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

Percentage Accurate: 68.7% → 99.8%

Time: 6.2s

Alternatives: 15

Speedup: 0.8×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 68.7% 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

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 61.7%

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

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

   3. add-sqr-sqrt N/A

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

   4. associate-*r* N/A

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

   5. lower-*.f64 N/A

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

   6. lower-*.f64 N/A

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

   7. lower-sqrt.f64 N/A

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

   8. lower-sqrt.f64 30.5

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

4. Applied rewrites 30.5%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. associate-*l* N/A

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

   5. lift-sqrt.f64 N/A

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

   6. lift-sqrt.f64 N/A

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

   7. add-sqr-sqrt N/A

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

   8. *-commutative N/A

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

   9. lift-*.f64 N/A

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

   10. lift-*.f64 N/A

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

   11. associate-*l* N/A

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

   12. times-frac N/A

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

   13. lower-*.f64 N/A

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

   14. lower-/.f64 N/A

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

   15. lower-/.f64 N/A

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

   16. lower-*.f64 92.5

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

6. Applied rewrites 92.5%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-*l/ N/A

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

   4. lower-/.f64 N/A

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

   5. lower-*.f64 92.5

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

   6. lift-/.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. associate-/r* N/A

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

   9. lift-/.f64 N/A

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

   10. lower-/.f64 99.8

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

   11. lift-+.f64 N/A

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

   12. +-commutative N/A

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

   13. lower-+.f64 99.8

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

   14. lift-+.f64 N/A

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

   15. +-commutative N/A

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

   16. lower-+.f64 99.8

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

   17. lift-+.f64 N/A

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

   18. +-commutative N/A

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

   19. lower-+.f64 99.8

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

   20. lift-+.f64 N/A

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

   21. +-commutative N/A

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

   22. lower-+.f64 99.8

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

8. Applied rewrites 99.8%

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

Alternative 2: 63.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(x + y\right) + 1\\ \mathbf{if}\;y \leq 1.9 \cdot 10^{-235}:\\ \;\;\;\;\frac{\frac{y}{1 + x}}{y + x}\\ \mathbf{elif}\;y \leq 1.25 \cdot 10^{-112}:\\ \;\;\;\;1 \cdot \frac{x}{\left(x + y\right) \cdot t\_0}\\ \mathbf{elif}\;y \leq 2.8 \cdot 10^{+92}:\\ \;\;\;\;\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \frac{1}{1 + \left(y + x\right)}}{y + x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ (+ x y) 1.0)))
   (if (<= y 1.9e-235)
     (/ (/ y (+ 1.0 x)) (+ y x))
     (if (<= y 1.25e-112)
       (* 1.0 (/ x (* (+ x y) t_0)))
       (if (<= y 2.8e+92)
         (/ (* x y) (* (* (+ x y) (+ x y)) t_0))
         (/ (* x (/ 1.0 (+ 1.0 (+ y x)))) (+ y x)))))))

C

double code(double x, double y) {
	double t_0 = (x + y) + 1.0;
	double tmp;
	if (y <= 1.9e-235) {
		tmp = (y / (1.0 + x)) / (y + x);
	} else if (y <= 1.25e-112) {
		tmp = 1.0 * (x / ((x + y) * t_0));
	} else if (y <= 2.8e+92) {
		tmp = (x * y) / (((x + y) * (x + y)) * t_0);
	} else {
		tmp = (x * (1.0 / (1.0 + (y + x)))) / (y + x);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (x + y) + 1.0d0
    if (y <= 1.9d-235) then
        tmp = (y / (1.0d0 + x)) / (y + x)
    else if (y <= 1.25d-112) then
        tmp = 1.0d0 * (x / ((x + y) * t_0))
    else if (y <= 2.8d+92) then
        tmp = (x * y) / (((x + y) * (x + y)) * t_0)
    else
        tmp = (x * (1.0d0 / (1.0d0 + (y + x)))) / (y + x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = (x + y) + 1.0;
	double tmp;
	if (y <= 1.9e-235) {
		tmp = (y / (1.0 + x)) / (y + x);
	} else if (y <= 1.25e-112) {
		tmp = 1.0 * (x / ((x + y) * t_0));
	} else if (y <= 2.8e+92) {
		tmp = (x * y) / (((x + y) * (x + y)) * t_0);
	} else {
		tmp = (x * (1.0 / (1.0 + (y + x)))) / (y + x);
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (x + y) + 1.0
	tmp = 0
	if y <= 1.9e-235:
		tmp = (y / (1.0 + x)) / (y + x)
	elif y <= 1.25e-112:
		tmp = 1.0 * (x / ((x + y) * t_0))
	elif y <= 2.8e+92:
		tmp = (x * y) / (((x + y) * (x + y)) * t_0)
	else:
		tmp = (x * (1.0 / (1.0 + (y + x)))) / (y + x)
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(x + y) + 1.0)
	tmp = 0.0
	if (y <= 1.9e-235)
		tmp = Float64(Float64(y / Float64(1.0 + x)) / Float64(y + x));
	elseif (y <= 1.25e-112)
		tmp = Float64(1.0 * Float64(x / Float64(Float64(x + y) * t_0)));
	elseif (y <= 2.8e+92)
		tmp = Float64(Float64(x * y) / Float64(Float64(Float64(x + y) * Float64(x + y)) * t_0));
	else
		tmp = Float64(Float64(x * Float64(1.0 / Float64(1.0 + Float64(y + x)))) / Float64(y + x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (x + y) + 1.0;
	tmp = 0.0;
	if (y <= 1.9e-235)
		tmp = (y / (1.0 + x)) / (y + x);
	elseif (y <= 1.25e-112)
		tmp = 1.0 * (x / ((x + y) * t_0));
	elseif (y <= 2.8e+92)
		tmp = (x * y) / (((x + y) * (x + y)) * t_0);
	else
		tmp = (x * (1.0 / (1.0 + (y + x)))) / (y + x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(x + y), $MachinePrecision] + 1.0), $MachinePrecision]}, If[LessEqual[y, 1.9e-235], N[(N[(y / N[(1.0 + x), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.25e-112], N[(1.0 * N[(x / N[(N[(x + y), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2.8e+92], N[(N[(x * y), $MachinePrecision] / N[(N[(N[(x + y), $MachinePrecision] * N[(x + y), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(x * N[(1.0 / N[(1.0 + N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if y < 1.90000000000000013e-235

   1. Initial program 63.7%

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

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

      3. add-sqr-sqrt N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-sqrt.f64 28.4

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

   4. Applied rewrites 28.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift-sqrt.f64 N/A

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

      7. add-sqr-sqrt N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*l* N/A

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

      12. times-frac N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lower-*.f64 96.5

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

   6. Applied rewrites 96.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 96.5

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-/r* N/A

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

      9. lift-/.f64 N/A

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

      10. lower-/.f64 99.9

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 99.9

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

      14. lift-+.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 99.9

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

      17. lift-+.f64 N/A

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

      18. +-commutative N/A

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

      19. lower-+.f64 99.9

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

      20. lift-+.f64 N/A

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

      21. +-commutative N/A

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

      22. lower-+.f64 99.9

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

   8. Applied rewrites 99.9%

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

   9. Taylor expanded in y around 0

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

   11. Applied rewrites 53.6%

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

   if 1.90000000000000013e-235 < y < 1.25000000000000011e-112

   1. Initial program 67.1%

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

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

      3. add-sqr-sqrt N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-sqrt.f64 41.7

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

   4. Applied rewrites 41.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift-sqrt.f64 N/A

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

      7. add-sqr-sqrt N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*l* N/A

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

      11. times-frac N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-*.f64 99.9

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

   6. Applied rewrites 99.9%

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

   7. Taylor expanded in x around 0

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

   9. Applied rewrites 59.9%

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

   if 1.25000000000000011e-112 < y < 2.80000000000000001e92

   1. Initial program 91.6%

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

   if 2.80000000000000001e92 < y

   1. Initial program 36.5%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing
   3. 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. *-commutative N/A

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

      3. add-sqr-sqrt N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-sqrt.f64 20.4

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

   4. Applied rewrites 20.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift-sqrt.f64 N/A

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

      7. add-sqr-sqrt N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*l* N/A

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

      12. times-frac N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lower-*.f64 74.9

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

   6. Applied rewrites 74.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 74.9

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-/r* N/A

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

      9. lift-/.f64 N/A

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

      10. lower-/.f64 99.7

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 99.7

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

      14. lift-+.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 99.7

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

      17. lift-+.f64 N/A

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

      18. +-commutative N/A

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

      19. lower-+.f64 99.7

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

      20. lift-+.f64 N/A

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

      21. +-commutative N/A

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

      22. lower-+.f64 99.7

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

   8. Applied rewrites 99.7%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 73.2%

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

4. Final simplification 63.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.9 \cdot 10^{-235}:\\ \;\;\;\;\frac{\frac{y}{1 + x}}{y + x}\\ \mathbf{elif}\;y \leq 1.25 \cdot 10^{-112}:\\ \;\;\;\;1 \cdot \frac{x}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)}\\ \mathbf{elif}\;y \leq 2.8 \cdot 10^{+92}:\\ \;\;\;\;\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \frac{1}{1 + \left(y + x\right)}}{y + x}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 73.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5.3 \cdot 10^{-6}:\\ \;\;\;\;1 \cdot \frac{y}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)}\\ \mathbf{elif}\;x \leq 1.75 \cdot 10^{-79}:\\ \;\;\;\;\frac{y}{x + y} \cdot \frac{x}{\left(x + y\right) \cdot \left(y + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -5.3e-6)
   (* 1.0 (/ y (* (+ x y) (+ (+ x y) 1.0))))
   (if (<= x 1.75e-79)
     (* (/ y (+ x y)) (/ x (* (+ x y) (+ y 1.0))))
     (/ (/ x y) (+ y x)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -5.3e-6) {
		tmp = 1.0 * (y / ((x + y) * ((x + y) + 1.0)));
	} else if (x <= 1.75e-79) {
		tmp = (y / (x + y)) * (x / ((x + y) * (y + 1.0)));
	} else {
		tmp = (x / y) / (y + x);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-5.3d-6)) then
        tmp = 1.0d0 * (y / ((x + y) * ((x + y) + 1.0d0)))
    else if (x <= 1.75d-79) then
        tmp = (y / (x + y)) * (x / ((x + y) * (y + 1.0d0)))
    else
        tmp = (x / y) / (y + x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -5.3e-6) {
		tmp = 1.0 * (y / ((x + y) * ((x + y) + 1.0)));
	} else if (x <= 1.75e-79) {
		tmp = (y / (x + y)) * (x / ((x + y) * (y + 1.0)));
	} else {
		tmp = (x / y) / (y + x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -5.3e-6:
		tmp = 1.0 * (y / ((x + y) * ((x + y) + 1.0)))
	elif x <= 1.75e-79:
		tmp = (y / (x + y)) * (x / ((x + y) * (y + 1.0)))
	else:
		tmp = (x / y) / (y + x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -5.3e-6)
		tmp = Float64(1.0 * Float64(y / Float64(Float64(x + y) * Float64(Float64(x + y) + 1.0))));
	elseif (x <= 1.75e-79)
		tmp = Float64(Float64(y / Float64(x + y)) * Float64(x / Float64(Float64(x + y) * Float64(y + 1.0))));
	else
		tmp = Float64(Float64(x / y) / Float64(y + x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -5.3e-6)
		tmp = 1.0 * (y / ((x + y) * ((x + y) + 1.0)));
	elseif (x <= 1.75e-79)
		tmp = (y / (x + y)) * (x / ((x + y) * (y + 1.0)));
	else
		tmp = (x / y) / (y + x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -5.3e-6], N[(1.0 * N[(y / N[(N[(x + y), $MachinePrecision] * N[(N[(x + y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.75e-79], N[(N[(y / N[(x + y), $MachinePrecision]), $MachinePrecision] * N[(x / N[(N[(x + y), $MachinePrecision] * N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -5.3000000000000001e-6

   1. Initial program 54.3%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. add-sqr-sqrt N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-sqrt.f64 0.0

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

   4. Applied rewrites 0.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift-sqrt.f64 N/A

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

      7. add-sqr-sqrt N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*l* N/A

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

      12. times-frac N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lower-*.f64 86.2

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

   6. Applied rewrites 86.2%

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

   7. Taylor expanded in x around inf

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

   9. Applied rewrites 80.9%

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

   if -5.3000000000000001e-6 < x < 1.75000000000000015e-79

   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. *-commutative N/A

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

      3. add-sqr-sqrt N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-sqrt.f64 28.3

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

   4. Applied rewrites 28.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift-sqrt.f64 N/A

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

      7. add-sqr-sqrt N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*l* N/A

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

      11. times-frac N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-*.f64 99.9

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

   6. Applied rewrites 99.9%

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

   7. Taylor expanded in x around 0

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

   9. Applied rewrites 99.9%

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

   if 1.75000000000000015e-79 < x

   1. Initial program 59.2%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. add-sqr-sqrt N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-sqrt.f64 59.3

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

   4. Applied rewrites 59.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift-sqrt.f64 N/A

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

      7. add-sqr-sqrt N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*l* N/A

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

      12. times-frac N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lower-*.f64 88.3

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

   6. Applied rewrites 88.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 88.3

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-/r* N/A

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

      9. lift-/.f64 N/A

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

      10. lower-/.f64 99.7

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 99.7

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

      14. lift-+.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 99.7

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

      17. lift-+.f64 N/A

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

      18. +-commutative N/A

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

      19. lower-+.f64 99.7

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

      20. lift-+.f64 N/A

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

      21. +-commutative N/A

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

      22. lower-+.f64 99.7

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

   8. Applied rewrites 99.7%

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

   9. Taylor expanded in y around inf

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

   11. Applied rewrites 37.4%

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

Alternative 4: 94.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.25 \cdot 10^{+162}:\\ \;\;\;\;\frac{x}{x + y} \cdot \frac{y}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \frac{1}{1 + \left(y + x\right)}}{y + x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 1.25e+162)
   (* (/ x (+ x y)) (/ y (* (+ x y) (+ (+ x y) 1.0))))
   (/ (* x (/ 1.0 (+ 1.0 (+ y x)))) (+ y x))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 1.25e+162) {
		tmp = (x / (x + y)) * (y / ((x + y) * ((x + y) + 1.0)));
	} else {
		tmp = (x * (1.0 / (1.0 + (y + x)))) / (y + x);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 1.25d+162) then
        tmp = (x / (x + y)) * (y / ((x + y) * ((x + y) + 1.0d0)))
    else
        tmp = (x * (1.0d0 / (1.0d0 + (y + x)))) / (y + x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 1.25e+162) {
		tmp = (x / (x + y)) * (y / ((x + y) * ((x + y) + 1.0)));
	} else {
		tmp = (x * (1.0 / (1.0 + (y + x)))) / (y + x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 1.25e+162:
		tmp = (x / (x + y)) * (y / ((x + y) * ((x + y) + 1.0)))
	else:
		tmp = (x * (1.0 / (1.0 + (y + x)))) / (y + x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 1.25e+162)
		tmp = Float64(Float64(x / Float64(x + y)) * Float64(y / Float64(Float64(x + y) * Float64(Float64(x + y) + 1.0))));
	else
		tmp = Float64(Float64(x * Float64(1.0 / Float64(1.0 + Float64(y + x)))) / Float64(y + x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 1.25e+162)
		tmp = (x / (x + y)) * (y / ((x + y) * ((x + y) + 1.0)));
	else
		tmp = (x * (1.0 / (1.0 + (y + x)))) / (y + x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 1.25e+162], N[(N[(x / N[(x + y), $MachinePrecision]), $MachinePrecision] * N[(y / N[(N[(x + y), $MachinePrecision] * N[(N[(x + y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * N[(1.0 / N[(1.0 + N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.2499999999999999e162

   1. Initial program 67.0%

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

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

      3. add-sqr-sqrt N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-sqrt.f64 33.3

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

   4. Applied rewrites 33.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift-sqrt.f64 N/A

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

      7. add-sqr-sqrt N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*l* N/A

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

      12. times-frac N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lower-*.f64 95.8

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

   6. Applied rewrites 95.8%

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

   if 1.2499999999999999e162 < y

   1. Initial program 35.1%

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

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

      3. add-sqr-sqrt N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-sqrt.f64 16.6

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

   4. Applied rewrites 16.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift-sqrt.f64 N/A

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

      7. add-sqr-sqrt N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*l* N/A

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

      12. times-frac N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lower-*.f64 76.1

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

   6. Applied rewrites 76.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 76.1

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-/r* N/A

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

      9. lift-/.f64 N/A

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

      10. lower-/.f64 99.8

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 99.8

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

      14. lift-+.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 99.8

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

      17. lift-+.f64 N/A

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

      18. +-commutative N/A

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

      19. lower-+.f64 99.8

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

      20. lift-+.f64 N/A

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

      21. +-commutative N/A

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

      22. lower-+.f64 99.8

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

   8. Applied rewrites 99.8%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 80.3%

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

Alternative 5: 99.8% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 61.7%

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

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

   3. add-sqr-sqrt N/A

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

   4. associate-*r* N/A

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

   5. lower-*.f64 N/A

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

   6. lower-*.f64 N/A

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

   7. lower-sqrt.f64 N/A

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

   8. lower-sqrt.f64 30.5

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

4. Applied rewrites 30.5%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. associate-*l* N/A

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

   5. lift-sqrt.f64 N/A

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

   6. lift-sqrt.f64 N/A

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

   7. add-sqr-sqrt N/A

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

   8. *-commutative N/A

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

   9. lift-*.f64 N/A

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

   10. lift-*.f64 N/A

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

   11. associate-*l* N/A

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

   12. times-frac N/A

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

   13. lower-*.f64 N/A

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

   14. lower-/.f64 N/A

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

   15. lower-/.f64 N/A

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

   16. lower-*.f64 92.5

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

6. Applied rewrites 92.5%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-*r/ N/A

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

   4. lift-*.f64 N/A

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

   5. times-frac N/A

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

   6. lower-*.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. lift-+.f64 N/A

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

   9. +-commutative N/A

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

   10. lower-+.f64 N/A

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

   11. lift-+.f64 N/A

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

   12. +-commutative N/A

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

   13. lower-+.f64 N/A

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

   14. lower-/.f64 99.8

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

   15. lift-+.f64 N/A

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

   16. +-commutative N/A

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

   17. lower-+.f64 99.8

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

   18. lift-+.f64 N/A

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

   19. +-commutative N/A

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

   20. lower-+.f64 99.8

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

8. Applied rewrites 99.8%

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

Alternative 6: 58.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 7.2 \cdot 10^{-146}:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(x, x, x\right)}\\ \mathbf{elif}\;y \leq 170000000000:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(y, y, y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 7.2e-146)
   (/ y (fma x x x))
   (if (<= y 170000000000.0) (/ x (fma y y y)) (/ (/ x y) (+ y x)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 7.2e-146) {
		tmp = y / fma(x, x, x);
	} else if (y <= 170000000000.0) {
		tmp = x / fma(y, y, y);
	} else {
		tmp = (x / y) / (y + x);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 7.2e-146)
		tmp = Float64(y / fma(x, x, x));
	elseif (y <= 170000000000.0)
		tmp = Float64(x / fma(y, y, y));
	else
		tmp = Float64(Float64(x / y) / Float64(y + x));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, 7.2e-146], N[(y / N[(x * x + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 170000000000.0], N[(x / N[(y * y + y), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 7.19999999999999957e-146

   1. Initial program 65.5%

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

   5. Applied rewrites 55.7%

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

   if 7.19999999999999957e-146 < y < 1.7e11

   1. Initial program 80.2%

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

   5. Applied rewrites 54.6%

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

   if 1.7e11 < y

   1. Initial program 44.5%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing
   3. 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. *-commutative N/A

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

      3. add-sqr-sqrt N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-sqrt.f64 27.9

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

   4. Applied rewrites 27.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift-sqrt.f64 N/A

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

      7. add-sqr-sqrt N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*l* N/A

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

      12. times-frac N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lower-*.f64 78.6

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

   6. Applied rewrites 78.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 78.6

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-/r* N/A

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

      9. lift-/.f64 N/A

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

      10. lower-/.f64 99.7

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 99.7

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

      14. lift-+.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 99.7

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

      17. lift-+.f64 N/A

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

      18. +-commutative N/A

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

      19. lower-+.f64 99.7

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

      20. lift-+.f64 N/A

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

      21. +-commutative N/A

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

      22. lower-+.f64 99.7

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

   8. Applied rewrites 99.7%

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

   9. Taylor expanded in y around inf

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

   11. Applied rewrites 70.1%

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

Alternative 7: 61.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.2 \cdot 10^{-257}:\\ \;\;\;\;1 \cdot \frac{y}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{1 + y}}{y + x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -2.2e-257)
   (* 1.0 (/ y (* (+ x y) (+ (+ x y) 1.0))))
   (/ (/ x (+ 1.0 y)) (+ y x))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -2.2e-257) {
		tmp = 1.0 * (y / ((x + y) * ((x + y) + 1.0)));
	} else {
		tmp = (x / (1.0 + y)) / (y + x);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-2.2d-257)) then
        tmp = 1.0d0 * (y / ((x + y) * ((x + y) + 1.0d0)))
    else
        tmp = (x / (1.0d0 + y)) / (y + x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -2.2e-257) {
		tmp = 1.0 * (y / ((x + y) * ((x + y) + 1.0)));
	} else {
		tmp = (x / (1.0 + y)) / (y + x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -2.2e-257:
		tmp = 1.0 * (y / ((x + y) * ((x + y) + 1.0)))
	else:
		tmp = (x / (1.0 + y)) / (y + x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -2.2e-257)
		tmp = Float64(1.0 * Float64(y / Float64(Float64(x + y) * Float64(Float64(x + y) + 1.0))));
	else
		tmp = Float64(Float64(x / Float64(1.0 + y)) / Float64(y + x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -2.2e-257)
		tmp = 1.0 * (y / ((x + y) * ((x + y) + 1.0)));
	else
		tmp = (x / (1.0 + y)) / (y + x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -2.2e-257], N[(1.0 * N[(y / N[(N[(x + y), $MachinePrecision] * N[(N[(x + y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(1.0 + y), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2.19999999999999988e-257

   1. Initial program 62.5%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing
   3. 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. *-commutative N/A

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

      3. add-sqr-sqrt N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-sqrt.f64 0.0

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

   4. Applied rewrites 0.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift-sqrt.f64 N/A

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

      7. add-sqr-sqrt N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*l* N/A

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

      12. times-frac N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lower-*.f64 92.3

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

   6. Applied rewrites 92.3%

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

   7. Taylor expanded in x around inf

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

   9. Applied rewrites 71.1%

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

   if -2.19999999999999988e-257 < x

   1. Initial program 60.9%

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

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

      3. add-sqr-sqrt N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-sqrt.f64 60.0

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

   4. Applied rewrites 60.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift-sqrt.f64 N/A

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

      7. add-sqr-sqrt N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*l* N/A

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

      12. times-frac N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lower-*.f64 92.7

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

   6. Applied rewrites 92.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 92.7

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-/r* N/A

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

      9. lift-/.f64 N/A

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

      10. lower-/.f64 99.8

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 99.8

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

      14. lift-+.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 99.8

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

      17. lift-+.f64 N/A

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

      18. +-commutative N/A

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

      19. lower-+.f64 99.8

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

      20. lift-+.f64 N/A

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

      21. +-commutative N/A

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

      22. lower-+.f64 99.8

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

   8. Applied rewrites 99.8%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 54.5%

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

Alternative 8: 54.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -12000000:\\ \;\;\;\;\frac{y}{x \cdot x}\\ \mathbf{elif}\;x \leq -4.2 \cdot 10^{-184} \lor \neg \left(x \leq 10^{-151}\right):\\ \;\;\;\;\frac{x}{y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -12000000.0)
   (/ y (* x x))
   (if (or (<= x -4.2e-184) (not (<= x 1e-151))) (/ x (* y y)) (/ x y))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -12000000.0) {
		tmp = y / (x * x);
	} else if ((x <= -4.2e-184) || !(x <= 1e-151)) {
		tmp = x / (y * y);
	} else {
		tmp = x / y;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-12000000.0d0)) then
        tmp = y / (x * x)
    else if ((x <= (-4.2d-184)) .or. (.not. (x <= 1d-151))) then
        tmp = x / (y * y)
    else
        tmp = x / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -12000000.0) {
		tmp = y / (x * x);
	} else if ((x <= -4.2e-184) || !(x <= 1e-151)) {
		tmp = x / (y * y);
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -12000000.0:
		tmp = y / (x * x)
	elif (x <= -4.2e-184) or not (x <= 1e-151):
		tmp = x / (y * y)
	else:
		tmp = x / y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -12000000.0)
		tmp = Float64(y / Float64(x * x));
	elseif ((x <= -4.2e-184) || !(x <= 1e-151))
		tmp = Float64(x / Float64(y * y));
	else
		tmp = Float64(x / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -12000000.0)
		tmp = y / (x * x);
	elseif ((x <= -4.2e-184) || ~((x <= 1e-151)))
		tmp = x / (y * y);
	else
		tmp = x / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -12000000.0], N[(y / N[(x * x), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[x, -4.2e-184], N[Not[LessEqual[x, 1e-151]], $MachinePrecision]], N[(x / N[(y * y), $MachinePrecision]), $MachinePrecision], N[(x / y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.2e7

   1. Initial program 50.8%

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

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 65.5%

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

   if -1.2e7 < x < -4.1999999999999998e-184 or 9.9999999999999994e-152 < x

   1. Initial program 72.2%

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

   3. Taylor expanded in y around inf

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

   5. Applied rewrites 42.9%

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

   if -4.1999999999999998e-184 < x < 9.9999999999999994e-152

   1. Initial program 46.7%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 79.3%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 71.7%

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

4. Final simplification 54.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -12000000:\\ \;\;\;\;\frac{y}{x \cdot x}\\ \mathbf{elif}\;x \leq -4.2 \cdot 10^{-184} \lor \neg \left(x \leq 10^{-151}\right):\\ \;\;\;\;\frac{x}{y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 58.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 7.2 \cdot 10^{-146}:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(x, x, x\right)}\\ \mathbf{elif}\;y \leq 6.5 \cdot 10^{+19}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(y, y, y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 7.2e-146)
   (/ y (fma x x x))
   (if (<= y 6.5e+19) (/ x (fma y y y)) (/ (/ x y) y))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 7.2e-146) {
		tmp = y / fma(x, x, x);
	} else if (y <= 6.5e+19) {
		tmp = x / fma(y, y, y);
	} else {
		tmp = (x / y) / y;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 7.2e-146)
		tmp = Float64(y / fma(x, x, x));
	elseif (y <= 6.5e+19)
		tmp = Float64(x / fma(y, y, y));
	else
		tmp = Float64(Float64(x / y) / y);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, 7.2e-146], N[(y / N[(x * x + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 6.5e+19], N[(x / N[(y * y + y), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 7.19999999999999957e-146

   1. Initial program 65.5%

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

   5. Applied rewrites 55.7%

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

   if 7.19999999999999957e-146 < y < 6.5e19

   1. Initial program 81.5%

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

   5. Applied rewrites 54.4%

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

   if 6.5e19 < y

   1. Initial program 42.8%

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

   3. Taylor expanded in y around inf

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

   5. Applied rewrites 67.7%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 70.0%

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

Alternative 10: 59.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2.6 \cdot 10^{-154}:\\ \;\;\;\;\frac{\frac{y}{1 + x}}{y + x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{1 + y}}{y + x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 2.6e-154) (/ (/ y (+ 1.0 x)) (+ y x)) (/ (/ x (+ 1.0 y)) (+ y x))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 2.6e-154) {
		tmp = (y / (1.0 + x)) / (y + x);
	} else {
		tmp = (x / (1.0 + y)) / (y + x);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 2.6d-154) then
        tmp = (y / (1.0d0 + x)) / (y + x)
    else
        tmp = (x / (1.0d0 + y)) / (y + x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 2.6e-154) {
		tmp = (y / (1.0 + x)) / (y + x);
	} else {
		tmp = (x / (1.0 + y)) / (y + x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 2.6e-154:
		tmp = (y / (1.0 + x)) / (y + x)
	else:
		tmp = (x / (1.0 + y)) / (y + x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 2.6e-154)
		tmp = Float64(Float64(y / Float64(1.0 + x)) / Float64(y + x));
	else
		tmp = Float64(Float64(x / Float64(1.0 + y)) / Float64(y + x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 2.6e-154)
		tmp = (y / (1.0 + x)) / (y + x);
	else
		tmp = (x / (1.0 + y)) / (y + x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 2.6e-154], N[(N[(y / N[(1.0 + x), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(1.0 + y), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2.6e-154

   1. Initial program 65.5%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing
   3. 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. *-commutative N/A

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

      3. add-sqr-sqrt N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-sqrt.f64 30.5

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

   4. Applied rewrites 30.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift-sqrt.f64 N/A

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

      7. add-sqr-sqrt N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*l* N/A

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

      12. times-frac N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lower-*.f64 96.9

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

   6. Applied rewrites 96.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 96.9

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-/r* N/A

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

      9. lift-/.f64 N/A

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

      10. lower-/.f64 99.9

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 99.9

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

      14. lift-+.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 99.9

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

      17. lift-+.f64 N/A

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

      18. +-commutative N/A

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

      19. lower-+.f64 99.9

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

      20. lift-+.f64 N/A

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

      21. +-commutative N/A

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

      22. lower-+.f64 99.9

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

   8. Applied rewrites 99.9%

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

   9. Taylor expanded in y around 0

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

   11. Applied rewrites 56.7%

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

   if 2.6e-154 < y

   1. Initial program 55.1%

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

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

      3. add-sqr-sqrt N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-sqrt.f64 30.4

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

   4. Applied rewrites 30.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift-sqrt.f64 N/A

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

      7. add-sqr-sqrt N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*l* N/A

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

      12. times-frac N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lower-*.f64 84.9

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

   6. Applied rewrites 84.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 84.9

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-/r* N/A

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

      9. lift-/.f64 N/A

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

      10. lower-/.f64 99.7

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 99.7

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

      14. lift-+.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 99.7

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

      17. lift-+.f64 N/A

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

      18. +-commutative N/A

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

      19. lower-+.f64 99.7

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

      20. lift-+.f64 N/A

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

      21. +-commutative N/A

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

      22. lower-+.f64 99.7

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

   8. Applied rewrites 99.7%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 65.6%

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

Alternative 11: 58.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 7.2 \cdot 10^{-146}:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(x, x, x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{1 + y}}{y + x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 7.2e-146) (/ y (fma x x x)) (/ (/ x (+ 1.0 y)) (+ y x))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 7.2e-146) {
		tmp = y / fma(x, x, x);
	} else {
		tmp = (x / (1.0 + y)) / (y + x);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 7.2e-146)
		tmp = Float64(y / fma(x, x, x));
	else
		tmp = Float64(Float64(x / Float64(1.0 + y)) / Float64(y + x));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, 7.2e-146], N[(y / N[(x * x + x), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(1.0 + y), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 7.19999999999999957e-146

   1. Initial program 65.5%

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

   5. Applied rewrites 55.7%

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

   if 7.19999999999999957e-146 < y

   1. Initial program 55.1%

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

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

      3. add-sqr-sqrt N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-sqrt.f64 30.4

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

   4. Applied rewrites 30.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift-sqrt.f64 N/A

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

      7. add-sqr-sqrt N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*l* N/A

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

      12. times-frac N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lower-*.f64 84.9

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

   6. Applied rewrites 84.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 84.9

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-/r* N/A

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

      9. lift-/.f64 N/A

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

      10. lower-/.f64 99.7

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 99.7

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

      14. lift-+.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 99.7

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

      17. lift-+.f64 N/A

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

      18. +-commutative N/A

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

      19. lower-+.f64 99.7

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

      20. lift-+.f64 N/A

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

      21. +-commutative N/A

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

      22. lower-+.f64 99.7

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

   8. Applied rewrites 99.7%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 65.6%

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

Alternative 12: 57.9% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 7.2 \cdot 10^{-146}:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(x, x, x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(y, y, y\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 7.2e-146) (/ y (fma x x x)) (/ x (fma y y y))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 7.2e-146) {
		tmp = y / fma(x, x, x);
	} else {
		tmp = x / fma(y, y, y);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 7.2e-146)
		tmp = Float64(y / fma(x, x, x));
	else
		tmp = Float64(x / fma(y, y, y));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, 7.2e-146], N[(y / N[(x * x + x), $MachinePrecision]), $MachinePrecision], N[(x / N[(y * y + y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 7.19999999999999957e-146

   1. Initial program 65.5%

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

   5. Applied rewrites 55.7%

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

   if 7.19999999999999957e-146 < y

   1. Initial program 55.1%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 63.4%

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

Alternative 13: 61.6% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -12000000:\\ \;\;\;\;\frac{y}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(y, y, y\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -12000000.0) (/ y (* x x)) (/ x (fma y y y))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -12000000.0) {
		tmp = y / (x * x);
	} else {
		tmp = x / fma(y, y, y);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -12000000.0)
		tmp = Float64(y / Float64(x * x));
	else
		tmp = Float64(x / fma(y, y, y));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, -12000000.0], N[(y / N[(x * x), $MachinePrecision]), $MachinePrecision], N[(x / N[(y * y + y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.2e7

   1. Initial program 50.8%

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

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 65.5%

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

   if -1.2e7 < x

   1. Initial program 65.3%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 57.6%

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

Alternative 14: 37.4% 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 67.1%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 45.6%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 24.1%

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

   if 1 < y

   1. Initial program 46.9%

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

   3. Taylor expanded in y around inf

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

   5. Applied rewrites 65.2%

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

Alternative 15: 26.7% accurate, 3.3× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 61.7%

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

3. Taylor expanded in x around 0

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

5. Applied rewrites 51.0%

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

6. Taylor expanded in y around 0

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

8. Applied rewrites 23.7%

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

Developer Target 1: 99.8% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2025021 
(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))))