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

Percentage Accurate: 69.5% → 99.8%

Time: 5.7s

Alternatives: 18

Speedup: 1.0×

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: 69.5% 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, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 69.5%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. lift-*.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. associate-*l* N/A

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

   7. times-frac N/A

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

   8. lower-*.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. lift-+.f64 N/A

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

   11. +-commutative N/A

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

   12. lower-+.f64 N/A

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

   13. lower-/.f64 N/A

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

   14. *-commutative N/A

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

   15. lower-*.f64 93.8

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

   16. lift-+.f64 N/A

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

   17. add-flip N/A

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

   18. lower--.f64 N/A

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

   19. lift-+.f64 N/A

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

   20. +-commutative N/A

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

   21. lower-+.f64 N/A

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

   22. metadata-eval 93.8

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

   23. lift-+.f64 N/A

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

   24. +-commutative N/A

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

   25. lower-+.f64 93.8

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

3. Applied rewrites 93.8%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-*r/ N/A

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

   4. lift-*.f64 N/A

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

   5. times-frac N/A

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

   6. lower-*.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. lift-+.f64 N/A

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

   9. +-commutative N/A

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

   10. lift-+.f64 N/A

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

   11. lift-+.f64 N/A

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

   12. +-commutative N/A

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

   13. lift-+.f64 N/A

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

   14. lower-/.f64 99.8

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

   15. lift-+.f64 N/A

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

   16. +-commutative N/A

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

   17. lift-+.f64 99.8

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

5. Applied rewrites 99.8%

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

Alternative 2: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 69.5%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. lift-*.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. associate-*l* N/A

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

   7. times-frac N/A

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

   8. lower-*.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. lift-+.f64 N/A

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

   11. +-commutative N/A

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

   12. lower-+.f64 N/A

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

   13. lower-/.f64 N/A

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

   14. *-commutative N/A

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

   15. lower-*.f64 93.8

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

   16. lift-+.f64 N/A

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

   17. add-flip N/A

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

   18. lower--.f64 N/A

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

   19. lift-+.f64 N/A

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

   20. +-commutative N/A

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

   21. lower-+.f64 N/A

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

   22. metadata-eval 93.8

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

   23. lift-+.f64 N/A

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

   24. +-commutative N/A

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

   25. lower-+.f64 93.8

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

3. Applied rewrites 93.8%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-*r/ N/A

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

   4. lift-*.f64 N/A

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

   5. times-frac N/A

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

   6. lower-*.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. lift-+.f64 N/A

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

   9. +-commutative N/A

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

   10. lift-+.f64 N/A

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

   11. lift-+.f64 N/A

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

   12. +-commutative N/A

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

   13. lift-+.f64 N/A

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

   14. lower-/.f64 99.8

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

   15. lift-+.f64 N/A

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

   16. +-commutative N/A

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

   17. lift-+.f64 99.8

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

5. Applied rewrites 99.8%

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

   1. lift-/.f64 N/A

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

   2. mult-flip N/A

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

   3. lift-/.f64 N/A

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

   4. associate-*l/ N/A

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

   5. mult-flip N/A

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

   6. lift--.f64 N/A

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

   7. metadata-eval N/A

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

   8. add-flip N/A

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

   9. lift-+.f64 N/A

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

   10. lower-/.f64 N/A

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

7. Applied rewrites 99.8%

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

Alternative 3: 80.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{x + y}\\ \mathbf{if}\;x \leq -4.6 \cdot 10^{+172}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + y} \cdot t\_0\\ \mathbf{elif}\;x \leq 1.7 \cdot 10^{+25}:\\ \;\;\;\;\frac{\frac{y}{x + y} \cdot x}{\left(x - \left(-1 - y\right)\right) \cdot \left(x + y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(x + y\right) - -1} \cdot t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ x (+ x y))))
   (if (<= x -4.6e+172)
     (* (/ (/ y x) (+ x y)) t_0)
     (if (<= x 1.7e+25)
       (/ (* (/ y (+ x y)) x) (* (- x (- -1.0 y)) (+ x y)))
       (* (/ 1.0 (- (+ x y) -1.0)) t_0)))))

C

double code(double x, double y) {
	double t_0 = x / (x + y);
	double tmp;
	if (x <= -4.6e+172) {
		tmp = ((y / x) / (x + y)) * t_0;
	} else if (x <= 1.7e+25) {
		tmp = ((y / (x + y)) * x) / ((x - (-1.0 - y)) * (x + y));
	} else {
		tmp = (1.0 / ((x + y) - -1.0)) * 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 / (x + y)
    if (x <= (-4.6d+172)) then
        tmp = ((y / x) / (x + y)) * t_0
    else if (x <= 1.7d+25) then
        tmp = ((y / (x + y)) * x) / ((x - ((-1.0d0) - y)) * (x + y))
    else
        tmp = (1.0d0 / ((x + y) - (-1.0d0))) * t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = x / (x + y);
	double tmp;
	if (x <= -4.6e+172) {
		tmp = ((y / x) / (x + y)) * t_0;
	} else if (x <= 1.7e+25) {
		tmp = ((y / (x + y)) * x) / ((x - (-1.0 - y)) * (x + y));
	} else {
		tmp = (1.0 / ((x + y) - -1.0)) * t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = x / (x + y)
	tmp = 0
	if x <= -4.6e+172:
		tmp = ((y / x) / (x + y)) * t_0
	elif x <= 1.7e+25:
		tmp = ((y / (x + y)) * x) / ((x - (-1.0 - y)) * (x + y))
	else:
		tmp = (1.0 / ((x + y) - -1.0)) * t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(x / Float64(x + y))
	tmp = 0.0
	if (x <= -4.6e+172)
		tmp = Float64(Float64(Float64(y / x) / Float64(x + y)) * t_0);
	elseif (x <= 1.7e+25)
		tmp = Float64(Float64(Float64(y / Float64(x + y)) * x) / Float64(Float64(x - Float64(-1.0 - y)) * Float64(x + y)));
	else
		tmp = Float64(Float64(1.0 / Float64(Float64(x + y) - -1.0)) * t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = x / (x + y);
	tmp = 0.0;
	if (x <= -4.6e+172)
		tmp = ((y / x) / (x + y)) * t_0;
	elseif (x <= 1.7e+25)
		tmp = ((y / (x + y)) * x) / ((x - (-1.0 - y)) * (x + y));
	else
		tmp = (1.0 / ((x + y) - -1.0)) * t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(x / N[(x + y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -4.6e+172], N[(N[(N[(y / x), $MachinePrecision] / N[(x + y), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision], If[LessEqual[x, 1.7e+25], N[(N[(N[(y / N[(x + y), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision] / N[(N[(x - N[(-1.0 - y), $MachinePrecision]), $MachinePrecision] * N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[(N[(x + y), $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -4.6000000000000002e172

   1. Initial program 69.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 93.8

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

      16. lift-+.f64 N/A

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

      17. add-flip N/A

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

      18. lower--.f64 N/A

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

      19. lift-+.f64 N/A

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

      20. +-commutative N/A

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

      21. lower-+.f64 N/A

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

      22. metadata-eval 93.8

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

      23. lift-+.f64 N/A

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

      24. +-commutative N/A

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

      25. lower-+.f64 93.8

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

   3. Applied rewrites 93.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lift-*.f64 N/A

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

      5. times-frac N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-+.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lift-+.f64 N/A

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

      14. lower-/.f64 99.8

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

      15. lift-+.f64 N/A

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

      16. +-commutative N/A

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

      17. lift-+.f64 99.8

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

   5. Applied rewrites 99.8%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*l/ N/A

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

      5. mult-flip N/A

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

      6. lift--.f64 N/A

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

      7. metadata-eval N/A

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

      8. add-flip N/A

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

      9. lift-+.f64 N/A

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

      10. lower-/.f64 N/A

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

   7. Applied rewrites 99.8%

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

   8. Taylor expanded in x around inf

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

      1. lower-/.f64 37.9

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

   10. Applied rewrites 37.9%

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

   if -4.6000000000000002e172 < x < 1.69999999999999992e25

   1. Initial program 69.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 93.8

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

      16. lift-+.f64 N/A

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

      17. add-flip N/A

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

      18. lower--.f64 N/A

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

      19. lift-+.f64 N/A

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

      20. +-commutative N/A

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

      21. lower-+.f64 N/A

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

      22. metadata-eval 93.8

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

      23. lift-+.f64 N/A

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

      24. +-commutative N/A

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

      25. lower-+.f64 93.8

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

   3. Applied rewrites 93.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lift-*.f64 N/A

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

      5. times-frac N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-+.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lift-+.f64 N/A

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

      14. lower-/.f64 99.8

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

      15. lift-+.f64 N/A

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

      16. +-commutative N/A

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

      17. lift-+.f64 99.8

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

   5. Applied rewrites 99.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. frac-times N/A

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

      5. *-commutative N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 93.8

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

      10. lift--.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. associate--l+ N/A

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

      13. lift--.f64 N/A

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

      14. add-flip N/A

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

      15. lower--.f64 N/A

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

      16. lift--.f64 N/A

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

      17. sub-negate-rev N/A

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

      18. lower--.f64 93.8

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

   7. Applied rewrites 93.8%

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

   if 1.69999999999999992e25 < x

   1. Initial program 69.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 93.8

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

      16. lift-+.f64 N/A

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

      17. add-flip N/A

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

      18. lower--.f64 N/A

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

      19. lift-+.f64 N/A

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

      20. +-commutative N/A

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

      21. lower-+.f64 N/A

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

      22. metadata-eval 93.8

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

      23. lift-+.f64 N/A

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

      24. +-commutative N/A

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

      25. lower-+.f64 93.8

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

   3. Applied rewrites 93.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lift-*.f64 N/A

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

      5. times-frac N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-+.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lift-+.f64 N/A

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

      14. lower-/.f64 99.8

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

      15. lift-+.f64 N/A

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

      16. +-commutative N/A

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

      17. lift-+.f64 99.8

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

   5. Applied rewrites 99.8%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 52.9%

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

Alternative 4: 76.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3.65 \cdot 10^{-57}:\\ \;\;\;\;\frac{\frac{y}{x}}{\left(x + y\right) - -1} \cdot \frac{x}{x + y}\\ \mathbf{elif}\;y \leq 3 \cdot 10^{-24}:\\ \;\;\;\;\frac{y}{x + y} \cdot \frac{x}{\left(x - -1\right) \cdot \left(x + y\right)}\\ \mathbf{elif}\;y \leq 1.35 \cdot 10^{+127}:\\ \;\;\;\;\frac{\frac{x}{x - \left(-1 - y\right)} \cdot y}{\left(x + y\right) \cdot \left(x + y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{y} \cdot x}{x + y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -3.65e-57)
   (* (/ (/ y x) (- (+ x y) -1.0)) (/ x (+ x y)))
   (if (<= y 3e-24)
     (* (/ y (+ x y)) (/ x (* (- x -1.0) (+ x y))))
     (if (<= y 1.35e+127)
       (/ (* (/ x (- x (- -1.0 y))) y) (* (+ x y) (+ x y)))
       (/ (* (/ 1.0 y) x) (+ x y))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -3.65e-57) {
		tmp = ((y / x) / ((x + y) - -1.0)) * (x / (x + y));
	} else if (y <= 3e-24) {
		tmp = (y / (x + y)) * (x / ((x - -1.0) * (x + y)));
	} else if (y <= 1.35e+127) {
		tmp = ((x / (x - (-1.0 - y))) * y) / ((x + y) * (x + y));
	} else {
		tmp = ((1.0 / y) * x) / (x + 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 <= (-3.65d-57)) then
        tmp = ((y / x) / ((x + y) - (-1.0d0))) * (x / (x + y))
    else if (y <= 3d-24) then
        tmp = (y / (x + y)) * (x / ((x - (-1.0d0)) * (x + y)))
    else if (y <= 1.35d+127) then
        tmp = ((x / (x - ((-1.0d0) - y))) * y) / ((x + y) * (x + y))
    else
        tmp = ((1.0d0 / y) * x) / (x + y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -3.65e-57) {
		tmp = ((y / x) / ((x + y) - -1.0)) * (x / (x + y));
	} else if (y <= 3e-24) {
		tmp = (y / (x + y)) * (x / ((x - -1.0) * (x + y)));
	} else if (y <= 1.35e+127) {
		tmp = ((x / (x - (-1.0 - y))) * y) / ((x + y) * (x + y));
	} else {
		tmp = ((1.0 / y) * x) / (x + y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -3.65e-57:
		tmp = ((y / x) / ((x + y) - -1.0)) * (x / (x + y))
	elif y <= 3e-24:
		tmp = (y / (x + y)) * (x / ((x - -1.0) * (x + y)))
	elif y <= 1.35e+127:
		tmp = ((x / (x - (-1.0 - y))) * y) / ((x + y) * (x + y))
	else:
		tmp = ((1.0 / y) * x) / (x + y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -3.65e-57)
		tmp = Float64(Float64(Float64(y / x) / Float64(Float64(x + y) - -1.0)) * Float64(x / Float64(x + y)));
	elseif (y <= 3e-24)
		tmp = Float64(Float64(y / Float64(x + y)) * Float64(x / Float64(Float64(x - -1.0) * Float64(x + y))));
	elseif (y <= 1.35e+127)
		tmp = Float64(Float64(Float64(x / Float64(x - Float64(-1.0 - y))) * y) / Float64(Float64(x + y) * Float64(x + y)));
	else
		tmp = Float64(Float64(Float64(1.0 / y) * x) / Float64(x + y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -3.65e-57)
		tmp = ((y / x) / ((x + y) - -1.0)) * (x / (x + y));
	elseif (y <= 3e-24)
		tmp = (y / (x + y)) * (x / ((x - -1.0) * (x + y)));
	elseif (y <= 1.35e+127)
		tmp = ((x / (x - (-1.0 - y))) * y) / ((x + y) * (x + y));
	else
		tmp = ((1.0 / y) * x) / (x + y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -3.65e-57], N[(N[(N[(y / x), $MachinePrecision] / N[(N[(x + y), $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision] * N[(x / N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 3e-24], N[(N[(y / N[(x + y), $MachinePrecision]), $MachinePrecision] * N[(x / N[(N[(x - -1.0), $MachinePrecision] * N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.35e+127], N[(N[(N[(x / N[(x - N[(-1.0 - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision] / N[(N[(x + y), $MachinePrecision] * N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 / y), $MachinePrecision] * x), $MachinePrecision] / N[(x + y), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if y < -3.65000000000000011e-57

   1. Initial program 69.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 93.8

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

      16. lift-+.f64 N/A

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

      17. add-flip N/A

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

      18. lower--.f64 N/A

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

      19. lift-+.f64 N/A

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

      20. +-commutative N/A

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

      21. lower-+.f64 N/A

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

      22. metadata-eval 93.8

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

      23. lift-+.f64 N/A

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

      24. +-commutative N/A

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

      25. lower-+.f64 93.8

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

   3. Applied rewrites 93.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lift-*.f64 N/A

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

      5. times-frac N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-+.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lift-+.f64 N/A

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

      14. lower-/.f64 99.8

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

      15. lift-+.f64 N/A

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

      16. +-commutative N/A

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

      17. lift-+.f64 99.8

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

   5. Applied rewrites 99.8%

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

   6. Taylor expanded in x around inf

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

      1. lower-/.f64 49.6

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

   8. Applied rewrites 49.6%

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

   if -3.65000000000000011e-57 < y < 2.99999999999999995e-24

   1. Initial program 69.5%

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

   2. Taylor expanded in y around 0

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

      1. lower-+.f64 59.1

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

   4. Applied rewrites 59.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-+.f64 N/A

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

      10. times-frac N/A

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

      11. lift-/.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lift-+.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. lift-+.f64 N/A

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

      18. +-commutative N/A

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

      19. lift-+.f64 N/A

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

   6. Applied rewrites 75.5%

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

   if 2.99999999999999995e-24 < y < 1.3500000000000001e127

   1. Initial program 69.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 93.8

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

      16. lift-+.f64 N/A

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

      17. add-flip N/A

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

      18. lower--.f64 N/A

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

      19. lift-+.f64 N/A

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

      20. +-commutative N/A

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

      21. lower-+.f64 N/A

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

      22. metadata-eval 93.8

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

      23. lift-+.f64 N/A

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

      24. +-commutative N/A

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

      25. lower-+.f64 93.8

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

   3. Applied rewrites 93.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lift-*.f64 N/A

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

      5. times-frac N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-+.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lift-+.f64 N/A

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

      14. lower-/.f64 99.8

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

      15. lift-+.f64 N/A

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

      16. +-commutative N/A

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

      17. lift-+.f64 99.8

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

   5. Applied rewrites 99.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. frac-times N/A

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

      5. associate-/l* N/A

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

      6. lift-/.f64 N/A

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

      7. associate-/l/ N/A

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

      8. lift-/.f64 N/A

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

      9. times-frac N/A

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

      10. lift-*.f64 N/A

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

      11. lower-/.f64 N/A

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

   7. Applied rewrites 82.8%

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

   if 1.3500000000000001e127 < y

   1. Initial program 69.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 93.8

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

      16. lift-+.f64 N/A

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

      17. add-flip N/A

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

      18. lower--.f64 N/A

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

      19. lift-+.f64 N/A

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

      20. +-commutative N/A

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

      21. lower-+.f64 N/A

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

      22. metadata-eval 93.8

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

      23. lift-+.f64 N/A

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

      24. +-commutative N/A

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

      25. lower-+.f64 93.8

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

   3. Applied rewrites 93.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lift-*.f64 N/A

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

      5. times-frac N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-+.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lift-+.f64 N/A

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

      14. lower-/.f64 99.8

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

      15. lift-+.f64 N/A

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

      16. +-commutative N/A

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

      17. lift-+.f64 99.8

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

   5. Applied rewrites 99.8%

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

   6. Taylor expanded in y around inf

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

      1. lower-/.f64 39.7

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

   8. Applied rewrites 39.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 39.6

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

   10. Applied rewrites 39.6%

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

Alternative 5: 74.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3.65 \cdot 10^{-57}:\\ \;\;\;\;\frac{\frac{y}{x}}{\left(x + y\right) - -1} \cdot \frac{x}{x + y}\\ \mathbf{elif}\;y \leq 3400:\\ \;\;\;\;\frac{y}{x + y} \cdot \frac{x}{\left(x - -1\right) \cdot \left(x + y\right)}\\ \mathbf{elif}\;y \leq 8.2 \cdot 10^{+82}:\\ \;\;\;\;\frac{y}{\left(\left(y + x\right) - -1\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{y} \cdot x}{x + y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -3.65e-57)
   (* (/ (/ y x) (- (+ x y) -1.0)) (/ x (+ x y)))
   (if (<= y 3400.0)
     (* (/ y (+ x y)) (/ x (* (- x -1.0) (+ x y))))
     (if (<= y 8.2e+82)
       (* (/ y (* (- (+ y x) -1.0) (* (+ y x) (+ y x)))) x)
       (/ (* (/ 1.0 y) x) (+ x y))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -3.65e-57) {
		tmp = ((y / x) / ((x + y) - -1.0)) * (x / (x + y));
	} else if (y <= 3400.0) {
		tmp = (y / (x + y)) * (x / ((x - -1.0) * (x + y)));
	} else if (y <= 8.2e+82) {
		tmp = (y / (((y + x) - -1.0) * ((y + x) * (y + x)))) * x;
	} else {
		tmp = ((1.0 / y) * x) / (x + 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 <= (-3.65d-57)) then
        tmp = ((y / x) / ((x + y) - (-1.0d0))) * (x / (x + y))
    else if (y <= 3400.0d0) then
        tmp = (y / (x + y)) * (x / ((x - (-1.0d0)) * (x + y)))
    else if (y <= 8.2d+82) then
        tmp = (y / (((y + x) - (-1.0d0)) * ((y + x) * (y + x)))) * x
    else
        tmp = ((1.0d0 / y) * x) / (x + y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -3.65e-57) {
		tmp = ((y / x) / ((x + y) - -1.0)) * (x / (x + y));
	} else if (y <= 3400.0) {
		tmp = (y / (x + y)) * (x / ((x - -1.0) * (x + y)));
	} else if (y <= 8.2e+82) {
		tmp = (y / (((y + x) - -1.0) * ((y + x) * (y + x)))) * x;
	} else {
		tmp = ((1.0 / y) * x) / (x + y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -3.65e-57:
		tmp = ((y / x) / ((x + y) - -1.0)) * (x / (x + y))
	elif y <= 3400.0:
		tmp = (y / (x + y)) * (x / ((x - -1.0) * (x + y)))
	elif y <= 8.2e+82:
		tmp = (y / (((y + x) - -1.0) * ((y + x) * (y + x)))) * x
	else:
		tmp = ((1.0 / y) * x) / (x + y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -3.65e-57)
		tmp = Float64(Float64(Float64(y / x) / Float64(Float64(x + y) - -1.0)) * Float64(x / Float64(x + y)));
	elseif (y <= 3400.0)
		tmp = Float64(Float64(y / Float64(x + y)) * Float64(x / Float64(Float64(x - -1.0) * Float64(x + y))));
	elseif (y <= 8.2e+82)
		tmp = Float64(Float64(y / Float64(Float64(Float64(y + x) - -1.0) * Float64(Float64(y + x) * Float64(y + x)))) * x);
	else
		tmp = Float64(Float64(Float64(1.0 / y) * x) / Float64(x + y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -3.65e-57)
		tmp = ((y / x) / ((x + y) - -1.0)) * (x / (x + y));
	elseif (y <= 3400.0)
		tmp = (y / (x + y)) * (x / ((x - -1.0) * (x + y)));
	elseif (y <= 8.2e+82)
		tmp = (y / (((y + x) - -1.0) * ((y + x) * (y + x)))) * x;
	else
		tmp = ((1.0 / y) * x) / (x + y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -3.65e-57], N[(N[(N[(y / x), $MachinePrecision] / N[(N[(x + y), $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision] * N[(x / N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 3400.0], N[(N[(y / N[(x + y), $MachinePrecision]), $MachinePrecision] * N[(x / N[(N[(x - -1.0), $MachinePrecision] * N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 8.2e+82], N[(N[(y / N[(N[(N[(y + x), $MachinePrecision] - -1.0), $MachinePrecision] * N[(N[(y + x), $MachinePrecision] * N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision], N[(N[(N[(1.0 / y), $MachinePrecision] * x), $MachinePrecision] / N[(x + y), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if y < -3.65000000000000011e-57

   1. Initial program 69.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 93.8

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

      16. lift-+.f64 N/A

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

      17. add-flip N/A

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

      18. lower--.f64 N/A

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

      19. lift-+.f64 N/A

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

      20. +-commutative N/A

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

      21. lower-+.f64 N/A

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

      22. metadata-eval 93.8

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

      23. lift-+.f64 N/A

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

      24. +-commutative N/A

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

      25. lower-+.f64 93.8

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

   3. Applied rewrites 93.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lift-*.f64 N/A

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

      5. times-frac N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-+.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lift-+.f64 N/A

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

      14. lower-/.f64 99.8

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

      15. lift-+.f64 N/A

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

      16. +-commutative N/A

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

      17. lift-+.f64 99.8

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

   5. Applied rewrites 99.8%

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

   6. Taylor expanded in x around inf

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

      1. lower-/.f64 49.6

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

   8. Applied rewrites 49.6%

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

   if -3.65000000000000011e-57 < y < 3400

   1. Initial program 69.5%

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

   2. Taylor expanded in y around 0

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

      1. lower-+.f64 59.1

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

   4. Applied rewrites 59.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-+.f64 N/A

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

      10. times-frac N/A

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

      11. lift-/.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lift-+.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. lift-+.f64 N/A

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

      18. +-commutative N/A

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

      19. lift-+.f64 N/A

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

   6. Applied rewrites 75.5%

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

   if 3400 < y < 8.1999999999999999e82

   1. Initial program 69.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

   3. Applied rewrites 82.0%

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

   if 8.1999999999999999e82 < y

   1. Initial program 69.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 93.8

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

      16. lift-+.f64 N/A

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

      17. add-flip N/A

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

      18. lower--.f64 N/A

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

      19. lift-+.f64 N/A

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

      20. +-commutative N/A

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

      21. lower-+.f64 N/A

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

      22. metadata-eval 93.8

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

      23. lift-+.f64 N/A

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

      24. +-commutative N/A

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

      25. lower-+.f64 93.8

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

   3. Applied rewrites 93.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lift-*.f64 N/A

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

      5. times-frac N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-+.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lift-+.f64 N/A

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

      14. lower-/.f64 99.8

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

      15. lift-+.f64 N/A

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

      16. +-commutative N/A

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

      17. lift-+.f64 99.8

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

   5. Applied rewrites 99.8%

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

   6. Taylor expanded in y around inf

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

      1. lower-/.f64 39.7

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

   8. Applied rewrites 39.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 39.6

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

   10. Applied rewrites 39.6%

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

Alternative 6: 74.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3.65 \cdot 10^{-57}:\\ \;\;\;\;\frac{\frac{y}{x}}{\left(x + y\right) - -1} \cdot \frac{x}{x + y}\\ \mathbf{elif}\;y \leq 3400:\\ \;\;\;\;\frac{y}{x + y} \cdot \frac{x}{\left(x - -1\right) \cdot \left(x + y\right)}\\ \mathbf{elif}\;y \leq 8.2 \cdot 10^{+82}:\\ \;\;\;\;\frac{y}{\left(\left(x - \left(-1 - y\right)\right) \cdot \left(x + y\right)\right) \cdot \left(x + y\right)} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{y} \cdot x}{x + y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -3.65e-57)
   (* (/ (/ y x) (- (+ x y) -1.0)) (/ x (+ x y)))
   (if (<= y 3400.0)
     (* (/ y (+ x y)) (/ x (* (- x -1.0) (+ x y))))
     (if (<= y 8.2e+82)
       (* (/ y (* (* (- x (- -1.0 y)) (+ x y)) (+ x y))) x)
       (/ (* (/ 1.0 y) x) (+ x y))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -3.65e-57) {
		tmp = ((y / x) / ((x + y) - -1.0)) * (x / (x + y));
	} else if (y <= 3400.0) {
		tmp = (y / (x + y)) * (x / ((x - -1.0) * (x + y)));
	} else if (y <= 8.2e+82) {
		tmp = (y / (((x - (-1.0 - y)) * (x + y)) * (x + y))) * x;
	} else {
		tmp = ((1.0 / y) * x) / (x + 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 <= (-3.65d-57)) then
        tmp = ((y / x) / ((x + y) - (-1.0d0))) * (x / (x + y))
    else if (y <= 3400.0d0) then
        tmp = (y / (x + y)) * (x / ((x - (-1.0d0)) * (x + y)))
    else if (y <= 8.2d+82) then
        tmp = (y / (((x - ((-1.0d0) - y)) * (x + y)) * (x + y))) * x
    else
        tmp = ((1.0d0 / y) * x) / (x + y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -3.65e-57) {
		tmp = ((y / x) / ((x + y) - -1.0)) * (x / (x + y));
	} else if (y <= 3400.0) {
		tmp = (y / (x + y)) * (x / ((x - -1.0) * (x + y)));
	} else if (y <= 8.2e+82) {
		tmp = (y / (((x - (-1.0 - y)) * (x + y)) * (x + y))) * x;
	} else {
		tmp = ((1.0 / y) * x) / (x + y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -3.65e-57:
		tmp = ((y / x) / ((x + y) - -1.0)) * (x / (x + y))
	elif y <= 3400.0:
		tmp = (y / (x + y)) * (x / ((x - -1.0) * (x + y)))
	elif y <= 8.2e+82:
		tmp = (y / (((x - (-1.0 - y)) * (x + y)) * (x + y))) * x
	else:
		tmp = ((1.0 / y) * x) / (x + y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -3.65e-57)
		tmp = Float64(Float64(Float64(y / x) / Float64(Float64(x + y) - -1.0)) * Float64(x / Float64(x + y)));
	elseif (y <= 3400.0)
		tmp = Float64(Float64(y / Float64(x + y)) * Float64(x / Float64(Float64(x - -1.0) * Float64(x + y))));
	elseif (y <= 8.2e+82)
		tmp = Float64(Float64(y / Float64(Float64(Float64(x - Float64(-1.0 - y)) * Float64(x + y)) * Float64(x + y))) * x);
	else
		tmp = Float64(Float64(Float64(1.0 / y) * x) / Float64(x + y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -3.65e-57)
		tmp = ((y / x) / ((x + y) - -1.0)) * (x / (x + y));
	elseif (y <= 3400.0)
		tmp = (y / (x + y)) * (x / ((x - -1.0) * (x + y)));
	elseif (y <= 8.2e+82)
		tmp = (y / (((x - (-1.0 - y)) * (x + y)) * (x + y))) * x;
	else
		tmp = ((1.0 / y) * x) / (x + y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -3.65e-57], N[(N[(N[(y / x), $MachinePrecision] / N[(N[(x + y), $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision] * N[(x / N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 3400.0], N[(N[(y / N[(x + y), $MachinePrecision]), $MachinePrecision] * N[(x / N[(N[(x - -1.0), $MachinePrecision] * N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 8.2e+82], N[(N[(y / N[(N[(N[(x - N[(-1.0 - y), $MachinePrecision]), $MachinePrecision] * N[(x + y), $MachinePrecision]), $MachinePrecision] * N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision], N[(N[(N[(1.0 / y), $MachinePrecision] * x), $MachinePrecision] / N[(x + y), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if y < -3.65000000000000011e-57

   1. Initial program 69.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 93.8

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

      16. lift-+.f64 N/A

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

      17. add-flip N/A

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

      18. lower--.f64 N/A

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

      19. lift-+.f64 N/A

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

      20. +-commutative N/A

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

      21. lower-+.f64 N/A

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

      22. metadata-eval 93.8

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

      23. lift-+.f64 N/A

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

      24. +-commutative N/A

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

      25. lower-+.f64 93.8

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

   3. Applied rewrites 93.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lift-*.f64 N/A

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

      5. times-frac N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-+.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lift-+.f64 N/A

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

      14. lower-/.f64 99.8

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

      15. lift-+.f64 N/A

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

      16. +-commutative N/A

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

      17. lift-+.f64 99.8

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

   5. Applied rewrites 99.8%

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

   6. Taylor expanded in x around inf

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

      1. lower-/.f64 49.6

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

   8. Applied rewrites 49.6%

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

   if -3.65000000000000011e-57 < y < 3400

   1. Initial program 69.5%

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

   2. Taylor expanded in y around 0

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

      1. lower-+.f64 59.1

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

   4. Applied rewrites 59.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-+.f64 N/A

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

      10. times-frac N/A

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

      11. lift-/.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lift-+.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. lift-+.f64 N/A

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

      18. +-commutative N/A

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

      19. lift-+.f64 N/A

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

   6. Applied rewrites 75.5%

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

   if 3400 < y < 8.1999999999999999e82

   1. Initial program 69.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 93.8

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

      16. lift-+.f64 N/A

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

      17. add-flip N/A

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

      18. lower--.f64 N/A

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

      19. lift-+.f64 N/A

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

      20. +-commutative N/A

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

      21. lower-+.f64 N/A

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

      22. metadata-eval 93.8

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

      23. lift-+.f64 N/A

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

      24. +-commutative N/A

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

      25. lower-+.f64 93.8

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

   3. Applied rewrites 93.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lift-*.f64 N/A

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

      5. times-frac N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-+.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lift-+.f64 N/A

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

      14. lower-/.f64 99.8

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

      15. lift-+.f64 N/A

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

      16. +-commutative N/A

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

      17. lift-+.f64 99.8

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

   5. Applied rewrites 99.8%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. lift-/.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. times-frac N/A

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

      8. lift-*.f64 N/A

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

      9. associate-/r* N/A

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

      10. lift--.f64 N/A

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

      11. metadata-eval N/A

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

      12. add-flip N/A

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

      13. lift-*.f64 N/A

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

      14. lift-+.f64 N/A

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

      15. lift-+.f64 N/A

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

      16. lift-+.f64 N/A

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

      17. associate-/l* N/A

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

   7. Applied rewrites 82.0%

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

   if 8.1999999999999999e82 < y

   1. Initial program 69.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 93.8

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

      16. lift-+.f64 N/A

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

      17. add-flip N/A

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

      18. lower--.f64 N/A

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

      19. lift-+.f64 N/A

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

      20. +-commutative N/A

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

      21. lower-+.f64 N/A

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

      22. metadata-eval 93.8

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

      23. lift-+.f64 N/A

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

      24. +-commutative N/A

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

      25. lower-+.f64 93.8

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

   3. Applied rewrites 93.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lift-*.f64 N/A

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

      5. times-frac N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-+.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lift-+.f64 N/A

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

      14. lower-/.f64 99.8

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

      15. lift-+.f64 N/A

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

      16. +-commutative N/A

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

      17. lift-+.f64 99.8

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

   5. Applied rewrites 99.8%

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

   6. Taylor expanded in y around inf

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

      1. lower-/.f64 39.7

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

   8. Applied rewrites 39.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 39.6

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

   10. Applied rewrites 39.6%

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

Alternative 7: 74.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{x + y}\\ \mathbf{if}\;y \leq -3.65 \cdot 10^{-57}:\\ \;\;\;\;\frac{\frac{y}{x}}{\left(x + y\right) - -1} \cdot t\_0\\ \mathbf{elif}\;y \leq 5200:\\ \;\;\;\;\frac{y}{x + y} \cdot \frac{x}{\left(x - -1\right) \cdot \left(x + y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + y} \cdot t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ x (+ x y))))
   (if (<= y -3.65e-57)
     (* (/ (/ y x) (- (+ x y) -1.0)) t_0)
     (if (<= y 5200.0)
       (* (/ y (+ x y)) (/ x (* (- x -1.0) (+ x y))))
       (* (/ 1.0 (+ 1.0 y)) t_0)))))

C

double code(double x, double y) {
	double t_0 = x / (x + y);
	double tmp;
	if (y <= -3.65e-57) {
		tmp = ((y / x) / ((x + y) - -1.0)) * t_0;
	} else if (y <= 5200.0) {
		tmp = (y / (x + y)) * (x / ((x - -1.0) * (x + y)));
	} else {
		tmp = (1.0 / (1.0 + y)) * 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 / (x + y)
    if (y <= (-3.65d-57)) then
        tmp = ((y / x) / ((x + y) - (-1.0d0))) * t_0
    else if (y <= 5200.0d0) then
        tmp = (y / (x + y)) * (x / ((x - (-1.0d0)) * (x + y)))
    else
        tmp = (1.0d0 / (1.0d0 + y)) * t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = x / (x + y);
	double tmp;
	if (y <= -3.65e-57) {
		tmp = ((y / x) / ((x + y) - -1.0)) * t_0;
	} else if (y <= 5200.0) {
		tmp = (y / (x + y)) * (x / ((x - -1.0) * (x + y)));
	} else {
		tmp = (1.0 / (1.0 + y)) * t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = x / (x + y)
	tmp = 0
	if y <= -3.65e-57:
		tmp = ((y / x) / ((x + y) - -1.0)) * t_0
	elif y <= 5200.0:
		tmp = (y / (x + y)) * (x / ((x - -1.0) * (x + y)))
	else:
		tmp = (1.0 / (1.0 + y)) * t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(x / Float64(x + y))
	tmp = 0.0
	if (y <= -3.65e-57)
		tmp = Float64(Float64(Float64(y / x) / Float64(Float64(x + y) - -1.0)) * t_0);
	elseif (y <= 5200.0)
		tmp = Float64(Float64(y / Float64(x + y)) * Float64(x / Float64(Float64(x - -1.0) * Float64(x + y))));
	else
		tmp = Float64(Float64(1.0 / Float64(1.0 + y)) * t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = x / (x + y);
	tmp = 0.0;
	if (y <= -3.65e-57)
		tmp = ((y / x) / ((x + y) - -1.0)) * t_0;
	elseif (y <= 5200.0)
		tmp = (y / (x + y)) * (x / ((x - -1.0) * (x + y)));
	else
		tmp = (1.0 / (1.0 + y)) * t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(x / N[(x + y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -3.65e-57], N[(N[(N[(y / x), $MachinePrecision] / N[(N[(x + y), $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision], If[LessEqual[y, 5200.0], N[(N[(y / N[(x + y), $MachinePrecision]), $MachinePrecision] * N[(x / N[(N[(x - -1.0), $MachinePrecision] * N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[(1.0 + y), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < -3.65000000000000011e-57

   1. Initial program 69.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. times-frac N/A

         \[\leadsto \color{blue}{\frac{y}{x + y} \cdot \frac{x}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      8. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{y}{x + y} \cdot \frac{x}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      9. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{y}{x + y}} \cdot \frac{x}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      10. lift-+.f64 N/A

          \[\leadsto \frac{y}{\color{blue}{x + y}} \cdot \frac{x}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      11. +-commutative N/A

          \[\leadsto \frac{y}{\color{blue}{y + x}} \cdot \frac{x}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      12. lower-+.f64 N/A

          \[\leadsto \frac{y}{\color{blue}{y + x}} \cdot \frac{x}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      13. lower-/.f64 N/A

          \[\leadsto \frac{y}{y + x} \cdot \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      14. *-commutative N/A

          \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\color{blue}{\left(\left(x + y\right) + 1\right) \cdot \left(x + y\right)}} \]

      15. lower-*.f64 93.8

          \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\color{blue}{\left(\left(x + y\right) + 1\right) \cdot \left(x + y\right)}} \]

      16. lift-+.f64 N/A

          \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\color{blue}{\left(\left(x + y\right) + 1\right)} \cdot \left(x + y\right)} \]

      17. add-flip N/A

          \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\color{blue}{\left(\left(x + y\right) - \left(\mathsf{neg}\left(1\right)\right)\right)} \cdot \left(x + y\right)} \]

      18. lower--.f64 N/A

          \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\color{blue}{\left(\left(x + y\right) - \left(\mathsf{neg}\left(1\right)\right)\right)} \cdot \left(x + y\right)} \]

      19. lift-+.f64 N/A

          \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\left(\color{blue}{\left(x + y\right)} - \left(\mathsf{neg}\left(1\right)\right)\right) \cdot \left(x + y\right)} \]

      20. +-commutative N/A

          \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\left(\color{blue}{\left(y + x\right)} - \left(\mathsf{neg}\left(1\right)\right)\right) \cdot \left(x + y\right)} \]

      21. lower-+.f64 N/A

          \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\left(\color{blue}{\left(y + x\right)} - \left(\mathsf{neg}\left(1\right)\right)\right) \cdot \left(x + y\right)} \]

      22. metadata-eval 93.8

          \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\left(\left(y + x\right) - \color{blue}{-1}\right) \cdot \left(x + y\right)} \]

      23. lift-+.f64 N/A

          \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\left(\left(y + x\right) - -1\right) \cdot \color{blue}{\left(x + y\right)}} \]

      24. +-commutative N/A

          \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\left(\left(y + x\right) - -1\right) \cdot \color{blue}{\left(y + x\right)}} \]

      25. lower-+.f64 93.8

          \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\left(\left(y + x\right) - -1\right) \cdot \color{blue}{\left(y + x\right)}} \]

   3. Applied rewrites 93.8%

      \[\leadsto \color{blue}{\frac{y}{y + x} \cdot \frac{x}{\left(\left(y + x\right) - -1\right) \cdot \left(y + x\right)}} \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{y}{y + x} \cdot \frac{x}{\left(\left(y + x\right) - -1\right) \cdot \left(y + x\right)}} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{y}{y + x} \cdot \color{blue}{\frac{x}{\left(\left(y + x\right) - -1\right) \cdot \left(y + x\right)}} \]

      3. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{\frac{y}{y + x} \cdot x}{\left(\left(y + x\right) - -1\right) \cdot \left(y + x\right)}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\frac{y}{y + x} \cdot x}{\color{blue}{\left(\left(y + x\right) - -1\right) \cdot \left(y + x\right)}} \]

      5. times-frac N/A

         \[\leadsto \color{blue}{\frac{\frac{y}{y + x}}{\left(y + x\right) - -1} \cdot \frac{x}{y + x}} \]

      6. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{y}{y + x}}{\left(y + x\right) - -1} \cdot \frac{x}{y + x}} \]

      7. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{y}{y + x}}{\left(y + x\right) - -1}} \cdot \frac{x}{y + x} \]

      8. lift-+.f64 N/A

         \[\leadsto \frac{\frac{y}{\color{blue}{y + x}}}{\left(y + x\right) - -1} \cdot \frac{x}{y + x} \]

      9. +-commutative N/A

         \[\leadsto \frac{\frac{y}{\color{blue}{x + y}}}{\left(y + x\right) - -1} \cdot \frac{x}{y + x} \]

      10. lift-+.f64 N/A

          \[\leadsto \frac{\frac{y}{\color{blue}{x + y}}}{\left(y + x\right) - -1} \cdot \frac{x}{y + x} \]

      11. lift-+.f64 N/A

          \[\leadsto \frac{\frac{y}{x + y}}{\color{blue}{\left(y + x\right)} - -1} \cdot \frac{x}{y + x} \]

      12. +-commutative N/A

          \[\leadsto \frac{\frac{y}{x + y}}{\color{blue}{\left(x + y\right)} - -1} \cdot \frac{x}{y + x} \]

      13. lift-+.f64 N/A

          \[\leadsto \frac{\frac{y}{x + y}}{\color{blue}{\left(x + y\right)} - -1} \cdot \frac{x}{y + x} \]

      14. lower-/.f64 99.8

          \[\leadsto \frac{\frac{y}{x + y}}{\left(x + y\right) - -1} \cdot \color{blue}{\frac{x}{y + x}} \]

      15. lift-+.f64 N/A

          \[\leadsto \frac{\frac{y}{x + y}}{\left(x + y\right) - -1} \cdot \frac{x}{\color{blue}{y + x}} \]

      16. +-commutative N/A

          \[\leadsto \frac{\frac{y}{x + y}}{\left(x + y\right) - -1} \cdot \frac{x}{\color{blue}{x + y}} \]

      17. lift-+.f64 99.8

          \[\leadsto \frac{\frac{y}{x + y}}{\left(x + y\right) - -1} \cdot \frac{x}{\color{blue}{x + y}} \]

   5. Applied rewrites 99.8%

      \[\leadsto \color{blue}{\frac{\frac{y}{x + y}}{\left(x + y\right) - -1} \cdot \frac{x}{x + y}} \]

   6. Taylor expanded in x around inf

      \[\leadsto \frac{\color{blue}{\frac{y}{x}}}{\left(x + y\right) - -1} \cdot \frac{x}{x + y} \]
   7. Step-by-step derivation

      1. lower-/.f64 49.6

         \[\leadsto \frac{\frac{y}{\color{blue}{x}}}{\left(x + y\right) - -1} \cdot \frac{x}{x + y} \]

   8. Applied rewrites 49.6%

      \[\leadsto \frac{\color{blue}{\frac{y}{x}}}{\left(x + y\right) - -1} \cdot \frac{x}{x + y} \]

   if -3.65000000000000011e-57 < y < 5200

   1. Initial program 69.5%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

   2. Taylor expanded in y around 0

      \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \color{blue}{\left(1 + x\right)}} \]
   3. Step-by-step derivation

      1. lower-+.f64 59.1

         \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(1 + \color{blue}{x}\right)} \]

   4. Applied rewrites 59.1%

      \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \color{blue}{\left(1 + x\right)}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(1 + x\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot y}}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(1 + x\right)} \]

      3. *-commutative N/A

         \[\leadsto \frac{\color{blue}{y \cdot x}}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(1 + x\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(1 + x\right)}} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right)} \cdot \left(1 + x\right)} \]

      6. associate-*l* N/A

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(1 + x\right)\right)}} \]

      7. lift-+.f64 N/A

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(x + y\right)} \cdot \left(\left(x + y\right) \cdot \left(1 + x\right)\right)} \]

      8. +-commutative N/A

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(y + x\right)} \cdot \left(\left(x + y\right) \cdot \left(1 + x\right)\right)} \]

      9. lift-+.f64 N/A

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(y + x\right)} \cdot \left(\left(x + y\right) \cdot \left(1 + x\right)\right)} \]

      10. times-frac N/A

          \[\leadsto \color{blue}{\frac{y}{y + x} \cdot \frac{x}{\left(x + y\right) \cdot \left(1 + x\right)}} \]

      11. lift-/.f64 N/A

          \[\leadsto \color{blue}{\frac{y}{y + x}} \cdot \frac{x}{\left(x + y\right) \cdot \left(1 + x\right)} \]

      12. lower-*.f64 N/A

          \[\leadsto \color{blue}{\frac{y}{y + x} \cdot \frac{x}{\left(x + y\right) \cdot \left(1 + x\right)}} \]

      13. lift-+.f64 N/A

          \[\leadsto \frac{y}{\color{blue}{y + x}} \cdot \frac{x}{\left(x + y\right) \cdot \left(1 + x\right)} \]

      14. +-commutative N/A

          \[\leadsto \frac{y}{\color{blue}{x + y}} \cdot \frac{x}{\left(x + y\right) \cdot \left(1 + x\right)} \]

      15. lift-+.f64 N/A

          \[\leadsto \frac{y}{\color{blue}{x + y}} \cdot \frac{x}{\left(x + y\right) \cdot \left(1 + x\right)} \]

      16. lower-/.f64 N/A

          \[\leadsto \frac{y}{x + y} \cdot \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(1 + x\right)}} \]

      17. lift-+.f64 N/A

          \[\leadsto \frac{y}{x + y} \cdot \frac{x}{\color{blue}{\left(x + y\right)} \cdot \left(1 + x\right)} \]

      18. +-commutative N/A

          \[\leadsto \frac{y}{x + y} \cdot \frac{x}{\color{blue}{\left(y + x\right)} \cdot \left(1 + x\right)} \]

      19. lift-+.f64 N/A

          \[\leadsto \frac{y}{x + y} \cdot \frac{x}{\color{blue}{\left(y + x\right)} \cdot \left(1 + x\right)} \]

   6. Applied rewrites 75.5%

      \[\leadsto \color{blue}{\frac{y}{x + y} \cdot \frac{x}{\left(x - -1\right) \cdot \left(x + y\right)}} \]

   if 5200 < y

   1. Initial program 69.5%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot y}}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      3. *-commutative N/A

         \[\leadsto \frac{\color{blue}{y \cdot x}}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right)} \cdot \left(\left(x + y\right) + 1\right)} \]

      6. associate-*l* N/A

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}} \]

      7. times-frac N/A

         \[\leadsto \color{blue}{\frac{y}{x + y} \cdot \frac{x}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      8. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{y}{x + y} \cdot \frac{x}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      9. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{y}{x + y}} \cdot \frac{x}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      10. lift-+.f64 N/A

          \[\leadsto \frac{y}{\color{blue}{x + y}} \cdot \frac{x}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      11. +-commutative N/A

          \[\leadsto \frac{y}{\color{blue}{y + x}} \cdot \frac{x}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      12. lower-+.f64 N/A

          \[\leadsto \frac{y}{\color{blue}{y + x}} \cdot \frac{x}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      13. lower-/.f64 N/A

          \[\leadsto \frac{y}{y + x} \cdot \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      14. *-commutative N/A

          \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\color{blue}{\left(\left(x + y\right) + 1\right) \cdot \left(x + y\right)}} \]

      15. lower-*.f64 93.8

          \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\color{blue}{\left(\left(x + y\right) + 1\right) \cdot \left(x + y\right)}} \]

      16. lift-+.f64 N/A

          \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\color{blue}{\left(\left(x + y\right) + 1\right)} \cdot \left(x + y\right)} \]

      17. add-flip N/A

          \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\color{blue}{\left(\left(x + y\right) - \left(\mathsf{neg}\left(1\right)\right)\right)} \cdot \left(x + y\right)} \]

      18. lower--.f64 N/A

          \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\color{blue}{\left(\left(x + y\right) - \left(\mathsf{neg}\left(1\right)\right)\right)} \cdot \left(x + y\right)} \]

      19. lift-+.f64 N/A

          \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\left(\color{blue}{\left(x + y\right)} - \left(\mathsf{neg}\left(1\right)\right)\right) \cdot \left(x + y\right)} \]

      20. +-commutative N/A

          \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\left(\color{blue}{\left(y + x\right)} - \left(\mathsf{neg}\left(1\right)\right)\right) \cdot \left(x + y\right)} \]

      21. lower-+.f64 N/A

          \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\left(\color{blue}{\left(y + x\right)} - \left(\mathsf{neg}\left(1\right)\right)\right) \cdot \left(x + y\right)} \]

      22. metadata-eval 93.8

          \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\left(\left(y + x\right) - \color{blue}{-1}\right) \cdot \left(x + y\right)} \]

      23. lift-+.f64 N/A

          \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\left(\left(y + x\right) - -1\right) \cdot \color{blue}{\left(x + y\right)}} \]

      24. +-commutative N/A

          \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\left(\left(y + x\right) - -1\right) \cdot \color{blue}{\left(y + x\right)}} \]

      25. lower-+.f64 93.8

          \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\left(\left(y + x\right) - -1\right) \cdot \color{blue}{\left(y + x\right)}} \]

   3. Applied rewrites 93.8%

      \[\leadsto \color{blue}{\frac{y}{y + x} \cdot \frac{x}{\left(\left(y + x\right) - -1\right) \cdot \left(y + x\right)}} \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{y}{y + x} \cdot \frac{x}{\left(\left(y + x\right) - -1\right) \cdot \left(y + x\right)}} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{y}{y + x} \cdot \color{blue}{\frac{x}{\left(\left(y + x\right) - -1\right) \cdot \left(y + x\right)}} \]

      3. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{\frac{y}{y + x} \cdot x}{\left(\left(y + x\right) - -1\right) \cdot \left(y + x\right)}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\frac{y}{y + x} \cdot x}{\color{blue}{\left(\left(y + x\right) - -1\right) \cdot \left(y + x\right)}} \]

      5. times-frac N/A

         \[\leadsto \color{blue}{\frac{\frac{y}{y + x}}{\left(y + x\right) - -1} \cdot \frac{x}{y + x}} \]

      6. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{y}{y + x}}{\left(y + x\right) - -1} \cdot \frac{x}{y + x}} \]

      7. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{y}{y + x}}{\left(y + x\right) - -1}} \cdot \frac{x}{y + x} \]

      8. lift-+.f64 N/A

         \[\leadsto \frac{\frac{y}{\color{blue}{y + x}}}{\left(y + x\right) - -1} \cdot \frac{x}{y + x} \]

      9. +-commutative N/A

         \[\leadsto \frac{\frac{y}{\color{blue}{x + y}}}{\left(y + x\right) - -1} \cdot \frac{x}{y + x} \]

      10. lift-+.f64 N/A

          \[\leadsto \frac{\frac{y}{\color{blue}{x + y}}}{\left(y + x\right) - -1} \cdot \frac{x}{y + x} \]

      11. lift-+.f64 N/A

          \[\leadsto \frac{\frac{y}{x + y}}{\color{blue}{\left(y + x\right)} - -1} \cdot \frac{x}{y + x} \]

      12. +-commutative N/A

          \[\leadsto \frac{\frac{y}{x + y}}{\color{blue}{\left(x + y\right)} - -1} \cdot \frac{x}{y + x} \]

      13. lift-+.f64 N/A

          \[\leadsto \frac{\frac{y}{x + y}}{\color{blue}{\left(x + y\right)} - -1} \cdot \frac{x}{y + x} \]

      14. lower-/.f64 99.8

          \[\leadsto \frac{\frac{y}{x + y}}{\left(x + y\right) - -1} \cdot \color{blue}{\frac{x}{y + x}} \]

      15. lift-+.f64 N/A

          \[\leadsto \frac{\frac{y}{x + y}}{\left(x + y\right) - -1} \cdot \frac{x}{\color{blue}{y + x}} \]

      16. +-commutative N/A

          \[\leadsto \frac{\frac{y}{x + y}}{\left(x + y\right) - -1} \cdot \frac{x}{\color{blue}{x + y}} \]

      17. lift-+.f64 99.8

          \[\leadsto \frac{\frac{y}{x + y}}{\left(x + y\right) - -1} \cdot \frac{x}{\color{blue}{x + y}} \]

   5. Applied rewrites 99.8%

      \[\leadsto \color{blue}{\frac{\frac{y}{x + y}}{\left(x + y\right) - -1} \cdot \frac{x}{x + y}} \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{1 + y}} \cdot \frac{x}{x + y} \]
   7. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{1}{\color{blue}{1 + y}} \cdot \frac{x}{x + y} \]

      2. lower-+.f64 52.2

         \[\leadsto \frac{1}{1 + \color{blue}{y}} \cdot \frac{x}{x + y} \]

   8. Applied rewrites 52.2%

      \[\leadsto \color{blue}{\frac{1}{1 + y}} \cdot \frac{x}{x + y} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 8: 72.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{x + y}\\ \mathbf{if}\;y \leq -2100000000:\\ \;\;\;\;\frac{\frac{y}{x}}{x + y} \cdot t\_0\\ \mathbf{elif}\;y \leq 5200:\\ \;\;\;\;\frac{y}{x + y} \cdot \frac{x}{\left(x - -1\right) \cdot \left(x + y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + y} \cdot t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ x (+ x y))))
   (if (<= y -2100000000.0)
     (* (/ (/ y x) (+ x y)) t_0)
     (if (<= y 5200.0)
       (* (/ y (+ x y)) (/ x (* (- x -1.0) (+ x y))))
       (* (/ 1.0 (+ 1.0 y)) t_0)))))

C

double code(double x, double y) {
	double t_0 = x / (x + y);
	double tmp;
	if (y <= -2100000000.0) {
		tmp = ((y / x) / (x + y)) * t_0;
	} else if (y <= 5200.0) {
		tmp = (y / (x + y)) * (x / ((x - -1.0) * (x + y)));
	} else {
		tmp = (1.0 / (1.0 + y)) * 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 / (x + y)
    if (y <= (-2100000000.0d0)) then
        tmp = ((y / x) / (x + y)) * t_0
    else if (y <= 5200.0d0) then
        tmp = (y / (x + y)) * (x / ((x - (-1.0d0)) * (x + y)))
    else
        tmp = (1.0d0 / (1.0d0 + y)) * t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = x / (x + y);
	double tmp;
	if (y <= -2100000000.0) {
		tmp = ((y / x) / (x + y)) * t_0;
	} else if (y <= 5200.0) {
		tmp = (y / (x + y)) * (x / ((x - -1.0) * (x + y)));
	} else {
		tmp = (1.0 / (1.0 + y)) * t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = x / (x + y)
	tmp = 0
	if y <= -2100000000.0:
		tmp = ((y / x) / (x + y)) * t_0
	elif y <= 5200.0:
		tmp = (y / (x + y)) * (x / ((x - -1.0) * (x + y)))
	else:
		tmp = (1.0 / (1.0 + y)) * t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(x / Float64(x + y))
	tmp = 0.0
	if (y <= -2100000000.0)
		tmp = Float64(Float64(Float64(y / x) / Float64(x + y)) * t_0);
	elseif (y <= 5200.0)
		tmp = Float64(Float64(y / Float64(x + y)) * Float64(x / Float64(Float64(x - -1.0) * Float64(x + y))));
	else
		tmp = Float64(Float64(1.0 / Float64(1.0 + y)) * t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = x / (x + y);
	tmp = 0.0;
	if (y <= -2100000000.0)
		tmp = ((y / x) / (x + y)) * t_0;
	elseif (y <= 5200.0)
		tmp = (y / (x + y)) * (x / ((x - -1.0) * (x + y)));
	else
		tmp = (1.0 / (1.0 + y)) * t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(x / N[(x + y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -2100000000.0], N[(N[(N[(y / x), $MachinePrecision] / N[(x + y), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision], If[LessEqual[y, 5200.0], N[(N[(y / N[(x + y), $MachinePrecision]), $MachinePrecision] * N[(x / N[(N[(x - -1.0), $MachinePrecision] * N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[(1.0 + y), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.1e9

   1. Initial program 69.5%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot y}}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      3. *-commutative N/A

         \[\leadsto \frac{\color{blue}{y \cdot x}}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right)} \cdot \left(\left(x + y\right) + 1\right)} \]

      6. associate-*l* N/A

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}} \]

      7. times-frac N/A

         \[\leadsto \color{blue}{\frac{y}{x + y} \cdot \frac{x}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      8. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{y}{x + y} \cdot \frac{x}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      9. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{y}{x + y}} \cdot \frac{x}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      10. lift-+.f64 N/A

          \[\leadsto \frac{y}{\color{blue}{x + y}} \cdot \frac{x}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      11. +-commutative N/A

          \[\leadsto \frac{y}{\color{blue}{y + x}} \cdot \frac{x}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      12. lower-+.f64 N/A

          \[\leadsto \frac{y}{\color{blue}{y + x}} \cdot \frac{x}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      13. lower-/.f64 N/A

          \[\leadsto \frac{y}{y + x} \cdot \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      14. *-commutative N/A

          \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\color{blue}{\left(\left(x + y\right) + 1\right) \cdot \left(x + y\right)}} \]

      15. lower-*.f64 93.8

          \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\color{blue}{\left(\left(x + y\right) + 1\right) \cdot \left(x + y\right)}} \]

      16. lift-+.f64 N/A

          \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\color{blue}{\left(\left(x + y\right) + 1\right)} \cdot \left(x + y\right)} \]

      17. add-flip N/A

          \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\color{blue}{\left(\left(x + y\right) - \left(\mathsf{neg}\left(1\right)\right)\right)} \cdot \left(x + y\right)} \]

      18. lower--.f64 N/A

          \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\color{blue}{\left(\left(x + y\right) - \left(\mathsf{neg}\left(1\right)\right)\right)} \cdot \left(x + y\right)} \]

      19. lift-+.f64 N/A

          \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\left(\color{blue}{\left(x + y\right)} - \left(\mathsf{neg}\left(1\right)\right)\right) \cdot \left(x + y\right)} \]

      20. +-commutative N/A

          \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\left(\color{blue}{\left(y + x\right)} - \left(\mathsf{neg}\left(1\right)\right)\right) \cdot \left(x + y\right)} \]

      21. lower-+.f64 N/A

          \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\left(\color{blue}{\left(y + x\right)} - \left(\mathsf{neg}\left(1\right)\right)\right) \cdot \left(x + y\right)} \]

      22. metadata-eval 93.8

          \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\left(\left(y + x\right) - \color{blue}{-1}\right) \cdot \left(x + y\right)} \]

      23. lift-+.f64 N/A

          \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\left(\left(y + x\right) - -1\right) \cdot \color{blue}{\left(x + y\right)}} \]

      24. +-commutative N/A

          \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\left(\left(y + x\right) - -1\right) \cdot \color{blue}{\left(y + x\right)}} \]

      25. lower-+.f64 93.8

          \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\left(\left(y + x\right) - -1\right) \cdot \color{blue}{\left(y + x\right)}} \]

   3. Applied rewrites 93.8%

      \[\leadsto \color{blue}{\frac{y}{y + x} \cdot \frac{x}{\left(\left(y + x\right) - -1\right) \cdot \left(y + x\right)}} \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{y}{y + x} \cdot \frac{x}{\left(\left(y + x\right) - -1\right) \cdot \left(y + x\right)}} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{y}{y + x} \cdot \color{blue}{\frac{x}{\left(\left(y + x\right) - -1\right) \cdot \left(y + x\right)}} \]

      3. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{\frac{y}{y + x} \cdot x}{\left(\left(y + x\right) - -1\right) \cdot \left(y + x\right)}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\frac{y}{y + x} \cdot x}{\color{blue}{\left(\left(y + x\right) - -1\right) \cdot \left(y + x\right)}} \]

      5. times-frac N/A

         \[\leadsto \color{blue}{\frac{\frac{y}{y + x}}{\left(y + x\right) - -1} \cdot \frac{x}{y + x}} \]

      6. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{y}{y + x}}{\left(y + x\right) - -1} \cdot \frac{x}{y + x}} \]

      7. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{y}{y + x}}{\left(y + x\right) - -1}} \cdot \frac{x}{y + x} \]

      8. lift-+.f64 N/A

         \[\leadsto \frac{\frac{y}{\color{blue}{y + x}}}{\left(y + x\right) - -1} \cdot \frac{x}{y + x} \]

      9. +-commutative N/A

         \[\leadsto \frac{\frac{y}{\color{blue}{x + y}}}{\left(y + x\right) - -1} \cdot \frac{x}{y + x} \]

      10. lift-+.f64 N/A

          \[\leadsto \frac{\frac{y}{\color{blue}{x + y}}}{\left(y + x\right) - -1} \cdot \frac{x}{y + x} \]

      11. lift-+.f64 N/A

          \[\leadsto \frac{\frac{y}{x + y}}{\color{blue}{\left(y + x\right)} - -1} \cdot \frac{x}{y + x} \]

      12. +-commutative N/A

          \[\leadsto \frac{\frac{y}{x + y}}{\color{blue}{\left(x + y\right)} - -1} \cdot \frac{x}{y + x} \]

      13. lift-+.f64 N/A

          \[\leadsto \frac{\frac{y}{x + y}}{\color{blue}{\left(x + y\right)} - -1} \cdot \frac{x}{y + x} \]

      14. lower-/.f64 99.8

          \[\leadsto \frac{\frac{y}{x + y}}{\left(x + y\right) - -1} \cdot \color{blue}{\frac{x}{y + x}} \]

      15. lift-+.f64 N/A

          \[\leadsto \frac{\frac{y}{x + y}}{\left(x + y\right) - -1} \cdot \frac{x}{\color{blue}{y + x}} \]

      16. +-commutative N/A

          \[\leadsto \frac{\frac{y}{x + y}}{\left(x + y\right) - -1} \cdot \frac{x}{\color{blue}{x + y}} \]

      17. lift-+.f64 99.8

          \[\leadsto \frac{\frac{y}{x + y}}{\left(x + y\right) - -1} \cdot \frac{x}{\color{blue}{x + y}} \]

   5. Applied rewrites 99.8%

      \[\leadsto \color{blue}{\frac{\frac{y}{x + y}}{\left(x + y\right) - -1} \cdot \frac{x}{x + y}} \]
   6. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{y}{x + y}}{\left(x + y\right) - -1}} \cdot \frac{x}{x + y} \]

      2. mult-flip N/A

         \[\leadsto \color{blue}{\left(\frac{y}{x + y} \cdot \frac{1}{\left(x + y\right) - -1}\right)} \cdot \frac{x}{x + y} \]

      3. lift-/.f64 N/A

         \[\leadsto \left(\color{blue}{\frac{y}{x + y}} \cdot \frac{1}{\left(x + y\right) - -1}\right) \cdot \frac{x}{x + y} \]

      4. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{y \cdot \frac{1}{\left(x + y\right) - -1}}{x + y}} \cdot \frac{x}{x + y} \]

      5. mult-flip N/A

         \[\leadsto \frac{\color{blue}{\frac{y}{\left(x + y\right) - -1}}}{x + y} \cdot \frac{x}{x + y} \]

      6. lift--.f64 N/A

         \[\leadsto \frac{\frac{y}{\color{blue}{\left(x + y\right) - -1}}}{x + y} \cdot \frac{x}{x + y} \]

      7. metadata-eval N/A

         \[\leadsto \frac{\frac{y}{\left(x + y\right) - \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}}}{x + y} \cdot \frac{x}{x + y} \]

      8. add-flip N/A

         \[\leadsto \frac{\frac{y}{\color{blue}{\left(x + y\right) + 1}}}{x + y} \cdot \frac{x}{x + y} \]

      9. lift-+.f64 N/A

         \[\leadsto \frac{\frac{y}{\color{blue}{\left(x + y\right)} + 1}}{x + y} \cdot \frac{x}{x + y} \]

      10. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{\frac{y}{\left(x + y\right) + 1}}{x + y}} \cdot \frac{x}{x + y} \]

   7. Applied rewrites 99.8%

      \[\leadsto \color{blue}{\frac{\frac{y}{x - \left(-1 - y\right)}}{x + y}} \cdot \frac{x}{x + y} \]

   8. Taylor expanded in x around inf

      \[\leadsto \frac{\color{blue}{\frac{y}{x}}}{x + y} \cdot \frac{x}{x + y} \]
   9. Step-by-step derivation

      1. lower-/.f64 37.9

         \[\leadsto \frac{\frac{y}{\color{blue}{x}}}{x + y} \cdot \frac{x}{x + y} \]

   10. Applied rewrites 37.9%

       \[\leadsto \frac{\color{blue}{\frac{y}{x}}}{x + y} \cdot \frac{x}{x + y} \]

   if -2.1e9 < y < 5200

   1. Initial program 69.5%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

   2. Taylor expanded in y around 0

      \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \color{blue}{\left(1 + x\right)}} \]
   3. Step-by-step derivation

      1. lower-+.f64 59.1

         \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(1 + \color{blue}{x}\right)} \]

   4. Applied rewrites 59.1%

      \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \color{blue}{\left(1 + x\right)}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(1 + x\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot y}}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(1 + x\right)} \]

      3. *-commutative N/A

         \[\leadsto \frac{\color{blue}{y \cdot x}}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(1 + x\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(1 + x\right)}} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right)} \cdot \left(1 + x\right)} \]

      6. associate-*l* N/A

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(1 + x\right)\right)}} \]

      7. lift-+.f64 N/A

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(x + y\right)} \cdot \left(\left(x + y\right) \cdot \left(1 + x\right)\right)} \]

      8. +-commutative N/A

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(y + x\right)} \cdot \left(\left(x + y\right) \cdot \left(1 + x\right)\right)} \]

      9. lift-+.f64 N/A

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(y + x\right)} \cdot \left(\left(x + y\right) \cdot \left(1 + x\right)\right)} \]

      10. times-frac N/A

          \[\leadsto \color{blue}{\frac{y}{y + x} \cdot \frac{x}{\left(x + y\right) \cdot \left(1 + x\right)}} \]

      11. lift-/.f64 N/A

          \[\leadsto \color{blue}{\frac{y}{y + x}} \cdot \frac{x}{\left(x + y\right) \cdot \left(1 + x\right)} \]

      12. lower-*.f64 N/A

          \[\leadsto \color{blue}{\frac{y}{y + x} \cdot \frac{x}{\left(x + y\right) \cdot \left(1 + x\right)}} \]

      13. lift-+.f64 N/A

          \[\leadsto \frac{y}{\color{blue}{y + x}} \cdot \frac{x}{\left(x + y\right) \cdot \left(1 + x\right)} \]

      14. +-commutative N/A

          \[\leadsto \frac{y}{\color{blue}{x + y}} \cdot \frac{x}{\left(x + y\right) \cdot \left(1 + x\right)} \]

      15. lift-+.f64 N/A

          \[\leadsto \frac{y}{\color{blue}{x + y}} \cdot \frac{x}{\left(x + y\right) \cdot \left(1 + x\right)} \]

      16. lower-/.f64 N/A

          \[\leadsto \frac{y}{x + y} \cdot \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(1 + x\right)}} \]

      17. lift-+.f64 N/A

          \[\leadsto \frac{y}{x + y} \cdot \frac{x}{\color{blue}{\left(x + y\right)} \cdot \left(1 + x\right)} \]

      18. +-commutative N/A

          \[\leadsto \frac{y}{x + y} \cdot \frac{x}{\color{blue}{\left(y + x\right)} \cdot \left(1 + x\right)} \]

      19. lift-+.f64 N/A

          \[\leadsto \frac{y}{x + y} \cdot \frac{x}{\color{blue}{\left(y + x\right)} \cdot \left(1 + x\right)} \]

   6. Applied rewrites 75.5%

      \[\leadsto \color{blue}{\frac{y}{x + y} \cdot \frac{x}{\left(x - -1\right) \cdot \left(x + y\right)}} \]

   if 5200 < y

   1. Initial program 69.5%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot y}}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      3. *-commutative N/A

         \[\leadsto \frac{\color{blue}{y \cdot x}}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right)} \cdot \left(\left(x + y\right) + 1\right)} \]

      6. associate-*l* N/A

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}} \]

      7. times-frac N/A

         \[\leadsto \color{blue}{\frac{y}{x + y} \cdot \frac{x}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      8. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{y}{x + y} \cdot \frac{x}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      9. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{y}{x + y}} \cdot \frac{x}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      10. lift-+.f64 N/A

          \[\leadsto \frac{y}{\color{blue}{x + y}} \cdot \frac{x}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      11. +-commutative N/A

          \[\leadsto \frac{y}{\color{blue}{y + x}} \cdot \frac{x}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      12. lower-+.f64 N/A

          \[\leadsto \frac{y}{\color{blue}{y + x}} \cdot \frac{x}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      13. lower-/.f64 N/A

          \[\leadsto \frac{y}{y + x} \cdot \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      14. *-commutative N/A

          \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\color{blue}{\left(\left(x + y\right) + 1\right) \cdot \left(x + y\right)}} \]

      15. lower-*.f64 93.8

          \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\color{blue}{\left(\left(x + y\right) + 1\right) \cdot \left(x + y\right)}} \]

      16. lift-+.f64 N/A

          \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\color{blue}{\left(\left(x + y\right) + 1\right)} \cdot \left(x + y\right)} \]

      17. add-flip N/A

          \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\color{blue}{\left(\left(x + y\right) - \left(\mathsf{neg}\left(1\right)\right)\right)} \cdot \left(x + y\right)} \]

      18. lower--.f64 N/A

          \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\color{blue}{\left(\left(x + y\right) - \left(\mathsf{neg}\left(1\right)\right)\right)} \cdot \left(x + y\right)} \]

      19. lift-+.f64 N/A

          \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\left(\color{blue}{\left(x + y\right)} - \left(\mathsf{neg}\left(1\right)\right)\right) \cdot \left(x + y\right)} \]

      20. +-commutative N/A

          \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\left(\color{blue}{\left(y + x\right)} - \left(\mathsf{neg}\left(1\right)\right)\right) \cdot \left(x + y\right)} \]

      21. lower-+.f64 N/A

          \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\left(\color{blue}{\left(y + x\right)} - \left(\mathsf{neg}\left(1\right)\right)\right) \cdot \left(x + y\right)} \]

      22. metadata-eval 93.8

          \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\left(\left(y + x\right) - \color{blue}{-1}\right) \cdot \left(x + y\right)} \]

      23. lift-+.f64 N/A

          \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\left(\left(y + x\right) - -1\right) \cdot \color{blue}{\left(x + y\right)}} \]

      24. +-commutative N/A

          \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\left(\left(y + x\right) - -1\right) \cdot \color{blue}{\left(y + x\right)}} \]

      25. lower-+.f64 93.8

          \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\left(\left(y + x\right) - -1\right) \cdot \color{blue}{\left(y + x\right)}} \]

   3. Applied rewrites 93.8%

      \[\leadsto \color{blue}{\frac{y}{y + x} \cdot \frac{x}{\left(\left(y + x\right) - -1\right) \cdot \left(y + x\right)}} \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{y}{y + x} \cdot \frac{x}{\left(\left(y + x\right) - -1\right) \cdot \left(y + x\right)}} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{y}{y + x} \cdot \color{blue}{\frac{x}{\left(\left(y + x\right) - -1\right) \cdot \left(y + x\right)}} \]

      3. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{\frac{y}{y + x} \cdot x}{\left(\left(y + x\right) - -1\right) \cdot \left(y + x\right)}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\frac{y}{y + x} \cdot x}{\color{blue}{\left(\left(y + x\right) - -1\right) \cdot \left(y + x\right)}} \]

      5. times-frac N/A

         \[\leadsto \color{blue}{\frac{\frac{y}{y + x}}{\left(y + x\right) - -1} \cdot \frac{x}{y + x}} \]

      6. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{y}{y + x}}{\left(y + x\right) - -1} \cdot \frac{x}{y + x}} \]

      7. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{y}{y + x}}{\left(y + x\right) - -1}} \cdot \frac{x}{y + x} \]

      8. lift-+.f64 N/A

         \[\leadsto \frac{\frac{y}{\color{blue}{y + x}}}{\left(y + x\right) - -1} \cdot \frac{x}{y + x} \]

      9. +-commutative N/A

         \[\leadsto \frac{\frac{y}{\color{blue}{x + y}}}{\left(y + x\right) - -1} \cdot \frac{x}{y + x} \]

      10. lift-+.f64 N/A

          \[\leadsto \frac{\frac{y}{\color{blue}{x + y}}}{\left(y + x\right) - -1} \cdot \frac{x}{y + x} \]

      11. lift-+.f64 N/A

          \[\leadsto \frac{\frac{y}{x + y}}{\color{blue}{\left(y + x\right)} - -1} \cdot \frac{x}{y + x} \]

      12. +-commutative N/A

          \[\leadsto \frac{\frac{y}{x + y}}{\color{blue}{\left(x + y\right)} - -1} \cdot \frac{x}{y + x} \]

      13. lift-+.f64 N/A

          \[\leadsto \frac{\frac{y}{x + y}}{\color{blue}{\left(x + y\right)} - -1} \cdot \frac{x}{y + x} \]

      14. lower-/.f64 99.8

          \[\leadsto \frac{\frac{y}{x + y}}{\left(x + y\right) - -1} \cdot \color{blue}{\frac{x}{y + x}} \]

      15. lift-+.f64 N/A

          \[\leadsto \frac{\frac{y}{x + y}}{\left(x + y\right) - -1} \cdot \frac{x}{\color{blue}{y + x}} \]

      16. +-commutative N/A

          \[\leadsto \frac{\frac{y}{x + y}}{\left(x + y\right) - -1} \cdot \frac{x}{\color{blue}{x + y}} \]

      17. lift-+.f64 99.8

          \[\leadsto \frac{\frac{y}{x + y}}{\left(x + y\right) - -1} \cdot \frac{x}{\color{blue}{x + y}} \]

   5. Applied rewrites 99.8%

      \[\leadsto \color{blue}{\frac{\frac{y}{x + y}}{\left(x + y\right) - -1} \cdot \frac{x}{x + y}} \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{1 + y}} \cdot \frac{x}{x + y} \]
   7. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{1}{\color{blue}{1 + y}} \cdot \frac{x}{x + y} \]

      2. lower-+.f64 52.2

         \[\leadsto \frac{1}{1 + \color{blue}{y}} \cdot \frac{x}{x + y} \]

   8. Applied rewrites 52.2%

      \[\leadsto \color{blue}{\frac{1}{1 + y}} \cdot \frac{x}{x + y} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 9: 66.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 8.5 \cdot 10^{-217}:\\ \;\;\;\;\frac{y}{y + x} \cdot \frac{1}{1 + x}\\ \mathbf{elif}\;y \leq 2.1 \cdot 10^{-19}:\\ \;\;\;\;\frac{x}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(x - -1\right)} \cdot y\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + y} \cdot \frac{x}{x + y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 8.5e-217)
   (* (/ y (+ y x)) (/ 1.0 (+ 1.0 x)))
   (if (<= y 2.1e-19)
     (* (/ x (* (* (+ x y) (+ x y)) (- x -1.0))) y)
     (* (/ 1.0 (+ 1.0 y)) (/ x (+ x y))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 8.5e-217) {
		tmp = (y / (y + x)) * (1.0 / (1.0 + x));
	} else if (y <= 2.1e-19) {
		tmp = (x / (((x + y) * (x + y)) * (x - -1.0))) * y;
	} else {
		tmp = (1.0 / (1.0 + y)) * (x / (x + 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 <= 8.5d-217) then
        tmp = (y / (y + x)) * (1.0d0 / (1.0d0 + x))
    else if (y <= 2.1d-19) then
        tmp = (x / (((x + y) * (x + y)) * (x - (-1.0d0)))) * y
    else
        tmp = (1.0d0 / (1.0d0 + y)) * (x / (x + y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 8.5e-217) {
		tmp = (y / (y + x)) * (1.0 / (1.0 + x));
	} else if (y <= 2.1e-19) {
		tmp = (x / (((x + y) * (x + y)) * (x - -1.0))) * y;
	} else {
		tmp = (1.0 / (1.0 + y)) * (x / (x + y));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 8.5e-217:
		tmp = (y / (y + x)) * (1.0 / (1.0 + x))
	elif y <= 2.1e-19:
		tmp = (x / (((x + y) * (x + y)) * (x - -1.0))) * y
	else:
		tmp = (1.0 / (1.0 + y)) * (x / (x + y))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 8.5e-217)
		tmp = Float64(Float64(y / Float64(y + x)) * Float64(1.0 / Float64(1.0 + x)));
	elseif (y <= 2.1e-19)
		tmp = Float64(Float64(x / Float64(Float64(Float64(x + y) * Float64(x + y)) * Float64(x - -1.0))) * y);
	else
		tmp = Float64(Float64(1.0 / Float64(1.0 + y)) * Float64(x / Float64(x + y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 8.5e-217)
		tmp = (y / (y + x)) * (1.0 / (1.0 + x));
	elseif (y <= 2.1e-19)
		tmp = (x / (((x + y) * (x + y)) * (x - -1.0))) * y;
	else
		tmp = (1.0 / (1.0 + y)) * (x / (x + y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 8.5e-217], N[(N[(y / N[(y + x), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2.1e-19], N[(N[(x / N[(N[(N[(x + y), $MachinePrecision] * N[(x + y), $MachinePrecision]), $MachinePrecision] * N[(x - -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision], N[(N[(1.0 / N[(1.0 + y), $MachinePrecision]), $MachinePrecision] * N[(x / N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 8.4999999999999994e-217

   1. Initial program 69.5%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot y}}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      3. *-commutative N/A

         \[\leadsto \frac{\color{blue}{y \cdot x}}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right)} \cdot \left(\left(x + y\right) + 1\right)} \]

      6. associate-*l* N/A

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}} \]

      7. times-frac N/A

         \[\leadsto \color{blue}{\frac{y}{x + y} \cdot \frac{x}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      8. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{y}{x + y} \cdot \frac{x}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      9. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{y}{x + y}} \cdot \frac{x}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      10. lift-+.f64 N/A

          \[\leadsto \frac{y}{\color{blue}{x + y}} \cdot \frac{x}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      11. +-commutative N/A

          \[\leadsto \frac{y}{\color{blue}{y + x}} \cdot \frac{x}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      12. lower-+.f64 N/A

          \[\leadsto \frac{y}{\color{blue}{y + x}} \cdot \frac{x}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      13. lower-/.f64 N/A

          \[\leadsto \frac{y}{y + x} \cdot \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      14. *-commutative N/A

          \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\color{blue}{\left(\left(x + y\right) + 1\right) \cdot \left(x + y\right)}} \]

      15. lower-*.f64 93.8

          \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\color{blue}{\left(\left(x + y\right) + 1\right) \cdot \left(x + y\right)}} \]

      16. lift-+.f64 N/A

          \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\color{blue}{\left(\left(x + y\right) + 1\right)} \cdot \left(x + y\right)} \]

      17. add-flip N/A

          \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\color{blue}{\left(\left(x + y\right) - \left(\mathsf{neg}\left(1\right)\right)\right)} \cdot \left(x + y\right)} \]

      18. lower--.f64 N/A

          \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\color{blue}{\left(\left(x + y\right) - \left(\mathsf{neg}\left(1\right)\right)\right)} \cdot \left(x + y\right)} \]

      19. lift-+.f64 N/A

          \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\left(\color{blue}{\left(x + y\right)} - \left(\mathsf{neg}\left(1\right)\right)\right) \cdot \left(x + y\right)} \]

      20. +-commutative N/A

          \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\left(\color{blue}{\left(y + x\right)} - \left(\mathsf{neg}\left(1\right)\right)\right) \cdot \left(x + y\right)} \]

      21. lower-+.f64 N/A

          \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\left(\color{blue}{\left(y + x\right)} - \left(\mathsf{neg}\left(1\right)\right)\right) \cdot \left(x + y\right)} \]

      22. metadata-eval 93.8

          \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\left(\left(y + x\right) - \color{blue}{-1}\right) \cdot \left(x + y\right)} \]

      23. lift-+.f64 N/A

          \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\left(\left(y + x\right) - -1\right) \cdot \color{blue}{\left(x + y\right)}} \]

      24. +-commutative N/A

          \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\left(\left(y + x\right) - -1\right) \cdot \color{blue}{\left(y + x\right)}} \]

      25. lower-+.f64 93.8

          \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\left(\left(y + x\right) - -1\right) \cdot \color{blue}{\left(y + x\right)}} \]

   3. Applied rewrites 93.8%

      \[\leadsto \color{blue}{\frac{y}{y + x} \cdot \frac{x}{\left(\left(y + x\right) - -1\right) \cdot \left(y + x\right)}} \]

   4. Taylor expanded in y around 0

      \[\leadsto \frac{y}{y + x} \cdot \color{blue}{\frac{1}{1 + x}} \]
   5. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{y}{y + x} \cdot \frac{1}{\color{blue}{1 + x}} \]

      2. lower-+.f64 49.6

         \[\leadsto \frac{y}{y + x} \cdot \frac{1}{1 + \color{blue}{x}} \]

   6. Applied rewrites 49.6%

      \[\leadsto \frac{y}{y + x} \cdot \color{blue}{\frac{1}{1 + x}} \]

   if 8.4999999999999994e-217 < y < 2.0999999999999999e-19

   1. Initial program 69.5%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

   2. Taylor expanded in y around 0

      \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \color{blue}{\left(1 + x\right)}} \]
   3. Step-by-step derivation

      1. lower-+.f64 59.1

         \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(1 + \color{blue}{x}\right)} \]

   4. Applied rewrites 59.1%

      \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \color{blue}{\left(1 + x\right)}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(1 + x\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot y}}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(1 + x\right)} \]

      3. *-commutative N/A

         \[\leadsto \frac{\color{blue}{y \cdot x}}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(1 + x\right)} \]

      4. associate-/l* N/A

         \[\leadsto \color{blue}{y \cdot \frac{x}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(1 + x\right)}} \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{y \cdot \frac{x}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(1 + x\right)}} \]

      6. lower-/.f64 75.9

         \[\leadsto y \cdot \color{blue}{\frac{x}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(1 + x\right)}} \]

      7. lift-*.f64 N/A

         \[\leadsto y \cdot \frac{x}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(1 + x\right)}} \]

      8. *-commutative N/A

         \[\leadsto y \cdot \frac{x}{\color{blue}{\left(1 + x\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}} \]

      9. lower-*.f64 75.9

         \[\leadsto y \cdot \frac{x}{\color{blue}{\left(1 + x\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}} \]

      10. lift-+.f64 N/A

          \[\leadsto y \cdot \frac{x}{\left(1 + \color{blue}{x}\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)} \]

      11. +-commutative N/A

          \[\leadsto y \cdot \frac{x}{\left(x + \color{blue}{1}\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)} \]

      12. add-flip-rev N/A

          \[\leadsto y \cdot \frac{x}{\left(x - \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)} \]

      13. metadata-eval N/A

          \[\leadsto y \cdot \frac{x}{\left(x - -1\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)} \]

      14. lower--.f64 75.9

          \[\leadsto y \cdot \frac{x}{\left(x - \color{blue}{-1}\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)} \]

   6. Applied rewrites 75.9%

      \[\leadsto \color{blue}{y \cdot \frac{x}{\left(x - -1\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{y \cdot \frac{x}{\left(x - -1\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\frac{x}{\left(x - -1\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)} \cdot y} \]

      3. lower-*.f64 75.9

         \[\leadsto \color{blue}{\frac{x}{\left(x - -1\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)} \cdot y} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{\left(x - -1\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}} \cdot y \]

      5. *-commutative N/A

         \[\leadsto \frac{x}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(x - -1\right)}} \cdot y \]

      6. lower-*.f64 75.9

         \[\leadsto \frac{x}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(x - -1\right)}} \cdot y \]

   8. Applied rewrites 75.9%

      \[\leadsto \color{blue}{\frac{x}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(x - -1\right)} \cdot y} \]

   if 2.0999999999999999e-19 < y

   1. Initial program 69.5%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot y}}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      3. *-commutative N/A

         \[\leadsto \frac{\color{blue}{y \cdot x}}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right)} \cdot \left(\left(x + y\right) + 1\right)} \]

      6. associate-*l* N/A

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}} \]

      7. times-frac N/A

         \[\leadsto \color{blue}{\frac{y}{x + y} \cdot \frac{x}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      8. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{y}{x + y} \cdot \frac{x}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      9. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{y}{x + y}} \cdot \frac{x}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      10. lift-+.f64 N/A

          \[\leadsto \frac{y}{\color{blue}{x + y}} \cdot \frac{x}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      11. +-commutative N/A

          \[\leadsto \frac{y}{\color{blue}{y + x}} \cdot \frac{x}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      12. lower-+.f64 N/A

          \[\leadsto \frac{y}{\color{blue}{y + x}} \cdot \frac{x}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      13. lower-/.f64 N/A

          \[\leadsto \frac{y}{y + x} \cdot \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      14. *-commutative N/A

          \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\color{blue}{\left(\left(x + y\right) + 1\right) \cdot \left(x + y\right)}} \]

      15. lower-*.f64 93.8

          \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\color{blue}{\left(\left(x + y\right) + 1\right) \cdot \left(x + y\right)}} \]

      16. lift-+.f64 N/A

          \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\color{blue}{\left(\left(x + y\right) + 1\right)} \cdot \left(x + y\right)} \]

      17. add-flip N/A

          \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\color{blue}{\left(\left(x + y\right) - \left(\mathsf{neg}\left(1\right)\right)\right)} \cdot \left(x + y\right)} \]

      18. lower--.f64 N/A

          \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\color{blue}{\left(\left(x + y\right) - \left(\mathsf{neg}\left(1\right)\right)\right)} \cdot \left(x + y\right)} \]

      19. lift-+.f64 N/A

          \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\left(\color{blue}{\left(x + y\right)} - \left(\mathsf{neg}\left(1\right)\right)\right) \cdot \left(x + y\right)} \]

      20. +-commutative N/A

          \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\left(\color{blue}{\left(y + x\right)} - \left(\mathsf{neg}\left(1\right)\right)\right) \cdot \left(x + y\right)} \]

      21. lower-+.f64 N/A

          \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\left(\color{blue}{\left(y + x\right)} - \left(\mathsf{neg}\left(1\right)\right)\right) \cdot \left(x + y\right)} \]

      22. metadata-eval 93.8

          \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\left(\left(y + x\right) - \color{blue}{-1}\right) \cdot \left(x + y\right)} \]

      23. lift-+.f64 N/A

          \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\left(\left(y + x\right) - -1\right) \cdot \color{blue}{\left(x + y\right)}} \]

      24. +-commutative N/A

          \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\left(\left(y + x\right) - -1\right) \cdot \color{blue}{\left(y + x\right)}} \]

      25. lower-+.f64 93.8

          \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\left(\left(y + x\right) - -1\right) \cdot \color{blue}{\left(y + x\right)}} \]

   3. Applied rewrites 93.8%

      \[\leadsto \color{blue}{\frac{y}{y + x} \cdot \frac{x}{\left(\left(y + x\right) - -1\right) \cdot \left(y + x\right)}} \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{y}{y + x} \cdot \frac{x}{\left(\left(y + x\right) - -1\right) \cdot \left(y + x\right)}} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{y}{y + x} \cdot \color{blue}{\frac{x}{\left(\left(y + x\right) - -1\right) \cdot \left(y + x\right)}} \]

      3. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{\frac{y}{y + x} \cdot x}{\left(\left(y + x\right) - -1\right) \cdot \left(y + x\right)}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\frac{y}{y + x} \cdot x}{\color{blue}{\left(\left(y + x\right) - -1\right) \cdot \left(y + x\right)}} \]

      5. times-frac N/A

         \[\leadsto \color{blue}{\frac{\frac{y}{y + x}}{\left(y + x\right) - -1} \cdot \frac{x}{y + x}} \]

      6. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{y}{y + x}}{\left(y + x\right) - -1} \cdot \frac{x}{y + x}} \]

      7. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{y}{y + x}}{\left(y + x\right) - -1}} \cdot \frac{x}{y + x} \]

      8. lift-+.f64 N/A

         \[\leadsto \frac{\frac{y}{\color{blue}{y + x}}}{\left(y + x\right) - -1} \cdot \frac{x}{y + x} \]

      9. +-commutative N/A

         \[\leadsto \frac{\frac{y}{\color{blue}{x + y}}}{\left(y + x\right) - -1} \cdot \frac{x}{y + x} \]

      10. lift-+.f64 N/A

          \[\leadsto \frac{\frac{y}{\color{blue}{x + y}}}{\left(y + x\right) - -1} \cdot \frac{x}{y + x} \]

      11. lift-+.f64 N/A

          \[\leadsto \frac{\frac{y}{x + y}}{\color{blue}{\left(y + x\right)} - -1} \cdot \frac{x}{y + x} \]

      12. +-commutative N/A

          \[\leadsto \frac{\frac{y}{x + y}}{\color{blue}{\left(x + y\right)} - -1} \cdot \frac{x}{y + x} \]

      13. lift-+.f64 N/A

          \[\leadsto \frac{\frac{y}{x + y}}{\color{blue}{\left(x + y\right)} - -1} \cdot \frac{x}{y + x} \]

      14. lower-/.f64 99.8

          \[\leadsto \frac{\frac{y}{x + y}}{\left(x + y\right) - -1} \cdot \color{blue}{\frac{x}{y + x}} \]

      15. lift-+.f64 N/A

          \[\leadsto \frac{\frac{y}{x + y}}{\left(x + y\right) - -1} \cdot \frac{x}{\color{blue}{y + x}} \]

      16. +-commutative N/A

          \[\leadsto \frac{\frac{y}{x + y}}{\left(x + y\right) - -1} \cdot \frac{x}{\color{blue}{x + y}} \]

      17. lift-+.f64 99.8

          \[\leadsto \frac{\frac{y}{x + y}}{\left(x + y\right) - -1} \cdot \frac{x}{\color{blue}{x + y}} \]

   5. Applied rewrites 99.8%

      \[\leadsto \color{blue}{\frac{\frac{y}{x + y}}{\left(x + y\right) - -1} \cdot \frac{x}{x + y}} \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{1 + y}} \cdot \frac{x}{x + y} \]
   7. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{1}{\color{blue}{1 + y}} \cdot \frac{x}{x + y} \]

      2. lower-+.f64 52.2

         \[\leadsto \frac{1}{1 + \color{blue}{y}} \cdot \frac{x}{x + y} \]

   8. Applied rewrites 52.2%

      \[\leadsto \color{blue}{\frac{1}{1 + y}} \cdot \frac{x}{x + y} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 10: 64.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2.3 \cdot 10^{-128}:\\ \;\;\;\;\frac{y}{y + x} \cdot \frac{1}{1 + x}\\ \mathbf{elif}\;y \leq 4.9 \cdot 10^{+81}:\\ \;\;\;\;1 \cdot \frac{x}{\left(\left(y + x\right) - -1\right) \cdot \left(y + x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{y} \cdot x}{x + y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 2.3e-128)
   (* (/ y (+ y x)) (/ 1.0 (+ 1.0 x)))
   (if (<= y 4.9e+81)
     (* 1.0 (/ x (* (- (+ y x) -1.0) (+ y x))))
     (/ (* (/ 1.0 y) x) (+ x y)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 2.3e-128) {
		tmp = (y / (y + x)) * (1.0 / (1.0 + x));
	} else if (y <= 4.9e+81) {
		tmp = 1.0 * (x / (((y + x) - -1.0) * (y + x)));
	} else {
		tmp = ((1.0 / y) * x) / (x + 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 <= 2.3d-128) then
        tmp = (y / (y + x)) * (1.0d0 / (1.0d0 + x))
    else if (y <= 4.9d+81) then
        tmp = 1.0d0 * (x / (((y + x) - (-1.0d0)) * (y + x)))
    else
        tmp = ((1.0d0 / y) * x) / (x + y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 2.3e-128) {
		tmp = (y / (y + x)) * (1.0 / (1.0 + x));
	} else if (y <= 4.9e+81) {
		tmp = 1.0 * (x / (((y + x) - -1.0) * (y + x)));
	} else {
		tmp = ((1.0 / y) * x) / (x + y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 2.3e-128:
		tmp = (y / (y + x)) * (1.0 / (1.0 + x))
	elif y <= 4.9e+81:
		tmp = 1.0 * (x / (((y + x) - -1.0) * (y + x)))
	else:
		tmp = ((1.0 / y) * x) / (x + y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 2.3e-128)
		tmp = Float64(Float64(y / Float64(y + x)) * Float64(1.0 / Float64(1.0 + x)));
	elseif (y <= 4.9e+81)
		tmp = Float64(1.0 * Float64(x / Float64(Float64(Float64(y + x) - -1.0) * Float64(y + x))));
	else
		tmp = Float64(Float64(Float64(1.0 / y) * x) / Float64(x + y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 2.3e-128)
		tmp = (y / (y + x)) * (1.0 / (1.0 + x));
	elseif (y <= 4.9e+81)
		tmp = 1.0 * (x / (((y + x) - -1.0) * (y + x)));
	else
		tmp = ((1.0 / y) * x) / (x + y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 2.3e-128], N[(N[(y / N[(y + x), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 4.9e+81], N[(1.0 * N[(x / N[(N[(N[(y + x), $MachinePrecision] - -1.0), $MachinePrecision] * N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 / y), $MachinePrecision] * x), $MachinePrecision] / N[(x + y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 2.3000000000000001e-128

   1. Initial program 69.5%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot y}}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      3. *-commutative N/A

         \[\leadsto \frac{\color{blue}{y \cdot x}}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right)} \cdot \left(\left(x + y\right) + 1\right)} \]

      6. associate-*l* N/A

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}} \]

      7. times-frac N/A

         \[\leadsto \color{blue}{\frac{y}{x + y} \cdot \frac{x}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      8. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{y}{x + y} \cdot \frac{x}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      9. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{y}{x + y}} \cdot \frac{x}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      10. lift-+.f64 N/A

          \[\leadsto \frac{y}{\color{blue}{x + y}} \cdot \frac{x}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      11. +-commutative N/A

          \[\leadsto \frac{y}{\color{blue}{y + x}} \cdot \frac{x}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      12. lower-+.f64 N/A

          \[\leadsto \frac{y}{\color{blue}{y + x}} \cdot \frac{x}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      13. lower-/.f64 N/A

          \[\leadsto \frac{y}{y + x} \cdot \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      14. *-commutative N/A

          \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\color{blue}{\left(\left(x + y\right) + 1\right) \cdot \left(x + y\right)}} \]

      15. lower-*.f64 93.8

          \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\color{blue}{\left(\left(x + y\right) + 1\right) \cdot \left(x + y\right)}} \]

      16. lift-+.f64 N/A

          \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\color{blue}{\left(\left(x + y\right) + 1\right)} \cdot \left(x + y\right)} \]

      17. add-flip N/A

          \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\color{blue}{\left(\left(x + y\right) - \left(\mathsf{neg}\left(1\right)\right)\right)} \cdot \left(x + y\right)} \]

      18. lower--.f64 N/A

          \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\color{blue}{\left(\left(x + y\right) - \left(\mathsf{neg}\left(1\right)\right)\right)} \cdot \left(x + y\right)} \]

      19. lift-+.f64 N/A

          \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\left(\color{blue}{\left(x + y\right)} - \left(\mathsf{neg}\left(1\right)\right)\right) \cdot \left(x + y\right)} \]

      20. +-commutative N/A

          \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\left(\color{blue}{\left(y + x\right)} - \left(\mathsf{neg}\left(1\right)\right)\right) \cdot \left(x + y\right)} \]

      21. lower-+.f64 N/A

          \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\left(\color{blue}{\left(y + x\right)} - \left(\mathsf{neg}\left(1\right)\right)\right) \cdot \left(x + y\right)} \]

      22. metadata-eval 93.8

          \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\left(\left(y + x\right) - \color{blue}{-1}\right) \cdot \left(x + y\right)} \]

      23. lift-+.f64 N/A

          \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\left(\left(y + x\right) - -1\right) \cdot \color{blue}{\left(x + y\right)}} \]

      24. +-commutative N/A

          \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\left(\left(y + x\right) - -1\right) \cdot \color{blue}{\left(y + x\right)}} \]

      25. lower-+.f64 93.8

          \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\left(\left(y + x\right) - -1\right) \cdot \color{blue}{\left(y + x\right)}} \]

   3. Applied rewrites 93.8%

      \[\leadsto \color{blue}{\frac{y}{y + x} \cdot \frac{x}{\left(\left(y + x\right) - -1\right) \cdot \left(y + x\right)}} \]

   4. Taylor expanded in y around 0

      \[\leadsto \frac{y}{y + x} \cdot \color{blue}{\frac{1}{1 + x}} \]
   5. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{y}{y + x} \cdot \frac{1}{\color{blue}{1 + x}} \]

      2. lower-+.f64 49.6

         \[\leadsto \frac{y}{y + x} \cdot \frac{1}{1 + \color{blue}{x}} \]

   6. Applied rewrites 49.6%

      \[\leadsto \frac{y}{y + x} \cdot \color{blue}{\frac{1}{1 + x}} \]

   if 2.3000000000000001e-128 < y < 4.90000000000000023e81

   1. Initial program 69.5%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot y}}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      3. *-commutative N/A

         \[\leadsto \frac{\color{blue}{y \cdot x}}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right)} \cdot \left(\left(x + y\right) + 1\right)} \]

      6. associate-*l* N/A

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}} \]

      7. times-frac N/A

         \[\leadsto \color{blue}{\frac{y}{x + y} \cdot \frac{x}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      8. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{y}{x + y} \cdot \frac{x}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      9. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{y}{x + y}} \cdot \frac{x}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      10. lift-+.f64 N/A

          \[\leadsto \frac{y}{\color{blue}{x + y}} \cdot \frac{x}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      11. +-commutative N/A

          \[\leadsto \frac{y}{\color{blue}{y + x}} \cdot \frac{x}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      12. lower-+.f64 N/A

          \[\leadsto \frac{y}{\color{blue}{y + x}} \cdot \frac{x}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      13. lower-/.f64 N/A

          \[\leadsto \frac{y}{y + x} \cdot \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      14. *-commutative N/A

          \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\color{blue}{\left(\left(x + y\right) + 1\right) \cdot \left(x + y\right)}} \]

      15. lower-*.f64 93.8

          \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\color{blue}{\left(\left(x + y\right) + 1\right) \cdot \left(x + y\right)}} \]

      16. lift-+.f64 N/A

          \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\color{blue}{\left(\left(x + y\right) + 1\right)} \cdot \left(x + y\right)} \]

      17. add-flip N/A

          \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\color{blue}{\left(\left(x + y\right) - \left(\mathsf{neg}\left(1\right)\right)\right)} \cdot \left(x + y\right)} \]

      18. lower--.f64 N/A

          \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\color{blue}{\left(\left(x + y\right) - \left(\mathsf{neg}\left(1\right)\right)\right)} \cdot \left(x + y\right)} \]

      19. lift-+.f64 N/A

          \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\left(\color{blue}{\left(x + y\right)} - \left(\mathsf{neg}\left(1\right)\right)\right) \cdot \left(x + y\right)} \]

      20. +-commutative N/A

          \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\left(\color{blue}{\left(y + x\right)} - \left(\mathsf{neg}\left(1\right)\right)\right) \cdot \left(x + y\right)} \]

      21. lower-+.f64 N/A

          \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\left(\color{blue}{\left(y + x\right)} - \left(\mathsf{neg}\left(1\right)\right)\right) \cdot \left(x + y\right)} \]

      22. metadata-eval 93.8

          \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\left(\left(y + x\right) - \color{blue}{-1}\right) \cdot \left(x + y\right)} \]

      23. lift-+.f64 N/A

          \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\left(\left(y + x\right) - -1\right) \cdot \color{blue}{\left(x + y\right)}} \]

      24. +-commutative N/A

          \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\left(\left(y + x\right) - -1\right) \cdot \color{blue}{\left(y + x\right)}} \]

      25. lower-+.f64 93.8

          \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\left(\left(y + x\right) - -1\right) \cdot \color{blue}{\left(y + x\right)}} \]

   3. Applied rewrites 93.8%

      \[\leadsto \color{blue}{\frac{y}{y + x} \cdot \frac{x}{\left(\left(y + x\right) - -1\right) \cdot \left(y + x\right)}} \]

   4. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1} \cdot \frac{x}{\left(\left(y + x\right) - -1\right) \cdot \left(y + x\right)} \]
   5. Step-by-step derivation

   6. Applied rewrites 66.8%

      \[\leadsto \color{blue}{1} \cdot \frac{x}{\left(\left(y + x\right) - -1\right) \cdot \left(y + x\right)} \]

   if 4.90000000000000023e81 < y

   1. Initial program 69.5%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot y}}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      3. *-commutative N/A

         \[\leadsto \frac{\color{blue}{y \cdot x}}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right)} \cdot \left(\left(x + y\right) + 1\right)} \]

      6. associate-*l* N/A

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}} \]

      7. times-frac N/A

         \[\leadsto \color{blue}{\frac{y}{x + y} \cdot \frac{x}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      8. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{y}{x + y} \cdot \frac{x}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      9. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{y}{x + y}} \cdot \frac{x}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      10. lift-+.f64 N/A

          \[\leadsto \frac{y}{\color{blue}{x + y}} \cdot \frac{x}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      11. +-commutative N/A

          \[\leadsto \frac{y}{\color{blue}{y + x}} \cdot \frac{x}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      12. lower-+.f64 N/A

          \[\leadsto \frac{y}{\color{blue}{y + x}} \cdot \frac{x}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      13. lower-/.f64 N/A

          \[\leadsto \frac{y}{y + x} \cdot \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      14. *-commutative N/A

          \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\color{blue}{\left(\left(x + y\right) + 1\right) \cdot \left(x + y\right)}} \]

      15. lower-*.f64 93.8

          \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\color{blue}{\left(\left(x + y\right) + 1\right) \cdot \left(x + y\right)}} \]

      16. lift-+.f64 N/A

          \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\color{blue}{\left(\left(x + y\right) + 1\right)} \cdot \left(x + y\right)} \]

      17. add-flip N/A

          \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\color{blue}{\left(\left(x + y\right) - \left(\mathsf{neg}\left(1\right)\right)\right)} \cdot \left(x + y\right)} \]

      18. lower--.f64 N/A

          \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\color{blue}{\left(\left(x + y\right) - \left(\mathsf{neg}\left(1\right)\right)\right)} \cdot \left(x + y\right)} \]

      19. lift-+.f64 N/A

          \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\left(\color{blue}{\left(x + y\right)} - \left(\mathsf{neg}\left(1\right)\right)\right) \cdot \left(x + y\right)} \]

      20. +-commutative N/A

          \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\left(\color{blue}{\left(y + x\right)} - \left(\mathsf{neg}\left(1\right)\right)\right) \cdot \left(x + y\right)} \]

      21. lower-+.f64 N/A

          \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\left(\color{blue}{\left(y + x\right)} - \left(\mathsf{neg}\left(1\right)\right)\right) \cdot \left(x + y\right)} \]

      22. metadata-eval 93.8

          \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\left(\left(y + x\right) - \color{blue}{-1}\right) \cdot \left(x + y\right)} \]

      23. lift-+.f64 N/A

          \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\left(\left(y + x\right) - -1\right) \cdot \color{blue}{\left(x + y\right)}} \]

      24. +-commutative N/A

          \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\left(\left(y + x\right) - -1\right) \cdot \color{blue}{\left(y + x\right)}} \]

      25. lower-+.f64 93.8

          \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\left(\left(y + x\right) - -1\right) \cdot \color{blue}{\left(y + x\right)}} \]

   3. Applied rewrites 93.8%

      \[\leadsto \color{blue}{\frac{y}{y + x} \cdot \frac{x}{\left(\left(y + x\right) - -1\right) \cdot \left(y + x\right)}} \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{y}{y + x} \cdot \frac{x}{\left(\left(y + x\right) - -1\right) \cdot \left(y + x\right)}} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{y}{y + x} \cdot \color{blue}{\frac{x}{\left(\left(y + x\right) - -1\right) \cdot \left(y + x\right)}} \]

      3. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{\frac{y}{y + x} \cdot x}{\left(\left(y + x\right) - -1\right) \cdot \left(y + x\right)}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\frac{y}{y + x} \cdot x}{\color{blue}{\left(\left(y + x\right) - -1\right) \cdot \left(y + x\right)}} \]

      5. times-frac N/A

         \[\leadsto \color{blue}{\frac{\frac{y}{y + x}}{\left(y + x\right) - -1} \cdot \frac{x}{y + x}} \]

      6. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{y}{y + x}}{\left(y + x\right) - -1} \cdot \frac{x}{y + x}} \]

      7. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{y}{y + x}}{\left(y + x\right) - -1}} \cdot \frac{x}{y + x} \]

      8. lift-+.f64 N/A

         \[\leadsto \frac{\frac{y}{\color{blue}{y + x}}}{\left(y + x\right) - -1} \cdot \frac{x}{y + x} \]

      9. +-commutative N/A

         \[\leadsto \frac{\frac{y}{\color{blue}{x + y}}}{\left(y + x\right) - -1} \cdot \frac{x}{y + x} \]

      10. lift-+.f64 N/A

          \[\leadsto \frac{\frac{y}{\color{blue}{x + y}}}{\left(y + x\right) - -1} \cdot \frac{x}{y + x} \]

      11. lift-+.f64 N/A

          \[\leadsto \frac{\frac{y}{x + y}}{\color{blue}{\left(y + x\right)} - -1} \cdot \frac{x}{y + x} \]

      12. +-commutative N/A

          \[\leadsto \frac{\frac{y}{x + y}}{\color{blue}{\left(x + y\right)} - -1} \cdot \frac{x}{y + x} \]

      13. lift-+.f64 N/A

          \[\leadsto \frac{\frac{y}{x + y}}{\color{blue}{\left(x + y\right)} - -1} \cdot \frac{x}{y + x} \]

      14. lower-/.f64 99.8

          \[\leadsto \frac{\frac{y}{x + y}}{\left(x + y\right) - -1} \cdot \color{blue}{\frac{x}{y + x}} \]

      15. lift-+.f64 N/A

          \[\leadsto \frac{\frac{y}{x + y}}{\left(x + y\right) - -1} \cdot \frac{x}{\color{blue}{y + x}} \]

      16. +-commutative N/A

          \[\leadsto \frac{\frac{y}{x + y}}{\left(x + y\right) - -1} \cdot \frac{x}{\color{blue}{x + y}} \]

      17. lift-+.f64 99.8

          \[\leadsto \frac{\frac{y}{x + y}}{\left(x + y\right) - -1} \cdot \frac{x}{\color{blue}{x + y}} \]

   5. Applied rewrites 99.8%

      \[\leadsto \color{blue}{\frac{\frac{y}{x + y}}{\left(x + y\right) - -1} \cdot \frac{x}{x + y}} \]

   6. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\frac{1}{y}} \cdot \frac{x}{x + y} \]
   7. Step-by-step derivation

      1. lower-/.f64 39.7

         \[\leadsto \frac{1}{\color{blue}{y}} \cdot \frac{x}{x + y} \]

   8. Applied rewrites 39.7%

      \[\leadsto \color{blue}{\frac{1}{y}} \cdot \frac{x}{x + y} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{y} \cdot \frac{x}{x + y}} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{1}{y} \cdot \color{blue}{\frac{x}{x + y}} \]

      3. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{\frac{1}{y} \cdot x}{x + y}} \]

      4. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{1}{y} \cdot x}{x + y}} \]

      5. lower-*.f64 39.6

         \[\leadsto \frac{\color{blue}{\frac{1}{y} \cdot x}}{x + y} \]

   10. Applied rewrites 39.6%

       \[\leadsto \color{blue}{\frac{\frac{1}{y} \cdot x}{x + y}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 11: 61.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1850000000:\\ \;\;\;\;\frac{y}{y + x} \cdot \frac{1}{x}\\ \mathbf{elif}\;y \leq 1.08 \cdot 10^{-94}:\\ \;\;\;\;\frac{y}{x \cdot \left(1 + x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + y} \cdot \frac{x}{x + y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1850000000.0)
   (* (/ y (+ y x)) (/ 1.0 x))
   (if (<= y 1.08e-94)
     (/ y (* x (+ 1.0 x)))
     (* (/ 1.0 (+ 1.0 y)) (/ x (+ x y))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1850000000.0) {
		tmp = (y / (y + x)) * (1.0 / x);
	} else if (y <= 1.08e-94) {
		tmp = y / (x * (1.0 + x));
	} else {
		tmp = (1.0 / (1.0 + y)) * (x / (x + 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 <= (-1850000000.0d0)) then
        tmp = (y / (y + x)) * (1.0d0 / x)
    else if (y <= 1.08d-94) then
        tmp = y / (x * (1.0d0 + x))
    else
        tmp = (1.0d0 / (1.0d0 + y)) * (x / (x + y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -1850000000.0) {
		tmp = (y / (y + x)) * (1.0 / x);
	} else if (y <= 1.08e-94) {
		tmp = y / (x * (1.0 + x));
	} else {
		tmp = (1.0 / (1.0 + y)) * (x / (x + y));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1850000000.0:
		tmp = (y / (y + x)) * (1.0 / x)
	elif y <= 1.08e-94:
		tmp = y / (x * (1.0 + x))
	else:
		tmp = (1.0 / (1.0 + y)) * (x / (x + y))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1850000000.0)
		tmp = Float64(Float64(y / Float64(y + x)) * Float64(1.0 / x));
	elseif (y <= 1.08e-94)
		tmp = Float64(y / Float64(x * Float64(1.0 + x)));
	else
		tmp = Float64(Float64(1.0 / Float64(1.0 + y)) * Float64(x / Float64(x + y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -1850000000.0)
		tmp = (y / (y + x)) * (1.0 / x);
	elseif (y <= 1.08e-94)
		tmp = y / (x * (1.0 + x));
	else
		tmp = (1.0 / (1.0 + y)) * (x / (x + y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1850000000.0], N[(N[(y / N[(y + x), $MachinePrecision]), $MachinePrecision] * N[(1.0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.08e-94], N[(y / N[(x * N[(1.0 + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[(1.0 + y), $MachinePrecision]), $MachinePrecision] * N[(x / N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.85e9

   1. Initial program 69.5%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot y}}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      3. *-commutative N/A

         \[\leadsto \frac{\color{blue}{y \cdot x}}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right)} \cdot \left(\left(x + y\right) + 1\right)} \]

      6. associate-*l* N/A

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}} \]

      7. times-frac N/A

         \[\leadsto \color{blue}{\frac{y}{x + y} \cdot \frac{x}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      8. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{y}{x + y} \cdot \frac{x}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      9. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{y}{x + y}} \cdot \frac{x}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      10. lift-+.f64 N/A

          \[\leadsto \frac{y}{\color{blue}{x + y}} \cdot \frac{x}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      11. +-commutative N/A

          \[\leadsto \frac{y}{\color{blue}{y + x}} \cdot \frac{x}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      12. lower-+.f64 N/A

          \[\leadsto \frac{y}{\color{blue}{y + x}} \cdot \frac{x}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      13. lower-/.f64 N/A

          \[\leadsto \frac{y}{y + x} \cdot \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      14. *-commutative N/A

          \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\color{blue}{\left(\left(x + y\right) + 1\right) \cdot \left(x + y\right)}} \]

      15. lower-*.f64 93.8

          \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\color{blue}{\left(\left(x + y\right) + 1\right) \cdot \left(x + y\right)}} \]

      16. lift-+.f64 N/A

          \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\color{blue}{\left(\left(x + y\right) + 1\right)} \cdot \left(x + y\right)} \]

      17. add-flip N/A

          \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\color{blue}{\left(\left(x + y\right) - \left(\mathsf{neg}\left(1\right)\right)\right)} \cdot \left(x + y\right)} \]

      18. lower--.f64 N/A

          \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\color{blue}{\left(\left(x + y\right) - \left(\mathsf{neg}\left(1\right)\right)\right)} \cdot \left(x + y\right)} \]

      19. lift-+.f64 N/A

          \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\left(\color{blue}{\left(x + y\right)} - \left(\mathsf{neg}\left(1\right)\right)\right) \cdot \left(x + y\right)} \]

      20. +-commutative N/A

          \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\left(\color{blue}{\left(y + x\right)} - \left(\mathsf{neg}\left(1\right)\right)\right) \cdot \left(x + y\right)} \]

      21. lower-+.f64 N/A

          \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\left(\color{blue}{\left(y + x\right)} - \left(\mathsf{neg}\left(1\right)\right)\right) \cdot \left(x + y\right)} \]

      22. metadata-eval 93.8

          \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\left(\left(y + x\right) - \color{blue}{-1}\right) \cdot \left(x + y\right)} \]

      23. lift-+.f64 N/A

          \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\left(\left(y + x\right) - -1\right) \cdot \color{blue}{\left(x + y\right)}} \]

      24. +-commutative N/A

          \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\left(\left(y + x\right) - -1\right) \cdot \color{blue}{\left(y + x\right)}} \]

      25. lower-+.f64 93.8

          \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\left(\left(y + x\right) - -1\right) \cdot \color{blue}{\left(y + x\right)}} \]

   3. Applied rewrites 93.8%

      \[\leadsto \color{blue}{\frac{y}{y + x} \cdot \frac{x}{\left(\left(y + x\right) - -1\right) \cdot \left(y + x\right)}} \]

   4. Taylor expanded in x around inf

      \[\leadsto \frac{y}{y + x} \cdot \color{blue}{\frac{1}{x}} \]
   5. Step-by-step derivation

      1. lower-/.f64 37.6

         \[\leadsto \frac{y}{y + x} \cdot \frac{1}{\color{blue}{x}} \]

   6. Applied rewrites 37.6%

      \[\leadsto \frac{y}{y + x} \cdot \color{blue}{\frac{1}{x}} \]

   if -1.85e9 < y < 1.08e-94

   1. Initial program 69.5%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

   2. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{y}{x \cdot \left(1 + x\right)}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{y}{\color{blue}{x \cdot \left(1 + x\right)}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{y}{x \cdot \color{blue}{\left(1 + x\right)}} \]

      3. lower-+.f64 47.9

         \[\leadsto \frac{y}{x \cdot \left(1 + \color{blue}{x}\right)} \]

   4. Applied rewrites 47.9%

      \[\leadsto \color{blue}{\frac{y}{x \cdot \left(1 + x\right)}} \]

   if 1.08e-94 < y

   1. Initial program 69.5%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot y}}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      3. *-commutative N/A

         \[\leadsto \frac{\color{blue}{y \cdot x}}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right)} \cdot \left(\left(x + y\right) + 1\right)} \]

      6. associate-*l* N/A

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}} \]

      7. times-frac N/A

         \[\leadsto \color{blue}{\frac{y}{x + y} \cdot \frac{x}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      8. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{y}{x + y} \cdot \frac{x}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      9. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{y}{x + y}} \cdot \frac{x}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      10. lift-+.f64 N/A

          \[\leadsto \frac{y}{\color{blue}{x + y}} \cdot \frac{x}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      11. +-commutative N/A

          \[\leadsto \frac{y}{\color{blue}{y + x}} \cdot \frac{x}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      12. lower-+.f64 N/A

          \[\leadsto \frac{y}{\color{blue}{y + x}} \cdot \frac{x}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      13. lower-/.f64 N/A

          \[\leadsto \frac{y}{y + x} \cdot \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      14. *-commutative N/A

          \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\color{blue}{\left(\left(x + y\right) + 1\right) \cdot \left(x + y\right)}} \]

      15. lower-*.f64 93.8

          \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\color{blue}{\left(\left(x + y\right) + 1\right) \cdot \left(x + y\right)}} \]

      16. lift-+.f64 N/A

          \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\color{blue}{\left(\left(x + y\right) + 1\right)} \cdot \left(x + y\right)} \]

      17. add-flip N/A

          \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\color{blue}{\left(\left(x + y\right) - \left(\mathsf{neg}\left(1\right)\right)\right)} \cdot \left(x + y\right)} \]

      18. lower--.f64 N/A

          \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\color{blue}{\left(\left(x + y\right) - \left(\mathsf{neg}\left(1\right)\right)\right)} \cdot \left(x + y\right)} \]

      19. lift-+.f64 N/A

          \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\left(\color{blue}{\left(x + y\right)} - \left(\mathsf{neg}\left(1\right)\right)\right) \cdot \left(x + y\right)} \]

      20. +-commutative N/A

          \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\left(\color{blue}{\left(y + x\right)} - \left(\mathsf{neg}\left(1\right)\right)\right) \cdot \left(x + y\right)} \]

      21. lower-+.f64 N/A

          \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\left(\color{blue}{\left(y + x\right)} - \left(\mathsf{neg}\left(1\right)\right)\right) \cdot \left(x + y\right)} \]

      22. metadata-eval 93.8

          \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\left(\left(y + x\right) - \color{blue}{-1}\right) \cdot \left(x + y\right)} \]

      23. lift-+.f64 N/A

          \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\left(\left(y + x\right) - -1\right) \cdot \color{blue}{\left(x + y\right)}} \]

      24. +-commutative N/A

          \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\left(\left(y + x\right) - -1\right) \cdot \color{blue}{\left(y + x\right)}} \]

      25. lower-+.f64 93.8

          \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\left(\left(y + x\right) - -1\right) \cdot \color{blue}{\left(y + x\right)}} \]

   3. Applied rewrites 93.8%

      \[\leadsto \color{blue}{\frac{y}{y + x} \cdot \frac{x}{\left(\left(y + x\right) - -1\right) \cdot \left(y + x\right)}} \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{y}{y + x} \cdot \frac{x}{\left(\left(y + x\right) - -1\right) \cdot \left(y + x\right)}} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{y}{y + x} \cdot \color{blue}{\frac{x}{\left(\left(y + x\right) - -1\right) \cdot \left(y + x\right)}} \]

      3. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{\frac{y}{y + x} \cdot x}{\left(\left(y + x\right) - -1\right) \cdot \left(y + x\right)}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\frac{y}{y + x} \cdot x}{\color{blue}{\left(\left(y + x\right) - -1\right) \cdot \left(y + x\right)}} \]

      5. times-frac N/A

         \[\leadsto \color{blue}{\frac{\frac{y}{y + x}}{\left(y + x\right) - -1} \cdot \frac{x}{y + x}} \]

      6. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{y}{y + x}}{\left(y + x\right) - -1} \cdot \frac{x}{y + x}} \]

      7. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{y}{y + x}}{\left(y + x\right) - -1}} \cdot \frac{x}{y + x} \]

      8. lift-+.f64 N/A

         \[\leadsto \frac{\frac{y}{\color{blue}{y + x}}}{\left(y + x\right) - -1} \cdot \frac{x}{y + x} \]

      9. +-commutative N/A

         \[\leadsto \frac{\frac{y}{\color{blue}{x + y}}}{\left(y + x\right) - -1} \cdot \frac{x}{y + x} \]

      10. lift-+.f64 N/A

          \[\leadsto \frac{\frac{y}{\color{blue}{x + y}}}{\left(y + x\right) - -1} \cdot \frac{x}{y + x} \]

      11. lift-+.f64 N/A

          \[\leadsto \frac{\frac{y}{x + y}}{\color{blue}{\left(y + x\right)} - -1} \cdot \frac{x}{y + x} \]

      12. +-commutative N/A

          \[\leadsto \frac{\frac{y}{x + y}}{\color{blue}{\left(x + y\right)} - -1} \cdot \frac{x}{y + x} \]

      13. lift-+.f64 N/A

          \[\leadsto \frac{\frac{y}{x + y}}{\color{blue}{\left(x + y\right)} - -1} \cdot \frac{x}{y + x} \]

      14. lower-/.f64 99.8

          \[\leadsto \frac{\frac{y}{x + y}}{\left(x + y\right) - -1} \cdot \color{blue}{\frac{x}{y + x}} \]

      15. lift-+.f64 N/A

          \[\leadsto \frac{\frac{y}{x + y}}{\left(x + y\right) - -1} \cdot \frac{x}{\color{blue}{y + x}} \]

      16. +-commutative N/A

          \[\leadsto \frac{\frac{y}{x + y}}{\left(x + y\right) - -1} \cdot \frac{x}{\color{blue}{x + y}} \]

      17. lift-+.f64 99.8

          \[\leadsto \frac{\frac{y}{x + y}}{\left(x + y\right) - -1} \cdot \frac{x}{\color{blue}{x + y}} \]

   5. Applied rewrites 99.8%

      \[\leadsto \color{blue}{\frac{\frac{y}{x + y}}{\left(x + y\right) - -1} \cdot \frac{x}{x + y}} \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{1 + y}} \cdot \frac{x}{x + y} \]
   7. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{1}{\color{blue}{1 + y}} \cdot \frac{x}{x + y} \]

      2. lower-+.f64 52.2

         \[\leadsto \frac{1}{1 + \color{blue}{y}} \cdot \frac{x}{x + y} \]

   8. Applied rewrites 52.2%

      \[\leadsto \color{blue}{\frac{1}{1 + y}} \cdot \frac{x}{x + y} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 12: 61.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.08 \cdot 10^{-94}:\\ \;\;\;\;\frac{y}{y + x} \cdot \frac{1}{1 + x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + y} \cdot \frac{x}{x + y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 1.08e-94)
   (* (/ y (+ y x)) (/ 1.0 (+ 1.0 x)))
   (* (/ 1.0 (+ 1.0 y)) (/ x (+ x y)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 1.08e-94) {
		tmp = (y / (y + x)) * (1.0 / (1.0 + x));
	} else {
		tmp = (1.0 / (1.0 + y)) * (x / (x + 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 <= 1.08d-94) then
        tmp = (y / (y + x)) * (1.0d0 / (1.0d0 + x))
    else
        tmp = (1.0d0 / (1.0d0 + y)) * (x / (x + y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 1.08e-94) {
		tmp = (y / (y + x)) * (1.0 / (1.0 + x));
	} else {
		tmp = (1.0 / (1.0 + y)) * (x / (x + y));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 1.08e-94:
		tmp = (y / (y + x)) * (1.0 / (1.0 + x))
	else:
		tmp = (1.0 / (1.0 + y)) * (x / (x + y))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 1.08e-94)
		tmp = Float64(Float64(y / Float64(y + x)) * Float64(1.0 / Float64(1.0 + x)));
	else
		tmp = Float64(Float64(1.0 / Float64(1.0 + y)) * Float64(x / Float64(x + y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 1.08e-94)
		tmp = (y / (y + x)) * (1.0 / (1.0 + x));
	else
		tmp = (1.0 / (1.0 + y)) * (x / (x + y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 1.08e-94], N[(N[(y / N[(y + x), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[(1.0 + y), $MachinePrecision]), $MachinePrecision] * N[(x / N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.08e-94

   1. Initial program 69.5%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot y}}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      3. *-commutative N/A

         \[\leadsto \frac{\color{blue}{y \cdot x}}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right)} \cdot \left(\left(x + y\right) + 1\right)} \]

      6. associate-*l* N/A

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}} \]

      7. times-frac N/A

         \[\leadsto \color{blue}{\frac{y}{x + y} \cdot \frac{x}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      8. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{y}{x + y} \cdot \frac{x}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      9. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{y}{x + y}} \cdot \frac{x}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      10. lift-+.f64 N/A

          \[\leadsto \frac{y}{\color{blue}{x + y}} \cdot \frac{x}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      11. +-commutative N/A

          \[\leadsto \frac{y}{\color{blue}{y + x}} \cdot \frac{x}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      12. lower-+.f64 N/A

          \[\leadsto \frac{y}{\color{blue}{y + x}} \cdot \frac{x}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      13. lower-/.f64 N/A

          \[\leadsto \frac{y}{y + x} \cdot \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      14. *-commutative N/A

          \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\color{blue}{\left(\left(x + y\right) + 1\right) \cdot \left(x + y\right)}} \]

      15. lower-*.f64 93.8

          \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\color{blue}{\left(\left(x + y\right) + 1\right) \cdot \left(x + y\right)}} \]

      16. lift-+.f64 N/A

          \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\color{blue}{\left(\left(x + y\right) + 1\right)} \cdot \left(x + y\right)} \]

      17. add-flip N/A

          \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\color{blue}{\left(\left(x + y\right) - \left(\mathsf{neg}\left(1\right)\right)\right)} \cdot \left(x + y\right)} \]

      18. lower--.f64 N/A

          \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\color{blue}{\left(\left(x + y\right) - \left(\mathsf{neg}\left(1\right)\right)\right)} \cdot \left(x + y\right)} \]

      19. lift-+.f64 N/A

          \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\left(\color{blue}{\left(x + y\right)} - \left(\mathsf{neg}\left(1\right)\right)\right) \cdot \left(x + y\right)} \]

      20. +-commutative N/A

          \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\left(\color{blue}{\left(y + x\right)} - \left(\mathsf{neg}\left(1\right)\right)\right) \cdot \left(x + y\right)} \]

      21. lower-+.f64 N/A

          \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\left(\color{blue}{\left(y + x\right)} - \left(\mathsf{neg}\left(1\right)\right)\right) \cdot \left(x + y\right)} \]

      22. metadata-eval 93.8

          \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\left(\left(y + x\right) - \color{blue}{-1}\right) \cdot \left(x + y\right)} \]

      23. lift-+.f64 N/A

          \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\left(\left(y + x\right) - -1\right) \cdot \color{blue}{\left(x + y\right)}} \]

      24. +-commutative N/A

          \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\left(\left(y + x\right) - -1\right) \cdot \color{blue}{\left(y + x\right)}} \]

      25. lower-+.f64 93.8

          \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\left(\left(y + x\right) - -1\right) \cdot \color{blue}{\left(y + x\right)}} \]

   3. Applied rewrites 93.8%

      \[\leadsto \color{blue}{\frac{y}{y + x} \cdot \frac{x}{\left(\left(y + x\right) - -1\right) \cdot \left(y + x\right)}} \]

   4. Taylor expanded in y around 0

      \[\leadsto \frac{y}{y + x} \cdot \color{blue}{\frac{1}{1 + x}} \]
   5. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{y}{y + x} \cdot \frac{1}{\color{blue}{1 + x}} \]

      2. lower-+.f64 49.6

         \[\leadsto \frac{y}{y + x} \cdot \frac{1}{1 + \color{blue}{x}} \]

   6. Applied rewrites 49.6%

      \[\leadsto \frac{y}{y + x} \cdot \color{blue}{\frac{1}{1 + x}} \]

   if 1.08e-94 < y

   1. Initial program 69.5%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot y}}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      3. *-commutative N/A

         \[\leadsto \frac{\color{blue}{y \cdot x}}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right)} \cdot \left(\left(x + y\right) + 1\right)} \]

      6. associate-*l* N/A

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}} \]

      7. times-frac N/A

         \[\leadsto \color{blue}{\frac{y}{x + y} \cdot \frac{x}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      8. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{y}{x + y} \cdot \frac{x}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      9. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{y}{x + y}} \cdot \frac{x}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      10. lift-+.f64 N/A

          \[\leadsto \frac{y}{\color{blue}{x + y}} \cdot \frac{x}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      11. +-commutative N/A

          \[\leadsto \frac{y}{\color{blue}{y + x}} \cdot \frac{x}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      12. lower-+.f64 N/A

          \[\leadsto \frac{y}{\color{blue}{y + x}} \cdot \frac{x}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      13. lower-/.f64 N/A

          \[\leadsto \frac{y}{y + x} \cdot \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      14. *-commutative N/A

          \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\color{blue}{\left(\left(x + y\right) + 1\right) \cdot \left(x + y\right)}} \]

      15. lower-*.f64 93.8

          \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\color{blue}{\left(\left(x + y\right) + 1\right) \cdot \left(x + y\right)}} \]

      16. lift-+.f64 N/A

          \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\color{blue}{\left(\left(x + y\right) + 1\right)} \cdot \left(x + y\right)} \]

      17. add-flip N/A

          \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\color{blue}{\left(\left(x + y\right) - \left(\mathsf{neg}\left(1\right)\right)\right)} \cdot \left(x + y\right)} \]

      18. lower--.f64 N/A

          \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\color{blue}{\left(\left(x + y\right) - \left(\mathsf{neg}\left(1\right)\right)\right)} \cdot \left(x + y\right)} \]

      19. lift-+.f64 N/A

          \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\left(\color{blue}{\left(x + y\right)} - \left(\mathsf{neg}\left(1\right)\right)\right) \cdot \left(x + y\right)} \]

      20. +-commutative N/A

          \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\left(\color{blue}{\left(y + x\right)} - \left(\mathsf{neg}\left(1\right)\right)\right) \cdot \left(x + y\right)} \]

      21. lower-+.f64 N/A

          \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\left(\color{blue}{\left(y + x\right)} - \left(\mathsf{neg}\left(1\right)\right)\right) \cdot \left(x + y\right)} \]

      22. metadata-eval 93.8

          \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\left(\left(y + x\right) - \color{blue}{-1}\right) \cdot \left(x + y\right)} \]

      23. lift-+.f64 N/A

          \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\left(\left(y + x\right) - -1\right) \cdot \color{blue}{\left(x + y\right)}} \]

      24. +-commutative N/A

          \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\left(\left(y + x\right) - -1\right) \cdot \color{blue}{\left(y + x\right)}} \]

      25. lower-+.f64 93.8

          \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\left(\left(y + x\right) - -1\right) \cdot \color{blue}{\left(y + x\right)}} \]

   3. Applied rewrites 93.8%

      \[\leadsto \color{blue}{\frac{y}{y + x} \cdot \frac{x}{\left(\left(y + x\right) - -1\right) \cdot \left(y + x\right)}} \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{y}{y + x} \cdot \frac{x}{\left(\left(y + x\right) - -1\right) \cdot \left(y + x\right)}} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{y}{y + x} \cdot \color{blue}{\frac{x}{\left(\left(y + x\right) - -1\right) \cdot \left(y + x\right)}} \]

      3. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{\frac{y}{y + x} \cdot x}{\left(\left(y + x\right) - -1\right) \cdot \left(y + x\right)}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\frac{y}{y + x} \cdot x}{\color{blue}{\left(\left(y + x\right) - -1\right) \cdot \left(y + x\right)}} \]

      5. times-frac N/A

         \[\leadsto \color{blue}{\frac{\frac{y}{y + x}}{\left(y + x\right) - -1} \cdot \frac{x}{y + x}} \]

      6. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{y}{y + x}}{\left(y + x\right) - -1} \cdot \frac{x}{y + x}} \]

      7. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{y}{y + x}}{\left(y + x\right) - -1}} \cdot \frac{x}{y + x} \]

      8. lift-+.f64 N/A

         \[\leadsto \frac{\frac{y}{\color{blue}{y + x}}}{\left(y + x\right) - -1} \cdot \frac{x}{y + x} \]

      9. +-commutative N/A

         \[\leadsto \frac{\frac{y}{\color{blue}{x + y}}}{\left(y + x\right) - -1} \cdot \frac{x}{y + x} \]

      10. lift-+.f64 N/A

          \[\leadsto \frac{\frac{y}{\color{blue}{x + y}}}{\left(y + x\right) - -1} \cdot \frac{x}{y + x} \]

      11. lift-+.f64 N/A

          \[\leadsto \frac{\frac{y}{x + y}}{\color{blue}{\left(y + x\right)} - -1} \cdot \frac{x}{y + x} \]

      12. +-commutative N/A

          \[\leadsto \frac{\frac{y}{x + y}}{\color{blue}{\left(x + y\right)} - -1} \cdot \frac{x}{y + x} \]

      13. lift-+.f64 N/A

          \[\leadsto \frac{\frac{y}{x + y}}{\color{blue}{\left(x + y\right)} - -1} \cdot \frac{x}{y + x} \]

      14. lower-/.f64 99.8

          \[\leadsto \frac{\frac{y}{x + y}}{\left(x + y\right) - -1} \cdot \color{blue}{\frac{x}{y + x}} \]

      15. lift-+.f64 N/A

          \[\leadsto \frac{\frac{y}{x + y}}{\left(x + y\right) - -1} \cdot \frac{x}{\color{blue}{y + x}} \]

      16. +-commutative N/A

          \[\leadsto \frac{\frac{y}{x + y}}{\left(x + y\right) - -1} \cdot \frac{x}{\color{blue}{x + y}} \]

      17. lift-+.f64 99.8

          \[\leadsto \frac{\frac{y}{x + y}}{\left(x + y\right) - -1} \cdot \frac{x}{\color{blue}{x + y}} \]

   5. Applied rewrites 99.8%

      \[\leadsto \color{blue}{\frac{\frac{y}{x + y}}{\left(x + y\right) - -1} \cdot \frac{x}{x + y}} \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{1 + y}} \cdot \frac{x}{x + y} \]
   7. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{1}{\color{blue}{1 + y}} \cdot \frac{x}{x + y} \]

      2. lower-+.f64 52.2

         \[\leadsto \frac{1}{1 + \color{blue}{y}} \cdot \frac{x}{x + y} \]

   8. Applied rewrites 52.2%

      \[\leadsto \color{blue}{\frac{1}{1 + y}} \cdot \frac{x}{x + y} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 13: 61.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1850000000:\\ \;\;\;\;\frac{y}{y + x} \cdot \frac{1}{x}\\ \mathbf{elif}\;y \leq 1.2 \cdot 10^{-99}:\\ \;\;\;\;\frac{y}{x \cdot \left(1 + x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y - -1}}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1850000000.0)
   (* (/ y (+ y x)) (/ 1.0 x))
   (if (<= y 1.2e-99) (/ y (* x (+ 1.0 x))) (/ (/ x (- y -1.0)) y))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1850000000.0) {
		tmp = (y / (y + x)) * (1.0 / x);
	} else if (y <= 1.2e-99) {
		tmp = y / (x * (1.0 + x));
	} else {
		tmp = (x / (y - -1.0)) / 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 <= (-1850000000.0d0)) then
        tmp = (y / (y + x)) * (1.0d0 / x)
    else if (y <= 1.2d-99) then
        tmp = y / (x * (1.0d0 + x))
    else
        tmp = (x / (y - (-1.0d0))) / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -1850000000.0) {
		tmp = (y / (y + x)) * (1.0 / x);
	} else if (y <= 1.2e-99) {
		tmp = y / (x * (1.0 + x));
	} else {
		tmp = (x / (y - -1.0)) / y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1850000000.0:
		tmp = (y / (y + x)) * (1.0 / x)
	elif y <= 1.2e-99:
		tmp = y / (x * (1.0 + x))
	else:
		tmp = (x / (y - -1.0)) / y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1850000000.0)
		tmp = Float64(Float64(y / Float64(y + x)) * Float64(1.0 / x));
	elseif (y <= 1.2e-99)
		tmp = Float64(y / Float64(x * Float64(1.0 + x)));
	else
		tmp = Float64(Float64(x / Float64(y - -1.0)) / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -1850000000.0)
		tmp = (y / (y + x)) * (1.0 / x);
	elseif (y <= 1.2e-99)
		tmp = y / (x * (1.0 + x));
	else
		tmp = (x / (y - -1.0)) / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1850000000.0], N[(N[(y / N[(y + x), $MachinePrecision]), $MachinePrecision] * N[(1.0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.2e-99], N[(y / N[(x * N[(1.0 + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(y - -1.0), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.85e9

   1. Initial program 69.5%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot y}}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      3. *-commutative N/A

         \[\leadsto \frac{\color{blue}{y \cdot x}}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right)} \cdot \left(\left(x + y\right) + 1\right)} \]

      6. associate-*l* N/A

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}} \]

      7. times-frac N/A

         \[\leadsto \color{blue}{\frac{y}{x + y} \cdot \frac{x}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      8. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{y}{x + y} \cdot \frac{x}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      9. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{y}{x + y}} \cdot \frac{x}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      10. lift-+.f64 N/A

          \[\leadsto \frac{y}{\color{blue}{x + y}} \cdot \frac{x}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      11. +-commutative N/A

          \[\leadsto \frac{y}{\color{blue}{y + x}} \cdot \frac{x}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      12. lower-+.f64 N/A

          \[\leadsto \frac{y}{\color{blue}{y + x}} \cdot \frac{x}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      13. lower-/.f64 N/A

          \[\leadsto \frac{y}{y + x} \cdot \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      14. *-commutative N/A

          \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\color{blue}{\left(\left(x + y\right) + 1\right) \cdot \left(x + y\right)}} \]

      15. lower-*.f64 93.8

          \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\color{blue}{\left(\left(x + y\right) + 1\right) \cdot \left(x + y\right)}} \]

      16. lift-+.f64 N/A

          \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\color{blue}{\left(\left(x + y\right) + 1\right)} \cdot \left(x + y\right)} \]

      17. add-flip N/A

          \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\color{blue}{\left(\left(x + y\right) - \left(\mathsf{neg}\left(1\right)\right)\right)} \cdot \left(x + y\right)} \]

      18. lower--.f64 N/A

          \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\color{blue}{\left(\left(x + y\right) - \left(\mathsf{neg}\left(1\right)\right)\right)} \cdot \left(x + y\right)} \]

      19. lift-+.f64 N/A

          \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\left(\color{blue}{\left(x + y\right)} - \left(\mathsf{neg}\left(1\right)\right)\right) \cdot \left(x + y\right)} \]

      20. +-commutative N/A

          \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\left(\color{blue}{\left(y + x\right)} - \left(\mathsf{neg}\left(1\right)\right)\right) \cdot \left(x + y\right)} \]

      21. lower-+.f64 N/A

          \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\left(\color{blue}{\left(y + x\right)} - \left(\mathsf{neg}\left(1\right)\right)\right) \cdot \left(x + y\right)} \]

      22. metadata-eval 93.8

          \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\left(\left(y + x\right) - \color{blue}{-1}\right) \cdot \left(x + y\right)} \]

      23. lift-+.f64 N/A

          \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\left(\left(y + x\right) - -1\right) \cdot \color{blue}{\left(x + y\right)}} \]

      24. +-commutative N/A

          \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\left(\left(y + x\right) - -1\right) \cdot \color{blue}{\left(y + x\right)}} \]

      25. lower-+.f64 93.8

          \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\left(\left(y + x\right) - -1\right) \cdot \color{blue}{\left(y + x\right)}} \]

   3. Applied rewrites 93.8%

      \[\leadsto \color{blue}{\frac{y}{y + x} \cdot \frac{x}{\left(\left(y + x\right) - -1\right) \cdot \left(y + x\right)}} \]

   4. Taylor expanded in x around inf

      \[\leadsto \frac{y}{y + x} \cdot \color{blue}{\frac{1}{x}} \]
   5. Step-by-step derivation

      1. lower-/.f64 37.6

         \[\leadsto \frac{y}{y + x} \cdot \frac{1}{\color{blue}{x}} \]

   6. Applied rewrites 37.6%

      \[\leadsto \frac{y}{y + x} \cdot \color{blue}{\frac{1}{x}} \]

   if -1.85e9 < y < 1.2e-99

   1. Initial program 69.5%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

   2. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{y}{x \cdot \left(1 + x\right)}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{y}{\color{blue}{x \cdot \left(1 + x\right)}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{y}{x \cdot \color{blue}{\left(1 + x\right)}} \]

      3. lower-+.f64 47.9

         \[\leadsto \frac{y}{x \cdot \left(1 + \color{blue}{x}\right)} \]

   4. Applied rewrites 47.9%

      \[\leadsto \color{blue}{\frac{y}{x \cdot \left(1 + x\right)}} \]

   if 1.2e-99 < y

   1. Initial program 69.5%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

   2. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{y \cdot \left(1 + y\right)}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{x}{y \cdot \color{blue}{\left(1 + y\right)}} \]

      3. lower-+.f64 50.0

         \[\leadsto \frac{x}{y \cdot \left(1 + \color{blue}{y}\right)} \]

   4. Applied rewrites 50.0%

      \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{y \cdot \left(1 + y\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{x}{y \cdot \color{blue}{\left(1 + y\right)}} \]

      3. *-commutative N/A

         \[\leadsto \frac{x}{\left(1 + y\right) \cdot \color{blue}{y}} \]

      4. associate-/r* N/A

         \[\leadsto \frac{\frac{x}{1 + y}}{\color{blue}{y}} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{\frac{x}{1 + y}}{\color{blue}{y}} \]

      6. lower-/.f64 51.7

         \[\leadsto \frac{\frac{x}{1 + y}}{y} \]

      7. lift-+.f64 N/A

         \[\leadsto \frac{\frac{x}{1 + y}}{y} \]

      8. +-commutative N/A

         \[\leadsto \frac{\frac{x}{y + 1}}{y} \]

      9. add-flip N/A

         \[\leadsto \frac{\frac{x}{y - \left(\mathsf{neg}\left(1\right)\right)}}{y} \]

      10. metadata-eval N/A

          \[\leadsto \frac{\frac{x}{y - -1}}{y} \]

      11. lower--.f64 51.7

          \[\leadsto \frac{\frac{x}{y - -1}}{y} \]

   6. Applied rewrites 51.7%

      \[\leadsto \frac{\frac{x}{y - -1}}{\color{blue}{y}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 14: 60.3% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.2 \cdot 10^{-99}:\\ \;\;\;\;\frac{y}{x \cdot \left(1 + x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y - -1}}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 1.2e-99) (/ y (* x (+ 1.0 x))) (/ (/ x (- y -1.0)) y)))

C

double code(double x, double y) {
	double tmp;
	if (y <= 1.2e-99) {
		tmp = y / (x * (1.0 + x));
	} else {
		tmp = (x / (y - -1.0)) / 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 <= 1.2d-99) then
        tmp = y / (x * (1.0d0 + x))
    else
        tmp = (x / (y - (-1.0d0))) / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 1.2e-99) {
		tmp = y / (x * (1.0 + x));
	} else {
		tmp = (x / (y - -1.0)) / y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 1.2e-99:
		tmp = y / (x * (1.0 + x))
	else:
		tmp = (x / (y - -1.0)) / y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 1.2e-99)
		tmp = Float64(y / Float64(x * Float64(1.0 + x)));
	else
		tmp = Float64(Float64(x / Float64(y - -1.0)) / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 1.2e-99)
		tmp = y / (x * (1.0 + x));
	else
		tmp = (x / (y - -1.0)) / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 1.2e-99], N[(y / N[(x * N[(1.0 + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(y - -1.0), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.2e-99

   1. Initial program 69.5%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

   2. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{y}{x \cdot \left(1 + x\right)}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{y}{\color{blue}{x \cdot \left(1 + x\right)}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{y}{x \cdot \color{blue}{\left(1 + x\right)}} \]

      3. lower-+.f64 47.9

         \[\leadsto \frac{y}{x \cdot \left(1 + \color{blue}{x}\right)} \]

   4. Applied rewrites 47.9%

      \[\leadsto \color{blue}{\frac{y}{x \cdot \left(1 + x\right)}} \]

   if 1.2e-99 < y

   1. Initial program 69.5%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

   2. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{y \cdot \left(1 + y\right)}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{x}{y \cdot \color{blue}{\left(1 + y\right)}} \]

      3. lower-+.f64 50.0

         \[\leadsto \frac{x}{y \cdot \left(1 + \color{blue}{y}\right)} \]

   4. Applied rewrites 50.0%

      \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{y \cdot \left(1 + y\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{x}{y \cdot \color{blue}{\left(1 + y\right)}} \]

      3. *-commutative N/A

         \[\leadsto \frac{x}{\left(1 + y\right) \cdot \color{blue}{y}} \]

      4. associate-/r* N/A

         \[\leadsto \frac{\frac{x}{1 + y}}{\color{blue}{y}} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{\frac{x}{1 + y}}{\color{blue}{y}} \]

      6. lower-/.f64 51.7

         \[\leadsto \frac{\frac{x}{1 + y}}{y} \]

      7. lift-+.f64 N/A

         \[\leadsto \frac{\frac{x}{1 + y}}{y} \]

      8. +-commutative N/A

         \[\leadsto \frac{\frac{x}{y + 1}}{y} \]

      9. add-flip N/A

         \[\leadsto \frac{\frac{x}{y - \left(\mathsf{neg}\left(1\right)\right)}}{y} \]

      10. metadata-eval N/A

          \[\leadsto \frac{\frac{x}{y - -1}}{y} \]

      11. lower--.f64 51.7

          \[\leadsto \frac{\frac{x}{y - -1}}{y} \]

   6. Applied rewrites 51.7%

      \[\leadsto \frac{\frac{x}{y - -1}}{\color{blue}{y}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 15: 59.4% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.2 \cdot 10^{-99}:\\ \;\;\;\;\frac{y}{x \cdot \left(1 + x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(y, y, y\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 1.2e-99) (/ y (* x (+ 1.0 x))) (/ x (fma y y y))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 1.2e-99) {
		tmp = y / (x * (1.0 + x));
	} else {
		tmp = x / fma(y, y, y);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 1.2e-99)
		tmp = Float64(y / Float64(x * Float64(1.0 + x)));
	else
		tmp = Float64(x / fma(y, y, y));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, 1.2e-99], N[(y / N[(x * N[(1.0 + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x / N[(y * y + y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.2e-99

   1. Initial program 69.5%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

   2. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{y}{x \cdot \left(1 + x\right)}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{y}{\color{blue}{x \cdot \left(1 + x\right)}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{y}{x \cdot \color{blue}{\left(1 + x\right)}} \]

      3. lower-+.f64 47.9

         \[\leadsto \frac{y}{x \cdot \left(1 + \color{blue}{x}\right)} \]

   4. Applied rewrites 47.9%

      \[\leadsto \color{blue}{\frac{y}{x \cdot \left(1 + x\right)}} \]

   if 1.2e-99 < y

   1. Initial program 69.5%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

   2. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{y \cdot \left(1 + y\right)}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{x}{y \cdot \color{blue}{\left(1 + y\right)}} \]

      3. lower-+.f64 50.0

         \[\leadsto \frac{x}{y \cdot \left(1 + \color{blue}{y}\right)} \]

   4. Applied rewrites 50.0%

      \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{x}{y \cdot \color{blue}{\left(1 + y\right)}} \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{x}{y \cdot \left(1 + \color{blue}{y}\right)} \]

      3. +-commutative N/A

         \[\leadsto \frac{x}{y \cdot \left(y + \color{blue}{1}\right)} \]

      4. distribute-rgt-in N/A

         \[\leadsto \frac{x}{y \cdot y + \color{blue}{1 \cdot y}} \]

      5. *-lft-identity N/A

         \[\leadsto \frac{x}{y \cdot y + y} \]

      6. lower-fma.f64 50.0

         \[\leadsto \frac{x}{\mathsf{fma}\left(y, \color{blue}{y}, y\right)} \]

   6. Applied rewrites 50.0%

      \[\leadsto \color{blue}{\frac{x}{\mathsf{fma}\left(y, y, y\right)}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 16: 50.0% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \frac{x}{\mathsf{fma}\left(y, y, y\right)} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (/ x (fma y y y)))

C

double code(double x, double y) {
	return x / fma(y, y, y);
}

Julia

function code(x, y)
	return Float64(x / fma(y, y, y))
end

Wolfram

code[x_, y_] := N[(x / N[(y * y + y), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 69.5%

   \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

2. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]
3. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \frac{x}{\color{blue}{y \cdot \left(1 + y\right)}} \]

   2. lower-*.f64 N/A

      \[\leadsto \frac{x}{y \cdot \color{blue}{\left(1 + y\right)}} \]

   3. lower-+.f64 50.0

      \[\leadsto \frac{x}{y \cdot \left(1 + \color{blue}{y}\right)} \]

4. Applied rewrites 50.0%

   \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]
5. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \frac{x}{y \cdot \color{blue}{\left(1 + y\right)}} \]

   2. lift-+.f64 N/A

      \[\leadsto \frac{x}{y \cdot \left(1 + \color{blue}{y}\right)} \]

   3. +-commutative N/A

      \[\leadsto \frac{x}{y \cdot \left(y + \color{blue}{1}\right)} \]

   4. distribute-rgt-in N/A

      \[\leadsto \frac{x}{y \cdot y + \color{blue}{1 \cdot y}} \]

   5. *-lft-identity N/A

      \[\leadsto \frac{x}{y \cdot y + y} \]

   6. lower-fma.f64 50.0

      \[\leadsto \frac{x}{\mathsf{fma}\left(y, \color{blue}{y}, y\right)} \]

6. Applied rewrites 50.0%

   \[\leadsto \color{blue}{\frac{x}{\mathsf{fma}\left(y, y, y\right)}} \]
7. Add Preprocessing

Alternative 17: 27.4% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\frac{y}{x}} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (/ 1.0 (/ y x)))

C

double code(double x, double y) {
	return 1.0 / (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 = 1.0d0 / (y / x)
end function

Java

public static double code(double x, double y) {
	return 1.0 / (y / x);
}

Python

def code(x, y):
	return 1.0 / (y / x)

Julia

function code(x, y)
	return Float64(1.0 / Float64(y / x))
end

MATLAB

function tmp = code(x, y)
	tmp = 1.0 / (y / x);
end

Wolfram

code[x_, y_] := N[(1.0 / N[(y / x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 69.5%

   \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

2. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]
3. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \frac{x}{\color{blue}{y \cdot \left(1 + y\right)}} \]

   2. lower-*.f64 N/A

      \[\leadsto \frac{x}{y \cdot \color{blue}{\left(1 + y\right)}} \]

   3. lower-+.f64 50.0

      \[\leadsto \frac{x}{y \cdot \left(1 + \color{blue}{y}\right)} \]

4. Applied rewrites 50.0%

   \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]

5. Taylor expanded in y around 0

   \[\leadsto \frac{x}{\color{blue}{y}} \]
6. Step-by-step derivation

   1. lower-/.f64 26.9

      \[\leadsto \frac{x}{y} \]

7. Applied rewrites 26.9%

   \[\leadsto \frac{x}{\color{blue}{y}} \]
8. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \frac{x}{y} \]

   2. div-flip N/A

      \[\leadsto \frac{1}{\frac{y}{\color{blue}{x}}} \]

   3. lower-/.f64 N/A

      \[\leadsto \frac{1}{\frac{y}{\color{blue}{x}}} \]

   4. lower-/.f64 27.4

      \[\leadsto \frac{1}{\frac{y}{x}} \]

9. Applied rewrites 27.4%

   \[\leadsto \frac{1}{\frac{y}{\color{blue}{x}}} \]
10. Add Preprocessing

Alternative 18: 26.9% accurate, 5.5× 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 69.5%

   \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

2. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]
3. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \frac{x}{\color{blue}{y \cdot \left(1 + y\right)}} \]

   2. lower-*.f64 N/A

      \[\leadsto \frac{x}{y \cdot \color{blue}{\left(1 + y\right)}} \]

   3. lower-+.f64 50.0

      \[\leadsto \frac{x}{y \cdot \left(1 + \color{blue}{y}\right)} \]

4. Applied rewrites 50.0%

   \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]

5. Taylor expanded in y around 0

   \[\leadsto \frac{x}{\color{blue}{y}} \]
6. Step-by-step derivation

   1. lower-/.f64 26.9

      \[\leadsto \frac{x}{y} \]

7. Applied rewrites 26.9%

   \[\leadsto \frac{x}{\color{blue}{y}} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2025142 
(FPCore (x y)
  :name "Numeric.SpecFunctions:incompleteBetaApprox from math-functions-0.1.5.2, A"
  :precision binary64
  (/ (* x y) (* (* (+ x y) (+ x y)) (+ (+ x y) 1.0))))