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

Percentage Accurate: 68.4% → 99.8%

Time: 4.7s

Alternatives: 18

Speedup: 1.5×

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: 68.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 69.0%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-+.f64 N/A

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

   5. lift-+.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. lift-+.f64 N/A

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

   8. lift-+.f64 N/A

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

   9. times-frac N/A

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

   10. lower-*.f64 N/A

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

   11. lower-/.f64 N/A

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

   12. pow2 N/A

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

   13. lower-pow.f64 N/A

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

   14. +-commutative N/A

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

   15. lower-+.f64 N/A

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

   16. lower-/.f64 N/A

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

   17. lift-+.f64 N/A

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

   18. +-commutative N/A

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

   19. lower-+.f64 88.3

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

4. Applied rewrites 88.3%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lower-pow.f64 N/A

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

   4. pow2 N/A

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

   5. lift-*.f64 N/A

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

   6. +-commutative N/A

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

   7. lift-+.f64 N/A

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

   8. +-commutative N/A

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

   9. lift-+.f64 88.3

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

6. Applied rewrites 88.3%

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

   1. lift-/.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. lift-+.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. associate-/r* N/A

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

   6. +-commutative N/A

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

   7. lower-/.f64 N/A

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

   8. +-commutative N/A

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

   9. lower-/.f64 N/A

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

   10. +-commutative N/A

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

   11. lift-+.f64 N/A

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

   12. +-commutative N/A

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

   13. lift-+.f64 99.8

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

8. Applied rewrites 99.8%

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

   1. lift-*.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. lift-+.f64 N/A

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

   5. lift-/.f64 N/A

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

   6. lift-/.f64 N/A

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

   7. lift-+.f64 N/A

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

   8. lift-+.f64 N/A

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

   9. associate-*l/ N/A

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

   10. lower-/.f64 N/A

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

   11. lower-*.f64 N/A

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

   12. lift-/.f64 N/A

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

   13. lift-+.f64 N/A

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

   14. lift-+.f64 N/A

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

   15. lift-+.f64 N/A

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

   16. lift-/.f64 N/A

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

   17. lift-+.f64 99.8

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

10. Applied rewrites 99.8%

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

Alternative 2: 83.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_0 := \left(y + x\right) + 1\\ \mathbf{if}\;y \leq 2.8 \cdot 10^{-259}:\\ \;\;\;\;\frac{1}{y + x} \cdot \frac{y}{t\_0}\\ \mathbf{elif}\;y \leq 1.1 \cdot 10^{-161}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;y \leq 2.3 \cdot 10^{+102}:\\ \;\;\;\;\frac{x \cdot y}{\left(y + x\right) \cdot \left(\left(y + x\right) \cdot t\_0\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + x}}{y + x} \cdot 1\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ (+ y x) 1.0)))
   (if (<= y 2.8e-259)
     (* (/ 1.0 (+ y x)) (/ y t_0))
     (if (<= y 1.1e-161)
       (/ x y)
       (if (<= y 2.3e+102)
         (/ (* x y) (* (+ y x) (* (+ y x) t_0)))
         (* (/ (/ x (+ y x)) (+ y x)) 1.0))))))

C

assert(x < y);
double code(double x, double y) {
	double t_0 = (y + x) + 1.0;
	double tmp;
	if (y <= 2.8e-259) {
		tmp = (1.0 / (y + x)) * (y / t_0);
	} else if (y <= 1.1e-161) {
		tmp = x / y;
	} else if (y <= 2.3e+102) {
		tmp = (x * y) / ((y + x) * ((y + x) * t_0));
	} else {
		tmp = ((x / (y + x)) / (y + x)) * 1.0;
	}
	return tmp;
}

Fortran

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

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

real(8) function code(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (y + x) + 1.0d0
    if (y <= 2.8d-259) then
        tmp = (1.0d0 / (y + x)) * (y / t_0)
    else if (y <= 1.1d-161) then
        tmp = x / y
    else if (y <= 2.3d+102) then
        tmp = (x * y) / ((y + x) * ((y + x) * t_0))
    else
        tmp = ((x / (y + x)) / (y + x)) * 1.0d0
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double t_0 = (y + x) + 1.0;
	double tmp;
	if (y <= 2.8e-259) {
		tmp = (1.0 / (y + x)) * (y / t_0);
	} else if (y <= 1.1e-161) {
		tmp = x / y;
	} else if (y <= 2.3e+102) {
		tmp = (x * y) / ((y + x) * ((y + x) * t_0));
	} else {
		tmp = ((x / (y + x)) / (y + x)) * 1.0;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	t_0 = (y + x) + 1.0
	tmp = 0
	if y <= 2.8e-259:
		tmp = (1.0 / (y + x)) * (y / t_0)
	elif y <= 1.1e-161:
		tmp = x / y
	elif y <= 2.3e+102:
		tmp = (x * y) / ((y + x) * ((y + x) * t_0))
	else:
		tmp = ((x / (y + x)) / (y + x)) * 1.0
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	t_0 = Float64(Float64(y + x) + 1.0)
	tmp = 0.0
	if (y <= 2.8e-259)
		tmp = Float64(Float64(1.0 / Float64(y + x)) * Float64(y / t_0));
	elseif (y <= 1.1e-161)
		tmp = Float64(x / y);
	elseif (y <= 2.3e+102)
		tmp = Float64(Float64(x * y) / Float64(Float64(y + x) * Float64(Float64(y + x) * t_0)));
	else
		tmp = Float64(Float64(Float64(x / Float64(y + x)) / Float64(y + x)) * 1.0);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	t_0 = (y + x) + 1.0;
	tmp = 0.0;
	if (y <= 2.8e-259)
		tmp = (1.0 / (y + x)) * (y / t_0);
	elseif (y <= 1.1e-161)
		tmp = x / y;
	elseif (y <= 2.3e+102)
		tmp = (x * y) / ((y + x) * ((y + x) * t_0));
	else
		tmp = ((x / (y + x)) / (y + x)) * 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := Block[{t$95$0 = N[(N[(y + x), $MachinePrecision] + 1.0), $MachinePrecision]}, If[LessEqual[y, 2.8e-259], N[(N[(1.0 / N[(y + x), $MachinePrecision]), $MachinePrecision] * N[(y / t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.1e-161], N[(x / y), $MachinePrecision], If[LessEqual[y, 2.3e+102], N[(N[(x * y), $MachinePrecision] / N[(N[(y + x), $MachinePrecision] * N[(N[(y + x), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x / N[(y + x), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if y < 2.8e-259

   1. Initial program 69.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. times-frac N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. pow2 N/A

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

      13. lower-pow.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. lift-+.f64 N/A

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

      18. +-commutative N/A

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

      19. lower-+.f64 89.4

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

   4. Applied rewrites 89.4%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-pow.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-+.f64 89.4

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

   6. Applied rewrites 89.4%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/r* N/A

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

      6. +-commutative N/A

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

      7. lower-/.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-/.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lift-+.f64 99.8

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

   8. Applied rewrites 99.8%

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

   9. Taylor expanded in x around inf

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

   11. Applied rewrites 53.1%

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

   if 2.8e-259 < y < 1.10000000000000001e-161

   1. Initial program 62.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. times-frac N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. pow2 N/A

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

      13. lower-pow.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. lift-+.f64 N/A

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

      18. +-commutative N/A

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

      19. lower-+.f64 66.4

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

   4. Applied rewrites 66.4%

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

   5. Taylor expanded in x around 0

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

      1. frac-times N/A

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

      2. unpow2 N/A

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

      3. +-commutative N/A

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

      4. distribute-lft-out N/A

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

      5. +-commutative N/A

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

      6. lower-/.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lift-+.f64 24.1

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

   7. Applied rewrites 24.1%

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

   8. Taylor expanded in y around 0

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

   10. Applied rewrites 24.1%

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

   if 1.10000000000000001e-161 < y < 2.2999999999999999e102

   1. Initial program 87.9%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 N/A

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

      14. lift-+.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 87.9

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

   4. Applied rewrites 87.9%

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

   if 2.2999999999999999e102 < y

   1. Initial program 54.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. times-frac N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. pow2 N/A

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

      13. lower-pow.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. lift-+.f64 N/A

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

      18. +-commutative N/A

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

      19. lower-+.f64 88.6

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

   4. Applied rewrites 88.6%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-pow.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-+.f64 88.6

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

   6. Applied rewrites 88.6%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/r* N/A

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

      6. +-commutative N/A

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

      7. lower-/.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-/.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lift-+.f64 99.8

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

   8. Applied rewrites 99.8%

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

   9. Taylor expanded in y around inf

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

   11. Applied rewrites 84.6%

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

4. Final simplification 62.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.8 \cdot 10^{-259}:\\ \;\;\;\;\frac{1}{y + x} \cdot \frac{y}{\left(y + x\right) + 1}\\ \mathbf{elif}\;y \leq 1.1 \cdot 10^{-161}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;y \leq 2.3 \cdot 10^{+102}:\\ \;\;\;\;\frac{x \cdot y}{\left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(\left(y + x\right) + 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + x}}{y + x} \cdot 1\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 83.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 2.8 \cdot 10^{-259}:\\ \;\;\;\;\frac{1}{y + x} \cdot \frac{y}{\left(y + x\right) + 1}\\ \mathbf{elif}\;y \leq 1.1 \cdot 10^{-161}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;y \leq 2.3 \cdot 10^{+102}:\\ \;\;\;\;\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + x}}{y + x} \cdot 1\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 2.8e-259)
   (* (/ 1.0 (+ y x)) (/ y (+ (+ y x) 1.0)))
   (if (<= y 1.1e-161)
     (/ x y)
     (if (<= y 2.3e+102)
       (/ (* x y) (* (* (+ x y) (+ x y)) (+ (+ x y) 1.0)))
       (* (/ (/ x (+ y x)) (+ y x)) 1.0)))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 2.8e-259) {
		tmp = (1.0 / (y + x)) * (y / ((y + x) + 1.0));
	} else if (y <= 1.1e-161) {
		tmp = x / y;
	} else if (y <= 2.3e+102) {
		tmp = (x * y) / (((x + y) * (x + y)) * ((x + y) + 1.0));
	} else {
		tmp = ((x / (y + x)) / (y + x)) * 1.0;
	}
	return tmp;
}

Fortran

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

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

real(8) function code(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 2.8d-259) then
        tmp = (1.0d0 / (y + x)) * (y / ((y + x) + 1.0d0))
    else if (y <= 1.1d-161) then
        tmp = x / y
    else if (y <= 2.3d+102) then
        tmp = (x * y) / (((x + y) * (x + y)) * ((x + y) + 1.0d0))
    else
        tmp = ((x / (y + x)) / (y + x)) * 1.0d0
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (y <= 2.8e-259) {
		tmp = (1.0 / (y + x)) * (y / ((y + x) + 1.0));
	} else if (y <= 1.1e-161) {
		tmp = x / y;
	} else if (y <= 2.3e+102) {
		tmp = (x * y) / (((x + y) * (x + y)) * ((x + y) + 1.0));
	} else {
		tmp = ((x / (y + x)) / (y + x)) * 1.0;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if y <= 2.8e-259:
		tmp = (1.0 / (y + x)) * (y / ((y + x) + 1.0))
	elif y <= 1.1e-161:
		tmp = x / y
	elif y <= 2.3e+102:
		tmp = (x * y) / (((x + y) * (x + y)) * ((x + y) + 1.0))
	else:
		tmp = ((x / (y + x)) / (y + x)) * 1.0
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= 2.8e-259)
		tmp = Float64(Float64(1.0 / Float64(y + x)) * Float64(y / Float64(Float64(y + x) + 1.0)));
	elseif (y <= 1.1e-161)
		tmp = Float64(x / y);
	elseif (y <= 2.3e+102)
		tmp = Float64(Float64(x * y) / Float64(Float64(Float64(x + y) * Float64(x + y)) * Float64(Float64(x + y) + 1.0)));
	else
		tmp = Float64(Float64(Float64(x / Float64(y + x)) / Float64(y + x)) * 1.0);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 2.8e-259)
		tmp = (1.0 / (y + x)) * (y / ((y + x) + 1.0));
	elseif (y <= 1.1e-161)
		tmp = x / y;
	elseif (y <= 2.3e+102)
		tmp = (x * y) / (((x + y) * (x + y)) * ((x + y) + 1.0));
	else
		tmp = ((x / (y + x)) / (y + x)) * 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[y, 2.8e-259], N[(N[(1.0 / N[(y + x), $MachinePrecision]), $MachinePrecision] * N[(y / N[(N[(y + x), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.1e-161], N[(x / y), $MachinePrecision], If[LessEqual[y, 2.3e+102], N[(N[(x * y), $MachinePrecision] / N[(N[(N[(x + y), $MachinePrecision] * N[(x + y), $MachinePrecision]), $MachinePrecision] * N[(N[(x + y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x / N[(y + x), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if y < 2.8e-259

   1. Initial program 69.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. times-frac N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. pow2 N/A

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

      13. lower-pow.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. lift-+.f64 N/A

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

      18. +-commutative N/A

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

      19. lower-+.f64 89.4

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

   4. Applied rewrites 89.4%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-pow.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-+.f64 89.4

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

   6. Applied rewrites 89.4%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/r* N/A

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

      6. +-commutative N/A

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

      7. lower-/.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-/.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lift-+.f64 99.8

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

   8. Applied rewrites 99.8%

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

   9. Taylor expanded in x around inf

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

   11. Applied rewrites 53.1%

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

   if 2.8e-259 < y < 1.10000000000000001e-161

   1. Initial program 62.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. times-frac N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. pow2 N/A

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

      13. lower-pow.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. lift-+.f64 N/A

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

      18. +-commutative N/A

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

      19. lower-+.f64 66.4

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

   4. Applied rewrites 66.4%

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

   5. Taylor expanded in x around 0

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

      1. frac-times N/A

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

      2. unpow2 N/A

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

      3. +-commutative N/A

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

      4. distribute-lft-out N/A

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

      5. +-commutative N/A

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

      6. lower-/.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lift-+.f64 24.1

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

   7. Applied rewrites 24.1%

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

   8. Taylor expanded in y around 0

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

   10. Applied rewrites 24.1%

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

   if 1.10000000000000001e-161 < y < 2.2999999999999999e102

   1. Initial program 87.9%

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

   if 2.2999999999999999e102 < y

   1. Initial program 54.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. times-frac N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. pow2 N/A

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

      13. lower-pow.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. lift-+.f64 N/A

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

      18. +-commutative N/A

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

      19. lower-+.f64 88.6

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

   4. Applied rewrites 88.6%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-pow.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-+.f64 88.6

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

   6. Applied rewrites 88.6%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/r* N/A

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

      6. +-commutative N/A

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

      7. lower-/.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-/.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lift-+.f64 99.8

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

   8. Applied rewrites 99.8%

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

   9. Taylor expanded in y around inf

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

   11. Applied rewrites 84.6%

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

4. Final simplification 62.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.8 \cdot 10^{-259}:\\ \;\;\;\;\frac{1}{y + x} \cdot \frac{y}{\left(y + x\right) + 1}\\ \mathbf{elif}\;y \leq 1.1 \cdot 10^{-161}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;y \leq 2.3 \cdot 10^{+102}:\\ \;\;\;\;\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + x}}{y + x} \cdot 1\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 92.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_0 := \frac{y}{\left(y + x\right) + 1}\\ \mathbf{if}\;x \leq -5.7 \cdot 10^{+155}:\\ \;\;\;\;\frac{1}{y + x} \cdot t\_0\\ \mathbf{elif}\;x \leq -2.15 \cdot 10^{-159}:\\ \;\;\;\;\frac{x}{\left(y + x\right) \cdot \left(y + x\right)} \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\left(1 + y\right) \cdot y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ y (+ (+ y x) 1.0))))
   (if (<= x -5.7e+155)
     (* (/ 1.0 (+ y x)) t_0)
     (if (<= x -2.15e-159)
       (* (/ x (* (+ y x) (+ y x))) t_0)
       (/ x (* (+ 1.0 y) y))))))

C

assert(x < y);
double code(double x, double y) {
	double t_0 = y / ((y + x) + 1.0);
	double tmp;
	if (x <= -5.7e+155) {
		tmp = (1.0 / (y + x)) * t_0;
	} else if (x <= -2.15e-159) {
		tmp = (x / ((y + x) * (y + x))) * t_0;
	} else {
		tmp = x / ((1.0 + y) * y);
	}
	return tmp;
}

Fortran

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

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

real(8) function code(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = y / ((y + x) + 1.0d0)
    if (x <= (-5.7d+155)) then
        tmp = (1.0d0 / (y + x)) * t_0
    else if (x <= (-2.15d-159)) then
        tmp = (x / ((y + x) * (y + x))) * t_0
    else
        tmp = x / ((1.0d0 + y) * y)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double t_0 = y / ((y + x) + 1.0);
	double tmp;
	if (x <= -5.7e+155) {
		tmp = (1.0 / (y + x)) * t_0;
	} else if (x <= -2.15e-159) {
		tmp = (x / ((y + x) * (y + x))) * t_0;
	} else {
		tmp = x / ((1.0 + y) * y);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	t_0 = y / ((y + x) + 1.0)
	tmp = 0
	if x <= -5.7e+155:
		tmp = (1.0 / (y + x)) * t_0
	elif x <= -2.15e-159:
		tmp = (x / ((y + x) * (y + x))) * t_0
	else:
		tmp = x / ((1.0 + y) * y)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	t_0 = Float64(y / Float64(Float64(y + x) + 1.0))
	tmp = 0.0
	if (x <= -5.7e+155)
		tmp = Float64(Float64(1.0 / Float64(y + x)) * t_0);
	elseif (x <= -2.15e-159)
		tmp = Float64(Float64(x / Float64(Float64(y + x) * Float64(y + x))) * t_0);
	else
		tmp = Float64(x / Float64(Float64(1.0 + y) * y));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	t_0 = y / ((y + x) + 1.0);
	tmp = 0.0;
	if (x <= -5.7e+155)
		tmp = (1.0 / (y + x)) * t_0;
	elseif (x <= -2.15e-159)
		tmp = (x / ((y + x) * (y + x))) * t_0;
	else
		tmp = x / ((1.0 + y) * y);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := Block[{t$95$0 = N[(y / N[(N[(y + x), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -5.7e+155], N[(N[(1.0 / N[(y + x), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision], If[LessEqual[x, -2.15e-159], N[(N[(x / N[(N[(y + x), $MachinePrecision] * N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision], N[(x / N[(N[(1.0 + y), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -5.6999999999999996e155

   1. Initial program 56.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. times-frac N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. pow2 N/A

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

      13. lower-pow.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. lift-+.f64 N/A

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

      18. +-commutative N/A

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

      19. lower-+.f64 84.0

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

   4. Applied rewrites 84.0%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-pow.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-+.f64 84.0

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

   6. Applied rewrites 84.0%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/r* N/A

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

      6. +-commutative N/A

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

      7. lower-/.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-/.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lift-+.f64 99.8

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

   8. Applied rewrites 99.8%

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

   9. Taylor expanded in x around inf

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

   11. Applied rewrites 81.6%

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

   if -5.6999999999999996e155 < x < -2.15e-159

   1. Initial program 73.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. times-frac N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. pow2 N/A

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

      13. lower-pow.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. lift-+.f64 N/A

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

      18. +-commutative N/A

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

      19. lower-+.f64 97.1

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

   4. Applied rewrites 97.1%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-pow.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-+.f64 97.1

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

   6. Applied rewrites 97.1%

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

   if -2.15e-159 < x

   1. Initial program 69.5%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-+.f64 56.2

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

   5. Applied rewrites 56.2%

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

Alternative 5: 88.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -0.095:\\ \;\;\;\;\frac{1}{y + x} \cdot \frac{y}{\left(y + x\right) + 1}\\ \mathbf{elif}\;x \leq -2.15 \cdot 10^{-159}:\\ \;\;\;\;\frac{x}{\left(y + x\right) \cdot \left(y + x\right)} \cdot \frac{y}{y + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\left(1 + y\right) \cdot y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -0.095)
   (* (/ 1.0 (+ y x)) (/ y (+ (+ y x) 1.0)))
   (if (<= x -2.15e-159)
     (* (/ x (* (+ y x) (+ y x))) (/ y (+ y 1.0)))
     (/ x (* (+ 1.0 y) y)))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -0.095) {
		tmp = (1.0 / (y + x)) * (y / ((y + x) + 1.0));
	} else if (x <= -2.15e-159) {
		tmp = (x / ((y + x) * (y + x))) * (y / (y + 1.0));
	} else {
		tmp = x / ((1.0 + y) * y);
	}
	return tmp;
}

Fortran

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

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

real(8) function code(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-0.095d0)) then
        tmp = (1.0d0 / (y + x)) * (y / ((y + x) + 1.0d0))
    else if (x <= (-2.15d-159)) then
        tmp = (x / ((y + x) * (y + x))) * (y / (y + 1.0d0))
    else
        tmp = x / ((1.0d0 + y) * y)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (x <= -0.095) {
		tmp = (1.0 / (y + x)) * (y / ((y + x) + 1.0));
	} else if (x <= -2.15e-159) {
		tmp = (x / ((y + x) * (y + x))) * (y / (y + 1.0));
	} else {
		tmp = x / ((1.0 + y) * y);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -0.095:
		tmp = (1.0 / (y + x)) * (y / ((y + x) + 1.0))
	elif x <= -2.15e-159:
		tmp = (x / ((y + x) * (y + x))) * (y / (y + 1.0))
	else:
		tmp = x / ((1.0 + y) * y)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -0.095)
		tmp = Float64(Float64(1.0 / Float64(y + x)) * Float64(y / Float64(Float64(y + x) + 1.0)));
	elseif (x <= -2.15e-159)
		tmp = Float64(Float64(x / Float64(Float64(y + x) * Float64(y + x))) * Float64(y / Float64(y + 1.0)));
	else
		tmp = Float64(x / Float64(Float64(1.0 + y) * y));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -0.095)
		tmp = (1.0 / (y + x)) * (y / ((y + x) + 1.0));
	elseif (x <= -2.15e-159)
		tmp = (x / ((y + x) * (y + x))) * (y / (y + 1.0));
	else
		tmp = x / ((1.0 + y) * y);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[x, -0.095], N[(N[(1.0 / N[(y + x), $MachinePrecision]), $MachinePrecision] * N[(y / N[(N[(y + x), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -2.15e-159], N[(N[(x / N[(N[(y + x), $MachinePrecision] * N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(y / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x / N[(N[(1.0 + y), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.095000000000000001

   1. Initial program 59.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. times-frac N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. pow2 N/A

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

      13. lower-pow.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. lift-+.f64 N/A

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

      18. +-commutative N/A

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

      19. lower-+.f64 88.6

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

   4. Applied rewrites 88.6%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-pow.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-+.f64 88.6

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

   6. Applied rewrites 88.6%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/r* N/A

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

      6. +-commutative N/A

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

      7. lower-/.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-/.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lift-+.f64 99.7

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

   8. Applied rewrites 99.7%

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

   9. Taylor expanded in x around inf

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

   11. Applied rewrites 71.7%

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

   if -0.095000000000000001 < x < -2.15e-159

   1. Initial program 81.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. times-frac N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. pow2 N/A

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

      13. lower-pow.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. lift-+.f64 N/A

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

      18. +-commutative N/A

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

      19. lower-+.f64 99.8

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

   4. Applied rewrites 99.8%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-pow.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-+.f64 99.8

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

   6. Applied rewrites 99.8%

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

   7. Taylor expanded in x around 0

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

      1. +-commutative 99.2

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

   9. Applied rewrites 99.2%

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

   if -2.15e-159 < x

   1. Initial program 69.5%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-+.f64 56.2

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

   5. Applied rewrites 56.2%

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

Alternative 6: 96.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_0 := \frac{x}{y + x}\\ \mathbf{if}\;x \leq -5.4 \cdot 10^{+155}:\\ \;\;\;\;\frac{t\_0}{y + x} \cdot \frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0 \cdot y}{\left(\left(y + x\right) + 1\right) \cdot \left(y + x\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ x (+ y x))))
   (if (<= x -5.4e+155)
     (* (/ t_0 (+ y x)) (/ y x))
     (/ (* t_0 y) (* (+ (+ y x) 1.0) (+ y x))))))

C

assert(x < y);
double code(double x, double y) {
	double t_0 = x / (y + x);
	double tmp;
	if (x <= -5.4e+155) {
		tmp = (t_0 / (y + x)) * (y / x);
	} else {
		tmp = (t_0 * y) / (((y + x) + 1.0) * (y + x));
	}
	return tmp;
}

Fortran

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

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

real(8) function code(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x / (y + x)
    if (x <= (-5.4d+155)) then
        tmp = (t_0 / (y + x)) * (y / x)
    else
        tmp = (t_0 * y) / (((y + x) + 1.0d0) * (y + x))
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double t_0 = x / (y + x);
	double tmp;
	if (x <= -5.4e+155) {
		tmp = (t_0 / (y + x)) * (y / x);
	} else {
		tmp = (t_0 * y) / (((y + x) + 1.0) * (y + x));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	t_0 = x / (y + x)
	tmp = 0
	if x <= -5.4e+155:
		tmp = (t_0 / (y + x)) * (y / x)
	else:
		tmp = (t_0 * y) / (((y + x) + 1.0) * (y + x))
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	t_0 = Float64(x / Float64(y + x))
	tmp = 0.0
	if (x <= -5.4e+155)
		tmp = Float64(Float64(t_0 / Float64(y + x)) * Float64(y / x));
	else
		tmp = Float64(Float64(t_0 * y) / Float64(Float64(Float64(y + x) + 1.0) * Float64(y + x)));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	t_0 = x / (y + x);
	tmp = 0.0;
	if (x <= -5.4e+155)
		tmp = (t_0 / (y + x)) * (y / x);
	else
		tmp = (t_0 * y) / (((y + x) + 1.0) * (y + x));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := Block[{t$95$0 = N[(x / N[(y + x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -5.4e+155], N[(N[(t$95$0 / N[(y + x), $MachinePrecision]), $MachinePrecision] * N[(y / x), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$0 * y), $MachinePrecision] / N[(N[(N[(y + x), $MachinePrecision] + 1.0), $MachinePrecision] * N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -5.39999999999999987e155

   1. Initial program 56.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. times-frac N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. pow2 N/A

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

      13. lower-pow.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. lift-+.f64 N/A

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

      18. +-commutative N/A

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

      19. lower-+.f64 84.0

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

   4. Applied rewrites 84.0%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-pow.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-+.f64 84.0

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

   6. Applied rewrites 84.0%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/r* N/A

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

      6. +-commutative N/A

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

      7. lower-/.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-/.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lift-+.f64 99.8

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

   8. Applied rewrites 99.8%

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

   9. Taylor expanded in x around inf

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

   11. Applied rewrites 87.7%

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

   if -5.39999999999999987e155 < x

   1. Initial program 70.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. times-frac N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. pow2 N/A

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

      13. lower-pow.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. lift-+.f64 N/A

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

      18. +-commutative N/A

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

      19. lower-+.f64 88.9

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

   4. Applied rewrites 88.9%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-pow.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-+.f64 88.9

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

   6. Applied rewrites 88.9%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/r* N/A

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

      6. +-commutative N/A

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

      7. lower-/.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-/.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lift-+.f64 99.8

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

   8. Applied rewrites 99.8%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. frac-times N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lift-/.f64 N/A

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

      13. lift-+.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. lift-+.f64 N/A

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

      17. lift-+.f64 N/A

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

      18. lift-+.f64 96.3

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

   10. Applied rewrites 96.3%

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

Alternative 7: 96.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_0 := \frac{x}{y + x}\\ \mathbf{if}\;x \leq -5.4 \cdot 10^{+155}:\\ \;\;\;\;\frac{t\_0}{y + x} \cdot \frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \frac{y}{\left(\left(y + x\right) + 1\right) \cdot \left(y + x\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ x (+ y x))))
   (if (<= x -5.4e+155)
     (* (/ t_0 (+ y x)) (/ y x))
     (* t_0 (/ y (* (+ (+ y x) 1.0) (+ y x)))))))

C

assert(x < y);
double code(double x, double y) {
	double t_0 = x / (y + x);
	double tmp;
	if (x <= -5.4e+155) {
		tmp = (t_0 / (y + x)) * (y / x);
	} else {
		tmp = t_0 * (y / (((y + x) + 1.0) * (y + x)));
	}
	return tmp;
}

Fortran

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

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

real(8) function code(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x / (y + x)
    if (x <= (-5.4d+155)) then
        tmp = (t_0 / (y + x)) * (y / x)
    else
        tmp = t_0 * (y / (((y + x) + 1.0d0) * (y + x)))
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double t_0 = x / (y + x);
	double tmp;
	if (x <= -5.4e+155) {
		tmp = (t_0 / (y + x)) * (y / x);
	} else {
		tmp = t_0 * (y / (((y + x) + 1.0) * (y + x)));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	t_0 = x / (y + x)
	tmp = 0
	if x <= -5.4e+155:
		tmp = (t_0 / (y + x)) * (y / x)
	else:
		tmp = t_0 * (y / (((y + x) + 1.0) * (y + x)))
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	t_0 = Float64(x / Float64(y + x))
	tmp = 0.0
	if (x <= -5.4e+155)
		tmp = Float64(Float64(t_0 / Float64(y + x)) * Float64(y / x));
	else
		tmp = Float64(t_0 * Float64(y / Float64(Float64(Float64(y + x) + 1.0) * Float64(y + x))));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	t_0 = x / (y + x);
	tmp = 0.0;
	if (x <= -5.4e+155)
		tmp = (t_0 / (y + x)) * (y / x);
	else
		tmp = t_0 * (y / (((y + x) + 1.0) * (y + x)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := Block[{t$95$0 = N[(x / N[(y + x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -5.4e+155], N[(N[(t$95$0 / N[(y + x), $MachinePrecision]), $MachinePrecision] * N[(y / x), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(y / N[(N[(N[(y + x), $MachinePrecision] + 1.0), $MachinePrecision] * N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -5.39999999999999987e155

   1. Initial program 56.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. times-frac N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. pow2 N/A

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

      13. lower-pow.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. lift-+.f64 N/A

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

      18. +-commutative N/A

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

      19. lower-+.f64 84.0

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

   4. Applied rewrites 84.0%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-pow.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-+.f64 84.0

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

   6. Applied rewrites 84.0%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/r* N/A

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

      6. +-commutative N/A

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

      7. lower-/.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-/.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lift-+.f64 99.8

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

   8. Applied rewrites 99.8%

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

   9. Taylor expanded in x around inf

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

   11. Applied rewrites 87.7%

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

   if -5.39999999999999987e155 < x

   1. Initial program 70.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. times-frac N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. pow2 N/A

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

      13. lower-pow.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. lift-+.f64 N/A

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

      18. +-commutative N/A

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

      19. lower-+.f64 88.9

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

   4. Applied rewrites 88.9%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-pow.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-+.f64 88.9

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

   6. Applied rewrites 88.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. frac-times N/A

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

      10. associate-*r* N/A

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

      11. times-frac N/A

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

      12. lower-*.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-/.f64 N/A

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

      15. +-commutative N/A

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

      16. lift-+.f64 N/A

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

      17. lower-/.f64 N/A

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

      18. +-commutative N/A

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

      19. *-commutative N/A

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

      20. lower-*.f64 N/A

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

   8. Applied rewrites 96.3%

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

Alternative 8: 95.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_0 := \frac{x}{y + x}\\ \mathbf{if}\;y \leq 1.25 \cdot 10^{+130}:\\ \;\;\;\;t\_0 \cdot \frac{y}{\left(\left(y + x\right) + 1\right) \cdot \left(y + x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0}{y + x} \cdot 1\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ x (+ y x))))
   (if (<= y 1.25e+130)
     (* t_0 (/ y (* (+ (+ y x) 1.0) (+ y x))))
     (* (/ t_0 (+ y x)) 1.0))))

C

assert(x < y);
double code(double x, double y) {
	double t_0 = x / (y + x);
	double tmp;
	if (y <= 1.25e+130) {
		tmp = t_0 * (y / (((y + x) + 1.0) * (y + x)));
	} else {
		tmp = (t_0 / (y + x)) * 1.0;
	}
	return tmp;
}

Fortran

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

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

real(8) function code(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x / (y + x)
    if (y <= 1.25d+130) then
        tmp = t_0 * (y / (((y + x) + 1.0d0) * (y + x)))
    else
        tmp = (t_0 / (y + x)) * 1.0d0
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double t_0 = x / (y + x);
	double tmp;
	if (y <= 1.25e+130) {
		tmp = t_0 * (y / (((y + x) + 1.0) * (y + x)));
	} else {
		tmp = (t_0 / (y + x)) * 1.0;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	t_0 = x / (y + x)
	tmp = 0
	if y <= 1.25e+130:
		tmp = t_0 * (y / (((y + x) + 1.0) * (y + x)))
	else:
		tmp = (t_0 / (y + x)) * 1.0
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	t_0 = Float64(x / Float64(y + x))
	tmp = 0.0
	if (y <= 1.25e+130)
		tmp = Float64(t_0 * Float64(y / Float64(Float64(Float64(y + x) + 1.0) * Float64(y + x))));
	else
		tmp = Float64(Float64(t_0 / Float64(y + x)) * 1.0);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	t_0 = x / (y + x);
	tmp = 0.0;
	if (y <= 1.25e+130)
		tmp = t_0 * (y / (((y + x) + 1.0) * (y + x)));
	else
		tmp = (t_0 / (y + x)) * 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := Block[{t$95$0 = N[(x / N[(y + x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, 1.25e+130], N[(t$95$0 * N[(y / N[(N[(N[(y + x), $MachinePrecision] + 1.0), $MachinePrecision] * N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$0 / N[(y + x), $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < 1.2499999999999999e130

   1. Initial program 70.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. times-frac N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. pow2 N/A

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

      13. lower-pow.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. lift-+.f64 N/A

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

      18. +-commutative N/A

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

      19. lower-+.f64 88.7

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

   4. Applied rewrites 88.7%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-pow.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-+.f64 88.7

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

   6. Applied rewrites 88.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. frac-times N/A

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

      10. associate-*r* N/A

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

      11. times-frac N/A

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

      12. lower-*.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-/.f64 N/A

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

      15. +-commutative N/A

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

      16. lift-+.f64 N/A

          \[\leadsto \frac{x}{\color{blue}{y + x}} \cdot \frac{y}{\left(y + x\right) \cdot \left(\left(y + x\right) + 1\right)} \]

      17. lower-/.f64 N/A

          \[\leadsto \frac{x}{y + x} \cdot \color{blue}{\frac{y}{\left(y + x\right) \cdot \left(\left(y + x\right) + 1\right)}} \]

      18. +-commutative N/A

          \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\color{blue}{\left(x + y\right)} \cdot \left(\left(y + x\right) + 1\right)} \]

      19. *-commutative N/A

          \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\color{blue}{\left(\left(y + x\right) + 1\right) \cdot \left(x + y\right)}} \]

      20. lower-*.f64 N/A

          \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\color{blue}{\left(\left(y + x\right) + 1\right) \cdot \left(x + y\right)}} \]

   8. Applied rewrites 96.7%

      \[\leadsto \color{blue}{\frac{x}{y + x} \cdot \frac{y}{\left(\left(y + x\right) + 1\right) \cdot \left(y + x\right)}} \]

   if 1.2499999999999999e130 < y

   1. Initial program 60.7%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot y}}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      2. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      4. lift-+.f64 N/A

         \[\leadsto \frac{x \cdot y}{\left(\color{blue}{\left(x + y\right)} \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      5. lift-+.f64 N/A

         \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \color{blue}{\left(x + y\right)}\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right)} \cdot \left(\left(x + y\right) + 1\right)} \]

      7. lift-+.f64 N/A

         \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\color{blue}{\left(x + y\right)} + 1\right)} \]

      8. lift-+.f64 N/A

         \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \color{blue}{\left(\left(x + y\right) + 1\right)}} \]

      9. times-frac N/A

         \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}} \]

      10. lower-*.f64 N/A

          \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}} \]

      11. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)}} \cdot \frac{y}{\left(x + y\right) + 1} \]

      12. pow2 N/A

          \[\leadsto \frac{x}{\color{blue}{{\left(x + y\right)}^{2}}} \cdot \frac{y}{\left(x + y\right) + 1} \]

      13. lower-pow.f64 N/A

          \[\leadsto \frac{x}{\color{blue}{{\left(x + y\right)}^{2}}} \cdot \frac{y}{\left(x + y\right) + 1} \]

      14. +-commutative N/A

          \[\leadsto \frac{x}{{\color{blue}{\left(y + x\right)}}^{2}} \cdot \frac{y}{\left(x + y\right) + 1} \]

      15. lower-+.f64 N/A

          \[\leadsto \frac{x}{{\color{blue}{\left(y + x\right)}}^{2}} \cdot \frac{y}{\left(x + y\right) + 1} \]

      16. lower-/.f64 N/A

          \[\leadsto \frac{x}{{\left(y + x\right)}^{2}} \cdot \color{blue}{\frac{y}{\left(x + y\right) + 1}} \]

      17. lift-+.f64 N/A

          \[\leadsto \frac{x}{{\left(y + x\right)}^{2}} \cdot \frac{y}{\color{blue}{\left(x + y\right) + 1}} \]

      18. +-commutative N/A

          \[\leadsto \frac{x}{{\left(y + x\right)}^{2}} \cdot \frac{y}{\color{blue}{\left(y + x\right)} + 1} \]

      19. lower-+.f64 86.7

          \[\leadsto \frac{x}{{\left(y + x\right)}^{2}} \cdot \frac{y}{\color{blue}{\left(y + x\right)} + 1} \]

   4. Applied rewrites 86.7%

      \[\leadsto \color{blue}{\frac{x}{{\left(y + x\right)}^{2}} \cdot \frac{y}{\left(y + x\right) + 1}} \]
   5. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{x}{{\color{blue}{\left(y + x\right)}}^{2}} \cdot \frac{y}{\left(y + x\right) + 1} \]

      2. +-commutative N/A

         \[\leadsto \frac{x}{{\color{blue}{\left(x + y\right)}}^{2}} \cdot \frac{y}{\left(y + x\right) + 1} \]

      3. lower-pow.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{{\left(x + y\right)}^{2}}} \cdot \frac{y}{\left(y + x\right) + 1} \]

      4. pow2 N/A

         \[\leadsto \frac{x}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right)}} \cdot \frac{y}{\left(y + x\right) + 1} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right)}} \cdot \frac{y}{\left(y + x\right) + 1} \]

      6. +-commutative N/A

         \[\leadsto \frac{x}{\color{blue}{\left(y + x\right)} \cdot \left(x + y\right)} \cdot \frac{y}{\left(y + x\right) + 1} \]

      7. lift-+.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{\left(y + x\right)} \cdot \left(x + y\right)} \cdot \frac{y}{\left(y + x\right) + 1} \]

      8. +-commutative N/A

         \[\leadsto \frac{x}{\left(y + x\right) \cdot \color{blue}{\left(y + x\right)}} \cdot \frac{y}{\left(y + x\right) + 1} \]

      9. lift-+.f64 86.7

         \[\leadsto \frac{x}{\left(y + x\right) \cdot \color{blue}{\left(y + x\right)}} \cdot \frac{y}{\left(y + x\right) + 1} \]

   6. Applied rewrites 86.7%

      \[\leadsto \frac{x}{\color{blue}{\left(y + x\right) \cdot \left(y + x\right)}} \cdot \frac{y}{\left(y + x\right) + 1} \]
   7. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{\left(y + x\right) \cdot \left(y + x\right)}} \cdot \frac{y}{\left(y + x\right) + 1} \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{\left(y + x\right)} \cdot \left(y + x\right)} \cdot \frac{y}{\left(y + x\right) + 1} \]

      3. lift-+.f64 N/A

         \[\leadsto \frac{x}{\left(y + x\right) \cdot \color{blue}{\left(y + x\right)}} \cdot \frac{y}{\left(y + x\right) + 1} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{\left(y + x\right) \cdot \left(y + x\right)}} \cdot \frac{y}{\left(y + x\right) + 1} \]

      5. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{x}{y + x}}{y + x}} \cdot \frac{y}{\left(y + x\right) + 1} \]

      6. +-commutative N/A

         \[\leadsto \frac{\frac{x}{y + x}}{\color{blue}{x + y}} \cdot \frac{y}{\left(y + x\right) + 1} \]

      7. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{x}{y + x}}{x + y}} \cdot \frac{y}{\left(y + x\right) + 1} \]

      8. +-commutative N/A

         \[\leadsto \frac{\frac{x}{\color{blue}{x + y}}}{x + y} \cdot \frac{y}{\left(y + x\right) + 1} \]

      9. lower-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{x}{x + y}}}{x + y} \cdot \frac{y}{\left(y + x\right) + 1} \]

      10. +-commutative N/A

          \[\leadsto \frac{\frac{x}{\color{blue}{y + x}}}{x + y} \cdot \frac{y}{\left(y + x\right) + 1} \]

      11. lift-+.f64 N/A

          \[\leadsto \frac{\frac{x}{\color{blue}{y + x}}}{x + y} \cdot \frac{y}{\left(y + x\right) + 1} \]

      12. +-commutative N/A

          \[\leadsto \frac{\frac{x}{y + x}}{\color{blue}{y + x}} \cdot \frac{y}{\left(y + x\right) + 1} \]

      13. lift-+.f64 99.8

          \[\leadsto \frac{\frac{x}{y + x}}{\color{blue}{y + x}} \cdot \frac{y}{\left(y + x\right) + 1} \]

   8. Applied rewrites 99.8%

      \[\leadsto \color{blue}{\frac{\frac{x}{y + x}}{y + x}} \cdot \frac{y}{\left(y + x\right) + 1} \]

   9. Taylor expanded in y around inf

      \[\leadsto \frac{\frac{x}{y + x}}{y + x} \cdot \color{blue}{1} \]
   10. Step-by-step derivation

   11. Applied rewrites 85.7%

       \[\leadsto \frac{\frac{x}{y + x}}{y + x} \cdot \color{blue}{1} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 99.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \frac{\frac{x}{y + x}}{y + x} \cdot \frac{y}{\left(y + x\right) + 1} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (* (/ (/ x (+ y x)) (+ y x)) (/ y (+ (+ y x) 1.0))))

C

assert(x < y);
double code(double x, double y) {
	return ((x / (y + x)) / (y + x)) * (y / ((y + x) + 1.0));
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = ((x / (y + x)) / (y + x)) * (y / ((y + x) + 1.0d0))
end function

Java

assert x < y;
public static double code(double x, double y) {
	return ((x / (y + x)) / (y + x)) * (y / ((y + x) + 1.0));
}

Python

[x, y] = sort([x, y])
def code(x, y):
	return ((x / (y + x)) / (y + x)) * (y / ((y + x) + 1.0))

Julia

x, y = sort([x, y])
function code(x, y)
	return Float64(Float64(Float64(x / Float64(y + x)) / Float64(y + x)) * Float64(y / Float64(Float64(y + x) + 1.0)))
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp = code(x, y)
	tmp = ((x / (y + x)) / (y + x)) * (y / ((y + x) + 1.0));
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := N[(N[(N[(x / N[(y + x), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision] * N[(y / N[(N[(y + x), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 69.0%

   \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \frac{\color{blue}{x \cdot y}}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

   2. lift-/.f64 N/A

      \[\leadsto \color{blue}{\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

   3. lift-*.f64 N/A

      \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

   4. lift-+.f64 N/A

      \[\leadsto \frac{x \cdot y}{\left(\color{blue}{\left(x + y\right)} \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

   5. lift-+.f64 N/A

      \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \color{blue}{\left(x + y\right)}\right) \cdot \left(\left(x + y\right) + 1\right)} \]

   6. lift-*.f64 N/A

      \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right)} \cdot \left(\left(x + y\right) + 1\right)} \]

   7. lift-+.f64 N/A

      \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\color{blue}{\left(x + y\right)} + 1\right)} \]

   8. lift-+.f64 N/A

      \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \color{blue}{\left(\left(x + y\right) + 1\right)}} \]

   9. times-frac N/A

      \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}} \]

   10. lower-*.f64 N/A

       \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}} \]

   11. lower-/.f64 N/A

       \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)}} \cdot \frac{y}{\left(x + y\right) + 1} \]

   12. pow2 N/A

       \[\leadsto \frac{x}{\color{blue}{{\left(x + y\right)}^{2}}} \cdot \frac{y}{\left(x + y\right) + 1} \]

   13. lower-pow.f64 N/A

       \[\leadsto \frac{x}{\color{blue}{{\left(x + y\right)}^{2}}} \cdot \frac{y}{\left(x + y\right) + 1} \]

   14. +-commutative N/A

       \[\leadsto \frac{x}{{\color{blue}{\left(y + x\right)}}^{2}} \cdot \frac{y}{\left(x + y\right) + 1} \]

   15. lower-+.f64 N/A

       \[\leadsto \frac{x}{{\color{blue}{\left(y + x\right)}}^{2}} \cdot \frac{y}{\left(x + y\right) + 1} \]

   16. lower-/.f64 N/A

       \[\leadsto \frac{x}{{\left(y + x\right)}^{2}} \cdot \color{blue}{\frac{y}{\left(x + y\right) + 1}} \]

   17. lift-+.f64 N/A

       \[\leadsto \frac{x}{{\left(y + x\right)}^{2}} \cdot \frac{y}{\color{blue}{\left(x + y\right) + 1}} \]

   18. +-commutative N/A

       \[\leadsto \frac{x}{{\left(y + x\right)}^{2}} \cdot \frac{y}{\color{blue}{\left(y + x\right)} + 1} \]

   19. lower-+.f64 88.3

       \[\leadsto \frac{x}{{\left(y + x\right)}^{2}} \cdot \frac{y}{\color{blue}{\left(y + x\right)} + 1} \]

4. Applied rewrites 88.3%

   \[\leadsto \color{blue}{\frac{x}{{\left(y + x\right)}^{2}} \cdot \frac{y}{\left(y + x\right) + 1}} \]
5. Step-by-step derivation

   1. lift-+.f64 N/A

      \[\leadsto \frac{x}{{\color{blue}{\left(y + x\right)}}^{2}} \cdot \frac{y}{\left(y + x\right) + 1} \]

   2. +-commutative N/A

      \[\leadsto \frac{x}{{\color{blue}{\left(x + y\right)}}^{2}} \cdot \frac{y}{\left(y + x\right) + 1} \]

   3. lower-pow.f64 N/A

      \[\leadsto \frac{x}{\color{blue}{{\left(x + y\right)}^{2}}} \cdot \frac{y}{\left(y + x\right) + 1} \]

   4. pow2 N/A

      \[\leadsto \frac{x}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right)}} \cdot \frac{y}{\left(y + x\right) + 1} \]

   5. lift-*.f64 N/A

      \[\leadsto \frac{x}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right)}} \cdot \frac{y}{\left(y + x\right) + 1} \]

   6. +-commutative N/A

      \[\leadsto \frac{x}{\color{blue}{\left(y + x\right)} \cdot \left(x + y\right)} \cdot \frac{y}{\left(y + x\right) + 1} \]

   7. lift-+.f64 N/A

      \[\leadsto \frac{x}{\color{blue}{\left(y + x\right)} \cdot \left(x + y\right)} \cdot \frac{y}{\left(y + x\right) + 1} \]

   8. +-commutative N/A

      \[\leadsto \frac{x}{\left(y + x\right) \cdot \color{blue}{\left(y + x\right)}} \cdot \frac{y}{\left(y + x\right) + 1} \]

   9. lift-+.f64 88.3

      \[\leadsto \frac{x}{\left(y + x\right) \cdot \color{blue}{\left(y + x\right)}} \cdot \frac{y}{\left(y + x\right) + 1} \]

6. Applied rewrites 88.3%

   \[\leadsto \frac{x}{\color{blue}{\left(y + x\right) \cdot \left(y + x\right)}} \cdot \frac{y}{\left(y + x\right) + 1} \]
7. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \color{blue}{\frac{x}{\left(y + x\right) \cdot \left(y + x\right)}} \cdot \frac{y}{\left(y + x\right) + 1} \]

   2. lift-+.f64 N/A

      \[\leadsto \frac{x}{\color{blue}{\left(y + x\right)} \cdot \left(y + x\right)} \cdot \frac{y}{\left(y + x\right) + 1} \]

   3. lift-+.f64 N/A

      \[\leadsto \frac{x}{\left(y + x\right) \cdot \color{blue}{\left(y + x\right)}} \cdot \frac{y}{\left(y + x\right) + 1} \]

   4. lift-*.f64 N/A

      \[\leadsto \frac{x}{\color{blue}{\left(y + x\right) \cdot \left(y + x\right)}} \cdot \frac{y}{\left(y + x\right) + 1} \]

   5. associate-/r* N/A

      \[\leadsto \color{blue}{\frac{\frac{x}{y + x}}{y + x}} \cdot \frac{y}{\left(y + x\right) + 1} \]

   6. +-commutative N/A

      \[\leadsto \frac{\frac{x}{y + x}}{\color{blue}{x + y}} \cdot \frac{y}{\left(y + x\right) + 1} \]

   7. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\frac{x}{y + x}}{x + y}} \cdot \frac{y}{\left(y + x\right) + 1} \]

   8. +-commutative N/A

      \[\leadsto \frac{\frac{x}{\color{blue}{x + y}}}{x + y} \cdot \frac{y}{\left(y + x\right) + 1} \]

   9. lower-/.f64 N/A

      \[\leadsto \frac{\color{blue}{\frac{x}{x + y}}}{x + y} \cdot \frac{y}{\left(y + x\right) + 1} \]

   10. +-commutative N/A

       \[\leadsto \frac{\frac{x}{\color{blue}{y + x}}}{x + y} \cdot \frac{y}{\left(y + x\right) + 1} \]

   11. lift-+.f64 N/A

       \[\leadsto \frac{\frac{x}{\color{blue}{y + x}}}{x + y} \cdot \frac{y}{\left(y + x\right) + 1} \]

   12. +-commutative N/A

       \[\leadsto \frac{\frac{x}{y + x}}{\color{blue}{y + x}} \cdot \frac{y}{\left(y + x\right) + 1} \]

   13. lift-+.f64 99.8

       \[\leadsto \frac{\frac{x}{y + x}}{\color{blue}{y + x}} \cdot \frac{y}{\left(y + x\right) + 1} \]

8. Applied rewrites 99.8%

   \[\leadsto \color{blue}{\frac{\frac{x}{y + x}}{y + x}} \cdot \frac{y}{\left(y + x\right) + 1} \]
9. Add Preprocessing

Alternative 10: 85.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -0.085:\\ \;\;\;\;\frac{1}{y + x} \cdot \frac{y}{\left(y + x\right) + 1}\\ \mathbf{elif}\;x \leq -3.3 \cdot 10^{-159}:\\ \;\;\;\;\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(y + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\left(1 + y\right) \cdot y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -0.085)
   (* (/ 1.0 (+ y x)) (/ y (+ (+ y x) 1.0)))
   (if (<= x -3.3e-159)
     (/ (* x y) (* (* (+ x y) (+ x y)) (+ y 1.0)))
     (/ x (* (+ 1.0 y) y)))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -0.085) {
		tmp = (1.0 / (y + x)) * (y / ((y + x) + 1.0));
	} else if (x <= -3.3e-159) {
		tmp = (x * y) / (((x + y) * (x + y)) * (y + 1.0));
	} else {
		tmp = x / ((1.0 + y) * y);
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-0.085d0)) then
        tmp = (1.0d0 / (y + x)) * (y / ((y + x) + 1.0d0))
    else if (x <= (-3.3d-159)) then
        tmp = (x * y) / (((x + y) * (x + y)) * (y + 1.0d0))
    else
        tmp = x / ((1.0d0 + y) * y)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (x <= -0.085) {
		tmp = (1.0 / (y + x)) * (y / ((y + x) + 1.0));
	} else if (x <= -3.3e-159) {
		tmp = (x * y) / (((x + y) * (x + y)) * (y + 1.0));
	} else {
		tmp = x / ((1.0 + y) * y);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -0.085:
		tmp = (1.0 / (y + x)) * (y / ((y + x) + 1.0))
	elif x <= -3.3e-159:
		tmp = (x * y) / (((x + y) * (x + y)) * (y + 1.0))
	else:
		tmp = x / ((1.0 + y) * y)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -0.085)
		tmp = Float64(Float64(1.0 / Float64(y + x)) * Float64(y / Float64(Float64(y + x) + 1.0)));
	elseif (x <= -3.3e-159)
		tmp = Float64(Float64(x * y) / Float64(Float64(Float64(x + y) * Float64(x + y)) * Float64(y + 1.0)));
	else
		tmp = Float64(x / Float64(Float64(1.0 + y) * y));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -0.085)
		tmp = (1.0 / (y + x)) * (y / ((y + x) + 1.0));
	elseif (x <= -3.3e-159)
		tmp = (x * y) / (((x + y) * (x + y)) * (y + 1.0));
	else
		tmp = x / ((1.0 + y) * y);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[x, -0.085], N[(N[(1.0 / N[(y + x), $MachinePrecision]), $MachinePrecision] * N[(y / N[(N[(y + x), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -3.3e-159], N[(N[(x * y), $MachinePrecision] / N[(N[(N[(x + y), $MachinePrecision] * N[(x + y), $MachinePrecision]), $MachinePrecision] * N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x / N[(N[(1.0 + y), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.0850000000000000061

   1. Initial program 59.2%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot y}}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      2. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      4. lift-+.f64 N/A

         \[\leadsto \frac{x \cdot y}{\left(\color{blue}{\left(x + y\right)} \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      5. lift-+.f64 N/A

         \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \color{blue}{\left(x + y\right)}\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right)} \cdot \left(\left(x + y\right) + 1\right)} \]

      7. lift-+.f64 N/A

         \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\color{blue}{\left(x + y\right)} + 1\right)} \]

      8. lift-+.f64 N/A

         \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \color{blue}{\left(\left(x + y\right) + 1\right)}} \]

      9. times-frac N/A

         \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}} \]

      10. lower-*.f64 N/A

          \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}} \]

      11. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)}} \cdot \frac{y}{\left(x + y\right) + 1} \]

      12. pow2 N/A

          \[\leadsto \frac{x}{\color{blue}{{\left(x + y\right)}^{2}}} \cdot \frac{y}{\left(x + y\right) + 1} \]

      13. lower-pow.f64 N/A

          \[\leadsto \frac{x}{\color{blue}{{\left(x + y\right)}^{2}}} \cdot \frac{y}{\left(x + y\right) + 1} \]

      14. +-commutative N/A

          \[\leadsto \frac{x}{{\color{blue}{\left(y + x\right)}}^{2}} \cdot \frac{y}{\left(x + y\right) + 1} \]

      15. lower-+.f64 N/A

          \[\leadsto \frac{x}{{\color{blue}{\left(y + x\right)}}^{2}} \cdot \frac{y}{\left(x + y\right) + 1} \]

      16. lower-/.f64 N/A

          \[\leadsto \frac{x}{{\left(y + x\right)}^{2}} \cdot \color{blue}{\frac{y}{\left(x + y\right) + 1}} \]

      17. lift-+.f64 N/A

          \[\leadsto \frac{x}{{\left(y + x\right)}^{2}} \cdot \frac{y}{\color{blue}{\left(x + y\right) + 1}} \]

      18. +-commutative N/A

          \[\leadsto \frac{x}{{\left(y + x\right)}^{2}} \cdot \frac{y}{\color{blue}{\left(y + x\right)} + 1} \]

      19. lower-+.f64 88.6

          \[\leadsto \frac{x}{{\left(y + x\right)}^{2}} \cdot \frac{y}{\color{blue}{\left(y + x\right)} + 1} \]

   4. Applied rewrites 88.6%

      \[\leadsto \color{blue}{\frac{x}{{\left(y + x\right)}^{2}} \cdot \frac{y}{\left(y + x\right) + 1}} \]
   5. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{x}{{\color{blue}{\left(y + x\right)}}^{2}} \cdot \frac{y}{\left(y + x\right) + 1} \]

      2. +-commutative N/A

         \[\leadsto \frac{x}{{\color{blue}{\left(x + y\right)}}^{2}} \cdot \frac{y}{\left(y + x\right) + 1} \]

      3. lower-pow.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{{\left(x + y\right)}^{2}}} \cdot \frac{y}{\left(y + x\right) + 1} \]

      4. pow2 N/A

         \[\leadsto \frac{x}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right)}} \cdot \frac{y}{\left(y + x\right) + 1} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right)}} \cdot \frac{y}{\left(y + x\right) + 1} \]

      6. +-commutative N/A

         \[\leadsto \frac{x}{\color{blue}{\left(y + x\right)} \cdot \left(x + y\right)} \cdot \frac{y}{\left(y + x\right) + 1} \]

      7. lift-+.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{\left(y + x\right)} \cdot \left(x + y\right)} \cdot \frac{y}{\left(y + x\right) + 1} \]

      8. +-commutative N/A

         \[\leadsto \frac{x}{\left(y + x\right) \cdot \color{blue}{\left(y + x\right)}} \cdot \frac{y}{\left(y + x\right) + 1} \]

      9. lift-+.f64 88.6

         \[\leadsto \frac{x}{\left(y + x\right) \cdot \color{blue}{\left(y + x\right)}} \cdot \frac{y}{\left(y + x\right) + 1} \]

   6. Applied rewrites 88.6%

      \[\leadsto \frac{x}{\color{blue}{\left(y + x\right) \cdot \left(y + x\right)}} \cdot \frac{y}{\left(y + x\right) + 1} \]
   7. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{\left(y + x\right) \cdot \left(y + x\right)}} \cdot \frac{y}{\left(y + x\right) + 1} \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{\left(y + x\right)} \cdot \left(y + x\right)} \cdot \frac{y}{\left(y + x\right) + 1} \]

      3. lift-+.f64 N/A

         \[\leadsto \frac{x}{\left(y + x\right) \cdot \color{blue}{\left(y + x\right)}} \cdot \frac{y}{\left(y + x\right) + 1} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{\left(y + x\right) \cdot \left(y + x\right)}} \cdot \frac{y}{\left(y + x\right) + 1} \]

      5. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{x}{y + x}}{y + x}} \cdot \frac{y}{\left(y + x\right) + 1} \]

      6. +-commutative N/A

         \[\leadsto \frac{\frac{x}{y + x}}{\color{blue}{x + y}} \cdot \frac{y}{\left(y + x\right) + 1} \]

      7. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{x}{y + x}}{x + y}} \cdot \frac{y}{\left(y + x\right) + 1} \]

      8. +-commutative N/A

         \[\leadsto \frac{\frac{x}{\color{blue}{x + y}}}{x + y} \cdot \frac{y}{\left(y + x\right) + 1} \]

      9. lower-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{x}{x + y}}}{x + y} \cdot \frac{y}{\left(y + x\right) + 1} \]

      10. +-commutative N/A

          \[\leadsto \frac{\frac{x}{\color{blue}{y + x}}}{x + y} \cdot \frac{y}{\left(y + x\right) + 1} \]

      11. lift-+.f64 N/A

          \[\leadsto \frac{\frac{x}{\color{blue}{y + x}}}{x + y} \cdot \frac{y}{\left(y + x\right) + 1} \]

      12. +-commutative N/A

          \[\leadsto \frac{\frac{x}{y + x}}{\color{blue}{y + x}} \cdot \frac{y}{\left(y + x\right) + 1} \]

      13. lift-+.f64 99.7

          \[\leadsto \frac{\frac{x}{y + x}}{\color{blue}{y + x}} \cdot \frac{y}{\left(y + x\right) + 1} \]

   8. Applied rewrites 99.7%

      \[\leadsto \color{blue}{\frac{\frac{x}{y + x}}{y + x}} \cdot \frac{y}{\left(y + x\right) + 1} \]

   9. Taylor expanded in x around inf

      \[\leadsto \frac{\color{blue}{1}}{y + x} \cdot \frac{y}{\left(y + x\right) + 1} \]
   10. Step-by-step derivation

   11. Applied rewrites 71.7%

       \[\leadsto \frac{\color{blue}{1}}{y + x} \cdot \frac{y}{\left(y + x\right) + 1} \]

   if -0.0850000000000000061 < x < -3.3000000000000002e-159

   1. Initial program 81.4%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\color{blue}{y} + 1\right)} \]
   4. Step-by-step derivation

   5. Applied rewrites 81.4%

      \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\color{blue}{y} + 1\right)} \]

   if -3.3000000000000002e-159 < x

   1. Initial program 69.5%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{y \cdot \left(1 + y\right)}} \]

      2. *-commutative N/A

         \[\leadsto \frac{x}{\left(1 + y\right) \cdot \color{blue}{y}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{x}{\left(1 + y\right) \cdot \color{blue}{y}} \]

      4. lower-+.f64 56.2

         \[\leadsto \frac{x}{\left(1 + y\right) \cdot y} \]

   5. Applied rewrites 56.2%

      \[\leadsto \color{blue}{\frac{x}{\left(1 + y\right) \cdot y}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 11: 83.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -5.6 \cdot 10^{+24}:\\ \;\;\;\;\frac{\frac{y}{x}}{x}\\ \mathbf{elif}\;x \leq -3.3 \cdot 10^{-159}:\\ \;\;\;\;\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(y + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\left(1 + y\right) \cdot y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -5.6e+24)
   (/ (/ y x) x)
   (if (<= x -3.3e-159)
     (/ (* x y) (* (* (+ x y) (+ x y)) (+ y 1.0)))
     (/ x (* (+ 1.0 y) y)))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -5.6e+24) {
		tmp = (y / x) / x;
	} else if (x <= -3.3e-159) {
		tmp = (x * y) / (((x + y) * (x + y)) * (y + 1.0));
	} else {
		tmp = x / ((1.0 + y) * y);
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-5.6d+24)) then
        tmp = (y / x) / x
    else if (x <= (-3.3d-159)) then
        tmp = (x * y) / (((x + y) * (x + y)) * (y + 1.0d0))
    else
        tmp = x / ((1.0d0 + y) * y)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (x <= -5.6e+24) {
		tmp = (y / x) / x;
	} else if (x <= -3.3e-159) {
		tmp = (x * y) / (((x + y) * (x + y)) * (y + 1.0));
	} else {
		tmp = x / ((1.0 + y) * y);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -5.6e+24:
		tmp = (y / x) / x
	elif x <= -3.3e-159:
		tmp = (x * y) / (((x + y) * (x + y)) * (y + 1.0))
	else:
		tmp = x / ((1.0 + y) * y)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -5.6e+24)
		tmp = Float64(Float64(y / x) / x);
	elseif (x <= -3.3e-159)
		tmp = Float64(Float64(x * y) / Float64(Float64(Float64(x + y) * Float64(x + y)) * Float64(y + 1.0)));
	else
		tmp = Float64(x / Float64(Float64(1.0 + y) * y));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -5.6e+24)
		tmp = (y / x) / x;
	elseif (x <= -3.3e-159)
		tmp = (x * y) / (((x + y) * (x + y)) * (y + 1.0));
	else
		tmp = x / ((1.0 + y) * y);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[x, -5.6e+24], N[(N[(y / x), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[x, -3.3e-159], N[(N[(x * y), $MachinePrecision] / N[(N[(N[(x + y), $MachinePrecision] * N[(x + y), $MachinePrecision]), $MachinePrecision] * N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x / N[(N[(1.0 + y), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -5.6000000000000003e24

   1. Initial program 55.6%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot y}}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      2. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      4. lift-+.f64 N/A

         \[\leadsto \frac{x \cdot y}{\left(\color{blue}{\left(x + y\right)} \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      5. lift-+.f64 N/A

         \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \color{blue}{\left(x + y\right)}\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right)} \cdot \left(\left(x + y\right) + 1\right)} \]

      7. lift-+.f64 N/A

         \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\color{blue}{\left(x + y\right)} + 1\right)} \]

      8. lift-+.f64 N/A

         \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \color{blue}{\left(\left(x + y\right) + 1\right)}} \]

      9. times-frac N/A

         \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}} \]

      10. lower-*.f64 N/A

          \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}} \]

      11. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)}} \cdot \frac{y}{\left(x + y\right) + 1} \]

      12. pow2 N/A

          \[\leadsto \frac{x}{\color{blue}{{\left(x + y\right)}^{2}}} \cdot \frac{y}{\left(x + y\right) + 1} \]

      13. lower-pow.f64 N/A

          \[\leadsto \frac{x}{\color{blue}{{\left(x + y\right)}^{2}}} \cdot \frac{y}{\left(x + y\right) + 1} \]

      14. +-commutative N/A

          \[\leadsto \frac{x}{{\color{blue}{\left(y + x\right)}}^{2}} \cdot \frac{y}{\left(x + y\right) + 1} \]

      15. lower-+.f64 N/A

          \[\leadsto \frac{x}{{\color{blue}{\left(y + x\right)}}^{2}} \cdot \frac{y}{\left(x + y\right) + 1} \]

      16. lower-/.f64 N/A

          \[\leadsto \frac{x}{{\left(y + x\right)}^{2}} \cdot \color{blue}{\frac{y}{\left(x + y\right) + 1}} \]

      17. lift-+.f64 N/A

          \[\leadsto \frac{x}{{\left(y + x\right)}^{2}} \cdot \frac{y}{\color{blue}{\left(x + y\right) + 1}} \]

      18. +-commutative N/A

          \[\leadsto \frac{x}{{\left(y + x\right)}^{2}} \cdot \frac{y}{\color{blue}{\left(y + x\right)} + 1} \]

      19. lower-+.f64 87.7

          \[\leadsto \frac{x}{{\left(y + x\right)}^{2}} \cdot \frac{y}{\color{blue}{\left(y + x\right)} + 1} \]

   4. Applied rewrites 87.7%

      \[\leadsto \color{blue}{\frac{x}{{\left(y + x\right)}^{2}} \cdot \frac{y}{\left(y + x\right) + 1}} \]

   5. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{y}{{x}^{2}}} \]
   6. Step-by-step derivation

      1. frac-times N/A

         \[\leadsto \frac{\color{blue}{y}}{{x}^{2}} \]

      2. unpow2 N/A

         \[\leadsto \frac{y}{{x}^{2}} \]

      3. +-commutative N/A

         \[\leadsto \frac{y}{{x}^{2}} \]

      4. distribute-lft-out N/A

         \[\leadsto \frac{y}{{x}^{2}} \]

      5. +-commutative N/A

         \[\leadsto \frac{y}{{x}^{2}} \]

      6. lower-/.f64 N/A

         \[\leadsto \frac{y}{\color{blue}{{x}^{2}}} \]

      7. unpow2 N/A

         \[\leadsto \frac{y}{x \cdot \color{blue}{x}} \]

      8. lower-*.f64 75.3

         \[\leadsto \frac{y}{x \cdot \color{blue}{x}} \]

   7. Applied rewrites 75.3%

      \[\leadsto \color{blue}{\frac{y}{x \cdot x}} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{y}{x \cdot \color{blue}{x}} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{y}{\color{blue}{x \cdot x}} \]

      3. associate-/r* N/A

         \[\leadsto \frac{\frac{y}{x}}{\color{blue}{x}} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{\frac{y}{x}}{\color{blue}{x}} \]

      5. lower-/.f64 73.7

         \[\leadsto \frac{\frac{y}{x}}{x} \]

   9. Applied rewrites 73.7%

      \[\leadsto \frac{\frac{y}{x}}{\color{blue}{x}} \]

   if -5.6000000000000003e24 < x < -3.3000000000000002e-159

   1. Initial program 83.4%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\color{blue}{y} + 1\right)} \]
   4. Step-by-step derivation

   5. Applied rewrites 79.2%

      \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\color{blue}{y} + 1\right)} \]

   if -3.3000000000000002e-159 < x

   1. Initial program 69.5%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{y \cdot \left(1 + y\right)}} \]

      2. *-commutative N/A

         \[\leadsto \frac{x}{\left(1 + y\right) \cdot \color{blue}{y}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{x}{\left(1 + y\right) \cdot \color{blue}{y}} \]

      4. lower-+.f64 56.2

         \[\leadsto \frac{x}{\left(1 + y\right) \cdot y} \]

   5. Applied rewrites 56.2%

      \[\leadsto \color{blue}{\frac{x}{\left(1 + y\right) \cdot y}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 64.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.6 \cdot 10^{+24}:\\ \;\;\;\;\frac{\frac{y}{x}}{x}\\ \mathbf{elif}\;x \leq -3.3 \cdot 10^{-159}:\\ \;\;\;\;\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(y + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\left(1 + y\right) \cdot y}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 80.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 1.35 \cdot 10^{-64}:\\ \;\;\;\;\frac{y}{\left(1 + x\right) \cdot x}\\ \mathbf{elif}\;y \leq 1.5 \cdot 10^{+15}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(y, 1, y \cdot y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + x}}{y + x} \cdot 1\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 1.35e-64)
   (/ y (* (+ 1.0 x) x))
   (if (<= y 1.5e+15)
     (/ x (fma y 1.0 (* y y)))
     (* (/ (/ x (+ y x)) (+ y x)) 1.0))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 1.35e-64) {
		tmp = y / ((1.0 + x) * x);
	} else if (y <= 1.5e+15) {
		tmp = x / fma(y, 1.0, (y * y));
	} else {
		tmp = ((x / (y + x)) / (y + x)) * 1.0;
	}
	return tmp;
}

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= 1.35e-64)
		tmp = Float64(y / Float64(Float64(1.0 + x) * x));
	elseif (y <= 1.5e+15)
		tmp = Float64(x / fma(y, 1.0, Float64(y * y)));
	else
		tmp = Float64(Float64(Float64(x / Float64(y + x)) / Float64(y + x)) * 1.0);
	end
	return tmp
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[y, 1.35e-64], N[(y / N[(N[(1.0 + x), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.5e+15], N[(x / N[(y * 1.0 + N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x / N[(y + x), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 1.34999999999999993e-64

   1. Initial program 69.9%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{y}{x \cdot \left(1 + x\right)}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{y}{\color{blue}{x \cdot \left(1 + x\right)}} \]

      2. *-commutative N/A

         \[\leadsto \frac{y}{\left(1 + x\right) \cdot \color{blue}{x}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{y}{\left(1 + x\right) \cdot \color{blue}{x}} \]

      4. lower-+.f64 58.4

         \[\leadsto \frac{y}{\left(1 + x\right) \cdot x} \]

   5. Applied rewrites 58.4%

      \[\leadsto \color{blue}{\frac{y}{\left(1 + x\right) \cdot x}} \]

   if 1.34999999999999993e-64 < y < 1.5e15

   1. Initial program 85.1%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot y}}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      2. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      4. lift-+.f64 N/A

         \[\leadsto \frac{x \cdot y}{\left(\color{blue}{\left(x + y\right)} \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      5. lift-+.f64 N/A

         \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \color{blue}{\left(x + y\right)}\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right)} \cdot \left(\left(x + y\right) + 1\right)} \]

      7. lift-+.f64 N/A

         \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\color{blue}{\left(x + y\right)} + 1\right)} \]

      8. lift-+.f64 N/A

         \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \color{blue}{\left(\left(x + y\right) + 1\right)}} \]

      9. times-frac N/A

         \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}} \]

      10. lower-*.f64 N/A

          \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}} \]

      11. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)}} \cdot \frac{y}{\left(x + y\right) + 1} \]

      12. pow2 N/A

          \[\leadsto \frac{x}{\color{blue}{{\left(x + y\right)}^{2}}} \cdot \frac{y}{\left(x + y\right) + 1} \]

      13. lower-pow.f64 N/A

          \[\leadsto \frac{x}{\color{blue}{{\left(x + y\right)}^{2}}} \cdot \frac{y}{\left(x + y\right) + 1} \]

      14. +-commutative N/A

          \[\leadsto \frac{x}{{\color{blue}{\left(y + x\right)}}^{2}} \cdot \frac{y}{\left(x + y\right) + 1} \]

      15. lower-+.f64 N/A

          \[\leadsto \frac{x}{{\color{blue}{\left(y + x\right)}}^{2}} \cdot \frac{y}{\left(x + y\right) + 1} \]

      16. lower-/.f64 N/A

          \[\leadsto \frac{x}{{\left(y + x\right)}^{2}} \cdot \color{blue}{\frac{y}{\left(x + y\right) + 1}} \]

      17. lift-+.f64 N/A

          \[\leadsto \frac{x}{{\left(y + x\right)}^{2}} \cdot \frac{y}{\color{blue}{\left(x + y\right) + 1}} \]

      18. +-commutative N/A

          \[\leadsto \frac{x}{{\left(y + x\right)}^{2}} \cdot \frac{y}{\color{blue}{\left(y + x\right)} + 1} \]

      19. lower-+.f64 99.5

          \[\leadsto \frac{x}{{\left(y + x\right)}^{2}} \cdot \frac{y}{\color{blue}{\left(y + x\right)} + 1} \]

   4. Applied rewrites 99.5%

      \[\leadsto \color{blue}{\frac{x}{{\left(y + x\right)}^{2}} \cdot \frac{y}{\left(y + x\right) + 1}} \]

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]
   6. Step-by-step derivation

      1. frac-times N/A

         \[\leadsto \frac{\color{blue}{x}}{y \cdot \left(1 + y\right)} \]

      2. unpow2 N/A

         \[\leadsto \frac{x}{y \cdot \left(1 + y\right)} \]

      3. +-commutative N/A

         \[\leadsto \frac{x}{y \cdot \left(1 + y\right)} \]

      4. distribute-lft-out N/A

         \[\leadsto \frac{x}{y \cdot \left(1 + y\right)} \]

      5. +-commutative N/A

         \[\leadsto \frac{x}{y \cdot \left(1 + y\right)} \]

      6. lower-/.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{y \cdot \left(1 + y\right)}} \]

      7. *-commutative N/A

         \[\leadsto \frac{x}{\left(1 + y\right) \cdot \color{blue}{y}} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{x}{\left(1 + y\right) \cdot \color{blue}{y}} \]

      9. lift-+.f64 32.7

         \[\leadsto \frac{x}{\left(1 + y\right) \cdot y} \]

   7. Applied rewrites 32.7%

      \[\leadsto \color{blue}{\frac{x}{\left(1 + y\right) \cdot y}} \]
   8. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{x}{\left(1 + y\right) \cdot y} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{x}{\left(1 + y\right) \cdot \color{blue}{y}} \]

      3. *-commutative N/A

         \[\leadsto \frac{x}{y \cdot \color{blue}{\left(1 + y\right)}} \]

      4. distribute-lft-in N/A

         \[\leadsto \frac{x}{y \cdot 1 + \color{blue}{y \cdot y}} \]

      5. pow2 N/A

         \[\leadsto \frac{x}{y \cdot 1 + {y}^{\color{blue}{2}}} \]

      6. lower-fma.f64 N/A

         \[\leadsto \frac{x}{\mathsf{fma}\left(y, \color{blue}{1}, {y}^{2}\right)} \]

      7. pow2 N/A

         \[\leadsto \frac{x}{\mathsf{fma}\left(y, 1, y \cdot y\right)} \]

      8. lift-*.f64 32.7

         \[\leadsto \frac{x}{\mathsf{fma}\left(y, 1, y \cdot y\right)} \]

   9. Applied rewrites 32.7%

      \[\leadsto \frac{x}{\mathsf{fma}\left(y, \color{blue}{1}, y \cdot y\right)} \]

   if 1.5e15 < y

   1. Initial program 64.2%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot y}}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      2. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      4. lift-+.f64 N/A

         \[\leadsto \frac{x \cdot y}{\left(\color{blue}{\left(x + y\right)} \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      5. lift-+.f64 N/A

         \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \color{blue}{\left(x + y\right)}\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right)} \cdot \left(\left(x + y\right) + 1\right)} \]

      7. lift-+.f64 N/A

         \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\color{blue}{\left(x + y\right)} + 1\right)} \]

      8. lift-+.f64 N/A

         \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \color{blue}{\left(\left(x + y\right) + 1\right)}} \]

      9. times-frac N/A

         \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}} \]

      10. lower-*.f64 N/A

          \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}} \]

      11. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)}} \cdot \frac{y}{\left(x + y\right) + 1} \]

      12. pow2 N/A

          \[\leadsto \frac{x}{\color{blue}{{\left(x + y\right)}^{2}}} \cdot \frac{y}{\left(x + y\right) + 1} \]

      13. lower-pow.f64 N/A

          \[\leadsto \frac{x}{\color{blue}{{\left(x + y\right)}^{2}}} \cdot \frac{y}{\left(x + y\right) + 1} \]

      14. +-commutative N/A

          \[\leadsto \frac{x}{{\color{blue}{\left(y + x\right)}}^{2}} \cdot \frac{y}{\left(x + y\right) + 1} \]

      15. lower-+.f64 N/A

          \[\leadsto \frac{x}{{\color{blue}{\left(y + x\right)}}^{2}} \cdot \frac{y}{\left(x + y\right) + 1} \]

      16. lower-/.f64 N/A

          \[\leadsto \frac{x}{{\left(y + x\right)}^{2}} \cdot \color{blue}{\frac{y}{\left(x + y\right) + 1}} \]

      17. lift-+.f64 N/A

          \[\leadsto \frac{x}{{\left(y + x\right)}^{2}} \cdot \frac{y}{\color{blue}{\left(x + y\right) + 1}} \]

      18. +-commutative N/A

          \[\leadsto \frac{x}{{\left(y + x\right)}^{2}} \cdot \frac{y}{\color{blue}{\left(y + x\right)} + 1} \]

      19. lower-+.f64 91.5

          \[\leadsto \frac{x}{{\left(y + x\right)}^{2}} \cdot \frac{y}{\color{blue}{\left(y + x\right)} + 1} \]

   4. Applied rewrites 91.5%

      \[\leadsto \color{blue}{\frac{x}{{\left(y + x\right)}^{2}} \cdot \frac{y}{\left(y + x\right) + 1}} \]
   5. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{x}{{\color{blue}{\left(y + x\right)}}^{2}} \cdot \frac{y}{\left(y + x\right) + 1} \]

      2. +-commutative N/A

         \[\leadsto \frac{x}{{\color{blue}{\left(x + y\right)}}^{2}} \cdot \frac{y}{\left(y + x\right) + 1} \]

      3. lower-pow.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{{\left(x + y\right)}^{2}}} \cdot \frac{y}{\left(y + x\right) + 1} \]

      4. pow2 N/A

         \[\leadsto \frac{x}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right)}} \cdot \frac{y}{\left(y + x\right) + 1} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right)}} \cdot \frac{y}{\left(y + x\right) + 1} \]

      6. +-commutative N/A

         \[\leadsto \frac{x}{\color{blue}{\left(y + x\right)} \cdot \left(x + y\right)} \cdot \frac{y}{\left(y + x\right) + 1} \]

      7. lift-+.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{\left(y + x\right)} \cdot \left(x + y\right)} \cdot \frac{y}{\left(y + x\right) + 1} \]

      8. +-commutative N/A

         \[\leadsto \frac{x}{\left(y + x\right) \cdot \color{blue}{\left(y + x\right)}} \cdot \frac{y}{\left(y + x\right) + 1} \]

      9. lift-+.f64 91.5

         \[\leadsto \frac{x}{\left(y + x\right) \cdot \color{blue}{\left(y + x\right)}} \cdot \frac{y}{\left(y + x\right) + 1} \]

   6. Applied rewrites 91.5%

      \[\leadsto \frac{x}{\color{blue}{\left(y + x\right) \cdot \left(y + x\right)}} \cdot \frac{y}{\left(y + x\right) + 1} \]
   7. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{\left(y + x\right) \cdot \left(y + x\right)}} \cdot \frac{y}{\left(y + x\right) + 1} \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{\left(y + x\right)} \cdot \left(y + x\right)} \cdot \frac{y}{\left(y + x\right) + 1} \]

      3. lift-+.f64 N/A

         \[\leadsto \frac{x}{\left(y + x\right) \cdot \color{blue}{\left(y + x\right)}} \cdot \frac{y}{\left(y + x\right) + 1} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{\left(y + x\right) \cdot \left(y + x\right)}} \cdot \frac{y}{\left(y + x\right) + 1} \]

      5. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{x}{y + x}}{y + x}} \cdot \frac{y}{\left(y + x\right) + 1} \]

      6. +-commutative N/A

         \[\leadsto \frac{\frac{x}{y + x}}{\color{blue}{x + y}} \cdot \frac{y}{\left(y + x\right) + 1} \]

      7. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{x}{y + x}}{x + y}} \cdot \frac{y}{\left(y + x\right) + 1} \]

      8. +-commutative N/A

         \[\leadsto \frac{\frac{x}{\color{blue}{x + y}}}{x + y} \cdot \frac{y}{\left(y + x\right) + 1} \]

      9. lower-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{x}{x + y}}}{x + y} \cdot \frac{y}{\left(y + x\right) + 1} \]

      10. +-commutative N/A

          \[\leadsto \frac{\frac{x}{\color{blue}{y + x}}}{x + y} \cdot \frac{y}{\left(y + x\right) + 1} \]

      11. lift-+.f64 N/A

          \[\leadsto \frac{\frac{x}{\color{blue}{y + x}}}{x + y} \cdot \frac{y}{\left(y + x\right) + 1} \]

      12. +-commutative N/A

          \[\leadsto \frac{\frac{x}{y + x}}{\color{blue}{y + x}} \cdot \frac{y}{\left(y + x\right) + 1} \]

      13. lift-+.f64 99.8

          \[\leadsto \frac{\frac{x}{y + x}}{\color{blue}{y + x}} \cdot \frac{y}{\left(y + x\right) + 1} \]

   8. Applied rewrites 99.8%

      \[\leadsto \color{blue}{\frac{\frac{x}{y + x}}{y + x}} \cdot \frac{y}{\left(y + x\right) + 1} \]

   9. Taylor expanded in y around inf

      \[\leadsto \frac{\frac{x}{y + x}}{y + x} \cdot \color{blue}{1} \]
   10. Step-by-step derivation

   11. Applied rewrites 79.3%

       \[\leadsto \frac{\frac{x}{y + x}}{y + x} \cdot \color{blue}{1} \]
3. Recombined 3 regimes into one program.

4. Final simplification 63.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.35 \cdot 10^{-64}:\\ \;\;\;\;\frac{y}{\left(1 + x\right) \cdot x}\\ \mathbf{elif}\;y \leq 1.5 \cdot 10^{+15}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(y, 1, y \cdot y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + x}}{y + x} \cdot 1\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 69.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -7.5 \cdot 10^{+24}:\\ \;\;\;\;\frac{y}{x \cdot x}\\ \mathbf{elif}\;x \leq -4.2 \cdot 10^{-204} \lor \neg \left(x \leq 5.6 \cdot 10^{-187}\right):\\ \;\;\;\;\frac{x}{y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -7.5e+24)
   (/ y (* x x))
   (if (or (<= x -4.2e-204) (not (<= x 5.6e-187))) (/ x (* y y)) (/ x y))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -7.5e+24) {
		tmp = y / (x * x);
	} else if ((x <= -4.2e-204) || !(x <= 5.6e-187)) {
		tmp = x / (y * y);
	} else {
		tmp = x / y;
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-7.5d+24)) then
        tmp = y / (x * x)
    else if ((x <= (-4.2d-204)) .or. (.not. (x <= 5.6d-187))) then
        tmp = x / (y * y)
    else
        tmp = x / y
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (x <= -7.5e+24) {
		tmp = y / (x * x);
	} else if ((x <= -4.2e-204) || !(x <= 5.6e-187)) {
		tmp = x / (y * y);
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -7.5e+24:
		tmp = y / (x * x)
	elif (x <= -4.2e-204) or not (x <= 5.6e-187):
		tmp = x / (y * y)
	else:
		tmp = x / y
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -7.5e+24)
		tmp = Float64(y / Float64(x * x));
	elseif ((x <= -4.2e-204) || !(x <= 5.6e-187))
		tmp = Float64(x / Float64(y * y));
	else
		tmp = Float64(x / y);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -7.5e+24)
		tmp = y / (x * x);
	elseif ((x <= -4.2e-204) || ~((x <= 5.6e-187)))
		tmp = x / (y * y);
	else
		tmp = x / y;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[x, -7.5e+24], N[(y / N[(x * x), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[x, -4.2e-204], N[Not[LessEqual[x, 5.6e-187]], $MachinePrecision]], N[(x / N[(y * y), $MachinePrecision]), $MachinePrecision], N[(x / y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -7.50000000000000014e24

   1. Initial program 55.6%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{y}{{x}^{2}}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{y}{\color{blue}{{x}^{2}}} \]

      2. unpow2 N/A

         \[\leadsto \frac{y}{x \cdot \color{blue}{x}} \]

      3. lower-*.f64 75.3

         \[\leadsto \frac{y}{x \cdot \color{blue}{x}} \]

   5. Applied rewrites 75.3%

      \[\leadsto \color{blue}{\frac{y}{x \cdot x}} \]

   if -7.50000000000000014e24 < x < -4.20000000000000018e-204 or 5.6e-187 < x

   1. Initial program 78.0%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\frac{x}{{y}^{2}}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{{y}^{2}}} \]

      2. unpow2 N/A

         \[\leadsto \frac{x}{y \cdot \color{blue}{y}} \]

      3. lower-*.f64 44.2

         \[\leadsto \frac{x}{y \cdot \color{blue}{y}} \]

   5. Applied rewrites 44.2%

      \[\leadsto \color{blue}{\frac{x}{y \cdot y}} \]

   if -4.20000000000000018e-204 < x < 5.6e-187

   1. Initial program 51.0%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot y}}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      2. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      4. lift-+.f64 N/A

         \[\leadsto \frac{x \cdot y}{\left(\color{blue}{\left(x + y\right)} \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      5. lift-+.f64 N/A

         \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \color{blue}{\left(x + y\right)}\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right)} \cdot \left(\left(x + y\right) + 1\right)} \]

      7. lift-+.f64 N/A

         \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\color{blue}{\left(x + y\right)} + 1\right)} \]

      8. lift-+.f64 N/A

         \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \color{blue}{\left(\left(x + y\right) + 1\right)}} \]

      9. times-frac N/A

         \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}} \]

      10. lower-*.f64 N/A

          \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}} \]

      11. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)}} \cdot \frac{y}{\left(x + y\right) + 1} \]

      12. pow2 N/A

          \[\leadsto \frac{x}{\color{blue}{{\left(x + y\right)}^{2}}} \cdot \frac{y}{\left(x + y\right) + 1} \]

      13. lower-pow.f64 N/A

          \[\leadsto \frac{x}{\color{blue}{{\left(x + y\right)}^{2}}} \cdot \frac{y}{\left(x + y\right) + 1} \]

      14. +-commutative N/A

          \[\leadsto \frac{x}{{\color{blue}{\left(y + x\right)}}^{2}} \cdot \frac{y}{\left(x + y\right) + 1} \]

      15. lower-+.f64 N/A

          \[\leadsto \frac{x}{{\color{blue}{\left(y + x\right)}}^{2}} \cdot \frac{y}{\left(x + y\right) + 1} \]

      16. lower-/.f64 N/A

          \[\leadsto \frac{x}{{\left(y + x\right)}^{2}} \cdot \color{blue}{\frac{y}{\left(x + y\right) + 1}} \]

      17. lift-+.f64 N/A

          \[\leadsto \frac{x}{{\left(y + x\right)}^{2}} \cdot \frac{y}{\color{blue}{\left(x + y\right) + 1}} \]

      18. +-commutative N/A

          \[\leadsto \frac{x}{{\left(y + x\right)}^{2}} \cdot \frac{y}{\color{blue}{\left(y + x\right)} + 1} \]

      19. lower-+.f64 65.9

          \[\leadsto \frac{x}{{\left(y + x\right)}^{2}} \cdot \frac{y}{\color{blue}{\left(y + x\right)} + 1} \]

   4. Applied rewrites 65.9%

      \[\leadsto \color{blue}{\frac{x}{{\left(y + x\right)}^{2}} \cdot \frac{y}{\left(y + x\right) + 1}} \]

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]
   6. Step-by-step derivation

      1. frac-times N/A

         \[\leadsto \frac{\color{blue}{x}}{y \cdot \left(1 + y\right)} \]

      2. unpow2 N/A

         \[\leadsto \frac{x}{y \cdot \left(1 + y\right)} \]

      3. +-commutative N/A

         \[\leadsto \frac{x}{y \cdot \left(1 + y\right)} \]

      4. distribute-lft-out N/A

         \[\leadsto \frac{x}{y \cdot \left(1 + y\right)} \]

      5. +-commutative N/A

         \[\leadsto \frac{x}{y \cdot \left(1 + y\right)} \]

      6. lower-/.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{y \cdot \left(1 + y\right)}} \]

      7. *-commutative N/A

         \[\leadsto \frac{x}{\left(1 + y\right) \cdot \color{blue}{y}} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{x}{\left(1 + y\right) \cdot \color{blue}{y}} \]

      9. lift-+.f64 83.6

         \[\leadsto \frac{x}{\left(1 + y\right) \cdot y} \]

   7. Applied rewrites 83.6%

      \[\leadsto \color{blue}{\frac{x}{\left(1 + y\right) \cdot y}} \]

   8. Taylor expanded in y around 0

      \[\leadsto \frac{x}{y} \]
   9. Step-by-step derivation

   10. Applied rewrites 76.6%

       \[\leadsto \frac{x}{y} \]
3. Recombined 3 regimes into one program.

4. Final simplification 55.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -7.5 \cdot 10^{+24}:\\ \;\;\;\;\frac{y}{x \cdot x}\\ \mathbf{elif}\;x \leq -4.2 \cdot 10^{-204} \lor \neg \left(x \leq 5.6 \cdot 10^{-187}\right):\\ \;\;\;\;\frac{x}{y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 80.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 1.35 \cdot 10^{-64}:\\ \;\;\;\;\frac{y}{\left(1 + x\right) \cdot x}\\ \mathbf{elif}\;y \leq 10^{+47}:\\ \;\;\;\;\frac{x}{\left(1 + y\right) \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 1.35e-64)
   (/ y (* (+ 1.0 x) x))
   (if (<= y 1e+47) (/ x (* (+ 1.0 y) y)) (/ (/ x y) y))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 1.35e-64) {
		tmp = y / ((1.0 + x) * x);
	} else if (y <= 1e+47) {
		tmp = x / ((1.0 + y) * y);
	} else {
		tmp = (x / y) / y;
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 1.35d-64) then
        tmp = y / ((1.0d0 + x) * x)
    else if (y <= 1d+47) then
        tmp = x / ((1.0d0 + y) * y)
    else
        tmp = (x / y) / y
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (y <= 1.35e-64) {
		tmp = y / ((1.0 + x) * x);
	} else if (y <= 1e+47) {
		tmp = x / ((1.0 + y) * y);
	} else {
		tmp = (x / y) / y;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if y <= 1.35e-64:
		tmp = y / ((1.0 + x) * x)
	elif y <= 1e+47:
		tmp = x / ((1.0 + y) * y)
	else:
		tmp = (x / y) / y
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= 1.35e-64)
		tmp = Float64(y / Float64(Float64(1.0 + x) * x));
	elseif (y <= 1e+47)
		tmp = Float64(x / Float64(Float64(1.0 + y) * y));
	else
		tmp = Float64(Float64(x / y) / y);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 1.35e-64)
		tmp = y / ((1.0 + x) * x);
	elseif (y <= 1e+47)
		tmp = x / ((1.0 + y) * y);
	else
		tmp = (x / y) / y;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[y, 1.35e-64], N[(y / N[(N[(1.0 + x), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1e+47], N[(x / N[(N[(1.0 + y), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 1.34999999999999993e-64

   1. Initial program 69.9%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{y}{x \cdot \left(1 + x\right)}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{y}{\color{blue}{x \cdot \left(1 + x\right)}} \]

      2. *-commutative N/A

         \[\leadsto \frac{y}{\left(1 + x\right) \cdot \color{blue}{x}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{y}{\left(1 + x\right) \cdot \color{blue}{x}} \]

      4. lower-+.f64 58.4

         \[\leadsto \frac{y}{\left(1 + x\right) \cdot x} \]

   5. Applied rewrites 58.4%

      \[\leadsto \color{blue}{\frac{y}{\left(1 + x\right) \cdot x}} \]

   if 1.34999999999999993e-64 < y < 1e47

   1. Initial program 86.2%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{y \cdot \left(1 + y\right)}} \]

      2. *-commutative N/A

         \[\leadsto \frac{x}{\left(1 + y\right) \cdot \color{blue}{y}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{x}{\left(1 + y\right) \cdot \color{blue}{y}} \]

      4. lower-+.f64 39.9

         \[\leadsto \frac{x}{\left(1 + y\right) \cdot y} \]

   5. Applied rewrites 39.9%

      \[\leadsto \color{blue}{\frac{x}{\left(1 + y\right) \cdot y}} \]

   if 1e47 < y

   1. Initial program 61.4%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\frac{x}{{y}^{2}}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{{y}^{2}}} \]

      2. unpow2 N/A

         \[\leadsto \frac{x}{y \cdot \color{blue}{y}} \]

      3. lower-*.f64 82.5

         \[\leadsto \frac{x}{y \cdot \color{blue}{y}} \]

   5. Applied rewrites 82.5%

      \[\leadsto \color{blue}{\frac{x}{y \cdot y}} \]
   6. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{y \cdot y}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{x}{y \cdot \color{blue}{y}} \]

      3. associate-/r* N/A

         \[\leadsto \frac{\frac{x}{y}}{\color{blue}{y}} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{\frac{x}{y}}{\color{blue}{y}} \]

      5. lower-/.f64 81.6

         \[\leadsto \frac{\frac{x}{y}}{y} \]

   7. Applied rewrites 81.6%

      \[\leadsto \frac{\frac{x}{y}}{\color{blue}{y}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 15: 78.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 1.35 \cdot 10^{-64}:\\ \;\;\;\;\frac{y}{\left(1 + x\right) \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\left(1 + y\right) \cdot y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 1.35e-64) (/ y (* (+ 1.0 x) x)) (/ x (* (+ 1.0 y) y))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 1.35e-64) {
		tmp = y / ((1.0 + x) * x);
	} else {
		tmp = x / ((1.0 + y) * y);
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 1.35d-64) then
        tmp = y / ((1.0d0 + x) * x)
    else
        tmp = x / ((1.0d0 + y) * y)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (y <= 1.35e-64) {
		tmp = y / ((1.0 + x) * x);
	} else {
		tmp = x / ((1.0 + y) * y);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if y <= 1.35e-64:
		tmp = y / ((1.0 + x) * x)
	else:
		tmp = x / ((1.0 + y) * y)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= 1.35e-64)
		tmp = Float64(y / Float64(Float64(1.0 + x) * x));
	else
		tmp = Float64(x / Float64(Float64(1.0 + y) * y));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 1.35e-64)
		tmp = y / ((1.0 + x) * x);
	else
		tmp = x / ((1.0 + y) * y);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[y, 1.35e-64], N[(y / N[(N[(1.0 + x), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision], N[(x / N[(N[(1.0 + y), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.34999999999999993e-64

   1. Initial program 69.9%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{y}{x \cdot \left(1 + x\right)}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{y}{\color{blue}{x \cdot \left(1 + x\right)}} \]

      2. *-commutative N/A

         \[\leadsto \frac{y}{\left(1 + x\right) \cdot \color{blue}{x}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{y}{\left(1 + x\right) \cdot \color{blue}{x}} \]

      4. lower-+.f64 58.4

         \[\leadsto \frac{y}{\left(1 + x\right) \cdot x} \]

   5. Applied rewrites 58.4%

      \[\leadsto \color{blue}{\frac{y}{\left(1 + x\right) \cdot x}} \]

   if 1.34999999999999993e-64 < y

   1. Initial program 67.3%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{y \cdot \left(1 + y\right)}} \]

      2. *-commutative N/A

         \[\leadsto \frac{x}{\left(1 + y\right) \cdot \color{blue}{y}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{x}{\left(1 + y\right) \cdot \color{blue}{y}} \]

      4. lower-+.f64 72.4

         \[\leadsto \frac{x}{\left(1 + y\right) \cdot y} \]

   5. Applied rewrites 72.4%

      \[\leadsto \color{blue}{\frac{x}{\left(1 + y\right) \cdot y}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 16: 76.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -7.5 \cdot 10^{+24}:\\ \;\;\;\;\frac{y}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\left(1 + y\right) \cdot y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -7.5e+24) (/ y (* x x)) (/ x (* (+ 1.0 y) y))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -7.5e+24) {
		tmp = y / (x * x);
	} else {
		tmp = x / ((1.0 + y) * y);
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-7.5d+24)) then
        tmp = y / (x * x)
    else
        tmp = x / ((1.0d0 + y) * y)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (x <= -7.5e+24) {
		tmp = y / (x * x);
	} else {
		tmp = x / ((1.0 + y) * y);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -7.5e+24:
		tmp = y / (x * x)
	else:
		tmp = x / ((1.0 + y) * y)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -7.5e+24)
		tmp = Float64(y / Float64(x * x));
	else
		tmp = Float64(x / Float64(Float64(1.0 + y) * y));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -7.5e+24)
		tmp = y / (x * x);
	else
		tmp = x / ((1.0 + y) * y);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[x, -7.5e+24], N[(y / N[(x * x), $MachinePrecision]), $MachinePrecision], N[(x / N[(N[(1.0 + y), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -7.50000000000000014e24

   1. Initial program 55.6%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{y}{{x}^{2}}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{y}{\color{blue}{{x}^{2}}} \]

      2. unpow2 N/A

         \[\leadsto \frac{y}{x \cdot \color{blue}{x}} \]

      3. lower-*.f64 75.3

         \[\leadsto \frac{y}{x \cdot \color{blue}{x}} \]

   5. Applied rewrites 75.3%

      \[\leadsto \color{blue}{\frac{y}{x \cdot x}} \]

   if -7.50000000000000014e24 < x

   1. Initial program 72.7%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{y \cdot \left(1 + y\right)}} \]

      2. *-commutative N/A

         \[\leadsto \frac{x}{\left(1 + y\right) \cdot \color{blue}{y}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{x}{\left(1 + y\right) \cdot \color{blue}{y}} \]

      4. lower-+.f64 59.3

         \[\leadsto \frac{x}{\left(1 + y\right) \cdot y} \]

   5. Applied rewrites 59.3%

      \[\leadsto \color{blue}{\frac{x}{\left(1 + y\right) \cdot y}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 17: 47.7% accurate, 1.7× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 1:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y) :precision binary64 (if (<= y 1.0) (/ x y) (/ x (* y y))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 1.0) {
		tmp = x / y;
	} else {
		tmp = x / (y * y);
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 1.0d0) then
        tmp = x / y
    else
        tmp = x / (y * y)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (y <= 1.0) {
		tmp = x / y;
	} else {
		tmp = x / (y * y);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if y <= 1.0:
		tmp = x / y
	else:
		tmp = x / (y * y)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= 1.0)
		tmp = Float64(x / y);
	else
		tmp = Float64(x / Float64(y * y));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 1.0)
		tmp = x / y;
	else
		tmp = x / (y * y);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[y, 1.0], N[(x / y), $MachinePrecision], N[(x / N[(y * y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1

   1. Initial program 70.8%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot y}}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      2. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      4. lift-+.f64 N/A

         \[\leadsto \frac{x \cdot y}{\left(\color{blue}{\left(x + y\right)} \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      5. lift-+.f64 N/A

         \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \color{blue}{\left(x + y\right)}\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right)} \cdot \left(\left(x + y\right) + 1\right)} \]

      7. lift-+.f64 N/A

         \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\color{blue}{\left(x + y\right)} + 1\right)} \]

      8. lift-+.f64 N/A

         \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \color{blue}{\left(\left(x + y\right) + 1\right)}} \]

      9. times-frac N/A

         \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}} \]

      10. lower-*.f64 N/A

          \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}} \]

      11. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)}} \cdot \frac{y}{\left(x + y\right) + 1} \]

      12. pow2 N/A

          \[\leadsto \frac{x}{\color{blue}{{\left(x + y\right)}^{2}}} \cdot \frac{y}{\left(x + y\right) + 1} \]

      13. lower-pow.f64 N/A

          \[\leadsto \frac{x}{\color{blue}{{\left(x + y\right)}^{2}}} \cdot \frac{y}{\left(x + y\right) + 1} \]

      14. +-commutative N/A

          \[\leadsto \frac{x}{{\color{blue}{\left(y + x\right)}}^{2}} \cdot \frac{y}{\left(x + y\right) + 1} \]

      15. lower-+.f64 N/A

          \[\leadsto \frac{x}{{\color{blue}{\left(y + x\right)}}^{2}} \cdot \frac{y}{\left(x + y\right) + 1} \]

      16. lower-/.f64 N/A

          \[\leadsto \frac{x}{{\left(y + x\right)}^{2}} \cdot \color{blue}{\frac{y}{\left(x + y\right) + 1}} \]

      17. lift-+.f64 N/A

          \[\leadsto \frac{x}{{\left(y + x\right)}^{2}} \cdot \frac{y}{\color{blue}{\left(x + y\right) + 1}} \]

      18. +-commutative N/A

          \[\leadsto \frac{x}{{\left(y + x\right)}^{2}} \cdot \frac{y}{\color{blue}{\left(y + x\right)} + 1} \]

      19. lower-+.f64 86.8

          \[\leadsto \frac{x}{{\left(y + x\right)}^{2}} \cdot \frac{y}{\color{blue}{\left(y + x\right)} + 1} \]

   4. Applied rewrites 86.8%

      \[\leadsto \color{blue}{\frac{x}{{\left(y + x\right)}^{2}} \cdot \frac{y}{\left(y + x\right) + 1}} \]

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]
   6. Step-by-step derivation

      1. frac-times N/A

         \[\leadsto \frac{\color{blue}{x}}{y \cdot \left(1 + y\right)} \]

      2. unpow2 N/A

         \[\leadsto \frac{x}{y \cdot \left(1 + y\right)} \]

      3. +-commutative N/A

         \[\leadsto \frac{x}{y \cdot \left(1 + y\right)} \]

      4. distribute-lft-out N/A

         \[\leadsto \frac{x}{y \cdot \left(1 + y\right)} \]

      5. +-commutative N/A

         \[\leadsto \frac{x}{y \cdot \left(1 + y\right)} \]

      6. lower-/.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{y \cdot \left(1 + y\right)}} \]

      7. *-commutative N/A

         \[\leadsto \frac{x}{\left(1 + y\right) \cdot \color{blue}{y}} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{x}{\left(1 + y\right) \cdot \color{blue}{y}} \]

      9. lift-+.f64 39.6

         \[\leadsto \frac{x}{\left(1 + y\right) \cdot y} \]

   7. Applied rewrites 39.6%

      \[\leadsto \color{blue}{\frac{x}{\left(1 + y\right) \cdot y}} \]

   8. Taylor expanded in y around 0

      \[\leadsto \frac{x}{y} \]
   9. Step-by-step derivation

   10. Applied rewrites 25.4%

       \[\leadsto \frac{x}{y} \]

   if 1 < y

   1. Initial program 64.7%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\frac{x}{{y}^{2}}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{{y}^{2}}} \]

      2. unpow2 N/A

         \[\leadsto \frac{x}{y \cdot \color{blue}{y}} \]

      3. lower-*.f64 78.2

         \[\leadsto \frac{x}{y \cdot \color{blue}{y}} \]

   5. Applied rewrites 78.2%

      \[\leadsto \color{blue}{\frac{x}{y \cdot y}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 41.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot y}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 26.2% accurate, 3.3× 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 69.0%

   \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \frac{\color{blue}{x \cdot y}}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

   2. lift-/.f64 N/A

      \[\leadsto \color{blue}{\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

   3. lift-*.f64 N/A

      \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

   4. lift-+.f64 N/A

      \[\leadsto \frac{x \cdot y}{\left(\color{blue}{\left(x + y\right)} \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

   5. lift-+.f64 N/A

      \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \color{blue}{\left(x + y\right)}\right) \cdot \left(\left(x + y\right) + 1\right)} \]

   6. lift-*.f64 N/A

      \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right)} \cdot \left(\left(x + y\right) + 1\right)} \]

   7. lift-+.f64 N/A

      \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\color{blue}{\left(x + y\right)} + 1\right)} \]

   8. lift-+.f64 N/A

      \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \color{blue}{\left(\left(x + y\right) + 1\right)}} \]

   9. times-frac N/A

      \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}} \]

   10. lower-*.f64 N/A

       \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}} \]

   11. lower-/.f64 N/A

       \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)}} \cdot \frac{y}{\left(x + y\right) + 1} \]

   12. pow2 N/A

       \[\leadsto \frac{x}{\color{blue}{{\left(x + y\right)}^{2}}} \cdot \frac{y}{\left(x + y\right) + 1} \]

   13. lower-pow.f64 N/A

       \[\leadsto \frac{x}{\color{blue}{{\left(x + y\right)}^{2}}} \cdot \frac{y}{\left(x + y\right) + 1} \]

   14. +-commutative N/A

       \[\leadsto \frac{x}{{\color{blue}{\left(y + x\right)}}^{2}} \cdot \frac{y}{\left(x + y\right) + 1} \]

   15. lower-+.f64 N/A

       \[\leadsto \frac{x}{{\color{blue}{\left(y + x\right)}}^{2}} \cdot \frac{y}{\left(x + y\right) + 1} \]

   16. lower-/.f64 N/A

       \[\leadsto \frac{x}{{\left(y + x\right)}^{2}} \cdot \color{blue}{\frac{y}{\left(x + y\right) + 1}} \]

   17. lift-+.f64 N/A

       \[\leadsto \frac{x}{{\left(y + x\right)}^{2}} \cdot \frac{y}{\color{blue}{\left(x + y\right) + 1}} \]

   18. +-commutative N/A

       \[\leadsto \frac{x}{{\left(y + x\right)}^{2}} \cdot \frac{y}{\color{blue}{\left(y + x\right)} + 1} \]

   19. lower-+.f64 88.3

       \[\leadsto \frac{x}{{\left(y + x\right)}^{2}} \cdot \frac{y}{\color{blue}{\left(y + x\right)} + 1} \]

4. Applied rewrites 88.3%

   \[\leadsto \color{blue}{\frac{x}{{\left(y + x\right)}^{2}} \cdot \frac{y}{\left(y + x\right) + 1}} \]

5. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]
6. Step-by-step derivation

   1. frac-times N/A

      \[\leadsto \frac{\color{blue}{x}}{y \cdot \left(1 + y\right)} \]

   2. unpow2 N/A

      \[\leadsto \frac{x}{y \cdot \left(1 + y\right)} \]

   3. +-commutative N/A

      \[\leadsto \frac{x}{y \cdot \left(1 + y\right)} \]

   4. distribute-lft-out N/A

      \[\leadsto \frac{x}{y \cdot \left(1 + y\right)} \]

   5. +-commutative N/A

      \[\leadsto \frac{x}{y \cdot \left(1 + y\right)} \]

   6. lower-/.f64 N/A

      \[\leadsto \frac{x}{\color{blue}{y \cdot \left(1 + y\right)}} \]

   7. *-commutative N/A

      \[\leadsto \frac{x}{\left(1 + y\right) \cdot \color{blue}{y}} \]

   8. lower-*.f64 N/A

      \[\leadsto \frac{x}{\left(1 + y\right) \cdot \color{blue}{y}} \]

   9. lift-+.f64 51.2

      \[\leadsto \frac{x}{\left(1 + y\right) \cdot y} \]

7. Applied rewrites 51.2%

   \[\leadsto \color{blue}{\frac{x}{\left(1 + y\right) \cdot y}} \]

8. Taylor expanded in y around 0

   \[\leadsto \frac{x}{y} \]
9. Step-by-step derivation

10. Applied rewrites 26.4%

    \[\leadsto \frac{x}{y} \]

11. Final simplification 26.4%

    \[\leadsto \frac{x}{y} \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2025085 
(FPCore (x y)
  :name "Numeric.SpecFunctions:incompleteBetaApprox from math-functions-0.1.5.2, A"
  :precision binary64

  :alt
  (! :herbie-platform default (/ (/ (/ x (+ (+ y 1) x)) (+ y x)) (/ 1 (/ y (+ y x)))))

  (/ (* x y) (* (* (+ x y) (+ x y)) (+ (+ x y) 1.0))))