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

Percentage Accurate: 69.1% → 95.6%

Time: 4.2s

Alternatives: 15

Speedup: 1.5×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 69.1% 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: 95.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_0 := \frac{x}{y + x}\\ \mathbf{if}\;y \leq 2.7 \cdot 10^{+141}:\\ \;\;\;\;t\_0 \cdot \frac{y}{\left(\left(y + x\right) + 1\right) \cdot \left(y + x\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(2, x, 1\right)}{y}, -1, 1\right)}{y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ x (+ y x))))
   (if (<= y 2.7e+141)
     (* t_0 (/ y (* (+ (+ y x) 1.0) (+ y x))))
     (* t_0 (/ (fma (/ (fma 2.0 x 1.0) y) -1.0 1.0) y)))))

C

assert(x < y);
double code(double x, double y) {
	double t_0 = x / (y + x);
	double tmp;
	if (y <= 2.7e+141) {
		tmp = t_0 * (y / (((y + x) + 1.0) * (y + x)));
	} else {
		tmp = t_0 * (fma((fma(2.0, x, 1.0) / y), -1.0, 1.0) / y);
	}
	return tmp;
}

Julia

x, y = sort([x, y])
function code(x, y)
	t_0 = Float64(x / Float64(y + x))
	tmp = 0.0
	if (y <= 2.7e+141)
		tmp = Float64(t_0 * Float64(y / Float64(Float64(Float64(y + x) + 1.0) * Float64(y + x))));
	else
		tmp = Float64(t_0 * Float64(fma(Float64(fma(2.0, x, 1.0) / y), -1.0, 1.0) / y));
	end
	return tmp
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := Block[{t$95$0 = N[(x / N[(y + x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, 2.7e+141], N[(t$95$0 * N[(y / N[(N[(N[(y + x), $MachinePrecision] + 1.0), $MachinePrecision] * N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(N[(N[(N[(2.0 * x + 1.0), $MachinePrecision] / y), $MachinePrecision] * -1.0 + 1.0), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < 2.7000000000000001e141

   1. Initial program 73.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. times-frac N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. pow2 N/A

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

      13. lower-pow.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. lift-+.f64 N/A

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

      18. +-commutative N/A

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

      19. lower-+.f64 91.6

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

   4. Applied rewrites 91.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. frac-times N/A

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

      9. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. times-frac N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lift-+.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 N/A

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

      18. lift-+.f64 N/A

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

      19. lift-+.f64 N/A

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

      20. lift-+.f64 96.7

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

   6. Applied rewrites 96.7%

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

   if 2.7000000000000001e141 < y

   1. Initial program 67.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. times-frac N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. pow2 N/A

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

      13. lower-pow.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. lift-+.f64 N/A

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

      18. +-commutative N/A

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

      19. lower-+.f64 84.8

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

   4. Applied rewrites 84.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. frac-times N/A

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

      9. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. times-frac N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lift-+.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 N/A

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

      18. lift-+.f64 N/A

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

      19. lift-+.f64 N/A

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

      20. lift-+.f64 84.8

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

   6. Applied rewrites 84.8%

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

   7. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 84.8

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

   9. Applied rewrites 84.8%

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

   10. Taylor expanded in y around inf

       \[\leadsto \frac{x}{y + x} \cdot \color{blue}{\frac{1 + -1 \cdot \frac{1 + 2 \cdot x}{y}}{y}} \]
   11. Step-by-step derivation

       1. lower-/.f64 N/A

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

       2. +-commutative N/A

          \[\leadsto \frac{x}{y + x} \cdot \frac{-1 \cdot \frac{1 + 2 \cdot x}{y} + 1}{y} \]

       3. *-commutative N/A

          \[\leadsto \frac{x}{y + x} \cdot \frac{\frac{1 + 2 \cdot x}{y} \cdot -1 + 1}{y} \]

       4. div-add N/A

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

       5. associate-*r/ N/A

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

       6. +-commutative N/A

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

       7. lower-fma.f64 N/A

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

       8. +-commutative N/A

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

       9. associate-*r/ N/A

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

       10. div-add N/A

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

       11. lower-/.f64 N/A

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

       12. +-commutative N/A

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

       13. lower-fma.f64 89.9

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

   12. Applied rewrites 89.9%

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

Alternative 2: 87.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 8.2 \cdot 10^{-138}:\\ \;\;\;\;\frac{\frac{y}{1 + x}}{x}\\ \mathbf{elif}\;y \leq 2.3 \cdot 10^{-113}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;y \leq 1.65 \cdot 10^{+96}:\\ \;\;\;\;\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 8.2e-138)
   (/ (/ y (+ 1.0 x)) x)
   (if (<= y 2.3e-113)
     (/ x y)
     (if (<= y 1.65e+96)
       (/ (* x y) (* (* (+ x y) (+ x y)) (+ (+ x y) 1.0)))
       (/ (/ x y) y)))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 8.2e-138) {
		tmp = (y / (1.0 + x)) / x;
	} else if (y <= 2.3e-113) {
		tmp = x / y;
	} else if (y <= 1.65e+96) {
		tmp = (x * y) / (((x + y) * (x + y)) * ((x + y) + 1.0));
	} else {
		tmp = (x / y) / y;
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 8.2d-138) then
        tmp = (y / (1.0d0 + x)) / x
    else if (y <= 2.3d-113) then
        tmp = x / y
    else if (y <= 1.65d+96) then
        tmp = (x * y) / (((x + y) * (x + y)) * ((x + y) + 1.0d0))
    else
        tmp = (x / y) / y
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (y <= 8.2e-138) {
		tmp = (y / (1.0 + x)) / x;
	} else if (y <= 2.3e-113) {
		tmp = x / y;
	} else if (y <= 1.65e+96) {
		tmp = (x * y) / (((x + y) * (x + y)) * ((x + y) + 1.0));
	} else {
		tmp = (x / y) / y;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if y <= 8.2e-138:
		tmp = (y / (1.0 + x)) / x
	elif y <= 2.3e-113:
		tmp = x / y
	elif y <= 1.65e+96:
		tmp = (x * y) / (((x + y) * (x + y)) * ((x + y) + 1.0))
	else:
		tmp = (x / y) / y
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= 8.2e-138)
		tmp = Float64(Float64(y / Float64(1.0 + x)) / x);
	elseif (y <= 2.3e-113)
		tmp = Float64(x / y);
	elseif (y <= 1.65e+96)
		tmp = Float64(Float64(x * y) / Float64(Float64(Float64(x + y) * Float64(x + y)) * Float64(Float64(x + y) + 1.0)));
	else
		tmp = Float64(Float64(x / y) / y);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 8.2e-138)
		tmp = (y / (1.0 + x)) / x;
	elseif (y <= 2.3e-113)
		tmp = x / y;
	elseif (y <= 1.65e+96)
		tmp = (x * y) / (((x + y) * (x + y)) * ((x + y) + 1.0));
	else
		tmp = (x / y) / y;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[y, 8.2e-138], N[(N[(y / N[(1.0 + x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[y, 2.3e-113], N[(x / y), $MachinePrecision], If[LessEqual[y, 1.65e+96], N[(N[(x * y), $MachinePrecision] / N[(N[(N[(x + y), $MachinePrecision] * N[(x + y), $MachinePrecision]), $MachinePrecision] * N[(N[(x + y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / y), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if y < 8.19999999999999998e-138

   1. Initial program 73.7%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-+.f64 57.7

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

   5. Applied rewrites 57.7%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lift-+.f64 58.8

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

   7. Applied rewrites 58.8%

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

   if 8.19999999999999998e-138 < y < 2.30000000000000008e-113

   1. Initial program 4.3%

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-+.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 100.0%

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

   if 2.30000000000000008e-113 < y < 1.64999999999999992e96

   1. Initial program 78.3%

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

   if 1.64999999999999992e96 < y

   1. Initial program 67.6%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 84.6

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

   5. Applied rewrites 84.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 87.8

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

   7. Applied rewrites 87.8%

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

Alternative 3: 95.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 2.7 \cdot 10^{+141}:\\ \;\;\;\;\frac{x}{y + x} \cdot \frac{y}{\left(\left(y + x\right) + 1\right) \cdot \left(y + x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 2.7e+141)
   (* (/ x (+ y x)) (/ y (* (+ (+ y x) 1.0) (+ y x))))
   (/ (/ x y) y)))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 2.7e+141) {
		tmp = (x / (y + x)) * (y / (((y + x) + 1.0) * (y + x)));
	} else {
		tmp = (x / y) / y;
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
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 <= 2.7d+141) then
        tmp = (x / (y + x)) * (y / (((y + x) + 1.0d0) * (y + x)))
    else
        tmp = (x / y) / y
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (y <= 2.7e+141) {
		tmp = (x / (y + x)) * (y / (((y + x) + 1.0) * (y + x)));
	} else {
		tmp = (x / y) / y;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if y <= 2.7e+141:
		tmp = (x / (y + x)) * (y / (((y + x) + 1.0) * (y + x)))
	else:
		tmp = (x / y) / y
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= 2.7e+141)
		tmp = Float64(Float64(x / Float64(y + x)) * Float64(y / Float64(Float64(Float64(y + x) + 1.0) * Float64(y + x))));
	else
		tmp = Float64(Float64(x / y) / y);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 2.7e+141)
		tmp = (x / (y + x)) * (y / (((y + x) + 1.0) * (y + x)));
	else
		tmp = (x / y) / y;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[y, 2.7e+141], N[(N[(x / N[(y + x), $MachinePrecision]), $MachinePrecision] * N[(y / N[(N[(N[(y + x), $MachinePrecision] + 1.0), $MachinePrecision] * N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2.7000000000000001e141

   1. Initial program 73.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. times-frac N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. pow2 N/A

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

      13. lower-pow.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. lift-+.f64 N/A

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

      18. +-commutative N/A

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

      19. lower-+.f64 91.6

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

   4. Applied rewrites 91.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. frac-times N/A

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

      9. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. times-frac N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lift-+.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 N/A

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

      18. lift-+.f64 N/A

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

      19. lift-+.f64 N/A

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

      20. lift-+.f64 96.7

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

   6. Applied rewrites 96.7%

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

   if 2.7000000000000001e141 < y

   1. Initial program 67.6%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 84.8

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

   5. Applied rewrites 84.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 89.2

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

   7. Applied rewrites 89.2%

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

Alternative 4: 91.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -3.05 \cdot 10^{-15}:\\ \;\;\;\;1 \cdot \frac{y}{\left(\left(y + x\right) + 1\right) \cdot \left(y + x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y + x} \cdot \frac{y}{\left(y + 1\right) \cdot \left(y + x\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -3.05e-15)
   (* 1.0 (/ y (* (+ (+ y x) 1.0) (+ y x))))
   (* (/ x (+ y x)) (/ y (* (+ y 1.0) (+ y x))))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -3.05e-15) {
		tmp = 1.0 * (y / (((y + x) + 1.0) * (y + x)));
	} else {
		tmp = (x / (y + x)) * (y / ((y + 1.0) * (y + x)));
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
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 <= (-3.05d-15)) then
        tmp = 1.0d0 * (y / (((y + x) + 1.0d0) * (y + x)))
    else
        tmp = (x / (y + x)) * (y / ((y + 1.0d0) * (y + x)))
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (x <= -3.05e-15) {
		tmp = 1.0 * (y / (((y + x) + 1.0) * (y + x)));
	} else {
		tmp = (x / (y + x)) * (y / ((y + 1.0) * (y + x)));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -3.05e-15:
		tmp = 1.0 * (y / (((y + x) + 1.0) * (y + x)))
	else:
		tmp = (x / (y + x)) * (y / ((y + 1.0) * (y + x)))
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -3.05e-15)
		tmp = Float64(1.0 * Float64(y / Float64(Float64(Float64(y + x) + 1.0) * Float64(y + x))));
	else
		tmp = Float64(Float64(x / Float64(y + x)) * Float64(y / Float64(Float64(y + 1.0) * Float64(y + x))));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -3.05e-15)
		tmp = 1.0 * (y / (((y + x) + 1.0) * (y + x)));
	else
		tmp = (x / (y + x)) * (y / ((y + 1.0) * (y + x)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[x, -3.05e-15], N[(1.0 * N[(y / N[(N[(N[(y + x), $MachinePrecision] + 1.0), $MachinePrecision] * N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(y + x), $MachinePrecision]), $MachinePrecision] * N[(y / N[(N[(y + 1.0), $MachinePrecision] * N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -3.04999999999999986e-15

   1. Initial program 68.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. times-frac N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. pow2 N/A

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

      13. lower-pow.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. lift-+.f64 N/A

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

      18. +-commutative N/A

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

      19. lower-+.f64 93.7

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

   4. Applied rewrites 93.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. frac-times N/A

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

      9. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. times-frac N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lift-+.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 N/A

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

      18. lift-+.f64 N/A

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

      19. lift-+.f64 N/A

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

      20. lift-+.f64 93.8

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

   6. Applied rewrites 93.8%

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

   7. Taylor expanded in x around inf

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

   9. Applied rewrites 87.5%

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

   if -3.04999999999999986e-15 < x

   1. Initial program 73.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. times-frac N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. pow2 N/A

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

      13. lower-pow.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. lift-+.f64 N/A

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

      18. +-commutative N/A

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

      19. lower-+.f64 89.6

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

   4. Applied rewrites 89.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. frac-times N/A

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

      9. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. times-frac N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lift-+.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 N/A

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

      18. lift-+.f64 N/A

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

      19. lift-+.f64 N/A

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

      20. lift-+.f64 95.4

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

   6. Applied rewrites 95.4%

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

   7. Taylor expanded in x around 0

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

   9. Applied rewrites 85.8%

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

Alternative 5: 83.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{-187}:\\ \;\;\;\;1 \cdot \frac{y}{\left(\left(y + x\right) + 1\right) \cdot \left(y + x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y + x} \cdot \frac{y}{y \cdot \left(1 + y\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -2e-187)
   (* 1.0 (/ y (* (+ (+ y x) 1.0) (+ y x))))
   (* (/ x (+ y x)) (/ y (* y (+ 1.0 y))))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -2e-187) {
		tmp = 1.0 * (y / (((y + x) + 1.0) * (y + x)));
	} else {
		tmp = (x / (y + x)) * (y / (y * (1.0 + y)));
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
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 <= (-2d-187)) then
        tmp = 1.0d0 * (y / (((y + x) + 1.0d0) * (y + x)))
    else
        tmp = (x / (y + x)) * (y / (y * (1.0d0 + y)))
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (x <= -2e-187) {
		tmp = 1.0 * (y / (((y + x) + 1.0) * (y + x)));
	} else {
		tmp = (x / (y + x)) * (y / (y * (1.0 + y)));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -2e-187:
		tmp = 1.0 * (y / (((y + x) + 1.0) * (y + x)))
	else:
		tmp = (x / (y + x)) * (y / (y * (1.0 + y)))
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -2e-187)
		tmp = Float64(1.0 * Float64(y / Float64(Float64(Float64(y + x) + 1.0) * Float64(y + x))));
	else
		tmp = Float64(Float64(x / Float64(y + x)) * Float64(y / Float64(y * Float64(1.0 + y))));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -2e-187)
		tmp = 1.0 * (y / (((y + x) + 1.0) * (y + x)));
	else
		tmp = (x / (y + x)) * (y / (y * (1.0 + y)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[x, -2e-187], N[(1.0 * N[(y / N[(N[(N[(y + x), $MachinePrecision] + 1.0), $MachinePrecision] * N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(y + x), $MachinePrecision]), $MachinePrecision] * N[(y / N[(y * N[(1.0 + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2e-187

   1. Initial program 75.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. times-frac N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. pow2 N/A

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

      13. lower-pow.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. lift-+.f64 N/A

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

      18. +-commutative N/A

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

      19. lower-+.f64 96.1

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

   4. Applied rewrites 96.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. frac-times N/A

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

      9. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. times-frac N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lift-+.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 N/A

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

      18. lift-+.f64 N/A

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

      19. lift-+.f64 N/A

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

      20. lift-+.f64 96.2

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

   6. Applied rewrites 96.2%

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

   7. Taylor expanded in x around inf

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

   9. Applied rewrites 77.4%

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

   if -2e-187 < x

   1. Initial program 69.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. times-frac N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. pow2 N/A

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

      13. lower-pow.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. lift-+.f64 N/A

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

      18. +-commutative N/A

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

      19. lower-+.f64 86.9

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

   4. Applied rewrites 86.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. frac-times N/A

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

      9. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. times-frac N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lift-+.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 N/A

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

      18. lift-+.f64 N/A

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

      19. lift-+.f64 N/A

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

      20. lift-+.f64 94.2

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

   6. Applied rewrites 94.2%

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

   7. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. lift-+.f64 62.0

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

   9. Applied rewrites 62.0%

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

Alternative 6: 83.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{-187}:\\ \;\;\;\;1 \cdot \frac{y}{\left(\left(y + x\right) + 1\right) \cdot \left(y + x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\left(1 + y\right) \cdot y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -2e-187)
   (* 1.0 (/ y (* (+ (+ y x) 1.0) (+ y x))))
   (/ x (* (+ 1.0 y) y))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -2e-187) {
		tmp = 1.0 * (y / (((y + x) + 1.0) * (y + x)));
	} else {
		tmp = x / ((1.0 + y) * y);
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
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 <= (-2d-187)) then
        tmp = 1.0d0 * (y / (((y + x) + 1.0d0) * (y + x)))
    else
        tmp = x / ((1.0d0 + y) * y)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (x <= -2e-187) {
		tmp = 1.0 * (y / (((y + x) + 1.0) * (y + x)));
	} else {
		tmp = x / ((1.0 + y) * y);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -2e-187:
		tmp = 1.0 * (y / (((y + x) + 1.0) * (y + x)))
	else:
		tmp = x / ((1.0 + y) * y)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -2e-187)
		tmp = Float64(1.0 * Float64(y / Float64(Float64(Float64(y + x) + 1.0) * Float64(y + x))));
	else
		tmp = Float64(x / Float64(Float64(1.0 + y) * y));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -2e-187)
		tmp = 1.0 * (y / (((y + x) + 1.0) * (y + x)));
	else
		tmp = x / ((1.0 + y) * y);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[x, -2e-187], N[(1.0 * N[(y / N[(N[(N[(y + x), $MachinePrecision] + 1.0), $MachinePrecision] * N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x / N[(N[(1.0 + y), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2e-187

   1. Initial program 75.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. times-frac N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. pow2 N/A

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

      13. lower-pow.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. lift-+.f64 N/A

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

      18. +-commutative N/A

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

      19. lower-+.f64 96.1

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

   4. Applied rewrites 96.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. frac-times N/A

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

      9. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. times-frac N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lift-+.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 N/A

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

      18. lift-+.f64 N/A

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

      19. lift-+.f64 N/A

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

      20. lift-+.f64 96.2

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

   6. Applied rewrites 96.2%

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

   7. Taylor expanded in x around inf

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

   9. Applied rewrites 77.4%

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

   if -2e-187 < x

   1. Initial program 69.7%

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-+.f64 61.8

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

   5. Applied rewrites 61.8%

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

Alternative 7: 72.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq -3.4 \cdot 10^{-129}:\\ \;\;\;\;\frac{y}{x \cdot x}\\ \mathbf{elif}\;y \leq 2 \cdot 10^{-147}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y -3.4e-129)
   (/ y (* x x))
   (if (<= y 2e-147) (/ y x) (if (<= y 1.0) (/ x y) (/ x (* y y))))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= -3.4e-129) {
		tmp = y / (x * x);
	} else if (y <= 2e-147) {
		tmp = y / x;
	} else if (y <= 1.0) {
		tmp = x / y;
	} else {
		tmp = x / (y * y);
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-3.4d-129)) then
        tmp = y / (x * x)
    else if (y <= 2d-147) then
        tmp = y / x
    else if (y <= 1.0d0) then
        tmp = x / y
    else
        tmp = x / (y * y)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (y <= -3.4e-129) {
		tmp = y / (x * x);
	} else if (y <= 2e-147) {
		tmp = y / x;
	} else if (y <= 1.0) {
		tmp = x / y;
	} else {
		tmp = x / (y * y);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if y <= -3.4e-129:
		tmp = y / (x * x)
	elif y <= 2e-147:
		tmp = y / x
	elif y <= 1.0:
		tmp = x / y
	else:
		tmp = x / (y * y)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= -3.4e-129)
		tmp = Float64(y / Float64(x * x));
	elseif (y <= 2e-147)
		tmp = Float64(y / x);
	elseif (y <= 1.0)
		tmp = Float64(x / y);
	else
		tmp = Float64(x / Float64(y * y));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -3.4e-129)
		tmp = y / (x * x);
	elseif (y <= 2e-147)
		tmp = y / x;
	elseif (y <= 1.0)
		tmp = x / y;
	else
		tmp = x / (y * y);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[y, -3.4e-129], N[(y / N[(x * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2e-147], N[(y / x), $MachinePrecision], If[LessEqual[y, 1.0], N[(x / y), $MachinePrecision], N[(x / N[(y * y), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if y < -3.40000000000000013e-129

   1. Initial program 78.5%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 32.3

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

   5. Applied rewrites 32.3%

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

   if -3.40000000000000013e-129 < y < 1.9999999999999999e-147

   1. Initial program 69.1%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-+.f64 88.3

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

   5. Applied rewrites 88.3%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 77.1%

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

   if 1.9999999999999999e-147 < y < 1

   1. Initial program 69.4%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-+.f64 64.3

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

   5. Applied rewrites 64.3%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 57.2%

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

   if 1 < y

   1. Initial program 68.6%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 75.4

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

   5. Applied rewrites 75.4%

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

Alternative 8: 82.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 8.2 \cdot 10^{-138}:\\ \;\;\;\;\frac{\frac{y}{1 + x}}{x}\\ \mathbf{elif}\;y \leq 220000000000:\\ \;\;\;\;\frac{x}{\left(1 + y\right) \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 8.2e-138)
   (/ (/ y (+ 1.0 x)) x)
   (if (<= y 220000000000.0) (/ x (* (+ 1.0 y) y)) (/ (/ x y) y))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 8.2e-138) {
		tmp = (y / (1.0 + x)) / x;
	} else if (y <= 220000000000.0) {
		tmp = x / ((1.0 + y) * y);
	} else {
		tmp = (x / y) / y;
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 8.2d-138) then
        tmp = (y / (1.0d0 + x)) / x
    else if (y <= 220000000000.0d0) then
        tmp = x / ((1.0d0 + y) * y)
    else
        tmp = (x / y) / y
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (y <= 8.2e-138) {
		tmp = (y / (1.0 + x)) / x;
	} else if (y <= 220000000000.0) {
		tmp = x / ((1.0 + y) * y);
	} else {
		tmp = (x / y) / y;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if y <= 8.2e-138:
		tmp = (y / (1.0 + x)) / x
	elif y <= 220000000000.0:
		tmp = x / ((1.0 + y) * y)
	else:
		tmp = (x / y) / y
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= 8.2e-138)
		tmp = Float64(Float64(y / Float64(1.0 + x)) / x);
	elseif (y <= 220000000000.0)
		tmp = Float64(x / Float64(Float64(1.0 + y) * y));
	else
		tmp = Float64(Float64(x / y) / y);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 8.2e-138)
		tmp = (y / (1.0 + x)) / x;
	elseif (y <= 220000000000.0)
		tmp = x / ((1.0 + y) * y);
	else
		tmp = (x / y) / y;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[y, 8.2e-138], N[(N[(y / N[(1.0 + x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[y, 220000000000.0], N[(x / N[(N[(1.0 + y), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 8.19999999999999998e-138

   1. Initial program 73.7%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-+.f64 57.7

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

   5. Applied rewrites 57.7%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lift-+.f64 58.8

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

   7. Applied rewrites 58.8%

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

   if 8.19999999999999998e-138 < y < 2.2e11

   1. Initial program 73.7%

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-+.f64 64.3

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

   5. Applied rewrites 64.3%

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

   if 2.2e11 < y

   1. Initial program 68.1%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 75.7

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

   5. Applied rewrites 75.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 77.9

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

   7. Applied rewrites 77.9%

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

Alternative 9: 80.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 8.2 \cdot 10^{-138}:\\ \;\;\;\;\frac{y}{\left(1 + x\right) \cdot x}\\ \mathbf{elif}\;y \leq 220000000000:\\ \;\;\;\;\frac{x}{\left(1 + y\right) \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 8.2e-138)
   (/ y (* (+ 1.0 x) x))
   (if (<= y 220000000000.0) (/ x (* (+ 1.0 y) y)) (/ (/ x y) y))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 8.2e-138) {
		tmp = y / ((1.0 + x) * x);
	} else if (y <= 220000000000.0) {
		tmp = x / ((1.0 + y) * y);
	} else {
		tmp = (x / y) / y;
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 8.2d-138) then
        tmp = y / ((1.0d0 + x) * x)
    else if (y <= 220000000000.0d0) then
        tmp = x / ((1.0d0 + y) * y)
    else
        tmp = (x / y) / y
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (y <= 8.2e-138) {
		tmp = y / ((1.0 + x) * x);
	} else if (y <= 220000000000.0) {
		tmp = x / ((1.0 + y) * y);
	} else {
		tmp = (x / y) / y;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if y <= 8.2e-138:
		tmp = y / ((1.0 + x) * x)
	elif y <= 220000000000.0:
		tmp = x / ((1.0 + y) * y)
	else:
		tmp = (x / y) / y
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= 8.2e-138)
		tmp = Float64(y / Float64(Float64(1.0 + x) * x));
	elseif (y <= 220000000000.0)
		tmp = Float64(x / Float64(Float64(1.0 + y) * y));
	else
		tmp = Float64(Float64(x / y) / y);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 8.2e-138)
		tmp = y / ((1.0 + x) * x);
	elseif (y <= 220000000000.0)
		tmp = x / ((1.0 + y) * y);
	else
		tmp = (x / y) / y;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[y, 8.2e-138], N[(y / N[(N[(1.0 + x), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 220000000000.0], N[(x / N[(N[(1.0 + y), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 8.19999999999999998e-138

   1. Initial program 73.7%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-+.f64 57.7

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

   5. Applied rewrites 57.7%

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

   if 8.19999999999999998e-138 < y < 2.2e11

   1. Initial program 73.7%

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-+.f64 64.3

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

   5. Applied rewrites 64.3%

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

   if 2.2e11 < y

   1. Initial program 68.1%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 75.7

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

   5. Applied rewrites 75.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 77.9

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

   7. Applied rewrites 77.9%

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

Alternative 10: 78.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{y}{x \cdot x}\\ \mathbf{elif}\;x \leq -1.55 \cdot 10^{-108}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-x, y, y\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\left(1 + y\right) \cdot y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -1.0)
   (/ y (* x x))
   (if (<= x -1.55e-108) (/ (fma (- x) y y) x) (/ x (* (+ 1.0 y) y)))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -1.0) {
		tmp = y / (x * x);
	} else if (x <= -1.55e-108) {
		tmp = fma(-x, y, y) / x;
	} else {
		tmp = x / ((1.0 + y) * y);
	}
	return tmp;
}

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -1.0)
		tmp = Float64(y / Float64(x * x));
	elseif (x <= -1.55e-108)
		tmp = Float64(fma(Float64(-x), y, y) / x);
	else
		tmp = Float64(x / Float64(Float64(1.0 + y) * y));
	end
	return tmp
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[x, -1.0], N[(y / N[(x * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -1.55e-108], N[(N[((-x) * y + y), $MachinePrecision] / x), $MachinePrecision], N[(x / N[(N[(1.0 + y), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1

   1. Initial program 67.8%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 78.9

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

   5. Applied rewrites 78.9%

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

   if -1 < x < -1.55000000000000007e-108

   1. Initial program 88.1%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-+.f64 65.7

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

   5. Applied rewrites 65.7%

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

   6. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-*r* N/A

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

      4. mul-1-neg N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-neg.f64 65.7

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

   8. Applied rewrites 65.7%

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

   if -1.55000000000000007e-108 < 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 x around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-+.f64 64.5

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

   5. Applied rewrites 64.5%

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

Alternative 11: 64.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 2 \cdot 10^{-147}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 2e-147) (/ y x) (if (<= y 1.0) (/ x y) (/ x (* y y)))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 2e-147) {
		tmp = y / x;
	} else if (y <= 1.0) {
		tmp = x / y;
	} else {
		tmp = x / (y * y);
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
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 <= 2d-147) then
        tmp = y / x
    else if (y <= 1.0d0) then
        tmp = x / y
    else
        tmp = x / (y * y)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (y <= 2e-147) {
		tmp = y / x;
	} else if (y <= 1.0) {
		tmp = x / y;
	} else {
		tmp = x / (y * y);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if y <= 2e-147:
		tmp = y / x
	elif y <= 1.0:
		tmp = x / y
	else:
		tmp = x / (y * y)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= 2e-147)
		tmp = Float64(y / x);
	elseif (y <= 1.0)
		tmp = Float64(x / y);
	else
		tmp = Float64(x / Float64(y * y));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 2e-147)
		tmp = y / x;
	elseif (y <= 1.0)
		tmp = x / y;
	else
		tmp = x / (y * y);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[y, 2e-147], N[(y / x), $MachinePrecision], If[LessEqual[y, 1.0], N[(x / y), $MachinePrecision], N[(x / N[(y * y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 1.9999999999999999e-147

   1. Initial program 74.2%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-+.f64 58.1

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

   5. Applied rewrites 58.1%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 37.7%

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

   if 1.9999999999999999e-147 < y < 1

   1. Initial program 69.4%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-+.f64 64.3

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

   5. Applied rewrites 64.3%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 57.2%

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

   if 1 < y

   1. Initial program 68.6%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 75.4

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

   5. Applied rewrites 75.4%

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

Alternative 12: 78.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 8.2 \cdot 10^{-138}:\\ \;\;\;\;\frac{y}{\left(1 + x\right) \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\left(1 + y\right) \cdot y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 8.2e-138) (/ y (* (+ 1.0 x) x)) (/ x (* (+ 1.0 y) y))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 8.2e-138) {
		tmp = y / ((1.0 + x) * x);
	} else {
		tmp = x / ((1.0 + y) * y);
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 8.2d-138) then
        tmp = y / ((1.0d0 + x) * x)
    else
        tmp = x / ((1.0d0 + y) * y)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (y <= 8.2e-138) {
		tmp = y / ((1.0 + x) * x);
	} else {
		tmp = x / ((1.0 + y) * y);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if y <= 8.2e-138:
		tmp = y / ((1.0 + x) * x)
	else:
		tmp = x / ((1.0 + y) * y)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= 8.2e-138)
		tmp = Float64(y / Float64(Float64(1.0 + x) * x));
	else
		tmp = Float64(x / Float64(Float64(1.0 + y) * y));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 8.2e-138)
		tmp = y / ((1.0 + x) * x);
	else
		tmp = x / ((1.0 + y) * y);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[y, 8.2e-138], N[(y / N[(N[(1.0 + x), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision], N[(x / N[(N[(1.0 + y), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 8.19999999999999998e-138

   1. Initial program 73.7%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-+.f64 57.7

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

   5. Applied rewrites 57.7%

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

   if 8.19999999999999998e-138 < y

   1. Initial program 69.5%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-+.f64 73.0

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

   5. Applied rewrites 73.0%

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

Alternative 13: 43.8% accurate, 2.2× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 2 \cdot 10^{-147}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y) :precision binary64 (if (<= y 2e-147) (/ y x) (/ x y)))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 2e-147) {
		tmp = y / x;
	} else {
		tmp = x / y;
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
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 <= 2d-147) then
        tmp = y / x
    else
        tmp = x / y
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (y <= 2e-147) {
		tmp = y / x;
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if y <= 2e-147:
		tmp = y / x
	else:
		tmp = x / y
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= 2e-147)
		tmp = Float64(y / x);
	else
		tmp = Float64(x / y);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 2e-147)
		tmp = y / x;
	else
		tmp = x / y;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[y, 2e-147], N[(y / x), $MachinePrecision], N[(x / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.9999999999999999e-147

   1. Initial program 74.2%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-+.f64 58.1

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

   5. Applied rewrites 58.1%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 37.7%

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

   if 1.9999999999999999e-147 < y

   1. Initial program 68.8%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-+.f64 73.3

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

   5. Applied rewrites 73.3%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 37.8%

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

Alternative 14: 26.4% accurate, 3.3× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \frac{x}{y} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y) :precision binary64 (/ x y))

C

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

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
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

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

Python

[x, y] = sort([x, y])
def code(x, y):
	return x / y

Julia

x, y = sort([x, y])
function code(x, y)
	return Float64(x / y)
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp = code(x, y)
	tmp = x / y;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := N[(x / y), $MachinePrecision]

Derivation

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 x around 0

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

   1. lower-/.f64 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 N/A

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

   4. lower-+.f64 52.9

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

5. Applied rewrites 52.9%

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

6. Taylor expanded in y around 0

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

8. Applied rewrites 30.0%

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

Alternative 15: 3.6% accurate, 6.5× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ y \cdot x \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y) :precision binary64 (* y x))

C

assert(x < y);
double code(double x, double y) {
	return y * x;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

assert x < y;
public static double code(double x, double y) {
	return y * x;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	return y * x

Julia

x, y = sort([x, y])
function code(x, y)
	return Float64(y * x)
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp = code(x, y)
	tmp = y * x;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := N[(y * x), $MachinePrecision]

Derivation

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 x around 0

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

   1. lower-/.f64 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 N/A

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

   4. lower-+.f64 52.9

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

5. Applied rewrites 52.9%

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

6. Taylor expanded in y around 0

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

   1. lower-/.f64 N/A

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

   2. lower-+.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower--.f64 N/A

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

   5. lift-*.f64 15.4

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

8. Applied rewrites 15.4%

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

9. Taylor expanded in y around inf

   \[\leadsto x \cdot y \]
10. Step-by-step derivation

    1. *-commutative N/A

       \[\leadsto y \cdot x \]

    2. lower-*.f64 3.6

       \[\leadsto y \cdot x \]

11. Applied rewrites 3.6%

    \[\leadsto y \cdot x \]
12. 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 2025064 
(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))))