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

Percentage Accurate: 69.0% → 99.8%

Time: 3.6s

Alternatives: 15

Speedup: 0.8×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 69.0% 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

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_] := N[(N[(N[(y / N[(N[(y + x), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision] * N[(x / N[(y + x), $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 88.0

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

4. Applied rewrites 88.0%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-+.f64 N/A

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

   4. lift-pow.f64 N/A

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

   5. lift-/.f64 N/A

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

   6. lift-+.f64 N/A

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

   7. lift-+.f64 N/A

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

   8. frac-times N/A

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

   9. unpow2 N/A

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

   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. lower-/.f64 N/A

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

   14. lift-+.f64 N/A

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

   15. lower-/.f64 N/A

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

   16. *-commutative N/A

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

   17. lower-*.f64 N/A

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

   18. lift-+.f64 N/A

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

   19. lift-+.f64 N/A

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

   20. lift-+.f64 94.0

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

6. Applied rewrites 94.0%

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

   1. lift-*.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. lift-/.f64 N/A

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

   5. lift-+.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. lift-+.f64 N/A

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

   8. lift-+.f64 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 N/A

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

   11. associate-/r* N/A

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

   12. lower-/.f64 N/A

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

   13. lower-/.f64 N/A

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

   14. lift-+.f64 N/A

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

   15. lift-+.f64 N/A

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

   16. lift-+.f64 N/A

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

   17. lift-/.f64 N/A

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

   18. lift-+.f64 99.8

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

8. Applied rewrites 99.8%

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

Alternative 2: 68.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(y + x\right) + 1\\ t_1 := t\_0 \cdot \left(y + x\right)\\ \mathbf{if}\;x \leq -1.2 \cdot 10^{+163}:\\ \;\;\;\;\frac{\frac{y}{t\_0}}{y + x} \cdot 1\\ \mathbf{elif}\;x \leq -1.2 \cdot 10^{-5}:\\ \;\;\;\;1 \cdot \frac{y}{t\_1}\\ \mathbf{elif}\;x \leq -8.8 \cdot 10^{-163}:\\ \;\;\;\;x \cdot \frac{y}{\left(y + x\right) \cdot t\_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\left(1 + y\right) \cdot y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ (+ y x) 1.0)) (t_1 (* t_0 (+ y x))))
   (if (<= x -1.2e+163)
     (* (/ (/ y t_0) (+ y x)) 1.0)
     (if (<= x -1.2e-5)
       (* 1.0 (/ y t_1))
       (if (<= x -8.8e-163)
         (* x (/ y (* (+ y x) t_1)))
         (/ x (* (+ 1.0 y) y)))))))

C

double code(double x, double y) {
	double t_0 = (y + x) + 1.0;
	double t_1 = t_0 * (y + x);
	double tmp;
	if (x <= -1.2e+163) {
		tmp = ((y / t_0) / (y + x)) * 1.0;
	} else if (x <= -1.2e-5) {
		tmp = 1.0 * (y / t_1);
	} else if (x <= -8.8e-163) {
		tmp = x * (y / ((y + x) * t_1));
	} else {
		tmp = x / ((1.0 + y) * y);
	}
	return tmp;
}

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
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (y + x) + 1.0d0
    t_1 = t_0 * (y + x)
    if (x <= (-1.2d+163)) then
        tmp = ((y / t_0) / (y + x)) * 1.0d0
    else if (x <= (-1.2d-5)) then
        tmp = 1.0d0 * (y / t_1)
    else if (x <= (-8.8d-163)) then
        tmp = x * (y / ((y + x) * t_1))
    else
        tmp = x / ((1.0d0 + y) * y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = (y + x) + 1.0;
	double t_1 = t_0 * (y + x);
	double tmp;
	if (x <= -1.2e+163) {
		tmp = ((y / t_0) / (y + x)) * 1.0;
	} else if (x <= -1.2e-5) {
		tmp = 1.0 * (y / t_1);
	} else if (x <= -8.8e-163) {
		tmp = x * (y / ((y + x) * t_1));
	} else {
		tmp = x / ((1.0 + y) * y);
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (y + x) + 1.0
	t_1 = t_0 * (y + x)
	tmp = 0
	if x <= -1.2e+163:
		tmp = ((y / t_0) / (y + x)) * 1.0
	elif x <= -1.2e-5:
		tmp = 1.0 * (y / t_1)
	elif x <= -8.8e-163:
		tmp = x * (y / ((y + x) * t_1))
	else:
		tmp = x / ((1.0 + y) * y)
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(y + x) + 1.0)
	t_1 = Float64(t_0 * Float64(y + x))
	tmp = 0.0
	if (x <= -1.2e+163)
		tmp = Float64(Float64(Float64(y / t_0) / Float64(y + x)) * 1.0);
	elseif (x <= -1.2e-5)
		tmp = Float64(1.0 * Float64(y / t_1));
	elseif (x <= -8.8e-163)
		tmp = Float64(x * Float64(y / Float64(Float64(y + x) * t_1)));
	else
		tmp = Float64(x / Float64(Float64(1.0 + y) * y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (y + x) + 1.0;
	t_1 = t_0 * (y + x);
	tmp = 0.0;
	if (x <= -1.2e+163)
		tmp = ((y / t_0) / (y + x)) * 1.0;
	elseif (x <= -1.2e-5)
		tmp = 1.0 * (y / t_1);
	elseif (x <= -8.8e-163)
		tmp = x * (y / ((y + x) * t_1));
	else
		tmp = x / ((1.0 + y) * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(y + x), $MachinePrecision] + 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * N[(y + x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.2e+163], N[(N[(N[(y / t$95$0), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision], If[LessEqual[x, -1.2e-5], N[(1.0 * N[(y / t$95$1), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -8.8e-163], N[(x * N[(y / N[(N[(y + x), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x / N[(N[(1.0 + y), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -1.1999999999999999e163

   1. Initial program 59.7%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   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 83.8

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

   4. Applied rewrites 83.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. frac-times N/A

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

      9. unpow2 N/A

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

      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. lower-/.f64 N/A

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

      14. lift-+.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 N/A

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

      18. lift-+.f64 N/A

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

      19. lift-+.f64 N/A

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

      20. lift-+.f64 83.8

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

   6. Applied rewrites 83.8%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. associate-/r* N/A

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

      12. lower-/.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lift-+.f64 N/A

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

      15. lift-+.f64 N/A

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

      16. lift-+.f64 N/A

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

      17. lift-/.f64 N/A

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

      18. lift-+.f64 99.9

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

   8. Applied rewrites 99.9%

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

   9. Taylor expanded in x around inf

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

   11. Applied rewrites 88.8%

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

   if -1.1999999999999999e163 < x < -1.2e-5

   1. Initial program 67.7%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   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 92.0

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

   4. Applied rewrites 92.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. frac-times N/A

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

      9. unpow2 N/A

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

      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. lower-/.f64 N/A

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

      14. lift-+.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 N/A

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

      18. lift-+.f64 N/A

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

      19. lift-+.f64 N/A

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

      20. lift-+.f64 92.1

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

   6. Applied rewrites 92.1%

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

   7. Taylor expanded in x around inf

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

   9. Applied rewrites 78.8%

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

   if -1.2e-5 < x < -8.80000000000000044e-163

   1. Initial program 86.8%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   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. associate-/l* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lift-+.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 N/A

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

      17. pow2 N/A

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

      18. lower-pow.f64 N/A

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

      19. +-commutative N/A

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

      20. lower-+.f64 97.4

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

   4. Applied rewrites 97.4%

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

      1. lift-+.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. +-commutative N/A

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

      4. pow2 N/A

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

      5. distribute-lft-in N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lift-+.f64 97.4

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

   6. Applied rewrites 97.4%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. +-commutative N/A

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

      5. lift-+.f64 N/A

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

      6. lift-fma.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. distribute-lft-out N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. associate-*l* N/A

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

      13. +-commutative N/A

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

      14. +-commutative N/A

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

      15. +-commutative N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 N/A

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

      18. lift-+.f64 N/A

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

      19. lift-+.f64 N/A

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

      20. lift-+.f64 N/A

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

      21. lift-*.f64 N/A

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

      22. lift-+.f64 97.4

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

   8. Applied rewrites 97.4%

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

   if -8.80000000000000044e-163 < x

   1. Initial program 67.5%

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

   3. 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 56.1

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

   5. Applied rewrites 56.1%

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

Alternative 3: 94.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(y + x\right) + 1\\ \mathbf{if}\;x \leq -1.2 \cdot 10^{+163}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{y}{x}, -2, 1\right)}{x} \cdot \frac{y}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + x} \cdot y}{t\_0 \cdot \left(y + x\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ (+ y x) 1.0)))
   (if (<= x -1.2e+163)
     (* (/ (fma (/ y x) -2.0 1.0) x) (/ y t_0))
     (/ (* (/ x (+ y x)) y) (* t_0 (+ y x))))))

C

double code(double x, double y) {
	double t_0 = (y + x) + 1.0;
	double tmp;
	if (x <= -1.2e+163) {
		tmp = (fma((y / x), -2.0, 1.0) / x) * (y / t_0);
	} else {
		tmp = ((x / (y + x)) * y) / (t_0 * (y + x));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(y + x) + 1.0)
	tmp = 0.0
	if (x <= -1.2e+163)
		tmp = Float64(Float64(fma(Float64(y / x), -2.0, 1.0) / x) * Float64(y / t_0));
	else
		tmp = Float64(Float64(Float64(x / Float64(y + x)) * y) / Float64(t_0 * Float64(y + x)));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(y + x), $MachinePrecision] + 1.0), $MachinePrecision]}, If[LessEqual[x, -1.2e+163], N[(N[(N[(N[(y / x), $MachinePrecision] * -2.0 + 1.0), $MachinePrecision] / x), $MachinePrecision] * N[(y / t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x / N[(y + x), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision] / N[(t$95$0 * N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.1999999999999999e163

   1. Initial program 59.7%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   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 83.8

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

   4. Applied rewrites 83.8%

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

   5. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 89.0

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

   7. Applied rewrites 89.0%

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

   if -1.1999999999999999e163 < x

   1. Initial program 70.2%

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

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

      2. lift-/.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. frac-times N/A

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

      9. unpow2 N/A

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

      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. lower-/.f64 N/A

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

      14. lift-+.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 N/A

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

      18. lift-+.f64 N/A

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

      19. lift-+.f64 N/A

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

      20. lift-+.f64 95.3

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

   6. Applied rewrites 95.3%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. associate-*r/ N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lift-/.f64 N/A

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

      13. lift-+.f64 N/A

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

      14. lift-+.f64 N/A

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

      15. lift-+.f64 N/A

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

      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 95.3

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

   8. Applied rewrites 95.3%

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

Alternative 4: 83.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(y + x\right) + 1\\ \mathbf{if}\;x \leq -1.2 \cdot 10^{+163}:\\ \;\;\;\;\frac{\frac{y}{t\_0}}{y + x} \cdot 1\\ \mathbf{elif}\;x \leq -2.8 \cdot 10^{-7}:\\ \;\;\;\;1 \cdot \frac{y}{t\_0 \cdot \left(y + x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y + x} \cdot \frac{y}{\left(y + 1\right) \cdot \left(y + x\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ (+ y x) 1.0)))
   (if (<= x -1.2e+163)
     (* (/ (/ y t_0) (+ y x)) 1.0)
     (if (<= x -2.8e-7)
       (* 1.0 (/ y (* t_0 (+ y x))))
       (* (/ x (+ y x)) (/ y (* (+ y 1.0) (+ y x))))))))

C

double code(double x, double y) {
	double t_0 = (y + x) + 1.0;
	double tmp;
	if (x <= -1.2e+163) {
		tmp = ((y / t_0) / (y + x)) * 1.0;
	} else if (x <= -2.8e-7) {
		tmp = 1.0 * (y / (t_0 * (y + x)));
	} else {
		tmp = (x / (y + x)) * (y / ((y + 1.0) * (y + x)));
	}
	return tmp;
}

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
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (y + x) + 1.0d0
    if (x <= (-1.2d+163)) then
        tmp = ((y / t_0) / (y + x)) * 1.0d0
    else if (x <= (-2.8d-7)) then
        tmp = 1.0d0 * (y / (t_0 * (y + x)))
    else
        tmp = (x / (y + x)) * (y / ((y + 1.0d0) * (y + x)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = (y + x) + 1.0;
	double tmp;
	if (x <= -1.2e+163) {
		tmp = ((y / t_0) / (y + x)) * 1.0;
	} else if (x <= -2.8e-7) {
		tmp = 1.0 * (y / (t_0 * (y + x)));
	} else {
		tmp = (x / (y + x)) * (y / ((y + 1.0) * (y + x)));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (y + x) + 1.0
	tmp = 0
	if x <= -1.2e+163:
		tmp = ((y / t_0) / (y + x)) * 1.0
	elif x <= -2.8e-7:
		tmp = 1.0 * (y / (t_0 * (y + x)))
	else:
		tmp = (x / (y + x)) * (y / ((y + 1.0) * (y + x)))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(y + x) + 1.0)
	tmp = 0.0
	if (x <= -1.2e+163)
		tmp = Float64(Float64(Float64(y / t_0) / Float64(y + x)) * 1.0);
	elseif (x <= -2.8e-7)
		tmp = Float64(1.0 * Float64(y / Float64(t_0 * Float64(y + x))));
	else
		tmp = Float64(Float64(x / Float64(y + x)) * Float64(y / Float64(Float64(y + 1.0) * Float64(y + x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (y + x) + 1.0;
	tmp = 0.0;
	if (x <= -1.2e+163)
		tmp = ((y / t_0) / (y + x)) * 1.0;
	elseif (x <= -2.8e-7)
		tmp = 1.0 * (y / (t_0 * (y + x)));
	else
		tmp = (x / (y + x)) * (y / ((y + 1.0) * (y + x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(y + x), $MachinePrecision] + 1.0), $MachinePrecision]}, If[LessEqual[x, -1.2e+163], N[(N[(N[(y / t$95$0), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision], If[LessEqual[x, -2.8e-7], N[(1.0 * N[(y / N[(t$95$0 * N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(y + x), $MachinePrecision]), $MachinePrecision] * N[(y / N[(N[(y + 1.0), $MachinePrecision] * N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.1999999999999999e163

   1. Initial program 59.7%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   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 83.8

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

   4. Applied rewrites 83.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. frac-times N/A

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

      9. unpow2 N/A

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

      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. lower-/.f64 N/A

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

      14. lift-+.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 N/A

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

      18. lift-+.f64 N/A

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

      19. lift-+.f64 N/A

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

      20. lift-+.f64 83.8

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

   6. Applied rewrites 83.8%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. associate-/r* N/A

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

      12. lower-/.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lift-+.f64 N/A

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

      15. lift-+.f64 N/A

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

      16. lift-+.f64 N/A

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

      17. lift-/.f64 N/A

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

      18. lift-+.f64 99.9

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

   8. Applied rewrites 99.9%

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

   9. Taylor expanded in x around inf

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

   11. Applied rewrites 88.8%

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

   if -1.1999999999999999e163 < x < -2.80000000000000019e-7

   1. Initial program 67.8%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   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 92.1

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

   4. Applied rewrites 92.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. frac-times N/A

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

      9. unpow2 N/A

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

      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. lower-/.f64 N/A

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

      14. lift-+.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 N/A

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

      18. lift-+.f64 N/A

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

      19. lift-+.f64 N/A

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

      20. lift-+.f64 92.2

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

   6. Applied rewrites 92.2%

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

   7. Taylor expanded in x around inf

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

   9. Applied rewrites 78.7%

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

   if -2.80000000000000019e-7 < x

   1. Initial program 70.7%

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

      1. lift-*.f64 N/A

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

      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 87.9

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

   4. Applied rewrites 87.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. frac-times N/A

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

      9. unpow2 N/A

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

      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. lower-/.f64 N/A

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

      14. lift-+.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 N/A

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

      18. lift-+.f64 N/A

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

      19. lift-+.f64 N/A

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

      20. lift-+.f64 95.9

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

   6. Applied rewrites 95.9%

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

   7. Taylor expanded in x around 0

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

   9. Applied rewrites 83.9%

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

Alternative 5: 94.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{y + x}\\ \mathbf{if}\;x \leq -1.2 \cdot 10^{+163}:\\ \;\;\;\;\frac{\frac{y}{x}}{y + x} \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0 \cdot y}{\left(\left(y + x\right) + 1\right) \cdot \left(y + x\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ x (+ y x))))
   (if (<= x -1.2e+163)
     (* (/ (/ y x) (+ y x)) t_0)
     (/ (* t_0 y) (* (+ (+ y x) 1.0) (+ y x))))))

C

double code(double x, double y) {
	double t_0 = x / (y + x);
	double tmp;
	if (x <= -1.2e+163) {
		tmp = ((y / x) / (y + x)) * t_0;
	} else {
		tmp = (t_0 * y) / (((y + x) + 1.0) * (y + x));
	}
	return tmp;
}

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
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x / (y + x)
    if (x <= (-1.2d+163)) then
        tmp = ((y / x) / (y + x)) * t_0
    else
        tmp = (t_0 * y) / (((y + x) + 1.0d0) * (y + x))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = x / (y + x);
	double tmp;
	if (x <= -1.2e+163) {
		tmp = ((y / x) / (y + x)) * t_0;
	} else {
		tmp = (t_0 * y) / (((y + x) + 1.0) * (y + x));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = x / (y + x)
	tmp = 0
	if x <= -1.2e+163:
		tmp = ((y / x) / (y + x)) * t_0
	else:
		tmp = (t_0 * y) / (((y + x) + 1.0) * (y + x))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(x / Float64(y + x))
	tmp = 0.0
	if (x <= -1.2e+163)
		tmp = Float64(Float64(Float64(y / x) / Float64(y + x)) * t_0);
	else
		tmp = Float64(Float64(t_0 * y) / Float64(Float64(Float64(y + x) + 1.0) * Float64(y + x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = x / (y + x);
	tmp = 0.0;
	if (x <= -1.2e+163)
		tmp = ((y / x) / (y + x)) * t_0;
	else
		tmp = (t_0 * y) / (((y + x) + 1.0) * (y + x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(x / N[(y + x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.2e+163], N[(N[(N[(y / x), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision] * t$95$0), $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 < -1.1999999999999999e163

   1. Initial program 59.7%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   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 83.8

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

   4. Applied rewrites 83.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. frac-times N/A

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

      9. unpow2 N/A

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

      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. lower-/.f64 N/A

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

      14. lift-+.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 N/A

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

      18. lift-+.f64 N/A

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

      19. lift-+.f64 N/A

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

      20. lift-+.f64 83.8

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

   6. Applied rewrites 83.8%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. associate-/r* N/A

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

      12. lower-/.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lift-+.f64 N/A

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

      15. lift-+.f64 N/A

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

      16. lift-+.f64 N/A

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

      17. lift-/.f64 N/A

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

      18. lift-+.f64 99.9

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

   8. Applied rewrites 99.9%

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

   9. Taylor expanded in x around inf

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

   11. Applied rewrites 88.8%

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

   if -1.1999999999999999e163 < x

   1. Initial program 70.2%

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

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

      2. lift-/.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. frac-times N/A

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

      9. unpow2 N/A

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

      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. lower-/.f64 N/A

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

      14. lift-+.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 N/A

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

      18. lift-+.f64 N/A

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

      19. lift-+.f64 N/A

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

      20. lift-+.f64 95.3

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

   6. Applied rewrites 95.3%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. associate-*r/ N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lift-/.f64 N/A

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

      13. lift-+.f64 N/A

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

      14. lift-+.f64 N/A

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

      15. lift-+.f64 N/A

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

      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 95.3

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

   8. Applied rewrites 95.3%

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

Alternative 6: 94.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{y + x}\\ \mathbf{if}\;x \leq -1.2 \cdot 10^{+163}:\\ \;\;\;\;\frac{\frac{y}{x}}{y + x} \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \frac{y}{\left(\left(y + x\right) + 1\right) \cdot \left(y + x\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ x (+ y x))))
   (if (<= x -1.2e+163)
     (* (/ (/ y x) (+ y x)) t_0)
     (* t_0 (/ y (* (+ (+ y x) 1.0) (+ y x)))))))

C

double code(double x, double y) {
	double t_0 = x / (y + x);
	double tmp;
	if (x <= -1.2e+163) {
		tmp = ((y / x) / (y + x)) * t_0;
	} else {
		tmp = t_0 * (y / (((y + x) + 1.0) * (y + x)));
	}
	return tmp;
}

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
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x / (y + x)
    if (x <= (-1.2d+163)) then
        tmp = ((y / x) / (y + x)) * t_0
    else
        tmp = t_0 * (y / (((y + x) + 1.0d0) * (y + x)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = x / (y + x);
	double tmp;
	if (x <= -1.2e+163) {
		tmp = ((y / x) / (y + x)) * t_0;
	} else {
		tmp = t_0 * (y / (((y + x) + 1.0) * (y + x)));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = x / (y + x)
	tmp = 0
	if x <= -1.2e+163:
		tmp = ((y / x) / (y + x)) * t_0
	else:
		tmp = t_0 * (y / (((y + x) + 1.0) * (y + x)))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(x / Float64(y + x))
	tmp = 0.0
	if (x <= -1.2e+163)
		tmp = Float64(Float64(Float64(y / x) / Float64(y + x)) * t_0);
	else
		tmp = Float64(t_0 * Float64(y / Float64(Float64(Float64(y + x) + 1.0) * Float64(y + x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = x / (y + x);
	tmp = 0.0;
	if (x <= -1.2e+163)
		tmp = ((y / x) / (y + x)) * t_0;
	else
		tmp = t_0 * (y / (((y + x) + 1.0) * (y + x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(x / N[(y + x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.2e+163], N[(N[(N[(y / x), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision] * t$95$0), $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 < -1.1999999999999999e163

   1. Initial program 59.7%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   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 83.8

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

   4. Applied rewrites 83.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. frac-times N/A

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

      9. unpow2 N/A

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

      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. lower-/.f64 N/A

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

      14. lift-+.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 N/A

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

      18. lift-+.f64 N/A

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

      19. lift-+.f64 N/A

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

      20. lift-+.f64 83.8

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

   6. Applied rewrites 83.8%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. associate-/r* N/A

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

      12. lower-/.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lift-+.f64 N/A

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

      15. lift-+.f64 N/A

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

      16. lift-+.f64 N/A

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

      17. lift-/.f64 N/A

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

      18. lift-+.f64 99.9

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

   8. Applied rewrites 99.9%

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

   9. Taylor expanded in x around inf

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

   11. Applied rewrites 88.8%

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

   if -1.1999999999999999e163 < x

   1. Initial program 70.2%

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

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

      2. lift-/.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. frac-times N/A

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

      9. unpow2 N/A

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

      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. lower-/.f64 N/A

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

      14. lift-+.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 N/A

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

      18. lift-+.f64 N/A

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

      19. lift-+.f64 N/A

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

      20. lift-+.f64 95.3

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

   6. Applied rewrites 95.3%

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

Alternative 7: 94.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(y + x\right) + 1\\ \mathbf{if}\;x \leq -1.2 \cdot 10^{+163}:\\ \;\;\;\;\frac{\frac{y}{t\_0}}{y + x} \cdot 1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y + x} \cdot \frac{y}{t\_0 \cdot \left(y + x\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ (+ y x) 1.0)))
   (if (<= x -1.2e+163)
     (* (/ (/ y t_0) (+ y x)) 1.0)
     (* (/ x (+ y x)) (/ y (* t_0 (+ y x)))))))

C

double code(double x, double y) {
	double t_0 = (y + x) + 1.0;
	double tmp;
	if (x <= -1.2e+163) {
		tmp = ((y / t_0) / (y + x)) * 1.0;
	} else {
		tmp = (x / (y + x)) * (y / (t_0 * (y + x)));
	}
	return tmp;
}

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
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (y + x) + 1.0d0
    if (x <= (-1.2d+163)) then
        tmp = ((y / t_0) / (y + x)) * 1.0d0
    else
        tmp = (x / (y + x)) * (y / (t_0 * (y + x)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = (y + x) + 1.0;
	double tmp;
	if (x <= -1.2e+163) {
		tmp = ((y / t_0) / (y + x)) * 1.0;
	} else {
		tmp = (x / (y + x)) * (y / (t_0 * (y + x)));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (y + x) + 1.0
	tmp = 0
	if x <= -1.2e+163:
		tmp = ((y / t_0) / (y + x)) * 1.0
	else:
		tmp = (x / (y + x)) * (y / (t_0 * (y + x)))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(y + x) + 1.0)
	tmp = 0.0
	if (x <= -1.2e+163)
		tmp = Float64(Float64(Float64(y / t_0) / Float64(y + x)) * 1.0);
	else
		tmp = Float64(Float64(x / Float64(y + x)) * Float64(y / Float64(t_0 * Float64(y + x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (y + x) + 1.0;
	tmp = 0.0;
	if (x <= -1.2e+163)
		tmp = ((y / t_0) / (y + x)) * 1.0;
	else
		tmp = (x / (y + x)) * (y / (t_0 * (y + x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(y + x), $MachinePrecision] + 1.0), $MachinePrecision]}, If[LessEqual[x, -1.2e+163], N[(N[(N[(y / t$95$0), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision], N[(N[(x / N[(y + x), $MachinePrecision]), $MachinePrecision] * N[(y / N[(t$95$0 * N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.1999999999999999e163

   1. Initial program 59.7%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   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 83.8

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

   4. Applied rewrites 83.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. frac-times N/A

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

      9. unpow2 N/A

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

      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. lower-/.f64 N/A

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

      14. lift-+.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 N/A

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

      18. lift-+.f64 N/A

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

      19. lift-+.f64 N/A

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

      20. lift-+.f64 83.8

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

   6. Applied rewrites 83.8%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. associate-/r* N/A

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

      12. lower-/.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lift-+.f64 N/A

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

      15. lift-+.f64 N/A

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

      16. lift-+.f64 N/A

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

      17. lift-/.f64 N/A

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

      18. lift-+.f64 99.9

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

   8. Applied rewrites 99.9%

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

   9. Taylor expanded in x around inf

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

   11. Applied rewrites 88.8%

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

   if -1.1999999999999999e163 < x

   1. Initial program 70.2%

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

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

      2. lift-/.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. frac-times N/A

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

      9. unpow2 N/A

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

      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. lower-/.f64 N/A

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

      14. lift-+.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 N/A

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

      18. lift-+.f64 N/A

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

      19. lift-+.f64 N/A

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

      20. lift-+.f64 95.3

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

   6. Applied rewrites 95.3%

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

Alternative 8: 63.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(y + x\right) + 1\\ \mathbf{if}\;x \leq -1.2 \cdot 10^{+163}:\\ \;\;\;\;\frac{\frac{y}{t\_0}}{y + x} \cdot 1\\ \mathbf{elif}\;x \leq -1.55 \cdot 10^{-194}:\\ \;\;\;\;1 \cdot \frac{y}{t\_0 \cdot \left(y + x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\left(1 + y\right) \cdot y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ (+ y x) 1.0)))
   (if (<= x -1.2e+163)
     (* (/ (/ y t_0) (+ y x)) 1.0)
     (if (<= x -1.55e-194)
       (* 1.0 (/ y (* t_0 (+ y x))))
       (/ x (* (+ 1.0 y) y))))))

C

double code(double x, double y) {
	double t_0 = (y + x) + 1.0;
	double tmp;
	if (x <= -1.2e+163) {
		tmp = ((y / t_0) / (y + x)) * 1.0;
	} else if (x <= -1.55e-194) {
		tmp = 1.0 * (y / (t_0 * (y + x)));
	} else {
		tmp = x / ((1.0 + y) * y);
	}
	return tmp;
}

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
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (y + x) + 1.0d0
    if (x <= (-1.2d+163)) then
        tmp = ((y / t_0) / (y + x)) * 1.0d0
    else if (x <= (-1.55d-194)) then
        tmp = 1.0d0 * (y / (t_0 * (y + x)))
    else
        tmp = x / ((1.0d0 + y) * y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = (y + x) + 1.0;
	double tmp;
	if (x <= -1.2e+163) {
		tmp = ((y / t_0) / (y + x)) * 1.0;
	} else if (x <= -1.55e-194) {
		tmp = 1.0 * (y / (t_0 * (y + x)));
	} else {
		tmp = x / ((1.0 + y) * y);
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (y + x) + 1.0
	tmp = 0
	if x <= -1.2e+163:
		tmp = ((y / t_0) / (y + x)) * 1.0
	elif x <= -1.55e-194:
		tmp = 1.0 * (y / (t_0 * (y + x)))
	else:
		tmp = x / ((1.0 + y) * y)
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(y + x) + 1.0)
	tmp = 0.0
	if (x <= -1.2e+163)
		tmp = Float64(Float64(Float64(y / t_0) / Float64(y + x)) * 1.0);
	elseif (x <= -1.55e-194)
		tmp = Float64(1.0 * Float64(y / Float64(t_0 * Float64(y + x))));
	else
		tmp = Float64(x / Float64(Float64(1.0 + y) * y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (y + x) + 1.0;
	tmp = 0.0;
	if (x <= -1.2e+163)
		tmp = ((y / t_0) / (y + x)) * 1.0;
	elseif (x <= -1.55e-194)
		tmp = 1.0 * (y / (t_0 * (y + x)));
	else
		tmp = x / ((1.0 + y) * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(y + x), $MachinePrecision] + 1.0), $MachinePrecision]}, If[LessEqual[x, -1.2e+163], N[(N[(N[(y / t$95$0), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision], If[LessEqual[x, -1.55e-194], N[(1.0 * N[(y / N[(t$95$0 * N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x / N[(N[(1.0 + y), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.1999999999999999e163

   1. Initial program 59.7%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   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 83.8

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

   4. Applied rewrites 83.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. frac-times N/A

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

      9. unpow2 N/A

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

      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. lower-/.f64 N/A

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

      14. lift-+.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 N/A

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

      18. lift-+.f64 N/A

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

      19. lift-+.f64 N/A

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

      20. lift-+.f64 83.8

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

   6. Applied rewrites 83.8%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. associate-/r* N/A

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

      12. lower-/.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lift-+.f64 N/A

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

      15. lift-+.f64 N/A

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

      16. lift-+.f64 N/A

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

      17. lift-/.f64 N/A

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

      18. lift-+.f64 99.9

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

   8. Applied rewrites 99.9%

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

   9. Taylor expanded in x around inf

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

   11. Applied rewrites 88.8%

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

   if -1.1999999999999999e163 < x < -1.55000000000000005e-194

   1. Initial program 75.7%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   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 93.2

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

   4. Applied rewrites 93.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. frac-times N/A

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

      9. unpow2 N/A

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

      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. lower-/.f64 N/A

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

      14. lift-+.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 N/A

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

      18. lift-+.f64 N/A

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

      19. lift-+.f64 N/A

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

      20. lift-+.f64 96.0

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

   6. Applied rewrites 96.0%

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

   7. Taylor expanded in x around inf

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

   9. Applied rewrites 68.8%

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

   if -1.55000000000000005e-194 < x

   1. Initial program 67.5%

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

   3. 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 55.3

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

   5. Applied rewrites 55.3%

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

Alternative 9: 63.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.2 \cdot 10^{+163}:\\ \;\;\;\;\frac{\frac{y}{x}}{x}\\ \mathbf{elif}\;x \leq -1.55 \cdot 10^{-194}:\\ \;\;\;\;1 \cdot \frac{y}{\left(\left(y + x\right) + 1\right) \cdot \left(y + x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\left(1 + y\right) \cdot y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1.2e+163)
   (/ (/ y x) x)
   (if (<= x -1.55e-194)
     (* 1.0 (/ y (* (+ (+ y x) 1.0) (+ y x))))
     (/ x (* (+ 1.0 y) y)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.2e+163) {
		tmp = (y / x) / x;
	} else if (x <= -1.55e-194) {
		tmp = 1.0 * (y / (((y + x) + 1.0) * (y + x)));
	} else {
		tmp = x / ((1.0 + y) * y);
	}
	return tmp;
}

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
    real(8) :: tmp
    if (x <= (-1.2d+163)) then
        tmp = (y / x) / x
    else if (x <= (-1.55d-194)) then
        tmp = 1.0d0 * (y / (((y + x) + 1.0d0) * (y + x)))
    else
        tmp = x / ((1.0d0 + y) * y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -1.2e+163) {
		tmp = (y / x) / x;
	} else if (x <= -1.55e-194) {
		tmp = 1.0 * (y / (((y + x) + 1.0) * (y + x)));
	} else {
		tmp = x / ((1.0 + y) * y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -1.2e+163:
		tmp = (y / x) / x
	elif x <= -1.55e-194:
		tmp = 1.0 * (y / (((y + x) + 1.0) * (y + x)))
	else:
		tmp = x / ((1.0 + y) * y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1.2e+163)
		tmp = Float64(Float64(y / x) / x);
	elseif (x <= -1.55e-194)
		tmp = Float64(1.0 * Float64(y / Float64(Float64(Float64(y + x) + 1.0) * Float64(y + x))));
	else
		tmp = Float64(x / Float64(Float64(1.0 + y) * y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1.2e+163)
		tmp = (y / x) / x;
	elseif (x <= -1.55e-194)
		tmp = 1.0 * (y / (((y + x) + 1.0) * (y + x)));
	else
		tmp = x / ((1.0 + y) * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1.2e+163], N[(N[(y / x), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[x, -1.55e-194], N[(1.0 * N[(y / N[(N[(N[(y + x), $MachinePrecision] + 1.0), $MachinePrecision] * N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x / N[(N[(1.0 + y), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.1999999999999999e163

   1. Initial program 59.7%

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

   3. Taylor expanded in x around 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 83.8

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

   5. Applied rewrites 83.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      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 88.3

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

   7. Applied rewrites 88.3%

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

   if -1.1999999999999999e163 < x < -1.55000000000000005e-194

   1. Initial program 75.7%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   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 93.2

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

   4. Applied rewrites 93.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. frac-times N/A

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

      9. unpow2 N/A

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

      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. lower-/.f64 N/A

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

      14. lift-+.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 N/A

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

      18. lift-+.f64 N/A

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

      19. lift-+.f64 N/A

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

      20. lift-+.f64 96.0

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

   6. Applied rewrites 96.0%

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

   7. Taylor expanded in x around inf

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

   9. Applied rewrites 68.8%

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

   if -1.55000000000000005e-194 < x

   1. Initial program 67.5%

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

   3. 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 55.3

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

   5. Applied rewrites 55.3%

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

Alternative 10: 54.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{y \cdot y}\\ \mathbf{if}\;x \leq -240000:\\ \;\;\;\;\frac{y}{x \cdot x}\\ \mathbf{elif}\;x \leq -7.2 \cdot 10^{-94}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 4 \cdot 10^{-103}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ x (* y y))))
   (if (<= x -240000.0)
     (/ y (* x x))
     (if (<= x -7.2e-94) t_0 (if (<= x 4e-103) (/ x y) t_0)))))

C

double code(double x, double y) {
	double t_0 = x / (y * y);
	double tmp;
	if (x <= -240000.0) {
		tmp = y / (x * x);
	} else if (x <= -7.2e-94) {
		tmp = t_0;
	} else if (x <= 4e-103) {
		tmp = x / y;
	} else {
		tmp = t_0;
	}
	return tmp;
}

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
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x / (y * y)
    if (x <= (-240000.0d0)) then
        tmp = y / (x * x)
    else if (x <= (-7.2d-94)) then
        tmp = t_0
    else if (x <= 4d-103) then
        tmp = x / y
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = x / (y * y);
	double tmp;
	if (x <= -240000.0) {
		tmp = y / (x * x);
	} else if (x <= -7.2e-94) {
		tmp = t_0;
	} else if (x <= 4e-103) {
		tmp = x / y;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = x / (y * y)
	tmp = 0
	if x <= -240000.0:
		tmp = y / (x * x)
	elif x <= -7.2e-94:
		tmp = t_0
	elif x <= 4e-103:
		tmp = x / y
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(x / Float64(y * y))
	tmp = 0.0
	if (x <= -240000.0)
		tmp = Float64(y / Float64(x * x));
	elseif (x <= -7.2e-94)
		tmp = t_0;
	elseif (x <= 4e-103)
		tmp = Float64(x / y);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = x / (y * y);
	tmp = 0.0;
	if (x <= -240000.0)
		tmp = y / (x * x);
	elseif (x <= -7.2e-94)
		tmp = t_0;
	elseif (x <= 4e-103)
		tmp = x / y;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(x / N[(y * y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -240000.0], N[(y / N[(x * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -7.2e-94], t$95$0, If[LessEqual[x, 4e-103], N[(x / y), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.4e5

   1. Initial program 63.5%

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

   3. Taylor expanded in x around 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 72.5

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

   5. Applied rewrites 72.5%

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

   if -2.4e5 < x < -7.2e-94 or 3.99999999999999983e-103 < x

   1. Initial program 73.4%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   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 34.7

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

   5. Applied rewrites 34.7%

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

   if -7.2e-94 < x < 3.99999999999999983e-103

   1. Initial program 67.8%

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

   3. Taylor expanded in x around 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 81.3

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

   5. Applied rewrites 81.3%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 64.2%

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

Alternative 11: 60.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.2 \cdot 10^{+163}:\\ \;\;\;\;\frac{\frac{y}{x}}{x}\\ \mathbf{elif}\;x \leq -2.2 \cdot 10^{-102}:\\ \;\;\;\;\frac{y}{\left(1 + x\right) \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\left(1 + y\right) \cdot y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1.2e+163)
   (/ (/ y x) x)
   (if (<= x -2.2e-102) (/ y (* (+ 1.0 x) x)) (/ x (* (+ 1.0 y) y)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.2e+163) {
		tmp = (y / x) / x;
	} else if (x <= -2.2e-102) {
		tmp = y / ((1.0 + x) * x);
	} else {
		tmp = x / ((1.0 + y) * y);
	}
	return tmp;
}

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
    real(8) :: tmp
    if (x <= (-1.2d+163)) then
        tmp = (y / x) / x
    else if (x <= (-2.2d-102)) then
        tmp = y / ((1.0d0 + x) * x)
    else
        tmp = x / ((1.0d0 + y) * y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -1.2e+163) {
		tmp = (y / x) / x;
	} else if (x <= -2.2e-102) {
		tmp = y / ((1.0 + x) * x);
	} else {
		tmp = x / ((1.0 + y) * y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -1.2e+163:
		tmp = (y / x) / x
	elif x <= -2.2e-102:
		tmp = y / ((1.0 + x) * x)
	else:
		tmp = x / ((1.0 + y) * y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1.2e+163)
		tmp = Float64(Float64(y / x) / x);
	elseif (x <= -2.2e-102)
		tmp = Float64(y / Float64(Float64(1.0 + x) * x));
	else
		tmp = Float64(x / Float64(Float64(1.0 + y) * y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1.2e+163)
		tmp = (y / x) / x;
	elseif (x <= -2.2e-102)
		tmp = y / ((1.0 + x) * x);
	else
		tmp = x / ((1.0 + y) * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1.2e+163], N[(N[(y / x), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[x, -2.2e-102], 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 3 regimes

2. if x < -1.1999999999999999e163

   1. Initial program 59.7%

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

   3. Taylor expanded in x around 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 83.8

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

   5. Applied rewrites 83.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      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 88.3

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

   7. Applied rewrites 88.3%

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

   if -1.1999999999999999e163 < x < -2.20000000000000013e-102

   1. Initial program 75.2%

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

   3. Taylor expanded in 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 56.0

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

   5. Applied rewrites 56.0%

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

   if -2.20000000000000013e-102 < x

   1. Initial program 68.6%

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

   3. Taylor expanded in 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 57.4

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

   5. Applied rewrites 57.4%

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

Alternative 12: 60.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.2 \cdot 10^{-102}:\\ \;\;\;\;\frac{y}{\left(1 + x\right) \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\left(1 + y\right) \cdot y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -2.2e-102) (/ y (* (+ 1.0 x) x)) (/ x (* (+ 1.0 y) y))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -2.2e-102) {
		tmp = y / ((1.0 + x) * x);
	} else {
		tmp = x / ((1.0 + y) * y);
	}
	return tmp;
}

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
    real(8) :: tmp
    if (x <= (-2.2d-102)) then
        tmp = y / ((1.0d0 + x) * x)
    else
        tmp = x / ((1.0d0 + y) * y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -2.2e-102) {
		tmp = y / ((1.0 + x) * x);
	} else {
		tmp = x / ((1.0 + y) * y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -2.2e-102:
		tmp = y / ((1.0 + x) * x)
	else:
		tmp = x / ((1.0 + y) * y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -2.2e-102)
		tmp = Float64(y / Float64(Float64(1.0 + x) * x));
	else
		tmp = Float64(x / Float64(Float64(1.0 + y) * y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -2.2e-102)
		tmp = y / ((1.0 + x) * x);
	else
		tmp = x / ((1.0 + y) * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -2.2e-102], 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 x < -2.20000000000000013e-102

   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. 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 65.7

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

   5. Applied rewrites 65.7%

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

   if -2.20000000000000013e-102 < x

   1. Initial program 68.6%

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

   3. Taylor expanded in 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 57.4

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

   5. Applied rewrites 57.4%

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

Alternative 13: 61.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -240000:\\ \;\;\;\;\frac{y}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\left(1 + y\right) \cdot y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -240000.0) (/ y (* x x)) (/ x (* (+ 1.0 y) y))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -240000.0) {
		tmp = y / (x * x);
	} else {
		tmp = x / ((1.0 + y) * y);
	}
	return tmp;
}

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
    real(8) :: tmp
    if (x <= (-240000.0d0)) then
        tmp = y / (x * x)
    else
        tmp = x / ((1.0d0 + y) * y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -240000.0) {
		tmp = y / (x * x);
	} else {
		tmp = x / ((1.0 + y) * y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -240000.0:
		tmp = y / (x * x)
	else:
		tmp = x / ((1.0 + y) * y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -240000.0)
		tmp = Float64(y / Float64(x * x));
	else
		tmp = Float64(x / Float64(Float64(1.0 + y) * y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -240000.0)
		tmp = y / (x * x);
	else
		tmp = x / ((1.0 + y) * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -240000.0], 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 < -2.4e5

   1. Initial program 63.5%

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

   3. Taylor expanded in x around 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 72.5

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

   5. Applied rewrites 72.5%

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

   if -2.4e5 < x

   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. 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 57.5

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

   5. Applied rewrites 57.5%

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

Alternative 14: 36.3% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= y 1.0) (/ x y) (/ x (* y y))))

C

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

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
    real(8) :: tmp
    if (y <= 1.0d0) then
        tmp = x / y
    else
        tmp = x / (y * y)
    end if
    code = tmp
end function

Java

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

def code(x, y):
	tmp = 0
	if y <= 1.0:
		tmp = x / y
	else:
		tmp = x / (y * y)
	return tmp

Julia

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

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

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 70.6%

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

   3. Taylor expanded in x around 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 40.6

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

   5. Applied rewrites 40.6%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 24.8%

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

   if 1 < 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. 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 71.4

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

   5. Applied rewrites 71.4%

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

Alternative 15: 25.7% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ \frac{x}{y} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (/ x y))

C

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

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
end function

Java

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

Python

def code(x, y):
	return x / y

Julia

function code(x, y)
	return Float64(x / y)
end

MATLAB

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

Wolfram

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. 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 48.5

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

5. Applied rewrites 48.5%

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

6. Taylor expanded in y around 0

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

8. Applied rewrites 25.7%

   \[\leadsto \frac{x}{y} \]
9. Add Preprocessing

Developer Target 1: 99.8% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2025091 
(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))))