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

Percentage Accurate: 69.0% → 90.1%

Time: 4.2s

Alternatives: 11

Speedup: 1.5×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 69.0% 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: 90.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -2.25 \cdot 10^{+45}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\left(3 \cdot y\right) \cdot \frac{y}{x}, -1, y\right)}{x}}{x}\\ \mathbf{elif}\;x \leq -1.6 \cdot 10^{-164}:\\ \;\;\;\;x \cdot \frac{y}{\left(\left(y + x\right) + 1\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{1 + y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -2.25e+45)
   (/ (/ (fma (* (* 3.0 y) (/ y x)) -1.0 y) x) x)
   (if (<= x -1.6e-164)
     (* x (/ y (* (+ (+ y x) 1.0) (* (+ y x) (+ y x)))))
     (/ (/ x y) (+ 1.0 y)))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -2.25e+45) {
		tmp = (fma(((3.0 * y) * (y / x)), -1.0, y) / x) / x;
	} else if (x <= -1.6e-164) {
		tmp = x * (y / (((y + x) + 1.0) * ((y + x) * (y + x))));
	} else {
		tmp = (x / y) / (1.0 + y);
	}
	return tmp;
}

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -2.25e+45)
		tmp = Float64(Float64(fma(Float64(Float64(3.0 * y) * Float64(y / x)), -1.0, y) / x) / x);
	elseif (x <= -1.6e-164)
		tmp = Float64(x * Float64(y / Float64(Float64(Float64(y + x) + 1.0) * Float64(Float64(y + x) * Float64(y + x)))));
	else
		tmp = Float64(Float64(x / y) / Float64(1.0 + y));
	end
	return tmp
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[x, -2.25e+45], N[(N[(N[(N[(N[(3.0 * y), $MachinePrecision] * N[(y / x), $MachinePrecision]), $MachinePrecision] * -1.0 + y), $MachinePrecision] / x), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[x, -1.6e-164], N[(x * N[(y / N[(N[(N[(y + x), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[(y + x), $MachinePrecision] * N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / N[(1.0 + y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.2499999999999999e45

   1. Initial program 59.4%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. distribute-rgt1-in N/A

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

      10. metadata-eval N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(3 \cdot y + 1\right) \cdot y}{x}, -1, y\right)}{{x}^{2}} \]

      11. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(3, y, 1\right) \cdot y}{x}, -1, y\right)}{{x}^{2}} \]

      12. unpow2 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(3, y, 1\right) \cdot y}{x}, -1, y\right)}{x \cdot \color{blue}{x}} \]

      13. lower-*.f64 63.1

          \[\leadsto \frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(3, y, 1\right) \cdot y}{x}, -1, y\right)}{x \cdot \color{blue}{x}} \]

   5. Applied rewrites 63.1%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(3, y, 1\right) \cdot y}{x}, -1, y\right)}{x \cdot x}} \]

   6. Taylor expanded in y around inf

      \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(3 \cdot y\right) \cdot y}{x}, -1, y\right)}{x \cdot x} \]
   7. Step-by-step derivation

      1. lower-*.f64 63.1

         \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(3 \cdot y\right) \cdot y}{x}, -1, y\right)}{x \cdot x} \]

   8. Applied rewrites 63.1%

      \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(3 \cdot y\right) \cdot y}{x}, -1, y\right)}{x \cdot x} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(3 \cdot y\right) \cdot y}{x}, -1, y\right)}{x \cdot \color{blue}{x}} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(3 \cdot y\right) \cdot y}{x}, -1, y\right)}{\color{blue}{x \cdot x}} \]

      3. associate-/r* N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left(\frac{\left(3 \cdot y\right) \cdot y}{x}, -1, y\right)}{x}}{\color{blue}{x}} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left(\frac{\left(3 \cdot y\right) \cdot y}{x}, -1, y\right)}{x}}{\color{blue}{x}} \]

      5. lower-/.f64 74.5

         \[\leadsto \frac{\frac{\mathsf{fma}\left(\frac{\left(3 \cdot y\right) \cdot y}{x}, -1, y\right)}{x}}{x} \]

      6. lift-/.f64 N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left(\frac{\left(3 \cdot y\right) \cdot y}{x}, -1, y\right)}{x}}{x} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left(\frac{\left(3 \cdot y\right) \cdot y}{x}, -1, y\right)}{x}}{x} \]

      8. associate-/l* N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left(\left(3 \cdot y\right) \cdot \frac{y}{x}, -1, y\right)}{x}}{x} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left(\left(3 \cdot y\right) \cdot \frac{y}{x}, -1, y\right)}{x}}{x} \]

      10. lower-/.f64 80.9

          \[\leadsto \frac{\frac{\mathsf{fma}\left(\left(3 \cdot y\right) \cdot \frac{y}{x}, -1, y\right)}{x}}{x} \]

   10. Applied rewrites 80.9%

       \[\leadsto \frac{\frac{\mathsf{fma}\left(\left(3 \cdot y\right) \cdot \frac{y}{x}, -1, y\right)}{x}}{\color{blue}{x}} \]

   if -2.2499999999999999e45 < x < -1.6e-164

   1. Initial program 88.7%

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

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lift-+.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 N/A

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

      17. pow2 N/A

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

      18. lower-pow.f64 N/A

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

      19. +-commutative N/A

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

      20. lower-+.f64 93.5

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

   4. Applied rewrites 93.5%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-pow.f64 N/A

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

      4. pow2 N/A

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

      5. lower-*.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-+.f64 93.5

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

   6. Applied rewrites 93.5%

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

   if -1.6e-164 < x

   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 54.6

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

   5. Applied rewrites 54.6%

      \[\leadsto \color{blue}{\frac{x}{\left(1 + y\right) \cdot y}} \]
   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. lower-/.f64 N/A

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

      8. lift-+.f64 57.0

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

   7. Applied rewrites 57.0%

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

Alternative 2: 89.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -2.25 \cdot 10^{+45}:\\ \;\;\;\;\frac{\frac{y}{x}}{x}\\ \mathbf{elif}\;x \leq -1.6 \cdot 10^{-164}:\\ \;\;\;\;x \cdot \frac{y}{\left(\left(y + x\right) + 1\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{1 + y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -2.25e+45)
   (/ (/ y x) x)
   (if (<= x -1.6e-164)
     (* x (/ y (* (+ (+ y x) 1.0) (* (+ y x) (+ y x)))))
     (/ (/ x y) (+ 1.0 y)))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -2.25e+45) {
		tmp = (y / x) / x;
	} else if (x <= -1.6e-164) {
		tmp = x * (y / (((y + x) + 1.0) * ((y + x) * (y + 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 (x <= (-2.25d+45)) then
        tmp = (y / x) / x
    else if (x <= (-1.6d-164)) then
        tmp = x * (y / (((y + x) + 1.0d0) * ((y + x) * (y + 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 (x <= -2.25e+45) {
		tmp = (y / x) / x;
	} else if (x <= -1.6e-164) {
		tmp = x * (y / (((y + x) + 1.0) * ((y + x) * (y + x))));
	} else {
		tmp = (x / y) / (1.0 + y);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -2.25e+45:
		tmp = (y / x) / x
	elif x <= -1.6e-164:
		tmp = x * (y / (((y + x) + 1.0) * ((y + x) * (y + x))))
	else:
		tmp = (x / y) / (1.0 + y)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -2.25e+45)
		tmp = Float64(Float64(y / x) / x);
	elseif (x <= -1.6e-164)
		tmp = Float64(x * Float64(y / Float64(Float64(Float64(y + x) + 1.0) * Float64(Float64(y + x) * Float64(y + 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 (x <= -2.25e+45)
		tmp = (y / x) / x;
	elseif (x <= -1.6e-164)
		tmp = x * (y / (((y + x) + 1.0) * ((y + x) * (y + 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[x, -2.25e+45], N[(N[(y / x), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[x, -1.6e-164], N[(x * N[(y / N[(N[(N[(y + x), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[(y + x), $MachinePrecision] * N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / N[(1.0 + y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.2499999999999999e45

   1. Initial program 59.4%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. distribute-rgt1-in N/A

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

      10. metadata-eval N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(3 \cdot y + 1\right) \cdot y}{x}, -1, y\right)}{{x}^{2}} \]

      11. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(3, y, 1\right) \cdot y}{x}, -1, y\right)}{{x}^{2}} \]

      12. unpow2 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(3, y, 1\right) \cdot y}{x}, -1, y\right)}{x \cdot \color{blue}{x}} \]

      13. lower-*.f64 63.1

          \[\leadsto \frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(3, y, 1\right) \cdot y}{x}, -1, y\right)}{x \cdot \color{blue}{x}} \]

   5. Applied rewrites 63.1%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(3, y, 1\right) \cdot y}{x}, -1, y\right)}{x \cdot x}} \]

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 73.4%

      \[\leadsto \frac{y}{\color{blue}{x} \cdot x} \]
   9. 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 81.3

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

   10. Applied rewrites 81.3%

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

   if -2.2499999999999999e45 < x < -1.6e-164

   1. Initial program 88.7%

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

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lift-+.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 N/A

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

      17. pow2 N/A

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

      18. lower-pow.f64 N/A

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

      19. +-commutative N/A

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

      20. lower-+.f64 93.5

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

   4. Applied rewrites 93.5%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-pow.f64 N/A

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

      4. pow2 N/A

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

      5. lower-*.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-+.f64 93.5

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

   6. Applied rewrites 93.5%

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

   if -1.6e-164 < x

   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 54.6

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

   5. Applied rewrites 54.6%

      \[\leadsto \color{blue}{\frac{x}{\left(1 + y\right) \cdot y}} \]
   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. lower-/.f64 N/A

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

      8. lift-+.f64 57.0

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

   7. Applied rewrites 57.0%

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

Alternative 3: 87.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -2.8 \cdot 10^{+22}:\\ \;\;\;\;\frac{\frac{y}{x}}{x}\\ \mathbf{elif}\;x \leq -1.6 \cdot 10^{-164}:\\ \;\;\;\;x \cdot \frac{y}{\left(y + 1\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{1 + y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -2.8e+22)
   (/ (/ y x) x)
   (if (<= x -1.6e-164)
     (* x (/ y (* (+ y 1.0) (* (+ y x) (+ y x)))))
     (/ (/ x y) (+ 1.0 y)))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -2.8e+22) {
		tmp = (y / x) / x;
	} else if (x <= -1.6e-164) {
		tmp = x * (y / ((y + 1.0) * ((y + x) * (y + 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 (x <= (-2.8d+22)) then
        tmp = (y / x) / x
    else if (x <= (-1.6d-164)) then
        tmp = x * (y / ((y + 1.0d0) * ((y + x) * (y + 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 (x <= -2.8e+22) {
		tmp = (y / x) / x;
	} else if (x <= -1.6e-164) {
		tmp = x * (y / ((y + 1.0) * ((y + x) * (y + x))));
	} else {
		tmp = (x / y) / (1.0 + y);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -2.8e+22:
		tmp = (y / x) / x
	elif x <= -1.6e-164:
		tmp = x * (y / ((y + 1.0) * ((y + x) * (y + x))))
	else:
		tmp = (x / y) / (1.0 + y)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -2.8e+22)
		tmp = Float64(Float64(y / x) / x);
	elseif (x <= -1.6e-164)
		tmp = Float64(x * Float64(y / Float64(Float64(y + 1.0) * Float64(Float64(y + x) * Float64(y + 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 (x <= -2.8e+22)
		tmp = (y / x) / x;
	elseif (x <= -1.6e-164)
		tmp = x * (y / ((y + 1.0) * ((y + x) * (y + 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[x, -2.8e+22], N[(N[(y / x), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[x, -1.6e-164], N[(x * N[(y / N[(N[(y + 1.0), $MachinePrecision] * N[(N[(y + x), $MachinePrecision] * N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / N[(1.0 + y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.8e22

   1. Initial program 61.1%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. distribute-rgt1-in N/A

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

      10. metadata-eval N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(3 \cdot y + 1\right) \cdot y}{x}, -1, y\right)}{{x}^{2}} \]

      11. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(3, y, 1\right) \cdot y}{x}, -1, y\right)}{{x}^{2}} \]

      12. unpow2 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(3, y, 1\right) \cdot y}{x}, -1, y\right)}{x \cdot \color{blue}{x}} \]

      13. lower-*.f64 64.4

          \[\leadsto \frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(3, y, 1\right) \cdot y}{x}, -1, y\right)}{x \cdot \color{blue}{x}} \]

   5. Applied rewrites 64.4%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(3, y, 1\right) \cdot y}{x}, -1, y\right)}{x \cdot x}} \]

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 73.9%

      \[\leadsto \frac{y}{\color{blue}{x} \cdot x} \]
   9. 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 81.3

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

   10. Applied rewrites 81.3%

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

   if -2.8e22 < x < -1.6e-164

   1. Initial program 89.7%

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

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lift-+.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 N/A

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

      17. pow2 N/A

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

      18. lower-pow.f64 N/A

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

      19. +-commutative N/A

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

      20. lower-+.f64 95.2

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

   4. Applied rewrites 95.2%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-pow.f64 N/A

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

      4. pow2 N/A

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

      5. lower-*.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-+.f64 95.2

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

   6. Applied rewrites 95.2%

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

   7. Taylor expanded in x around 0

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

   9. Applied rewrites 93.9%

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

   if -1.6e-164 < x

   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 54.6

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

   5. Applied rewrites 54.6%

      \[\leadsto \color{blue}{\frac{x}{\left(1 + y\right) \cdot y}} \]
   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. lower-/.f64 N/A

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

      8. lift-+.f64 57.0

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

   7. Applied rewrites 57.0%

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

Alternative 4: 79.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -2.45 \cdot 10^{+31}:\\ \;\;\;\;\frac{\frac{y}{x}}{x}\\ \mathbf{elif}\;x \leq -0.0105:\\ \;\;\;\;\frac{x}{y \cdot y}\\ \mathbf{elif}\;x \leq -3.05 \cdot 10^{-64}:\\ \;\;\;\;\frac{y}{\left(1 + x\right) \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{1 + y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -2.45e+31)
   (/ (/ y x) x)
   (if (<= x -0.0105)
     (/ x (* y y))
     (if (<= x -3.05e-64) (/ y (* (+ 1.0 x) x)) (/ (/ x y) (+ 1.0 y))))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -2.45e+31) {
		tmp = (y / x) / x;
	} else if (x <= -0.0105) {
		tmp = x / (y * y);
	} else if (x <= -3.05e-64) {
		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 (x <= (-2.45d+31)) then
        tmp = (y / x) / x
    else if (x <= (-0.0105d0)) then
        tmp = x / (y * y)
    else if (x <= (-3.05d-64)) 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 (x <= -2.45e+31) {
		tmp = (y / x) / x;
	} else if (x <= -0.0105) {
		tmp = x / (y * y);
	} else if (x <= -3.05e-64) {
		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 x <= -2.45e+31:
		tmp = (y / x) / x
	elif x <= -0.0105:
		tmp = x / (y * y)
	elif x <= -3.05e-64:
		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 (x <= -2.45e+31)
		tmp = Float64(Float64(y / x) / x);
	elseif (x <= -0.0105)
		tmp = Float64(x / Float64(y * y));
	elseif (x <= -3.05e-64)
		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 (x <= -2.45e+31)
		tmp = (y / x) / x;
	elseif (x <= -0.0105)
		tmp = x / (y * y);
	elseif (x <= -3.05e-64)
		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[x, -2.45e+31], N[(N[(y / x), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[x, -0.0105], N[(x / N[(y * y), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -3.05e-64], 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 4 regimes

2. if x < -2.44999999999999998e31

   1. Initial program 61.3%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. distribute-rgt1-in N/A

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

      10. metadata-eval N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(3 \cdot y + 1\right) \cdot y}{x}, -1, y\right)}{{x}^{2}} \]

      11. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(3, y, 1\right) \cdot y}{x}, -1, y\right)}{{x}^{2}} \]

      12. unpow2 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(3, y, 1\right) \cdot y}{x}, -1, y\right)}{x \cdot \color{blue}{x}} \]

      13. lower-*.f64 64.8

          \[\leadsto \frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(3, y, 1\right) \cdot y}{x}, -1, y\right)}{x \cdot \color{blue}{x}} \]

   5. Applied rewrites 64.8%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(3, y, 1\right) \cdot y}{x}, -1, y\right)}{x \cdot x}} \]

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 74.6%

      \[\leadsto \frac{y}{\color{blue}{x} \cdot x} \]
   9. 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 82.2

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

   10. Applied rewrites 82.2%

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

   if -2.44999999999999998e31 < x < -0.0105000000000000007

   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 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 83.6

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

   5. Applied rewrites 83.6%

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

   if -0.0105000000000000007 < x < -3.0499999999999998e-64

   1. Initial program 83.0%

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

   3. Taylor expanded in 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 67.4

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

   5. Applied rewrites 67.4%

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

   if -3.0499999999999998e-64 < x

   1. Initial program 71.2%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   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 59.3

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

   5. Applied rewrites 59.3%

      \[\leadsto \color{blue}{\frac{x}{\left(1 + y\right) \cdot y}} \]
   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. lower-/.f64 N/A

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

      8. lift-+.f64 61.3

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

   7. Applied rewrites 61.3%

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

Alternative 5: 78.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -2.45 \cdot 10^{+31}:\\ \;\;\;\;\frac{\frac{y}{x}}{x}\\ \mathbf{elif}\;x \leq -0.0105:\\ \;\;\;\;\frac{x}{y \cdot y}\\ \mathbf{elif}\;x \leq -3.05 \cdot 10^{-64}:\\ \;\;\;\;\frac{y}{\left(1 + x\right) \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(y, 1, y \cdot y\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -2.45e+31)
   (/ (/ y x) x)
   (if (<= x -0.0105)
     (/ x (* y y))
     (if (<= x -3.05e-64) (/ y (* (+ 1.0 x) x)) (/ x (fma y 1.0 (* y y)))))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -2.45e+31) {
		tmp = (y / x) / x;
	} else if (x <= -0.0105) {
		tmp = x / (y * y);
	} else if (x <= -3.05e-64) {
		tmp = y / ((1.0 + x) * x);
	} else {
		tmp = x / fma(y, 1.0, (y * y));
	}
	return tmp;
}

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -2.45e+31)
		tmp = Float64(Float64(y / x) / x);
	elseif (x <= -0.0105)
		tmp = Float64(x / Float64(y * y));
	elseif (x <= -3.05e-64)
		tmp = Float64(y / Float64(Float64(1.0 + x) * x));
	else
		tmp = Float64(x / fma(y, 1.0, Float64(y * y)));
	end
	return tmp
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[x, -2.45e+31], N[(N[(y / x), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[x, -0.0105], N[(x / N[(y * y), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -3.05e-64], N[(y / N[(N[(1.0 + x), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision], N[(x / N[(y * 1.0 + N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < -2.44999999999999998e31

   1. Initial program 61.3%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. distribute-rgt1-in N/A

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

      10. metadata-eval N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(3 \cdot y + 1\right) \cdot y}{x}, -1, y\right)}{{x}^{2}} \]

      11. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(3, y, 1\right) \cdot y}{x}, -1, y\right)}{{x}^{2}} \]

      12. unpow2 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(3, y, 1\right) \cdot y}{x}, -1, y\right)}{x \cdot \color{blue}{x}} \]

      13. lower-*.f64 64.8

          \[\leadsto \frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(3, y, 1\right) \cdot y}{x}, -1, y\right)}{x \cdot \color{blue}{x}} \]

   5. Applied rewrites 64.8%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(3, y, 1\right) \cdot y}{x}, -1, y\right)}{x \cdot x}} \]

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 74.6%

      \[\leadsto \frac{y}{\color{blue}{x} \cdot x} \]
   9. 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 82.2

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

   10. Applied rewrites 82.2%

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

   if -2.44999999999999998e31 < x < -0.0105000000000000007

   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 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 83.6

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

   5. Applied rewrites 83.6%

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

   if -0.0105000000000000007 < x < -3.0499999999999998e-64

   1. Initial program 83.0%

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

   3. Taylor expanded in 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 67.4

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

   5. Applied rewrites 67.4%

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

   if -3.0499999999999998e-64 < x

   1. Initial program 71.2%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   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 59.3

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

   5. Applied rewrites 59.3%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. distribute-lft-in N/A

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

      5. unpow2 N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 59.3

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

   7. Applied rewrites 59.3%

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

Alternative 6: 67.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -2.45 \cdot 10^{+31}:\\ \;\;\;\;\frac{y}{x \cdot x}\\ \mathbf{elif}\;x \leq -8.2 \cdot 10^{-202} \lor \neg \left(x \leq 4.9 \cdot 10^{-55}\right):\\ \;\;\;\;\frac{x}{y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -2.45e+31)
   (/ y (* x x))
   (if (or (<= x -8.2e-202) (not (<= x 4.9e-55))) (/ x (* y y)) (/ x y))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -2.45e+31) {
		tmp = y / (x * x);
	} else if ((x <= -8.2e-202) || !(x <= 4.9e-55)) {
		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 <= (-2.45d+31)) then
        tmp = y / (x * x)
    else if ((x <= (-8.2d-202)) .or. (.not. (x <= 4.9d-55))) 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 <= -2.45e+31) {
		tmp = y / (x * x);
	} else if ((x <= -8.2e-202) || !(x <= 4.9e-55)) {
		tmp = x / (y * y);
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -2.45e+31:
		tmp = y / (x * x)
	elif (x <= -8.2e-202) or not (x <= 4.9e-55):
		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 <= -2.45e+31)
		tmp = Float64(y / Float64(x * x));
	elseif ((x <= -8.2e-202) || !(x <= 4.9e-55))
		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 <= -2.45e+31)
		tmp = y / (x * x);
	elseif ((x <= -8.2e-202) || ~((x <= 4.9e-55)))
		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, -2.45e+31], N[(y / N[(x * x), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[x, -8.2e-202], N[Not[LessEqual[x, 4.9e-55]], $MachinePrecision]], N[(x / N[(y * y), $MachinePrecision]), $MachinePrecision], N[(x / y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.44999999999999998e31

   1. Initial program 61.3%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   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 74.6

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

   5. Applied rewrites 74.6%

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

   if -2.44999999999999998e31 < x < -8.2000000000000008e-202 or 4.90000000000000035e-55 < x

   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. 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 38.0

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

   5. Applied rewrites 38.0%

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

   if -8.2000000000000008e-202 < x < 4.90000000000000035e-55

   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 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 82.8

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

   5. Applied rewrites 82.8%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 69.8%

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

4. Final simplification 56.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.45 \cdot 10^{+31}:\\ \;\;\;\;\frac{y}{x \cdot x}\\ \mathbf{elif}\;x \leq -8.2 \cdot 10^{-202} \lor \neg \left(x \leq 4.9 \cdot 10^{-55}\right):\\ \;\;\;\;\frac{x}{y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 78.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -3.05 \cdot 10^{-64}:\\ \;\;\;\;\frac{y}{\left(1 + x\right) \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(y, 1, y \cdot y\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -3.05e-64) (/ y (* (+ 1.0 x) x)) (/ x (fma y 1.0 (* y y)))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -3.05e-64) {
		tmp = y / ((1.0 + x) * x);
	} else {
		tmp = x / fma(y, 1.0, (y * y));
	}
	return tmp;
}

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -3.05e-64)
		tmp = Float64(y / Float64(Float64(1.0 + x) * x));
	else
		tmp = Float64(x / fma(y, 1.0, Float64(y * y)));
	end
	return tmp
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[x, -3.05e-64], N[(y / N[(N[(1.0 + x), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision], N[(x / N[(y * 1.0 + N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -3.0499999999999998e-64

   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. 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 69.6

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

   5. Applied rewrites 69.6%

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

   if -3.0499999999999998e-64 < x

   1. Initial program 71.2%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   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 59.3

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

   5. Applied rewrites 59.3%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. distribute-lft-in N/A

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

      5. unpow2 N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 59.3

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

   7. Applied rewrites 59.3%

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

Alternative 8: 78.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -3.05 \cdot 10^{-64}:\\ \;\;\;\;\frac{y}{\left(1 + x\right) \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\left(1 + y\right) \cdot y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -3.05e-64) (/ y (* (+ 1.0 x) x)) (/ x (* (+ 1.0 y) y))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -3.05e-64) {
		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 <= (-3.05d-64)) 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 <= -3.05e-64) {
		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 <= -3.05e-64:
		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 <= -3.05e-64)
		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 <= -3.05e-64)
		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, -3.05e-64], 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 < -3.0499999999999998e-64

   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. 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 69.6

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

   5. Applied rewrites 69.6%

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

   if -3.0499999999999998e-64 < x

   1. Initial program 71.2%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   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 59.3

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

   5. Applied rewrites 59.3%

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

Alternative 9: 74.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -2.45 \cdot 10^{+31}:\\ \;\;\;\;\frac{y}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\left(1 + y\right) \cdot y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -2.45e+31) (/ y (* x x)) (/ x (* (+ 1.0 y) y))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -2.45e+31) {
		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 <= (-2.45d+31)) 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 <= -2.45e+31) {
		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 <= -2.45e+31:
		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 <= -2.45e+31)
		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 <= -2.45e+31)
		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, -2.45e+31], N[(y / N[(x * x), $MachinePrecision]), $MachinePrecision], N[(x / N[(N[(1.0 + y), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2.44999999999999998e31

   1. Initial program 61.3%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   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 74.6

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

   5. Applied rewrites 74.6%

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

   if -2.44999999999999998e31 < x

   1. Initial program 72.4%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-+.f64 58.6

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

   5. Applied rewrites 58.6%

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

Alternative 10: 46.5% 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 71.8%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-+.f64 43.4

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

   5. Applied rewrites 43.4%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 25.8%

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

   if 1 < y

   1. Initial program 61.9%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 66.7

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

   5. Applied rewrites 66.7%

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

Alternative 11: 26.5% 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 69.6%

   \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
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 48.7

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

5. Applied rewrites 48.7%

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

6. Taylor expanded in y around 0

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

8. Applied rewrites 25.1%

   \[\leadsto \frac{x}{y} \]
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 2025037 
(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))))