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

Percentage Accurate: 69.5% → 99.8%

Time: 4.0s

Alternatives: 17

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

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

   2. lift-/.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-+.f64 N/A

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

   6. lift-+.f64 N/A

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

   7. lift-+.f64 N/A

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

   8. lift-+.f64 N/A

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

   9. associate-/r* N/A

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

   10. lower-/.f64 N/A

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

   11. lower-/.f64 N/A

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

   12. *-commutative N/A

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

   13. lower-*.f64 N/A

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

   14. lower-*.f64 N/A

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

   15. +-commutative N/A

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

   16. lower-+.f64 N/A

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

   17. +-commutative N/A

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

   18. lower-+.f64 N/A

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

   19. lift-+.f64 N/A

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

   20. +-commutative N/A

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

   21. lower-+.f64 74.4

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

3. Applied rewrites 74.4%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-+.f64 N/A

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

   5. lift-+.f64 N/A

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

   6. times-frac N/A

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

   7. lower-*.f64 N/A

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

   8. lower-/.f64 N/A

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

   9. lift-+.f64 N/A

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

   10. lower-/.f64 N/A

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

   11. lift-+.f64 99.8

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

5. Applied rewrites 99.8%

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

Alternative 2: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 69.5%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-+.f64 N/A

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

   6. lift-+.f64 N/A

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

   7. lift-+.f64 N/A

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

   8. lift-+.f64 N/A

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

   9. times-frac N/A

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

   10. lower-*.f64 N/A

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

   11. lower-/.f64 N/A

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

   12. lower-*.f64 N/A

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

   13. +-commutative N/A

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

   14. lower-+.f64 N/A

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

   15. +-commutative N/A

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

   16. lower-+.f64 N/A

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

   17. lower-/.f64 N/A

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

   18. lift-+.f64 N/A

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

   19. +-commutative N/A

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

   20. lower-+.f64 88.1

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

3. Applied rewrites 88.1%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-+.f64 N/A

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

   4. lift-+.f64 N/A

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

   5. associate-/r* N/A

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

   6. lower-/.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. lift-+.f64 N/A

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

   9. lift-+.f64 99.8

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

5. Applied rewrites 99.8%

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

Alternative 3: 84.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(y + x\right) + 1\\ \mathbf{if}\;y \leq 2 \cdot 10^{-20}:\\ \;\;\;\;\frac{\frac{y}{y + x} \cdot \frac{x}{y + x}}{1 + x}\\ \mathbf{elif}\;y \leq 5.9 \cdot 10^{+128}:\\ \;\;\;\;\frac{x}{\left(y + x\right) \cdot \left(y + x\right)} \cdot \frac{y}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ (+ y x) 1.0)))
   (if (<= y 2e-20)
     (/ (* (/ y (+ y x)) (/ x (+ y x))) (+ 1.0 x))
     (if (<= y 5.9e+128)
       (* (/ x (* (+ y x) (+ y x))) (/ y t_0))
       (/ (/ x y) t_0)))))

C

double code(double x, double y) {
	double t_0 = (y + x) + 1.0;
	double tmp;
	if (y <= 2e-20) {
		tmp = ((y / (y + x)) * (x / (y + x))) / (1.0 + x);
	} else if (y <= 5.9e+128) {
		tmp = (x / ((y + x) * (y + x))) * (y / t_0);
	} else {
		tmp = (x / 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 = (y + x) + 1.0d0
    if (y <= 2d-20) then
        tmp = ((y / (y + x)) * (x / (y + x))) / (1.0d0 + x)
    else if (y <= 5.9d+128) then
        tmp = (x / ((y + x) * (y + x))) * (y / t_0)
    else
        tmp = (x / y) / t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = (y + x) + 1.0;
	double tmp;
	if (y <= 2e-20) {
		tmp = ((y / (y + x)) * (x / (y + x))) / (1.0 + x);
	} else if (y <= 5.9e+128) {
		tmp = (x / ((y + x) * (y + x))) * (y / t_0);
	} else {
		tmp = (x / y) / t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (y + x) + 1.0
	tmp = 0
	if y <= 2e-20:
		tmp = ((y / (y + x)) * (x / (y + x))) / (1.0 + x)
	elif y <= 5.9e+128:
		tmp = (x / ((y + x) * (y + x))) * (y / t_0)
	else:
		tmp = (x / y) / t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(y + x) + 1.0)
	tmp = 0.0
	if (y <= 2e-20)
		tmp = Float64(Float64(Float64(y / Float64(y + x)) * Float64(x / Float64(y + x))) / Float64(1.0 + x));
	elseif (y <= 5.9e+128)
		tmp = Float64(Float64(x / Float64(Float64(y + x) * Float64(y + x))) * Float64(y / t_0));
	else
		tmp = Float64(Float64(x / y) / t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (y + x) + 1.0;
	tmp = 0.0;
	if (y <= 2e-20)
		tmp = ((y / (y + x)) * (x / (y + x))) / (1.0 + x);
	elseif (y <= 5.9e+128)
		tmp = (x / ((y + x) * (y + x))) * (y / t_0);
	else
		tmp = (x / y) / t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(y + x), $MachinePrecision] + 1.0), $MachinePrecision]}, If[LessEqual[y, 2e-20], N[(N[(N[(y / N[(y + x), $MachinePrecision]), $MachinePrecision] * N[(x / N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 5.9e+128], N[(N[(x / N[(N[(y + x), $MachinePrecision] * N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(y / t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < 1.99999999999999989e-20

   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

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. associate-/r* N/A

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

      10. lower-/.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 N/A

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

      17. +-commutative N/A

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

      18. lower-+.f64 N/A

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

      19. lift-+.f64 N/A

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

      20. +-commutative N/A

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

      21. lower-+.f64 74.4

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

   3. Applied rewrites 74.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift-+.f64 99.8

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

   5. Applied rewrites 99.8%

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

   6. Taylor expanded in y around 0

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

      1. lower-+.f64 74.5

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

   8. Applied rewrites 74.5%

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

   if 1.99999999999999989e-20 < y < 5.89999999999999987e128

   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

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. times-frac N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-+.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 N/A

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

      17. lower-/.f64 N/A

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

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

      20. lower-+.f64 88.1

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

   3. Applied rewrites 88.1%

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

   if 5.89999999999999987e128 < 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

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. associate-/r* N/A

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

      10. lower-/.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 N/A

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

      17. +-commutative N/A

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

      18. lower-+.f64 N/A

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

      19. lift-+.f64 N/A

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

      20. +-commutative N/A

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

      21. lower-+.f64 74.4

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

   3. Applied rewrites 74.4%

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

   4. Taylor expanded in x around 0

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

      1. lower-/.f64 52.0

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

   6. Applied rewrites 52.0%

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

Alternative 4: 83.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(y + x\right) + 1\\ \mathbf{if}\;y \leq 3400:\\ \;\;\;\;\frac{\frac{y}{y + x} \cdot \frac{x}{y + x}}{1 + x}\\ \mathbf{elif}\;y \leq 6.5 \cdot 10^{+95}:\\ \;\;\;\;x \cdot \frac{y}{t\_0 \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ (+ y x) 1.0)))
   (if (<= y 3400.0)
     (/ (* (/ y (+ y x)) (/ x (+ y x))) (+ 1.0 x))
     (if (<= y 6.5e+95)
       (* x (/ y (* t_0 (* (+ y x) (+ y x)))))
       (/ (/ x y) t_0)))))

C

double code(double x, double y) {
	double t_0 = (y + x) + 1.0;
	double tmp;
	if (y <= 3400.0) {
		tmp = ((y / (y + x)) * (x / (y + x))) / (1.0 + x);
	} else if (y <= 6.5e+95) {
		tmp = x * (y / (t_0 * ((y + x) * (y + x))));
	} else {
		tmp = (x / 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 = (y + x) + 1.0d0
    if (y <= 3400.0d0) then
        tmp = ((y / (y + x)) * (x / (y + x))) / (1.0d0 + x)
    else if (y <= 6.5d+95) then
        tmp = x * (y / (t_0 * ((y + x) * (y + x))))
    else
        tmp = (x / y) / t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = (y + x) + 1.0;
	double tmp;
	if (y <= 3400.0) {
		tmp = ((y / (y + x)) * (x / (y + x))) / (1.0 + x);
	} else if (y <= 6.5e+95) {
		tmp = x * (y / (t_0 * ((y + x) * (y + x))));
	} else {
		tmp = (x / y) / t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (y + x) + 1.0
	tmp = 0
	if y <= 3400.0:
		tmp = ((y / (y + x)) * (x / (y + x))) / (1.0 + x)
	elif y <= 6.5e+95:
		tmp = x * (y / (t_0 * ((y + x) * (y + x))))
	else:
		tmp = (x / y) / t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(y + x) + 1.0)
	tmp = 0.0
	if (y <= 3400.0)
		tmp = Float64(Float64(Float64(y / Float64(y + x)) * Float64(x / Float64(y + x))) / Float64(1.0 + x));
	elseif (y <= 6.5e+95)
		tmp = Float64(x * Float64(y / Float64(t_0 * Float64(Float64(y + x) * Float64(y + x)))));
	else
		tmp = Float64(Float64(x / y) / t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (y + x) + 1.0;
	tmp = 0.0;
	if (y <= 3400.0)
		tmp = ((y / (y + x)) * (x / (y + x))) / (1.0 + x);
	elseif (y <= 6.5e+95)
		tmp = x * (y / (t_0 * ((y + x) * (y + x))));
	else
		tmp = (x / y) / t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(y + x), $MachinePrecision] + 1.0), $MachinePrecision]}, If[LessEqual[y, 3400.0], N[(N[(N[(y / N[(y + x), $MachinePrecision]), $MachinePrecision] * N[(x / N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 6.5e+95], N[(x * N[(y / N[(t$95$0 * N[(N[(y + x), $MachinePrecision] * N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if 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. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. associate-/r* N/A

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

      10. lower-/.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 N/A

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

      17. +-commutative N/A

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

      18. lower-+.f64 N/A

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

      19. lift-+.f64 N/A

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

      20. +-commutative N/A

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

      21. lower-+.f64 74.4

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

   3. Applied rewrites 74.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift-+.f64 99.8

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

   5. Applied rewrites 99.8%

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

   6. Taylor expanded in y around 0

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

      1. lower-+.f64 74.5

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

   8. Applied rewrites 74.5%

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

   if 3400 < y < 6.5e95

   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

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. associate-/l* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lift-+.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 N/A

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

      17. lower-*.f64 N/A

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

      18. +-commutative N/A

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

      19. lower-+.f64 N/A

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

      20. +-commutative N/A

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

      21. lower-+.f64 82.0

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

   3. Applied rewrites 82.0%

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

   if 6.5e95 < 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

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. associate-/r* N/A

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

      10. lower-/.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 N/A

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

      17. +-commutative N/A

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

      18. lower-+.f64 N/A

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

      19. lift-+.f64 N/A

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

      20. +-commutative N/A

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

      21. lower-+.f64 74.4

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

   3. Applied rewrites 74.4%

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

   4. Taylor expanded in x around 0

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

      1. lower-/.f64 52.0

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

   6. Applied rewrites 52.0%

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

Alternative 5: 80.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 6500:\\ \;\;\;\;\frac{\frac{y}{y + x} \cdot \frac{x}{y + x}}{1 + x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{\left(y + x\right) + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 6500.0)
   (/ (* (/ y (+ y x)) (/ x (+ y x))) (+ 1.0 x))
   (/ (/ x y) (+ (+ y x) 1.0))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 6500.0) {
		tmp = ((y / (y + x)) * (x / (y + x))) / (1.0 + x);
	} else {
		tmp = (x / y) / ((y + x) + 1.0);
	}
	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) :: tmp
    if (y <= 6500.0d0) then
        tmp = ((y / (y + x)) * (x / (y + x))) / (1.0d0 + x)
    else
        tmp = (x / y) / ((y + x) + 1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 6500.0) {
		tmp = ((y / (y + x)) * (x / (y + x))) / (1.0 + x);
	} else {
		tmp = (x / y) / ((y + x) + 1.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 6500.0:
		tmp = ((y / (y + x)) * (x / (y + x))) / (1.0 + x)
	else:
		tmp = (x / y) / ((y + x) + 1.0)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 6500.0)
		tmp = Float64(Float64(Float64(y / Float64(y + x)) * Float64(x / Float64(y + x))) / Float64(1.0 + x));
	else
		tmp = Float64(Float64(x / y) / Float64(Float64(y + x) + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 6500.0)
		tmp = ((y / (y + x)) * (x / (y + x))) / (1.0 + x);
	else
		tmp = (x / y) / ((y + x) + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 6500.0], N[(N[(N[(y / N[(y + x), $MachinePrecision]), $MachinePrecision] * N[(x / N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + x), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / N[(N[(y + x), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 6500

   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

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. associate-/r* N/A

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

      10. lower-/.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 N/A

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

      17. +-commutative N/A

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

      18. lower-+.f64 N/A

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

      19. lift-+.f64 N/A

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

      20. +-commutative N/A

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

      21. lower-+.f64 74.4

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

   3. Applied rewrites 74.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift-+.f64 99.8

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

   5. Applied rewrites 99.8%

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

   6. Taylor expanded in y around 0

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

      1. lower-+.f64 74.5

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

   8. Applied rewrites 74.5%

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

   if 6500 < 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

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. associate-/r* N/A

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

      10. lower-/.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 N/A

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

      17. +-commutative N/A

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

      18. lower-+.f64 N/A

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

      19. lift-+.f64 N/A

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

      20. +-commutative N/A

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

      21. lower-+.f64 74.4

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

   3. Applied rewrites 74.4%

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

   4. Taylor expanded in x around 0

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

      1. lower-/.f64 52.0

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

   6. Applied rewrites 52.0%

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

Alternative 6: 70.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -30000000000:\\ \;\;\;\;\frac{\frac{x}{x + y}}{x + y} \cdot \frac{y}{x}\\ \mathbf{elif}\;x \leq -8.5 \cdot 10^{-153}:\\ \;\;\;\;\frac{x}{\left(y + x\right) \cdot \left(y + x\right)} \cdot \frac{y}{1 + y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + x}}{\left(y + x\right) + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -30000000000.0)
   (* (/ (/ x (+ x y)) (+ x y)) (/ y x))
   (if (<= x -8.5e-153)
     (* (/ x (* (+ y x) (+ y x))) (/ y (+ 1.0 y)))
     (/ (/ x (+ y x)) (+ (+ y x) 1.0)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -30000000000.0) {
		tmp = ((x / (x + y)) / (x + y)) * (y / x);
	} else if (x <= -8.5e-153) {
		tmp = (x / ((y + x) * (y + x))) * (y / (1.0 + y));
	} else {
		tmp = (x / (y + x)) / ((y + x) + 1.0);
	}
	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) :: tmp
    if (x <= (-30000000000.0d0)) then
        tmp = ((x / (x + y)) / (x + y)) * (y / x)
    else if (x <= (-8.5d-153)) then
        tmp = (x / ((y + x) * (y + x))) * (y / (1.0d0 + y))
    else
        tmp = (x / (y + x)) / ((y + x) + 1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -30000000000.0) {
		tmp = ((x / (x + y)) / (x + y)) * (y / x);
	} else if (x <= -8.5e-153) {
		tmp = (x / ((y + x) * (y + x))) * (y / (1.0 + y));
	} else {
		tmp = (x / (y + x)) / ((y + x) + 1.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -30000000000.0:
		tmp = ((x / (x + y)) / (x + y)) * (y / x)
	elif x <= -8.5e-153:
		tmp = (x / ((y + x) * (y + x))) * (y / (1.0 + y))
	else:
		tmp = (x / (y + x)) / ((y + x) + 1.0)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -30000000000.0)
		tmp = Float64(Float64(Float64(x / Float64(x + y)) / Float64(x + y)) * Float64(y / x));
	elseif (x <= -8.5e-153)
		tmp = Float64(Float64(x / Float64(Float64(y + x) * Float64(y + x))) * Float64(y / Float64(1.0 + y)));
	else
		tmp = Float64(Float64(x / Float64(y + x)) / Float64(Float64(y + x) + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -30000000000.0)
		tmp = ((x / (x + y)) / (x + y)) * (y / x);
	elseif (x <= -8.5e-153)
		tmp = (x / ((y + x) * (y + x))) * (y / (1.0 + y));
	else
		tmp = (x / (y + x)) / ((y + x) + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -30000000000.0], N[(N[(N[(x / N[(x + y), $MachinePrecision]), $MachinePrecision] / N[(x + y), $MachinePrecision]), $MachinePrecision] * N[(y / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -8.5e-153], N[(N[(x / N[(N[(y + x), $MachinePrecision] * N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(y / N[(1.0 + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(y + x), $MachinePrecision]), $MachinePrecision] / N[(N[(y + x), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -3e10

   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

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. times-frac N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-+.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 N/A

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

      17. lower-/.f64 N/A

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

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

      20. lower-+.f64 88.1

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

   3. Applied rewrites 88.1%

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

   4. Taylor expanded in x around inf

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

      1. lower-/.f64 46.7

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

   6. Applied rewrites 46.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. associate-/r* N/A

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

      6. lower-/.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 49.0

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

   8. Applied rewrites 49.0%

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

   if -3e10 < x < -8.4999999999999996e-153

   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

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. times-frac N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-+.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 N/A

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

      17. lower-/.f64 N/A

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

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

      20. lower-+.f64 88.1

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

   3. Applied rewrites 88.1%

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

   4. Taylor expanded in x around 0

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

      1. lower-+.f64 75.8

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

   6. Applied rewrites 75.8%

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

   if -8.4999999999999996e-153 < 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

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. associate-/r* N/A

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

      10. lower-/.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 N/A

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

      17. +-commutative N/A

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

      18. lower-+.f64 N/A

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

      19. lift-+.f64 N/A

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

      20. +-commutative N/A

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

      21. lower-+.f64 74.4

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

   3. Applied rewrites 74.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift-+.f64 99.8

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

   5. Applied rewrites 99.8%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. associate-*l/ N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. lift-+.f64 99.8

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

   7. Applied rewrites 99.8%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 52.9%

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

Alternative 7: 68.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(y + x\right) + 1\\ \mathbf{if}\;y \leq 3.5 \cdot 10^{-157}:\\ \;\;\;\;\frac{\frac{y}{x}}{t\_0}\\ \mathbf{elif}\;y \leq 2.1 \cdot 10^{-19}:\\ \;\;\;\;x \cdot \frac{y}{\left(1 + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ (+ y x) 1.0)))
   (if (<= y 3.5e-157)
     (/ (/ y x) t_0)
     (if (<= y 2.1e-19)
       (* x (/ y (* (+ 1.0 x) (* (+ y x) (+ y x)))))
       (/ (/ x y) t_0)))))

C

double code(double x, double y) {
	double t_0 = (y + x) + 1.0;
	double tmp;
	if (y <= 3.5e-157) {
		tmp = (y / x) / t_0;
	} else if (y <= 2.1e-19) {
		tmp = x * (y / ((1.0 + x) * ((y + x) * (y + x))));
	} else {
		tmp = (x / 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 = (y + x) + 1.0d0
    if (y <= 3.5d-157) then
        tmp = (y / x) / t_0
    else if (y <= 2.1d-19) then
        tmp = x * (y / ((1.0d0 + x) * ((y + x) * (y + x))))
    else
        tmp = (x / y) / t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = (y + x) + 1.0;
	double tmp;
	if (y <= 3.5e-157) {
		tmp = (y / x) / t_0;
	} else if (y <= 2.1e-19) {
		tmp = x * (y / ((1.0 + x) * ((y + x) * (y + x))));
	} else {
		tmp = (x / y) / t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (y + x) + 1.0
	tmp = 0
	if y <= 3.5e-157:
		tmp = (y / x) / t_0
	elif y <= 2.1e-19:
		tmp = x * (y / ((1.0 + x) * ((y + x) * (y + x))))
	else:
		tmp = (x / y) / t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(y + x) + 1.0)
	tmp = 0.0
	if (y <= 3.5e-157)
		tmp = Float64(Float64(y / x) / t_0);
	elseif (y <= 2.1e-19)
		tmp = Float64(x * Float64(y / Float64(Float64(1.0 + x) * Float64(Float64(y + x) * Float64(y + x)))));
	else
		tmp = Float64(Float64(x / y) / t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (y + x) + 1.0;
	tmp = 0.0;
	if (y <= 3.5e-157)
		tmp = (y / x) / t_0;
	elseif (y <= 2.1e-19)
		tmp = x * (y / ((1.0 + x) * ((y + x) * (y + x))));
	else
		tmp = (x / y) / t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(y + x), $MachinePrecision] + 1.0), $MachinePrecision]}, If[LessEqual[y, 3.5e-157], N[(N[(y / x), $MachinePrecision] / t$95$0), $MachinePrecision], If[LessEqual[y, 2.1e-19], N[(x * N[(y / N[(N[(1.0 + x), $MachinePrecision] * N[(N[(y + x), $MachinePrecision] * N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < 3.5000000000000002e-157

   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

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. associate-/r* N/A

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

      10. lower-/.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 N/A

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

      17. +-commutative N/A

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

      18. lower-+.f64 N/A

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

      19. lift-+.f64 N/A

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

      20. +-commutative N/A

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

      21. lower-+.f64 74.4

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

   3. Applied rewrites 74.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift-+.f64 99.8

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

   5. Applied rewrites 99.8%

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

   6. Taylor expanded in x around inf

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

      1. lower-/.f64 49.4

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

   8. Applied rewrites 49.4%

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

   if 3.5000000000000002e-157 < 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. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. associate-/l* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lift-+.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 N/A

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

      17. lower-*.f64 N/A

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

      18. +-commutative N/A

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

      19. lower-+.f64 N/A

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

      20. +-commutative N/A

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

      21. lower-+.f64 82.0

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

   3. Applied rewrites 82.0%

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

   4. Taylor expanded in y around 0

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

      1. lower-+.f64 71.7

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

   6. Applied rewrites 71.7%

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

   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

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. associate-/r* N/A

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

      10. lower-/.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 N/A

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

      17. +-commutative N/A

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

      18. lower-+.f64 N/A

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

      19. lift-+.f64 N/A

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

      20. +-commutative N/A

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

      21. lower-+.f64 74.4

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

   3. Applied rewrites 74.4%

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

   4. Taylor expanded in x around 0

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

      1. lower-/.f64 52.0

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

   6. Applied rewrites 52.0%

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

Alternative 8: 65.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -26000000000:\\ \;\;\;\;\frac{\frac{x}{x + y}}{x + y} \cdot \frac{y}{x}\\ \mathbf{elif}\;x \leq -1.35 \cdot 10^{-152}:\\ \;\;\;\;\frac{y \cdot x}{\left(\left(y + x\right) \cdot \left(y + x\right)\right) \cdot \left(1 + y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + x}}{\left(y + x\right) + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -26000000000.0)
   (* (/ (/ x (+ x y)) (+ x y)) (/ y x))
   (if (<= x -1.35e-152)
     (/ (* y x) (* (* (+ y x) (+ y x)) (+ 1.0 y)))
     (/ (/ x (+ y x)) (+ (+ y x) 1.0)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -26000000000.0) {
		tmp = ((x / (x + y)) / (x + y)) * (y / x);
	} else if (x <= -1.35e-152) {
		tmp = (y * x) / (((y + x) * (y + x)) * (1.0 + y));
	} else {
		tmp = (x / (y + x)) / ((y + x) + 1.0);
	}
	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) :: tmp
    if (x <= (-26000000000.0d0)) then
        tmp = ((x / (x + y)) / (x + y)) * (y / x)
    else if (x <= (-1.35d-152)) then
        tmp = (y * x) / (((y + x) * (y + x)) * (1.0d0 + y))
    else
        tmp = (x / (y + x)) / ((y + x) + 1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -26000000000.0) {
		tmp = ((x / (x + y)) / (x + y)) * (y / x);
	} else if (x <= -1.35e-152) {
		tmp = (y * x) / (((y + x) * (y + x)) * (1.0 + y));
	} else {
		tmp = (x / (y + x)) / ((y + x) + 1.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -26000000000.0:
		tmp = ((x / (x + y)) / (x + y)) * (y / x)
	elif x <= -1.35e-152:
		tmp = (y * x) / (((y + x) * (y + x)) * (1.0 + y))
	else:
		tmp = (x / (y + x)) / ((y + x) + 1.0)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -26000000000.0)
		tmp = Float64(Float64(Float64(x / Float64(x + y)) / Float64(x + y)) * Float64(y / x));
	elseif (x <= -1.35e-152)
		tmp = Float64(Float64(y * x) / Float64(Float64(Float64(y + x) * Float64(y + x)) * Float64(1.0 + y)));
	else
		tmp = Float64(Float64(x / Float64(y + x)) / Float64(Float64(y + x) + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -26000000000.0)
		tmp = ((x / (x + y)) / (x + y)) * (y / x);
	elseif (x <= -1.35e-152)
		tmp = (y * x) / (((y + x) * (y + x)) * (1.0 + y));
	else
		tmp = (x / (y + x)) / ((y + x) + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -26000000000.0], N[(N[(N[(x / N[(x + y), $MachinePrecision]), $MachinePrecision] / N[(x + y), $MachinePrecision]), $MachinePrecision] * N[(y / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -1.35e-152], N[(N[(y * x), $MachinePrecision] / N[(N[(N[(y + x), $MachinePrecision] * N[(y + x), $MachinePrecision]), $MachinePrecision] * N[(1.0 + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(y + x), $MachinePrecision]), $MachinePrecision] / N[(N[(y + x), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.6e10

   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

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. times-frac N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-+.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 N/A

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

      17. lower-/.f64 N/A

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

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

      20. lower-+.f64 88.1

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

   3. Applied rewrites 88.1%

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

   4. Taylor expanded in x around inf

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

      1. lower-/.f64 46.7

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

   6. Applied rewrites 46.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. associate-/r* N/A

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

      6. lower-/.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 49.0

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

   8. Applied rewrites 49.0%

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

   if -2.6e10 < x < -1.34999999999999999e-152

   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

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. associate-/r* N/A

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

      10. lower-/.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 N/A

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

      17. +-commutative N/A

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

      18. lower-+.f64 N/A

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

      19. lift-+.f64 N/A

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

      20. +-commutative N/A

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

      21. lower-+.f64 74.4

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

   3. Applied rewrites 74.4%

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

   4. Taylor expanded in x around 0

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

      1. lower-+.f64 62.1

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

   6. Applied rewrites 62.1%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. associate-/l/ N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. lift-*.f64 59.7

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

   8. Applied rewrites 59.7%

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

   if -1.34999999999999999e-152 < 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

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. associate-/r* N/A

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

      10. lower-/.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 N/A

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

      17. +-commutative N/A

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

      18. lower-+.f64 N/A

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

      19. lift-+.f64 N/A

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

      20. +-commutative N/A

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

      21. lower-+.f64 74.4

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

   3. Applied rewrites 74.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift-+.f64 99.8

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

   5. Applied rewrites 99.8%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. associate-*l/ N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. lift-+.f64 99.8

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

   7. Applied rewrites 99.8%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 52.9%

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

Alternative 9: 63.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(y + x\right) + 1\\ \mathbf{if}\;y \leq 1.65 \cdot 10^{-129}:\\ \;\;\;\;\frac{\frac{y}{x}}{t\_0}\\ \mathbf{elif}\;y \leq 7.2 \cdot 10^{-56}:\\ \;\;\;\;\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(1 + x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ (+ y x) 1.0)))
   (if (<= y 1.65e-129)
     (/ (/ y x) t_0)
     (if (<= y 7.2e-56)
       (/ (* x y) (* (* (+ x y) (+ x y)) (+ 1.0 x)))
       (/ (/ x y) t_0)))))

C

double code(double x, double y) {
	double t_0 = (y + x) + 1.0;
	double tmp;
	if (y <= 1.65e-129) {
		tmp = (y / x) / t_0;
	} else if (y <= 7.2e-56) {
		tmp = (x * y) / (((x + y) * (x + y)) * (1.0 + x));
	} else {
		tmp = (x / 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 = (y + x) + 1.0d0
    if (y <= 1.65d-129) then
        tmp = (y / x) / t_0
    else if (y <= 7.2d-56) then
        tmp = (x * y) / (((x + y) * (x + y)) * (1.0d0 + x))
    else
        tmp = (x / y) / t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = (y + x) + 1.0;
	double tmp;
	if (y <= 1.65e-129) {
		tmp = (y / x) / t_0;
	} else if (y <= 7.2e-56) {
		tmp = (x * y) / (((x + y) * (x + y)) * (1.0 + x));
	} else {
		tmp = (x / y) / t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (y + x) + 1.0
	tmp = 0
	if y <= 1.65e-129:
		tmp = (y / x) / t_0
	elif y <= 7.2e-56:
		tmp = (x * y) / (((x + y) * (x + y)) * (1.0 + x))
	else:
		tmp = (x / y) / t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(y + x) + 1.0)
	tmp = 0.0
	if (y <= 1.65e-129)
		tmp = Float64(Float64(y / x) / t_0);
	elseif (y <= 7.2e-56)
		tmp = Float64(Float64(x * y) / Float64(Float64(Float64(x + y) * Float64(x + y)) * Float64(1.0 + x)));
	else
		tmp = Float64(Float64(x / y) / t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (y + x) + 1.0;
	tmp = 0.0;
	if (y <= 1.65e-129)
		tmp = (y / x) / t_0;
	elseif (y <= 7.2e-56)
		tmp = (x * y) / (((x + y) * (x + y)) * (1.0 + x));
	else
		tmp = (x / y) / t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(y + x), $MachinePrecision] + 1.0), $MachinePrecision]}, If[LessEqual[y, 1.65e-129], N[(N[(y / x), $MachinePrecision] / t$95$0), $MachinePrecision], If[LessEqual[y, 7.2e-56], N[(N[(x * y), $MachinePrecision] / N[(N[(N[(x + y), $MachinePrecision] * N[(x + y), $MachinePrecision]), $MachinePrecision] * N[(1.0 + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < 1.64999999999999994e-129

   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

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. associate-/r* N/A

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

      10. lower-/.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 N/A

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

      17. +-commutative N/A

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

      18. lower-+.f64 N/A

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

      19. lift-+.f64 N/A

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

      20. +-commutative N/A

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

      21. lower-+.f64 74.4

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

   3. Applied rewrites 74.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift-+.f64 99.8

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

   5. Applied rewrites 99.8%

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

   6. Taylor expanded in x around inf

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

      1. lower-/.f64 49.4

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

   8. Applied rewrites 49.4%

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

   if 1.64999999999999994e-129 < y < 7.19999999999999956e-56

   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%

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

   if 7.19999999999999956e-56 < 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

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. associate-/r* N/A

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

      10. lower-/.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 N/A

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

      17. +-commutative N/A

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

      18. lower-+.f64 N/A

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

      19. lift-+.f64 N/A

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

      20. +-commutative N/A

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

      21. lower-+.f64 74.4

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

   3. Applied rewrites 74.4%

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

   4. Taylor expanded in x around 0

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

      1. lower-/.f64 52.0

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

   6. Applied rewrites 52.0%

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

Alternative 10: 62.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(y + x\right) + 1\\ \mathbf{if}\;y \leq 3.6 \cdot 10^{-100}:\\ \;\;\;\;\frac{\frac{y}{x}}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ (+ y x) 1.0)))
   (if (<= y 3.6e-100) (/ (/ y x) t_0) (/ (/ x y) t_0))))

C

double code(double x, double y) {
	double t_0 = (y + x) + 1.0;
	double tmp;
	if (y <= 3.6e-100) {
		tmp = (y / x) / t_0;
	} else {
		tmp = (x / 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 = (y + x) + 1.0d0
    if (y <= 3.6d-100) then
        tmp = (y / x) / t_0
    else
        tmp = (x / y) / t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = (y + x) + 1.0;
	double tmp;
	if (y <= 3.6e-100) {
		tmp = (y / x) / t_0;
	} else {
		tmp = (x / y) / t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (y + x) + 1.0
	tmp = 0
	if y <= 3.6e-100:
		tmp = (y / x) / t_0
	else:
		tmp = (x / y) / t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(y + x) + 1.0)
	tmp = 0.0
	if (y <= 3.6e-100)
		tmp = Float64(Float64(y / x) / t_0);
	else
		tmp = Float64(Float64(x / y) / t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (y + x) + 1.0;
	tmp = 0.0;
	if (y <= 3.6e-100)
		tmp = (y / x) / t_0;
	else
		tmp = (x / y) / t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(y + x), $MachinePrecision] + 1.0), $MachinePrecision]}, If[LessEqual[y, 3.6e-100], N[(N[(y / x), $MachinePrecision] / t$95$0), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < 3.5999999999999999e-100

   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

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. associate-/r* N/A

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

      10. lower-/.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 N/A

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

      17. +-commutative N/A

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

      18. lower-+.f64 N/A

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

      19. lift-+.f64 N/A

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

      20. +-commutative N/A

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

      21. lower-+.f64 74.4

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

   3. Applied rewrites 74.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift-+.f64 99.8

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

   5. Applied rewrites 99.8%

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

   6. Taylor expanded in x around inf

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

      1. lower-/.f64 49.4

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

   8. Applied rewrites 49.4%

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

   if 3.5999999999999999e-100 < 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

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. associate-/r* N/A

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

      10. lower-/.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 N/A

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

      17. +-commutative N/A

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

      18. lower-+.f64 N/A

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

      19. lift-+.f64 N/A

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

      20. +-commutative N/A

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

      21. lower-+.f64 74.4

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

   3. Applied rewrites 74.4%

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

   4. Taylor expanded in x around 0

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

      1. lower-/.f64 52.0

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

   6. Applied rewrites 52.0%

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

Alternative 11: 61.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2.55 \cdot 10^{-70}:\\ \;\;\;\;\frac{\frac{y}{x}}{\left(y + x\right) + 1}\\ \mathbf{elif}\;y \leq 7.5 \cdot 10^{+155}:\\ \;\;\;\;\frac{x}{\left(1 + y\right) \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 2.55e-70)
   (/ (/ y x) (+ (+ y x) 1.0))
   (if (<= y 7.5e+155) (/ x (* (+ 1.0 y) y)) (/ (/ x y) y))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 2.55e-70) {
		tmp = (y / x) / ((y + x) + 1.0);
	} else if (y <= 7.5e+155) {
		tmp = x / ((1.0 + y) * y);
	} else {
		tmp = (x / y) / y;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 2.55d-70) then
        tmp = (y / x) / ((y + x) + 1.0d0)
    else if (y <= 7.5d+155) then
        tmp = x / ((1.0d0 + y) * y)
    else
        tmp = (x / y) / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 2.55e-70) {
		tmp = (y / x) / ((y + x) + 1.0);
	} else if (y <= 7.5e+155) {
		tmp = x / ((1.0 + y) * y);
	} else {
		tmp = (x / y) / y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 2.55e-70:
		tmp = (y / x) / ((y + x) + 1.0)
	elif y <= 7.5e+155:
		tmp = x / ((1.0 + y) * y)
	else:
		tmp = (x / y) / y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 2.55e-70)
		tmp = Float64(Float64(y / x) / Float64(Float64(y + x) + 1.0));
	elseif (y <= 7.5e+155)
		tmp = Float64(x / Float64(Float64(1.0 + y) * y));
	else
		tmp = Float64(Float64(x / y) / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 2.55e-70)
		tmp = (y / x) / ((y + x) + 1.0);
	elseif (y <= 7.5e+155)
		tmp = x / ((1.0 + y) * y);
	else
		tmp = (x / y) / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 2.55e-70], N[(N[(y / x), $MachinePrecision] / N[(N[(y + x), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 7.5e+155], N[(x / N[(N[(1.0 + y), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 2.55000000000000013e-70

   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

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. associate-/r* N/A

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

      10. lower-/.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 N/A

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

      17. +-commutative N/A

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

      18. lower-+.f64 N/A

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

      19. lift-+.f64 N/A

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

      20. +-commutative N/A

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

      21. lower-+.f64 74.4

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

   3. Applied rewrites 74.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift-+.f64 99.8

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

   5. Applied rewrites 99.8%

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

   6. Taylor expanded in x around inf

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

      1. lower-/.f64 49.4

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

   8. Applied rewrites 49.4%

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

   if 2.55000000000000013e-70 < y < 7.4999999999999999e155

   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. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-+.f64 50.0

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

   4. Applied rewrites 50.0%

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

   if 7.4999999999999999e155 < 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 y around inf

      \[\leadsto \color{blue}{\frac{x}{{y}^{2}}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{{y}^{2}}} \]

      2. unpow2 N/A

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

      3. lower-*.f64 37.5

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

   4. Applied rewrites 37.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

         \[\leadsto \frac{\frac{x}{y}}{\color{blue}{y}} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{\frac{x}{y}}{\color{blue}{y}} \]

      5. lower-/.f64 39.3

         \[\leadsto \frac{\frac{x}{y}}{y} \]

   6. Applied rewrites 39.3%

      \[\leadsto \frac{\frac{x}{y}}{\color{blue}{y}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 12: 61.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -24000000000:\\ \;\;\;\;\frac{\frac{y}{x}}{x}\\ \mathbf{elif}\;y \leq 3.6 \cdot 10^{-100}:\\ \;\;\;\;\frac{y}{\left(1 + x\right) \cdot x}\\ \mathbf{elif}\;y \leq 7.5 \cdot 10^{+155}:\\ \;\;\;\;\frac{x}{\left(1 + y\right) \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -24000000000.0)
   (/ (/ y x) x)
   (if (<= y 3.6e-100)
     (/ y (* (+ 1.0 x) x))
     (if (<= y 7.5e+155) (/ x (* (+ 1.0 y) y)) (/ (/ x y) y)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -24000000000.0) {
		tmp = (y / x) / x;
	} else if (y <= 3.6e-100) {
		tmp = y / ((1.0 + x) * x);
	} else if (y <= 7.5e+155) {
		tmp = x / ((1.0 + y) * y);
	} else {
		tmp = (x / y) / y;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-24000000000.0d0)) then
        tmp = (y / x) / x
    else if (y <= 3.6d-100) then
        tmp = y / ((1.0d0 + x) * x)
    else if (y <= 7.5d+155) then
        tmp = x / ((1.0d0 + y) * y)
    else
        tmp = (x / y) / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -24000000000.0) {
		tmp = (y / x) / x;
	} else if (y <= 3.6e-100) {
		tmp = y / ((1.0 + x) * x);
	} else if (y <= 7.5e+155) {
		tmp = x / ((1.0 + y) * y);
	} else {
		tmp = (x / y) / y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -24000000000.0:
		tmp = (y / x) / x
	elif y <= 3.6e-100:
		tmp = y / ((1.0 + x) * x)
	elif y <= 7.5e+155:
		tmp = x / ((1.0 + y) * y)
	else:
		tmp = (x / y) / y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -24000000000.0)
		tmp = Float64(Float64(y / x) / x);
	elseif (y <= 3.6e-100)
		tmp = Float64(y / Float64(Float64(1.0 + x) * x));
	elseif (y <= 7.5e+155)
		tmp = Float64(x / Float64(Float64(1.0 + y) * y));
	else
		tmp = Float64(Float64(x / y) / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -24000000000.0)
		tmp = (y / x) / x;
	elseif (y <= 3.6e-100)
		tmp = y / ((1.0 + x) * x);
	elseif (y <= 7.5e+155)
		tmp = x / ((1.0 + y) * y);
	else
		tmp = (x / y) / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -24000000000.0], N[(N[(y / x), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[y, 3.6e-100], N[(y / N[(N[(1.0 + x), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 7.5e+155], N[(x / N[(N[(1.0 + y), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / y), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if y < -2.4e10

   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 inf

      \[\leadsto \color{blue}{\frac{y}{{x}^{2}}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{y}{\color{blue}{{x}^{2}}} \]

      2. unpow2 N/A

         \[\leadsto \frac{y}{x \cdot \color{blue}{x}} \]

      3. lower-*.f64 35.9

         \[\leadsto \frac{y}{x \cdot \color{blue}{x}} \]

   4. Applied rewrites 35.9%

      \[\leadsto \color{blue}{\frac{y}{x \cdot x}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{y}{x \cdot \color{blue}{x}} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{y}{\color{blue}{x \cdot x}} \]

      3. associate-/r* N/A

         \[\leadsto \frac{\frac{y}{x}}{\color{blue}{x}} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{\frac{y}{x}}{\color{blue}{x}} \]

      5. lower-/.f64 37.1

         \[\leadsto \frac{\frac{y}{x}}{x} \]

   6. Applied rewrites 37.1%

      \[\leadsto \frac{\frac{y}{x}}{\color{blue}{x}} \]

   if -2.4e10 < y < 3.5999999999999999e-100

   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. *-commutative N/A

         \[\leadsto \frac{y}{\left(1 + x\right) \cdot \color{blue}{x}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{y}{\left(1 + x\right) \cdot \color{blue}{x}} \]

      4. lower-+.f64 47.9

         \[\leadsto \frac{y}{\left(1 + x\right) \cdot x} \]

   4. Applied rewrites 47.9%

      \[\leadsto \color{blue}{\frac{y}{\left(1 + x\right) \cdot x}} \]

   if 3.5999999999999999e-100 < y < 7.4999999999999999e155

   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. *-commutative N/A

         \[\leadsto \frac{x}{\left(1 + y\right) \cdot \color{blue}{y}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{x}{\left(1 + y\right) \cdot \color{blue}{y}} \]

      4. lower-+.f64 50.0

         \[\leadsto \frac{x}{\left(1 + y\right) \cdot y} \]

   4. Applied rewrites 50.0%

      \[\leadsto \color{blue}{\frac{x}{\left(1 + y\right) \cdot y}} \]

   if 7.4999999999999999e155 < 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 y around inf

      \[\leadsto \color{blue}{\frac{x}{{y}^{2}}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{{y}^{2}}} \]

      2. unpow2 N/A

         \[\leadsto \frac{x}{y \cdot \color{blue}{y}} \]

      3. lower-*.f64 37.5

         \[\leadsto \frac{x}{y \cdot \color{blue}{y}} \]

   4. Applied rewrites 37.5%

      \[\leadsto \color{blue}{\frac{x}{y \cdot y}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{y \cdot y}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{x}{y \cdot \color{blue}{y}} \]

      3. associate-/r* N/A

         \[\leadsto \frac{\frac{x}{y}}{\color{blue}{y}} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{\frac{x}{y}}{\color{blue}{y}} \]

      5. lower-/.f64 39.3

         \[\leadsto \frac{\frac{x}{y}}{y} \]

   6. Applied rewrites 39.3%

      \[\leadsto \frac{\frac{x}{y}}{\color{blue}{y}} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 13: 60.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -7.2 \cdot 10^{+46}:\\ \;\;\;\;\frac{\frac{y}{x}}{x}\\ \mathbf{elif}\;x \leq 2.1 \cdot 10^{+23}:\\ \;\;\;\;\frac{x}{\left(1 + y\right) \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -7.2e+46)
   (/ (/ y x) x)
   (if (<= x 2.1e+23) (/ x (* (+ 1.0 y) y)) (/ (/ x y) y))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -7.2e+46) {
		tmp = (y / x) / x;
	} else if (x <= 2.1e+23) {
		tmp = x / ((1.0 + y) * y);
	} else {
		tmp = (x / y) / y;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-7.2d+46)) then
        tmp = (y / x) / x
    else if (x <= 2.1d+23) then
        tmp = x / ((1.0d0 + y) * y)
    else
        tmp = (x / y) / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -7.2e+46) {
		tmp = (y / x) / x;
	} else if (x <= 2.1e+23) {
		tmp = x / ((1.0 + y) * y);
	} else {
		tmp = (x / y) / y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -7.2e+46:
		tmp = (y / x) / x
	elif x <= 2.1e+23:
		tmp = x / ((1.0 + y) * y)
	else:
		tmp = (x / y) / y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -7.2e+46)
		tmp = Float64(Float64(y / x) / x);
	elseif (x <= 2.1e+23)
		tmp = Float64(x / Float64(Float64(1.0 + y) * y));
	else
		tmp = Float64(Float64(x / y) / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -7.2e+46)
		tmp = (y / x) / x;
	elseif (x <= 2.1e+23)
		tmp = x / ((1.0 + y) * y);
	else
		tmp = (x / y) / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -7.2e+46], N[(N[(y / x), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[x, 2.1e+23], N[(x / N[(N[(1.0 + y), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -7.1999999999999997e46

   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 inf

      \[\leadsto \color{blue}{\frac{y}{{x}^{2}}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{y}{\color{blue}{{x}^{2}}} \]

      2. unpow2 N/A

         \[\leadsto \frac{y}{x \cdot \color{blue}{x}} \]

      3. lower-*.f64 35.9

         \[\leadsto \frac{y}{x \cdot \color{blue}{x}} \]

   4. Applied rewrites 35.9%

      \[\leadsto \color{blue}{\frac{y}{x \cdot x}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{y}{x \cdot \color{blue}{x}} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{y}{\color{blue}{x \cdot x}} \]

      3. associate-/r* N/A

         \[\leadsto \frac{\frac{y}{x}}{\color{blue}{x}} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{\frac{y}{x}}{\color{blue}{x}} \]

      5. lower-/.f64 37.1

         \[\leadsto \frac{\frac{y}{x}}{x} \]

   6. Applied rewrites 37.1%

      \[\leadsto \frac{\frac{y}{x}}{\color{blue}{x}} \]

   if -7.1999999999999997e46 < x < 2.1000000000000001e23

   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. *-commutative N/A

         \[\leadsto \frac{x}{\left(1 + y\right) \cdot \color{blue}{y}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{x}{\left(1 + y\right) \cdot \color{blue}{y}} \]

      4. lower-+.f64 50.0

         \[\leadsto \frac{x}{\left(1 + y\right) \cdot y} \]

   4. Applied rewrites 50.0%

      \[\leadsto \color{blue}{\frac{x}{\left(1 + y\right) \cdot y}} \]

   if 2.1000000000000001e23 < 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. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\frac{x}{{y}^{2}}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{{y}^{2}}} \]

      2. unpow2 N/A

         \[\leadsto \frac{x}{y \cdot \color{blue}{y}} \]

      3. lower-*.f64 37.5

         \[\leadsto \frac{x}{y \cdot \color{blue}{y}} \]

   4. Applied rewrites 37.5%

      \[\leadsto \color{blue}{\frac{x}{y \cdot y}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{y \cdot y}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{x}{y \cdot \color{blue}{y}} \]

      3. associate-/r* N/A

         \[\leadsto \frac{\frac{x}{y}}{\color{blue}{y}} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{\frac{x}{y}}{\color{blue}{y}} \]

      5. lower-/.f64 39.3

         \[\leadsto \frac{\frac{x}{y}}{y} \]

   6. Applied rewrites 39.3%

      \[\leadsto \frac{\frac{x}{y}}{\color{blue}{y}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 14: 50.1% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 52000:\\ \;\;\;\;\frac{\frac{y}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 52000.0) (/ (/ y x) x) (/ (/ x y) y)))

C

double code(double x, double y) {
	double tmp;
	if (y <= 52000.0) {
		tmp = (y / x) / x;
	} else {
		tmp = (x / y) / y;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 52000.0d0) then
        tmp = (y / x) / x
    else
        tmp = (x / y) / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 52000.0) {
		tmp = (y / x) / x;
	} else {
		tmp = (x / y) / y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 52000.0:
		tmp = (y / x) / x
	else:
		tmp = (x / y) / y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 52000.0)
		tmp = Float64(Float64(y / x) / x);
	else
		tmp = Float64(Float64(x / y) / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 52000.0)
		tmp = (y / x) / x;
	else
		tmp = (x / y) / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 52000.0], N[(N[(y / x), $MachinePrecision] / x), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 52000

   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 inf

      \[\leadsto \color{blue}{\frac{y}{{x}^{2}}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{y}{\color{blue}{{x}^{2}}} \]

      2. unpow2 N/A

         \[\leadsto \frac{y}{x \cdot \color{blue}{x}} \]

      3. lower-*.f64 35.9

         \[\leadsto \frac{y}{x \cdot \color{blue}{x}} \]

   4. Applied rewrites 35.9%

      \[\leadsto \color{blue}{\frac{y}{x \cdot x}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{y}{x \cdot \color{blue}{x}} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{y}{\color{blue}{x \cdot x}} \]

      3. associate-/r* N/A

         \[\leadsto \frac{\frac{y}{x}}{\color{blue}{x}} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{\frac{y}{x}}{\color{blue}{x}} \]

      5. lower-/.f64 37.1

         \[\leadsto \frac{\frac{y}{x}}{x} \]

   6. Applied rewrites 37.1%

      \[\leadsto \frac{\frac{y}{x}}{\color{blue}{x}} \]

   if 52000 < 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 y around inf

      \[\leadsto \color{blue}{\frac{x}{{y}^{2}}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{{y}^{2}}} \]

      2. unpow2 N/A

         \[\leadsto \frac{x}{y \cdot \color{blue}{y}} \]

      3. lower-*.f64 37.5

         \[\leadsto \frac{x}{y \cdot \color{blue}{y}} \]

   4. Applied rewrites 37.5%

      \[\leadsto \color{blue}{\frac{x}{y \cdot y}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{y \cdot y}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{x}{y \cdot \color{blue}{y}} \]

      3. associate-/r* N/A

         \[\leadsto \frac{\frac{x}{y}}{\color{blue}{y}} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{\frac{x}{y}}{\color{blue}{y}} \]

      5. lower-/.f64 39.3

         \[\leadsto \frac{\frac{x}{y}}{y} \]

   6. Applied rewrites 39.3%

      \[\leadsto \frac{\frac{x}{y}}{\color{blue}{y}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 15: 49.5% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 52000:\\ \;\;\;\;\frac{y}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 52000.0) (/ y (* x x)) (/ (/ x y) y)))

C

double code(double x, double y) {
	double tmp;
	if (y <= 52000.0) {
		tmp = y / (x * x);
	} else {
		tmp = (x / y) / y;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 52000.0d0) then
        tmp = y / (x * x)
    else
        tmp = (x / y) / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 52000.0) {
		tmp = y / (x * x);
	} else {
		tmp = (x / y) / y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 52000.0:
		tmp = y / (x * x)
	else:
		tmp = (x / y) / y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 52000.0)
		tmp = Float64(y / Float64(x * x));
	else
		tmp = Float64(Float64(x / y) / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 52000.0)
		tmp = y / (x * x);
	else
		tmp = (x / y) / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 52000.0], N[(y / N[(x * x), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 52000

   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 inf

      \[\leadsto \color{blue}{\frac{y}{{x}^{2}}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{y}{\color{blue}{{x}^{2}}} \]

      2. unpow2 N/A

         \[\leadsto \frac{y}{x \cdot \color{blue}{x}} \]

      3. lower-*.f64 35.9

         \[\leadsto \frac{y}{x \cdot \color{blue}{x}} \]

   4. Applied rewrites 35.9%

      \[\leadsto \color{blue}{\frac{y}{x \cdot x}} \]

   if 52000 < 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 y around inf

      \[\leadsto \color{blue}{\frac{x}{{y}^{2}}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{{y}^{2}}} \]

      2. unpow2 N/A

         \[\leadsto \frac{x}{y \cdot \color{blue}{y}} \]

      3. lower-*.f64 37.5

         \[\leadsto \frac{x}{y \cdot \color{blue}{y}} \]

   4. Applied rewrites 37.5%

      \[\leadsto \color{blue}{\frac{x}{y \cdot y}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{y \cdot y}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{x}{y \cdot \color{blue}{y}} \]

      3. associate-/r* N/A

         \[\leadsto \frac{\frac{x}{y}}{\color{blue}{y}} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{\frac{x}{y}}{\color{blue}{y}} \]

      5. lower-/.f64 39.3

         \[\leadsto \frac{\frac{x}{y}}{y} \]

   6. Applied rewrites 39.3%

      \[\leadsto \frac{\frac{x}{y}}{\color{blue}{y}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 16: 48.7% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 52000:\\ \;\;\;\;\frac{y}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 52000.0) (/ y (* x x)) (/ x (* y y))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 52000.0) {
		tmp = y / (x * x);
	} else {
		tmp = x / (y * y);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 52000.0d0) then
        tmp = y / (x * x)
    else
        tmp = x / (y * y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 52000.0) {
		tmp = y / (x * x);
	} else {
		tmp = x / (y * y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 52000.0:
		tmp = y / (x * x)
	else:
		tmp = x / (y * y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 52000.0)
		tmp = Float64(y / Float64(x * x));
	else
		tmp = Float64(x / Float64(y * y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 52000.0)
		tmp = y / (x * x);
	else
		tmp = x / (y * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 52000.0], N[(y / N[(x * x), $MachinePrecision]), $MachinePrecision], N[(x / N[(y * y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 52000

   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 inf

      \[\leadsto \color{blue}{\frac{y}{{x}^{2}}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{y}{\color{blue}{{x}^{2}}} \]

      2. unpow2 N/A

         \[\leadsto \frac{y}{x \cdot \color{blue}{x}} \]

      3. lower-*.f64 35.9

         \[\leadsto \frac{y}{x \cdot \color{blue}{x}} \]

   4. Applied rewrites 35.9%

      \[\leadsto \color{blue}{\frac{y}{x \cdot x}} \]

   if 52000 < 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 y around inf

      \[\leadsto \color{blue}{\frac{x}{{y}^{2}}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{{y}^{2}}} \]

      2. unpow2 N/A

         \[\leadsto \frac{x}{y \cdot \color{blue}{y}} \]

      3. lower-*.f64 37.5

         \[\leadsto \frac{x}{y \cdot \color{blue}{y}} \]

   4. Applied rewrites 37.5%

      \[\leadsto \color{blue}{\frac{x}{y \cdot y}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 17: 37.5% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ \frac{x}{y \cdot y} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (/ x (* y y)))

C

double code(double x, double y) {
	return x / (y * 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 * y)
end function

Java

public static double code(double x, double y) {
	return x / (y * y);
}

Python

def code(x, y):
	return x / (y * y)

Julia

function code(x, y)
	return Float64(x / Float64(y * y))
end

MATLAB

function tmp = code(x, y)
	tmp = x / (y * y);
end

Wolfram

code[x_, y_] := N[(x / N[(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 y around inf

   \[\leadsto \color{blue}{\frac{x}{{y}^{2}}} \]
3. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \frac{x}{\color{blue}{{y}^{2}}} \]

   2. unpow2 N/A

      \[\leadsto \frac{x}{y \cdot \color{blue}{y}} \]

   3. lower-*.f64 37.5

      \[\leadsto \frac{x}{y \cdot \color{blue}{y}} \]

4. Applied rewrites 37.5%

   \[\leadsto \color{blue}{\frac{x}{y \cdot y}} \]
5. 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))))