(y, y, y);
	}
	return tmp;
}

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -2.7e-6)
		tmp = Float64(y / Float64(x * x));
	else
		tmp = Float64(x / fma(y, y, y));
	end
	return tmp
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[x, -2.7e-6], N[(y / N[(x * x), $MachinePrecision]), $MachinePrecision], N[(x / N[(y * y + y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2.69999999999999998e-6

   1. Initial program 53.8%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

      1. unpow2 N/A

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

      2. associate-/r* N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 67.0

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

   5. Applied rewrites 67.0%

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

   7. Applied rewrites 65.8%

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

   if -2.69999999999999998e-6 < x

   1. Initial program 70.7%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. distribute-lft-in N/A

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

      4. *-rgt-identity N/A

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

      5. lower-fma.f64 53.4

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

   5. Applied rewrites 53.4%

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

Alternative 18: 64.4% accurate, 1.7× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -2.7 \cdot 10^{-6}:\\ \;\;\;\;\frac{y}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -2.7e-6) (/ y (* x x)) (/ x (* y y))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -2.7e-6) {
		tmp = y / (x * x);
	} else {
		tmp = x / (y * y);
	}
	return tmp;
}

Fortran

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

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

real(8) function code(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-2.7d-6)) then
        tmp = y / (x * x)
    else
        tmp = x / (y * y)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (x <= -2.7e-6) {
		tmp = y / (x * x);
	} else {
		tmp = x / (y * y);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -2.7e-6:
		tmp = y / (x * x)
	else:
		tmp = x / (y * y)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -2.7e-6)
		tmp = Float64(y / Float64(x * x));
	else
		tmp = Float64(x / Float64(y * y));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -2.7e-6)
		tmp = y / (x * x);
	else
		tmp = x / (y * y);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[x, -2.7e-6], N[(y / N[(x * x), $MachinePrecision]), $MachinePrecision], N[(x / N[(y * y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2.69999999999999998e-6

   1. Initial program 53.8%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

      1. unpow2 N/A

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

      2. associate-/r* N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 67.0

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

   5. Applied rewrites 67.0%

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

   7. Applied rewrites 65.8%

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

   if -2.69999999999999998e-6 < x

   1. Initial program 70.7%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing
   3. 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 92.7

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

   4. Applied rewrites 92.7%

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

   5. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 36.1

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

   7. Applied rewrites 36.1%

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

Alternative 19: 36.4% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

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

real(8) function code(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = x / (y * y)
end function

Java

assert x < y;
public static double code(double x, double y) {
	return x / (y * y);
}

Python

[x, y] = sort([x, y])
def code(x, y):
	return x / (y * y)

Julia

x, y = sort([x, y])
function code(x, y)
	return Float64(x / Float64(y * y))
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp = code(x, y)
	tmp = x / (y * y);
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := N[(x / N[(y * y), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 66.9%

   \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
2. Add Preprocessing
3. 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 91.4

       \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\color{blue}{\left(\left(x + y\right) + 1\right) \cdot \left(x + y\right)}} \]

4. Applied rewrites 91.4%

   \[\leadsto \color{blue}{\frac{y}{y + x} \cdot \frac{x}{\left(\left(y + x\right) - -1\right) \cdot \left(y + x\right)}} \]

5. Taylor expanded in y around inf

   \[\leadsto \color{blue}{\frac{x}{{y}^{2}}} \]
6. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{x}{{y}^{2}}} \]

   2. unpow2 N/A

      \[\leadsto \frac{x}{\color{blue}{y \cdot y}} \]

   3. lower-*.f64 32.8

      \[\leadsto \frac{x}{\color{blue}{y \cdot y}} \]

7. Applied rewrites 32.8%

   \[\leadsto \color{blue}{\frac{x}{y \cdot y}} \]
8. Add Preprocessing

Developer Target 1: 99.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{\frac{x}{\left(y + 1\right) + x}}{y + x}}{\frac{1}{\frac{y}{y + x}}} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (/ (/ (/ x (+ (+ y 1.0) x)) (+ y x)) (/ 1.0 (/ y (+ y x)))))

C

double code(double x, double y) {
	return ((x / ((y + 1.0) + x)) / (y + x)) / (1.0 / (y / (y + x)));
}

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 + 1.0d0) + x)) / (y + x)) / (1.0d0 / (y / (y + x)))
end function

Java

public static double code(double x, double y) {
	return ((x / ((y + 1.0) + x)) / (y + x)) / (1.0 / (y / (y + x)));
}

Python

def code(x, y):
	return ((x / ((y + 1.0) + x)) / (y + x)) / (1.0 / (y / (y + x)))

Julia

function code(x, y)
	return Float64(Float64(Float64(x / Float64(Float64(y + 1.0) + x)) / Float64(y + x)) / Float64(1.0 / Float64(y / Float64(y + x))))
end

MATLAB

function tmp = code(x, y)
	tmp = ((x / ((y + 1.0) + x)) / (y + x)) / (1.0 / (y / (y + x)));
end

Wolfram

code[x_, y_] := N[(N[(N[(x / N[(N[(y + 1.0), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision] / N[(1.0 / N[(y / N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2025015 
(FPCore (x y)
  :name "Numeric.SpecFunctions:incompleteBetaApprox from math-functions-0.1.5.2, A"
  :precision binary64

  :alt
  (! :herbie-platform default (/ (/ (/ x (+ (+ y 1) x)) (+ y x)) (/ 1 (/ y (+ y x)))))

  (/ (* x y) (* (* (+ x y) (+ x y)) (+ (+ x y) 1.0))))