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

Percentage Accurate: 69.1% → 99.3%

Time: 3.5s

Alternatives: 15

Speedup: 1.7×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 69.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.3% accurate, 1.0× speedup

Math

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

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 (- -1.0 y)))))

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 69.1%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. lift-*.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. associate-*l* N/A

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

   7. times-frac N/A

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

   8. lower-*.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. lift-+.f64 N/A

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

   11. +-commutative N/A

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

   12. lower-+.f64 N/A

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

   13. lower-/.f64 N/A

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

   14. *-commutative N/A

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

   15. lower-*.f64 93.6

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

   16. lift-+.f64 N/A

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

   17. add-flip N/A

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

   18. lower--.f64 N/A

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

   19. lift-+.f64 N/A

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

   20. +-commutative N/A

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

   21. lower-+.f64 N/A

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

   22. metadata-eval 93.6

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

   23. lift-+.f64 N/A

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

   24. +-commutative N/A

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

   25. lower-+.f64 93.6

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

3. Applied rewrites 93.6%

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

   1. lift-*.f64 N/A

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

   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. div-flip N/A

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

   4. mult-flip-rev N/A

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

   5. lower-/.f64 N/A

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

   6. lift-+.f64 N/A

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

   7. +-commutative N/A

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

   8. lift-+.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. associate-/l* N/A

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

   11. lower-*.f64 N/A

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

   12. lift-+.f64 N/A

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

   13. +-commutative N/A

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

   14. lift-+.f64 N/A

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

   15. lower-/.f64 99.3

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

   16. lift-+.f64 N/A

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

   17. +-commutative N/A

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

   18. lift-+.f64 99.3

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

5. Applied rewrites 99.3%

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

   1. lift-*.f64 N/A

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

   2. lift--.f64 N/A

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

   3. metadata-eval N/A

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

   4. add-flip N/A

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

   5. lift-+.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

   8. associate-+l+ N/A

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

   9. metadata-eval N/A

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

   10. sub-flip N/A

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

   11. sub-negate-rev N/A

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

   12. sub-flip-reverse N/A

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

   13. lower--.f64 N/A

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

   14. lower--.f64 99.3

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

7. Applied rewrites 99.3%

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

Alternative 2: 96.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_0 := x - \left(-1 - y\right)\\ t_1 := \frac{y}{x + y}\\ \mathbf{if}\;x \leq -2.35 \cdot 10^{+166}:\\ \;\;\;\;\frac{t\_1}{\frac{x + y}{x} \cdot \left(x - -1\right)}\\ \mathbf{elif}\;x \leq 5.5 \cdot 10^{-90}:\\ \;\;\;\;\frac{t\_1 \cdot x}{t\_0 \cdot \left(x + y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 \cdot \frac{x}{t\_0}}{x + y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- x (- -1.0 y))) (t_1 (/ y (+ x y))))
   (if (<= x -2.35e+166)
     (/ t_1 (* (/ (+ x y) x) (- x -1.0)))
     (if (<= x 5.5e-90)
       (/ (* t_1 x) (* t_0 (+ x y)))
       (/ (* 1.0 (/ x t_0)) (+ x y))))))

C

assert(x < y);
double code(double x, double y) {
	double t_0 = x - (-1.0 - y);
	double t_1 = y / (x + y);
	double tmp;
	if (x <= -2.35e+166) {
		tmp = t_1 / (((x + y) / x) * (x - -1.0));
	} else if (x <= 5.5e-90) {
		tmp = (t_1 * x) / (t_0 * (x + y));
	} else {
		tmp = (1.0 * (x / t_0)) / (x + y);
	}
	return tmp;
}

Fortran

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

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

real(8) function code(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = x - ((-1.0d0) - y)
    t_1 = y / (x + y)
    if (x <= (-2.35d+166)) then
        tmp = t_1 / (((x + y) / x) * (x - (-1.0d0)))
    else if (x <= 5.5d-90) then
        tmp = (t_1 * x) / (t_0 * (x + y))
    else
        tmp = (1.0d0 * (x / t_0)) / (x + y)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double t_0 = x - (-1.0 - y);
	double t_1 = y / (x + y);
	double tmp;
	if (x <= -2.35e+166) {
		tmp = t_1 / (((x + y) / x) * (x - -1.0));
	} else if (x <= 5.5e-90) {
		tmp = (t_1 * x) / (t_0 * (x + y));
	} else {
		tmp = (1.0 * (x / t_0)) / (x + y);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	t_0 = x - (-1.0 - y)
	t_1 = y / (x + y)
	tmp = 0
	if x <= -2.35e+166:
		tmp = t_1 / (((x + y) / x) * (x - -1.0))
	elif x <= 5.5e-90:
		tmp = (t_1 * x) / (t_0 * (x + y))
	else:
		tmp = (1.0 * (x / t_0)) / (x + y)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	t_0 = Float64(x - Float64(-1.0 - y))
	t_1 = Float64(y / Float64(x + y))
	tmp = 0.0
	if (x <= -2.35e+166)
		tmp = Float64(t_1 / Float64(Float64(Float64(x + y) / x) * Float64(x - -1.0)));
	elseif (x <= 5.5e-90)
		tmp = Float64(Float64(t_1 * x) / Float64(t_0 * Float64(x + y)));
	else
		tmp = Float64(Float64(1.0 * Float64(x / t_0)) / Float64(x + y));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	t_0 = x - (-1.0 - y);
	t_1 = y / (x + y);
	tmp = 0.0;
	if (x <= -2.35e+166)
		tmp = t_1 / (((x + y) / x) * (x - -1.0));
	elseif (x <= 5.5e-90)
		tmp = (t_1 * x) / (t_0 * (x + y));
	else
		tmp = (1.0 * (x / t_0)) / (x + y);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := Block[{t$95$0 = N[(x - N[(-1.0 - y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(y / N[(x + y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -2.35e+166], N[(t$95$1 / N[(N[(N[(x + y), $MachinePrecision] / x), $MachinePrecision] * N[(x - -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 5.5e-90], N[(N[(t$95$1 * x), $MachinePrecision] / N[(t$95$0 * N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 * N[(x / t$95$0), $MachinePrecision]), $MachinePrecision] / N[(x + y), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.35e166

   1. Initial program 69.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 93.6

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

      16. lift-+.f64 N/A

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

      17. add-flip N/A

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

      18. lower--.f64 N/A

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

      19. lift-+.f64 N/A

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

      20. +-commutative N/A

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

      21. lower-+.f64 N/A

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

      22. metadata-eval 93.6

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

      23. lift-+.f64 N/A

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

      24. +-commutative N/A

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

      25. lower-+.f64 93.6

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

   3. Applied rewrites 93.6%

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

      1. lift-*.f64 N/A

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

      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. div-flip N/A

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

      4. mult-flip-rev N/A

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

      5. lower-/.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. lift-+.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-/l* N/A

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

      11. lower-*.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. +-commutative N/A

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

      14. lift-+.f64 N/A

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

      15. lower-/.f64 99.3

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

      16. lift-+.f64 N/A

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

      17. +-commutative N/A

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

      18. lift-+.f64 99.3

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

   5. Applied rewrites 99.3%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. metadata-eval N/A

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

      4. add-flip N/A

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

      5. lift-+.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. associate-+l+ N/A

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

      9. metadata-eval N/A

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

      10. sub-flip N/A

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

      11. sub-negate-rev N/A

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

      12. sub-flip-reverse N/A

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

      13. lower--.f64 N/A

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

      14. lower--.f64 99.3

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

   7. Applied rewrites 99.3%

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

   8. Taylor expanded in y around 0

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

   10. Applied rewrites 76.5%

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

   if -2.35e166 < x < 5.5000000000000003e-90

   1. Initial program 69.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 93.6

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

      16. lift-+.f64 N/A

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

      17. add-flip N/A

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

      18. lower--.f64 N/A

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

      19. lift-+.f64 N/A

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

      20. +-commutative N/A

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

      21. lower-+.f64 N/A

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

      22. metadata-eval 93.6

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

      23. lift-+.f64 N/A

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

      24. +-commutative N/A

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

      25. lower-+.f64 93.6

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

   3. Applied rewrites 93.6%

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

      1. lift-*.f64 N/A

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

      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. div-flip N/A

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

      4. mult-flip-rev N/A

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

      5. lower-/.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. lift-+.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-/l* N/A

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

      11. lower-*.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. +-commutative N/A

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

      14. lift-+.f64 N/A

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

      15. lower-/.f64 99.3

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

      16. lift-+.f64 N/A

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

      17. +-commutative N/A

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

      18. lift-+.f64 99.3

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

   5. Applied rewrites 99.3%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. metadata-eval N/A

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

      6. add-flip N/A

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

      7. lift-+.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. associate-*r/ N/A

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

      10. +-commutative N/A

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

      11. metadata-eval N/A

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

      12. sub-flip N/A

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

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. div-flip N/A

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

      16. associate-*r/ N/A

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

   7. Applied rewrites 93.7%

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

   if 5.5000000000000003e-90 < x

   1. Initial program 69.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 93.6

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

      16. lift-+.f64 N/A

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

      17. add-flip N/A

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

      18. lower--.f64 N/A

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

      19. lift-+.f64 N/A

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

      20. +-commutative N/A

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

      21. lower-+.f64 N/A

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

      22. metadata-eval 93.6

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

      23. lift-+.f64 N/A

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

      24. +-commutative N/A

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

      25. lower-+.f64 93.6

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

   3. Applied rewrites 93.6%

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

      1. lift-*.f64 N/A

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

      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. div-flip N/A

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

      4. mult-flip-rev N/A

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

      5. lower-/.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. lift-+.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-/l* N/A

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

      11. lower-*.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. +-commutative N/A

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

      14. lift-+.f64 N/A

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

      15. lower-/.f64 99.3

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

      16. lift-+.f64 N/A

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

      17. +-commutative N/A

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

      18. lift-+.f64 99.3

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

   5. Applied rewrites 99.3%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 50.9%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. metadata-eval N/A

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

      6. add-flip N/A

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

      7. lift-+.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. associate-*r/ N/A

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

      10. +-commutative N/A

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

      11. metadata-eval N/A

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

      12. sub-flip N/A

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

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. div-flip N/A

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

      16. associate-/r* N/A

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

      17. +-commutative N/A

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

      18. lift-+.f64 N/A

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

   10. Applied rewrites 51.5%

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

Alternative 3: 93.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 230000:\\ \;\;\;\;\frac{\frac{y}{x + y}}{\frac{x + y}{x} \cdot \left(x - -1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 \cdot \frac{x}{x - \left(-1 - y\right)}}{x + y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 230000.0)
   (/ (/ y (+ x y)) (* (/ (+ x y) x) (- x -1.0)))
   (/ (* 1.0 (/ x (- x (- -1.0 y)))) (+ x y))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 230000.0) {
		tmp = (y / (x + y)) / (((x + y) / x) * (x - -1.0));
	} else {
		tmp = (1.0 * (x / (x - (-1.0 - y)))) / (x + y);
	}
	return tmp;
}

Fortran

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

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

real(8) function code(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 230000.0d0) then
        tmp = (y / (x + y)) / (((x + y) / x) * (x - (-1.0d0)))
    else
        tmp = (1.0d0 * (x / (x - ((-1.0d0) - y)))) / (x + y)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (y <= 230000.0) {
		tmp = (y / (x + y)) / (((x + y) / x) * (x - -1.0));
	} else {
		tmp = (1.0 * (x / (x - (-1.0 - y)))) / (x + y);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if y <= 230000.0:
		tmp = (y / (x + y)) / (((x + y) / x) * (x - -1.0))
	else:
		tmp = (1.0 * (x / (x - (-1.0 - y)))) / (x + y)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= 230000.0)
		tmp = Float64(Float64(y / Float64(x + y)) / Float64(Float64(Float64(x + y) / x) * Float64(x - -1.0)));
	else
		tmp = Float64(Float64(1.0 * Float64(x / Float64(x - Float64(-1.0 - y)))) / Float64(x + y));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 230000.0)
		tmp = (y / (x + y)) / (((x + y) / x) * (x - -1.0));
	else
		tmp = (1.0 * (x / (x - (-1.0 - y)))) / (x + y);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[y, 230000.0], N[(N[(y / N[(x + y), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(x + y), $MachinePrecision] / x), $MachinePrecision] * N[(x - -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 * N[(x / N[(x - N[(-1.0 - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x + y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2.3e5

   1. Initial program 69.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 93.6

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

      16. lift-+.f64 N/A

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

      17. add-flip N/A

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

      18. lower--.f64 N/A

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

      19. lift-+.f64 N/A

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

      20. +-commutative N/A

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

      21. lower-+.f64 N/A

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

      22. metadata-eval 93.6

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

      23. lift-+.f64 N/A

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

      24. +-commutative N/A

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

      25. lower-+.f64 93.6

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

   3. Applied rewrites 93.6%

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

      1. lift-*.f64 N/A

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

      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. div-flip N/A

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

      4. mult-flip-rev N/A

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

      5. lower-/.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. lift-+.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-/l* N/A

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

      11. lower-*.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. +-commutative N/A

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

      14. lift-+.f64 N/A

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

      15. lower-/.f64 99.3

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

      16. lift-+.f64 N/A

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

      17. +-commutative N/A

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

      18. lift-+.f64 99.3

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

   5. Applied rewrites 99.3%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. metadata-eval N/A

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

      4. add-flip N/A

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

      5. lift-+.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. associate-+l+ N/A

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

      9. metadata-eval N/A

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

      10. sub-flip N/A

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

      11. sub-negate-rev N/A

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

      12. sub-flip-reverse N/A

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

      13. lower--.f64 N/A

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

      14. lower--.f64 99.3

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

   7. Applied rewrites 99.3%

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

   8. Taylor expanded in y around 0

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

   10. Applied rewrites 76.5%

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

   if 2.3e5 < y

   1. Initial program 69.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 93.6

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

      16. lift-+.f64 N/A

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

      17. add-flip N/A

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

      18. lower--.f64 N/A

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

      19. lift-+.f64 N/A

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

      20. +-commutative N/A

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

      21. lower-+.f64 N/A

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

      22. metadata-eval 93.6

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

      23. lift-+.f64 N/A

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

      24. +-commutative N/A

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

      25. lower-+.f64 93.6

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

   3. Applied rewrites 93.6%

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

      1. lift-*.f64 N/A

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

      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. div-flip N/A

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

      4. mult-flip-rev N/A

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

      5. lower-/.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. lift-+.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-/l* N/A

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

      11. lower-*.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. +-commutative N/A

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

      14. lift-+.f64 N/A

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

      15. lower-/.f64 99.3

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

      16. lift-+.f64 N/A

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

      17. +-commutative N/A

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

      18. lift-+.f64 99.3

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

   5. Applied rewrites 99.3%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 50.9%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. metadata-eval N/A

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

      6. add-flip N/A

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

      7. lift-+.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. associate-*r/ N/A

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

      10. +-commutative N/A

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

      11. metadata-eval N/A

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

      12. sub-flip N/A

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

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. div-flip N/A

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

      16. associate-/r* N/A

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

      17. +-commutative N/A

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

      18. lift-+.f64 N/A

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

   10. Applied rewrites 51.5%

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

Alternative 4: 92.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_0 := \frac{y}{x + y}\\ \mathbf{if}\;x \leq -2.5 \cdot 10^{-5}:\\ \;\;\;\;\frac{t\_0}{1 + x}\\ \mathbf{elif}\;x \leq 5.5 \cdot 10^{-90}:\\ \;\;\;\;\frac{t\_0 \cdot x}{\left(y - -1\right) \cdot \left(x + y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 \cdot \frac{x}{x - \left(-1 - y\right)}}{x + y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ y (+ x y))))
   (if (<= x -2.5e-5)
     (/ t_0 (+ 1.0 x))
     (if (<= x 5.5e-90)
       (/ (* t_0 x) (* (- y -1.0) (+ x y)))
       (/ (* 1.0 (/ x (- x (- -1.0 y)))) (+ x y))))))

C

assert(x < y);
double code(double x, double y) {
	double t_0 = y / (x + y);
	double tmp;
	if (x <= -2.5e-5) {
		tmp = t_0 / (1.0 + x);
	} else if (x <= 5.5e-90) {
		tmp = (t_0 * x) / ((y - -1.0) * (x + y));
	} else {
		tmp = (1.0 * (x / (x - (-1.0 - y)))) / (x + y);
	}
	return tmp;
}

Fortran

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

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

real(8) function code(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = y / (x + y)
    if (x <= (-2.5d-5)) then
        tmp = t_0 / (1.0d0 + x)
    else if (x <= 5.5d-90) then
        tmp = (t_0 * x) / ((y - (-1.0d0)) * (x + y))
    else
        tmp = (1.0d0 * (x / (x - ((-1.0d0) - y)))) / (x + y)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double t_0 = y / (x + y);
	double tmp;
	if (x <= -2.5e-5) {
		tmp = t_0 / (1.0 + x);
	} else if (x <= 5.5e-90) {
		tmp = (t_0 * x) / ((y - -1.0) * (x + y));
	} else {
		tmp = (1.0 * (x / (x - (-1.0 - y)))) / (x + y);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	t_0 = y / (x + y)
	tmp = 0
	if x <= -2.5e-5:
		tmp = t_0 / (1.0 + x)
	elif x <= 5.5e-90:
		tmp = (t_0 * x) / ((y - -1.0) * (x + y))
	else:
		tmp = (1.0 * (x / (x - (-1.0 - y)))) / (x + y)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	t_0 = Float64(y / Float64(x + y))
	tmp = 0.0
	if (x <= -2.5e-5)
		tmp = Float64(t_0 / Float64(1.0 + x));
	elseif (x <= 5.5e-90)
		tmp = Float64(Float64(t_0 * x) / Float64(Float64(y - -1.0) * Float64(x + y)));
	else
		tmp = Float64(Float64(1.0 * Float64(x / Float64(x - Float64(-1.0 - y)))) / Float64(x + y));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	t_0 = y / (x + y);
	tmp = 0.0;
	if (x <= -2.5e-5)
		tmp = t_0 / (1.0 + x);
	elseif (x <= 5.5e-90)
		tmp = (t_0 * x) / ((y - -1.0) * (x + y));
	else
		tmp = (1.0 * (x / (x - (-1.0 - y)))) / (x + y);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := Block[{t$95$0 = N[(y / N[(x + y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -2.5e-5], N[(t$95$0 / N[(1.0 + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 5.5e-90], N[(N[(t$95$0 * x), $MachinePrecision] / N[(N[(y - -1.0), $MachinePrecision] * N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 * N[(x / N[(x - N[(-1.0 - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x + y), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.50000000000000012e-5

   1. Initial program 69.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 93.6

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

      16. lift-+.f64 N/A

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

      17. add-flip N/A

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

      18. lower--.f64 N/A

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

      19. lift-+.f64 N/A

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

      20. +-commutative N/A

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

      21. lower-+.f64 N/A

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

      22. metadata-eval 93.6

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

      23. lift-+.f64 N/A

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

      24. +-commutative N/A

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

      25. lower-+.f64 93.6

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

   3. Applied rewrites 93.6%

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

      1. lift-*.f64 N/A

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

      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. div-flip N/A

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

      4. mult-flip-rev N/A

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

      5. lower-/.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. lift-+.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-/l* N/A

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

      11. lower-*.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. +-commutative N/A

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

      14. lift-+.f64 N/A

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

      15. lower-/.f64 99.3

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

      16. lift-+.f64 N/A

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

      17. +-commutative N/A

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

      18. lift-+.f64 99.3

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

   5. Applied rewrites 99.3%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. metadata-eval N/A

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

      4. add-flip N/A

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

      5. lift-+.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. associate-+l+ N/A

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

      9. metadata-eval N/A

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

      10. sub-flip N/A

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

      11. sub-negate-rev N/A

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

      12. sub-flip-reverse N/A

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

      13. lower--.f64 N/A

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

      14. lower--.f64 99.3

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

   7. Applied rewrites 99.3%

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

   8. Taylor expanded in y around 0

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

      1. lower-+.f64 51.3

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

   10. Applied rewrites 51.3%

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

   if -2.50000000000000012e-5 < x < 5.5000000000000003e-90

   1. Initial program 69.1%

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

   2. Taylor expanded in x around 0

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

      1. lower-+.f64 58.7

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

   4. Applied rewrites 58.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-+.f64 N/A

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

      8. associate-/r* N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. associate-*l/ N/A

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

      12. lift-/.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lift-+.f64 N/A

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

      16. +-commutative N/A

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

      17. lift-+.f64 N/A

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

   6. Applied rewrites 75.8%

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

   if 5.5000000000000003e-90 < x

   1. Initial program 69.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 93.6

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

      16. lift-+.f64 N/A

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

      17. add-flip N/A

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

      18. lower--.f64 N/A

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

      19. lift-+.f64 N/A

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

      20. +-commutative N/A

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

      21. lower-+.f64 N/A

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

      22. metadata-eval 93.6

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

      23. lift-+.f64 N/A

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

      24. +-commutative N/A

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

      25. lower-+.f64 93.6

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

   3. Applied rewrites 93.6%

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

      1. lift-*.f64 N/A

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

      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. div-flip N/A

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

      4. mult-flip-rev N/A

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

      5. lower-/.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. lift-+.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-/l* N/A

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

      11. lower-*.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. +-commutative N/A

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

      14. lift-+.f64 N/A

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

      15. lower-/.f64 99.3

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

      16. lift-+.f64 N/A

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

      17. +-commutative N/A

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

      18. lift-+.f64 99.3

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

   5. Applied rewrites 99.3%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 50.9%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. metadata-eval N/A

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

      6. add-flip N/A

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

      7. lift-+.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. associate-*r/ N/A

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

      10. +-commutative N/A

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

      11. metadata-eval N/A

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

      12. sub-flip N/A

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

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. div-flip N/A

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

      16. associate-/r* N/A

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

      17. +-commutative N/A

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

      18. lift-+.f64 N/A

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

   10. Applied rewrites 51.5%

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

Alternative 5: 88.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 3.6 \cdot 10^{-167}:\\ \;\;\;\;\frac{\frac{y}{x + y}}{1 + x}\\ \mathbf{elif}\;y \leq 3500:\\ \;\;\;\;\frac{y}{\left(1 + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{1 \cdot \frac{x}{x - \left(-1 - y\right)}}{x + y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 3.6e-167)
   (/ (/ y (+ x y)) (+ 1.0 x))
   (if (<= y 3500.0)
     (* (/ y (* (+ 1.0 x) (* (+ y x) (+ y x)))) x)
     (/ (* 1.0 (/ x (- x (- -1.0 y)))) (+ x y)))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 3.6e-167) {
		tmp = (y / (x + y)) / (1.0 + x);
	} else if (y <= 3500.0) {
		tmp = (y / ((1.0 + x) * ((y + x) * (y + x)))) * x;
	} else {
		tmp = (1.0 * (x / (x - (-1.0 - y)))) / (x + y);
	}
	return tmp;
}

Fortran

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

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

real(8) function code(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 3.6d-167) then
        tmp = (y / (x + y)) / (1.0d0 + x)
    else if (y <= 3500.0d0) then
        tmp = (y / ((1.0d0 + x) * ((y + x) * (y + x)))) * x
    else
        tmp = (1.0d0 * (x / (x - ((-1.0d0) - y)))) / (x + y)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (y <= 3.6e-167) {
		tmp = (y / (x + y)) / (1.0 + x);
	} else if (y <= 3500.0) {
		tmp = (y / ((1.0 + x) * ((y + x) * (y + x)))) * x;
	} else {
		tmp = (1.0 * (x / (x - (-1.0 - y)))) / (x + y);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if y <= 3.6e-167:
		tmp = (y / (x + y)) / (1.0 + x)
	elif y <= 3500.0:
		tmp = (y / ((1.0 + x) * ((y + x) * (y + x)))) * x
	else:
		tmp = (1.0 * (x / (x - (-1.0 - y)))) / (x + y)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= 3.6e-167)
		tmp = Float64(Float64(y / Float64(x + y)) / Float64(1.0 + x));
	elseif (y <= 3500.0)
		tmp = Float64(Float64(y / Float64(Float64(1.0 + x) * Float64(Float64(y + x) * Float64(y + x)))) * x);
	else
		tmp = Float64(Float64(1.0 * Float64(x / Float64(x - Float64(-1.0 - y)))) / Float64(x + y));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 3.6e-167)
		tmp = (y / (x + y)) / (1.0 + x);
	elseif (y <= 3500.0)
		tmp = (y / ((1.0 + x) * ((y + x) * (y + x)))) * x;
	else
		tmp = (1.0 * (x / (x - (-1.0 - y)))) / (x + y);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[y, 3.6e-167], N[(N[(y / N[(x + y), $MachinePrecision]), $MachinePrecision] / N[(1.0 + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 3500.0], N[(N[(y / N[(N[(1.0 + x), $MachinePrecision] * N[(N[(y + x), $MachinePrecision] * N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision], N[(N[(1.0 * N[(x / N[(x - N[(-1.0 - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x + y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 3.6000000000000001e-167

   1. Initial program 69.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 93.6

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

      16. lift-+.f64 N/A

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

      17. add-flip N/A

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

      18. lower--.f64 N/A

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

      19. lift-+.f64 N/A

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

      20. +-commutative N/A

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

      21. lower-+.f64 N/A

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

      22. metadata-eval 93.6

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

      23. lift-+.f64 N/A

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

      24. +-commutative N/A

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

      25. lower-+.f64 93.6

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

   3. Applied rewrites 93.6%

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

      1. lift-*.f64 N/A

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

      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. div-flip N/A

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

      4. mult-flip-rev N/A

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

      5. lower-/.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. lift-+.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-/l* N/A

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

      11. lower-*.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. +-commutative N/A

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

      14. lift-+.f64 N/A

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

      15. lower-/.f64 99.3

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

      16. lift-+.f64 N/A

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

      17. +-commutative N/A

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

      18. lift-+.f64 99.3

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

   5. Applied rewrites 99.3%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. metadata-eval N/A

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

      4. add-flip N/A

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

      5. lift-+.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. associate-+l+ N/A

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

      9. metadata-eval N/A

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

      10. sub-flip N/A

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

      11. sub-negate-rev N/A

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

      12. sub-flip-reverse N/A

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

      13. lower--.f64 N/A

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

      14. lower--.f64 99.3

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

   7. Applied rewrites 99.3%

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

   8. Taylor expanded in y around 0

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

      1. lower-+.f64 51.3

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

   10. Applied rewrites 51.3%

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

   if 3.6000000000000001e-167 < y < 3500

   1. Initial program 69.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

   3. Applied rewrites 81.8%

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

   4. Taylor expanded in y around 0

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

      1. lower-+.f64 72.1

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

   6. Applied rewrites 72.1%

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

   if 3500 < y

   1. Initial program 69.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 93.6

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

      16. lift-+.f64 N/A

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

      17. add-flip N/A

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

      18. lower--.f64 N/A

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

      19. lift-+.f64 N/A

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

      20. +-commutative N/A

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

      21. lower-+.f64 N/A

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

      22. metadata-eval 93.6

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

      23. lift-+.f64 N/A

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

      24. +-commutative N/A

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

      25. lower-+.f64 93.6

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

   3. Applied rewrites 93.6%

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

      1. lift-*.f64 N/A

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

      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. div-flip N/A

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

      4. mult-flip-rev N/A

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

      5. lower-/.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. lift-+.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-/l* N/A

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

      11. lower-*.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. +-commutative N/A

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

      14. lift-+.f64 N/A

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

      15. lower-/.f64 99.3

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

      16. lift-+.f64 N/A

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

      17. +-commutative N/A

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

      18. lift-+.f64 99.3

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

   5. Applied rewrites 99.3%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 50.9%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. metadata-eval N/A

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

      6. add-flip N/A

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

      7. lift-+.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. associate-*r/ N/A

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

      10. +-commutative N/A

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

      11. metadata-eval N/A

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

      12. sub-flip N/A

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

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. div-flip N/A

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

      16. associate-/r* N/A

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

      17. +-commutative N/A

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

      18. lift-+.f64 N/A

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

   10. Applied rewrites 51.5%

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

Alternative 6: 86.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_0 := x - \left(-1 - y\right)\\ \mathbf{if}\;y \leq 4.4 \cdot 10^{-195}:\\ \;\;\;\;\frac{\frac{y}{x + y}}{1 + x}\\ \mathbf{elif}\;y \leq 1.8 \cdot 10^{+156}:\\ \;\;\;\;\frac{1 \cdot x}{t\_0 \cdot \left(x + y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 \cdot \frac{x}{t\_0}}{x + y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- x (- -1.0 y))))
   (if (<= y 4.4e-195)
     (/ (/ y (+ x y)) (+ 1.0 x))
     (if (<= y 1.8e+156)
       (/ (* 1.0 x) (* t_0 (+ x y)))
       (/ (* 1.0 (/ x t_0)) (+ x y))))))

C

assert(x < y);
double code(double x, double y) {
	double t_0 = x - (-1.0 - y);
	double tmp;
	if (y <= 4.4e-195) {
		tmp = (y / (x + y)) / (1.0 + x);
	} else if (y <= 1.8e+156) {
		tmp = (1.0 * x) / (t_0 * (x + y));
	} else {
		tmp = (1.0 * (x / t_0)) / (x + y);
	}
	return tmp;
}

Fortran

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

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

real(8) function code(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x - ((-1.0d0) - y)
    if (y <= 4.4d-195) then
        tmp = (y / (x + y)) / (1.0d0 + x)
    else if (y <= 1.8d+156) then
        tmp = (1.0d0 * x) / (t_0 * (x + y))
    else
        tmp = (1.0d0 * (x / t_0)) / (x + y)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double t_0 = x - (-1.0 - y);
	double tmp;
	if (y <= 4.4e-195) {
		tmp = (y / (x + y)) / (1.0 + x);
	} else if (y <= 1.8e+156) {
		tmp = (1.0 * x) / (t_0 * (x + y));
	} else {
		tmp = (1.0 * (x / t_0)) / (x + y);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	t_0 = x - (-1.0 - y)
	tmp = 0
	if y <= 4.4e-195:
		tmp = (y / (x + y)) / (1.0 + x)
	elif y <= 1.8e+156:
		tmp = (1.0 * x) / (t_0 * (x + y))
	else:
		tmp = (1.0 * (x / t_0)) / (x + y)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	t_0 = Float64(x - Float64(-1.0 - y))
	tmp = 0.0
	if (y <= 4.4e-195)
		tmp = Float64(Float64(y / Float64(x + y)) / Float64(1.0 + x));
	elseif (y <= 1.8e+156)
		tmp = Float64(Float64(1.0 * x) / Float64(t_0 * Float64(x + y)));
	else
		tmp = Float64(Float64(1.0 * Float64(x / t_0)) / Float64(x + y));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	t_0 = x - (-1.0 - y);
	tmp = 0.0;
	if (y <= 4.4e-195)
		tmp = (y / (x + y)) / (1.0 + x);
	elseif (y <= 1.8e+156)
		tmp = (1.0 * x) / (t_0 * (x + y));
	else
		tmp = (1.0 * (x / t_0)) / (x + y);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := Block[{t$95$0 = N[(x - N[(-1.0 - y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, 4.4e-195], N[(N[(y / N[(x + y), $MachinePrecision]), $MachinePrecision] / N[(1.0 + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.8e+156], N[(N[(1.0 * x), $MachinePrecision] / N[(t$95$0 * N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 * N[(x / t$95$0), $MachinePrecision]), $MachinePrecision] / N[(x + y), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < 4.40000000000000011e-195

   1. Initial program 69.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 93.6

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

      16. lift-+.f64 N/A

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

      17. add-flip N/A

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

      18. lower--.f64 N/A

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

      19. lift-+.f64 N/A

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

      20. +-commutative N/A

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

      21. lower-+.f64 N/A

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

      22. metadata-eval 93.6

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

      23. lift-+.f64 N/A

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

      24. +-commutative N/A

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

      25. lower-+.f64 93.6

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

   3. Applied rewrites 93.6%

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

      1. lift-*.f64 N/A

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

      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. div-flip N/A

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

      4. mult-flip-rev N/A

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

      5. lower-/.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. lift-+.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-/l* N/A

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

      11. lower-*.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. +-commutative N/A

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

      14. lift-+.f64 N/A

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

      15. lower-/.f64 99.3

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

      16. lift-+.f64 N/A

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

      17. +-commutative N/A

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

      18. lift-+.f64 99.3

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

   5. Applied rewrites 99.3%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. metadata-eval N/A

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

      4. add-flip N/A

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

      5. lift-+.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. associate-+l+ N/A

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

      9. metadata-eval N/A

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

      10. sub-flip N/A

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

      11. sub-negate-rev N/A

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

      12. sub-flip-reverse N/A

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

      13. lower--.f64 N/A

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

      14. lower--.f64 99.3

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

   7. Applied rewrites 99.3%

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

   8. Taylor expanded in y around 0

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

      1. lower-+.f64 51.3

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

   10. Applied rewrites 51.3%

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

   if 4.40000000000000011e-195 < y < 1.79999999999999989e156

   1. Initial program 69.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 93.6

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

      16. lift-+.f64 N/A

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

      17. add-flip N/A

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

      18. lower--.f64 N/A

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

      19. lift-+.f64 N/A

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

      20. +-commutative N/A

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

      21. lower-+.f64 N/A

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

      22. metadata-eval 93.6

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

      23. lift-+.f64 N/A

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

      24. +-commutative N/A

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

      25. lower-+.f64 93.6

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

   3. Applied rewrites 93.6%

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

      1. lift-*.f64 N/A

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

      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. div-flip N/A

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

      4. mult-flip-rev N/A

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

      5. lower-/.f64 N/A

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

      6. lift-+.f64 N/A

         \[\leadsto \frac{\frac{y}{\color{blue}{y + x}}}{\frac{\left(\left(y + x\right) - -1\right) \cdot \left(y + x\right)}{x}} \]

      7. +-commutative N/A

         \[\leadsto \frac{\frac{y}{\color{blue}{x + y}}}{\frac{\left(\left(y + x\right) - -1\right) \cdot \left(y + x\right)}{x}} \]

      8. lift-+.f64 N/A

         \[\leadsto \frac{\frac{y}{\color{blue}{x + y}}}{\frac{\left(\left(y + x\right) - -1\right) \cdot \left(y + x\right)}{x}} \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{\frac{y}{x + y}}{\frac{\color{blue}{\left(\left(y + x\right) - -1\right) \cdot \left(y + x\right)}}{x}} \]

      10. associate-/l* N/A

          \[\leadsto \frac{\frac{y}{x + y}}{\color{blue}{\left(\left(y + x\right) - -1\right) \cdot \frac{y + x}{x}}} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{\frac{y}{x + y}}{\color{blue}{\left(\left(y + x\right) - -1\right) \cdot \frac{y + x}{x}}} \]

      12. lift-+.f64 N/A

          \[\leadsto \frac{\frac{y}{x + y}}{\left(\color{blue}{\left(y + x\right)} - -1\right) \cdot \frac{y + x}{x}} \]

      13. +-commutative N/A

          \[\leadsto \frac{\frac{y}{x + y}}{\left(\color{blue}{\left(x + y\right)} - -1\right) \cdot \frac{y + x}{x}} \]

      14. lift-+.f64 N/A

          \[\leadsto \frac{\frac{y}{x + y}}{\left(\color{blue}{\left(x + y\right)} - -1\right) \cdot \frac{y + x}{x}} \]

      15. lower-/.f64 99.3

          \[\leadsto \frac{\frac{y}{x + y}}{\left(\left(x + y\right) - -1\right) \cdot \color{blue}{\frac{y + x}{x}}} \]

      16. lift-+.f64 N/A

          \[\leadsto \frac{\frac{y}{x + y}}{\left(\left(x + y\right) - -1\right) \cdot \frac{\color{blue}{y + x}}{x}} \]

      17. +-commutative N/A

          \[\leadsto \frac{\frac{y}{x + y}}{\left(\left(x + y\right) - -1\right) \cdot \frac{\color{blue}{x + y}}{x}} \]

      18. lift-+.f64 99.3

          \[\leadsto \frac{\frac{y}{x + y}}{\left(\left(x + y\right) - -1\right) \cdot \frac{\color{blue}{x + y}}{x}} \]

   5. Applied rewrites 99.3%

      \[\leadsto \color{blue}{\frac{\frac{y}{x + y}}{\left(\left(x + y\right) - -1\right) \cdot \frac{x + y}{x}}} \]

   6. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{1}}{\left(\left(x + y\right) - -1\right) \cdot \frac{x + y}{x}} \]
   7. Step-by-step derivation

   8. Applied rewrites 50.9%

      \[\leadsto \frac{\color{blue}{1}}{\left(\left(x + y\right) - -1\right) \cdot \frac{x + y}{x}} \]
   9. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{\left(\left(x + y\right) - -1\right) \cdot \frac{x + y}{x}}} \]

      2. mult-flip N/A

         \[\leadsto \color{blue}{1 \cdot \frac{1}{\left(\left(x + y\right) - -1\right) \cdot \frac{x + y}{x}}} \]

      3. lift-*.f64 N/A

         \[\leadsto 1 \cdot \frac{1}{\color{blue}{\left(\left(x + y\right) - -1\right) \cdot \frac{x + y}{x}}} \]

      4. lift--.f64 N/A

         \[\leadsto 1 \cdot \frac{1}{\color{blue}{\left(\left(x + y\right) - -1\right)} \cdot \frac{x + y}{x}} \]

      5. metadata-eval N/A

         \[\leadsto 1 \cdot \frac{1}{\left(\left(x + y\right) - \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right) \cdot \frac{x + y}{x}} \]

      6. add-flip N/A

         \[\leadsto 1 \cdot \frac{1}{\color{blue}{\left(\left(x + y\right) + 1\right)} \cdot \frac{x + y}{x}} \]

      7. lift-+.f64 N/A

         \[\leadsto 1 \cdot \frac{1}{\left(\color{blue}{\left(x + y\right)} + 1\right) \cdot \frac{x + y}{x}} \]

      8. lift-/.f64 N/A

         \[\leadsto 1 \cdot \frac{1}{\left(\left(x + y\right) + 1\right) \cdot \color{blue}{\frac{x + y}{x}}} \]

      9. associate-*r/ N/A

         \[\leadsto 1 \cdot \frac{1}{\color{blue}{\frac{\left(\left(x + y\right) + 1\right) \cdot \left(x + y\right)}{x}}} \]

      10. +-commutative N/A

          \[\leadsto 1 \cdot \frac{1}{\frac{\left(\color{blue}{\left(y + x\right)} + 1\right) \cdot \left(x + y\right)}{x}} \]

      11. metadata-eval N/A

          \[\leadsto 1 \cdot \frac{1}{\frac{\left(\left(y + x\right) + \color{blue}{\left(\mathsf{neg}\left(-1\right)\right)}\right) \cdot \left(x + y\right)}{x}} \]

      12. sub-flip N/A

          \[\leadsto 1 \cdot \frac{1}{\frac{\color{blue}{\left(\left(y + x\right) - -1\right)} \cdot \left(x + y\right)}{x}} \]

      13. lift-+.f64 N/A

          \[\leadsto 1 \cdot \frac{1}{\frac{\left(\left(y + x\right) - -1\right) \cdot \color{blue}{\left(x + y\right)}}{x}} \]

      14. +-commutative N/A

          \[\leadsto 1 \cdot \frac{1}{\frac{\left(\left(y + x\right) - -1\right) \cdot \color{blue}{\left(y + x\right)}}{x}} \]

      15. div-flip N/A

          \[\leadsto 1 \cdot \color{blue}{\frac{x}{\left(\left(y + x\right) - -1\right) \cdot \left(y + x\right)}} \]

      16. associate-*r/ N/A

          \[\leadsto \color{blue}{\frac{1 \cdot x}{\left(\left(y + x\right) - -1\right) \cdot \left(y + x\right)}} \]

   10. Applied rewrites 65.9%

       \[\leadsto \color{blue}{\frac{1 \cdot x}{\left(x - \left(-1 - y\right)\right) \cdot \left(x + y\right)}} \]

   if 1.79999999999999989e156 < y

   1. Initial program 69.1%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot y}}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      3. *-commutative N/A

         \[\leadsto \frac{\color{blue}{y \cdot x}}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right)} \cdot \left(\left(x + y\right) + 1\right)} \]

      6. associate-*l* N/A

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}} \]

      7. times-frac N/A

         \[\leadsto \color{blue}{\frac{y}{x + y} \cdot \frac{x}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      8. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{y}{x + y} \cdot \frac{x}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      9. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{y}{x + y}} \cdot \frac{x}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      10. lift-+.f64 N/A

          \[\leadsto \frac{y}{\color{blue}{x + y}} \cdot \frac{x}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      11. +-commutative N/A

          \[\leadsto \frac{y}{\color{blue}{y + x}} \cdot \frac{x}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      12. lower-+.f64 N/A

          \[\leadsto \frac{y}{\color{blue}{y + x}} \cdot \frac{x}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      13. lower-/.f64 N/A

          \[\leadsto \frac{y}{y + x} \cdot \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      14. *-commutative N/A

          \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\color{blue}{\left(\left(x + y\right) + 1\right) \cdot \left(x + y\right)}} \]

      15. lower-*.f64 93.6

          \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\color{blue}{\left(\left(x + y\right) + 1\right) \cdot \left(x + y\right)}} \]

      16. lift-+.f64 N/A

          \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\color{blue}{\left(\left(x + y\right) + 1\right)} \cdot \left(x + y\right)} \]

      17. add-flip N/A

          \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\color{blue}{\left(\left(x + y\right) - \left(\mathsf{neg}\left(1\right)\right)\right)} \cdot \left(x + y\right)} \]

      18. lower--.f64 N/A

          \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\color{blue}{\left(\left(x + y\right) - \left(\mathsf{neg}\left(1\right)\right)\right)} \cdot \left(x + y\right)} \]

      19. lift-+.f64 N/A

          \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\left(\color{blue}{\left(x + y\right)} - \left(\mathsf{neg}\left(1\right)\right)\right) \cdot \left(x + y\right)} \]

      20. +-commutative N/A

          \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\left(\color{blue}{\left(y + x\right)} - \left(\mathsf{neg}\left(1\right)\right)\right) \cdot \left(x + y\right)} \]

      21. lower-+.f64 N/A

          \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\left(\color{blue}{\left(y + x\right)} - \left(\mathsf{neg}\left(1\right)\right)\right) \cdot \left(x + y\right)} \]

      22. metadata-eval 93.6

          \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\left(\left(y + x\right) - \color{blue}{-1}\right) \cdot \left(x + y\right)} \]

      23. lift-+.f64 N/A

          \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\left(\left(y + x\right) - -1\right) \cdot \color{blue}{\left(x + y\right)}} \]

      24. +-commutative N/A

          \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\left(\left(y + x\right) - -1\right) \cdot \color{blue}{\left(y + x\right)}} \]

      25. lower-+.f64 93.6

          \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\left(\left(y + x\right) - -1\right) \cdot \color{blue}{\left(y + x\right)}} \]

   3. Applied rewrites 93.6%

      \[\leadsto \color{blue}{\frac{y}{y + x} \cdot \frac{x}{\left(\left(y + x\right) - -1\right) \cdot \left(y + x\right)}} \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{y}{y + x} \cdot \frac{x}{\left(\left(y + x\right) - -1\right) \cdot \left(y + x\right)}} \]

      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. div-flip N/A

         \[\leadsto \frac{y}{y + x} \cdot \color{blue}{\frac{1}{\frac{\left(\left(y + x\right) - -1\right) \cdot \left(y + x\right)}{x}}} \]

      4. mult-flip-rev N/A

         \[\leadsto \color{blue}{\frac{\frac{y}{y + x}}{\frac{\left(\left(y + x\right) - -1\right) \cdot \left(y + x\right)}{x}}} \]

      5. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{y}{y + x}}{\frac{\left(\left(y + x\right) - -1\right) \cdot \left(y + x\right)}{x}}} \]

      6. lift-+.f64 N/A

         \[\leadsto \frac{\frac{y}{\color{blue}{y + x}}}{\frac{\left(\left(y + x\right) - -1\right) \cdot \left(y + x\right)}{x}} \]

      7. +-commutative N/A

         \[\leadsto \frac{\frac{y}{\color{blue}{x + y}}}{\frac{\left(\left(y + x\right) - -1\right) \cdot \left(y + x\right)}{x}} \]

      8. lift-+.f64 N/A

         \[\leadsto \frac{\frac{y}{\color{blue}{x + y}}}{\frac{\left(\left(y + x\right) - -1\right) \cdot \left(y + x\right)}{x}} \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{\frac{y}{x + y}}{\frac{\color{blue}{\left(\left(y + x\right) - -1\right) \cdot \left(y + x\right)}}{x}} \]

      10. associate-/l* N/A

          \[\leadsto \frac{\frac{y}{x + y}}{\color{blue}{\left(\left(y + x\right) - -1\right) \cdot \frac{y + x}{x}}} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{\frac{y}{x + y}}{\color{blue}{\left(\left(y + x\right) - -1\right) \cdot \frac{y + x}{x}}} \]

      12. lift-+.f64 N/A

          \[\leadsto \frac{\frac{y}{x + y}}{\left(\color{blue}{\left(y + x\right)} - -1\right) \cdot \frac{y + x}{x}} \]

      13. +-commutative N/A

          \[\leadsto \frac{\frac{y}{x + y}}{\left(\color{blue}{\left(x + y\right)} - -1\right) \cdot \frac{y + x}{x}} \]

      14. lift-+.f64 N/A

          \[\leadsto \frac{\frac{y}{x + y}}{\left(\color{blue}{\left(x + y\right)} - -1\right) \cdot \frac{y + x}{x}} \]

      15. lower-/.f64 99.3

          \[\leadsto \frac{\frac{y}{x + y}}{\left(\left(x + y\right) - -1\right) \cdot \color{blue}{\frac{y + x}{x}}} \]

      16. lift-+.f64 N/A

          \[\leadsto \frac{\frac{y}{x + y}}{\left(\left(x + y\right) - -1\right) \cdot \frac{\color{blue}{y + x}}{x}} \]

      17. +-commutative N/A

          \[\leadsto \frac{\frac{y}{x + y}}{\left(\left(x + y\right) - -1\right) \cdot \frac{\color{blue}{x + y}}{x}} \]

      18. lift-+.f64 99.3

          \[\leadsto \frac{\frac{y}{x + y}}{\left(\left(x + y\right) - -1\right) \cdot \frac{\color{blue}{x + y}}{x}} \]

   5. Applied rewrites 99.3%

      \[\leadsto \color{blue}{\frac{\frac{y}{x + y}}{\left(\left(x + y\right) - -1\right) \cdot \frac{x + y}{x}}} \]

   6. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{1}}{\left(\left(x + y\right) - -1\right) \cdot \frac{x + y}{x}} \]
   7. Step-by-step derivation

   8. Applied rewrites 50.9%

      \[\leadsto \frac{\color{blue}{1}}{\left(\left(x + y\right) - -1\right) \cdot \frac{x + y}{x}} \]
   9. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{\left(\left(x + y\right) - -1\right) \cdot \frac{x + y}{x}}} \]

      2. mult-flip N/A

         \[\leadsto \color{blue}{1 \cdot \frac{1}{\left(\left(x + y\right) - -1\right) \cdot \frac{x + y}{x}}} \]

      3. lift-*.f64 N/A

         \[\leadsto 1 \cdot \frac{1}{\color{blue}{\left(\left(x + y\right) - -1\right) \cdot \frac{x + y}{x}}} \]

      4. lift--.f64 N/A

         \[\leadsto 1 \cdot \frac{1}{\color{blue}{\left(\left(x + y\right) - -1\right)} \cdot \frac{x + y}{x}} \]

      5. metadata-eval N/A

         \[\leadsto 1 \cdot \frac{1}{\left(\left(x + y\right) - \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right) \cdot \frac{x + y}{x}} \]

      6. add-flip N/A

         \[\leadsto 1 \cdot \frac{1}{\color{blue}{\left(\left(x + y\right) + 1\right)} \cdot \frac{x + y}{x}} \]

      7. lift-+.f64 N/A

         \[\leadsto 1 \cdot \frac{1}{\left(\color{blue}{\left(x + y\right)} + 1\right) \cdot \frac{x + y}{x}} \]

      8. lift-/.f64 N/A

         \[\leadsto 1 \cdot \frac{1}{\left(\left(x + y\right) + 1\right) \cdot \color{blue}{\frac{x + y}{x}}} \]

      9. associate-*r/ N/A

         \[\leadsto 1 \cdot \frac{1}{\color{blue}{\frac{\left(\left(x + y\right) + 1\right) \cdot \left(x + y\right)}{x}}} \]

      10. +-commutative N/A

          \[\leadsto 1 \cdot \frac{1}{\frac{\left(\color{blue}{\left(y + x\right)} + 1\right) \cdot \left(x + y\right)}{x}} \]

      11. metadata-eval N/A

          \[\leadsto 1 \cdot \frac{1}{\frac{\left(\left(y + x\right) + \color{blue}{\left(\mathsf{neg}\left(-1\right)\right)}\right) \cdot \left(x + y\right)}{x}} \]

      12. sub-flip N/A

          \[\leadsto 1 \cdot \frac{1}{\frac{\color{blue}{\left(\left(y + x\right) - -1\right)} \cdot \left(x + y\right)}{x}} \]

      13. lift-+.f64 N/A

          \[\leadsto 1 \cdot \frac{1}{\frac{\left(\left(y + x\right) - -1\right) \cdot \color{blue}{\left(x + y\right)}}{x}} \]

      14. +-commutative N/A

          \[\leadsto 1 \cdot \frac{1}{\frac{\left(\left(y + x\right) - -1\right) \cdot \color{blue}{\left(y + x\right)}}{x}} \]

      15. div-flip N/A

          \[\leadsto 1 \cdot \color{blue}{\frac{x}{\left(\left(y + x\right) - -1\right) \cdot \left(y + x\right)}} \]

      16. associate-/r* N/A

          \[\leadsto 1 \cdot \color{blue}{\frac{\frac{x}{\left(y + x\right) - -1}}{y + x}} \]

      17. +-commutative N/A

          \[\leadsto 1 \cdot \frac{\frac{x}{\left(y + x\right) - -1}}{\color{blue}{x + y}} \]

      18. lift-+.f64 N/A

          \[\leadsto 1 \cdot \frac{\frac{x}{\left(y + x\right) - -1}}{\color{blue}{x + y}} \]

   10. Applied rewrites 51.5%

       \[\leadsto \color{blue}{\frac{1 \cdot \frac{x}{x - \left(-1 - y\right)}}{x + y}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 7: 86.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_0 := x - \left(-1 - y\right)\\ \mathbf{if}\;y \leq 4.4 \cdot 10^{-195}:\\ \;\;\;\;\frac{\frac{y}{x + y}}{1 + x}\\ \mathbf{elif}\;y \leq 1.8 \cdot 10^{+156}:\\ \;\;\;\;\frac{1 \cdot x}{t\_0 \cdot \left(x + y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{t\_0} \cdot \frac{x}{x + y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- x (- -1.0 y))))
   (if (<= y 4.4e-195)
     (/ (/ y (+ x y)) (+ 1.0 x))
     (if (<= y 1.8e+156)
       (/ (* 1.0 x) (* t_0 (+ x y)))
       (* (/ 1.0 t_0) (/ x (+ x y)))))))

C

assert(x < y);
double code(double x, double y) {
	double t_0 = x - (-1.0 - y);
	double tmp;
	if (y <= 4.4e-195) {
		tmp = (y / (x + y)) / (1.0 + x);
	} else if (y <= 1.8e+156) {
		tmp = (1.0 * x) / (t_0 * (x + y));
	} else {
		tmp = (1.0 / t_0) * (x / (x + y));
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x - ((-1.0d0) - y)
    if (y <= 4.4d-195) then
        tmp = (y / (x + y)) / (1.0d0 + x)
    else if (y <= 1.8d+156) then
        tmp = (1.0d0 * x) / (t_0 * (x + y))
    else
        tmp = (1.0d0 / t_0) * (x / (x + y))
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double t_0 = x - (-1.0 - y);
	double tmp;
	if (y <= 4.4e-195) {
		tmp = (y / (x + y)) / (1.0 + x);
	} else if (y <= 1.8e+156) {
		tmp = (1.0 * x) / (t_0 * (x + y));
	} else {
		tmp = (1.0 / t_0) * (x / (x + y));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	t_0 = x - (-1.0 - y)
	tmp = 0
	if y <= 4.4e-195:
		tmp = (y / (x + y)) / (1.0 + x)
	elif y <= 1.8e+156:
		tmp = (1.0 * x) / (t_0 * (x + y))
	else:
		tmp = (1.0 / t_0) * (x / (x + y))
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	t_0 = Float64(x - Float64(-1.0 - y))
	tmp = 0.0
	if (y <= 4.4e-195)
		tmp = Float64(Float64(y / Float64(x + y)) / Float64(1.0 + x));
	elseif (y <= 1.8e+156)
		tmp = Float64(Float64(1.0 * x) / Float64(t_0 * Float64(x + y)));
	else
		tmp = Float64(Float64(1.0 / t_0) * Float64(x / Float64(x + y)));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	t_0 = x - (-1.0 - y);
	tmp = 0.0;
	if (y <= 4.4e-195)
		tmp = (y / (x + y)) / (1.0 + x);
	elseif (y <= 1.8e+156)
		tmp = (1.0 * x) / (t_0 * (x + y));
	else
		tmp = (1.0 / t_0) * (x / (x + y));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := Block[{t$95$0 = N[(x - N[(-1.0 - y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, 4.4e-195], N[(N[(y / N[(x + y), $MachinePrecision]), $MachinePrecision] / N[(1.0 + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.8e+156], N[(N[(1.0 * x), $MachinePrecision] / N[(t$95$0 * N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / t$95$0), $MachinePrecision] * N[(x / N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < 4.40000000000000011e-195

   1. Initial program 69.1%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot y}}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      3. *-commutative N/A

         \[\leadsto \frac{\color{blue}{y \cdot x}}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right)} \cdot \left(\left(x + y\right) + 1\right)} \]

      6. associate-*l* N/A

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}} \]

      7. times-frac N/A

         \[\leadsto \color{blue}{\frac{y}{x + y} \cdot \frac{x}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      8. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{y}{x + y} \cdot \frac{x}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      9. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{y}{x + y}} \cdot \frac{x}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      10. lift-+.f64 N/A

          \[\leadsto \frac{y}{\color{blue}{x + y}} \cdot \frac{x}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      11. +-commutative N/A

          \[\leadsto \frac{y}{\color{blue}{y + x}} \cdot \frac{x}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      12. lower-+.f64 N/A

          \[\leadsto \frac{y}{\color{blue}{y + x}} \cdot \frac{x}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      13. lower-/.f64 N/A

          \[\leadsto \frac{y}{y + x} \cdot \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      14. *-commutative N/A

          \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\color{blue}{\left(\left(x + y\right) + 1\right) \cdot \left(x + y\right)}} \]

      15. lower-*.f64 93.6

          \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\color{blue}{\left(\left(x + y\right) + 1\right) \cdot \left(x + y\right)}} \]

      16. lift-+.f64 N/A

          \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\color{blue}{\left(\left(x + y\right) + 1\right)} \cdot \left(x + y\right)} \]

      17. add-flip N/A

          \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\color{blue}{\left(\left(x + y\right) - \left(\mathsf{neg}\left(1\right)\right)\right)} \cdot \left(x + y\right)} \]

      18. lower--.f64 N/A

          \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\color{blue}{\left(\left(x + y\right) - \left(\mathsf{neg}\left(1\right)\right)\right)} \cdot \left(x + y\right)} \]

      19. lift-+.f64 N/A

          \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\left(\color{blue}{\left(x + y\right)} - \left(\mathsf{neg}\left(1\right)\right)\right) \cdot \left(x + y\right)} \]

      20. +-commutative N/A

          \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\left(\color{blue}{\left(y + x\right)} - \left(\mathsf{neg}\left(1\right)\right)\right) \cdot \left(x + y\right)} \]

      21. lower-+.f64 N/A

          \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\left(\color{blue}{\left(y + x\right)} - \left(\mathsf{neg}\left(1\right)\right)\right) \cdot \left(x + y\right)} \]

      22. metadata-eval 93.6

          \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\left(\left(y + x\right) - \color{blue}{-1}\right) \cdot \left(x + y\right)} \]

      23. lift-+.f64 N/A

          \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\left(\left(y + x\right) - -1\right) \cdot \color{blue}{\left(x + y\right)}} \]

      24. +-commutative N/A

          \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\left(\left(y + x\right) - -1\right) \cdot \color{blue}{\left(y + x\right)}} \]

      25. lower-+.f64 93.6

          \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\left(\left(y + x\right) - -1\right) \cdot \color{blue}{\left(y + x\right)}} \]

   3. Applied rewrites 93.6%

      \[\leadsto \color{blue}{\frac{y}{y + x} \cdot \frac{x}{\left(\left(y + x\right) - -1\right) \cdot \left(y + x\right)}} \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{y}{y + x} \cdot \frac{x}{\left(\left(y + x\right) - -1\right) \cdot \left(y + x\right)}} \]

      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. div-flip N/A

         \[\leadsto \frac{y}{y + x} \cdot \color{blue}{\frac{1}{\frac{\left(\left(y + x\right) - -1\right) \cdot \left(y + x\right)}{x}}} \]

      4. mult-flip-rev N/A

         \[\leadsto \color{blue}{\frac{\frac{y}{y + x}}{\frac{\left(\left(y + x\right) - -1\right) \cdot \left(y + x\right)}{x}}} \]

      5. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{y}{y + x}}{\frac{\left(\left(y + x\right) - -1\right) \cdot \left(y + x\right)}{x}}} \]

      6. lift-+.f64 N/A

         \[\leadsto \frac{\frac{y}{\color{blue}{y + x}}}{\frac{\left(\left(y + x\right) - -1\right) \cdot \left(y + x\right)}{x}} \]

      7. +-commutative N/A

         \[\leadsto \frac{\frac{y}{\color{blue}{x + y}}}{\frac{\left(\left(y + x\right) - -1\right) \cdot \left(y + x\right)}{x}} \]

      8. lift-+.f64 N/A

         \[\leadsto \frac{\frac{y}{\color{blue}{x + y}}}{\frac{\left(\left(y + x\right) - -1\right) \cdot \left(y + x\right)}{x}} \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{\frac{y}{x + y}}{\frac{\color{blue}{\left(\left(y + x\right) - -1\right) \cdot \left(y + x\right)}}{x}} \]

      10. associate-/l* N/A

          \[\leadsto \frac{\frac{y}{x + y}}{\color{blue}{\left(\left(y + x\right) - -1\right) \cdot \frac{y + x}{x}}} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{\frac{y}{x + y}}{\color{blue}{\left(\left(y + x\right) - -1\right) \cdot \frac{y + x}{x}}} \]

      12. lift-+.f64 N/A

          \[\leadsto \frac{\frac{y}{x + y}}{\left(\color{blue}{\left(y + x\right)} - -1\right) \cdot \frac{y + x}{x}} \]

      13. +-commutative N/A

          \[\leadsto \frac{\frac{y}{x + y}}{\left(\color{blue}{\left(x + y\right)} - -1\right) \cdot \frac{y + x}{x}} \]

      14. lift-+.f64 N/A

          \[\leadsto \frac{\frac{y}{x + y}}{\left(\color{blue}{\left(x + y\right)} - -1\right) \cdot \frac{y + x}{x}} \]

      15. lower-/.f64 99.3

          \[\leadsto \frac{\frac{y}{x + y}}{\left(\left(x + y\right) - -1\right) \cdot \color{blue}{\frac{y + x}{x}}} \]

      16. lift-+.f64 N/A

          \[\leadsto \frac{\frac{y}{x + y}}{\left(\left(x + y\right) - -1\right) \cdot \frac{\color{blue}{y + x}}{x}} \]

      17. +-commutative N/A

          \[\leadsto \frac{\frac{y}{x + y}}{\left(\left(x + y\right) - -1\right) \cdot \frac{\color{blue}{x + y}}{x}} \]

      18. lift-+.f64 99.3

          \[\leadsto \frac{\frac{y}{x + y}}{\left(\left(x + y\right) - -1\right) \cdot \frac{\color{blue}{x + y}}{x}} \]

   5. Applied rewrites 99.3%

      \[\leadsto \color{blue}{\frac{\frac{y}{x + y}}{\left(\left(x + y\right) - -1\right) \cdot \frac{x + y}{x}}} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\frac{y}{x + y}}{\color{blue}{\left(\left(x + y\right) - -1\right) \cdot \frac{x + y}{x}}} \]

      2. lift--.f64 N/A

         \[\leadsto \frac{\frac{y}{x + y}}{\color{blue}{\left(\left(x + y\right) - -1\right)} \cdot \frac{x + y}{x}} \]

      3. metadata-eval N/A

         \[\leadsto \frac{\frac{y}{x + y}}{\left(\left(x + y\right) - \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right) \cdot \frac{x + y}{x}} \]

      4. add-flip N/A

         \[\leadsto \frac{\frac{y}{x + y}}{\color{blue}{\left(\left(x + y\right) + 1\right)} \cdot \frac{x + y}{x}} \]

      5. lift-+.f64 N/A

         \[\leadsto \frac{\frac{y}{x + y}}{\left(\color{blue}{\left(x + y\right)} + 1\right) \cdot \frac{x + y}{x}} \]

      6. *-commutative N/A

         \[\leadsto \frac{\frac{y}{x + y}}{\color{blue}{\frac{x + y}{x} \cdot \left(\left(x + y\right) + 1\right)}} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{\frac{y}{x + y}}{\color{blue}{\frac{x + y}{x} \cdot \left(\left(x + y\right) + 1\right)}} \]

      8. associate-+l+ N/A

         \[\leadsto \frac{\frac{y}{x + y}}{\frac{x + y}{x} \cdot \color{blue}{\left(x + \left(y + 1\right)\right)}} \]

      9. metadata-eval N/A

         \[\leadsto \frac{\frac{y}{x + y}}{\frac{x + y}{x} \cdot \left(x + \left(y + \color{blue}{\left(\mathsf{neg}\left(-1\right)\right)}\right)\right)} \]

      10. sub-flip N/A

          \[\leadsto \frac{\frac{y}{x + y}}{\frac{x + y}{x} \cdot \left(x + \color{blue}{\left(y - -1\right)}\right)} \]

      11. sub-negate-rev N/A

          \[\leadsto \frac{\frac{y}{x + y}}{\frac{x + y}{x} \cdot \left(x + \color{blue}{\left(\mathsf{neg}\left(\left(-1 - y\right)\right)\right)}\right)} \]

      12. sub-flip-reverse N/A

          \[\leadsto \frac{\frac{y}{x + y}}{\frac{x + y}{x} \cdot \color{blue}{\left(x - \left(-1 - y\right)\right)}} \]

      13. lower--.f64 N/A

          \[\leadsto \frac{\frac{y}{x + y}}{\frac{x + y}{x} \cdot \color{blue}{\left(x - \left(-1 - y\right)\right)}} \]

      14. lower--.f64 99.3

          \[\leadsto \frac{\frac{y}{x + y}}{\frac{x + y}{x} \cdot \left(x - \color{blue}{\left(-1 - y\right)}\right)} \]

   7. Applied rewrites 99.3%

      \[\leadsto \frac{\frac{y}{x + y}}{\color{blue}{\frac{x + y}{x} \cdot \left(x - \left(-1 - y\right)\right)}} \]

   8. Taylor expanded in y around 0

      \[\leadsto \frac{\frac{y}{x + y}}{\color{blue}{1 + x}} \]
   9. Step-by-step derivation

      1. lower-+.f64 51.3

         \[\leadsto \frac{\frac{y}{x + y}}{1 + \color{blue}{x}} \]

   10. Applied rewrites 51.3%

       \[\leadsto \frac{\frac{y}{x + y}}{\color{blue}{1 + x}} \]

   if 4.40000000000000011e-195 < y < 1.79999999999999989e156

   1. Initial program 69.1%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot y}}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      3. *-commutative N/A

         \[\leadsto \frac{\color{blue}{y \cdot x}}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right)} \cdot \left(\left(x + y\right) + 1\right)} \]

      6. associate-*l* N/A

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}} \]

      7. times-frac N/A

         \[\leadsto \color{blue}{\frac{y}{x + y} \cdot \frac{x}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      8. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{y}{x + y} \cdot \frac{x}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      9. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{y}{x + y}} \cdot \frac{x}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      10. lift-+.f64 N/A

          \[\leadsto \frac{y}{\color{blue}{x + y}} \cdot \frac{x}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      11. +-commutative N/A

          \[\leadsto \frac{y}{\color{blue}{y + x}} \cdot \frac{x}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      12. lower-+.f64 N/A

          \[\leadsto \frac{y}{\color{blue}{y + x}} \cdot \frac{x}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      13. lower-/.f64 N/A

          \[\leadsto \frac{y}{y + x} \cdot \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      14. *-commutative N/A

          \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\color{blue}{\left(\left(x + y\right) + 1\right) \cdot \left(x + y\right)}} \]

      15. lower-*.f64 93.6

          \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\color{blue}{\left(\left(x + y\right) + 1\right) \cdot \left(x + y\right)}} \]

      16. lift-+.f64 N/A

          \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\color{blue}{\left(\left(x + y\right) + 1\right)} \cdot \left(x + y\right)} \]

      17. add-flip N/A

          \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\color{blue}{\left(\left(x + y\right) - \left(\mathsf{neg}\left(1\right)\right)\right)} \cdot \left(x + y\right)} \]

      18. lower--.f64 N/A

          \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\color{blue}{\left(\left(x + y\right) - \left(\mathsf{neg}\left(1\right)\right)\right)} \cdot \left(x + y\right)} \]

      19. lift-+.f64 N/A

          \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\left(\color{blue}{\left(x + y\right)} - \left(\mathsf{neg}\left(1\right)\right)\right) \cdot \left(x + y\right)} \]

      20. +-commutative N/A

          \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\left(\color{blue}{\left(y + x\right)} - \left(\mathsf{neg}\left(1\right)\right)\right) \cdot \left(x + y\right)} \]

      21. lower-+.f64 N/A

          \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\left(\color{blue}{\left(y + x\right)} - \left(\mathsf{neg}\left(1\right)\right)\right) \cdot \left(x + y\right)} \]

      22. metadata-eval 93.6

          \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\left(\left(y + x\right) - \color{blue}{-1}\right) \cdot \left(x + y\right)} \]

      23. lift-+.f64 N/A

          \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\left(\left(y + x\right) - -1\right) \cdot \color{blue}{\left(x + y\right)}} \]

      24. +-commutative N/A

          \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\left(\left(y + x\right) - -1\right) \cdot \color{blue}{\left(y + x\right)}} \]

      25. lower-+.f64 93.6

          \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\left(\left(y + x\right) - -1\right) \cdot \color{blue}{\left(y + x\right)}} \]

   3. Applied rewrites 93.6%

      \[\leadsto \color{blue}{\frac{y}{y + x} \cdot \frac{x}{\left(\left(y + x\right) - -1\right) \cdot \left(y + x\right)}} \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{y}{y + x} \cdot \frac{x}{\left(\left(y + x\right) - -1\right) \cdot \left(y + x\right)}} \]

      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. div-flip N/A

         \[\leadsto \frac{y}{y + x} \cdot \color{blue}{\frac{1}{\frac{\left(\left(y + x\right) - -1\right) \cdot \left(y + x\right)}{x}}} \]

      4. mult-flip-rev N/A

         \[\leadsto \color{blue}{\frac{\frac{y}{y + x}}{\frac{\left(\left(y + x\right) - -1\right) \cdot \left(y + x\right)}{x}}} \]

      5. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{y}{y + x}}{\frac{\left(\left(y + x\right) - -1\right) \cdot \left(y + x\right)}{x}}} \]

      6. lift-+.f64 N/A

         \[\leadsto \frac{\frac{y}{\color{blue}{y + x}}}{\frac{\left(\left(y + x\right) - -1\right) \cdot \left(y + x\right)}{x}} \]

      7. +-commutative N/A

         \[\leadsto \frac{\frac{y}{\color{blue}{x + y}}}{\frac{\left(\left(y + x\right) - -1\right) \cdot \left(y + x\right)}{x}} \]

      8. lift-+.f64 N/A

         \[\leadsto \frac{\frac{y}{\color{blue}{x + y}}}{\frac{\left(\left(y + x\right) - -1\right) \cdot \left(y + x\right)}{x}} \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{\frac{y}{x + y}}{\frac{\color{blue}{\left(\left(y + x\right) - -1\right) \cdot \left(y + x\right)}}{x}} \]

      10. associate-/l* N/A

          \[\leadsto \frac{\frac{y}{x + y}}{\color{blue}{\left(\left(y + x\right) - -1\right) \cdot \frac{y + x}{x}}} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{\frac{y}{x + y}}{\color{blue}{\left(\left(y + x\right) - -1\right) \cdot \frac{y + x}{x}}} \]

      12. lift-+.f64 N/A

          \[\leadsto \frac{\frac{y}{x + y}}{\left(\color{blue}{\left(y + x\right)} - -1\right) \cdot \frac{y + x}{x}} \]

      13. +-commutative N/A

          \[\leadsto \frac{\frac{y}{x + y}}{\left(\color{blue}{\left(x + y\right)} - -1\right) \cdot \frac{y + x}{x}} \]

      14. lift-+.f64 N/A

          \[\leadsto \frac{\frac{y}{x + y}}{\left(\color{blue}{\left(x + y\right)} - -1\right) \cdot \frac{y + x}{x}} \]

      15. lower-/.f64 99.3

          \[\leadsto \frac{\frac{y}{x + y}}{\left(\left(x + y\right) - -1\right) \cdot \color{blue}{\frac{y + x}{x}}} \]

      16. lift-+.f64 N/A

          \[\leadsto \frac{\frac{y}{x + y}}{\left(\left(x + y\right) - -1\right) \cdot \frac{\color{blue}{y + x}}{x}} \]

      17. +-commutative N/A

          \[\leadsto \frac{\frac{y}{x + y}}{\left(\left(x + y\right) - -1\right) \cdot \frac{\color{blue}{x + y}}{x}} \]

      18. lift-+.f64 99.3

          \[\leadsto \frac{\frac{y}{x + y}}{\left(\left(x + y\right) - -1\right) \cdot \frac{\color{blue}{x + y}}{x}} \]

   5. Applied rewrites 99.3%

      \[\leadsto \color{blue}{\frac{\frac{y}{x + y}}{\left(\left(x + y\right) - -1\right) \cdot \frac{x + y}{x}}} \]

   6. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{1}}{\left(\left(x + y\right) - -1\right) \cdot \frac{x + y}{x}} \]
   7. Step-by-step derivation

   8. Applied rewrites 50.9%

      \[\leadsto \frac{\color{blue}{1}}{\left(\left(x + y\right) - -1\right) \cdot \frac{x + y}{x}} \]
   9. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{\left(\left(x + y\right) - -1\right) \cdot \frac{x + y}{x}}} \]

      2. mult-flip N/A

         \[\leadsto \color{blue}{1 \cdot \frac{1}{\left(\left(x + y\right) - -1\right) \cdot \frac{x + y}{x}}} \]

      3. lift-*.f64 N/A

         \[\leadsto 1 \cdot \frac{1}{\color{blue}{\left(\left(x + y\right) - -1\right) \cdot \frac{x + y}{x}}} \]

      4. lift--.f64 N/A

         \[\leadsto 1 \cdot \frac{1}{\color{blue}{\left(\left(x + y\right) - -1\right)} \cdot \frac{x + y}{x}} \]

      5. metadata-eval N/A

         \[\leadsto 1 \cdot \frac{1}{\left(\left(x + y\right) - \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right) \cdot \frac{x + y}{x}} \]

      6. add-flip N/A

         \[\leadsto 1 \cdot \frac{1}{\color{blue}{\left(\left(x + y\right) + 1\right)} \cdot \frac{x + y}{x}} \]

      7. lift-+.f64 N/A

         \[\leadsto 1 \cdot \frac{1}{\left(\color{blue}{\left(x + y\right)} + 1\right) \cdot \frac{x + y}{x}} \]

      8. lift-/.f64 N/A

         \[\leadsto 1 \cdot \frac{1}{\left(\left(x + y\right) + 1\right) \cdot \color{blue}{\frac{x + y}{x}}} \]

      9. associate-*r/ N/A

         \[\leadsto 1 \cdot \frac{1}{\color{blue}{\frac{\left(\left(x + y\right) + 1\right) \cdot \left(x + y\right)}{x}}} \]

      10. +-commutative N/A

          \[\leadsto 1 \cdot \frac{1}{\frac{\left(\color{blue}{\left(y + x\right)} + 1\right) \cdot \left(x + y\right)}{x}} \]

      11. metadata-eval N/A

          \[\leadsto 1 \cdot \frac{1}{\frac{\left(\left(y + x\right) + \color{blue}{\left(\mathsf{neg}\left(-1\right)\right)}\right) \cdot \left(x + y\right)}{x}} \]

      12. sub-flip N/A

          \[\leadsto 1 \cdot \frac{1}{\frac{\color{blue}{\left(\left(y + x\right) - -1\right)} \cdot \left(x + y\right)}{x}} \]

      13. lift-+.f64 N/A

          \[\leadsto 1 \cdot \frac{1}{\frac{\left(\left(y + x\right) - -1\right) \cdot \color{blue}{\left(x + y\right)}}{x}} \]

      14. +-commutative N/A

          \[\leadsto 1 \cdot \frac{1}{\frac{\left(\left(y + x\right) - -1\right) \cdot \color{blue}{\left(y + x\right)}}{x}} \]

      15. div-flip N/A

          \[\leadsto 1 \cdot \color{blue}{\frac{x}{\left(\left(y + x\right) - -1\right) \cdot \left(y + x\right)}} \]

      16. associate-*r/ N/A

          \[\leadsto \color{blue}{\frac{1 \cdot x}{\left(\left(y + x\right) - -1\right) \cdot \left(y + x\right)}} \]

   10. Applied rewrites 65.9%

       \[\leadsto \color{blue}{\frac{1 \cdot x}{\left(x - \left(-1 - y\right)\right) \cdot \left(x + y\right)}} \]

   if 1.79999999999999989e156 < y

   1. Initial program 69.1%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot y}}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      3. *-commutative N/A

         \[\leadsto \frac{\color{blue}{y \cdot x}}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right)} \cdot \left(\left(x + y\right) + 1\right)} \]

      6. associate-*l* N/A

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}} \]

      7. times-frac N/A

         \[\leadsto \color{blue}{\frac{y}{x + y} \cdot \frac{x}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      8. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{y}{x + y} \cdot \frac{x}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      9. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{y}{x + y}} \cdot \frac{x}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      10. lift-+.f64 N/A

          \[\leadsto \frac{y}{\color{blue}{x + y}} \cdot \frac{x}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      11. +-commutative N/A

          \[\leadsto \frac{y}{\color{blue}{y + x}} \cdot \frac{x}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      12. lower-+.f64 N/A

          \[\leadsto \frac{y}{\color{blue}{y + x}} \cdot \frac{x}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      13. lower-/.f64 N/A

          \[\leadsto \frac{y}{y + x} \cdot \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      14. *-commutative N/A

          \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\color{blue}{\left(\left(x + y\right) + 1\right) \cdot \left(x + y\right)}} \]

      15. lower-*.f64 93.6

          \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\color{blue}{\left(\left(x + y\right) + 1\right) \cdot \left(x + y\right)}} \]

      16. lift-+.f64 N/A

          \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\color{blue}{\left(\left(x + y\right) + 1\right)} \cdot \left(x + y\right)} \]

      17. add-flip N/A

          \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\color{blue}{\left(\left(x + y\right) - \left(\mathsf{neg}\left(1\right)\right)\right)} \cdot \left(x + y\right)} \]

      18. lower--.f64 N/A

          \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\color{blue}{\left(\left(x + y\right) - \left(\mathsf{neg}\left(1\right)\right)\right)} \cdot \left(x + y\right)} \]

      19. lift-+.f64 N/A

          \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\left(\color{blue}{\left(x + y\right)} - \left(\mathsf{neg}\left(1\right)\right)\right) \cdot \left(x + y\right)} \]

      20. +-commutative N/A

          \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\left(\color{blue}{\left(y + x\right)} - \left(\mathsf{neg}\left(1\right)\right)\right) \cdot \left(x + y\right)} \]

      21. lower-+.f64 N/A

          \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\left(\color{blue}{\left(y + x\right)} - \left(\mathsf{neg}\left(1\right)\right)\right) \cdot \left(x + y\right)} \]

      22. metadata-eval 93.6

          \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\left(\left(y + x\right) - \color{blue}{-1}\right) \cdot \left(x + y\right)} \]

      23. lift-+.f64 N/A

          \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\left(\left(y + x\right) - -1\right) \cdot \color{blue}{\left(x + y\right)}} \]

      24. +-commutative N/A

          \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\left(\left(y + x\right) - -1\right) \cdot \color{blue}{\left(y + x\right)}} \]

      25. lower-+.f64 93.6

          \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\left(\left(y + x\right) - -1\right) \cdot \color{blue}{\left(y + x\right)}} \]

   3. Applied rewrites 93.6%

      \[\leadsto \color{blue}{\frac{y}{y + x} \cdot \frac{x}{\left(\left(y + x\right) - -1\right) \cdot \left(y + x\right)}} \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{y}{y + x} \cdot \frac{x}{\left(\left(y + x\right) - -1\right) \cdot \left(y + x\right)}} \]

      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. div-flip N/A

         \[\leadsto \frac{y}{y + x} \cdot \color{blue}{\frac{1}{\frac{\left(\left(y + x\right) - -1\right) \cdot \left(y + x\right)}{x}}} \]

      4. mult-flip-rev N/A

         \[\leadsto \color{blue}{\frac{\frac{y}{y + x}}{\frac{\left(\left(y + x\right) - -1\right) \cdot \left(y + x\right)}{x}}} \]

      5. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{y}{y + x}}{\frac{\left(\left(y + x\right) - -1\right) \cdot \left(y + x\right)}{x}}} \]

      6. lift-+.f64 N/A

         \[\leadsto \frac{\frac{y}{\color{blue}{y + x}}}{\frac{\left(\left(y + x\right) - -1\right) \cdot \left(y + x\right)}{x}} \]

      7. +-commutative N/A

         \[\leadsto \frac{\frac{y}{\color{blue}{x + y}}}{\frac{\left(\left(y + x\right) - -1\right) \cdot \left(y + x\right)}{x}} \]

      8. lift-+.f64 N/A

         \[\leadsto \frac{\frac{y}{\color{blue}{x + y}}}{\frac{\left(\left(y + x\right) - -1\right) \cdot \left(y + x\right)}{x}} \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{\frac{y}{x + y}}{\frac{\color{blue}{\left(\left(y + x\right) - -1\right) \cdot \left(y + x\right)}}{x}} \]

      10. associate-/l* N/A

          \[\leadsto \frac{\frac{y}{x + y}}{\color{blue}{\left(\left(y + x\right) - -1\right) \cdot \frac{y + x}{x}}} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{\frac{y}{x + y}}{\color{blue}{\left(\left(y + x\right) - -1\right) \cdot \frac{y + x}{x}}} \]

      12. lift-+.f64 N/A

          \[\leadsto \frac{\frac{y}{x + y}}{\left(\color{blue}{\left(y + x\right)} - -1\right) \cdot \frac{y + x}{x}} \]

      13. +-commutative N/A

          \[\leadsto \frac{\frac{y}{x + y}}{\left(\color{blue}{\left(x + y\right)} - -1\right) \cdot \frac{y + x}{x}} \]

      14. lift-+.f64 N/A

          \[\leadsto \frac{\frac{y}{x + y}}{\left(\color{blue}{\left(x + y\right)} - -1\right) \cdot \frac{y + x}{x}} \]

      15. lower-/.f64 99.3

          \[\leadsto \frac{\frac{y}{x + y}}{\left(\left(x + y\right) - -1\right) \cdot \color{blue}{\frac{y + x}{x}}} \]

      16. lift-+.f64 N/A

          \[\leadsto \frac{\frac{y}{x + y}}{\left(\left(x + y\right) - -1\right) \cdot \frac{\color{blue}{y + x}}{x}} \]

      17. +-commutative N/A

          \[\leadsto \frac{\frac{y}{x + y}}{\left(\left(x + y\right) - -1\right) \cdot \frac{\color{blue}{x + y}}{x}} \]

      18. lift-+.f64 99.3

          \[\leadsto \frac{\frac{y}{x + y}}{\left(\left(x + y\right) - -1\right) \cdot \frac{\color{blue}{x + y}}{x}} \]

   5. Applied rewrites 99.3%

      \[\leadsto \color{blue}{\frac{\frac{y}{x + y}}{\left(\left(x + y\right) - -1\right) \cdot \frac{x + y}{x}}} \]

   6. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{1}}{\left(\left(x + y\right) - -1\right) \cdot \frac{x + y}{x}} \]
   7. Step-by-step derivation

   8. Applied rewrites 50.9%

      \[\leadsto \frac{\color{blue}{1}}{\left(\left(x + y\right) - -1\right) \cdot \frac{x + y}{x}} \]
   9. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{\left(\left(x + y\right) - -1\right) \cdot \frac{x + y}{x}}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{1}{\color{blue}{\left(\left(x + y\right) - -1\right) \cdot \frac{x + y}{x}}} \]

      3. lift--.f64 N/A

         \[\leadsto \frac{1}{\color{blue}{\left(\left(x + y\right) - -1\right)} \cdot \frac{x + y}{x}} \]

      4. metadata-eval N/A

         \[\leadsto \frac{1}{\left(\left(x + y\right) - \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right) \cdot \frac{x + y}{x}} \]

      5. add-flip N/A

         \[\leadsto \frac{1}{\color{blue}{\left(\left(x + y\right) + 1\right)} \cdot \frac{x + y}{x}} \]

      6. lift-+.f64 N/A

         \[\leadsto \frac{1}{\left(\color{blue}{\left(x + y\right)} + 1\right) \cdot \frac{x + y}{x}} \]

      7. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{1}{\left(x + y\right) + 1}}{\frac{x + y}{x}}} \]

      8. mult-flip N/A

         \[\leadsto \color{blue}{\frac{1}{\left(x + y\right) + 1} \cdot \frac{1}{\frac{x + y}{x}}} \]

      9. lift-/.f64 N/A

         \[\leadsto \frac{1}{\left(x + y\right) + 1} \cdot \frac{1}{\color{blue}{\frac{x + y}{x}}} \]

      10. div-flip N/A

          \[\leadsto \frac{1}{\left(x + y\right) + 1} \cdot \color{blue}{\frac{x}{x + y}} \]

      11. lower-*.f64 N/A

          \[\leadsto \color{blue}{\frac{1}{\left(x + y\right) + 1} \cdot \frac{x}{x + y}} \]

   10. Applied rewrites 51.5%

       \[\leadsto \color{blue}{\frac{1}{x - \left(-1 - y\right)} \cdot \frac{x}{x + y}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 8: 86.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 4.4 \cdot 10^{-195}:\\ \;\;\;\;\frac{\frac{y}{x + y}}{1 + x}\\ \mathbf{elif}\;y \leq 1.8 \cdot 10^{+156}:\\ \;\;\;\;\frac{1 \cdot x}{\left(x - \left(-1 - y\right)\right) \cdot \left(x + y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y - -1}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 4.4e-195)
   (/ (/ y (+ x y)) (+ 1.0 x))
   (if (<= y 1.8e+156)
     (/ (* 1.0 x) (* (- x (- -1.0 y)) (+ x y)))
     (/ (/ x y) (- y -1.0)))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 4.4e-195) {
		tmp = (y / (x + y)) / (1.0 + x);
	} else if (y <= 1.8e+156) {
		tmp = (1.0 * x) / ((x - (-1.0 - y)) * (x + y));
	} else {
		tmp = (x / y) / (y - -1.0);
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 4.4d-195) then
        tmp = (y / (x + y)) / (1.0d0 + x)
    else if (y <= 1.8d+156) then
        tmp = (1.0d0 * x) / ((x - ((-1.0d0) - y)) * (x + y))
    else
        tmp = (x / y) / (y - (-1.0d0))
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (y <= 4.4e-195) {
		tmp = (y / (x + y)) / (1.0 + x);
	} else if (y <= 1.8e+156) {
		tmp = (1.0 * x) / ((x - (-1.0 - y)) * (x + y));
	} else {
		tmp = (x / y) / (y - -1.0);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if y <= 4.4e-195:
		tmp = (y / (x + y)) / (1.0 + x)
	elif y <= 1.8e+156:
		tmp = (1.0 * x) / ((x - (-1.0 - y)) * (x + y))
	else:
		tmp = (x / y) / (y - -1.0)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= 4.4e-195)
		tmp = Float64(Float64(y / Float64(x + y)) / Float64(1.0 + x));
	elseif (y <= 1.8e+156)
		tmp = Float64(Float64(1.0 * x) / Float64(Float64(x - Float64(-1.0 - y)) * Float64(x + y)));
	else
		tmp = Float64(Float64(x / y) / Float64(y - -1.0));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 4.4e-195)
		tmp = (y / (x + y)) / (1.0 + x);
	elseif (y <= 1.8e+156)
		tmp = (1.0 * x) / ((x - (-1.0 - y)) * (x + y));
	else
		tmp = (x / y) / (y - -1.0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[y, 4.4e-195], N[(N[(y / N[(x + y), $MachinePrecision]), $MachinePrecision] / N[(1.0 + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.8e+156], N[(N[(1.0 * x), $MachinePrecision] / N[(N[(x - N[(-1.0 - y), $MachinePrecision]), $MachinePrecision] * N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / N[(y - -1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 4.40000000000000011e-195

   1. Initial program 69.1%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot y}}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      3. *-commutative N/A

         \[\leadsto \frac{\color{blue}{y \cdot x}}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right)} \cdot \left(\left(x + y\right) + 1\right)} \]

      6. associate-*l* N/A

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}} \]

      7. times-frac N/A

         \[\leadsto \color{blue}{\frac{y}{x + y} \cdot \frac{x}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      8. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{y}{x + y} \cdot \frac{x}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      9. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{y}{x + y}} \cdot \frac{x}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      10. lift-+.f64 N/A

          \[\leadsto \frac{y}{\color{blue}{x + y}} \cdot \frac{x}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      11. +-commutative N/A

          \[\leadsto \frac{y}{\color{blue}{y + x}} \cdot \frac{x}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      12. lower-+.f64 N/A

          \[\leadsto \frac{y}{\color{blue}{y + x}} \cdot \frac{x}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      13. lower-/.f64 N/A

          \[\leadsto \frac{y}{y + x} \cdot \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      14. *-commutative N/A

          \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\color{blue}{\left(\left(x + y\right) + 1\right) \cdot \left(x + y\right)}} \]

      15. lower-*.f64 93.6

          \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\color{blue}{\left(\left(x + y\right) + 1\right) \cdot \left(x + y\right)}} \]

      16. lift-+.f64 N/A

          \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\color{blue}{\left(\left(x + y\right) + 1\right)} \cdot \left(x + y\right)} \]

      17. add-flip N/A

          \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\color{blue}{\left(\left(x + y\right) - \left(\mathsf{neg}\left(1\right)\right)\right)} \cdot \left(x + y\right)} \]

      18. lower--.f64 N/A

          \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\color{blue}{\left(\left(x + y\right) - \left(\mathsf{neg}\left(1\right)\right)\right)} \cdot \left(x + y\right)} \]

      19. lift-+.f64 N/A

          \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\left(\color{blue}{\left(x + y\right)} - \left(\mathsf{neg}\left(1\right)\right)\right) \cdot \left(x + y\right)} \]

      20. +-commutative N/A

          \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\left(\color{blue}{\left(y + x\right)} - \left(\mathsf{neg}\left(1\right)\right)\right) \cdot \left(x + y\right)} \]

      21. lower-+.f64 N/A

          \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\left(\color{blue}{\left(y + x\right)} - \left(\mathsf{neg}\left(1\right)\right)\right) \cdot \left(x + y\right)} \]

      22. metadata-eval 93.6

          \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\left(\left(y + x\right) - \color{blue}{-1}\right) \cdot \left(x + y\right)} \]

      23. lift-+.f64 N/A

          \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\left(\left(y + x\right) - -1\right) \cdot \color{blue}{\left(x + y\right)}} \]

      24. +-commutative N/A

          \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\left(\left(y + x\right) - -1\right) \cdot \color{blue}{\left(y + x\right)}} \]

      25. lower-+.f64 93.6

          \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\left(\left(y + x\right) - -1\right) \cdot \color{blue}{\left(y + x\right)}} \]

   3. Applied rewrites 93.6%

      \[\leadsto \color{blue}{\frac{y}{y + x} \cdot \frac{x}{\left(\left(y + x\right) - -1\right) \cdot \left(y + x\right)}} \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{y}{y + x} \cdot \frac{x}{\left(\left(y + x\right) - -1\right) \cdot \left(y + x\right)}} \]

      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. div-flip N/A

         \[\leadsto \frac{y}{y + x} \cdot \color{blue}{\frac{1}{\frac{\left(\left(y + x\right) - -1\right) \cdot \left(y + x\right)}{x}}} \]

      4. mult-flip-rev N/A

         \[\leadsto \color{blue}{\frac{\frac{y}{y + x}}{\frac{\left(\left(y + x\right) - -1\right) \cdot \left(y + x\right)}{x}}} \]

      5. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{y}{y + x}}{\frac{\left(\left(y + x\right) - -1\right) \cdot \left(y + x\right)}{x}}} \]

      6. lift-+.f64 N/A

         \[\leadsto \frac{\frac{y}{\color{blue}{y + x}}}{\frac{\left(\left(y + x\right) - -1\right) \cdot \left(y + x\right)}{x}} \]

      7. +-commutative N/A

         \[\leadsto \frac{\frac{y}{\color{blue}{x + y}}}{\frac{\left(\left(y + x\right) - -1\right) \cdot \left(y + x\right)}{x}} \]

      8. lift-+.f64 N/A

         \[\leadsto \frac{\frac{y}{\color{blue}{x + y}}}{\frac{\left(\left(y + x\right) - -1\right) \cdot \left(y + x\right)}{x}} \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{\frac{y}{x + y}}{\frac{\color{blue}{\left(\left(y + x\right) - -1\right) \cdot \left(y + x\right)}}{x}} \]

      10. associate-/l* N/A

          \[\leadsto \frac{\frac{y}{x + y}}{\color{blue}{\left(\left(y + x\right) - -1\right) \cdot \frac{y + x}{x}}} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{\frac{y}{x + y}}{\color{blue}{\left(\left(y + x\right) - -1\right) \cdot \frac{y + x}{x}}} \]

      12. lift-+.f64 N/A

          \[\leadsto \frac{\frac{y}{x + y}}{\left(\color{blue}{\left(y + x\right)} - -1\right) \cdot \frac{y + x}{x}} \]

      13. +-commutative N/A

          \[\leadsto \frac{\frac{y}{x + y}}{\left(\color{blue}{\left(x + y\right)} - -1\right) \cdot \frac{y + x}{x}} \]

      14. lift-+.f64 N/A

          \[\leadsto \frac{\frac{y}{x + y}}{\left(\color{blue}{\left(x + y\right)} - -1\right) \cdot \frac{y + x}{x}} \]

      15. lower-/.f64 99.3

          \[\leadsto \frac{\frac{y}{x + y}}{\left(\left(x + y\right) - -1\right) \cdot \color{blue}{\frac{y + x}{x}}} \]

      16. lift-+.f64 N/A

          \[\leadsto \frac{\frac{y}{x + y}}{\left(\left(x + y\right) - -1\right) \cdot \frac{\color{blue}{y + x}}{x}} \]

      17. +-commutative N/A

          \[\leadsto \frac{\frac{y}{x + y}}{\left(\left(x + y\right) - -1\right) \cdot \frac{\color{blue}{x + y}}{x}} \]

      18. lift-+.f64 99.3

          \[\leadsto \frac{\frac{y}{x + y}}{\left(\left(x + y\right) - -1\right) \cdot \frac{\color{blue}{x + y}}{x}} \]

   5. Applied rewrites 99.3%

      \[\leadsto \color{blue}{\frac{\frac{y}{x + y}}{\left(\left(x + y\right) - -1\right) \cdot \frac{x + y}{x}}} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\frac{y}{x + y}}{\color{blue}{\left(\left(x + y\right) - -1\right) \cdot \frac{x + y}{x}}} \]

      2. lift--.f64 N/A

         \[\leadsto \frac{\frac{y}{x + y}}{\color{blue}{\left(\left(x + y\right) - -1\right)} \cdot \frac{x + y}{x}} \]

      3. metadata-eval N/A

         \[\leadsto \frac{\frac{y}{x + y}}{\left(\left(x + y\right) - \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right) \cdot \frac{x + y}{x}} \]

      4. add-flip N/A

         \[\leadsto \frac{\frac{y}{x + y}}{\color{blue}{\left(\left(x + y\right) + 1\right)} \cdot \frac{x + y}{x}} \]

      5. lift-+.f64 N/A

         \[\leadsto \frac{\frac{y}{x + y}}{\left(\color{blue}{\left(x + y\right)} + 1\right) \cdot \frac{x + y}{x}} \]

      6. *-commutative N/A

         \[\leadsto \frac{\frac{y}{x + y}}{\color{blue}{\frac{x + y}{x} \cdot \left(\left(x + y\right) + 1\right)}} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{\frac{y}{x + y}}{\color{blue}{\frac{x + y}{x} \cdot \left(\left(x + y\right) + 1\right)}} \]

      8. associate-+l+ N/A

         \[\leadsto \frac{\frac{y}{x + y}}{\frac{x + y}{x} \cdot \color{blue}{\left(x + \left(y + 1\right)\right)}} \]

      9. metadata-eval N/A

         \[\leadsto \frac{\frac{y}{x + y}}{\frac{x + y}{x} \cdot \left(x + \left(y + \color{blue}{\left(\mathsf{neg}\left(-1\right)\right)}\right)\right)} \]

      10. sub-flip N/A

          \[\leadsto \frac{\frac{y}{x + y}}{\frac{x + y}{x} \cdot \left(x + \color{blue}{\left(y - -1\right)}\right)} \]

      11. sub-negate-rev N/A

          \[\leadsto \frac{\frac{y}{x + y}}{\frac{x + y}{x} \cdot \left(x + \color{blue}{\left(\mathsf{neg}\left(\left(-1 - y\right)\right)\right)}\right)} \]

      12. sub-flip-reverse N/A

          \[\leadsto \frac{\frac{y}{x + y}}{\frac{x + y}{x} \cdot \color{blue}{\left(x - \left(-1 - y\right)\right)}} \]

      13. lower--.f64 N/A

          \[\leadsto \frac{\frac{y}{x + y}}{\frac{x + y}{x} \cdot \color{blue}{\left(x - \left(-1 - y\right)\right)}} \]

      14. lower--.f64 99.3

          \[\leadsto \frac{\frac{y}{x + y}}{\frac{x + y}{x} \cdot \left(x - \color{blue}{\left(-1 - y\right)}\right)} \]

   7. Applied rewrites 99.3%

      \[\leadsto \frac{\frac{y}{x + y}}{\color{blue}{\frac{x + y}{x} \cdot \left(x - \left(-1 - y\right)\right)}} \]

   8. Taylor expanded in y around 0

      \[\leadsto \frac{\frac{y}{x + y}}{\color{blue}{1 + x}} \]
   9. Step-by-step derivation

      1. lower-+.f64 51.3

         \[\leadsto \frac{\frac{y}{x + y}}{1 + \color{blue}{x}} \]

   10. Applied rewrites 51.3%

       \[\leadsto \frac{\frac{y}{x + y}}{\color{blue}{1 + x}} \]

   if 4.40000000000000011e-195 < y < 1.79999999999999989e156

   1. Initial program 69.1%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot y}}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      3. *-commutative N/A

         \[\leadsto \frac{\color{blue}{y \cdot x}}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right)} \cdot \left(\left(x + y\right) + 1\right)} \]

      6. associate-*l* N/A

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}} \]

      7. times-frac N/A

         \[\leadsto \color{blue}{\frac{y}{x + y} \cdot \frac{x}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      8. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{y}{x + y} \cdot \frac{x}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      9. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{y}{x + y}} \cdot \frac{x}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      10. lift-+.f64 N/A

          \[\leadsto \frac{y}{\color{blue}{x + y}} \cdot \frac{x}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      11. +-commutative N/A

          \[\leadsto \frac{y}{\color{blue}{y + x}} \cdot \frac{x}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      12. lower-+.f64 N/A

          \[\leadsto \frac{y}{\color{blue}{y + x}} \cdot \frac{x}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      13. lower-/.f64 N/A

          \[\leadsto \frac{y}{y + x} \cdot \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      14. *-commutative N/A

          \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\color{blue}{\left(\left(x + y\right) + 1\right) \cdot \left(x + y\right)}} \]

      15. lower-*.f64 93.6

          \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\color{blue}{\left(\left(x + y\right) + 1\right) \cdot \left(x + y\right)}} \]

      16. lift-+.f64 N/A

          \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\color{blue}{\left(\left(x + y\right) + 1\right)} \cdot \left(x + y\right)} \]

      17. add-flip N/A

          \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\color{blue}{\left(\left(x + y\right) - \left(\mathsf{neg}\left(1\right)\right)\right)} \cdot \left(x + y\right)} \]

      18. lower--.f64 N/A

          \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\color{blue}{\left(\left(x + y\right) - \left(\mathsf{neg}\left(1\right)\right)\right)} \cdot \left(x + y\right)} \]

      19. lift-+.f64 N/A

          \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\left(\color{blue}{\left(x + y\right)} - \left(\mathsf{neg}\left(1\right)\right)\right) \cdot \left(x + y\right)} \]

      20. +-commutative N/A

          \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\left(\color{blue}{\left(y + x\right)} - \left(\mathsf{neg}\left(1\right)\right)\right) \cdot \left(x + y\right)} \]

      21. lower-+.f64 N/A

          \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\left(\color{blue}{\left(y + x\right)} - \left(\mathsf{neg}\left(1\right)\right)\right) \cdot \left(x + y\right)} \]

      22. metadata-eval 93.6

          \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\left(\left(y + x\right) - \color{blue}{-1}\right) \cdot \left(x + y\right)} \]

      23. lift-+.f64 N/A

          \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\left(\left(y + x\right) - -1\right) \cdot \color{blue}{\left(x + y\right)}} \]

      24. +-commutative N/A

          \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\left(\left(y + x\right) - -1\right) \cdot \color{blue}{\left(y + x\right)}} \]

      25. lower-+.f64 93.6

          \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\left(\left(y + x\right) - -1\right) \cdot \color{blue}{\left(y + x\right)}} \]

   3. Applied rewrites 93.6%

      \[\leadsto \color{blue}{\frac{y}{y + x} \cdot \frac{x}{\left(\left(y + x\right) - -1\right) \cdot \left(y + x\right)}} \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{y}{y + x} \cdot \frac{x}{\left(\left(y + x\right) - -1\right) \cdot \left(y + x\right)}} \]

      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. div-flip N/A

         \[\leadsto \frac{y}{y + x} \cdot \color{blue}{\frac{1}{\frac{\left(\left(y + x\right) - -1\right) \cdot \left(y + x\right)}{x}}} \]

      4. mult-flip-rev N/A

         \[\leadsto \color{blue}{\frac{\frac{y}{y + x}}{\frac{\left(\left(y + x\right) - -1\right) \cdot \left(y + x\right)}{x}}} \]

      5. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{y}{y + x}}{\frac{\left(\left(y + x\right) - -1\right) \cdot \left(y + x\right)}{x}}} \]

      6. lift-+.f64 N/A

         \[\leadsto \frac{\frac{y}{\color{blue}{y + x}}}{\frac{\left(\left(y + x\right) - -1\right) \cdot \left(y + x\right)}{x}} \]

      7. +-commutative N/A

         \[\leadsto \frac{\frac{y}{\color{blue}{x + y}}}{\frac{\left(\left(y + x\right) - -1\right) \cdot \left(y + x\right)}{x}} \]

      8. lift-+.f64 N/A

         \[\leadsto \frac{\frac{y}{\color{blue}{x + y}}}{\frac{\left(\left(y + x\right) - -1\right) \cdot \left(y + x\right)}{x}} \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{\frac{y}{x + y}}{\frac{\color{blue}{\left(\left(y + x\right) - -1\right) \cdot \left(y + x\right)}}{x}} \]

      10. associate-/l* N/A

          \[\leadsto \frac{\frac{y}{x + y}}{\color{blue}{\left(\left(y + x\right) - -1\right) \cdot \frac{y + x}{x}}} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{\frac{y}{x + y}}{\color{blue}{\left(\left(y + x\right) - -1\right) \cdot \frac{y + x}{x}}} \]

      12. lift-+.f64 N/A

          \[\leadsto \frac{\frac{y}{x + y}}{\left(\color{blue}{\left(y + x\right)} - -1\right) \cdot \frac{y + x}{x}} \]

      13. +-commutative N/A

          \[\leadsto \frac{\frac{y}{x + y}}{\left(\color{blue}{\left(x + y\right)} - -1\right) \cdot \frac{y + x}{x}} \]

      14. lift-+.f64 N/A

          \[\leadsto \frac{\frac{y}{x + y}}{\left(\color{blue}{\left(x + y\right)} - -1\right) \cdot \frac{y + x}{x}} \]

      15. lower-/.f64 99.3

          \[\leadsto \frac{\frac{y}{x + y}}{\left(\left(x + y\right) - -1\right) \cdot \color{blue}{\frac{y + x}{x}}} \]

      16. lift-+.f64 N/A

          \[\leadsto \frac{\frac{y}{x + y}}{\left(\left(x + y\right) - -1\right) \cdot \frac{\color{blue}{y + x}}{x}} \]

      17. +-commutative N/A

          \[\leadsto \frac{\frac{y}{x + y}}{\left(\left(x + y\right) - -1\right) \cdot \frac{\color{blue}{x + y}}{x}} \]

      18. lift-+.f64 99.3

          \[\leadsto \frac{\frac{y}{x + y}}{\left(\left(x + y\right) - -1\right) \cdot \frac{\color{blue}{x + y}}{x}} \]

   5. Applied rewrites 99.3%

      \[\leadsto \color{blue}{\frac{\frac{y}{x + y}}{\left(\left(x + y\right) - -1\right) \cdot \frac{x + y}{x}}} \]

   6. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{1}}{\left(\left(x + y\right) - -1\right) \cdot \frac{x + y}{x}} \]
   7. Step-by-step derivation

   8. Applied rewrites 50.9%

      \[\leadsto \frac{\color{blue}{1}}{\left(\left(x + y\right) - -1\right) \cdot \frac{x + y}{x}} \]
   9. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{\left(\left(x + y\right) - -1\right) \cdot \frac{x + y}{x}}} \]

      2. mult-flip N/A

         \[\leadsto \color{blue}{1 \cdot \frac{1}{\left(\left(x + y\right) - -1\right) \cdot \frac{x + y}{x}}} \]

      3. lift-*.f64 N/A

         \[\leadsto 1 \cdot \frac{1}{\color{blue}{\left(\left(x + y\right) - -1\right) \cdot \frac{x + y}{x}}} \]

      4. lift--.f64 N/A

         \[\leadsto 1 \cdot \frac{1}{\color{blue}{\left(\left(x + y\right) - -1\right)} \cdot \frac{x + y}{x}} \]

      5. metadata-eval N/A

         \[\leadsto 1 \cdot \frac{1}{\left(\left(x + y\right) - \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right) \cdot \frac{x + y}{x}} \]

      6. add-flip N/A

         \[\leadsto 1 \cdot \frac{1}{\color{blue}{\left(\left(x + y\right) + 1\right)} \cdot \frac{x + y}{x}} \]

      7. lift-+.f64 N/A

         \[\leadsto 1 \cdot \frac{1}{\left(\color{blue}{\left(x + y\right)} + 1\right) \cdot \frac{x + y}{x}} \]

      8. lift-/.f64 N/A

         \[\leadsto 1 \cdot \frac{1}{\left(\left(x + y\right) + 1\right) \cdot \color{blue}{\frac{x + y}{x}}} \]

      9. associate-*r/ N/A

         \[\leadsto 1 \cdot \frac{1}{\color{blue}{\frac{\left(\left(x + y\right) + 1\right) \cdot \left(x + y\right)}{x}}} \]

      10. +-commutative N/A

          \[\leadsto 1 \cdot \frac{1}{\frac{\left(\color{blue}{\left(y + x\right)} + 1\right) \cdot \left(x + y\right)}{x}} \]

      11. metadata-eval N/A

          \[\leadsto 1 \cdot \frac{1}{\frac{\left(\left(y + x\right) + \color{blue}{\left(\mathsf{neg}\left(-1\right)\right)}\right) \cdot \left(x + y\right)}{x}} \]

      12. sub-flip N/A

          \[\leadsto 1 \cdot \frac{1}{\frac{\color{blue}{\left(\left(y + x\right) - -1\right)} \cdot \left(x + y\right)}{x}} \]

      13. lift-+.f64 N/A

          \[\leadsto 1 \cdot \frac{1}{\frac{\left(\left(y + x\right) - -1\right) \cdot \color{blue}{\left(x + y\right)}}{x}} \]

      14. +-commutative N/A

          \[\leadsto 1 \cdot \frac{1}{\frac{\left(\left(y + x\right) - -1\right) \cdot \color{blue}{\left(y + x\right)}}{x}} \]

      15. div-flip N/A

          \[\leadsto 1 \cdot \color{blue}{\frac{x}{\left(\left(y + x\right) - -1\right) \cdot \left(y + x\right)}} \]

      16. associate-*r/ N/A

          \[\leadsto \color{blue}{\frac{1 \cdot x}{\left(\left(y + x\right) - -1\right) \cdot \left(y + x\right)}} \]

   10. Applied rewrites 65.9%

       \[\leadsto \color{blue}{\frac{1 \cdot x}{\left(x - \left(-1 - y\right)\right) \cdot \left(x + y\right)}} \]

   if 1.79999999999999989e156 < y

   1. Initial program 69.1%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

   2. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{y \cdot \left(1 + y\right)}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{x}{y \cdot \color{blue}{\left(1 + y\right)}} \]

      3. lower-+.f64 48.7

         \[\leadsto \frac{x}{y \cdot \left(1 + \color{blue}{y}\right)} \]

   4. Applied rewrites 48.7%

      \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{y \cdot \left(1 + y\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{x}{y \cdot \color{blue}{\left(1 + y\right)}} \]

      3. associate-/r* N/A

         \[\leadsto \frac{\frac{x}{y}}{\color{blue}{1 + y}} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{\frac{x}{y}}{\color{blue}{1 + y}} \]

      5. lower-/.f64 50.1

         \[\leadsto \frac{\frac{x}{y}}{\color{blue}{1} + y} \]

      6. lift-+.f64 N/A

         \[\leadsto \frac{\frac{x}{y}}{1 + \color{blue}{y}} \]

      7. +-commutative N/A

         \[\leadsto \frac{\frac{x}{y}}{y + \color{blue}{1}} \]

      8. add-flip N/A

         \[\leadsto \frac{\frac{x}{y}}{y - \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}} \]

      9. metadata-eval N/A

         \[\leadsto \frac{\frac{x}{y}}{y - -1} \]

      10. lower--.f64 50.1

          \[\leadsto \frac{\frac{x}{y}}{y - \color{blue}{-1}} \]

   6. Applied rewrites 50.1%

      \[\leadsto \frac{\frac{x}{y}}{\color{blue}{y - -1}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 9: 84.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 2.45 \cdot 10^{-83}:\\ \;\;\;\;\frac{\frac{y}{x + y}}{1 + x}\\ \mathbf{elif}\;y \leq 1.8 \cdot 10^{+156}:\\ \;\;\;\;\frac{1 \cdot x}{\left(y - -1\right) \cdot \left(x + y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y - -1}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 2.45e-83)
   (/ (/ y (+ x y)) (+ 1.0 x))
   (if (<= y 1.8e+156)
     (/ (* 1.0 x) (* (- y -1.0) (+ x y)))
     (/ (/ x y) (- y -1.0)))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 2.45e-83) {
		tmp = (y / (x + y)) / (1.0 + x);
	} else if (y <= 1.8e+156) {
		tmp = (1.0 * x) / ((y - -1.0) * (x + y));
	} else {
		tmp = (x / y) / (y - -1.0);
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 2.45d-83) then
        tmp = (y / (x + y)) / (1.0d0 + x)
    else if (y <= 1.8d+156) then
        tmp = (1.0d0 * x) / ((y - (-1.0d0)) * (x + y))
    else
        tmp = (x / y) / (y - (-1.0d0))
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (y <= 2.45e-83) {
		tmp = (y / (x + y)) / (1.0 + x);
	} else if (y <= 1.8e+156) {
		tmp = (1.0 * x) / ((y - -1.0) * (x + y));
	} else {
		tmp = (x / y) / (y - -1.0);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if y <= 2.45e-83:
		tmp = (y / (x + y)) / (1.0 + x)
	elif y <= 1.8e+156:
		tmp = (1.0 * x) / ((y - -1.0) * (x + y))
	else:
		tmp = (x / y) / (y - -1.0)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= 2.45e-83)
		tmp = Float64(Float64(y / Float64(x + y)) / Float64(1.0 + x));
	elseif (y <= 1.8e+156)
		tmp = Float64(Float64(1.0 * x) / Float64(Float64(y - -1.0) * Float64(x + y)));
	else
		tmp = Float64(Float64(x / y) / Float64(y - -1.0));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 2.45e-83)
		tmp = (y / (x + y)) / (1.0 + x);
	elseif (y <= 1.8e+156)
		tmp = (1.0 * x) / ((y - -1.0) * (x + y));
	else
		tmp = (x / y) / (y - -1.0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[y, 2.45e-83], N[(N[(y / N[(x + y), $MachinePrecision]), $MachinePrecision] / N[(1.0 + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.8e+156], N[(N[(1.0 * x), $MachinePrecision] / N[(N[(y - -1.0), $MachinePrecision] * N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / N[(y - -1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 2.45e-83

   1. Initial program 69.1%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot y}}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      3. *-commutative N/A

         \[\leadsto \frac{\color{blue}{y \cdot x}}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right)} \cdot \left(\left(x + y\right) + 1\right)} \]

      6. associate-*l* N/A

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}} \]

      7. times-frac N/A

         \[\leadsto \color{blue}{\frac{y}{x + y} \cdot \frac{x}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      8. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{y}{x + y} \cdot \frac{x}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      9. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{y}{x + y}} \cdot \frac{x}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      10. lift-+.f64 N/A

          \[\leadsto \frac{y}{\color{blue}{x + y}} \cdot \frac{x}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      11. +-commutative N/A

          \[\leadsto \frac{y}{\color{blue}{y + x}} \cdot \frac{x}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      12. lower-+.f64 N/A

          \[\leadsto \frac{y}{\color{blue}{y + x}} \cdot \frac{x}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      13. lower-/.f64 N/A

          \[\leadsto \frac{y}{y + x} \cdot \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      14. *-commutative N/A

          \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\color{blue}{\left(\left(x + y\right) + 1\right) \cdot \left(x + y\right)}} \]

      15. lower-*.f64 93.6

          \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\color{blue}{\left(\left(x + y\right) + 1\right) \cdot \left(x + y\right)}} \]

      16. lift-+.f64 N/A

          \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\color{blue}{\left(\left(x + y\right) + 1\right)} \cdot \left(x + y\right)} \]

      17. add-flip N/A

          \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\color{blue}{\left(\left(x + y\right) - \left(\mathsf{neg}\left(1\right)\right)\right)} \cdot \left(x + y\right)} \]

      18. lower--.f64 N/A

          \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\color{blue}{\left(\left(x + y\right) - \left(\mathsf{neg}\left(1\right)\right)\right)} \cdot \left(x + y\right)} \]

      19. lift-+.f64 N/A

          \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\left(\color{blue}{\left(x + y\right)} - \left(\mathsf{neg}\left(1\right)\right)\right) \cdot \left(x + y\right)} \]

      20. +-commutative N/A

          \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\left(\color{blue}{\left(y + x\right)} - \left(\mathsf{neg}\left(1\right)\right)\right) \cdot \left(x + y\right)} \]

      21. lower-+.f64 N/A

          \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\left(\color{blue}{\left(y + x\right)} - \left(\mathsf{neg}\left(1\right)\right)\right) \cdot \left(x + y\right)} \]

      22. metadata-eval 93.6

          \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\left(\left(y + x\right) - \color{blue}{-1}\right) \cdot \left(x + y\right)} \]

      23. lift-+.f64 N/A

          \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\left(\left(y + x\right) - -1\right) \cdot \color{blue}{\left(x + y\right)}} \]

      24. +-commutative N/A

          \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\left(\left(y + x\right) - -1\right) \cdot \color{blue}{\left(y + x\right)}} \]

      25. lower-+.f64 93.6

          \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\left(\left(y + x\right) - -1\right) \cdot \color{blue}{\left(y + x\right)}} \]

   3. Applied rewrites 93.6%

      \[\leadsto \color{blue}{\frac{y}{y + x} \cdot \frac{x}{\left(\left(y + x\right) - -1\right) \cdot \left(y + x\right)}} \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{y}{y + x} \cdot \frac{x}{\left(\left(y + x\right) - -1\right) \cdot \left(y + x\right)}} \]

      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. div-flip N/A

         \[\leadsto \frac{y}{y + x} \cdot \color{blue}{\frac{1}{\frac{\left(\left(y + x\right) - -1\right) \cdot \left(y + x\right)}{x}}} \]

      4. mult-flip-rev N/A

         \[\leadsto \color{blue}{\frac{\frac{y}{y + x}}{\frac{\left(\left(y + x\right) - -1\right) \cdot \left(y + x\right)}{x}}} \]

      5. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{y}{y + x}}{\frac{\left(\left(y + x\right) - -1\right) \cdot \left(y + x\right)}{x}}} \]

      6. lift-+.f64 N/A

         \[\leadsto \frac{\frac{y}{\color{blue}{y + x}}}{\frac{\left(\left(y + x\right) - -1\right) \cdot \left(y + x\right)}{x}} \]

      7. +-commutative N/A

         \[\leadsto \frac{\frac{y}{\color{blue}{x + y}}}{\frac{\left(\left(y + x\right) - -1\right) \cdot \left(y + x\right)}{x}} \]

      8. lift-+.f64 N/A

         \[\leadsto \frac{\frac{y}{\color{blue}{x + y}}}{\frac{\left(\left(y + x\right) - -1\right) \cdot \left(y + x\right)}{x}} \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{\frac{y}{x + y}}{\frac{\color{blue}{\left(\left(y + x\right) - -1\right) \cdot \left(y + x\right)}}{x}} \]

      10. associate-/l* N/A

          \[\leadsto \frac{\frac{y}{x + y}}{\color{blue}{\left(\left(y + x\right) - -1\right) \cdot \frac{y + x}{x}}} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{\frac{y}{x + y}}{\color{blue}{\left(\left(y + x\right) - -1\right) \cdot \frac{y + x}{x}}} \]

      12. lift-+.f64 N/A

          \[\leadsto \frac{\frac{y}{x + y}}{\left(\color{blue}{\left(y + x\right)} - -1\right) \cdot \frac{y + x}{x}} \]

      13. +-commutative N/A

          \[\leadsto \frac{\frac{y}{x + y}}{\left(\color{blue}{\left(x + y\right)} - -1\right) \cdot \frac{y + x}{x}} \]

      14. lift-+.f64 N/A

          \[\leadsto \frac{\frac{y}{x + y}}{\left(\color{blue}{\left(x + y\right)} - -1\right) \cdot \frac{y + x}{x}} \]

      15. lower-/.f64 99.3

          \[\leadsto \frac{\frac{y}{x + y}}{\left(\left(x + y\right) - -1\right) \cdot \color{blue}{\frac{y + x}{x}}} \]

      16. lift-+.f64 N/A

          \[\leadsto \frac{\frac{y}{x + y}}{\left(\left(x + y\right) - -1\right) \cdot \frac{\color{blue}{y + x}}{x}} \]

      17. +-commutative N/A

          \[\leadsto \frac{\frac{y}{x + y}}{\left(\left(x + y\right) - -1\right) \cdot \frac{\color{blue}{x + y}}{x}} \]

      18. lift-+.f64 99.3

          \[\leadsto \frac{\frac{y}{x + y}}{\left(\left(x + y\right) - -1\right) \cdot \frac{\color{blue}{x + y}}{x}} \]

   5. Applied rewrites 99.3%

      \[\leadsto \color{blue}{\frac{\frac{y}{x + y}}{\left(\left(x + y\right) - -1\right) \cdot \frac{x + y}{x}}} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\frac{y}{x + y}}{\color{blue}{\left(\left(x + y\right) - -1\right) \cdot \frac{x + y}{x}}} \]

      2. lift--.f64 N/A

         \[\leadsto \frac{\frac{y}{x + y}}{\color{blue}{\left(\left(x + y\right) - -1\right)} \cdot \frac{x + y}{x}} \]

      3. metadata-eval N/A

         \[\leadsto \frac{\frac{y}{x + y}}{\left(\left(x + y\right) - \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right) \cdot \frac{x + y}{x}} \]

      4. add-flip N/A

         \[\leadsto \frac{\frac{y}{x + y}}{\color{blue}{\left(\left(x + y\right) + 1\right)} \cdot \frac{x + y}{x}} \]

      5. lift-+.f64 N/A

         \[\leadsto \frac{\frac{y}{x + y}}{\left(\color{blue}{\left(x + y\right)} + 1\right) \cdot \frac{x + y}{x}} \]

      6. *-commutative N/A

         \[\leadsto \frac{\frac{y}{x + y}}{\color{blue}{\frac{x + y}{x} \cdot \left(\left(x + y\right) + 1\right)}} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{\frac{y}{x + y}}{\color{blue}{\frac{x + y}{x} \cdot \left(\left(x + y\right) + 1\right)}} \]

      8. associate-+l+ N/A

         \[\leadsto \frac{\frac{y}{x + y}}{\frac{x + y}{x} \cdot \color{blue}{\left(x + \left(y + 1\right)\right)}} \]

      9. metadata-eval N/A

         \[\leadsto \frac{\frac{y}{x + y}}{\frac{x + y}{x} \cdot \left(x + \left(y + \color{blue}{\left(\mathsf{neg}\left(-1\right)\right)}\right)\right)} \]

      10. sub-flip N/A

          \[\leadsto \frac{\frac{y}{x + y}}{\frac{x + y}{x} \cdot \left(x + \color{blue}{\left(y - -1\right)}\right)} \]

      11. sub-negate-rev N/A

          \[\leadsto \frac{\frac{y}{x + y}}{\frac{x + y}{x} \cdot \left(x + \color{blue}{\left(\mathsf{neg}\left(\left(-1 - y\right)\right)\right)}\right)} \]

      12. sub-flip-reverse N/A

          \[\leadsto \frac{\frac{y}{x + y}}{\frac{x + y}{x} \cdot \color{blue}{\left(x - \left(-1 - y\right)\right)}} \]

      13. lower--.f64 N/A

          \[\leadsto \frac{\frac{y}{x + y}}{\frac{x + y}{x} \cdot \color{blue}{\left(x - \left(-1 - y\right)\right)}} \]

      14. lower--.f64 99.3

          \[\leadsto \frac{\frac{y}{x + y}}{\frac{x + y}{x} \cdot \left(x - \color{blue}{\left(-1 - y\right)}\right)} \]

   7. Applied rewrites 99.3%

      \[\leadsto \frac{\frac{y}{x + y}}{\color{blue}{\frac{x + y}{x} \cdot \left(x - \left(-1 - y\right)\right)}} \]

   8. Taylor expanded in y around 0

      \[\leadsto \frac{\frac{y}{x + y}}{\color{blue}{1 + x}} \]
   9. Step-by-step derivation

      1. lower-+.f64 51.3

         \[\leadsto \frac{\frac{y}{x + y}}{1 + \color{blue}{x}} \]

   10. Applied rewrites 51.3%

       \[\leadsto \frac{\frac{y}{x + y}}{\color{blue}{1 + x}} \]

   if 2.45e-83 < y < 1.79999999999999989e156

   1. Initial program 69.1%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

   2. Taylor expanded in x around 0

      \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \color{blue}{\left(1 + y\right)}} \]
   3. Step-by-step derivation

      1. lower-+.f64 58.7

         \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(1 + \color{blue}{y}\right)} \]

   4. Applied rewrites 58.7%

      \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \color{blue}{\left(1 + y\right)}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(1 + y\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(1 + y\right)}} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right)} \cdot \left(1 + y\right)} \]

      4. associate-*l* N/A

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(1 + y\right)\right)}} \]

      5. lift-+.f64 N/A

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(x + y\right)} \cdot \left(\left(x + y\right) \cdot \left(1 + y\right)\right)} \]

      6. +-commutative N/A

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(y + x\right)} \cdot \left(\left(x + y\right) \cdot \left(1 + y\right)\right)} \]

      7. lift-+.f64 N/A

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(y + x\right)} \cdot \left(\left(x + y\right) \cdot \left(1 + y\right)\right)} \]

      8. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{x \cdot y}{y + x}}{\left(x + y\right) \cdot \left(1 + y\right)}} \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{\frac{\color{blue}{x \cdot y}}{y + x}}{\left(x + y\right) \cdot \left(1 + y\right)} \]

      10. *-commutative N/A

          \[\leadsto \frac{\frac{\color{blue}{y \cdot x}}{y + x}}{\left(x + y\right) \cdot \left(1 + y\right)} \]

      11. associate-*l/ N/A

          \[\leadsto \frac{\color{blue}{\frac{y}{y + x} \cdot x}}{\left(x + y\right) \cdot \left(1 + y\right)} \]

      12. lift-/.f64 N/A

          \[\leadsto \frac{\color{blue}{\frac{y}{y + x}} \cdot x}{\left(x + y\right) \cdot \left(1 + y\right)} \]

      13. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{\frac{y}{y + x} \cdot x}{\left(x + y\right) \cdot \left(1 + y\right)}} \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{\color{blue}{\frac{y}{y + x} \cdot x}}{\left(x + y\right) \cdot \left(1 + y\right)} \]

      15. lift-+.f64 N/A

          \[\leadsto \frac{\frac{y}{\color{blue}{y + x}} \cdot x}{\left(x + y\right) \cdot \left(1 + y\right)} \]

      16. +-commutative N/A

          \[\leadsto \frac{\frac{y}{\color{blue}{x + y}} \cdot x}{\left(x + y\right) \cdot \left(1 + y\right)} \]

      17. lift-+.f64 N/A

          \[\leadsto \frac{\frac{y}{\color{blue}{x + y}} \cdot x}{\left(x + y\right) \cdot \left(1 + y\right)} \]

   6. Applied rewrites 75.8%

      \[\leadsto \color{blue}{\frac{\frac{y}{x + y} \cdot x}{\left(y - -1\right) \cdot \left(x + y\right)}} \]

   7. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{1} \cdot x}{\left(y - -1\right) \cdot \left(x + y\right)} \]
   8. Step-by-step derivation

   9. Applied rewrites 51.8%

      \[\leadsto \frac{\color{blue}{1} \cdot x}{\left(y - -1\right) \cdot \left(x + y\right)} \]

   if 1.79999999999999989e156 < y

   1. Initial program 69.1%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

   2. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{y \cdot \left(1 + y\right)}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{x}{y \cdot \color{blue}{\left(1 + y\right)}} \]

      3. lower-+.f64 48.7

         \[\leadsto \frac{x}{y \cdot \left(1 + \color{blue}{y}\right)} \]

   4. Applied rewrites 48.7%

      \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{y \cdot \left(1 + y\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{x}{y \cdot \color{blue}{\left(1 + y\right)}} \]

      3. associate-/r* N/A

         \[\leadsto \frac{\frac{x}{y}}{\color{blue}{1 + y}} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{\frac{x}{y}}{\color{blue}{1 + y}} \]

      5. lower-/.f64 50.1

         \[\leadsto \frac{\frac{x}{y}}{\color{blue}{1} + y} \]

      6. lift-+.f64 N/A

         \[\leadsto \frac{\frac{x}{y}}{1 + \color{blue}{y}} \]

      7. +-commutative N/A

         \[\leadsto \frac{\frac{x}{y}}{y + \color{blue}{1}} \]

      8. add-flip N/A

         \[\leadsto \frac{\frac{x}{y}}{y - \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}} \]

      9. metadata-eval N/A

         \[\leadsto \frac{\frac{x}{y}}{y - -1} \]

      10. lower--.f64 50.1

          \[\leadsto \frac{\frac{x}{y}}{y - \color{blue}{-1}} \]

   6. Applied rewrites 50.1%

      \[\leadsto \frac{\frac{x}{y}}{\color{blue}{y - -1}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 10: 82.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 5.5 \cdot 10^{-83}:\\ \;\;\;\;\frac{\frac{y}{x + y}}{1 + x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y - -1}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 5.5e-83) (/ (/ y (+ x y)) (+ 1.0 x)) (/ (/ x y) (- y -1.0))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 5.5e-83) {
		tmp = (y / (x + y)) / (1.0 + x);
	} else {
		tmp = (x / y) / (y - -1.0);
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 5.5d-83) then
        tmp = (y / (x + y)) / (1.0d0 + x)
    else
        tmp = (x / y) / (y - (-1.0d0))
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (y <= 5.5e-83) {
		tmp = (y / (x + y)) / (1.0 + x);
	} else {
		tmp = (x / y) / (y - -1.0);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if y <= 5.5e-83:
		tmp = (y / (x + y)) / (1.0 + x)
	else:
		tmp = (x / y) / (y - -1.0)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= 5.5e-83)
		tmp = Float64(Float64(y / Float64(x + y)) / Float64(1.0 + x));
	else
		tmp = Float64(Float64(x / y) / Float64(y - -1.0));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 5.5e-83)
		tmp = (y / (x + y)) / (1.0 + x);
	else
		tmp = (x / y) / (y - -1.0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[y, 5.5e-83], N[(N[(y / N[(x + y), $MachinePrecision]), $MachinePrecision] / N[(1.0 + x), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / N[(y - -1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 5.49999999999999964e-83

   1. Initial program 69.1%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot y}}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      3. *-commutative N/A

         \[\leadsto \frac{\color{blue}{y \cdot x}}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right)} \cdot \left(\left(x + y\right) + 1\right)} \]

      6. associate-*l* N/A

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}} \]

      7. times-frac N/A

         \[\leadsto \color{blue}{\frac{y}{x + y} \cdot \frac{x}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      8. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{y}{x + y} \cdot \frac{x}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      9. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{y}{x + y}} \cdot \frac{x}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      10. lift-+.f64 N/A

          \[\leadsto \frac{y}{\color{blue}{x + y}} \cdot \frac{x}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      11. +-commutative N/A

          \[\leadsto \frac{y}{\color{blue}{y + x}} \cdot \frac{x}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      12. lower-+.f64 N/A

          \[\leadsto \frac{y}{\color{blue}{y + x}} \cdot \frac{x}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      13. lower-/.f64 N/A

          \[\leadsto \frac{y}{y + x} \cdot \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      14. *-commutative N/A

          \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\color{blue}{\left(\left(x + y\right) + 1\right) \cdot \left(x + y\right)}} \]

      15. lower-*.f64 93.6

          \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\color{blue}{\left(\left(x + y\right) + 1\right) \cdot \left(x + y\right)}} \]

      16. lift-+.f64 N/A

          \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\color{blue}{\left(\left(x + y\right) + 1\right)} \cdot \left(x + y\right)} \]

      17. add-flip N/A

          \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\color{blue}{\left(\left(x + y\right) - \left(\mathsf{neg}\left(1\right)\right)\right)} \cdot \left(x + y\right)} \]

      18. lower--.f64 N/A

          \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\color{blue}{\left(\left(x + y\right) - \left(\mathsf{neg}\left(1\right)\right)\right)} \cdot \left(x + y\right)} \]

      19. lift-+.f64 N/A

          \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\left(\color{blue}{\left(x + y\right)} - \left(\mathsf{neg}\left(1\right)\right)\right) \cdot \left(x + y\right)} \]

      20. +-commutative N/A

          \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\left(\color{blue}{\left(y + x\right)} - \left(\mathsf{neg}\left(1\right)\right)\right) \cdot \left(x + y\right)} \]

      21. lower-+.f64 N/A

          \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\left(\color{blue}{\left(y + x\right)} - \left(\mathsf{neg}\left(1\right)\right)\right) \cdot \left(x + y\right)} \]

      22. metadata-eval 93.6

          \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\left(\left(y + x\right) - \color{blue}{-1}\right) \cdot \left(x + y\right)} \]

      23. lift-+.f64 N/A

          \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\left(\left(y + x\right) - -1\right) \cdot \color{blue}{\left(x + y\right)}} \]

      24. +-commutative N/A

          \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\left(\left(y + x\right) - -1\right) \cdot \color{blue}{\left(y + x\right)}} \]

      25. lower-+.f64 93.6

          \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\left(\left(y + x\right) - -1\right) \cdot \color{blue}{\left(y + x\right)}} \]

   3. Applied rewrites 93.6%

      \[\leadsto \color{blue}{\frac{y}{y + x} \cdot \frac{x}{\left(\left(y + x\right) - -1\right) \cdot \left(y + x\right)}} \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{y}{y + x} \cdot \frac{x}{\left(\left(y + x\right) - -1\right) \cdot \left(y + x\right)}} \]

      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. div-flip N/A

         \[\leadsto \frac{y}{y + x} \cdot \color{blue}{\frac{1}{\frac{\left(\left(y + x\right) - -1\right) \cdot \left(y + x\right)}{x}}} \]

      4. mult-flip-rev N/A

         \[\leadsto \color{blue}{\frac{\frac{y}{y + x}}{\frac{\left(\left(y + x\right) - -1\right) \cdot \left(y + x\right)}{x}}} \]

      5. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{y}{y + x}}{\frac{\left(\left(y + x\right) - -1\right) \cdot \left(y + x\right)}{x}}} \]

      6. lift-+.f64 N/A

         \[\leadsto \frac{\frac{y}{\color{blue}{y + x}}}{\frac{\left(\left(y + x\right) - -1\right) \cdot \left(y + x\right)}{x}} \]

      7. +-commutative N/A

         \[\leadsto \frac{\frac{y}{\color{blue}{x + y}}}{\frac{\left(\left(y + x\right) - -1\right) \cdot \left(y + x\right)}{x}} \]

      8. lift-+.f64 N/A

         \[\leadsto \frac{\frac{y}{\color{blue}{x + y}}}{\frac{\left(\left(y + x\right) - -1\right) \cdot \left(y + x\right)}{x}} \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{\frac{y}{x + y}}{\frac{\color{blue}{\left(\left(y + x\right) - -1\right) \cdot \left(y + x\right)}}{x}} \]

      10. associate-/l* N/A

          \[\leadsto \frac{\frac{y}{x + y}}{\color{blue}{\left(\left(y + x\right) - -1\right) \cdot \frac{y + x}{x}}} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{\frac{y}{x + y}}{\color{blue}{\left(\left(y + x\right) - -1\right) \cdot \frac{y + x}{x}}} \]

      12. lift-+.f64 N/A

          \[\leadsto \frac{\frac{y}{x + y}}{\left(\color{blue}{\left(y + x\right)} - -1\right) \cdot \frac{y + x}{x}} \]

      13. +-commutative N/A

          \[\leadsto \frac{\frac{y}{x + y}}{\left(\color{blue}{\left(x + y\right)} - -1\right) \cdot \frac{y + x}{x}} \]

      14. lift-+.f64 N/A

          \[\leadsto \frac{\frac{y}{x + y}}{\left(\color{blue}{\left(x + y\right)} - -1\right) \cdot \frac{y + x}{x}} \]

      15. lower-/.f64 99.3

          \[\leadsto \frac{\frac{y}{x + y}}{\left(\left(x + y\right) - -1\right) \cdot \color{blue}{\frac{y + x}{x}}} \]

      16. lift-+.f64 N/A

          \[\leadsto \frac{\frac{y}{x + y}}{\left(\left(x + y\right) - -1\right) \cdot \frac{\color{blue}{y + x}}{x}} \]

      17. +-commutative N/A

          \[\leadsto \frac{\frac{y}{x + y}}{\left(\left(x + y\right) - -1\right) \cdot \frac{\color{blue}{x + y}}{x}} \]

      18. lift-+.f64 99.3

          \[\leadsto \frac{\frac{y}{x + y}}{\left(\left(x + y\right) - -1\right) \cdot \frac{\color{blue}{x + y}}{x}} \]

   5. Applied rewrites 99.3%

      \[\leadsto \color{blue}{\frac{\frac{y}{x + y}}{\left(\left(x + y\right) - -1\right) \cdot \frac{x + y}{x}}} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\frac{y}{x + y}}{\color{blue}{\left(\left(x + y\right) - -1\right) \cdot \frac{x + y}{x}}} \]

      2. lift--.f64 N/A

         \[\leadsto \frac{\frac{y}{x + y}}{\color{blue}{\left(\left(x + y\right) - -1\right)} \cdot \frac{x + y}{x}} \]

      3. metadata-eval N/A

         \[\leadsto \frac{\frac{y}{x + y}}{\left(\left(x + y\right) - \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right) \cdot \frac{x + y}{x}} \]

      4. add-flip N/A

         \[\leadsto \frac{\frac{y}{x + y}}{\color{blue}{\left(\left(x + y\right) + 1\right)} \cdot \frac{x + y}{x}} \]

      5. lift-+.f64 N/A

         \[\leadsto \frac{\frac{y}{x + y}}{\left(\color{blue}{\left(x + y\right)} + 1\right) \cdot \frac{x + y}{x}} \]

      6. *-commutative N/A

         \[\leadsto \frac{\frac{y}{x + y}}{\color{blue}{\frac{x + y}{x} \cdot \left(\left(x + y\right) + 1\right)}} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{\frac{y}{x + y}}{\color{blue}{\frac{x + y}{x} \cdot \left(\left(x + y\right) + 1\right)}} \]

      8. associate-+l+ N/A

         \[\leadsto \frac{\frac{y}{x + y}}{\frac{x + y}{x} \cdot \color{blue}{\left(x + \left(y + 1\right)\right)}} \]

      9. metadata-eval N/A

         \[\leadsto \frac{\frac{y}{x + y}}{\frac{x + y}{x} \cdot \left(x + \left(y + \color{blue}{\left(\mathsf{neg}\left(-1\right)\right)}\right)\right)} \]

      10. sub-flip N/A

          \[\leadsto \frac{\frac{y}{x + y}}{\frac{x + y}{x} \cdot \left(x + \color{blue}{\left(y - -1\right)}\right)} \]

      11. sub-negate-rev N/A

          \[\leadsto \frac{\frac{y}{x + y}}{\frac{x + y}{x} \cdot \left(x + \color{blue}{\left(\mathsf{neg}\left(\left(-1 - y\right)\right)\right)}\right)} \]

      12. sub-flip-reverse N/A

          \[\leadsto \frac{\frac{y}{x + y}}{\frac{x + y}{x} \cdot \color{blue}{\left(x - \left(-1 - y\right)\right)}} \]

      13. lower--.f64 N/A

          \[\leadsto \frac{\frac{y}{x + y}}{\frac{x + y}{x} \cdot \color{blue}{\left(x - \left(-1 - y\right)\right)}} \]

      14. lower--.f64 99.3

          \[\leadsto \frac{\frac{y}{x + y}}{\frac{x + y}{x} \cdot \left(x - \color{blue}{\left(-1 - y\right)}\right)} \]

   7. Applied rewrites 99.3%

      \[\leadsto \frac{\frac{y}{x + y}}{\color{blue}{\frac{x + y}{x} \cdot \left(x - \left(-1 - y\right)\right)}} \]

   8. Taylor expanded in y around 0

      \[\leadsto \frac{\frac{y}{x + y}}{\color{blue}{1 + x}} \]
   9. Step-by-step derivation

      1. lower-+.f64 51.3

         \[\leadsto \frac{\frac{y}{x + y}}{1 + \color{blue}{x}} \]

   10. Applied rewrites 51.3%

       \[\leadsto \frac{\frac{y}{x + y}}{\color{blue}{1 + x}} \]

   if 5.49999999999999964e-83 < y

   1. Initial program 69.1%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

   2. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{y \cdot \left(1 + y\right)}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{x}{y \cdot \color{blue}{\left(1 + y\right)}} \]

      3. lower-+.f64 48.7

         \[\leadsto \frac{x}{y \cdot \left(1 + \color{blue}{y}\right)} \]

   4. Applied rewrites 48.7%

      \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{y \cdot \left(1 + y\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{x}{y \cdot \color{blue}{\left(1 + y\right)}} \]

      3. associate-/r* N/A

         \[\leadsto \frac{\frac{x}{y}}{\color{blue}{1 + y}} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{\frac{x}{y}}{\color{blue}{1 + y}} \]

      5. lower-/.f64 50.1

         \[\leadsto \frac{\frac{x}{y}}{\color{blue}{1} + y} \]

      6. lift-+.f64 N/A

         \[\leadsto \frac{\frac{x}{y}}{1 + \color{blue}{y}} \]

      7. +-commutative N/A

         \[\leadsto \frac{\frac{x}{y}}{y + \color{blue}{1}} \]

      8. add-flip N/A

         \[\leadsto \frac{\frac{x}{y}}{y - \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}} \]

      9. metadata-eval N/A

         \[\leadsto \frac{\frac{x}{y}}{y - -1} \]

      10. lower--.f64 50.1

          \[\leadsto \frac{\frac{x}{y}}{y - \color{blue}{-1}} \]

   6. Applied rewrites 50.1%

      \[\leadsto \frac{\frac{x}{y}}{\color{blue}{y - -1}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 81.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 3.8 \cdot 10^{-67}:\\ \;\;\;\;\frac{y}{x \cdot \left(1 + x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y - -1}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 3.8e-67) (/ y (* x (+ 1.0 x))) (/ (/ x y) (- y -1.0))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 3.8e-67) {
		tmp = y / (x * (1.0 + x));
	} else {
		tmp = (x / y) / (y - -1.0);
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 3.8d-67) then
        tmp = y / (x * (1.0d0 + x))
    else
        tmp = (x / y) / (y - (-1.0d0))
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (y <= 3.8e-67) {
		tmp = y / (x * (1.0 + x));
	} else {
		tmp = (x / y) / (y - -1.0);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if y <= 3.8e-67:
		tmp = y / (x * (1.0 + x))
	else:
		tmp = (x / y) / (y - -1.0)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= 3.8e-67)
		tmp = Float64(y / Float64(x * Float64(1.0 + x)));
	else
		tmp = Float64(Float64(x / y) / Float64(y - -1.0));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 3.8e-67)
		tmp = y / (x * (1.0 + x));
	else
		tmp = (x / y) / (y - -1.0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[y, 3.8e-67], N[(y / N[(x * N[(1.0 + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / N[(y - -1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 3.79999999999999988e-67

   1. Initial program 69.1%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

   2. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{y}{x \cdot \left(1 + x\right)}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{y}{\color{blue}{x \cdot \left(1 + x\right)}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{y}{x \cdot \color{blue}{\left(1 + x\right)}} \]

      3. lower-+.f64 49.5

         \[\leadsto \frac{y}{x \cdot \left(1 + \color{blue}{x}\right)} \]

   4. Applied rewrites 49.5%

      \[\leadsto \color{blue}{\frac{y}{x \cdot \left(1 + x\right)}} \]

   if 3.79999999999999988e-67 < y

   1. Initial program 69.1%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

   2. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{y \cdot \left(1 + y\right)}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{x}{y \cdot \color{blue}{\left(1 + y\right)}} \]

      3. lower-+.f64 48.7

         \[\leadsto \frac{x}{y \cdot \left(1 + \color{blue}{y}\right)} \]

   4. Applied rewrites 48.7%

      \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{y \cdot \left(1 + y\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{x}{y \cdot \color{blue}{\left(1 + y\right)}} \]

      3. associate-/r* N/A

         \[\leadsto \frac{\frac{x}{y}}{\color{blue}{1 + y}} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{\frac{x}{y}}{\color{blue}{1 + y}} \]

      5. lower-/.f64 50.1

         \[\leadsto \frac{\frac{x}{y}}{\color{blue}{1} + y} \]

      6. lift-+.f64 N/A

         \[\leadsto \frac{\frac{x}{y}}{1 + \color{blue}{y}} \]

      7. +-commutative N/A

         \[\leadsto \frac{\frac{x}{y}}{y + \color{blue}{1}} \]

      8. add-flip N/A

         \[\leadsto \frac{\frac{x}{y}}{y - \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}} \]

      9. metadata-eval N/A

         \[\leadsto \frac{\frac{x}{y}}{y - -1} \]

      10. lower--.f64 50.1

          \[\leadsto \frac{\frac{x}{y}}{y - \color{blue}{-1}} \]

   6. Applied rewrites 50.1%

      \[\leadsto \frac{\frac{x}{y}}{\color{blue}{y - -1}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 78.9% accurate, 1.7× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 3.8 \cdot 10^{-67}:\\ \;\;\;\;\frac{y}{x \cdot \left(1 + x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(y, y, y\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 3.8e-67) (/ y (* x (+ 1.0 x))) (/ x (fma y y y))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 3.8e-67) {
		tmp = y / (x * (1.0 + x));
	} else {
		tmp = x / fma(y, y, y);
	}
	return tmp;
}

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= 3.8e-67)
		tmp = Float64(y / Float64(x * Float64(1.0 + x)));
	else
		tmp = Float64(x / fma(y, y, y));
	end
	return tmp
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[y, 3.8e-67], N[(y / N[(x * N[(1.0 + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x / N[(y * y + y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 3.79999999999999988e-67

   1. Initial program 69.1%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

   2. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{y}{x \cdot \left(1 + x\right)}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{y}{\color{blue}{x \cdot \left(1 + x\right)}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{y}{x \cdot \color{blue}{\left(1 + x\right)}} \]

      3. lower-+.f64 49.5

         \[\leadsto \frac{y}{x \cdot \left(1 + \color{blue}{x}\right)} \]

   4. Applied rewrites 49.5%

      \[\leadsto \color{blue}{\frac{y}{x \cdot \left(1 + x\right)}} \]

   if 3.79999999999999988e-67 < y

   1. Initial program 69.1%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

   2. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{y \cdot \left(1 + y\right)}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{x}{y \cdot \color{blue}{\left(1 + y\right)}} \]

      3. lower-+.f64 48.7

         \[\leadsto \frac{x}{y \cdot \left(1 + \color{blue}{y}\right)} \]

   4. Applied rewrites 48.7%

      \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{x}{y \cdot \color{blue}{\left(1 + y\right)}} \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{x}{y \cdot \left(1 + \color{blue}{y}\right)} \]

      3. +-commutative N/A

         \[\leadsto \frac{x}{y \cdot \left(y + \color{blue}{1}\right)} \]

      4. distribute-rgt-in N/A

         \[\leadsto \frac{x}{y \cdot y + \color{blue}{1 \cdot y}} \]

      5. *-lft-identity N/A

         \[\leadsto \frac{x}{y \cdot y + y} \]

      6. lower-fma.f64 48.7

         \[\leadsto \frac{x}{\mathsf{fma}\left(y, \color{blue}{y}, y\right)} \]

   6. Applied rewrites 48.7%

      \[\leadsto \color{blue}{\frac{x}{\mathsf{fma}\left(y, y, y\right)}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 13: 48.7% accurate, 2.6× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \frac{x}{\mathsf{fma}\left(y, y, y\right)} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y) :precision binary64 (/ x (fma y y y)))

C

assert(x < y);
double code(double x, double y) {
	return x / fma(y, y, y);
}

Julia

x, y = sort([x, y])
function code(x, y)
	return Float64(x / fma(y, y, y))
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := N[(x / N[(y * y + y), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 69.1%

   \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

2. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]
3. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \frac{x}{\color{blue}{y \cdot \left(1 + y\right)}} \]

   2. lower-*.f64 N/A

      \[\leadsto \frac{x}{y \cdot \color{blue}{\left(1 + y\right)}} \]

   3. lower-+.f64 48.7

      \[\leadsto \frac{x}{y \cdot \left(1 + \color{blue}{y}\right)} \]

4. Applied rewrites 48.7%

   \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]
5. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \frac{x}{y \cdot \color{blue}{\left(1 + y\right)}} \]

   2. lift-+.f64 N/A

      \[\leadsto \frac{x}{y \cdot \left(1 + \color{blue}{y}\right)} \]

   3. +-commutative N/A

      \[\leadsto \frac{x}{y \cdot \left(y + \color{blue}{1}\right)} \]

   4. distribute-rgt-in N/A

      \[\leadsto \frac{x}{y \cdot y + \color{blue}{1 \cdot y}} \]

   5. *-lft-identity N/A

      \[\leadsto \frac{x}{y \cdot y + y} \]

   6. lower-fma.f64 48.7

      \[\leadsto \frac{x}{\mathsf{fma}\left(y, \color{blue}{y}, y\right)} \]

6. Applied rewrites 48.7%

   \[\leadsto \color{blue}{\frac{x}{\mathsf{fma}\left(y, y, y\right)}} \]
7. Add Preprocessing

Alternative 14: 27.2% accurate, 3.1× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \frac{1}{\frac{y}{x}} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y) :precision binary64 (/ 1.0 (/ y x)))

C

assert(x < y);
double code(double x, double y) {
	return 1.0 / (y / x);
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = 1.0d0 / (y / x)
end function

Java

assert x < y;
public static double code(double x, double y) {
	return 1.0 / (y / x);
}

Python

[x, y] = sort([x, y])
def code(x, y):
	return 1.0 / (y / x)

Julia

x, y = sort([x, y])
function code(x, y)
	return Float64(1.0 / Float64(y / x))
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp = code(x, y)
	tmp = 1.0 / (y / x);
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := N[(1.0 / N[(y / x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 69.1%

   \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

2. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]
3. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \frac{x}{\color{blue}{y \cdot \left(1 + y\right)}} \]

   2. lower-*.f64 N/A

      \[\leadsto \frac{x}{y \cdot \color{blue}{\left(1 + y\right)}} \]

   3. lower-+.f64 48.7

      \[\leadsto \frac{x}{y \cdot \left(1 + \color{blue}{y}\right)} \]

4. Applied rewrites 48.7%

   \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]

5. Taylor expanded in y around 0

   \[\leadsto \frac{x}{\color{blue}{y}} \]
6. Step-by-step derivation

   1. lower-/.f64 26.7

      \[\leadsto \frac{x}{y} \]

7. Applied rewrites 26.7%

   \[\leadsto \frac{x}{\color{blue}{y}} \]
8. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \frac{x}{y} \]

   2. div-flip N/A

      \[\leadsto \frac{1}{\frac{y}{\color{blue}{x}}} \]

   3. lower-/.f64 N/A

      \[\leadsto \frac{1}{\frac{y}{\color{blue}{x}}} \]

   4. lower-/.f64 27.2

      \[\leadsto \frac{1}{\frac{y}{x}} \]

9. Applied rewrites 27.2%

   \[\leadsto \frac{1}{\frac{y}{\color{blue}{x}}} \]
10. Add Preprocessing

Alternative 15: 26.7% accurate, 5.5× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \frac{x}{y} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y) :precision binary64 (/ x y))

C

assert(x < y);
double code(double x, double y) {
	return x / y;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = x / y
end function

Java

assert x < y;
public static double code(double x, double y) {
	return x / y;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	return x / y

Julia

x, y = sort([x, y])
function code(x, y)
	return Float64(x / y)
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp = code(x, y)
	tmp = x / y;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := N[(x / y), $MachinePrecision]

Derivation

1. Initial program 69.1%

   \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

2. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]
3. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \frac{x}{\color{blue}{y \cdot \left(1 + y\right)}} \]

   2. lower-*.f64 N/A

      \[\leadsto \frac{x}{y \cdot \color{blue}{\left(1 + y\right)}} \]

   3. lower-+.f64 48.7

      \[\leadsto \frac{x}{y \cdot \left(1 + \color{blue}{y}\right)} \]

4. Applied rewrites 48.7%

   \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]

5. Taylor expanded in y around 0

   \[\leadsto \frac{x}{\color{blue}{y}} \]
6. Step-by-step derivation

   1. lower-/.f64 26.7

      \[\leadsto \frac{x}{y} \]

7. Applied rewrites 26.7%

   \[\leadsto \frac{x}{\color{blue}{y}} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2025156 
(FPCore (x y)
  :name "Numeric.SpecFunctions:incompleteBetaApprox from math-functions-0.1.5.2, A"
  :precision binary64
  (/ (* x y) (* (* (+ x y) (+ x y)) (+ (+ x y) 1.0))))