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

Percentage Accurate: 68.4% → 89.6%

Time: 4.1s

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: 68.4% 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: 89.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_0 := \left(y + x\right) + 1\\ \mathbf{if}\;x \leq -7.9 \cdot 10^{+146}:\\ \;\;\;\;\frac{\frac{y}{x}}{t\_0}\\ \mathbf{elif}\;x \leq -2.5 \cdot 10^{-161}:\\ \;\;\;\;\frac{\frac{y \cdot x}{{\left(y + x\right)}^{2}}}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(x \cdot \frac{x}{y}, -2, x\right)}{y}}{t\_0}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ (+ y x) 1.0)))
   (if (<= x -7.9e+146)
     (/ (/ y x) t_0)
     (if (<= x -2.5e-161)
       (/ (/ (* y x) (pow (+ y x) 2.0)) t_0)
       (/ (/ (fma (* x (/ x y)) -2.0 x) y) t_0)))))

C

assert(x < y);
double code(double x, double y) {
	double t_0 = (y + x) + 1.0;
	double tmp;
	if (x <= -7.9e+146) {
		tmp = (y / x) / t_0;
	} else if (x <= -2.5e-161) {
		tmp = ((y * x) / pow((y + x), 2.0)) / t_0;
	} else {
		tmp = (fma((x * (x / y)), -2.0, x) / y) / t_0;
	}
	return tmp;
}

Julia

x, y = sort([x, y])
function code(x, y)
	t_0 = Float64(Float64(y + x) + 1.0)
	tmp = 0.0
	if (x <= -7.9e+146)
		tmp = Float64(Float64(y / x) / t_0);
	elseif (x <= -2.5e-161)
		tmp = Float64(Float64(Float64(y * x) / (Float64(y + x) ^ 2.0)) / t_0);
	else
		tmp = Float64(Float64(fma(Float64(x * Float64(x / y)), -2.0, x) / y) / t_0);
	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[(N[(y + x), $MachinePrecision] + 1.0), $MachinePrecision]}, If[LessEqual[x, -7.9e+146], N[(N[(y / x), $MachinePrecision] / t$95$0), $MachinePrecision], If[LessEqual[x, -2.5e-161], N[(N[(N[(y * x), $MachinePrecision] / N[Power[N[(y + x), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision], N[(N[(N[(N[(x * N[(x / y), $MachinePrecision]), $MachinePrecision] * -2.0 + x), $MachinePrecision] / y), $MachinePrecision] / t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -7.8999999999999997e146

   1. Initial program 44.3%

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

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

      10. lower-/.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. pow2 N/A

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

      15. lower-pow.f64 N/A

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

      16. +-commutative N/A

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

      17. lower-+.f64 N/A

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

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

      20. lower-+.f64 44.3

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

   4. Applied rewrites 44.3%

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

   5. Taylor expanded in x around inf

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

      1. lower-/.f64 89.0

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

   7. Applied rewrites 89.0%

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

   if -7.8999999999999997e146 < x < -2.5e-161

   1. Initial program 74.7%

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

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

      10. lower-/.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. pow2 N/A

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

      15. lower-pow.f64 N/A

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

      16. +-commutative N/A

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

      17. lower-+.f64 N/A

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

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

      20. lower-+.f64 83.3

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

   4. Applied rewrites 83.3%

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

   if -2.5e-161 < x

   1. Initial program 64.9%

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

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

      10. lower-/.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. pow2 N/A

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

      15. lower-pow.f64 N/A

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

      16. +-commutative N/A

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

      17. lower-+.f64 N/A

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

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

      20. lower-+.f64 67.1

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

   4. Applied rewrites 67.1%

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

   5. Taylor expanded in x around 0

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

      1. lower-/.f64 51.2

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

   7. Applied rewrites 51.2%

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

   8. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. pow2 N/A

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

      6. associate-/l* N/A

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

      7. lower-*.f64 N/A

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

      8. lift-/.f64 51.0

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

   10. Applied rewrites 51.0%

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

Alternative 2: 88.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_0 := \left(y + x\right) + 1\\ \mathbf{if}\;x \leq -7.8 \cdot 10^{+105}:\\ \;\;\;\;\frac{\frac{y}{x}}{t\_0}\\ \mathbf{elif}\;x \leq -2.5 \cdot 10^{-161}:\\ \;\;\;\;\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(x \cdot \frac{x}{y}, -2, x\right)}{y}}{t\_0}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ (+ y x) 1.0)))
   (if (<= x -7.8e+105)
     (/ (/ y x) t_0)
     (if (<= x -2.5e-161)
       (/ (* x y) (* (* (+ x y) (+ x y)) (+ (+ x y) 1.0)))
       (/ (/ (fma (* x (/ x y)) -2.0 x) y) t_0)))))

C

assert(x < y);
double code(double x, double y) {
	double t_0 = (y + x) + 1.0;
	double tmp;
	if (x <= -7.8e+105) {
		tmp = (y / x) / t_0;
	} else if (x <= -2.5e-161) {
		tmp = (x * y) / (((x + y) * (x + y)) * ((x + y) + 1.0));
	} else {
		tmp = (fma((x * (x / y)), -2.0, x) / y) / t_0;
	}
	return tmp;
}

Julia

x, y = sort([x, y])
function code(x, y)
	t_0 = Float64(Float64(y + x) + 1.0)
	tmp = 0.0
	if (x <= -7.8e+105)
		tmp = Float64(Float64(y / x) / t_0);
	elseif (x <= -2.5e-161)
		tmp = Float64(Float64(x * y) / Float64(Float64(Float64(x + y) * Float64(x + y)) * Float64(Float64(x + y) + 1.0)));
	else
		tmp = Float64(Float64(fma(Float64(x * Float64(x / y)), -2.0, x) / y) / t_0);
	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[(N[(y + x), $MachinePrecision] + 1.0), $MachinePrecision]}, If[LessEqual[x, -7.8e+105], N[(N[(y / x), $MachinePrecision] / t$95$0), $MachinePrecision], If[LessEqual[x, -2.5e-161], N[(N[(x * y), $MachinePrecision] / N[(N[(N[(x + y), $MachinePrecision] * N[(x + y), $MachinePrecision]), $MachinePrecision] * N[(N[(x + y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(x * N[(x / y), $MachinePrecision]), $MachinePrecision] * -2.0 + x), $MachinePrecision] / y), $MachinePrecision] / t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -7.79999999999999957e105

   1. Initial program 39.6%

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

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

      10. lower-/.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. pow2 N/A

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

      15. lower-pow.f64 N/A

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

      16. +-commutative N/A

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

      17. lower-+.f64 N/A

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

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

      20. lower-+.f64 49.0

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

   4. Applied rewrites 49.0%

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

   5. Taylor expanded in x around inf

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

      1. lower-/.f64 88.0

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

   7. Applied rewrites 88.0%

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

   if -7.79999999999999957e105 < x < -2.5e-161

   1. Initial program 80.4%

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

   if -2.5e-161 < x

   1. Initial program 64.9%

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

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

      10. lower-/.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. pow2 N/A

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

      15. lower-pow.f64 N/A

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

      16. +-commutative N/A

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

      17. lower-+.f64 N/A

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

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

      20. lower-+.f64 67.1

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

   4. Applied rewrites 67.1%

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

   5. Taylor expanded in x around 0

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

      1. lower-/.f64 51.2

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

   7. Applied rewrites 51.2%

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

   8. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. pow2 N/A

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

      6. associate-/l* N/A

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

      7. lower-*.f64 N/A

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

      8. lift-/.f64 51.0

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

   10. Applied rewrites 51.0%

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

Alternative 3: 87.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_0 := \left(y + x\right) + 1\\ \mathbf{if}\;x \leq -7.8 \cdot 10^{+105}:\\ \;\;\;\;\frac{\frac{y}{x}}{t\_0}\\ \mathbf{elif}\;x \leq -3 \cdot 10^{-107}:\\ \;\;\;\;\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{t\_0}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ (+ y x) 1.0)))
   (if (<= x -7.8e+105)
     (/ (/ y x) t_0)
     (if (<= x -3e-107)
       (/ (* x y) (* (* (+ x y) (+ x y)) (+ (+ x y) 1.0)))
       (/ (/ x y) t_0)))))

C

assert(x < y);
double code(double x, double y) {
	double t_0 = (y + x) + 1.0;
	double tmp;
	if (x <= -7.8e+105) {
		tmp = (y / x) / t_0;
	} else if (x <= -3e-107) {
		tmp = (x * y) / (((x + y) * (x + y)) * ((x + y) + 1.0));
	} else {
		tmp = (x / y) / t_0;
	}
	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) :: t_0
    real(8) :: tmp
    t_0 = (y + x) + 1.0d0
    if (x <= (-7.8d+105)) then
        tmp = (y / x) / t_0
    else if (x <= (-3d-107)) then
        tmp = (x * y) / (((x + y) * (x + y)) * ((x + y) + 1.0d0))
    else
        tmp = (x / y) / t_0
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double t_0 = (y + x) + 1.0;
	double tmp;
	if (x <= -7.8e+105) {
		tmp = (y / x) / t_0;
	} else if (x <= -3e-107) {
		tmp = (x * y) / (((x + y) * (x + y)) * ((x + y) + 1.0));
	} else {
		tmp = (x / y) / t_0;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	t_0 = (y + x) + 1.0
	tmp = 0
	if x <= -7.8e+105:
		tmp = (y / x) / t_0
	elif x <= -3e-107:
		tmp = (x * y) / (((x + y) * (x + y)) * ((x + y) + 1.0))
	else:
		tmp = (x / y) / t_0
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	t_0 = Float64(Float64(y + x) + 1.0)
	tmp = 0.0
	if (x <= -7.8e+105)
		tmp = Float64(Float64(y / x) / t_0);
	elseif (x <= -3e-107)
		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) / t_0);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	t_0 = (y + x) + 1.0;
	tmp = 0.0;
	if (x <= -7.8e+105)
		tmp = (y / x) / t_0;
	elseif (x <= -3e-107)
		tmp = (x * y) / (((x + y) * (x + y)) * ((x + y) + 1.0));
	else
		tmp = (x / y) / t_0;
	end
	tmp_2 = 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[(N[(y + x), $MachinePrecision] + 1.0), $MachinePrecision]}, If[LessEqual[x, -7.8e+105], N[(N[(y / x), $MachinePrecision] / t$95$0), $MachinePrecision], If[LessEqual[x, -3e-107], 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] / t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -7.79999999999999957e105

   1. Initial program 39.6%

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

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

      10. lower-/.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. pow2 N/A

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

      15. lower-pow.f64 N/A

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

      16. +-commutative N/A

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

      17. lower-+.f64 N/A

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

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

      20. lower-+.f64 49.0

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

   4. Applied rewrites 49.0%

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

   5. Taylor expanded in x around inf

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

      1. lower-/.f64 88.0

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

   7. Applied rewrites 88.0%

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

   if -7.79999999999999957e105 < x < -2.9999999999999997e-107

   1. Initial program 78.6%

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

   if -2.9999999999999997e-107 < x

   1. Initial program 66.6%

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

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

      10. lower-/.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. pow2 N/A

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

      15. lower-pow.f64 N/A

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

      16. +-commutative N/A

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

      17. lower-+.f64 N/A

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

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

      20. lower-+.f64 68.6

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

   4. Applied rewrites 68.6%

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

   5. Taylor expanded in x around 0

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

      1. lower-/.f64 53.8

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

   7. Applied rewrites 53.8%

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

Alternative 4: 84.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_0 := \left(y + x\right) + 1\\ \mathbf{if}\;x \leq -1.65 \cdot 10^{-34}:\\ \;\;\;\;\frac{\frac{y}{x}}{t\_0}\\ \mathbf{elif}\;x \leq -3 \cdot 10^{-107}:\\ \;\;\;\;\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(y + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{t\_0}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ (+ y x) 1.0)))
   (if (<= x -1.65e-34)
     (/ (/ y x) t_0)
     (if (<= x -3e-107)
       (/ (* x y) (* (* (+ x y) (+ x y)) (+ y 1.0)))
       (/ (/ x y) t_0)))))

C

assert(x < y);
double code(double x, double y) {
	double t_0 = (y + x) + 1.0;
	double tmp;
	if (x <= -1.65e-34) {
		tmp = (y / x) / t_0;
	} else if (x <= -3e-107) {
		tmp = (x * y) / (((x + y) * (x + y)) * (y + 1.0));
	} else {
		tmp = (x / y) / t_0;
	}
	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) :: t_0
    real(8) :: tmp
    t_0 = (y + x) + 1.0d0
    if (x <= (-1.65d-34)) then
        tmp = (y / x) / t_0
    else if (x <= (-3d-107)) then
        tmp = (x * y) / (((x + y) * (x + y)) * (y + 1.0d0))
    else
        tmp = (x / y) / t_0
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double t_0 = (y + x) + 1.0;
	double tmp;
	if (x <= -1.65e-34) {
		tmp = (y / x) / t_0;
	} else if (x <= -3e-107) {
		tmp = (x * y) / (((x + y) * (x + y)) * (y + 1.0));
	} else {
		tmp = (x / y) / t_0;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	t_0 = (y + x) + 1.0
	tmp = 0
	if x <= -1.65e-34:
		tmp = (y / x) / t_0
	elif x <= -3e-107:
		tmp = (x * y) / (((x + y) * (x + y)) * (y + 1.0))
	else:
		tmp = (x / y) / t_0
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	t_0 = Float64(Float64(y + x) + 1.0)
	tmp = 0.0
	if (x <= -1.65e-34)
		tmp = Float64(Float64(y / x) / t_0);
	elseif (x <= -3e-107)
		tmp = Float64(Float64(x * y) / Float64(Float64(Float64(x + y) * Float64(x + y)) * Float64(y + 1.0)));
	else
		tmp = Float64(Float64(x / y) / t_0);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	t_0 = (y + x) + 1.0;
	tmp = 0.0;
	if (x <= -1.65e-34)
		tmp = (y / x) / t_0;
	elseif (x <= -3e-107)
		tmp = (x * y) / (((x + y) * (x + y)) * (y + 1.0));
	else
		tmp = (x / y) / t_0;
	end
	tmp_2 = 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[(N[(y + x), $MachinePrecision] + 1.0), $MachinePrecision]}, If[LessEqual[x, -1.65e-34], N[(N[(y / x), $MachinePrecision] / t$95$0), $MachinePrecision], If[LessEqual[x, -3e-107], N[(N[(x * y), $MachinePrecision] / N[(N[(N[(x + y), $MachinePrecision] * N[(x + y), $MachinePrecision]), $MachinePrecision] * N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.64999999999999991e-34

   1. Initial program 54.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. associate-/r* N/A

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

      10. lower-/.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. pow2 N/A

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

      15. lower-pow.f64 N/A

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

      16. +-commutative N/A

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

      17. lower-+.f64 N/A

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

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

      20. lower-+.f64 62.5

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

   4. Applied rewrites 62.5%

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

   5. Taylor expanded in x around inf

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

      1. lower-/.f64 76.2

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

   7. Applied rewrites 76.2%

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

   if -1.64999999999999991e-34 < x < -2.9999999999999997e-107

   1. Initial program 83.7%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 83.7%

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

   if -2.9999999999999997e-107 < x

   1. Initial program 66.6%

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

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

      10. lower-/.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. pow2 N/A

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

      15. lower-pow.f64 N/A

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

      16. +-commutative N/A

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

      17. lower-+.f64 N/A

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

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

      20. lower-+.f64 68.6

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

   4. Applied rewrites 68.6%

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

   5. Taylor expanded in x around 0

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

      1. lower-/.f64 53.8

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

   7. Applied rewrites 53.8%

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

Alternative 5: 83.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_0 := \left(y + x\right) + 1\\ \mathbf{if}\;x \leq -7.8 \cdot 10^{+105}:\\ \;\;\;\;\frac{\frac{y}{x}}{t\_0}\\ \mathbf{elif}\;x \leq -7.2 \cdot 10^{-87}:\\ \;\;\;\;\frac{x \cdot y}{\left(\left(x + y\right) \cdot x\right) \cdot \left(\left(x + y\right) + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{t\_0}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ (+ y x) 1.0)))
   (if (<= x -7.8e+105)
     (/ (/ y x) t_0)
     (if (<= x -7.2e-87)
       (/ (* x y) (* (* (+ x y) x) (+ (+ x y) 1.0)))
       (/ (/ x y) t_0)))))

C

assert(x < y);
double code(double x, double y) {
	double t_0 = (y + x) + 1.0;
	double tmp;
	if (x <= -7.8e+105) {
		tmp = (y / x) / t_0;
	} else if (x <= -7.2e-87) {
		tmp = (x * y) / (((x + y) * x) * ((x + y) + 1.0));
	} else {
		tmp = (x / y) / t_0;
	}
	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) :: t_0
    real(8) :: tmp
    t_0 = (y + x) + 1.0d0
    if (x <= (-7.8d+105)) then
        tmp = (y / x) / t_0
    else if (x <= (-7.2d-87)) then
        tmp = (x * y) / (((x + y) * x) * ((x + y) + 1.0d0))
    else
        tmp = (x / y) / t_0
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double t_0 = (y + x) + 1.0;
	double tmp;
	if (x <= -7.8e+105) {
		tmp = (y / x) / t_0;
	} else if (x <= -7.2e-87) {
		tmp = (x * y) / (((x + y) * x) * ((x + y) + 1.0));
	} else {
		tmp = (x / y) / t_0;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	t_0 = (y + x) + 1.0
	tmp = 0
	if x <= -7.8e+105:
		tmp = (y / x) / t_0
	elif x <= -7.2e-87:
		tmp = (x * y) / (((x + y) * x) * ((x + y) + 1.0))
	else:
		tmp = (x / y) / t_0
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	t_0 = Float64(Float64(y + x) + 1.0)
	tmp = 0.0
	if (x <= -7.8e+105)
		tmp = Float64(Float64(y / x) / t_0);
	elseif (x <= -7.2e-87)
		tmp = Float64(Float64(x * y) / Float64(Float64(Float64(x + y) * x) * Float64(Float64(x + y) + 1.0)));
	else
		tmp = Float64(Float64(x / y) / t_0);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	t_0 = (y + x) + 1.0;
	tmp = 0.0;
	if (x <= -7.8e+105)
		tmp = (y / x) / t_0;
	elseif (x <= -7.2e-87)
		tmp = (x * y) / (((x + y) * x) * ((x + y) + 1.0));
	else
		tmp = (x / y) / t_0;
	end
	tmp_2 = 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[(N[(y + x), $MachinePrecision] + 1.0), $MachinePrecision]}, If[LessEqual[x, -7.8e+105], N[(N[(y / x), $MachinePrecision] / t$95$0), $MachinePrecision], If[LessEqual[x, -7.2e-87], N[(N[(x * y), $MachinePrecision] / N[(N[(N[(x + y), $MachinePrecision] * x), $MachinePrecision] * N[(N[(x + y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -7.79999999999999957e105

   1. Initial program 39.6%

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

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

      10. lower-/.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. pow2 N/A

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

      15. lower-pow.f64 N/A

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

      16. +-commutative N/A

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

      17. lower-+.f64 N/A

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

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

      20. lower-+.f64 49.0

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

   4. Applied rewrites 49.0%

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

   5. Taylor expanded in x around inf

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

      1. lower-/.f64 88.0

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

   7. Applied rewrites 88.0%

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

   if -7.79999999999999957e105 < x < -7.19999999999999986e-87

   1. Initial program 82.7%

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

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 69.5%

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

   if -7.19999999999999986e-87 < x

   1. Initial program 66.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. associate-/r* N/A

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

      10. lower-/.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. pow2 N/A

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

      15. lower-pow.f64 N/A

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

      16. +-commutative N/A

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

      17. lower-+.f64 N/A

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

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

      20. lower-+.f64 68.1

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

   4. Applied rewrites 68.1%

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

   5. Taylor expanded in x around 0

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

      1. lower-/.f64 53.2

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

   7. Applied rewrites 53.2%

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

Alternative 6: 81.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq -2.7 \cdot 10^{+82}:\\ \;\;\;\;\frac{\frac{y}{x}}{x}\\ \mathbf{elif}\;y \leq 9 \cdot 10^{-54}:\\ \;\;\;\;\frac{y}{\left(1 + x\right) \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{\left(y + x\right) + 1}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y -2.7e+82)
   (/ (/ y x) x)
   (if (<= y 9e-54) (/ y (* (+ 1.0 x) x)) (/ (/ x y) (+ (+ y x) 1.0)))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= -2.7e+82) {
		tmp = (y / x) / x;
	} else if (y <= 9e-54) {
		tmp = y / ((1.0 + x) * x);
	} else {
		tmp = (x / y) / ((y + x) + 1.0);
	}
	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+82)) then
        tmp = (y / x) / x
    else if (y <= 9d-54) then
        tmp = y / ((1.0d0 + x) * x)
    else
        tmp = (x / y) / ((y + x) + 1.0d0)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (y <= -2.7e+82) {
		tmp = (y / x) / x;
	} else if (y <= 9e-54) {
		tmp = y / ((1.0 + x) * x);
	} else {
		tmp = (x / y) / ((y + x) + 1.0);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if y <= -2.7e+82:
		tmp = (y / x) / x
	elif y <= 9e-54:
		tmp = y / ((1.0 + x) * x)
	else:
		tmp = (x / y) / ((y + x) + 1.0)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= -2.7e+82)
		tmp = Float64(Float64(y / x) / x);
	elseif (y <= 9e-54)
		tmp = Float64(y / Float64(Float64(1.0 + x) * x));
	else
		tmp = Float64(Float64(x / y) / Float64(Float64(y + x) + 1.0));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -2.7e+82)
		tmp = (y / x) / x;
	elseif (y <= 9e-54)
		tmp = y / ((1.0 + x) * x);
	else
		tmp = (x / y) / ((y + x) + 1.0);
	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+82], N[(N[(y / x), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[y, 9e-54], N[(y / N[(N[(1.0 + x), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / N[(N[(y + x), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.6999999999999999e82

   1. Initial program 59.7%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 15.3

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

   5. Applied rewrites 15.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 25.6

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

   7. Applied rewrites 25.6%

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

   if -2.6999999999999999e82 < y < 8.9999999999999997e-54

   1. Initial program 68.4%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-+.f64 76.0

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

   5. Applied rewrites 76.0%

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

   if 8.9999999999999997e-54 < y

   1. Initial program 60.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. associate-/r* N/A

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

      10. lower-/.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. pow2 N/A

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

      15. lower-pow.f64 N/A

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

      16. +-commutative N/A

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

      17. lower-+.f64 N/A

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

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

      20. lower-+.f64 63.0

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

   4. Applied rewrites 63.0%

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

   5. Taylor expanded in x around 0

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

      1. lower-/.f64 70.9

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

   7. Applied rewrites 70.9%

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

Alternative 7: 81.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq -2.7 \cdot 10^{+82}:\\ \;\;\;\;\frac{\frac{y}{x}}{x}\\ \mathbf{elif}\;y \leq 9 \cdot 10^{-54}:\\ \;\;\;\;\frac{y}{\left(1 + x\right) \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{1 + y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y -2.7e+82)
   (/ (/ y x) x)
   (if (<= y 9e-54) (/ y (* (+ 1.0 x) x)) (/ (/ x y) (+ 1.0 y)))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= -2.7e+82) {
		tmp = (y / x) / x;
	} else if (y <= 9e-54) {
		tmp = y / ((1.0 + x) * x);
	} else {
		tmp = (x / 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 (y <= (-2.7d+82)) then
        tmp = (y / x) / x
    else if (y <= 9d-54) then
        tmp = y / ((1.0d0 + x) * x)
    else
        tmp = (x / 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 (y <= -2.7e+82) {
		tmp = (y / x) / x;
	} else if (y <= 9e-54) {
		tmp = y / ((1.0 + x) * x);
	} else {
		tmp = (x / y) / (1.0 + y);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if y <= -2.7e+82:
		tmp = (y / x) / x
	elif y <= 9e-54:
		tmp = y / ((1.0 + x) * x)
	else:
		tmp = (x / y) / (1.0 + y)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= -2.7e+82)
		tmp = Float64(Float64(y / x) / x);
	elseif (y <= 9e-54)
		tmp = Float64(y / Float64(Float64(1.0 + x) * x));
	else
		tmp = Float64(Float64(x / 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 (y <= -2.7e+82)
		tmp = (y / x) / x;
	elseif (y <= 9e-54)
		tmp = y / ((1.0 + x) * x);
	else
		tmp = (x / 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[y, -2.7e+82], N[(N[(y / x), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[y, 9e-54], N[(y / N[(N[(1.0 + x), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / N[(1.0 + y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.6999999999999999e82

   1. Initial program 59.7%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 15.3

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

   5. Applied rewrites 15.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 25.6

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

   7. Applied rewrites 25.6%

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

   if -2.6999999999999999e82 < y < 8.9999999999999997e-54

   1. Initial program 68.4%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-+.f64 76.0

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

   5. Applied rewrites 76.0%

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

   if 8.9999999999999997e-54 < y

   1. Initial program 60.2%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   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 69.5

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

   5. Applied rewrites 69.5%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. associate-/r* N/A

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

      6. lower-/.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-+.f64 70.5

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

   7. Applied rewrites 70.5%

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

Alternative 8: 65.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -1.05 \cdot 10^{+33}:\\ \;\;\;\;\frac{y}{x \cdot x}\\ \mathbf{elif}\;x \leq -2.1 \cdot 10^{-91} \lor \neg \left(x \leq 2.35 \cdot 10^{-303}\right):\\ \;\;\;\;\frac{x}{y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -1.05e+33)
   (/ y (* x x))
   (if (or (<= x -2.1e-91) (not (<= x 2.35e-303))) (/ x (* y y)) (/ x y))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -1.05e+33) {
		tmp = y / (x * x);
	} else if ((x <= -2.1e-91) || !(x <= 2.35e-303)) {
		tmp = x / (y * y);
	} 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 (x <= (-1.05d+33)) then
        tmp = y / (x * x)
    else if ((x <= (-2.1d-91)) .or. (.not. (x <= 2.35d-303))) then
        tmp = x / (y * y)
    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 (x <= -1.05e+33) {
		tmp = y / (x * x);
	} else if ((x <= -2.1e-91) || !(x <= 2.35e-303)) {
		tmp = x / (y * y);
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -1.05e+33:
		tmp = y / (x * x)
	elif (x <= -2.1e-91) or not (x <= 2.35e-303):
		tmp = x / (y * y)
	else:
		tmp = x / y
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -1.05e+33)
		tmp = Float64(y / Float64(x * x));
	elseif ((x <= -2.1e-91) || !(x <= 2.35e-303))
		tmp = Float64(x / Float64(y * y));
	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 (x <= -1.05e+33)
		tmp = y / (x * x);
	elseif ((x <= -2.1e-91) || ~((x <= 2.35e-303)))
		tmp = x / (y * y);
	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[x, -1.05e+33], N[(y / N[(x * x), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[x, -2.1e-91], N[Not[LessEqual[x, 2.35e-303]], $MachinePrecision]], N[(x / N[(y * y), $MachinePrecision]), $MachinePrecision], N[(x / y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.05e33

   1. Initial program 45.3%

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

   3. Taylor expanded in x around 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 70.7

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

   5. Applied rewrites 70.7%

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

   if -1.05e33 < x < -2.0999999999999999e-91 or 2.3499999999999999e-303 < x

   1. Initial program 72.2%

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

   3. Taylor expanded in y around inf

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

      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 39.9

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

   5. Applied rewrites 39.9%

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

   if -2.0999999999999999e-91 < x < 2.3499999999999999e-303

   1. Initial program 60.0%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-+.f64 79.7

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

   5. Applied rewrites 79.7%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 65.4%

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

4. Final simplification 50.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.05 \cdot 10^{+33}:\\ \;\;\;\;\frac{y}{x \cdot x}\\ \mathbf{elif}\;x \leq -2.1 \cdot 10^{-91} \lor \neg \left(x \leq 2.35 \cdot 10^{-303}\right):\\ \;\;\;\;\frac{x}{y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 80.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 9 \cdot 10^{-54}:\\ \;\;\;\;\frac{y}{\left(1 + x\right) \cdot x}\\ \mathbf{elif}\;y \leq 8 \cdot 10^{+17}:\\ \;\;\;\;\frac{x}{\left(1 + y\right) \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 9e-54)
   (/ y (* (+ 1.0 x) x))
   (if (<= y 8e+17) (/ x (* (+ 1.0 y) y)) (/ (/ x y) y))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 9e-54) {
		tmp = y / ((1.0 + x) * x);
	} else if (y <= 8e+17) {
		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 <= 9d-54) then
        tmp = y / ((1.0d0 + x) * x)
    else if (y <= 8d+17) 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 <= 9e-54) {
		tmp = y / ((1.0 + x) * x);
	} else if (y <= 8e+17) {
		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 <= 9e-54:
		tmp = y / ((1.0 + x) * x)
	elif y <= 8e+17:
		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 <= 9e-54)
		tmp = Float64(y / Float64(Float64(1.0 + x) * x));
	elseif (y <= 8e+17)
		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 <= 9e-54)
		tmp = y / ((1.0 + x) * x);
	elseif (y <= 8e+17)
		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, 9e-54], N[(y / N[(N[(1.0 + x), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 8e+17], 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 < 8.9999999999999997e-54

   1. Initial program 66.2%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   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 60.3

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

   5. Applied rewrites 60.3%

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

   if 8.9999999999999997e-54 < y < 8e17

   1. Initial program 92.5%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   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 64.9

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

   5. Applied rewrites 64.9%

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

   if 8e17 < y

   1. Initial program 51.6%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   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 70.7

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

   5. Applied rewrites 70.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 72.0

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

   7. Applied rewrites 72.0%

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

Alternative 10: 81.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_0 := \left(y + x\right) + 1\\ \mathbf{if}\;y \leq 9 \cdot 10^{-54}:\\ \;\;\;\;\frac{\frac{y}{x}}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{t\_0}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ (+ y x) 1.0)))
   (if (<= y 9e-54) (/ (/ y x) t_0) (/ (/ x y) t_0))))

C

assert(x < y);
double code(double x, double y) {
	double t_0 = (y + x) + 1.0;
	double tmp;
	if (y <= 9e-54) {
		tmp = (y / x) / t_0;
	} else {
		tmp = (x / y) / t_0;
	}
	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) :: t_0
    real(8) :: tmp
    t_0 = (y + x) + 1.0d0
    if (y <= 9d-54) then
        tmp = (y / x) / t_0
    else
        tmp = (x / y) / t_0
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double t_0 = (y + x) + 1.0;
	double tmp;
	if (y <= 9e-54) {
		tmp = (y / x) / t_0;
	} else {
		tmp = (x / y) / t_0;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	t_0 = (y + x) + 1.0
	tmp = 0
	if y <= 9e-54:
		tmp = (y / x) / t_0
	else:
		tmp = (x / y) / t_0
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	t_0 = Float64(Float64(y + x) + 1.0)
	tmp = 0.0
	if (y <= 9e-54)
		tmp = Float64(Float64(y / x) / t_0);
	else
		tmp = Float64(Float64(x / y) / t_0);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	t_0 = (y + x) + 1.0;
	tmp = 0.0;
	if (y <= 9e-54)
		tmp = (y / x) / t_0;
	else
		tmp = (x / y) / t_0;
	end
	tmp_2 = 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[(N[(y + x), $MachinePrecision] + 1.0), $MachinePrecision]}, If[LessEqual[y, 9e-54], N[(N[(y / x), $MachinePrecision] / t$95$0), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < 8.9999999999999997e-54

   1. Initial program 66.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. associate-/r* N/A

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

      10. lower-/.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. pow2 N/A

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

      15. lower-pow.f64 N/A

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

      16. +-commutative N/A

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

      17. lower-+.f64 N/A

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

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

      20. lower-+.f64 69.9

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

   4. Applied rewrites 69.9%

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

   5. Taylor expanded in x around inf

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

      1. lower-/.f64 63.3

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

   7. Applied rewrites 63.3%

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

   if 8.9999999999999997e-54 < y

   1. Initial program 60.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. associate-/r* N/A

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

      10. lower-/.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. pow2 N/A

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

      15. lower-pow.f64 N/A

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

      16. +-commutative N/A

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

      17. lower-+.f64 N/A

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

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

      20. lower-+.f64 63.0

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

   4. Applied rewrites 63.0%

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

   5. Taylor expanded in x around 0

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

      1. lower-/.f64 70.9

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

   7. Applied rewrites 70.9%

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

Alternative 11: 79.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -5.6 \cdot 10^{+200}:\\ \;\;\;\;\frac{\frac{y}{x}}{x}\\ \mathbf{elif}\;x \leq -8.5 \cdot 10^{-107}:\\ \;\;\;\;\frac{y}{\left(1 + x\right) \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\left(1 + y\right) \cdot y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -5.6e+200)
   (/ (/ y x) x)
   (if (<= x -8.5e-107) (/ y (* (+ 1.0 x) x)) (/ x (* (+ 1.0 y) y)))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -5.6e+200) {
		tmp = (y / x) / x;
	} else if (x <= -8.5e-107) {
		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 (x <= (-5.6d+200)) then
        tmp = (y / x) / x
    else if (x <= (-8.5d-107)) 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 (x <= -5.6e+200) {
		tmp = (y / x) / x;
	} else if (x <= -8.5e-107) {
		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 x <= -5.6e+200:
		tmp = (y / x) / x
	elif x <= -8.5e-107:
		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 (x <= -5.6e+200)
		tmp = Float64(Float64(y / x) / x);
	elseif (x <= -8.5e-107)
		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 (x <= -5.6e+200)
		tmp = (y / x) / x;
	elseif (x <= -8.5e-107)
		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[x, -5.6e+200], N[(N[(y / x), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[x, -8.5e-107], 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 3 regimes

2. if x < -5.59999999999999969e200

   1. Initial program 40.7%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 72.4

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

   5. Applied rewrites 72.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 92.9

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

   7. Applied rewrites 92.9%

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

   if -5.59999999999999969e200 < x < -8.49999999999999956e-107

   1. Initial program 69.8%

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

   3. Taylor expanded in 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 58.3

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

   5. Applied rewrites 58.3%

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

   if -8.49999999999999956e-107 < x

   1. Initial program 66.6%

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

   3. Taylor expanded in 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 54.1

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

   5. Applied rewrites 54.1%

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

Alternative 12: 78.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -8.5 \cdot 10^{-107}:\\ \;\;\;\;\frac{y}{\left(1 + x\right) \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\left(1 + y\right) \cdot y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -8.5e-107) (/ y (* (+ 1.0 x) x)) (/ x (* (+ 1.0 y) y))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -8.5e-107) {
		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 (x <= (-8.5d-107)) 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 (x <= -8.5e-107) {
		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 x <= -8.5e-107:
		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 (x <= -8.5e-107)
		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 (x <= -8.5e-107)
		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[x, -8.5e-107], N[(y / N[(N[(1.0 + x), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision], N[(x / N[(N[(1.0 + y), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -8.49999999999999956e-107

   1. Initial program 60.5%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-+.f64 62.8

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

   5. Applied rewrites 62.8%

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

   if -8.49999999999999956e-107 < x

   1. Initial program 66.6%

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

   3. Taylor expanded in 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 54.1

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

   5. Applied rewrites 54.1%

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

Alternative 13: 76.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -7.2 \cdot 10^{+17}:\\ \;\;\;\;\frac{y}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\left(1 + y\right) \cdot y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -7.2e+17) (/ y (* x x)) (/ x (* (+ 1.0 y) y))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -7.2e+17) {
		tmp = y / (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 (x <= (-7.2d+17)) then
        tmp = y / (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 (x <= -7.2e+17) {
		tmp = y / (x * x);
	} else {
		tmp = x / ((1.0 + y) * y);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -7.2e+17:
		tmp = y / (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 (x <= -7.2e+17)
		tmp = Float64(y / Float64(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 (x <= -7.2e+17)
		tmp = y / (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[x, -7.2e+17], N[(y / N[(x * x), $MachinePrecision]), $MachinePrecision], N[(x / N[(N[(1.0 + y), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -7.2e17

   1. Initial program 46.4%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 68.7

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

   5. Applied rewrites 68.7%

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

   if -7.2e17 < x

   1. Initial program 69.3%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   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 53.4

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

   5. Applied rewrites 53.4%

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

Alternative 14: 46.4% accurate, 1.7× speedup

Math

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

FPCore

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

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	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 <= 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 <= 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 <= 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 <= 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 <= 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, 1.0], N[(x / y), $MachinePrecision], N[(x / N[(y * y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1

   1. Initial program 68.2%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   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 39.5

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

   5. Applied rewrites 39.5%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 25.7%

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

   if 1 < y

   1. Initial program 51.5%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 69.5

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

   5. Applied rewrites 69.5%

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

Alternative 15: 26.0% 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 64.6%

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

3. Taylor expanded in 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 46.0

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

5. Applied rewrites 46.0%

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

6. Taylor expanded in y around 0

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

8. Applied rewrites 24.1%

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

Developer Target 1: 99.8% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2025040 
(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))))