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

Percentage Accurate: 69.2% → 99.8%

Time: 4.2s

Alternatives: 21

Speedup: 0.8×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 69.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 69.2%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-+.f64 N/A

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

   5. lift-+.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. lift-+.f64 N/A

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

   8. lift-+.f64 N/A

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

   9. times-frac N/A

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

   10. lower-*.f64 N/A

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

   11. lower-/.f64 N/A

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

   12. pow2 N/A

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

   13. lower-pow.f64 N/A

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

   14. +-commutative N/A

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

   15. lower-+.f64 N/A

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

   16. lower-/.f64 N/A

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

   17. lift-+.f64 N/A

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

   18. +-commutative N/A

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

   19. lower-+.f64 87.7

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

4. Applied rewrites 87.7%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lower-pow.f64 N/A

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

   4. pow2 N/A

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

   5. lift-*.f64 N/A

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

   6. +-commutative N/A

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

   7. lift-+.f64 N/A

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

   8. +-commutative N/A

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

   9. lift-+.f64 87.7

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

6. Applied rewrites 87.7%

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

   1. lift-/.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. lift-+.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. associate-/r* N/A

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

   6. +-commutative N/A

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

   7. +-commutative N/A

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

   8. lower-/.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. +-commutative N/A

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

   11. lift-+.f64 N/A

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

   12. +-commutative N/A

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

   13. lift-+.f64 99.7

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

8. Applied rewrites 99.7%

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

   1. lift-*.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. lift-+.f64 N/A

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

   5. lift-/.f64 N/A

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

   6. lift-/.f64 N/A

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

   7. lift-+.f64 N/A

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

   8. lift-+.f64 N/A

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

   9. associate-*l/ N/A

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

   10. lower-/.f64 N/A

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

10. Applied rewrites 99.8%

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

Alternative 2: 76.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y}{\left(y + x\right) + 1}\\ \mathbf{if}\;y \leq -2.3 \cdot 10^{-61}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{y}{x}, -2, 1\right)}{x} \cdot t\_0\\ \mathbf{elif}\;y \leq 6.1 \cdot 10^{+150}:\\ \;\;\;\;\frac{\frac{x}{y + x} \cdot y}{\left(y + x\right) \cdot \left(\left(1 + x\right) + y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + x} \cdot t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ y (+ (+ y x) 1.0))))
   (if (<= y -2.3e-61)
     (* (/ (fma (/ y x) -2.0 1.0) x) t_0)
     (if (<= y 6.1e+150)
       (/ (* (/ x (+ y x)) y) (* (+ y x) (+ (+ 1.0 x) y)))
       (* (/ (/ x y) (+ y x)) t_0)))))

C

double code(double x, double y) {
	double t_0 = y / ((y + x) + 1.0);
	double tmp;
	if (y <= -2.3e-61) {
		tmp = (fma((y / x), -2.0, 1.0) / x) * t_0;
	} else if (y <= 6.1e+150) {
		tmp = ((x / (y + x)) * y) / ((y + x) * ((1.0 + x) + y));
	} else {
		tmp = ((x / y) / (y + x)) * t_0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(y / Float64(Float64(y + x) + 1.0))
	tmp = 0.0
	if (y <= -2.3e-61)
		tmp = Float64(Float64(fma(Float64(y / x), -2.0, 1.0) / x) * t_0);
	elseif (y <= 6.1e+150)
		tmp = Float64(Float64(Float64(x / Float64(y + x)) * y) / Float64(Float64(y + x) * Float64(Float64(1.0 + x) + y)));
	else
		tmp = Float64(Float64(Float64(x / y) / Float64(y + x)) * t_0);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(y / N[(N[(y + x), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -2.3e-61], N[(N[(N[(N[(y / x), $MachinePrecision] * -2.0 + 1.0), $MachinePrecision] / x), $MachinePrecision] * t$95$0), $MachinePrecision], If[LessEqual[y, 6.1e+150], N[(N[(N[(x / N[(y + x), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision] / N[(N[(y + x), $MachinePrecision] * N[(N[(1.0 + x), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x / y), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.29999999999999992e-61

   1. Initial program 67.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. times-frac N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. pow2 N/A

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

      13. lower-pow.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. lift-+.f64 N/A

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

      18. +-commutative N/A

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

      19. lower-+.f64 89.8

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

   4. Applied rewrites 89.8%

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

   5. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 29.9

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

   7. Applied rewrites 29.9%

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

   if -2.29999999999999992e-61 < y < 6.10000000000000026e150

   1. Initial program 72.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. times-frac N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. pow2 N/A

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

      13. lower-pow.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. lift-+.f64 N/A

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

      18. +-commutative N/A

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

      19. lower-+.f64 88.2

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

   4. Applied rewrites 88.2%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-pow.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-+.f64 88.2

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

   6. Applied rewrites 88.2%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/r* N/A

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

      6. +-commutative N/A

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

      7. +-commutative N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lift-+.f64 99.7

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

   8. Applied rewrites 99.7%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. frac-times N/A

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

      10. +-commutative N/A

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

      11. +-commutative N/A

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

      12. lower-/.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lift-/.f64 N/A

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

      15. lift-+.f64 N/A

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

      16. +-commutative N/A

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

      17. lower-*.f64 N/A

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

      18. lift-+.f64 N/A

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

   10. Applied rewrites 98.3%

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

   if 6.10000000000000026e150 < y

   1. Initial program 57.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. times-frac N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. pow2 N/A

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

      13. lower-pow.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. lift-+.f64 N/A

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

      18. +-commutative N/A

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

      19. lower-+.f64 80.0

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

   4. Applied rewrites 80.0%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-pow.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-+.f64 80.0

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

   6. Applied rewrites 80.0%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/r* N/A

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

      6. +-commutative N/A

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

      7. +-commutative N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lift-+.f64 99.9

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

   8. Applied rewrites 99.9%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 86.7%

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

Alternative 3: 76.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y}{\left(y + x\right) + 1}\\ \mathbf{if}\;y \leq -2.3 \cdot 10^{-61}:\\ \;\;\;\;\frac{1}{y + x} \cdot t\_0\\ \mathbf{elif}\;y \leq 6.1 \cdot 10^{+150}:\\ \;\;\;\;\frac{\frac{x}{y + x} \cdot y}{\left(y + x\right) \cdot \left(\left(1 + x\right) + y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + x} \cdot t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ y (+ (+ y x) 1.0))))
   (if (<= y -2.3e-61)
     (* (/ 1.0 (+ y x)) t_0)
     (if (<= y 6.1e+150)
       (/ (* (/ x (+ y x)) y) (* (+ y x) (+ (+ 1.0 x) y)))
       (* (/ (/ x y) (+ y x)) t_0)))))

C

double code(double x, double y) {
	double t_0 = y / ((y + x) + 1.0);
	double tmp;
	if (y <= -2.3e-61) {
		tmp = (1.0 / (y + x)) * t_0;
	} else if (y <= 6.1e+150) {
		tmp = ((x / (y + x)) * y) / ((y + x) * ((1.0 + x) + y));
	} else {
		tmp = ((x / y) / (y + x)) * 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 / ((y + x) + 1.0d0)
    if (y <= (-2.3d-61)) then
        tmp = (1.0d0 / (y + x)) * t_0
    else if (y <= 6.1d+150) then
        tmp = ((x / (y + x)) * y) / ((y + x) * ((1.0d0 + x) + y))
    else
        tmp = ((x / y) / (y + x)) * t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = y / ((y + x) + 1.0);
	double tmp;
	if (y <= -2.3e-61) {
		tmp = (1.0 / (y + x)) * t_0;
	} else if (y <= 6.1e+150) {
		tmp = ((x / (y + x)) * y) / ((y + x) * ((1.0 + x) + y));
	} else {
		tmp = ((x / y) / (y + x)) * t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = y / ((y + x) + 1.0)
	tmp = 0
	if y <= -2.3e-61:
		tmp = (1.0 / (y + x)) * t_0
	elif y <= 6.1e+150:
		tmp = ((x / (y + x)) * y) / ((y + x) * ((1.0 + x) + y))
	else:
		tmp = ((x / y) / (y + x)) * t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(y / Float64(Float64(y + x) + 1.0))
	tmp = 0.0
	if (y <= -2.3e-61)
		tmp = Float64(Float64(1.0 / Float64(y + x)) * t_0);
	elseif (y <= 6.1e+150)
		tmp = Float64(Float64(Float64(x / Float64(y + x)) * y) / Float64(Float64(y + x) * Float64(Float64(1.0 + x) + y)));
	else
		tmp = Float64(Float64(Float64(x / y) / Float64(y + x)) * t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = y / ((y + x) + 1.0);
	tmp = 0.0;
	if (y <= -2.3e-61)
		tmp = (1.0 / (y + x)) * t_0;
	elseif (y <= 6.1e+150)
		tmp = ((x / (y + x)) * y) / ((y + x) * ((1.0 + x) + y));
	else
		tmp = ((x / y) / (y + x)) * t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(y / N[(N[(y + x), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -2.3e-61], N[(N[(1.0 / N[(y + x), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision], If[LessEqual[y, 6.1e+150], N[(N[(N[(x / N[(y + x), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision] / N[(N[(y + x), $MachinePrecision] * N[(N[(1.0 + x), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x / y), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.29999999999999992e-61

   1. Initial program 67.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. times-frac N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. pow2 N/A

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

      13. lower-pow.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. lift-+.f64 N/A

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

      18. +-commutative N/A

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

      19. lower-+.f64 89.8

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

   4. Applied rewrites 89.8%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-pow.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-+.f64 89.8

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

   6. Applied rewrites 89.8%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/r* N/A

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

      6. +-commutative N/A

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

      7. +-commutative N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lift-+.f64 99.8

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

   8. Applied rewrites 99.8%

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

   9. Taylor expanded in x around inf

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

   11. Applied rewrites 31.9%

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

   if -2.29999999999999992e-61 < y < 6.10000000000000026e150

   1. Initial program 72.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. times-frac N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. pow2 N/A

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

      13. lower-pow.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. lift-+.f64 N/A

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

      18. +-commutative N/A

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

      19. lower-+.f64 88.2

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

   4. Applied rewrites 88.2%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-pow.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-+.f64 88.2

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

   6. Applied rewrites 88.2%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/r* N/A

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

      6. +-commutative N/A

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

      7. +-commutative N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lift-+.f64 99.7

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

   8. Applied rewrites 99.7%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. frac-times N/A

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

      10. +-commutative N/A

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

      11. +-commutative N/A

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

      12. lower-/.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lift-/.f64 N/A

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

      15. lift-+.f64 N/A

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

      16. +-commutative N/A

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

      17. lower-*.f64 N/A

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

      18. lift-+.f64 N/A

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

   10. Applied rewrites 98.3%

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

   if 6.10000000000000026e150 < y

   1. Initial program 57.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. times-frac N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. pow2 N/A

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

      13. lower-pow.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. lift-+.f64 N/A

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

      18. +-commutative N/A

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

      19. lower-+.f64 80.0

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

   4. Applied rewrites 80.0%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-pow.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-+.f64 80.0

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

   6. Applied rewrites 80.0%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/r* N/A

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

      6. +-commutative N/A

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

      7. +-commutative N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lift-+.f64 99.9

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

   8. Applied rewrites 99.9%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 86.7%

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

Alternative 4: 64.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -132000:\\ \;\;\;\;\frac{\frac{y}{x}}{x}\\ \mathbf{elif}\;y \leq 8.2 \cdot 10^{-60}:\\ \;\;\;\;\frac{y}{\left(1 + x\right) \cdot x}\\ \mathbf{elif}\;y \leq 2 \cdot 10^{+16}:\\ \;\;\;\;\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(y + 1\right)}\\ \mathbf{elif}\;y \leq 2.1 \cdot 10^{+146}:\\ \;\;\;\;\frac{x}{\left(y + x\right) \cdot \left(y + x\right)} \cdot 1\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + x}}{y + x} \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -132000.0)
   (/ (/ y x) x)
   (if (<= y 8.2e-60)
     (/ y (* (+ 1.0 x) x))
     (if (<= y 2e+16)
       (/ (* x y) (* (* (+ x y) (+ x y)) (+ y 1.0)))
       (if (<= y 2.1e+146)
         (* (/ x (* (+ y x) (+ y x))) 1.0)
         (* (/ (/ x (+ y x)) (+ y x)) 1.0))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -132000.0) {
		tmp = (y / x) / x;
	} else if (y <= 8.2e-60) {
		tmp = y / ((1.0 + x) * x);
	} else if (y <= 2e+16) {
		tmp = (x * y) / (((x + y) * (x + y)) * (y + 1.0));
	} else if (y <= 2.1e+146) {
		tmp = (x / ((y + x) * (y + x))) * 1.0;
	} 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 (y <= (-132000.0d0)) then
        tmp = (y / x) / x
    else if (y <= 8.2d-60) then
        tmp = y / ((1.0d0 + x) * x)
    else if (y <= 2d+16) then
        tmp = (x * y) / (((x + y) * (x + y)) * (y + 1.0d0))
    else if (y <= 2.1d+146) then
        tmp = (x / ((y + x) * (y + x))) * 1.0d0
    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 (y <= -132000.0) {
		tmp = (y / x) / x;
	} else if (y <= 8.2e-60) {
		tmp = y / ((1.0 + x) * x);
	} else if (y <= 2e+16) {
		tmp = (x * y) / (((x + y) * (x + y)) * (y + 1.0));
	} else if (y <= 2.1e+146) {
		tmp = (x / ((y + x) * (y + x))) * 1.0;
	} else {
		tmp = ((x / (y + x)) / (y + x)) * 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -132000.0:
		tmp = (y / x) / x
	elif y <= 8.2e-60:
		tmp = y / ((1.0 + x) * x)
	elif y <= 2e+16:
		tmp = (x * y) / (((x + y) * (x + y)) * (y + 1.0))
	elif y <= 2.1e+146:
		tmp = (x / ((y + x) * (y + x))) * 1.0
	else:
		tmp = ((x / (y + x)) / (y + x)) * 1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -132000.0)
		tmp = Float64(Float64(y / x) / x);
	elseif (y <= 8.2e-60)
		tmp = Float64(y / Float64(Float64(1.0 + x) * x));
	elseif (y <= 2e+16)
		tmp = Float64(Float64(x * y) / Float64(Float64(Float64(x + y) * Float64(x + y)) * Float64(y + 1.0)));
	elseif (y <= 2.1e+146)
		tmp = Float64(Float64(x / Float64(Float64(y + x) * Float64(y + x))) * 1.0);
	else
		tmp = Float64(Float64(Float64(x / Float64(y + x)) / Float64(y + x)) * 1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -132000.0)
		tmp = (y / x) / x;
	elseif (y <= 8.2e-60)
		tmp = y / ((1.0 + x) * x);
	elseif (y <= 2e+16)
		tmp = (x * y) / (((x + y) * (x + y)) * (y + 1.0));
	elseif (y <= 2.1e+146)
		tmp = (x / ((y + x) * (y + x))) * 1.0;
	else
		tmp = ((x / (y + x)) / (y + x)) * 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -132000.0], N[(N[(y / x), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[y, 8.2e-60], N[(y / N[(N[(1.0 + x), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2e+16], N[(N[(x * y), $MachinePrecision] / N[(N[(N[(x + y), $MachinePrecision] * N[(x + y), $MachinePrecision]), $MachinePrecision] * N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2.1e+146], N[(N[(x / N[(N[(y + x), $MachinePrecision] * N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision], N[(N[(N[(x / N[(y + x), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if y < -132000

   1. Initial program 62.4%

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

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{y}{{x}^{2}}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{y}{\color{blue}{{x}^{2}}} \]

      2. unpow2 N/A

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

      3. lower-*.f64 22.6

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

   5. Applied rewrites 22.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 25.4

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

   7. Applied rewrites 25.4%

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

   if -132000 < y < 8.20000000000000025e-60

   1. Initial program 74.5%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-+.f64 76.5

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

   5. Applied rewrites 76.5%

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

   if 8.20000000000000025e-60 < y < 2e16

   1. Initial program 91.4%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 73.7%

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

   if 2e16 < y < 2.1000000000000001e146

   1. Initial program 63.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. times-frac N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. pow2 N/A

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

      13. lower-pow.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. lift-+.f64 N/A

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

      18. +-commutative N/A

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

      19. lower-+.f64 92.2

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

   4. Applied rewrites 92.2%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-pow.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-+.f64 92.2

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

   6. Applied rewrites 92.2%

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

   7. Taylor expanded in y around inf

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

   9. Applied rewrites 79.9%

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

   if 2.1000000000000001e146 < y

   1. Initial program 57.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. times-frac N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. pow2 N/A

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

      13. lower-pow.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. lift-+.f64 N/A

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

      18. +-commutative N/A

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

      19. lower-+.f64 80.1

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

   4. Applied rewrites 80.1%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-pow.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-+.f64 80.1

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

   6. Applied rewrites 80.1%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/r* N/A

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

      6. +-commutative N/A

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

      7. +-commutative N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lift-+.f64 99.9

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

   8. Applied rewrites 99.9%

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

   9. Taylor expanded in y around inf

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

       1. +-commutative 86.2

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

   11. Applied rewrites 86.2%

       \[\leadsto \frac{\frac{x}{y + x}}{y + x} \cdot \color{blue}{1} \]
3. Recombined 5 regimes into one program.
4. Add Preprocessing

Alternative 5: 76.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{y + x}\\ \mathbf{if}\;y \leq -2.3 \cdot 10^{-61}:\\ \;\;\;\;\frac{1}{y + x} \cdot \frac{y}{\left(y + x\right) + 1}\\ \mathbf{elif}\;y \leq 6.1 \cdot 10^{+150}:\\ \;\;\;\;\frac{t\_0 \cdot y}{\left(y + x\right) \cdot \left(\left(1 + x\right) + y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0}{y + x} \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ x (+ y x))))
   (if (<= y -2.3e-61)
     (* (/ 1.0 (+ y x)) (/ y (+ (+ y x) 1.0)))
     (if (<= y 6.1e+150)
       (/ (* t_0 y) (* (+ y x) (+ (+ 1.0 x) y)))
       (* (/ t_0 (+ y x)) 1.0)))))

C

double code(double x, double y) {
	double t_0 = x / (y + x);
	double tmp;
	if (y <= -2.3e-61) {
		tmp = (1.0 / (y + x)) * (y / ((y + x) + 1.0));
	} else if (y <= 6.1e+150) {
		tmp = (t_0 * y) / ((y + x) * ((1.0 + x) + y));
	} else {
		tmp = (t_0 / (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) :: t_0
    real(8) :: tmp
    t_0 = x / (y + x)
    if (y <= (-2.3d-61)) then
        tmp = (1.0d0 / (y + x)) * (y / ((y + x) + 1.0d0))
    else if (y <= 6.1d+150) then
        tmp = (t_0 * y) / ((y + x) * ((1.0d0 + x) + y))
    else
        tmp = (t_0 / (y + x)) * 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = x / (y + x);
	double tmp;
	if (y <= -2.3e-61) {
		tmp = (1.0 / (y + x)) * (y / ((y + x) + 1.0));
	} else if (y <= 6.1e+150) {
		tmp = (t_0 * y) / ((y + x) * ((1.0 + x) + y));
	} else {
		tmp = (t_0 / (y + x)) * 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = x / (y + x)
	tmp = 0
	if y <= -2.3e-61:
		tmp = (1.0 / (y + x)) * (y / ((y + x) + 1.0))
	elif y <= 6.1e+150:
		tmp = (t_0 * y) / ((y + x) * ((1.0 + x) + y))
	else:
		tmp = (t_0 / (y + x)) * 1.0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(x / Float64(y + x))
	tmp = 0.0
	if (y <= -2.3e-61)
		tmp = Float64(Float64(1.0 / Float64(y + x)) * Float64(y / Float64(Float64(y + x) + 1.0)));
	elseif (y <= 6.1e+150)
		tmp = Float64(Float64(t_0 * y) / Float64(Float64(y + x) * Float64(Float64(1.0 + x) + y)));
	else
		tmp = Float64(Float64(t_0 / Float64(y + x)) * 1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = x / (y + x);
	tmp = 0.0;
	if (y <= -2.3e-61)
		tmp = (1.0 / (y + x)) * (y / ((y + x) + 1.0));
	elseif (y <= 6.1e+150)
		tmp = (t_0 * y) / ((y + x) * ((1.0 + x) + y));
	else
		tmp = (t_0 / (y + x)) * 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(x / N[(y + x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -2.3e-61], N[(N[(1.0 / N[(y + x), $MachinePrecision]), $MachinePrecision] * N[(y / N[(N[(y + x), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 6.1e+150], N[(N[(t$95$0 * y), $MachinePrecision] / N[(N[(y + x), $MachinePrecision] * N[(N[(1.0 + x), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$0 / N[(y + x), $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.29999999999999992e-61

   1. Initial program 67.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. times-frac N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. pow2 N/A

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

      13. lower-pow.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. lift-+.f64 N/A

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

      18. +-commutative N/A

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

      19. lower-+.f64 89.8

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

   4. Applied rewrites 89.8%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-pow.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-+.f64 89.8

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

   6. Applied rewrites 89.8%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/r* N/A

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

      6. +-commutative N/A

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

      7. +-commutative N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lift-+.f64 99.8

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

   8. Applied rewrites 99.8%

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

   9. Taylor expanded in x around inf

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

   11. Applied rewrites 31.9%

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

   if -2.29999999999999992e-61 < y < 6.10000000000000026e150

   1. Initial program 72.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. times-frac N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. pow2 N/A

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

      13. lower-pow.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. lift-+.f64 N/A

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

      18. +-commutative N/A

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

      19. lower-+.f64 88.2

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

   4. Applied rewrites 88.2%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-pow.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-+.f64 88.2

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

   6. Applied rewrites 88.2%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/r* N/A

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

      6. +-commutative N/A

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

      7. +-commutative N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lift-+.f64 99.7

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

   8. Applied rewrites 99.7%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. frac-times N/A

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

      10. +-commutative N/A

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

      11. +-commutative N/A

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

      12. lower-/.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lift-/.f64 N/A

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

      15. lift-+.f64 N/A

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

      16. +-commutative N/A

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

      17. lower-*.f64 N/A

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

      18. lift-+.f64 N/A

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

   10. Applied rewrites 98.3%

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

   if 6.10000000000000026e150 < y

   1. Initial program 57.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. times-frac N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. pow2 N/A

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

      13. lower-pow.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. lift-+.f64 N/A

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

      18. +-commutative N/A

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

      19. lower-+.f64 80.0

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

   4. Applied rewrites 80.0%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-pow.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-+.f64 80.0

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

   6. Applied rewrites 80.0%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/r* N/A

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

      6. +-commutative N/A

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

      7. +-commutative N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lift-+.f64 99.9

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

   8. Applied rewrites 99.9%

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

   9. Taylor expanded in y around inf

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

       1. +-commutative 86.7

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

   11. Applied rewrites 86.7%

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

Alternative 6: 71.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y}{\left(y + x\right) + 1}\\ \mathbf{if}\;y \leq 2 \cdot 10^{-174}:\\ \;\;\;\;\frac{1}{y + x} \cdot t\_0\\ \mathbf{elif}\;y \leq 6.1 \cdot 10^{+150}:\\ \;\;\;\;\frac{x}{\left(y + x\right) \cdot \left(y + x\right)} \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + x}}{y + x} \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ y (+ (+ y x) 1.0))))
   (if (<= y 2e-174)
     (* (/ 1.0 (+ y x)) t_0)
     (if (<= y 6.1e+150)
       (* (/ x (* (+ y x) (+ y x))) t_0)
       (* (/ (/ x (+ y x)) (+ y x)) 1.0)))))

C

double code(double x, double y) {
	double t_0 = y / ((y + x) + 1.0);
	double tmp;
	if (y <= 2e-174) {
		tmp = (1.0 / (y + x)) * t_0;
	} else if (y <= 6.1e+150) {
		tmp = (x / ((y + x) * (y + x))) * t_0;
	} 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) :: t_0
    real(8) :: tmp
    t_0 = y / ((y + x) + 1.0d0)
    if (y <= 2d-174) then
        tmp = (1.0d0 / (y + x)) * t_0
    else if (y <= 6.1d+150) then
        tmp = (x / ((y + x) * (y + x))) * t_0
    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 t_0 = y / ((y + x) + 1.0);
	double tmp;
	if (y <= 2e-174) {
		tmp = (1.0 / (y + x)) * t_0;
	} else if (y <= 6.1e+150) {
		tmp = (x / ((y + x) * (y + x))) * t_0;
	} else {
		tmp = ((x / (y + x)) / (y + x)) * 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = y / ((y + x) + 1.0)
	tmp = 0
	if y <= 2e-174:
		tmp = (1.0 / (y + x)) * t_0
	elif y <= 6.1e+150:
		tmp = (x / ((y + x) * (y + x))) * t_0
	else:
		tmp = ((x / (y + x)) / (y + x)) * 1.0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(y / Float64(Float64(y + x) + 1.0))
	tmp = 0.0
	if (y <= 2e-174)
		tmp = Float64(Float64(1.0 / Float64(y + x)) * t_0);
	elseif (y <= 6.1e+150)
		tmp = Float64(Float64(x / Float64(Float64(y + x) * Float64(y + x))) * t_0);
	else
		tmp = Float64(Float64(Float64(x / Float64(y + x)) / Float64(y + x)) * 1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = y / ((y + x) + 1.0);
	tmp = 0.0;
	if (y <= 2e-174)
		tmp = (1.0 / (y + x)) * t_0;
	elseif (y <= 6.1e+150)
		tmp = (x / ((y + x) * (y + x))) * t_0;
	else
		tmp = ((x / (y + x)) / (y + x)) * 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(y / N[(N[(y + x), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, 2e-174], N[(N[(1.0 / N[(y + x), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision], If[LessEqual[y, 6.1e+150], N[(N[(x / N[(N[(y + x), $MachinePrecision] * N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision], N[(N[(N[(x / N[(y + x), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < 2e-174

   1. Initial program 68.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. times-frac N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. pow2 N/A

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

      13. lower-pow.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. lift-+.f64 N/A

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

      18. +-commutative N/A

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

      19. lower-+.f64 85.9

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

   4. Applied rewrites 85.9%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-pow.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-+.f64 85.9

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

   6. Applied rewrites 85.9%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/r* N/A

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

      6. +-commutative N/A

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

      7. +-commutative N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lift-+.f64 99.7

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

   8. Applied rewrites 99.7%

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

   9. Taylor expanded in x around inf

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

   11. Applied rewrites 57.6%

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

   if 2e-174 < y < 6.10000000000000026e150

   1. Initial program 76.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. times-frac N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. pow2 N/A

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

      13. lower-pow.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. lift-+.f64 N/A

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

      18. +-commutative N/A

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

      19. lower-+.f64 95.4

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

   4. Applied rewrites 95.4%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-pow.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-+.f64 95.4

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

   6. Applied rewrites 95.4%

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

   if 6.10000000000000026e150 < y

   1. Initial program 57.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. times-frac N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. pow2 N/A

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

      13. lower-pow.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. lift-+.f64 N/A

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

      18. +-commutative N/A

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

      19. lower-+.f64 80.0

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

   4. Applied rewrites 80.0%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-pow.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-+.f64 80.0

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

   6. Applied rewrites 80.0%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/r* N/A

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

      6. +-commutative N/A

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

      7. +-commutative N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lift-+.f64 99.9

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

   8. Applied rewrites 99.9%

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

   9. Taylor expanded in y around inf

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

       1. +-commutative 86.7

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

   11. Applied rewrites 86.7%

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

Alternative 7: 68.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(y + x\right) + 1\\ \mathbf{if}\;y \leq 2 \cdot 10^{-174}:\\ \;\;\;\;\frac{1}{y + x} \cdot \frac{y}{t\_0}\\ \mathbf{elif}\;y \leq 10^{+146}:\\ \;\;\;\;x \cdot \frac{y}{\mathsf{fma}\left(\mathsf{fma}\left(2, x, y\right), y, x \cdot x\right) \cdot t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + x}}{y + x} \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ (+ y x) 1.0)))
   (if (<= y 2e-174)
     (* (/ 1.0 (+ y x)) (/ y t_0))
     (if (<= y 1e+146)
       (* x (/ y (* (fma (fma 2.0 x y) y (* x x)) t_0)))
       (* (/ (/ x (+ y x)) (+ y x)) 1.0)))))

C

double code(double x, double y) {
	double t_0 = (y + x) + 1.0;
	double tmp;
	if (y <= 2e-174) {
		tmp = (1.0 / (y + x)) * (y / t_0);
	} else if (y <= 1e+146) {
		tmp = x * (y / (fma(fma(2.0, x, y), y, (x * x)) * t_0));
	} else {
		tmp = ((x / (y + x)) / (y + x)) * 1.0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(y + x) + 1.0)
	tmp = 0.0
	if (y <= 2e-174)
		tmp = Float64(Float64(1.0 / Float64(y + x)) * Float64(y / t_0));
	elseif (y <= 1e+146)
		tmp = Float64(x * Float64(y / Float64(fma(fma(2.0, x, y), y, Float64(x * x)) * t_0)));
	else
		tmp = Float64(Float64(Float64(x / Float64(y + x)) / Float64(y + x)) * 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(y + x), $MachinePrecision] + 1.0), $MachinePrecision]}, If[LessEqual[y, 2e-174], N[(N[(1.0 / N[(y + x), $MachinePrecision]), $MachinePrecision] * N[(y / t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1e+146], N[(x * N[(y / N[(N[(N[(2.0 * x + y), $MachinePrecision] * y + N[(x * x), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x / N[(y + x), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < 2e-174

   1. Initial program 68.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. times-frac N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. pow2 N/A

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

      13. lower-pow.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. lift-+.f64 N/A

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

      18. +-commutative N/A

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

      19. lower-+.f64 85.9

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

   4. Applied rewrites 85.9%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-pow.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-+.f64 85.9

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

   6. Applied rewrites 85.9%

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

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{\left(y + x\right) \cdot \left(y + x\right)}} \cdot \frac{y}{\left(y + x\right) + 1} \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{\left(y + x\right)} \cdot \left(y + x\right)} \cdot \frac{y}{\left(y + x\right) + 1} \]

      3. lift-+.f64 N/A

         \[\leadsto \frac{x}{\left(y + x\right) \cdot \color{blue}{\left(y + x\right)}} \cdot \frac{y}{\left(y + x\right) + 1} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{\left(y + x\right) \cdot \left(y + x\right)}} \cdot \frac{y}{\left(y + x\right) + 1} \]

      5. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{x}{y + x}}{y + x}} \cdot \frac{y}{\left(y + x\right) + 1} \]

      6. +-commutative N/A

         \[\leadsto \frac{\frac{x}{\color{blue}{x + y}}}{y + x} \cdot \frac{y}{\left(y + x\right) + 1} \]

      7. +-commutative N/A

         \[\leadsto \frac{\frac{x}{x + y}}{\color{blue}{x + y}} \cdot \frac{y}{\left(y + x\right) + 1} \]

      8. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{x}{x + y}}{x + y}} \cdot \frac{y}{\left(y + x\right) + 1} \]

      9. lower-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{x}{x + y}}}{x + y} \cdot \frac{y}{\left(y + x\right) + 1} \]

      10. +-commutative N/A

          \[\leadsto \frac{\frac{x}{\color{blue}{y + x}}}{x + y} \cdot \frac{y}{\left(y + x\right) + 1} \]

      11. lift-+.f64 N/A

          \[\leadsto \frac{\frac{x}{\color{blue}{y + x}}}{x + y} \cdot \frac{y}{\left(y + x\right) + 1} \]

      12. +-commutative N/A

          \[\leadsto \frac{\frac{x}{y + x}}{\color{blue}{y + x}} \cdot \frac{y}{\left(y + x\right) + 1} \]

      13. lift-+.f64 99.7

          \[\leadsto \frac{\frac{x}{y + x}}{\color{blue}{y + x}} \cdot \frac{y}{\left(y + x\right) + 1} \]

   8. Applied rewrites 99.7%

      \[\leadsto \color{blue}{\frac{\frac{x}{y + x}}{y + x}} \cdot \frac{y}{\left(y + x\right) + 1} \]

   9. Taylor expanded in x around inf

      \[\leadsto \frac{\color{blue}{1}}{y + x} \cdot \frac{y}{\left(y + x\right) + 1} \]
   10. Step-by-step derivation

   11. Applied rewrites 57.6%

       \[\leadsto \frac{\color{blue}{1}}{y + x} \cdot \frac{y}{\left(y + x\right) + 1} \]

   if 2e-174 < y < 9.99999999999999934e145

   1. Initial program 77.2%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \frac{x \cdot y}{\color{blue}{\left(y \cdot \left(y + 2 \cdot x\right) + {x}^{2}\right)} \cdot \left(\left(x + y\right) + 1\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{x \cdot y}{\left(\left(y + 2 \cdot x\right) \cdot y + {\color{blue}{x}}^{2}\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      2. lower-fma.f64 N/A

         \[\leadsto \frac{x \cdot y}{\mathsf{fma}\left(y + 2 \cdot x, \color{blue}{y}, {x}^{2}\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      3. +-commutative N/A

         \[\leadsto \frac{x \cdot y}{\mathsf{fma}\left(2 \cdot x + y, y, {x}^{2}\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{x \cdot y}{\mathsf{fma}\left(\mathsf{fma}\left(2, x, y\right), y, {x}^{2}\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      5. unpow2 N/A

         \[\leadsto \frac{x \cdot y}{\mathsf{fma}\left(\mathsf{fma}\left(2, x, y\right), y, x \cdot x\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      6. lower-*.f64 77.2

         \[\leadsto \frac{x \cdot y}{\mathsf{fma}\left(\mathsf{fma}\left(2, x, y\right), y, x \cdot x\right) \cdot \left(\left(x + y\right) + 1\right)} \]

   5. Applied rewrites 77.2%

      \[\leadsto \frac{x \cdot y}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(2, x, y\right), y, x \cdot x\right)} \cdot \left(\left(x + y\right) + 1\right)} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot y}}{\mathsf{fma}\left(\mathsf{fma}\left(2, x, y\right), y, x \cdot x\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      2. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x \cdot y}{\mathsf{fma}\left(\mathsf{fma}\left(2, x, y\right), y, x \cdot x\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      3. associate-/l* N/A

         \[\leadsto \color{blue}{x \cdot \frac{y}{\mathsf{fma}\left(\mathsf{fma}\left(2, x, y\right), y, x \cdot x\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{x \cdot \frac{y}{\mathsf{fma}\left(\mathsf{fma}\left(2, x, y\right), y, x \cdot x\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      5. lower-/.f64 85.7

         \[\leadsto x \cdot \color{blue}{\frac{y}{\mathsf{fma}\left(\mathsf{fma}\left(2, x, y\right), y, x \cdot x\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      6. lift-+.f64 N/A

         \[\leadsto x \cdot \frac{y}{\mathsf{fma}\left(\mathsf{fma}\left(2, x, y\right), y, x \cdot x\right) \cdot \left(\color{blue}{\left(x + y\right)} + 1\right)} \]

      7. +-commutative N/A

         \[\leadsto x \cdot \frac{y}{\mathsf{fma}\left(\mathsf{fma}\left(2, x, y\right), y, x \cdot x\right) \cdot \left(\color{blue}{\left(y + x\right)} + 1\right)} \]

      8. lift-+.f64 85.7

         \[\leadsto x \cdot \frac{y}{\mathsf{fma}\left(\mathsf{fma}\left(2, x, y\right), y, x \cdot x\right) \cdot \left(\color{blue}{\left(y + x\right)} + 1\right)} \]

   7. Applied rewrites 85.7%

      \[\leadsto \color{blue}{x \cdot \frac{y}{\mathsf{fma}\left(\mathsf{fma}\left(2, x, y\right), y, x \cdot x\right) \cdot \left(\left(y + x\right) + 1\right)}} \]

   if 9.99999999999999934e145 < y

   1. Initial program 57.7%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot y}}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      2. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      4. lift-+.f64 N/A

         \[\leadsto \frac{x \cdot y}{\left(\color{blue}{\left(x + y\right)} \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      5. lift-+.f64 N/A

         \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \color{blue}{\left(x + y\right)}\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right)} \cdot \left(\left(x + y\right) + 1\right)} \]

      7. lift-+.f64 N/A

         \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\color{blue}{\left(x + y\right)} + 1\right)} \]

      8. lift-+.f64 N/A

         \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \color{blue}{\left(\left(x + y\right) + 1\right)}} \]

      9. times-frac N/A

         \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}} \]

      10. lower-*.f64 N/A

          \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}} \]

      11. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)}} \cdot \frac{y}{\left(x + y\right) + 1} \]

      12. pow2 N/A

          \[\leadsto \frac{x}{\color{blue}{{\left(x + y\right)}^{2}}} \cdot \frac{y}{\left(x + y\right) + 1} \]

      13. lower-pow.f64 N/A

          \[\leadsto \frac{x}{\color{blue}{{\left(x + y\right)}^{2}}} \cdot \frac{y}{\left(x + y\right) + 1} \]

      14. +-commutative N/A

          \[\leadsto \frac{x}{{\color{blue}{\left(y + x\right)}}^{2}} \cdot \frac{y}{\left(x + y\right) + 1} \]

      15. lower-+.f64 N/A

          \[\leadsto \frac{x}{{\color{blue}{\left(y + x\right)}}^{2}} \cdot \frac{y}{\left(x + y\right) + 1} \]

      16. lower-/.f64 N/A

          \[\leadsto \frac{x}{{\left(y + x\right)}^{2}} \cdot \color{blue}{\frac{y}{\left(x + y\right) + 1}} \]

      17. lift-+.f64 N/A

          \[\leadsto \frac{x}{{\left(y + x\right)}^{2}} \cdot \frac{y}{\color{blue}{\left(x + y\right) + 1}} \]

      18. +-commutative N/A

          \[\leadsto \frac{x}{{\left(y + x\right)}^{2}} \cdot \frac{y}{\color{blue}{\left(y + x\right)} + 1} \]

      19. lower-+.f64 80.0

          \[\leadsto \frac{x}{{\left(y + x\right)}^{2}} \cdot \frac{y}{\color{blue}{\left(y + x\right)} + 1} \]

   4. Applied rewrites 80.0%

      \[\leadsto \color{blue}{\frac{x}{{\left(y + x\right)}^{2}} \cdot \frac{y}{\left(y + x\right) + 1}} \]
   5. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{x}{{\color{blue}{\left(y + x\right)}}^{2}} \cdot \frac{y}{\left(y + x\right) + 1} \]

      2. +-commutative N/A

         \[\leadsto \frac{x}{{\color{blue}{\left(x + y\right)}}^{2}} \cdot \frac{y}{\left(y + x\right) + 1} \]

      3. lower-pow.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{{\left(x + y\right)}^{2}}} \cdot \frac{y}{\left(y + x\right) + 1} \]

      4. pow2 N/A

         \[\leadsto \frac{x}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right)}} \cdot \frac{y}{\left(y + x\right) + 1} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right)}} \cdot \frac{y}{\left(y + x\right) + 1} \]

      6. +-commutative N/A

         \[\leadsto \frac{x}{\color{blue}{\left(y + x\right)} \cdot \left(x + y\right)} \cdot \frac{y}{\left(y + x\right) + 1} \]

      7. lift-+.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{\left(y + x\right)} \cdot \left(x + y\right)} \cdot \frac{y}{\left(y + x\right) + 1} \]

      8. +-commutative N/A

         \[\leadsto \frac{x}{\left(y + x\right) \cdot \color{blue}{\left(y + x\right)}} \cdot \frac{y}{\left(y + x\right) + 1} \]

      9. lift-+.f64 80.0

         \[\leadsto \frac{x}{\left(y + x\right) \cdot \color{blue}{\left(y + x\right)}} \cdot \frac{y}{\left(y + x\right) + 1} \]

   6. Applied rewrites 80.0%

      \[\leadsto \frac{x}{\color{blue}{\left(y + x\right) \cdot \left(y + x\right)}} \cdot \frac{y}{\left(y + x\right) + 1} \]
   7. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{\left(y + x\right) \cdot \left(y + x\right)}} \cdot \frac{y}{\left(y + x\right) + 1} \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{\left(y + x\right)} \cdot \left(y + x\right)} \cdot \frac{y}{\left(y + x\right) + 1} \]

      3. lift-+.f64 N/A

         \[\leadsto \frac{x}{\left(y + x\right) \cdot \color{blue}{\left(y + x\right)}} \cdot \frac{y}{\left(y + x\right) + 1} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{\left(y + x\right) \cdot \left(y + x\right)}} \cdot \frac{y}{\left(y + x\right) + 1} \]

      5. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{x}{y + x}}{y + x}} \cdot \frac{y}{\left(y + x\right) + 1} \]

      6. +-commutative N/A

         \[\leadsto \frac{\frac{x}{\color{blue}{x + y}}}{y + x} \cdot \frac{y}{\left(y + x\right) + 1} \]

      7. +-commutative N/A

         \[\leadsto \frac{\frac{x}{x + y}}{\color{blue}{x + y}} \cdot \frac{y}{\left(y + x\right) + 1} \]

      8. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{x}{x + y}}{x + y}} \cdot \frac{y}{\left(y + x\right) + 1} \]

      9. lower-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{x}{x + y}}}{x + y} \cdot \frac{y}{\left(y + x\right) + 1} \]

      10. +-commutative N/A

          \[\leadsto \frac{\frac{x}{\color{blue}{y + x}}}{x + y} \cdot \frac{y}{\left(y + x\right) + 1} \]

      11. lift-+.f64 N/A

          \[\leadsto \frac{\frac{x}{\color{blue}{y + x}}}{x + y} \cdot \frac{y}{\left(y + x\right) + 1} \]

      12. +-commutative N/A

          \[\leadsto \frac{\frac{x}{y + x}}{\color{blue}{y + x}} \cdot \frac{y}{\left(y + x\right) + 1} \]

      13. lift-+.f64 99.9

          \[\leadsto \frac{\frac{x}{y + x}}{\color{blue}{y + x}} \cdot \frac{y}{\left(y + x\right) + 1} \]

   8. Applied rewrites 99.9%

      \[\leadsto \color{blue}{\frac{\frac{x}{y + x}}{y + x}} \cdot \frac{y}{\left(y + x\right) + 1} \]

   9. Taylor expanded in y around inf

      \[\leadsto \frac{\frac{x}{y + x}}{y + x} \cdot \color{blue}{1} \]
   10. Step-by-step derivation

       1. +-commutative 86.1

          \[\leadsto \frac{\frac{x}{y + x}}{y + x} \cdot 1 \]

   11. Applied rewrites 86.1%

       \[\leadsto \frac{\frac{x}{y + x}}{y + x} \cdot \color{blue}{1} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 8: 66.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 4.4 \cdot 10^{-151}:\\ \;\;\;\;\frac{1}{y + x} \cdot \frac{y}{\left(y + x\right) + 1}\\ \mathbf{elif}\;y \leq 2.1 \cdot 10^{+146}:\\ \;\;\;\;\frac{x}{\left(y + x\right) \cdot \left(y + x\right)} \cdot \frac{y}{y + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + x}}{y + x} \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 4.4e-151)
   (* (/ 1.0 (+ y x)) (/ y (+ (+ y x) 1.0)))
   (if (<= y 2.1e+146)
     (* (/ x (* (+ y x) (+ y x))) (/ y (+ y 1.0)))
     (* (/ (/ x (+ y x)) (+ y x)) 1.0))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 4.4e-151) {
		tmp = (1.0 / (y + x)) * (y / ((y + x) + 1.0));
	} else if (y <= 2.1e+146) {
		tmp = (x / ((y + x) * (y + x))) * (y / (y + 1.0));
	} 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 (y <= 4.4d-151) then
        tmp = (1.0d0 / (y + x)) * (y / ((y + x) + 1.0d0))
    else if (y <= 2.1d+146) then
        tmp = (x / ((y + x) * (y + x))) * (y / (y + 1.0d0))
    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 (y <= 4.4e-151) {
		tmp = (1.0 / (y + x)) * (y / ((y + x) + 1.0));
	} else if (y <= 2.1e+146) {
		tmp = (x / ((y + x) * (y + x))) * (y / (y + 1.0));
	} else {
		tmp = ((x / (y + x)) / (y + x)) * 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 4.4e-151:
		tmp = (1.0 / (y + x)) * (y / ((y + x) + 1.0))
	elif y <= 2.1e+146:
		tmp = (x / ((y + x) * (y + x))) * (y / (y + 1.0))
	else:
		tmp = ((x / (y + x)) / (y + x)) * 1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 4.4e-151)
		tmp = Float64(Float64(1.0 / Float64(y + x)) * Float64(y / Float64(Float64(y + x) + 1.0)));
	elseif (y <= 2.1e+146)
		tmp = Float64(Float64(x / Float64(Float64(y + x) * Float64(y + x))) * Float64(y / Float64(y + 1.0)));
	else
		tmp = Float64(Float64(Float64(x / Float64(y + x)) / Float64(y + x)) * 1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 4.4e-151)
		tmp = (1.0 / (y + x)) * (y / ((y + x) + 1.0));
	elseif (y <= 2.1e+146)
		tmp = (x / ((y + x) * (y + x))) * (y / (y + 1.0));
	else
		tmp = ((x / (y + x)) / (y + x)) * 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 4.4e-151], N[(N[(1.0 / N[(y + x), $MachinePrecision]), $MachinePrecision] * N[(y / N[(N[(y + x), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2.1e+146], N[(N[(x / N[(N[(y + x), $MachinePrecision] * N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(y / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x / N[(y + x), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 4.3999999999999999e-151

   1. Initial program 68.3%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot y}}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      2. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      4. lift-+.f64 N/A

         \[\leadsto \frac{x \cdot y}{\left(\color{blue}{\left(x + y\right)} \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      5. lift-+.f64 N/A

         \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \color{blue}{\left(x + y\right)}\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right)} \cdot \left(\left(x + y\right) + 1\right)} \]

      7. lift-+.f64 N/A

         \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\color{blue}{\left(x + y\right)} + 1\right)} \]

      8. lift-+.f64 N/A

         \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \color{blue}{\left(\left(x + y\right) + 1\right)}} \]

      9. times-frac N/A

         \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}} \]

      10. lower-*.f64 N/A

          \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}} \]

      11. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)}} \cdot \frac{y}{\left(x + y\right) + 1} \]

      12. pow2 N/A

          \[\leadsto \frac{x}{\color{blue}{{\left(x + y\right)}^{2}}} \cdot \frac{y}{\left(x + y\right) + 1} \]

      13. lower-pow.f64 N/A

          \[\leadsto \frac{x}{\color{blue}{{\left(x + y\right)}^{2}}} \cdot \frac{y}{\left(x + y\right) + 1} \]

      14. +-commutative N/A

          \[\leadsto \frac{x}{{\color{blue}{\left(y + x\right)}}^{2}} \cdot \frac{y}{\left(x + y\right) + 1} \]

      15. lower-+.f64 N/A

          \[\leadsto \frac{x}{{\color{blue}{\left(y + x\right)}}^{2}} \cdot \frac{y}{\left(x + y\right) + 1} \]

      16. lower-/.f64 N/A

          \[\leadsto \frac{x}{{\left(y + x\right)}^{2}} \cdot \color{blue}{\frac{y}{\left(x + y\right) + 1}} \]

      17. lift-+.f64 N/A

          \[\leadsto \frac{x}{{\left(y + x\right)}^{2}} \cdot \frac{y}{\color{blue}{\left(x + y\right) + 1}} \]

      18. +-commutative N/A

          \[\leadsto \frac{x}{{\left(y + x\right)}^{2}} \cdot \frac{y}{\color{blue}{\left(y + x\right)} + 1} \]

      19. lower-+.f64 85.9

          \[\leadsto \frac{x}{{\left(y + x\right)}^{2}} \cdot \frac{y}{\color{blue}{\left(y + x\right)} + 1} \]

   4. Applied rewrites 85.9%

      \[\leadsto \color{blue}{\frac{x}{{\left(y + x\right)}^{2}} \cdot \frac{y}{\left(y + x\right) + 1}} \]
   5. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{x}{{\color{blue}{\left(y + x\right)}}^{2}} \cdot \frac{y}{\left(y + x\right) + 1} \]

      2. +-commutative N/A

         \[\leadsto \frac{x}{{\color{blue}{\left(x + y\right)}}^{2}} \cdot \frac{y}{\left(y + x\right) + 1} \]

      3. lower-pow.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{{\left(x + y\right)}^{2}}} \cdot \frac{y}{\left(y + x\right) + 1} \]

      4. pow2 N/A

         \[\leadsto \frac{x}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right)}} \cdot \frac{y}{\left(y + x\right) + 1} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right)}} \cdot \frac{y}{\left(y + x\right) + 1} \]

      6. +-commutative N/A

         \[\leadsto \frac{x}{\color{blue}{\left(y + x\right)} \cdot \left(x + y\right)} \cdot \frac{y}{\left(y + x\right) + 1} \]

      7. lift-+.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{\left(y + x\right)} \cdot \left(x + y\right)} \cdot \frac{y}{\left(y + x\right) + 1} \]

      8. +-commutative N/A

         \[\leadsto \frac{x}{\left(y + x\right) \cdot \color{blue}{\left(y + x\right)}} \cdot \frac{y}{\left(y + x\right) + 1} \]

      9. lift-+.f64 85.9

         \[\leadsto \frac{x}{\left(y + x\right) \cdot \color{blue}{\left(y + x\right)}} \cdot \frac{y}{\left(y + x\right) + 1} \]

   6. Applied rewrites 85.9%

      \[\leadsto \frac{x}{\color{blue}{\left(y + x\right) \cdot \left(y + x\right)}} \cdot \frac{y}{\left(y + x\right) + 1} \]
   7. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{\left(y + x\right) \cdot \left(y + x\right)}} \cdot \frac{y}{\left(y + x\right) + 1} \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{\left(y + x\right)} \cdot \left(y + x\right)} \cdot \frac{y}{\left(y + x\right) + 1} \]

      3. lift-+.f64 N/A

         \[\leadsto \frac{x}{\left(y + x\right) \cdot \color{blue}{\left(y + x\right)}} \cdot \frac{y}{\left(y + x\right) + 1} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{\left(y + x\right) \cdot \left(y + x\right)}} \cdot \frac{y}{\left(y + x\right) + 1} \]

      5. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{x}{y + x}}{y + x}} \cdot \frac{y}{\left(y + x\right) + 1} \]

      6. +-commutative N/A

         \[\leadsto \frac{\frac{x}{\color{blue}{x + y}}}{y + x} \cdot \frac{y}{\left(y + x\right) + 1} \]

      7. +-commutative N/A

         \[\leadsto \frac{\frac{x}{x + y}}{\color{blue}{x + y}} \cdot \frac{y}{\left(y + x\right) + 1} \]

      8. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{x}{x + y}}{x + y}} \cdot \frac{y}{\left(y + x\right) + 1} \]

      9. lower-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{x}{x + y}}}{x + y} \cdot \frac{y}{\left(y + x\right) + 1} \]

      10. +-commutative N/A

          \[\leadsto \frac{\frac{x}{\color{blue}{y + x}}}{x + y} \cdot \frac{y}{\left(y + x\right) + 1} \]

      11. lift-+.f64 N/A

          \[\leadsto \frac{\frac{x}{\color{blue}{y + x}}}{x + y} \cdot \frac{y}{\left(y + x\right) + 1} \]

      12. +-commutative N/A

          \[\leadsto \frac{\frac{x}{y + x}}{\color{blue}{y + x}} \cdot \frac{y}{\left(y + x\right) + 1} \]

      13. lift-+.f64 99.7

          \[\leadsto \frac{\frac{x}{y + x}}{\color{blue}{y + x}} \cdot \frac{y}{\left(y + x\right) + 1} \]

   8. Applied rewrites 99.7%

      \[\leadsto \color{blue}{\frac{\frac{x}{y + x}}{y + x}} \cdot \frac{y}{\left(y + x\right) + 1} \]

   9. Taylor expanded in x around inf

      \[\leadsto \frac{\color{blue}{1}}{y + x} \cdot \frac{y}{\left(y + x\right) + 1} \]
   10. Step-by-step derivation

   11. Applied rewrites 58.1%

       \[\leadsto \frac{\color{blue}{1}}{y + x} \cdot \frac{y}{\left(y + x\right) + 1} \]

   if 4.3999999999999999e-151 < y < 2.1000000000000001e146

   1. Initial program 77.3%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot y}}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      2. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      4. lift-+.f64 N/A

         \[\leadsto \frac{x \cdot y}{\left(\color{blue}{\left(x + y\right)} \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      5. lift-+.f64 N/A

         \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \color{blue}{\left(x + y\right)}\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right)} \cdot \left(\left(x + y\right) + 1\right)} \]

      7. lift-+.f64 N/A

         \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\color{blue}{\left(x + y\right)} + 1\right)} \]

      8. lift-+.f64 N/A

         \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \color{blue}{\left(\left(x + y\right) + 1\right)}} \]

      9. times-frac N/A

         \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}} \]

      10. lower-*.f64 N/A

          \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}} \]

      11. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)}} \cdot \frac{y}{\left(x + y\right) + 1} \]

      12. pow2 N/A

          \[\leadsto \frac{x}{\color{blue}{{\left(x + y\right)}^{2}}} \cdot \frac{y}{\left(x + y\right) + 1} \]

      13. lower-pow.f64 N/A

          \[\leadsto \frac{x}{\color{blue}{{\left(x + y\right)}^{2}}} \cdot \frac{y}{\left(x + y\right) + 1} \]

      14. +-commutative N/A

          \[\leadsto \frac{x}{{\color{blue}{\left(y + x\right)}}^{2}} \cdot \frac{y}{\left(x + y\right) + 1} \]

      15. lower-+.f64 N/A

          \[\leadsto \frac{x}{{\color{blue}{\left(y + x\right)}}^{2}} \cdot \frac{y}{\left(x + y\right) + 1} \]

      16. lower-/.f64 N/A

          \[\leadsto \frac{x}{{\left(y + x\right)}^{2}} \cdot \color{blue}{\frac{y}{\left(x + y\right) + 1}} \]

      17. lift-+.f64 N/A

          \[\leadsto \frac{x}{{\left(y + x\right)}^{2}} \cdot \frac{y}{\color{blue}{\left(x + y\right) + 1}} \]

      18. +-commutative N/A

          \[\leadsto \frac{x}{{\left(y + x\right)}^{2}} \cdot \frac{y}{\color{blue}{\left(y + x\right)} + 1} \]

      19. lower-+.f64 96.3

          \[\leadsto \frac{x}{{\left(y + x\right)}^{2}} \cdot \frac{y}{\color{blue}{\left(y + x\right)} + 1} \]

   4. Applied rewrites 96.3%

      \[\leadsto \color{blue}{\frac{x}{{\left(y + x\right)}^{2}} \cdot \frac{y}{\left(y + x\right) + 1}} \]
   5. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{x}{{\color{blue}{\left(y + x\right)}}^{2}} \cdot \frac{y}{\left(y + x\right) + 1} \]

      2. +-commutative N/A

         \[\leadsto \frac{x}{{\color{blue}{\left(x + y\right)}}^{2}} \cdot \frac{y}{\left(y + x\right) + 1} \]

      3. lower-pow.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{{\left(x + y\right)}^{2}}} \cdot \frac{y}{\left(y + x\right) + 1} \]

      4. pow2 N/A

         \[\leadsto \frac{x}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right)}} \cdot \frac{y}{\left(y + x\right) + 1} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right)}} \cdot \frac{y}{\left(y + x\right) + 1} \]

      6. +-commutative N/A

         \[\leadsto \frac{x}{\color{blue}{\left(y + x\right)} \cdot \left(x + y\right)} \cdot \frac{y}{\left(y + x\right) + 1} \]

      7. lift-+.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{\left(y + x\right)} \cdot \left(x + y\right)} \cdot \frac{y}{\left(y + x\right) + 1} \]

      8. +-commutative N/A

         \[\leadsto \frac{x}{\left(y + x\right) \cdot \color{blue}{\left(y + x\right)}} \cdot \frac{y}{\left(y + x\right) + 1} \]

      9. lift-+.f64 96.3

         \[\leadsto \frac{x}{\left(y + x\right) \cdot \color{blue}{\left(y + x\right)}} \cdot \frac{y}{\left(y + x\right) + 1} \]

   6. Applied rewrites 96.3%

      \[\leadsto \frac{x}{\color{blue}{\left(y + x\right) \cdot \left(y + x\right)}} \cdot \frac{y}{\left(y + x\right) + 1} \]

   7. Taylor expanded in x around 0

      \[\leadsto \frac{x}{\left(y + x\right) \cdot \left(y + x\right)} \cdot \frac{y}{\color{blue}{y} + 1} \]
   8. Step-by-step derivation

      1. +-commutative 78.0

         \[\leadsto \frac{x}{\left(y + x\right) \cdot \left(y + x\right)} \cdot \frac{y}{y + 1} \]

   9. Applied rewrites 78.0%

      \[\leadsto \frac{x}{\left(y + x\right) \cdot \left(y + x\right)} \cdot \frac{y}{\color{blue}{y} + 1} \]

   if 2.1000000000000001e146 < y

   1. Initial program 57.7%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot y}}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      2. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      4. lift-+.f64 N/A

         \[\leadsto \frac{x \cdot y}{\left(\color{blue}{\left(x + y\right)} \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      5. lift-+.f64 N/A

         \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \color{blue}{\left(x + y\right)}\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right)} \cdot \left(\left(x + y\right) + 1\right)} \]

      7. lift-+.f64 N/A

         \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\color{blue}{\left(x + y\right)} + 1\right)} \]

      8. lift-+.f64 N/A

         \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \color{blue}{\left(\left(x + y\right) + 1\right)}} \]

      9. times-frac N/A

         \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}} \]

      10. lower-*.f64 N/A

          \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}} \]

      11. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)}} \cdot \frac{y}{\left(x + y\right) + 1} \]

      12. pow2 N/A

          \[\leadsto \frac{x}{\color{blue}{{\left(x + y\right)}^{2}}} \cdot \frac{y}{\left(x + y\right) + 1} \]

      13. lower-pow.f64 N/A

          \[\leadsto \frac{x}{\color{blue}{{\left(x + y\right)}^{2}}} \cdot \frac{y}{\left(x + y\right) + 1} \]

      14. +-commutative N/A

          \[\leadsto \frac{x}{{\color{blue}{\left(y + x\right)}}^{2}} \cdot \frac{y}{\left(x + y\right) + 1} \]

      15. lower-+.f64 N/A

          \[\leadsto \frac{x}{{\color{blue}{\left(y + x\right)}}^{2}} \cdot \frac{y}{\left(x + y\right) + 1} \]

      16. lower-/.f64 N/A

          \[\leadsto \frac{x}{{\left(y + x\right)}^{2}} \cdot \color{blue}{\frac{y}{\left(x + y\right) + 1}} \]

      17. lift-+.f64 N/A

          \[\leadsto \frac{x}{{\left(y + x\right)}^{2}} \cdot \frac{y}{\color{blue}{\left(x + y\right) + 1}} \]

      18. +-commutative N/A

          \[\leadsto \frac{x}{{\left(y + x\right)}^{2}} \cdot \frac{y}{\color{blue}{\left(y + x\right)} + 1} \]

      19. lower-+.f64 80.1

          \[\leadsto \frac{x}{{\left(y + x\right)}^{2}} \cdot \frac{y}{\color{blue}{\left(y + x\right)} + 1} \]

   4. Applied rewrites 80.1%

      \[\leadsto \color{blue}{\frac{x}{{\left(y + x\right)}^{2}} \cdot \frac{y}{\left(y + x\right) + 1}} \]
   5. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{x}{{\color{blue}{\left(y + x\right)}}^{2}} \cdot \frac{y}{\left(y + x\right) + 1} \]

      2. +-commutative N/A

         \[\leadsto \frac{x}{{\color{blue}{\left(x + y\right)}}^{2}} \cdot \frac{y}{\left(y + x\right) + 1} \]

      3. lower-pow.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{{\left(x + y\right)}^{2}}} \cdot \frac{y}{\left(y + x\right) + 1} \]

      4. pow2 N/A

         \[\leadsto \frac{x}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right)}} \cdot \frac{y}{\left(y + x\right) + 1} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right)}} \cdot \frac{y}{\left(y + x\right) + 1} \]

      6. +-commutative N/A

         \[\leadsto \frac{x}{\color{blue}{\left(y + x\right)} \cdot \left(x + y\right)} \cdot \frac{y}{\left(y + x\right) + 1} \]

      7. lift-+.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{\left(y + x\right)} \cdot \left(x + y\right)} \cdot \frac{y}{\left(y + x\right) + 1} \]

      8. +-commutative N/A

         \[\leadsto \frac{x}{\left(y + x\right) \cdot \color{blue}{\left(y + x\right)}} \cdot \frac{y}{\left(y + x\right) + 1} \]

      9. lift-+.f64 80.1

         \[\leadsto \frac{x}{\left(y + x\right) \cdot \color{blue}{\left(y + x\right)}} \cdot \frac{y}{\left(y + x\right) + 1} \]

   6. Applied rewrites 80.1%

      \[\leadsto \frac{x}{\color{blue}{\left(y + x\right) \cdot \left(y + x\right)}} \cdot \frac{y}{\left(y + x\right) + 1} \]
   7. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{\left(y + x\right) \cdot \left(y + x\right)}} \cdot \frac{y}{\left(y + x\right) + 1} \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{\left(y + x\right)} \cdot \left(y + x\right)} \cdot \frac{y}{\left(y + x\right) + 1} \]

      3. lift-+.f64 N/A

         \[\leadsto \frac{x}{\left(y + x\right) \cdot \color{blue}{\left(y + x\right)}} \cdot \frac{y}{\left(y + x\right) + 1} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{\left(y + x\right) \cdot \left(y + x\right)}} \cdot \frac{y}{\left(y + x\right) + 1} \]

      5. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{x}{y + x}}{y + x}} \cdot \frac{y}{\left(y + x\right) + 1} \]

      6. +-commutative N/A

         \[\leadsto \frac{\frac{x}{\color{blue}{x + y}}}{y + x} \cdot \frac{y}{\left(y + x\right) + 1} \]

      7. +-commutative N/A

         \[\leadsto \frac{\frac{x}{x + y}}{\color{blue}{x + y}} \cdot \frac{y}{\left(y + x\right) + 1} \]

      8. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{x}{x + y}}{x + y}} \cdot \frac{y}{\left(y + x\right) + 1} \]

      9. lower-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{x}{x + y}}}{x + y} \cdot \frac{y}{\left(y + x\right) + 1} \]

      10. +-commutative N/A

          \[\leadsto \frac{\frac{x}{\color{blue}{y + x}}}{x + y} \cdot \frac{y}{\left(y + x\right) + 1} \]

      11. lift-+.f64 N/A

          \[\leadsto \frac{\frac{x}{\color{blue}{y + x}}}{x + y} \cdot \frac{y}{\left(y + x\right) + 1} \]

      12. +-commutative N/A

          \[\leadsto \frac{\frac{x}{y + x}}{\color{blue}{y + x}} \cdot \frac{y}{\left(y + x\right) + 1} \]

      13. lift-+.f64 99.9

          \[\leadsto \frac{\frac{x}{y + x}}{\color{blue}{y + x}} \cdot \frac{y}{\left(y + x\right) + 1} \]

   8. Applied rewrites 99.9%

      \[\leadsto \color{blue}{\frac{\frac{x}{y + x}}{y + x}} \cdot \frac{y}{\left(y + x\right) + 1} \]

   9. Taylor expanded in y around inf

      \[\leadsto \frac{\frac{x}{y + x}}{y + x} \cdot \color{blue}{1} \]
   10. Step-by-step derivation

       1. +-commutative 86.2

          \[\leadsto \frac{\frac{x}{y + x}}{y + x} \cdot 1 \]

   11. Applied rewrites 86.2%

       \[\leadsto \frac{\frac{x}{y + x}}{y + x} \cdot \color{blue}{1} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 9: 65.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 7.8 \cdot 10^{-60}:\\ \;\;\;\;\frac{1}{y + x} \cdot \frac{y}{\left(y + x\right) + 1}\\ \mathbf{elif}\;y \leq 2 \cdot 10^{+16}:\\ \;\;\;\;\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(y + 1\right)}\\ \mathbf{elif}\;y \leq 2.1 \cdot 10^{+146}:\\ \;\;\;\;\frac{x}{\left(y + x\right) \cdot \left(y + x\right)} \cdot 1\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + x}}{y + x} \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 7.8e-60)
   (* (/ 1.0 (+ y x)) (/ y (+ (+ y x) 1.0)))
   (if (<= y 2e+16)
     (/ (* x y) (* (* (+ x y) (+ x y)) (+ y 1.0)))
     (if (<= y 2.1e+146)
       (* (/ x (* (+ y x) (+ y x))) 1.0)
       (* (/ (/ x (+ y x)) (+ y x)) 1.0)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 7.8e-60) {
		tmp = (1.0 / (y + x)) * (y / ((y + x) + 1.0));
	} else if (y <= 2e+16) {
		tmp = (x * y) / (((x + y) * (x + y)) * (y + 1.0));
	} else if (y <= 2.1e+146) {
		tmp = (x / ((y + x) * (y + x))) * 1.0;
	} 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 (y <= 7.8d-60) then
        tmp = (1.0d0 / (y + x)) * (y / ((y + x) + 1.0d0))
    else if (y <= 2d+16) then
        tmp = (x * y) / (((x + y) * (x + y)) * (y + 1.0d0))
    else if (y <= 2.1d+146) then
        tmp = (x / ((y + x) * (y + x))) * 1.0d0
    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 (y <= 7.8e-60) {
		tmp = (1.0 / (y + x)) * (y / ((y + x) + 1.0));
	} else if (y <= 2e+16) {
		tmp = (x * y) / (((x + y) * (x + y)) * (y + 1.0));
	} else if (y <= 2.1e+146) {
		tmp = (x / ((y + x) * (y + x))) * 1.0;
	} else {
		tmp = ((x / (y + x)) / (y + x)) * 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 7.8e-60:
		tmp = (1.0 / (y + x)) * (y / ((y + x) + 1.0))
	elif y <= 2e+16:
		tmp = (x * y) / (((x + y) * (x + y)) * (y + 1.0))
	elif y <= 2.1e+146:
		tmp = (x / ((y + x) * (y + x))) * 1.0
	else:
		tmp = ((x / (y + x)) / (y + x)) * 1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 7.8e-60)
		tmp = Float64(Float64(1.0 / Float64(y + x)) * Float64(y / Float64(Float64(y + x) + 1.0)));
	elseif (y <= 2e+16)
		tmp = Float64(Float64(x * y) / Float64(Float64(Float64(x + y) * Float64(x + y)) * Float64(y + 1.0)));
	elseif (y <= 2.1e+146)
		tmp = Float64(Float64(x / Float64(Float64(y + x) * Float64(y + x))) * 1.0);
	else
		tmp = Float64(Float64(Float64(x / Float64(y + x)) / Float64(y + x)) * 1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 7.8e-60)
		tmp = (1.0 / (y + x)) * (y / ((y + x) + 1.0));
	elseif (y <= 2e+16)
		tmp = (x * y) / (((x + y) * (x + y)) * (y + 1.0));
	elseif (y <= 2.1e+146)
		tmp = (x / ((y + x) * (y + x))) * 1.0;
	else
		tmp = ((x / (y + x)) / (y + x)) * 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 7.8e-60], N[(N[(1.0 / N[(y + x), $MachinePrecision]), $MachinePrecision] * N[(y / N[(N[(y + x), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2e+16], N[(N[(x * y), $MachinePrecision] / N[(N[(N[(x + y), $MachinePrecision] * N[(x + y), $MachinePrecision]), $MachinePrecision] * N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2.1e+146], N[(N[(x / N[(N[(y + x), $MachinePrecision] * N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision], N[(N[(N[(x / N[(y + x), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if y < 7.8000000000000004e-60

   1. Initial program 70.1%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot y}}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      2. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      4. lift-+.f64 N/A

         \[\leadsto \frac{x \cdot y}{\left(\color{blue}{\left(x + y\right)} \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      5. lift-+.f64 N/A

         \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \color{blue}{\left(x + y\right)}\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right)} \cdot \left(\left(x + y\right) + 1\right)} \]

      7. lift-+.f64 N/A

         \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\color{blue}{\left(x + y\right)} + 1\right)} \]

      8. lift-+.f64 N/A

         \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \color{blue}{\left(\left(x + y\right) + 1\right)}} \]

      9. times-frac N/A

         \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}} \]

      10. lower-*.f64 N/A

          \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}} \]

      11. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)}} \cdot \frac{y}{\left(x + y\right) + 1} \]

      12. pow2 N/A

          \[\leadsto \frac{x}{\color{blue}{{\left(x + y\right)}^{2}}} \cdot \frac{y}{\left(x + y\right) + 1} \]

      13. lower-pow.f64 N/A

          \[\leadsto \frac{x}{\color{blue}{{\left(x + y\right)}^{2}}} \cdot \frac{y}{\left(x + y\right) + 1} \]

      14. +-commutative N/A

          \[\leadsto \frac{x}{{\color{blue}{\left(y + x\right)}}^{2}} \cdot \frac{y}{\left(x + y\right) + 1} \]

      15. lower-+.f64 N/A

          \[\leadsto \frac{x}{{\color{blue}{\left(y + x\right)}}^{2}} \cdot \frac{y}{\left(x + y\right) + 1} \]

      16. lower-/.f64 N/A

          \[\leadsto \frac{x}{{\left(y + x\right)}^{2}} \cdot \color{blue}{\frac{y}{\left(x + y\right) + 1}} \]

      17. lift-+.f64 N/A

          \[\leadsto \frac{x}{{\left(y + x\right)}^{2}} \cdot \frac{y}{\color{blue}{\left(x + y\right) + 1}} \]

      18. +-commutative N/A

          \[\leadsto \frac{x}{{\left(y + x\right)}^{2}} \cdot \frac{y}{\color{blue}{\left(y + x\right)} + 1} \]

      19. lower-+.f64 87.4

          \[\leadsto \frac{x}{{\left(y + x\right)}^{2}} \cdot \frac{y}{\color{blue}{\left(y + x\right)} + 1} \]

   4. Applied rewrites 87.4%

      \[\leadsto \color{blue}{\frac{x}{{\left(y + x\right)}^{2}} \cdot \frac{y}{\left(y + x\right) + 1}} \]
   5. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{x}{{\color{blue}{\left(y + x\right)}}^{2}} \cdot \frac{y}{\left(y + x\right) + 1} \]

      2. +-commutative N/A

         \[\leadsto \frac{x}{{\color{blue}{\left(x + y\right)}}^{2}} \cdot \frac{y}{\left(y + x\right) + 1} \]

      3. lower-pow.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{{\left(x + y\right)}^{2}}} \cdot \frac{y}{\left(y + x\right) + 1} \]

      4. pow2 N/A

         \[\leadsto \frac{x}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right)}} \cdot \frac{y}{\left(y + x\right) + 1} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right)}} \cdot \frac{y}{\left(y + x\right) + 1} \]

      6. +-commutative N/A

         \[\leadsto \frac{x}{\color{blue}{\left(y + x\right)} \cdot \left(x + y\right)} \cdot \frac{y}{\left(y + x\right) + 1} \]

      7. lift-+.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{\left(y + x\right)} \cdot \left(x + y\right)} \cdot \frac{y}{\left(y + x\right) + 1} \]

      8. +-commutative N/A

         \[\leadsto \frac{x}{\left(y + x\right) \cdot \color{blue}{\left(y + x\right)}} \cdot \frac{y}{\left(y + x\right) + 1} \]

      9. lift-+.f64 87.4

         \[\leadsto \frac{x}{\left(y + x\right) \cdot \color{blue}{\left(y + x\right)}} \cdot \frac{y}{\left(y + x\right) + 1} \]

   6. Applied rewrites 87.4%

      \[\leadsto \frac{x}{\color{blue}{\left(y + x\right) \cdot \left(y + x\right)}} \cdot \frac{y}{\left(y + x\right) + 1} \]
   7. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{\left(y + x\right) \cdot \left(y + x\right)}} \cdot \frac{y}{\left(y + x\right) + 1} \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{\left(y + x\right)} \cdot \left(y + x\right)} \cdot \frac{y}{\left(y + x\right) + 1} \]

      3. lift-+.f64 N/A

         \[\leadsto \frac{x}{\left(y + x\right) \cdot \color{blue}{\left(y + x\right)}} \cdot \frac{y}{\left(y + x\right) + 1} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{\left(y + x\right) \cdot \left(y + x\right)}} \cdot \frac{y}{\left(y + x\right) + 1} \]

      5. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{x}{y + x}}{y + x}} \cdot \frac{y}{\left(y + x\right) + 1} \]

      6. +-commutative N/A

         \[\leadsto \frac{\frac{x}{\color{blue}{x + y}}}{y + x} \cdot \frac{y}{\left(y + x\right) + 1} \]

      7. +-commutative N/A

         \[\leadsto \frac{\frac{x}{x + y}}{\color{blue}{x + y}} \cdot \frac{y}{\left(y + x\right) + 1} \]

      8. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{x}{x + y}}{x + y}} \cdot \frac{y}{\left(y + x\right) + 1} \]

      9. lower-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{x}{x + y}}}{x + y} \cdot \frac{y}{\left(y + x\right) + 1} \]

      10. +-commutative N/A

          \[\leadsto \frac{\frac{x}{\color{blue}{y + x}}}{x + y} \cdot \frac{y}{\left(y + x\right) + 1} \]

      11. lift-+.f64 N/A

          \[\leadsto \frac{\frac{x}{\color{blue}{y + x}}}{x + y} \cdot \frac{y}{\left(y + x\right) + 1} \]

      12. +-commutative N/A

          \[\leadsto \frac{\frac{x}{y + x}}{\color{blue}{y + x}} \cdot \frac{y}{\left(y + x\right) + 1} \]

      13. lift-+.f64 99.7

          \[\leadsto \frac{\frac{x}{y + x}}{\color{blue}{y + x}} \cdot \frac{y}{\left(y + x\right) + 1} \]

   8. Applied rewrites 99.7%

      \[\leadsto \color{blue}{\frac{\frac{x}{y + x}}{y + x}} \cdot \frac{y}{\left(y + x\right) + 1} \]

   9. Taylor expanded in x around inf

      \[\leadsto \frac{\color{blue}{1}}{y + x} \cdot \frac{y}{\left(y + x\right) + 1} \]
   10. Step-by-step derivation

   11. Applied rewrites 59.1%

       \[\leadsto \frac{\color{blue}{1}}{y + x} \cdot \frac{y}{\left(y + x\right) + 1} \]

   if 7.8000000000000004e-60 < y < 2e16

   1. Initial program 91.4%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\color{blue}{y} + 1\right)} \]
   4. Step-by-step derivation

   5. Applied rewrites 73.7%

      \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\color{blue}{y} + 1\right)} \]

   if 2e16 < y < 2.1000000000000001e146

   1. Initial program 63.2%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot y}}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      2. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      4. lift-+.f64 N/A

         \[\leadsto \frac{x \cdot y}{\left(\color{blue}{\left(x + y\right)} \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      5. lift-+.f64 N/A

         \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \color{blue}{\left(x + y\right)}\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right)} \cdot \left(\left(x + y\right) + 1\right)} \]

      7. lift-+.f64 N/A

         \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\color{blue}{\left(x + y\right)} + 1\right)} \]

      8. lift-+.f64 N/A

         \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \color{blue}{\left(\left(x + y\right) + 1\right)}} \]

      9. times-frac N/A

         \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}} \]

      10. lower-*.f64 N/A

          \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}} \]

      11. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)}} \cdot \frac{y}{\left(x + y\right) + 1} \]

      12. pow2 N/A

          \[\leadsto \frac{x}{\color{blue}{{\left(x + y\right)}^{2}}} \cdot \frac{y}{\left(x + y\right) + 1} \]

      13. lower-pow.f64 N/A

          \[\leadsto \frac{x}{\color{blue}{{\left(x + y\right)}^{2}}} \cdot \frac{y}{\left(x + y\right) + 1} \]

      14. +-commutative N/A

          \[\leadsto \frac{x}{{\color{blue}{\left(y + x\right)}}^{2}} \cdot \frac{y}{\left(x + y\right) + 1} \]

      15. lower-+.f64 N/A

          \[\leadsto \frac{x}{{\color{blue}{\left(y + x\right)}}^{2}} \cdot \frac{y}{\left(x + y\right) + 1} \]

      16. lower-/.f64 N/A

          \[\leadsto \frac{x}{{\left(y + x\right)}^{2}} \cdot \color{blue}{\frac{y}{\left(x + y\right) + 1}} \]

      17. lift-+.f64 N/A

          \[\leadsto \frac{x}{{\left(y + x\right)}^{2}} \cdot \frac{y}{\color{blue}{\left(x + y\right) + 1}} \]

      18. +-commutative N/A

          \[\leadsto \frac{x}{{\left(y + x\right)}^{2}} \cdot \frac{y}{\color{blue}{\left(y + x\right)} + 1} \]

      19. lower-+.f64 92.2

          \[\leadsto \frac{x}{{\left(y + x\right)}^{2}} \cdot \frac{y}{\color{blue}{\left(y + x\right)} + 1} \]

   4. Applied rewrites 92.2%

      \[\leadsto \color{blue}{\frac{x}{{\left(y + x\right)}^{2}} \cdot \frac{y}{\left(y + x\right) + 1}} \]
   5. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{x}{{\color{blue}{\left(y + x\right)}}^{2}} \cdot \frac{y}{\left(y + x\right) + 1} \]

      2. +-commutative N/A

         \[\leadsto \frac{x}{{\color{blue}{\left(x + y\right)}}^{2}} \cdot \frac{y}{\left(y + x\right) + 1} \]

      3. lower-pow.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{{\left(x + y\right)}^{2}}} \cdot \frac{y}{\left(y + x\right) + 1} \]

      4. pow2 N/A

         \[\leadsto \frac{x}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right)}} \cdot \frac{y}{\left(y + x\right) + 1} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right)}} \cdot \frac{y}{\left(y + x\right) + 1} \]

      6. +-commutative N/A

         \[\leadsto \frac{x}{\color{blue}{\left(y + x\right)} \cdot \left(x + y\right)} \cdot \frac{y}{\left(y + x\right) + 1} \]

      7. lift-+.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{\left(y + x\right)} \cdot \left(x + y\right)} \cdot \frac{y}{\left(y + x\right) + 1} \]

      8. +-commutative N/A

         \[\leadsto \frac{x}{\left(y + x\right) \cdot \color{blue}{\left(y + x\right)}} \cdot \frac{y}{\left(y + x\right) + 1} \]

      9. lift-+.f64 92.2

         \[\leadsto \frac{x}{\left(y + x\right) \cdot \color{blue}{\left(y + x\right)}} \cdot \frac{y}{\left(y + x\right) + 1} \]

   6. Applied rewrites 92.2%

      \[\leadsto \frac{x}{\color{blue}{\left(y + x\right) \cdot \left(y + x\right)}} \cdot \frac{y}{\left(y + x\right) + 1} \]

   7. Taylor expanded in y around inf

      \[\leadsto \frac{x}{\left(y + x\right) \cdot \left(y + x\right)} \cdot \color{blue}{1} \]
   8. Step-by-step derivation

   9. Applied rewrites 79.9%

      \[\leadsto \frac{x}{\left(y + x\right) \cdot \left(y + x\right)} \cdot \color{blue}{1} \]

   if 2.1000000000000001e146 < y

   1. Initial program 57.7%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot y}}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      2. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      4. lift-+.f64 N/A

         \[\leadsto \frac{x \cdot y}{\left(\color{blue}{\left(x + y\right)} \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      5. lift-+.f64 N/A

         \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \color{blue}{\left(x + y\right)}\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right)} \cdot \left(\left(x + y\right) + 1\right)} \]

      7. lift-+.f64 N/A

         \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\color{blue}{\left(x + y\right)} + 1\right)} \]

      8. lift-+.f64 N/A

         \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \color{blue}{\left(\left(x + y\right) + 1\right)}} \]

      9. times-frac N/A

         \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}} \]

      10. lower-*.f64 N/A

          \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}} \]

      11. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)}} \cdot \frac{y}{\left(x + y\right) + 1} \]

      12. pow2 N/A

          \[\leadsto \frac{x}{\color{blue}{{\left(x + y\right)}^{2}}} \cdot \frac{y}{\left(x + y\right) + 1} \]

      13. lower-pow.f64 N/A

          \[\leadsto \frac{x}{\color{blue}{{\left(x + y\right)}^{2}}} \cdot \frac{y}{\left(x + y\right) + 1} \]

      14. +-commutative N/A

          \[\leadsto \frac{x}{{\color{blue}{\left(y + x\right)}}^{2}} \cdot \frac{y}{\left(x + y\right) + 1} \]

      15. lower-+.f64 N/A

          \[\leadsto \frac{x}{{\color{blue}{\left(y + x\right)}}^{2}} \cdot \frac{y}{\left(x + y\right) + 1} \]

      16. lower-/.f64 N/A

          \[\leadsto \frac{x}{{\left(y + x\right)}^{2}} \cdot \color{blue}{\frac{y}{\left(x + y\right) + 1}} \]

      17. lift-+.f64 N/A

          \[\leadsto \frac{x}{{\left(y + x\right)}^{2}} \cdot \frac{y}{\color{blue}{\left(x + y\right) + 1}} \]

      18. +-commutative N/A

          \[\leadsto \frac{x}{{\left(y + x\right)}^{2}} \cdot \frac{y}{\color{blue}{\left(y + x\right)} + 1} \]

      19. lower-+.f64 80.1

          \[\leadsto \frac{x}{{\left(y + x\right)}^{2}} \cdot \frac{y}{\color{blue}{\left(y + x\right)} + 1} \]

   4. Applied rewrites 80.1%

      \[\leadsto \color{blue}{\frac{x}{{\left(y + x\right)}^{2}} \cdot \frac{y}{\left(y + x\right) + 1}} \]
   5. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{x}{{\color{blue}{\left(y + x\right)}}^{2}} \cdot \frac{y}{\left(y + x\right) + 1} \]

      2. +-commutative N/A

         \[\leadsto \frac{x}{{\color{blue}{\left(x + y\right)}}^{2}} \cdot \frac{y}{\left(y + x\right) + 1} \]

      3. lower-pow.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{{\left(x + y\right)}^{2}}} \cdot \frac{y}{\left(y + x\right) + 1} \]

      4. pow2 N/A

         \[\leadsto \frac{x}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right)}} \cdot \frac{y}{\left(y + x\right) + 1} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right)}} \cdot \frac{y}{\left(y + x\right) + 1} \]

      6. +-commutative N/A

         \[\leadsto \frac{x}{\color{blue}{\left(y + x\right)} \cdot \left(x + y\right)} \cdot \frac{y}{\left(y + x\right) + 1} \]

      7. lift-+.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{\left(y + x\right)} \cdot \left(x + y\right)} \cdot \frac{y}{\left(y + x\right) + 1} \]

      8. +-commutative N/A

         \[\leadsto \frac{x}{\left(y + x\right) \cdot \color{blue}{\left(y + x\right)}} \cdot \frac{y}{\left(y + x\right) + 1} \]

      9. lift-+.f64 80.1

         \[\leadsto \frac{x}{\left(y + x\right) \cdot \color{blue}{\left(y + x\right)}} \cdot \frac{y}{\left(y + x\right) + 1} \]

   6. Applied rewrites 80.1%

      \[\leadsto \frac{x}{\color{blue}{\left(y + x\right) \cdot \left(y + x\right)}} \cdot \frac{y}{\left(y + x\right) + 1} \]
   7. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{\left(y + x\right) \cdot \left(y + x\right)}} \cdot \frac{y}{\left(y + x\right) + 1} \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{\left(y + x\right)} \cdot \left(y + x\right)} \cdot \frac{y}{\left(y + x\right) + 1} \]

      3. lift-+.f64 N/A

         \[\leadsto \frac{x}{\left(y + x\right) \cdot \color{blue}{\left(y + x\right)}} \cdot \frac{y}{\left(y + x\right) + 1} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{\left(y + x\right) \cdot \left(y + x\right)}} \cdot \frac{y}{\left(y + x\right) + 1} \]

      5. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{x}{y + x}}{y + x}} \cdot \frac{y}{\left(y + x\right) + 1} \]

      6. +-commutative N/A

         \[\leadsto \frac{\frac{x}{\color{blue}{x + y}}}{y + x} \cdot \frac{y}{\left(y + x\right) + 1} \]

      7. +-commutative N/A

         \[\leadsto \frac{\frac{x}{x + y}}{\color{blue}{x + y}} \cdot \frac{y}{\left(y + x\right) + 1} \]

      8. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{x}{x + y}}{x + y}} \cdot \frac{y}{\left(y + x\right) + 1} \]

      9. lower-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{x}{x + y}}}{x + y} \cdot \frac{y}{\left(y + x\right) + 1} \]

      10. +-commutative N/A

          \[\leadsto \frac{\frac{x}{\color{blue}{y + x}}}{x + y} \cdot \frac{y}{\left(y + x\right) + 1} \]

      11. lift-+.f64 N/A

          \[\leadsto \frac{\frac{x}{\color{blue}{y + x}}}{x + y} \cdot \frac{y}{\left(y + x\right) + 1} \]

      12. +-commutative N/A

          \[\leadsto \frac{\frac{x}{y + x}}{\color{blue}{y + x}} \cdot \frac{y}{\left(y + x\right) + 1} \]

      13. lift-+.f64 99.9

          \[\leadsto \frac{\frac{x}{y + x}}{\color{blue}{y + x}} \cdot \frac{y}{\left(y + x\right) + 1} \]

   8. Applied rewrites 99.9%

      \[\leadsto \color{blue}{\frac{\frac{x}{y + x}}{y + x}} \cdot \frac{y}{\left(y + x\right) + 1} \]

   9. Taylor expanded in y around inf

      \[\leadsto \frac{\frac{x}{y + x}}{y + x} \cdot \color{blue}{1} \]
   10. Step-by-step derivation

       1. +-commutative 86.2

          \[\leadsto \frac{\frac{x}{y + x}}{y + x} \cdot 1 \]

   11. Applied rewrites 86.2%

       \[\leadsto \frac{\frac{x}{y + x}}{y + x} \cdot \color{blue}{1} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 10: 62.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -132000:\\ \;\;\;\;\frac{\frac{y}{x}}{x}\\ \mathbf{elif}\;y \leq 8 \cdot 10^{-55}:\\ \;\;\;\;\frac{y}{\left(1 + x\right) \cdot x}\\ \mathbf{elif}\;y \leq 2.1 \cdot 10^{+146}:\\ \;\;\;\;\frac{x}{\left(y + x\right) \cdot \left(y + x\right)} \cdot 1\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + x}}{y + x} \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -132000.0)
   (/ (/ y x) x)
   (if (<= y 8e-55)
     (/ y (* (+ 1.0 x) x))
     (if (<= y 2.1e+146)
       (* (/ x (* (+ y x) (+ y x))) 1.0)
       (* (/ (/ x (+ y x)) (+ y x)) 1.0)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -132000.0) {
		tmp = (y / x) / x;
	} else if (y <= 8e-55) {
		tmp = y / ((1.0 + x) * x);
	} else if (y <= 2.1e+146) {
		tmp = (x / ((y + x) * (y + x))) * 1.0;
	} 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 (y <= (-132000.0d0)) then
        tmp = (y / x) / x
    else if (y <= 8d-55) then
        tmp = y / ((1.0d0 + x) * x)
    else if (y <= 2.1d+146) then
        tmp = (x / ((y + x) * (y + x))) * 1.0d0
    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 (y <= -132000.0) {
		tmp = (y / x) / x;
	} else if (y <= 8e-55) {
		tmp = y / ((1.0 + x) * x);
	} else if (y <= 2.1e+146) {
		tmp = (x / ((y + x) * (y + x))) * 1.0;
	} else {
		tmp = ((x / (y + x)) / (y + x)) * 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -132000.0:
		tmp = (y / x) / x
	elif y <= 8e-55:
		tmp = y / ((1.0 + x) * x)
	elif y <= 2.1e+146:
		tmp = (x / ((y + x) * (y + x))) * 1.0
	else:
		tmp = ((x / (y + x)) / (y + x)) * 1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -132000.0)
		tmp = Float64(Float64(y / x) / x);
	elseif (y <= 8e-55)
		tmp = Float64(y / Float64(Float64(1.0 + x) * x));
	elseif (y <= 2.1e+146)
		tmp = Float64(Float64(x / Float64(Float64(y + x) * Float64(y + x))) * 1.0);
	else
		tmp = Float64(Float64(Float64(x / Float64(y + x)) / Float64(y + x)) * 1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -132000.0)
		tmp = (y / x) / x;
	elseif (y <= 8e-55)
		tmp = y / ((1.0 + x) * x);
	elseif (y <= 2.1e+146)
		tmp = (x / ((y + x) * (y + x))) * 1.0;
	else
		tmp = ((x / (y + x)) / (y + x)) * 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -132000.0], N[(N[(y / x), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[y, 8e-55], N[(y / N[(N[(1.0 + x), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2.1e+146], N[(N[(x / N[(N[(y + x), $MachinePrecision] * N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision], N[(N[(N[(x / N[(y + x), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if y < -132000

   1. Initial program 62.4%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{y}{{x}^{2}}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{y}{\color{blue}{{x}^{2}}} \]

      2. unpow2 N/A

         \[\leadsto \frac{y}{x \cdot \color{blue}{x}} \]

      3. lower-*.f64 22.6

         \[\leadsto \frac{y}{x \cdot \color{blue}{x}} \]

   5. Applied rewrites 22.6%

      \[\leadsto \color{blue}{\frac{y}{x \cdot x}} \]
   6. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{y}{\color{blue}{x \cdot x}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{y}{x \cdot \color{blue}{x}} \]

      3. associate-/r* N/A

         \[\leadsto \frac{\frac{y}{x}}{\color{blue}{x}} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{\frac{y}{x}}{\color{blue}{x}} \]

      5. lower-/.f64 25.4

         \[\leadsto \frac{\frac{y}{x}}{x} \]

   7. Applied rewrites 25.4%

      \[\leadsto \frac{\frac{y}{x}}{\color{blue}{x}} \]

   if -132000 < y < 7.99999999999999996e-55

   1. Initial program 74.6%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{y}{x \cdot \left(1 + x\right)}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{y}{\color{blue}{x \cdot \left(1 + x\right)}} \]

      2. *-commutative N/A

         \[\leadsto \frac{y}{\left(1 + x\right) \cdot \color{blue}{x}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{y}{\left(1 + x\right) \cdot \color{blue}{x}} \]

      4. lower-+.f64 76.3

         \[\leadsto \frac{y}{\left(1 + x\right) \cdot x} \]

   5. Applied rewrites 76.3%

      \[\leadsto \color{blue}{\frac{y}{\left(1 + x\right) \cdot x}} \]

   if 7.99999999999999996e-55 < y < 2.1000000000000001e146

   1. Initial program 73.3%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot y}}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      2. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      4. lift-+.f64 N/A

         \[\leadsto \frac{x \cdot y}{\left(\color{blue}{\left(x + y\right)} \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      5. lift-+.f64 N/A

         \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \color{blue}{\left(x + y\right)}\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right)} \cdot \left(\left(x + y\right) + 1\right)} \]

      7. lift-+.f64 N/A

         \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\color{blue}{\left(x + y\right)} + 1\right)} \]

      8. lift-+.f64 N/A

         \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \color{blue}{\left(\left(x + y\right) + 1\right)}} \]

      9. times-frac N/A

         \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}} \]

      10. lower-*.f64 N/A

          \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}} \]

      11. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)}} \cdot \frac{y}{\left(x + y\right) + 1} \]

      12. pow2 N/A

          \[\leadsto \frac{x}{\color{blue}{{\left(x + y\right)}^{2}}} \cdot \frac{y}{\left(x + y\right) + 1} \]

      13. lower-pow.f64 N/A

          \[\leadsto \frac{x}{\color{blue}{{\left(x + y\right)}^{2}}} \cdot \frac{y}{\left(x + y\right) + 1} \]

      14. +-commutative N/A

          \[\leadsto \frac{x}{{\color{blue}{\left(y + x\right)}}^{2}} \cdot \frac{y}{\left(x + y\right) + 1} \]

      15. lower-+.f64 N/A

          \[\leadsto \frac{x}{{\color{blue}{\left(y + x\right)}}^{2}} \cdot \frac{y}{\left(x + y\right) + 1} \]

      16. lower-/.f64 N/A

          \[\leadsto \frac{x}{{\left(y + x\right)}^{2}} \cdot \color{blue}{\frac{y}{\left(x + y\right) + 1}} \]

      17. lift-+.f64 N/A

          \[\leadsto \frac{x}{{\left(y + x\right)}^{2}} \cdot \frac{y}{\color{blue}{\left(x + y\right) + 1}} \]

      18. +-commutative N/A

          \[\leadsto \frac{x}{{\left(y + x\right)}^{2}} \cdot \frac{y}{\color{blue}{\left(y + x\right)} + 1} \]

      19. lower-+.f64 94.6

          \[\leadsto \frac{x}{{\left(y + x\right)}^{2}} \cdot \frac{y}{\color{blue}{\left(y + x\right)} + 1} \]

   4. Applied rewrites 94.6%

      \[\leadsto \color{blue}{\frac{x}{{\left(y + x\right)}^{2}} \cdot \frac{y}{\left(y + x\right) + 1}} \]
   5. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{x}{{\color{blue}{\left(y + x\right)}}^{2}} \cdot \frac{y}{\left(y + x\right) + 1} \]

      2. +-commutative N/A

         \[\leadsto \frac{x}{{\color{blue}{\left(x + y\right)}}^{2}} \cdot \frac{y}{\left(y + x\right) + 1} \]

      3. lower-pow.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{{\left(x + y\right)}^{2}}} \cdot \frac{y}{\left(y + x\right) + 1} \]

      4. pow2 N/A

         \[\leadsto \frac{x}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right)}} \cdot \frac{y}{\left(y + x\right) + 1} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right)}} \cdot \frac{y}{\left(y + x\right) + 1} \]

      6. +-commutative N/A

         \[\leadsto \frac{x}{\color{blue}{\left(y + x\right)} \cdot \left(x + y\right)} \cdot \frac{y}{\left(y + x\right) + 1} \]

      7. lift-+.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{\left(y + x\right)} \cdot \left(x + y\right)} \cdot \frac{y}{\left(y + x\right) + 1} \]

      8. +-commutative N/A

         \[\leadsto \frac{x}{\left(y + x\right) \cdot \color{blue}{\left(y + x\right)}} \cdot \frac{y}{\left(y + x\right) + 1} \]

      9. lift-+.f64 94.6

         \[\leadsto \frac{x}{\left(y + x\right) \cdot \color{blue}{\left(y + x\right)}} \cdot \frac{y}{\left(y + x\right) + 1} \]

   6. Applied rewrites 94.6%

      \[\leadsto \frac{x}{\color{blue}{\left(y + x\right) \cdot \left(y + x\right)}} \cdot \frac{y}{\left(y + x\right) + 1} \]

   7. Taylor expanded in y around inf

      \[\leadsto \frac{x}{\left(y + x\right) \cdot \left(y + x\right)} \cdot \color{blue}{1} \]
   8. Step-by-step derivation

   9. Applied rewrites 64.5%

      \[\leadsto \frac{x}{\left(y + x\right) \cdot \left(y + x\right)} \cdot \color{blue}{1} \]

   if 2.1000000000000001e146 < y

   1. Initial program 57.7%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot y}}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      2. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      4. lift-+.f64 N/A

         \[\leadsto \frac{x \cdot y}{\left(\color{blue}{\left(x + y\right)} \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      5. lift-+.f64 N/A

         \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \color{blue}{\left(x + y\right)}\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right)} \cdot \left(\left(x + y\right) + 1\right)} \]

      7. lift-+.f64 N/A

         \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\color{blue}{\left(x + y\right)} + 1\right)} \]

      8. lift-+.f64 N/A

         \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \color{blue}{\left(\left(x + y\right) + 1\right)}} \]

      9. times-frac N/A

         \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}} \]

      10. lower-*.f64 N/A

          \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}} \]

      11. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)}} \cdot \frac{y}{\left(x + y\right) + 1} \]

      12. pow2 N/A

          \[\leadsto \frac{x}{\color{blue}{{\left(x + y\right)}^{2}}} \cdot \frac{y}{\left(x + y\right) + 1} \]

      13. lower-pow.f64 N/A

          \[\leadsto \frac{x}{\color{blue}{{\left(x + y\right)}^{2}}} \cdot \frac{y}{\left(x + y\right) + 1} \]

      14. +-commutative N/A

          \[\leadsto \frac{x}{{\color{blue}{\left(y + x\right)}}^{2}} \cdot \frac{y}{\left(x + y\right) + 1} \]

      15. lower-+.f64 N/A

          \[\leadsto \frac{x}{{\color{blue}{\left(y + x\right)}}^{2}} \cdot \frac{y}{\left(x + y\right) + 1} \]

      16. lower-/.f64 N/A

          \[\leadsto \frac{x}{{\left(y + x\right)}^{2}} \cdot \color{blue}{\frac{y}{\left(x + y\right) + 1}} \]

      17. lift-+.f64 N/A

          \[\leadsto \frac{x}{{\left(y + x\right)}^{2}} \cdot \frac{y}{\color{blue}{\left(x + y\right) + 1}} \]

      18. +-commutative N/A

          \[\leadsto \frac{x}{{\left(y + x\right)}^{2}} \cdot \frac{y}{\color{blue}{\left(y + x\right)} + 1} \]

      19. lower-+.f64 80.1

          \[\leadsto \frac{x}{{\left(y + x\right)}^{2}} \cdot \frac{y}{\color{blue}{\left(y + x\right)} + 1} \]

   4. Applied rewrites 80.1%

      \[\leadsto \color{blue}{\frac{x}{{\left(y + x\right)}^{2}} \cdot \frac{y}{\left(y + x\right) + 1}} \]
   5. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{x}{{\color{blue}{\left(y + x\right)}}^{2}} \cdot \frac{y}{\left(y + x\right) + 1} \]

      2. +-commutative N/A

         \[\leadsto \frac{x}{{\color{blue}{\left(x + y\right)}}^{2}} \cdot \frac{y}{\left(y + x\right) + 1} \]

      3. lower-pow.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{{\left(x + y\right)}^{2}}} \cdot \frac{y}{\left(y + x\right) + 1} \]

      4. pow2 N/A

         \[\leadsto \frac{x}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right)}} \cdot \frac{y}{\left(y + x\right) + 1} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right)}} \cdot \frac{y}{\left(y + x\right) + 1} \]

      6. +-commutative N/A

         \[\leadsto \frac{x}{\color{blue}{\left(y + x\right)} \cdot \left(x + y\right)} \cdot \frac{y}{\left(y + x\right) + 1} \]

      7. lift-+.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{\left(y + x\right)} \cdot \left(x + y\right)} \cdot \frac{y}{\left(y + x\right) + 1} \]

      8. +-commutative N/A

         \[\leadsto \frac{x}{\left(y + x\right) \cdot \color{blue}{\left(y + x\right)}} \cdot \frac{y}{\left(y + x\right) + 1} \]

      9. lift-+.f64 80.1

         \[\leadsto \frac{x}{\left(y + x\right) \cdot \color{blue}{\left(y + x\right)}} \cdot \frac{y}{\left(y + x\right) + 1} \]

   6. Applied rewrites 80.1%

      \[\leadsto \frac{x}{\color{blue}{\left(y + x\right) \cdot \left(y + x\right)}} \cdot \frac{y}{\left(y + x\right) + 1} \]
   7. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{\left(y + x\right) \cdot \left(y + x\right)}} \cdot \frac{y}{\left(y + x\right) + 1} \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{\left(y + x\right)} \cdot \left(y + x\right)} \cdot \frac{y}{\left(y + x\right) + 1} \]

      3. lift-+.f64 N/A

         \[\leadsto \frac{x}{\left(y + x\right) \cdot \color{blue}{\left(y + x\right)}} \cdot \frac{y}{\left(y + x\right) + 1} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{\left(y + x\right) \cdot \left(y + x\right)}} \cdot \frac{y}{\left(y + x\right) + 1} \]

      5. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{x}{y + x}}{y + x}} \cdot \frac{y}{\left(y + x\right) + 1} \]

      6. +-commutative N/A

         \[\leadsto \frac{\frac{x}{\color{blue}{x + y}}}{y + x} \cdot \frac{y}{\left(y + x\right) + 1} \]

      7. +-commutative N/A

         \[\leadsto \frac{\frac{x}{x + y}}{\color{blue}{x + y}} \cdot \frac{y}{\left(y + x\right) + 1} \]

      8. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{x}{x + y}}{x + y}} \cdot \frac{y}{\left(y + x\right) + 1} \]

      9. lower-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{x}{x + y}}}{x + y} \cdot \frac{y}{\left(y + x\right) + 1} \]

      10. +-commutative N/A

          \[\leadsto \frac{\frac{x}{\color{blue}{y + x}}}{x + y} \cdot \frac{y}{\left(y + x\right) + 1} \]

      11. lift-+.f64 N/A

          \[\leadsto \frac{\frac{x}{\color{blue}{y + x}}}{x + y} \cdot \frac{y}{\left(y + x\right) + 1} \]

      12. +-commutative N/A

          \[\leadsto \frac{\frac{x}{y + x}}{\color{blue}{y + x}} \cdot \frac{y}{\left(y + x\right) + 1} \]

      13. lift-+.f64 99.9

          \[\leadsto \frac{\frac{x}{y + x}}{\color{blue}{y + x}} \cdot \frac{y}{\left(y + x\right) + 1} \]

   8. Applied rewrites 99.9%

      \[\leadsto \color{blue}{\frac{\frac{x}{y + x}}{y + x}} \cdot \frac{y}{\left(y + x\right) + 1} \]

   9. Taylor expanded in y around inf

      \[\leadsto \frac{\frac{x}{y + x}}{y + x} \cdot \color{blue}{1} \]
   10. Step-by-step derivation

       1. +-commutative 86.2

          \[\leadsto \frac{\frac{x}{y + x}}{y + x} \cdot 1 \]

   11. Applied rewrites 86.2%

       \[\leadsto \frac{\frac{x}{y + x}}{y + x} \cdot \color{blue}{1} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 11: 67.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 5.5 \cdot 10^{-158}:\\ \;\;\;\;\frac{1}{y + x} \cdot \frac{y}{\left(y + x\right) + 1}\\ \mathbf{elif}\;y \leq 4 \cdot 10^{+99}:\\ \;\;\;\;\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + x}}{y + x} \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 5.5e-158)
   (* (/ 1.0 (+ y x)) (/ y (+ (+ y x) 1.0)))
   (if (<= y 4e+99)
     (/ (* x y) (* (* (+ x y) (+ x y)) (+ (+ x y) 1.0)))
     (* (/ (/ x (+ y x)) (+ y x)) 1.0))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 5.5e-158) {
		tmp = (1.0 / (y + x)) * (y / ((y + x) + 1.0));
	} else if (y <= 4e+99) {
		tmp = (x * y) / (((x + y) * (x + y)) * ((x + y) + 1.0));
	} 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 (y <= 5.5d-158) then
        tmp = (1.0d0 / (y + x)) * (y / ((y + x) + 1.0d0))
    else if (y <= 4d+99) then
        tmp = (x * y) / (((x + y) * (x + y)) * ((x + y) + 1.0d0))
    else
        tmp = ((x / (y + x)) / (y + x)) * 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 5.5e-158) {
		tmp = (1.0 / (y + x)) * (y / ((y + x) + 1.0));
	} else if (y <= 4e+99) {
		tmp = (x * y) / (((x + y) * (x + y)) * ((x + y) + 1.0));
	} else {
		tmp = ((x / (y + x)) / (y + x)) * 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 5.5e-158:
		tmp = (1.0 / (y + x)) * (y / ((y + x) + 1.0))
	elif y <= 4e+99:
		tmp = (x * y) / (((x + y) * (x + y)) * ((x + y) + 1.0))
	else:
		tmp = ((x / (y + x)) / (y + x)) * 1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 5.5e-158)
		tmp = Float64(Float64(1.0 / Float64(y + x)) * Float64(y / Float64(Float64(y + x) + 1.0)));
	elseif (y <= 4e+99)
		tmp = Float64(Float64(x * y) / Float64(Float64(Float64(x + y) * Float64(x + y)) * Float64(Float64(x + y) + 1.0)));
	else
		tmp = Float64(Float64(Float64(x / Float64(y + x)) / Float64(y + x)) * 1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 5.5e-158)
		tmp = (1.0 / (y + x)) * (y / ((y + x) + 1.0));
	elseif (y <= 4e+99)
		tmp = (x * y) / (((x + y) * (x + y)) * ((x + y) + 1.0));
	else
		tmp = ((x / (y + x)) / (y + x)) * 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 5.5e-158], N[(N[(1.0 / N[(y + x), $MachinePrecision]), $MachinePrecision] * N[(y / N[(N[(y + x), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 4e+99], N[(N[(x * y), $MachinePrecision] / N[(N[(N[(x + y), $MachinePrecision] * N[(x + y), $MachinePrecision]), $MachinePrecision] * N[(N[(x + y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x / N[(y + x), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 5.50000000000000025e-158

   1. Initial program 68.3%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot y}}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      2. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      4. lift-+.f64 N/A

         \[\leadsto \frac{x \cdot y}{\left(\color{blue}{\left(x + y\right)} \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      5. lift-+.f64 N/A

         \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \color{blue}{\left(x + y\right)}\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right)} \cdot \left(\left(x + y\right) + 1\right)} \]

      7. lift-+.f64 N/A

         \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\color{blue}{\left(x + y\right)} + 1\right)} \]

      8. lift-+.f64 N/A

         \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \color{blue}{\left(\left(x + y\right) + 1\right)}} \]

      9. times-frac N/A

         \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}} \]

      10. lower-*.f64 N/A

          \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}} \]

      11. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)}} \cdot \frac{y}{\left(x + y\right) + 1} \]

      12. pow2 N/A

          \[\leadsto \frac{x}{\color{blue}{{\left(x + y\right)}^{2}}} \cdot \frac{y}{\left(x + y\right) + 1} \]

      13. lower-pow.f64 N/A

          \[\leadsto \frac{x}{\color{blue}{{\left(x + y\right)}^{2}}} \cdot \frac{y}{\left(x + y\right) + 1} \]

      14. +-commutative N/A

          \[\leadsto \frac{x}{{\color{blue}{\left(y + x\right)}}^{2}} \cdot \frac{y}{\left(x + y\right) + 1} \]

      15. lower-+.f64 N/A

          \[\leadsto \frac{x}{{\color{blue}{\left(y + x\right)}}^{2}} \cdot \frac{y}{\left(x + y\right) + 1} \]

      16. lower-/.f64 N/A

          \[\leadsto \frac{x}{{\left(y + x\right)}^{2}} \cdot \color{blue}{\frac{y}{\left(x + y\right) + 1}} \]

      17. lift-+.f64 N/A

          \[\leadsto \frac{x}{{\left(y + x\right)}^{2}} \cdot \frac{y}{\color{blue}{\left(x + y\right) + 1}} \]

      18. +-commutative N/A

          \[\leadsto \frac{x}{{\left(y + x\right)}^{2}} \cdot \frac{y}{\color{blue}{\left(y + x\right)} + 1} \]

      19. lower-+.f64 85.8

          \[\leadsto \frac{x}{{\left(y + x\right)}^{2}} \cdot \frac{y}{\color{blue}{\left(y + x\right)} + 1} \]

   4. Applied rewrites 85.8%

      \[\leadsto \color{blue}{\frac{x}{{\left(y + x\right)}^{2}} \cdot \frac{y}{\left(y + x\right) + 1}} \]
   5. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{x}{{\color{blue}{\left(y + x\right)}}^{2}} \cdot \frac{y}{\left(y + x\right) + 1} \]

      2. +-commutative N/A

         \[\leadsto \frac{x}{{\color{blue}{\left(x + y\right)}}^{2}} \cdot \frac{y}{\left(y + x\right) + 1} \]

      3. lower-pow.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{{\left(x + y\right)}^{2}}} \cdot \frac{y}{\left(y + x\right) + 1} \]

      4. pow2 N/A

         \[\leadsto \frac{x}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right)}} \cdot \frac{y}{\left(y + x\right) + 1} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right)}} \cdot \frac{y}{\left(y + x\right) + 1} \]

      6. +-commutative N/A

         \[\leadsto \frac{x}{\color{blue}{\left(y + x\right)} \cdot \left(x + y\right)} \cdot \frac{y}{\left(y + x\right) + 1} \]

      7. lift-+.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{\left(y + x\right)} \cdot \left(x + y\right)} \cdot \frac{y}{\left(y + x\right) + 1} \]

      8. +-commutative N/A

         \[\leadsto \frac{x}{\left(y + x\right) \cdot \color{blue}{\left(y + x\right)}} \cdot \frac{y}{\left(y + x\right) + 1} \]

      9. lift-+.f64 85.8

         \[\leadsto \frac{x}{\left(y + x\right) \cdot \color{blue}{\left(y + x\right)}} \cdot \frac{y}{\left(y + x\right) + 1} \]

   6. Applied rewrites 85.8%

      \[\leadsto \frac{x}{\color{blue}{\left(y + x\right) \cdot \left(y + x\right)}} \cdot \frac{y}{\left(y + x\right) + 1} \]
   7. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{\left(y + x\right) \cdot \left(y + x\right)}} \cdot \frac{y}{\left(y + x\right) + 1} \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{\left(y + x\right)} \cdot \left(y + x\right)} \cdot \frac{y}{\left(y + x\right) + 1} \]

      3. lift-+.f64 N/A

         \[\leadsto \frac{x}{\left(y + x\right) \cdot \color{blue}{\left(y + x\right)}} \cdot \frac{y}{\left(y + x\right) + 1} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{\left(y + x\right) \cdot \left(y + x\right)}} \cdot \frac{y}{\left(y + x\right) + 1} \]

      5. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{x}{y + x}}{y + x}} \cdot \frac{y}{\left(y + x\right) + 1} \]

      6. +-commutative N/A

         \[\leadsto \frac{\frac{x}{\color{blue}{x + y}}}{y + x} \cdot \frac{y}{\left(y + x\right) + 1} \]

      7. +-commutative N/A

         \[\leadsto \frac{\frac{x}{x + y}}{\color{blue}{x + y}} \cdot \frac{y}{\left(y + x\right) + 1} \]

      8. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{x}{x + y}}{x + y}} \cdot \frac{y}{\left(y + x\right) + 1} \]

      9. lower-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{x}{x + y}}}{x + y} \cdot \frac{y}{\left(y + x\right) + 1} \]

      10. +-commutative N/A

          \[\leadsto \frac{\frac{x}{\color{blue}{y + x}}}{x + y} \cdot \frac{y}{\left(y + x\right) + 1} \]

      11. lift-+.f64 N/A

          \[\leadsto \frac{\frac{x}{\color{blue}{y + x}}}{x + y} \cdot \frac{y}{\left(y + x\right) + 1} \]

      12. +-commutative N/A

          \[\leadsto \frac{\frac{x}{y + x}}{\color{blue}{y + x}} \cdot \frac{y}{\left(y + x\right) + 1} \]

      13. lift-+.f64 99.7

          \[\leadsto \frac{\frac{x}{y + x}}{\color{blue}{y + x}} \cdot \frac{y}{\left(y + x\right) + 1} \]

   8. Applied rewrites 99.7%

      \[\leadsto \color{blue}{\frac{\frac{x}{y + x}}{y + x}} \cdot \frac{y}{\left(y + x\right) + 1} \]

   9. Taylor expanded in x around inf

      \[\leadsto \frac{\color{blue}{1}}{y + x} \cdot \frac{y}{\left(y + x\right) + 1} \]
   10. Step-by-step derivation

   11. Applied rewrites 58.0%

       \[\leadsto \frac{\color{blue}{1}}{y + x} \cdot \frac{y}{\left(y + x\right) + 1} \]

   if 5.50000000000000025e-158 < y < 3.9999999999999999e99

   1. Initial program 84.2%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing

   if 3.9999999999999999e99 < y

   1. Initial program 53.0%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot y}}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      2. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      4. lift-+.f64 N/A

         \[\leadsto \frac{x \cdot y}{\left(\color{blue}{\left(x + y\right)} \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      5. lift-+.f64 N/A

         \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \color{blue}{\left(x + y\right)}\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right)} \cdot \left(\left(x + y\right) + 1\right)} \]

      7. lift-+.f64 N/A

         \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\color{blue}{\left(x + y\right)} + 1\right)} \]

      8. lift-+.f64 N/A

         \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \color{blue}{\left(\left(x + y\right) + 1\right)}} \]

      9. times-frac N/A

         \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}} \]

      10. lower-*.f64 N/A

          \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}} \]

      11. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)}} \cdot \frac{y}{\left(x + y\right) + 1} \]

      12. pow2 N/A

          \[\leadsto \frac{x}{\color{blue}{{\left(x + y\right)}^{2}}} \cdot \frac{y}{\left(x + y\right) + 1} \]

      13. lower-pow.f64 N/A

          \[\leadsto \frac{x}{\color{blue}{{\left(x + y\right)}^{2}}} \cdot \frac{y}{\left(x + y\right) + 1} \]

      14. +-commutative N/A

          \[\leadsto \frac{x}{{\color{blue}{\left(y + x\right)}}^{2}} \cdot \frac{y}{\left(x + y\right) + 1} \]

      15. lower-+.f64 N/A

          \[\leadsto \frac{x}{{\color{blue}{\left(y + x\right)}}^{2}} \cdot \frac{y}{\left(x + y\right) + 1} \]

      16. lower-/.f64 N/A

          \[\leadsto \frac{x}{{\left(y + x\right)}^{2}} \cdot \color{blue}{\frac{y}{\left(x + y\right) + 1}} \]

      17. lift-+.f64 N/A

          \[\leadsto \frac{x}{{\left(y + x\right)}^{2}} \cdot \frac{y}{\color{blue}{\left(x + y\right) + 1}} \]

      18. +-commutative N/A

          \[\leadsto \frac{x}{{\left(y + x\right)}^{2}} \cdot \frac{y}{\color{blue}{\left(y + x\right)} + 1} \]

      19. lower-+.f64 82.1

          \[\leadsto \frac{x}{{\left(y + x\right)}^{2}} \cdot \frac{y}{\color{blue}{\left(y + x\right)} + 1} \]

   4. Applied rewrites 82.1%

      \[\leadsto \color{blue}{\frac{x}{{\left(y + x\right)}^{2}} \cdot \frac{y}{\left(y + x\right) + 1}} \]
   5. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{x}{{\color{blue}{\left(y + x\right)}}^{2}} \cdot \frac{y}{\left(y + x\right) + 1} \]

      2. +-commutative N/A

         \[\leadsto \frac{x}{{\color{blue}{\left(x + y\right)}}^{2}} \cdot \frac{y}{\left(y + x\right) + 1} \]

      3. lower-pow.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{{\left(x + y\right)}^{2}}} \cdot \frac{y}{\left(y + x\right) + 1} \]

      4. pow2 N/A

         \[\leadsto \frac{x}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right)}} \cdot \frac{y}{\left(y + x\right) + 1} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right)}} \cdot \frac{y}{\left(y + x\right) + 1} \]

      6. +-commutative N/A

         \[\leadsto \frac{x}{\color{blue}{\left(y + x\right)} \cdot \left(x + y\right)} \cdot \frac{y}{\left(y + x\right) + 1} \]

      7. lift-+.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{\left(y + x\right)} \cdot \left(x + y\right)} \cdot \frac{y}{\left(y + x\right) + 1} \]

      8. +-commutative N/A

         \[\leadsto \frac{x}{\left(y + x\right) \cdot \color{blue}{\left(y + x\right)}} \cdot \frac{y}{\left(y + x\right) + 1} \]

      9. lift-+.f64 82.1

         \[\leadsto \frac{x}{\left(y + x\right) \cdot \color{blue}{\left(y + x\right)}} \cdot \frac{y}{\left(y + x\right) + 1} \]

   6. Applied rewrites 82.1%

      \[\leadsto \frac{x}{\color{blue}{\left(y + x\right) \cdot \left(y + x\right)}} \cdot \frac{y}{\left(y + x\right) + 1} \]
   7. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{\left(y + x\right) \cdot \left(y + x\right)}} \cdot \frac{y}{\left(y + x\right) + 1} \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{\left(y + x\right)} \cdot \left(y + x\right)} \cdot \frac{y}{\left(y + x\right) + 1} \]

      3. lift-+.f64 N/A

         \[\leadsto \frac{x}{\left(y + x\right) \cdot \color{blue}{\left(y + x\right)}} \cdot \frac{y}{\left(y + x\right) + 1} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{\left(y + x\right) \cdot \left(y + x\right)}} \cdot \frac{y}{\left(y + x\right) + 1} \]

      5. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{x}{y + x}}{y + x}} \cdot \frac{y}{\left(y + x\right) + 1} \]

      6. +-commutative N/A

         \[\leadsto \frac{\frac{x}{\color{blue}{x + y}}}{y + x} \cdot \frac{y}{\left(y + x\right) + 1} \]

      7. +-commutative N/A

         \[\leadsto \frac{\frac{x}{x + y}}{\color{blue}{x + y}} \cdot \frac{y}{\left(y + x\right) + 1} \]

      8. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{x}{x + y}}{x + y}} \cdot \frac{y}{\left(y + x\right) + 1} \]

      9. lower-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{x}{x + y}}}{x + y} \cdot \frac{y}{\left(y + x\right) + 1} \]

      10. +-commutative N/A

          \[\leadsto \frac{\frac{x}{\color{blue}{y + x}}}{x + y} \cdot \frac{y}{\left(y + x\right) + 1} \]

      11. lift-+.f64 N/A

          \[\leadsto \frac{\frac{x}{\color{blue}{y + x}}}{x + y} \cdot \frac{y}{\left(y + x\right) + 1} \]

      12. +-commutative N/A

          \[\leadsto \frac{\frac{x}{y + x}}{\color{blue}{y + x}} \cdot \frac{y}{\left(y + x\right) + 1} \]

      13. lift-+.f64 99.8

          \[\leadsto \frac{\frac{x}{y + x}}{\color{blue}{y + x}} \cdot \frac{y}{\left(y + x\right) + 1} \]

   8. Applied rewrites 99.8%

      \[\leadsto \color{blue}{\frac{\frac{x}{y + x}}{y + x}} \cdot \frac{y}{\left(y + x\right) + 1} \]

   9. Taylor expanded in y around inf

      \[\leadsto \frac{\frac{x}{y + x}}{y + x} \cdot \color{blue}{1} \]
   10. Step-by-step derivation

       1. +-commutative 82.3

          \[\leadsto \frac{\frac{x}{y + x}}{y + x} \cdot 1 \]

   11. Applied rewrites 82.3%

       \[\leadsto \frac{\frac{x}{y + x}}{y + x} \cdot \color{blue}{1} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 12: 99.7% accurate, 0.8× 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 69.2%

   \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \frac{\color{blue}{x \cdot y}}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

   2. lift-/.f64 N/A

      \[\leadsto \color{blue}{\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

   3. lift-*.f64 N/A

      \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

   4. lift-+.f64 N/A

      \[\leadsto \frac{x \cdot y}{\left(\color{blue}{\left(x + y\right)} \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

   5. lift-+.f64 N/A

      \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \color{blue}{\left(x + y\right)}\right) \cdot \left(\left(x + y\right) + 1\right)} \]

   6. lift-*.f64 N/A

      \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right)} \cdot \left(\left(x + y\right) + 1\right)} \]

   7. lift-+.f64 N/A

      \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\color{blue}{\left(x + y\right)} + 1\right)} \]

   8. lift-+.f64 N/A

      \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \color{blue}{\left(\left(x + y\right) + 1\right)}} \]

   9. times-frac N/A

      \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}} \]

   10. lower-*.f64 N/A

       \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}} \]

   11. lower-/.f64 N/A

       \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)}} \cdot \frac{y}{\left(x + y\right) + 1} \]

   12. pow2 N/A

       \[\leadsto \frac{x}{\color{blue}{{\left(x + y\right)}^{2}}} \cdot \frac{y}{\left(x + y\right) + 1} \]

   13. lower-pow.f64 N/A

       \[\leadsto \frac{x}{\color{blue}{{\left(x + y\right)}^{2}}} \cdot \frac{y}{\left(x + y\right) + 1} \]

   14. +-commutative N/A

       \[\leadsto \frac{x}{{\color{blue}{\left(y + x\right)}}^{2}} \cdot \frac{y}{\left(x + y\right) + 1} \]

   15. lower-+.f64 N/A

       \[\leadsto \frac{x}{{\color{blue}{\left(y + x\right)}}^{2}} \cdot \frac{y}{\left(x + y\right) + 1} \]

   16. lower-/.f64 N/A

       \[\leadsto \frac{x}{{\left(y + x\right)}^{2}} \cdot \color{blue}{\frac{y}{\left(x + y\right) + 1}} \]

   17. lift-+.f64 N/A

       \[\leadsto \frac{x}{{\left(y + x\right)}^{2}} \cdot \frac{y}{\color{blue}{\left(x + y\right) + 1}} \]

   18. +-commutative N/A

       \[\leadsto \frac{x}{{\left(y + x\right)}^{2}} \cdot \frac{y}{\color{blue}{\left(y + x\right)} + 1} \]

   19. lower-+.f64 87.7

       \[\leadsto \frac{x}{{\left(y + x\right)}^{2}} \cdot \frac{y}{\color{blue}{\left(y + x\right)} + 1} \]

4. Applied rewrites 87.7%

   \[\leadsto \color{blue}{\frac{x}{{\left(y + x\right)}^{2}} \cdot \frac{y}{\left(y + x\right) + 1}} \]
5. Step-by-step derivation

   1. lift-+.f64 N/A

      \[\leadsto \frac{x}{{\color{blue}{\left(y + x\right)}}^{2}} \cdot \frac{y}{\left(y + x\right) + 1} \]

   2. +-commutative N/A

      \[\leadsto \frac{x}{{\color{blue}{\left(x + y\right)}}^{2}} \cdot \frac{y}{\left(y + x\right) + 1} \]

   3. lower-pow.f64 N/A

      \[\leadsto \frac{x}{\color{blue}{{\left(x + y\right)}^{2}}} \cdot \frac{y}{\left(y + x\right) + 1} \]

   4. pow2 N/A

      \[\leadsto \frac{x}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right)}} \cdot \frac{y}{\left(y + x\right) + 1} \]

   5. lift-*.f64 N/A

      \[\leadsto \frac{x}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right)}} \cdot \frac{y}{\left(y + x\right) + 1} \]

   6. +-commutative N/A

      \[\leadsto \frac{x}{\color{blue}{\left(y + x\right)} \cdot \left(x + y\right)} \cdot \frac{y}{\left(y + x\right) + 1} \]

   7. lift-+.f64 N/A

      \[\leadsto \frac{x}{\color{blue}{\left(y + x\right)} \cdot \left(x + y\right)} \cdot \frac{y}{\left(y + x\right) + 1} \]

   8. +-commutative N/A

      \[\leadsto \frac{x}{\left(y + x\right) \cdot \color{blue}{\left(y + x\right)}} \cdot \frac{y}{\left(y + x\right) + 1} \]

   9. lift-+.f64 87.7

      \[\leadsto \frac{x}{\left(y + x\right) \cdot \color{blue}{\left(y + x\right)}} \cdot \frac{y}{\left(y + x\right) + 1} \]

6. Applied rewrites 87.7%

   \[\leadsto \frac{x}{\color{blue}{\left(y + x\right) \cdot \left(y + x\right)}} \cdot \frac{y}{\left(y + x\right) + 1} \]
7. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \color{blue}{\frac{x}{\left(y + x\right) \cdot \left(y + x\right)}} \cdot \frac{y}{\left(y + x\right) + 1} \]

   2. lift-+.f64 N/A

      \[\leadsto \frac{x}{\color{blue}{\left(y + x\right)} \cdot \left(y + x\right)} \cdot \frac{y}{\left(y + x\right) + 1} \]

   3. lift-+.f64 N/A

      \[\leadsto \frac{x}{\left(y + x\right) \cdot \color{blue}{\left(y + x\right)}} \cdot \frac{y}{\left(y + x\right) + 1} \]

   4. lift-*.f64 N/A

      \[\leadsto \frac{x}{\color{blue}{\left(y + x\right) \cdot \left(y + x\right)}} \cdot \frac{y}{\left(y + x\right) + 1} \]

   5. associate-/r* N/A

      \[\leadsto \color{blue}{\frac{\frac{x}{y + x}}{y + x}} \cdot \frac{y}{\left(y + x\right) + 1} \]

   6. +-commutative N/A

      \[\leadsto \frac{\frac{x}{\color{blue}{x + y}}}{y + x} \cdot \frac{y}{\left(y + x\right) + 1} \]

   7. +-commutative N/A

      \[\leadsto \frac{\frac{x}{x + y}}{\color{blue}{x + y}} \cdot \frac{y}{\left(y + x\right) + 1} \]

   8. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\frac{x}{x + y}}{x + y}} \cdot \frac{y}{\left(y + x\right) + 1} \]

   9. lower-/.f64 N/A

      \[\leadsto \frac{\color{blue}{\frac{x}{x + y}}}{x + y} \cdot \frac{y}{\left(y + x\right) + 1} \]

   10. +-commutative N/A

       \[\leadsto \frac{\frac{x}{\color{blue}{y + x}}}{x + y} \cdot \frac{y}{\left(y + x\right) + 1} \]

   11. lift-+.f64 N/A

       \[\leadsto \frac{\frac{x}{\color{blue}{y + x}}}{x + y} \cdot \frac{y}{\left(y + x\right) + 1} \]

   12. +-commutative N/A

       \[\leadsto \frac{\frac{x}{y + x}}{\color{blue}{y + x}} \cdot \frac{y}{\left(y + x\right) + 1} \]

   13. lift-+.f64 99.7

       \[\leadsto \frac{\frac{x}{y + x}}{\color{blue}{y + x}} \cdot \frac{y}{\left(y + x\right) + 1} \]

8. Applied rewrites 99.7%

   \[\leadsto \color{blue}{\frac{\frac{x}{y + x}}{y + x}} \cdot \frac{y}{\left(y + x\right) + 1} \]
9. Add Preprocessing

Alternative 13: 62.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -132000:\\ \;\;\;\;\frac{\frac{y}{x}}{x}\\ \mathbf{elif}\;y \leq 8 \cdot 10^{-55}:\\ \;\;\;\;\frac{y}{\left(1 + x\right) \cdot x}\\ \mathbf{elif}\;y \leq 6.1 \cdot 10^{+150}:\\ \;\;\;\;\frac{x}{\left(y + x\right) \cdot \left(y + x\right)} \cdot 1\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -132000.0)
   (/ (/ y x) x)
   (if (<= y 8e-55)
     (/ y (* (+ 1.0 x) x))
     (if (<= y 6.1e+150) (* (/ x (* (+ y x) (+ y x))) 1.0) (/ (/ x y) y)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -132000.0) {
		tmp = (y / x) / x;
	} else if (y <= 8e-55) {
		tmp = y / ((1.0 + x) * x);
	} else if (y <= 6.1e+150) {
		tmp = (x / ((y + x) * (y + x))) * 1.0;
	} 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 <= (-132000.0d0)) then
        tmp = (y / x) / x
    else if (y <= 8d-55) then
        tmp = y / ((1.0d0 + x) * x)
    else if (y <= 6.1d+150) then
        tmp = (x / ((y + x) * (y + x))) * 1.0d0
    else
        tmp = (x / y) / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -132000.0) {
		tmp = (y / x) / x;
	} else if (y <= 8e-55) {
		tmp = y / ((1.0 + x) * x);
	} else if (y <= 6.1e+150) {
		tmp = (x / ((y + x) * (y + x))) * 1.0;
	} else {
		tmp = (x / y) / y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -132000.0:
		tmp = (y / x) / x
	elif y <= 8e-55:
		tmp = y / ((1.0 + x) * x)
	elif y <= 6.1e+150:
		tmp = (x / ((y + x) * (y + x))) * 1.0
	else:
		tmp = (x / y) / y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -132000.0)
		tmp = Float64(Float64(y / x) / x);
	elseif (y <= 8e-55)
		tmp = Float64(y / Float64(Float64(1.0 + x) * x));
	elseif (y <= 6.1e+150)
		tmp = Float64(Float64(x / Float64(Float64(y + x) * Float64(y + x))) * 1.0);
	else
		tmp = Float64(Float64(x / y) / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -132000.0)
		tmp = (y / x) / x;
	elseif (y <= 8e-55)
		tmp = y / ((1.0 + x) * x);
	elseif (y <= 6.1e+150)
		tmp = (x / ((y + x) * (y + x))) * 1.0;
	else
		tmp = (x / y) / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -132000.0], N[(N[(y / x), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[y, 8e-55], N[(y / N[(N[(1.0 + x), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 6.1e+150], N[(N[(x / N[(N[(y + x), $MachinePrecision] * N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / y), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if y < -132000

   1. Initial program 62.4%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{y}{{x}^{2}}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{y}{\color{blue}{{x}^{2}}} \]

      2. unpow2 N/A

         \[\leadsto \frac{y}{x \cdot \color{blue}{x}} \]

      3. lower-*.f64 22.6

         \[\leadsto \frac{y}{x \cdot \color{blue}{x}} \]

   5. Applied rewrites 22.6%

      \[\leadsto \color{blue}{\frac{y}{x \cdot x}} \]
   6. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{y}{\color{blue}{x \cdot x}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{y}{x \cdot \color{blue}{x}} \]

      3. associate-/r* N/A

         \[\leadsto \frac{\frac{y}{x}}{\color{blue}{x}} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{\frac{y}{x}}{\color{blue}{x}} \]

      5. lower-/.f64 25.4

         \[\leadsto \frac{\frac{y}{x}}{x} \]

   7. Applied rewrites 25.4%

      \[\leadsto \frac{\frac{y}{x}}{\color{blue}{x}} \]

   if -132000 < y < 7.99999999999999996e-55

   1. Initial program 74.6%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{y}{x \cdot \left(1 + x\right)}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{y}{\color{blue}{x \cdot \left(1 + x\right)}} \]

      2. *-commutative N/A

         \[\leadsto \frac{y}{\left(1 + x\right) \cdot \color{blue}{x}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{y}{\left(1 + x\right) \cdot \color{blue}{x}} \]

      4. lower-+.f64 76.3

         \[\leadsto \frac{y}{\left(1 + x\right) \cdot x} \]

   5. Applied rewrites 76.3%

      \[\leadsto \color{blue}{\frac{y}{\left(1 + x\right) \cdot x}} \]

   if 7.99999999999999996e-55 < y < 6.10000000000000026e150

   1. Initial program 72.8%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot y}}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      2. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      4. lift-+.f64 N/A

         \[\leadsto \frac{x \cdot y}{\left(\color{blue}{\left(x + y\right)} \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      5. lift-+.f64 N/A

         \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \color{blue}{\left(x + y\right)}\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right)} \cdot \left(\left(x + y\right) + 1\right)} \]

      7. lift-+.f64 N/A

         \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\color{blue}{\left(x + y\right)} + 1\right)} \]

      8. lift-+.f64 N/A

         \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \color{blue}{\left(\left(x + y\right) + 1\right)}} \]

      9. times-frac N/A

         \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}} \]

      10. lower-*.f64 N/A

          \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}} \]

      11. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)}} \cdot \frac{y}{\left(x + y\right) + 1} \]

      12. pow2 N/A

          \[\leadsto \frac{x}{\color{blue}{{\left(x + y\right)}^{2}}} \cdot \frac{y}{\left(x + y\right) + 1} \]

      13. lower-pow.f64 N/A

          \[\leadsto \frac{x}{\color{blue}{{\left(x + y\right)}^{2}}} \cdot \frac{y}{\left(x + y\right) + 1} \]

      14. +-commutative N/A

          \[\leadsto \frac{x}{{\color{blue}{\left(y + x\right)}}^{2}} \cdot \frac{y}{\left(x + y\right) + 1} \]

      15. lower-+.f64 N/A

          \[\leadsto \frac{x}{{\color{blue}{\left(y + x\right)}}^{2}} \cdot \frac{y}{\left(x + y\right) + 1} \]

      16. lower-/.f64 N/A

          \[\leadsto \frac{x}{{\left(y + x\right)}^{2}} \cdot \color{blue}{\frac{y}{\left(x + y\right) + 1}} \]

      17. lift-+.f64 N/A

          \[\leadsto \frac{x}{{\left(y + x\right)}^{2}} \cdot \frac{y}{\color{blue}{\left(x + y\right) + 1}} \]

      18. +-commutative N/A

          \[\leadsto \frac{x}{{\left(y + x\right)}^{2}} \cdot \frac{y}{\color{blue}{\left(y + x\right)} + 1} \]

      19. lower-+.f64 94.4

          \[\leadsto \frac{x}{{\left(y + x\right)}^{2}} \cdot \frac{y}{\color{blue}{\left(y + x\right)} + 1} \]

   4. Applied rewrites 94.4%

      \[\leadsto \color{blue}{\frac{x}{{\left(y + x\right)}^{2}} \cdot \frac{y}{\left(y + x\right) + 1}} \]
   5. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{x}{{\color{blue}{\left(y + x\right)}}^{2}} \cdot \frac{y}{\left(y + x\right) + 1} \]

      2. +-commutative N/A

         \[\leadsto \frac{x}{{\color{blue}{\left(x + y\right)}}^{2}} \cdot \frac{y}{\left(y + x\right) + 1} \]

      3. lower-pow.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{{\left(x + y\right)}^{2}}} \cdot \frac{y}{\left(y + x\right) + 1} \]

      4. pow2 N/A

         \[\leadsto \frac{x}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right)}} \cdot \frac{y}{\left(y + x\right) + 1} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right)}} \cdot \frac{y}{\left(y + x\right) + 1} \]

      6. +-commutative N/A

         \[\leadsto \frac{x}{\color{blue}{\left(y + x\right)} \cdot \left(x + y\right)} \cdot \frac{y}{\left(y + x\right) + 1} \]

      7. lift-+.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{\left(y + x\right)} \cdot \left(x + y\right)} \cdot \frac{y}{\left(y + x\right) + 1} \]

      8. +-commutative N/A

         \[\leadsto \frac{x}{\left(y + x\right) \cdot \color{blue}{\left(y + x\right)}} \cdot \frac{y}{\left(y + x\right) + 1} \]

      9. lift-+.f64 94.4

         \[\leadsto \frac{x}{\left(y + x\right) \cdot \color{blue}{\left(y + x\right)}} \cdot \frac{y}{\left(y + x\right) + 1} \]

   6. Applied rewrites 94.4%

      \[\leadsto \frac{x}{\color{blue}{\left(y + x\right) \cdot \left(y + x\right)}} \cdot \frac{y}{\left(y + x\right) + 1} \]

   7. Taylor expanded in y around inf

      \[\leadsto \frac{x}{\left(y + x\right) \cdot \left(y + x\right)} \cdot \color{blue}{1} \]
   8. Step-by-step derivation

   9. Applied rewrites 64.8%

      \[\leadsto \frac{x}{\left(y + x\right) \cdot \left(y + x\right)} \cdot \color{blue}{1} \]

   if 6.10000000000000026e150 < y

   1. Initial program 57.9%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\frac{x}{{y}^{2}}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{{y}^{2}}} \]

      2. unpow2 N/A

         \[\leadsto \frac{x}{y \cdot \color{blue}{y}} \]

      3. lower-*.f64 79.7

         \[\leadsto \frac{x}{y \cdot \color{blue}{y}} \]

   5. Applied rewrites 79.7%

      \[\leadsto \color{blue}{\frac{x}{y \cdot y}} \]
   6. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{y \cdot y}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{x}{y \cdot \color{blue}{y}} \]

      3. associate-/r* N/A

         \[\leadsto \frac{\frac{x}{y}}{\color{blue}{y}} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{\frac{x}{y}}{\color{blue}{y}} \]

      5. lower-/.f64 86.2

         \[\leadsto \frac{\frac{x}{y}}{y} \]

   7. Applied rewrites 86.2%

      \[\leadsto \frac{\frac{x}{y}}{\color{blue}{y}} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 14: 54.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{y \cdot y}\\ \mathbf{if}\;x \leq -2.2 \cdot 10^{+27}:\\ \;\;\;\;\frac{y}{x \cdot x}\\ \mathbf{elif}\;x \leq -5.8 \cdot 10^{-190}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 9.5 \cdot 10^{-160}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ x (* y y))))
   (if (<= x -2.2e+27)
     (/ y (* x x))
     (if (<= x -5.8e-190) t_0 (if (<= x 9.5e-160) (/ x y) t_0)))))

C

double code(double x, double y) {
	double t_0 = x / (y * y);
	double tmp;
	if (x <= -2.2e+27) {
		tmp = y / (x * x);
	} else if (x <= -5.8e-190) {
		tmp = t_0;
	} else if (x <= 9.5e-160) {
		tmp = x / y;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x / (y * y)
    if (x <= (-2.2d+27)) then
        tmp = y / (x * x)
    else if (x <= (-5.8d-190)) then
        tmp = t_0
    else if (x <= 9.5d-160) then
        tmp = x / y
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = x / (y * y);
	double tmp;
	if (x <= -2.2e+27) {
		tmp = y / (x * x);
	} else if (x <= -5.8e-190) {
		tmp = t_0;
	} else if (x <= 9.5e-160) {
		tmp = x / y;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = x / (y * y)
	tmp = 0
	if x <= -2.2e+27:
		tmp = y / (x * x)
	elif x <= -5.8e-190:
		tmp = t_0
	elif x <= 9.5e-160:
		tmp = x / y
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(x / Float64(y * y))
	tmp = 0.0
	if (x <= -2.2e+27)
		tmp = Float64(y / Float64(x * x));
	elseif (x <= -5.8e-190)
		tmp = t_0;
	elseif (x <= 9.5e-160)
		tmp = Float64(x / y);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = x / (y * y);
	tmp = 0.0;
	if (x <= -2.2e+27)
		tmp = y / (x * x);
	elseif (x <= -5.8e-190)
		tmp = t_0;
	elseif (x <= 9.5e-160)
		tmp = x / y;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(x / N[(y * y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -2.2e+27], N[(y / N[(x * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -5.8e-190], t$95$0, If[LessEqual[x, 9.5e-160], N[(x / y), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.1999999999999999e27

   1. Initial program 59.7%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{y}{{x}^{2}}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{y}{\color{blue}{{x}^{2}}} \]

      2. unpow2 N/A

         \[\leadsto \frac{y}{x \cdot \color{blue}{x}} \]

      3. lower-*.f64 73.5

         \[\leadsto \frac{y}{x \cdot \color{blue}{x}} \]

   5. Applied rewrites 73.5%

      \[\leadsto \color{blue}{\frac{y}{x \cdot x}} \]

   if -2.1999999999999999e27 < x < -5.8000000000000004e-190 or 9.5000000000000002e-160 < x

   1. Initial program 75.5%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\frac{x}{{y}^{2}}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{{y}^{2}}} \]

      2. unpow2 N/A

         \[\leadsto \frac{x}{y \cdot \color{blue}{y}} \]

      3. lower-*.f64 37.9

         \[\leadsto \frac{x}{y \cdot \color{blue}{y}} \]

   5. Applied rewrites 37.9%

      \[\leadsto \color{blue}{\frac{x}{y \cdot y}} \]

   if -5.8000000000000004e-190 < x < 9.5000000000000002e-160

   1. Initial program 62.9%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{y \cdot \left(1 + y\right)}} \]

      2. *-commutative N/A

         \[\leadsto \frac{x}{\left(1 + y\right) \cdot \color{blue}{y}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{x}{\left(1 + y\right) \cdot \color{blue}{y}} \]

      4. lower-+.f64 87.5

         \[\leadsto \frac{x}{\left(1 + y\right) \cdot y} \]

   5. Applied rewrites 87.5%

      \[\leadsto \color{blue}{\frac{x}{\left(1 + y\right) \cdot y}} \]

   6. Taylor expanded in y around 0

      \[\leadsto \frac{x}{y} \]
   7. Step-by-step derivation

   8. Applied rewrites 75.3%

      \[\leadsto \frac{x}{y} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 15: 61.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -132000:\\ \;\;\;\;\frac{\frac{y}{x}}{x}\\ \mathbf{elif}\;y \leq 1200000000000:\\ \;\;\;\;\frac{y}{\left(1 + x\right) \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -132000.0)
   (/ (/ y x) x)
   (if (<= y 1200000000000.0) (/ y (* (+ 1.0 x) x)) (/ (/ x y) y))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -132000.0) {
		tmp = (y / x) / x;
	} else if (y <= 1200000000000.0) {
		tmp = y / ((1.0 + 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 <= (-132000.0d0)) then
        tmp = (y / x) / x
    else if (y <= 1200000000000.0d0) then
        tmp = y / ((1.0d0 + 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 <= -132000.0) {
		tmp = (y / x) / x;
	} else if (y <= 1200000000000.0) {
		tmp = y / ((1.0 + x) * x);
	} else {
		tmp = (x / y) / y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -132000.0:
		tmp = (y / x) / x
	elif y <= 1200000000000.0:
		tmp = y / ((1.0 + x) * x)
	else:
		tmp = (x / y) / y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -132000.0)
		tmp = Float64(Float64(y / x) / x);
	elseif (y <= 1200000000000.0)
		tmp = Float64(y / Float64(Float64(1.0 + 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 <= -132000.0)
		tmp = (y / x) / x;
	elseif (y <= 1200000000000.0)
		tmp = y / ((1.0 + x) * x);
	else
		tmp = (x / y) / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -132000.0], N[(N[(y / x), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[y, 1200000000000.0], N[(y / N[(N[(1.0 + x), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -132000

   1. Initial program 62.4%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{y}{{x}^{2}}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{y}{\color{blue}{{x}^{2}}} \]

      2. unpow2 N/A

         \[\leadsto \frac{y}{x \cdot \color{blue}{x}} \]

      3. lower-*.f64 22.6

         \[\leadsto \frac{y}{x \cdot \color{blue}{x}} \]

   5. Applied rewrites 22.6%

      \[\leadsto \color{blue}{\frac{y}{x \cdot x}} \]
   6. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{y}{\color{blue}{x \cdot x}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{y}{x \cdot \color{blue}{x}} \]

      3. associate-/r* N/A

         \[\leadsto \frac{\frac{y}{x}}{\color{blue}{x}} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{\frac{y}{x}}{\color{blue}{x}} \]

      5. lower-/.f64 25.4

         \[\leadsto \frac{\frac{y}{x}}{x} \]

   7. Applied rewrites 25.4%

      \[\leadsto \frac{\frac{y}{x}}{\color{blue}{x}} \]

   if -132000 < y < 1.2e12

   1. Initial program 76.4%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{y}{x \cdot \left(1 + x\right)}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{y}{\color{blue}{x \cdot \left(1 + x\right)}} \]

      2. *-commutative N/A

         \[\leadsto \frac{y}{\left(1 + x\right) \cdot \color{blue}{x}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{y}{\left(1 + x\right) \cdot \color{blue}{x}} \]

      4. lower-+.f64 73.9

         \[\leadsto \frac{y}{\left(1 + x\right) \cdot x} \]

   5. Applied rewrites 73.9%

      \[\leadsto \color{blue}{\frac{y}{\left(1 + x\right) \cdot x}} \]

   if 1.2e12 < y

   1. Initial program 60.7%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\frac{x}{{y}^{2}}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{{y}^{2}}} \]

      2. unpow2 N/A

         \[\leadsto \frac{x}{y \cdot \color{blue}{y}} \]

      3. lower-*.f64 70.8

         \[\leadsto \frac{x}{y \cdot \color{blue}{y}} \]

   5. Applied rewrites 70.8%

      \[\leadsto \color{blue}{\frac{x}{y \cdot y}} \]
   6. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{y \cdot y}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{x}{y \cdot \color{blue}{y}} \]

      3. associate-/r* N/A

         \[\leadsto \frac{\frac{x}{y}}{\color{blue}{y}} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{\frac{x}{y}}{\color{blue}{y}} \]

      5. lower-/.f64 74.2

         \[\leadsto \frac{\frac{x}{y}}{y} \]

   7. Applied rewrites 74.2%

      \[\leadsto \frac{\frac{x}{y}}{\color{blue}{y}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 16: 60.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1200000000000:\\ \;\;\;\;\frac{y}{\left(1 + x\right) \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 1200000000000.0) (/ y (* (+ 1.0 x) x)) (/ (/ x y) y)))

C

double code(double x, double y) {
	double tmp;
	if (y <= 1200000000000.0) {
		tmp = y / ((1.0 + 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 <= 1200000000000.0d0) then
        tmp = y / ((1.0d0 + 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 <= 1200000000000.0) {
		tmp = y / ((1.0 + x) * x);
	} else {
		tmp = (x / y) / y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 1200000000000.0:
		tmp = y / ((1.0 + x) * x)
	else:
		tmp = (x / y) / y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 1200000000000.0)
		tmp = Float64(y / Float64(Float64(1.0 + 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 <= 1200000000000.0)
		tmp = y / ((1.0 + x) * x);
	else
		tmp = (x / y) / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 1200000000000.0], N[(y / N[(N[(1.0 + x), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.2e12

   1. Initial program 71.7%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{y}{x \cdot \left(1 + x\right)}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{y}{\color{blue}{x \cdot \left(1 + x\right)}} \]

      2. *-commutative N/A

         \[\leadsto \frac{y}{\left(1 + x\right) \cdot \color{blue}{x}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{y}{\left(1 + x\right) \cdot \color{blue}{x}} \]

      4. lower-+.f64 56.9

         \[\leadsto \frac{y}{\left(1 + x\right) \cdot x} \]

   5. Applied rewrites 56.9%

      \[\leadsto \color{blue}{\frac{y}{\left(1 + x\right) \cdot x}} \]

   if 1.2e12 < y

   1. Initial program 60.7%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\frac{x}{{y}^{2}}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{{y}^{2}}} \]

      2. unpow2 N/A

         \[\leadsto \frac{x}{y \cdot \color{blue}{y}} \]

      3. lower-*.f64 70.8

         \[\leadsto \frac{x}{y \cdot \color{blue}{y}} \]

   5. Applied rewrites 70.8%

      \[\leadsto \color{blue}{\frac{x}{y \cdot y}} \]
   6. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{y \cdot y}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{x}{y \cdot \color{blue}{y}} \]

      3. associate-/r* N/A

         \[\leadsto \frac{\frac{x}{y}}{\color{blue}{y}} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{\frac{x}{y}}{\color{blue}{y}} \]

      5. lower-/.f64 74.2

         \[\leadsto \frac{\frac{x}{y}}{y} \]

   7. Applied rewrites 74.2%

      \[\leadsto \frac{\frac{x}{y}}{\color{blue}{y}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 17: 60.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -9.5 \cdot 10^{-80}:\\ \;\;\;\;\frac{y}{\left(1 + x\right) \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(y, 1, y \cdot y\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -9.5e-80) (/ y (* (+ 1.0 x) x)) (/ x (fma y 1.0 (* y y)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -9.5e-80) {
		tmp = y / ((1.0 + x) * x);
	} else {
		tmp = x / fma(y, 1.0, (y * y));
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -9.5e-80)
		tmp = Float64(y / Float64(Float64(1.0 + x) * x));
	else
		tmp = Float64(x / fma(y, 1.0, Float64(y * y)));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, -9.5e-80], N[(y / N[(N[(1.0 + x), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision], N[(x / N[(y * 1.0 + N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -9.5000000000000003e-80

   1. Initial program 67.9%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{y}{x \cdot \left(1 + x\right)}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{y}{\color{blue}{x \cdot \left(1 + x\right)}} \]

      2. *-commutative N/A

         \[\leadsto \frac{y}{\left(1 + x\right) \cdot \color{blue}{x}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{y}{\left(1 + x\right) \cdot \color{blue}{x}} \]

      4. lower-+.f64 65.9

         \[\leadsto \frac{y}{\left(1 + x\right) \cdot x} \]

   5. Applied rewrites 65.9%

      \[\leadsto \color{blue}{\frac{y}{\left(1 + x\right) \cdot x}} \]

   if -9.5000000000000003e-80 < x

   1. Initial program 69.8%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{y \cdot \left(1 + y\right)}} \]

      2. *-commutative N/A

         \[\leadsto \frac{x}{\left(1 + y\right) \cdot \color{blue}{y}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{x}{\left(1 + y\right) \cdot \color{blue}{y}} \]

      4. lower-+.f64 57.8

         \[\leadsto \frac{x}{\left(1 + y\right) \cdot y} \]

   5. Applied rewrites 57.8%

      \[\leadsto \color{blue}{\frac{x}{\left(1 + y\right) \cdot y}} \]
   6. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{x}{\left(1 + y\right) \cdot y} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{x}{\left(1 + y\right) \cdot \color{blue}{y}} \]

      3. *-commutative N/A

         \[\leadsto \frac{x}{y \cdot \color{blue}{\left(1 + y\right)}} \]

      4. distribute-lft-in N/A

         \[\leadsto \frac{x}{y \cdot 1 + \color{blue}{y \cdot y}} \]

      5. pow2 N/A

         \[\leadsto \frac{x}{y \cdot 1 + {y}^{\color{blue}{2}}} \]

      6. lower-fma.f64 N/A

         \[\leadsto \frac{x}{\mathsf{fma}\left(y, \color{blue}{1}, {y}^{2}\right)} \]

      7. pow2 N/A

         \[\leadsto \frac{x}{\mathsf{fma}\left(y, 1, y \cdot y\right)} \]

      8. lift-*.f64 57.8

         \[\leadsto \frac{x}{\mathsf{fma}\left(y, 1, y \cdot y\right)} \]

   7. Applied rewrites 57.8%

      \[\leadsto \frac{x}{\mathsf{fma}\left(y, \color{blue}{1}, y \cdot y\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 18: 60.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -9.5 \cdot 10^{-80}:\\ \;\;\;\;\frac{y}{\left(1 + x\right) \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\left(1 + y\right) \cdot y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -9.5e-80) (/ y (* (+ 1.0 x) x)) (/ x (* (+ 1.0 y) y))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -9.5e-80) {
		tmp = y / ((1.0 + x) * x);
	} else {
		tmp = x / ((1.0 + y) * y);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-9.5d-80)) then
        tmp = y / ((1.0d0 + x) * x)
    else
        tmp = x / ((1.0d0 + y) * y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -9.5e-80) {
		tmp = y / ((1.0 + x) * x);
	} else {
		tmp = x / ((1.0 + y) * y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -9.5e-80:
		tmp = y / ((1.0 + x) * x)
	else:
		tmp = x / ((1.0 + y) * y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -9.5e-80)
		tmp = Float64(y / Float64(Float64(1.0 + x) * x));
	else
		tmp = Float64(x / Float64(Float64(1.0 + y) * y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -9.5e-80)
		tmp = y / ((1.0 + x) * x);
	else
		tmp = x / ((1.0 + y) * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -9.5e-80], N[(y / N[(N[(1.0 + x), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision], N[(x / N[(N[(1.0 + y), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -9.5000000000000003e-80

   1. Initial program 67.9%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{y}{x \cdot \left(1 + x\right)}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{y}{\color{blue}{x \cdot \left(1 + x\right)}} \]

      2. *-commutative N/A

         \[\leadsto \frac{y}{\left(1 + x\right) \cdot \color{blue}{x}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{y}{\left(1 + x\right) \cdot \color{blue}{x}} \]

      4. lower-+.f64 65.9

         \[\leadsto \frac{y}{\left(1 + x\right) \cdot x} \]

   5. Applied rewrites 65.9%

      \[\leadsto \color{blue}{\frac{y}{\left(1 + x\right) \cdot x}} \]

   if -9.5000000000000003e-80 < x

   1. Initial program 69.8%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{y \cdot \left(1 + y\right)}} \]

      2. *-commutative N/A

         \[\leadsto \frac{x}{\left(1 + y\right) \cdot \color{blue}{y}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{x}{\left(1 + y\right) \cdot \color{blue}{y}} \]

      4. lower-+.f64 57.8

         \[\leadsto \frac{x}{\left(1 + y\right) \cdot y} \]

   5. Applied rewrites 57.8%

      \[\leadsto \color{blue}{\frac{x}{\left(1 + y\right) \cdot y}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 19: 61.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.2 \cdot 10^{+27}:\\ \;\;\;\;\frac{y}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\left(1 + y\right) \cdot y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -2.2e+27) (/ y (* x x)) (/ x (* (+ 1.0 y) y))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -2.2e+27) {
		tmp = y / (x * x);
	} else {
		tmp = x / ((1.0 + y) * y);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-2.2d+27)) then
        tmp = y / (x * x)
    else
        tmp = x / ((1.0d0 + y) * y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -2.2e+27) {
		tmp = y / (x * x);
	} else {
		tmp = x / ((1.0 + y) * y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -2.2e+27:
		tmp = y / (x * x)
	else:
		tmp = x / ((1.0 + y) * y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -2.2e+27)
		tmp = Float64(y / Float64(x * x));
	else
		tmp = Float64(x / Float64(Float64(1.0 + y) * y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -2.2e+27)
		tmp = y / (x * x);
	else
		tmp = x / ((1.0 + y) * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -2.2e+27], N[(y / N[(x * x), $MachinePrecision]), $MachinePrecision], N[(x / N[(N[(1.0 + y), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2.1999999999999999e27

   1. Initial program 59.7%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{y}{{x}^{2}}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{y}{\color{blue}{{x}^{2}}} \]

      2. unpow2 N/A

         \[\leadsto \frac{y}{x \cdot \color{blue}{x}} \]

      3. lower-*.f64 73.5

         \[\leadsto \frac{y}{x \cdot \color{blue}{x}} \]

   5. Applied rewrites 73.5%

      \[\leadsto \color{blue}{\frac{y}{x \cdot x}} \]

   if -2.1999999999999999e27 < x

   1. Initial program 72.0%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{y \cdot \left(1 + y\right)}} \]

      2. *-commutative N/A

         \[\leadsto \frac{x}{\left(1 + y\right) \cdot \color{blue}{y}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{x}{\left(1 + y\right) \cdot \color{blue}{y}} \]

      4. lower-+.f64 57.3

         \[\leadsto \frac{x}{\left(1 + y\right) \cdot y} \]

   5. Applied rewrites 57.3%

      \[\leadsto \color{blue}{\frac{x}{\left(1 + y\right) \cdot y}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 20: 36.1% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= y 1.0) (/ x y) (/ x (* y y))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 1.0) {
		tmp = x / y;
	} else {
		tmp = x / (y * y);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 1.0d0) then
        tmp = x / y
    else
        tmp = x / (y * y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 1.0) {
		tmp = x / y;
	} else {
		tmp = x / (y * y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 1.0:
		tmp = x / y
	else:
		tmp = x / (y * y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 1.0)
		tmp = Float64(x / y);
	else
		tmp = Float64(x / Float64(y * y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 1.0)
		tmp = x / y;
	else
		tmp = x / (y * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 1.0], N[(x / y), $MachinePrecision], N[(x / N[(y * y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1

   1. Initial program 71.5%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{y \cdot \left(1 + y\right)}} \]

      2. *-commutative N/A

         \[\leadsto \frac{x}{\left(1 + y\right) \cdot \color{blue}{y}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{x}{\left(1 + y\right) \cdot \color{blue}{y}} \]

      4. lower-+.f64 42.2

         \[\leadsto \frac{x}{\left(1 + y\right) \cdot y} \]

   5. Applied rewrites 42.2%

      \[\leadsto \color{blue}{\frac{x}{\left(1 + y\right) \cdot y}} \]

   6. Taylor expanded in y around 0

      \[\leadsto \frac{x}{y} \]
   7. Step-by-step derivation

   8. Applied rewrites 25.7%

      \[\leadsto \frac{x}{y} \]

   if 1 < y

   1. Initial program 62.1%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\frac{x}{{y}^{2}}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{{y}^{2}}} \]

      2. unpow2 N/A

         \[\leadsto \frac{x}{y \cdot \color{blue}{y}} \]

      3. lower-*.f64 68.5

         \[\leadsto \frac{x}{y \cdot \color{blue}{y}} \]

   5. Applied rewrites 68.5%

      \[\leadsto \color{blue}{\frac{x}{y \cdot y}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 21: 25.9% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ \frac{x}{y} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (/ x y))

C

double code(double x, double y) {
	return x / y;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = x / y
end function

Java

public static double code(double x, double y) {
	return x / y;
}

Python

def code(x, y):
	return x / y

Julia

function code(x, y)
	return Float64(x / y)
end

MATLAB

function tmp = code(x, y)
	tmp = x / y;
end

Wolfram

code[x_, y_] := N[(x / y), $MachinePrecision]

Derivation

1. Initial program 69.2%

   \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]
4. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \frac{x}{\color{blue}{y \cdot \left(1 + y\right)}} \]

   2. *-commutative N/A

      \[\leadsto \frac{x}{\left(1 + y\right) \cdot \color{blue}{y}} \]

   3. lower-*.f64 N/A

      \[\leadsto \frac{x}{\left(1 + y\right) \cdot \color{blue}{y}} \]

   4. lower-+.f64 48.9

      \[\leadsto \frac{x}{\left(1 + y\right) \cdot y} \]

5. Applied rewrites 48.9%

   \[\leadsto \color{blue}{\frac{x}{\left(1 + y\right) \cdot y}} \]

6. Taylor expanded in y around 0

   \[\leadsto \frac{x}{y} \]
7. Step-by-step derivation

8. Applied rewrites 25.9%

   \[\leadsto \frac{x}{y} \]
9. Add Preprocessing

Developer Target 1: 99.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{\frac{x}{\left(y + 1\right) + x}}{y + x}}{\frac{1}{\frac{y}{y + x}}} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (/ (/ (/ x (+ (+ y 1.0) x)) (+ y x)) (/ 1.0 (/ y (+ y x)))))

C

double code(double x, double y) {
	return ((x / ((y + 1.0) + x)) / (y + x)) / (1.0 / (y / (y + x)));
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = ((x / ((y + 1.0d0) + x)) / (y + x)) / (1.0d0 / (y / (y + x)))
end function

Java

public static double code(double x, double y) {
	return ((x / ((y + 1.0) + x)) / (y + x)) / (1.0 / (y / (y + x)));
}

Python

def code(x, y):
	return ((x / ((y + 1.0) + x)) / (y + x)) / (1.0 / (y / (y + x)))

Julia

function code(x, y)
	return Float64(Float64(Float64(x / Float64(Float64(y + 1.0) + x)) / Float64(y + x)) / Float64(1.0 / Float64(y / Float64(y + x))))
end

MATLAB

function tmp = code(x, y)
	tmp = ((x / ((y + 1.0) + x)) / (y + x)) / (1.0 / (y / (y + x)));
end

Wolfram

code[x_, y_] := N[(N[(N[(x / N[(N[(y + 1.0), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision] / N[(1.0 / N[(y / N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2025088 
(FPCore (x y)
  :name "Numeric.SpecFunctions:incompleteBetaApprox from math-functions-0.1.5.2, A"
  :precision binary64

  :alt
  (! :herbie-platform default (/ (/ (/ x (+ (+ y 1) x)) (+ y x)) (/ 1 (/ y (+ y x)))))

  (/ (* x y) (* (* (+ x y) (+ x y)) (+ (+ x y) 1.0))))