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

Percentage Accurate: 69.6% → 99.8%

Time: 5.4s

Alternatives: 17

Speedup: 1.6×

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: 69.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \frac{\frac{y}{\left(y + x\right) - -1} \cdot \frac{x}{y + x}}{y + x} \end{array} \]

FPCore

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

C

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

Fortran

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

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

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

Java

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

Python

[x, y] = sort([x, y])
def code(x, y):
	return ((y / ((y + x) - -1.0)) * (x / (y + x))) / (y + x)

Julia

x, y = sort([x, y])
function code(x, y)
	return Float64(Float64(Float64(y / Float64(Float64(y + x) - -1.0)) * Float64(x / Float64(y + x))) / Float64(y + x))
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp = code(x, y)
	tmp = ((y / ((y + x) - -1.0)) * (x / (y + x))) / (y + x);
end

Wolfram

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

Derivation

1. Initial program 68.9%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   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

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

   5. associate-*l* N/A

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

   6. times-frac N/A

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

   7. lower-*.f64 N/A

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

   8. lower-/.f64 N/A

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

   9. lift-+.f64 N/A

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

   10. +-commutative N/A

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

   11. lower-+.f64 N/A

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

   12. lower-/.f64 N/A

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

   13. *-commutative N/A

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

   14. lower-*.f64 93.0

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

   15. lift-+.f64 N/A

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

   16. metadata-eval N/A

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

   17. fp-cancel-sign-sub-inv N/A

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

   18. metadata-eval N/A

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

   19. metadata-eval N/A

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

   20. metadata-eval N/A

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

   21. lower--.f64 N/A

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

   22. lift-+.f64 N/A

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

   23. +-commutative N/A

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

   24. lower-+.f64 N/A

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

   25. metadata-eval 93.0

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

   26. lift-+.f64 N/A

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

4. Applied rewrites 93.0%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-/.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. associate-/r* N/A

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

   6. lift-+.f64 N/A

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

   7. +-commutative N/A

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

   8. lift-+.f64 N/A

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

   9. associate-*l/ N/A

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

   10. lower-/.f64 N/A

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

6. Applied rewrites 99.8%

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

Alternative 2: 95.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 9.5 \cdot 10^{+111}:\\ \;\;\;\;\frac{x}{y + x} \cdot \frac{y}{\left(\left(y + x\right) - -1\right) \cdot \left(y + x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(-x, \frac{\mathsf{fma}\left(2, x, 1\right)}{y}, x\right)}{y}}{y + x}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 9.5e+111)
   (* (/ x (+ y x)) (/ y (* (- (+ y x) -1.0) (+ y x))))
   (/ (/ (fma (- x) (/ (fma 2.0 x 1.0) y) x) y) (+ y x))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 9.5e+111) {
		tmp = (x / (y + x)) * (y / (((y + x) - -1.0) * (y + x)));
	} else {
		tmp = (fma(-x, (fma(2.0, x, 1.0) / y), x) / y) / (y + x);
	}
	return tmp;
}

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= 9.5e+111)
		tmp = Float64(Float64(x / Float64(y + x)) * Float64(y / Float64(Float64(Float64(y + x) - -1.0) * Float64(y + x))));
	else
		tmp = Float64(Float64(fma(Float64(-x), Float64(fma(2.0, x, 1.0) / y), x) / y) / Float64(y + x));
	end
	return tmp
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[y, 9.5e+111], N[(N[(x / N[(y + x), $MachinePrecision]), $MachinePrecision] * N[(y / N[(N[(N[(y + x), $MachinePrecision] - -1.0), $MachinePrecision] * N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[((-x) * N[(N[(2.0 * x + 1.0), $MachinePrecision] / y), $MachinePrecision] + x), $MachinePrecision] / y), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 9.50000000000000019e111

   1. Initial program 73.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      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

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

      5. associate-*l* N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 96.6

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

      15. lift-+.f64 N/A

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

      16. metadata-eval N/A

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

      17. fp-cancel-sign-sub-inv N/A

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

      18. metadata-eval N/A

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

      19. metadata-eval N/A

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

      20. metadata-eval N/A

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

      21. lower--.f64 N/A

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

      22. lift-+.f64 N/A

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

      23. +-commutative N/A

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

      24. lower-+.f64 N/A

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

      25. metadata-eval 96.6

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

      26. lift-+.f64 N/A

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

   4. Applied rewrites 96.6%

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

   if 9.50000000000000019e111 < y

   1. Initial program 50.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      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

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

      5. associate-*l* N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 78.7

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

      15. lift-+.f64 N/A

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

      16. metadata-eval N/A

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

      17. fp-cancel-sign-sub-inv N/A

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

      18. metadata-eval N/A

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

      19. metadata-eval N/A

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

      20. metadata-eval N/A

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

      21. lower--.f64 N/A

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

      22. lift-+.f64 N/A

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

      23. +-commutative N/A

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

      24. lower-+.f64 N/A

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

      25. metadata-eval 78.7

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

      26. lift-+.f64 N/A

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

   4. Applied rewrites 78.7%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/r* N/A

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. lift-+.f64 N/A

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

      9. associate-*l/ N/A

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

      10. lower-/.f64 N/A

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

   6. Applied rewrites 99.9%

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

   7. Taylor expanded in y around inf

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

   9. Applied rewrites 83.7%

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

Alternative 3: 92.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -2.4 \cdot 10^{-9}:\\ \;\;\;\;1 \cdot \frac{y}{\left(\left(y + x\right) - -1\right) \cdot \left(y + x\right)}\\ \mathbf{elif}\;x \leq 3.8 \cdot 10^{-63}:\\ \;\;\;\;\frac{y}{y + x} \cdot \frac{x}{\left(y - -1\right) \cdot \left(y + x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 \cdot \frac{x}{y + x}}{y + x}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -2.4e-9)
   (* 1.0 (/ y (* (- (+ y x) -1.0) (+ y x))))
   (if (<= x 3.8e-63)
     (* (/ y (+ y x)) (/ x (* (- y -1.0) (+ y x))))
     (/ (* 1.0 (/ x (+ y x))) (+ y x)))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -2.4e-9) {
		tmp = 1.0 * (y / (((y + x) - -1.0) * (y + x)));
	} else if (x <= 3.8e-63) {
		tmp = (y / (y + x)) * (x / ((y - -1.0) * (y + x)));
	} else {
		tmp = (1.0 * (x / (y + x))) / (y + x);
	}
	return tmp;
}

Fortran

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

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

real(8) function code(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-2.4d-9)) then
        tmp = 1.0d0 * (y / (((y + x) - (-1.0d0)) * (y + x)))
    else if (x <= 3.8d-63) then
        tmp = (y / (y + x)) * (x / ((y - (-1.0d0)) * (y + x)))
    else
        tmp = (1.0d0 * (x / (y + x))) / (y + x)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (x <= -2.4e-9) {
		tmp = 1.0 * (y / (((y + x) - -1.0) * (y + x)));
	} else if (x <= 3.8e-63) {
		tmp = (y / (y + x)) * (x / ((y - -1.0) * (y + x)));
	} else {
		tmp = (1.0 * (x / (y + x))) / (y + x);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -2.4e-9:
		tmp = 1.0 * (y / (((y + x) - -1.0) * (y + x)))
	elif x <= 3.8e-63:
		tmp = (y / (y + x)) * (x / ((y - -1.0) * (y + x)))
	else:
		tmp = (1.0 * (x / (y + x))) / (y + x)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -2.4e-9)
		tmp = Float64(1.0 * Float64(y / Float64(Float64(Float64(y + x) - -1.0) * Float64(y + x))));
	elseif (x <= 3.8e-63)
		tmp = Float64(Float64(y / Float64(y + x)) * Float64(x / Float64(Float64(y - -1.0) * Float64(y + x))));
	else
		tmp = Float64(Float64(1.0 * Float64(x / Float64(y + x))) / Float64(y + x));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -2.4e-9)
		tmp = 1.0 * (y / (((y + x) - -1.0) * (y + x)));
	elseif (x <= 3.8e-63)
		tmp = (y / (y + x)) * (x / ((y - -1.0) * (y + x)));
	else
		tmp = (1.0 * (x / (y + x))) / (y + x);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[x, -2.4e-9], N[(1.0 * N[(y / N[(N[(N[(y + x), $MachinePrecision] - -1.0), $MachinePrecision] * N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 3.8e-63], N[(N[(y / N[(y + x), $MachinePrecision]), $MachinePrecision] * N[(x / N[(N[(y - -1.0), $MachinePrecision] * N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 * N[(x / N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.4e-9

   1. Initial program 67.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      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

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

      5. associate-*l* N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 91.8

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

      15. lift-+.f64 N/A

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

      16. metadata-eval N/A

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

      17. fp-cancel-sign-sub-inv N/A

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

      18. metadata-eval N/A

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

      19. metadata-eval N/A

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

      20. metadata-eval N/A

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

      21. lower--.f64 N/A

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

      22. lift-+.f64 N/A

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

      23. +-commutative N/A

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

      24. lower-+.f64 N/A

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

      25. metadata-eval 91.8

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

      26. lift-+.f64 N/A

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

   4. Applied rewrites 91.8%

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

   5. Taylor expanded in x around inf

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

   7. Applied rewrites 87.3%

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

   if -2.4e-9 < x < 3.80000000000000017e-63

   1. Initial program 74.0%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 74.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lift-+.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 99.9

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

   7. Applied rewrites 99.9%

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

   if 3.80000000000000017e-63 < x

   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. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      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

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

      5. associate-*l* N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 82.6

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

      15. lift-+.f64 N/A

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

      16. metadata-eval N/A

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

      17. fp-cancel-sign-sub-inv N/A

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

      18. metadata-eval N/A

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

      19. metadata-eval N/A

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

      20. metadata-eval N/A

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

      21. lower--.f64 N/A

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

      22. lift-+.f64 N/A

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

      23. +-commutative N/A

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

      24. lower-+.f64 N/A

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

      25. metadata-eval 82.6

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

      26. lift-+.f64 N/A

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

   4. Applied rewrites 82.6%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/r* N/A

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. lift-+.f64 N/A

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

      9. associate-*l/ N/A

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

      10. lower-/.f64 N/A

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

   6. Applied rewrites 99.6%

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

   7. Taylor expanded in y around inf

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

   9. Applied rewrites 31.9%

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

Alternative 4: 89.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 4.2 \cdot 10^{-149}:\\ \;\;\;\;\frac{\frac{y}{x - -1}}{y + x}\\ \mathbf{elif}\;y \leq 8.5 \cdot 10^{+93}:\\ \;\;\;\;\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y + x} \cdot \frac{1}{y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 4.2e-149)
   (/ (/ y (- x -1.0)) (+ y x))
   (if (<= y 8.5e+93)
     (/ (* x y) (* (* (+ x y) (+ x y)) (+ (+ x y) 1.0)))
     (* (/ x (+ y x)) (/ 1.0 y)))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 4.2e-149) {
		tmp = (y / (x - -1.0)) / (y + x);
	} else if (y <= 8.5e+93) {
		tmp = (x * y) / (((x + y) * (x + y)) * ((x + y) + 1.0));
	} else {
		tmp = (x / (y + x)) * (1.0 / y);
	}
	return tmp;
}

Fortran

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

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

real(8) function code(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 4.2d-149) then
        tmp = (y / (x - (-1.0d0))) / (y + x)
    else if (y <= 8.5d+93) then
        tmp = (x * y) / (((x + y) * (x + y)) * ((x + y) + 1.0d0))
    else
        tmp = (x / (y + x)) * (1.0d0 / y)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (y <= 4.2e-149) {
		tmp = (y / (x - -1.0)) / (y + x);
	} else if (y <= 8.5e+93) {
		tmp = (x * y) / (((x + y) * (x + y)) * ((x + y) + 1.0));
	} else {
		tmp = (x / (y + x)) * (1.0 / y);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if y <= 4.2e-149:
		tmp = (y / (x - -1.0)) / (y + x)
	elif y <= 8.5e+93:
		tmp = (x * y) / (((x + y) * (x + y)) * ((x + y) + 1.0))
	else:
		tmp = (x / (y + x)) * (1.0 / y)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= 4.2e-149)
		tmp = Float64(Float64(y / Float64(x - -1.0)) / Float64(y + x));
	elseif (y <= 8.5e+93)
		tmp = Float64(Float64(x * y) / Float64(Float64(Float64(x + y) * Float64(x + y)) * Float64(Float64(x + y) + 1.0)));
	else
		tmp = Float64(Float64(x / Float64(y + x)) * Float64(1.0 / y));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 4.2e-149)
		tmp = (y / (x - -1.0)) / (y + x);
	elseif (y <= 8.5e+93)
		tmp = (x * y) / (((x + y) * (x + y)) * ((x + y) + 1.0));
	else
		tmp = (x / (y + x)) * (1.0 / y);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[y, 4.2e-149], N[(N[(y / N[(x - -1.0), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 8.5e+93], N[(N[(x * y), $MachinePrecision] / N[(N[(N[(x + y), $MachinePrecision] * N[(x + y), $MachinePrecision]), $MachinePrecision] * N[(N[(x + y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(y + x), $MachinePrecision]), $MachinePrecision] * N[(1.0 / y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 4.20000000000000022e-149

   1. Initial program 69.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      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

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

      5. associate-*l* N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 95.6

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

      15. lift-+.f64 N/A

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

      16. metadata-eval N/A

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

      17. fp-cancel-sign-sub-inv N/A

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

      18. metadata-eval N/A

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

      19. metadata-eval N/A

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

      20. metadata-eval N/A

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

      21. lower--.f64 N/A

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

      22. lift-+.f64 N/A

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

      23. +-commutative N/A

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

      24. lower-+.f64 N/A

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

      25. metadata-eval 95.6

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

      26. lift-+.f64 N/A

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

   4. Applied rewrites 95.6%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/r* N/A

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. lift-+.f64 N/A

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

      9. associate-*l/ N/A

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

      10. lower-/.f64 N/A

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

   6. Applied rewrites 99.8%

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

   7. Taylor expanded in y around 0

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

   9. Applied rewrites 54.4%

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

   if 4.20000000000000022e-149 < y < 8.5000000000000005e93

   1. Initial program 90.1%

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

   if 8.5000000000000005e93 < y

   1. Initial program 48.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      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

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

      5. associate-*l* N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 79.5

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

      15. lift-+.f64 N/A

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

      16. metadata-eval N/A

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

      17. fp-cancel-sign-sub-inv N/A

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

      18. metadata-eval N/A

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

      19. metadata-eval N/A

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

      20. metadata-eval N/A

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

      21. lower--.f64 N/A

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

      22. lift-+.f64 N/A

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

      23. +-commutative N/A

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

      24. lower-+.f64 N/A

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

      25. metadata-eval 79.5

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

      26. lift-+.f64 N/A

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

   4. Applied rewrites 79.5%

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

   5. Taylor expanded in y around inf

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

   7. Applied rewrites 81.1%

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

Alternative 5: 95.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_0 := \frac{x}{y + x}\\ \mathbf{if}\;y \leq 3.8 \cdot 10^{+130}:\\ \;\;\;\;t\_0 \cdot \frac{y}{\left(\left(y + x\right) - -1\right) \cdot \left(y + x\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \frac{1}{y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ x (+ y x))))
   (if (<= y 3.8e+130)
     (* t_0 (/ y (* (- (+ y x) -1.0) (+ y x))))
     (* t_0 (/ 1.0 y)))))

C

assert(x < y);
double code(double x, double y) {
	double t_0 = x / (y + x);
	double tmp;
	if (y <= 3.8e+130) {
		tmp = t_0 * (y / (((y + x) - -1.0) * (y + x)));
	} else {
		tmp = t_0 * (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) :: t_0
    real(8) :: tmp
    t_0 = x / (y + x)
    if (y <= 3.8d+130) then
        tmp = t_0 * (y / (((y + x) - (-1.0d0)) * (y + x)))
    else
        tmp = t_0 * (1.0d0 / y)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double t_0 = x / (y + x);
	double tmp;
	if (y <= 3.8e+130) {
		tmp = t_0 * (y / (((y + x) - -1.0) * (y + x)));
	} else {
		tmp = t_0 * (1.0 / y);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	t_0 = x / (y + x)
	tmp = 0
	if y <= 3.8e+130:
		tmp = t_0 * (y / (((y + x) - -1.0) * (y + x)))
	else:
		tmp = t_0 * (1.0 / y)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	t_0 = Float64(x / Float64(y + x))
	tmp = 0.0
	if (y <= 3.8e+130)
		tmp = Float64(t_0 * Float64(y / Float64(Float64(Float64(y + x) - -1.0) * Float64(y + x))));
	else
		tmp = Float64(t_0 * Float64(1.0 / y));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	t_0 = x / (y + x);
	tmp = 0.0;
	if (y <= 3.8e+130)
		tmp = t_0 * (y / (((y + x) - -1.0) * (y + x)));
	else
		tmp = t_0 * (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_] := Block[{t$95$0 = N[(x / N[(y + x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, 3.8e+130], N[(t$95$0 * N[(y / N[(N[(N[(y + x), $MachinePrecision] - -1.0), $MachinePrecision] * N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(1.0 / y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < 3.8000000000000002e130

   1. Initial program 73.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      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

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

      5. associate-*l* N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 96.2

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

      15. lift-+.f64 N/A

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

      16. metadata-eval N/A

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

      17. fp-cancel-sign-sub-inv N/A

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

      18. metadata-eval N/A

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

      19. metadata-eval N/A

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

      20. metadata-eval N/A

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

      21. lower--.f64 N/A

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

      22. lift-+.f64 N/A

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

      23. +-commutative N/A

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

      24. lower-+.f64 N/A

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

      25. metadata-eval 96.2

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

      26. lift-+.f64 N/A

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

   4. Applied rewrites 96.2%

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

   if 3.8000000000000002e130 < y

   1. Initial program 50.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      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

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

      5. associate-*l* N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 79.8

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

      15. lift-+.f64 N/A

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

      16. metadata-eval N/A

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

      17. fp-cancel-sign-sub-inv N/A

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

      18. metadata-eval N/A

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

      19. metadata-eval N/A

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

      20. metadata-eval N/A

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

      21. lower--.f64 N/A

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

      22. lift-+.f64 N/A

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

      23. +-commutative N/A

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

      24. lower-+.f64 N/A

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

      25. metadata-eval 79.8

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

      26. lift-+.f64 N/A

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

   4. Applied rewrites 79.8%

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

   5. Taylor expanded in y around inf

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

   7. Applied rewrites 85.2%

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

Alternative 6: 99.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \frac{y \cdot \frac{\frac{x}{x + y}}{\left(x + y\right) - -1}}{y + x} \end{array} \]

FPCore

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

C

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

Fortran

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

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

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

Java

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

Python

[x, y] = sort([x, y])
def code(x, y):
	return (y * ((x / (x + y)) / ((x + y) - -1.0))) / (y + x)

Julia

x, y = sort([x, y])
function code(x, y)
	return Float64(Float64(y * Float64(Float64(x / Float64(x + y)) / Float64(Float64(x + y) - -1.0))) / Float64(y + x))
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp = code(x, y)
	tmp = (y * ((x / (x + y)) / ((x + y) - -1.0))) / (y + x);
end

Wolfram

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

Derivation

1. Initial program 68.9%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   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

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

   5. associate-*l* N/A

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

   6. times-frac N/A

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

   7. lower-*.f64 N/A

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

   8. lower-/.f64 N/A

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

   9. lift-+.f64 N/A

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

   10. +-commutative N/A

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

   11. lower-+.f64 N/A

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

   12. lower-/.f64 N/A

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

   13. *-commutative N/A

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

   14. lower-*.f64 93.0

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

   15. lift-+.f64 N/A

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

   16. metadata-eval N/A

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

   17. fp-cancel-sign-sub-inv N/A

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

   18. metadata-eval N/A

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

   19. metadata-eval N/A

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

   20. metadata-eval N/A

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

   21. lower--.f64 N/A

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

   22. lift-+.f64 N/A

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

   23. +-commutative N/A

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

   24. lower-+.f64 N/A

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

   25. metadata-eval 93.0

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

   26. lift-+.f64 N/A

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

4. Applied rewrites 93.0%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-/.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. associate-/r* N/A

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

   6. lift-+.f64 N/A

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

   7. +-commutative N/A

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

   8. lift-+.f64 N/A

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

   9. associate-*l/ N/A

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

   10. lower-/.f64 N/A

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

6. Applied rewrites 99.8%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-*l/ N/A

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

   4. associate-/l* N/A

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

   5. lower-*.f64 N/A

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

   6. lower-/.f64 99.8

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

   7. lift-+.f64 N/A

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

   8. +-commutative N/A

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

   9. lower-+.f64 99.8

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

   10. lift-+.f64 N/A

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

   11. +-commutative N/A

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

   12. lower-+.f64 99.8

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

8. Applied rewrites 99.8%

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

Alternative 7: 85.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 4.6 \cdot 10^{-129}:\\ \;\;\;\;\frac{\frac{y}{x - -1}}{y + x}\\ \mathbf{elif}\;y \leq 9.4 \cdot 10^{+86}:\\ \;\;\;\;\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(1 + y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y + x} \cdot \frac{1}{y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 4.6e-129)
   (/ (/ y (- x -1.0)) (+ y x))
   (if (<= y 9.4e+86)
     (/ (* x y) (* (* (+ x y) (+ x y)) (+ 1.0 y)))
     (* (/ x (+ y x)) (/ 1.0 y)))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 4.6e-129) {
		tmp = (y / (x - -1.0)) / (y + x);
	} else if (y <= 9.4e+86) {
		tmp = (x * y) / (((x + y) * (x + y)) * (1.0 + y));
	} else {
		tmp = (x / (y + x)) * (1.0 / y);
	}
	return tmp;
}

Fortran

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

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

real(8) function code(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 4.6d-129) then
        tmp = (y / (x - (-1.0d0))) / (y + x)
    else if (y <= 9.4d+86) then
        tmp = (x * y) / (((x + y) * (x + y)) * (1.0d0 + y))
    else
        tmp = (x / (y + x)) * (1.0d0 / y)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (y <= 4.6e-129) {
		tmp = (y / (x - -1.0)) / (y + x);
	} else if (y <= 9.4e+86) {
		tmp = (x * y) / (((x + y) * (x + y)) * (1.0 + y));
	} else {
		tmp = (x / (y + x)) * (1.0 / y);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if y <= 4.6e-129:
		tmp = (y / (x - -1.0)) / (y + x)
	elif y <= 9.4e+86:
		tmp = (x * y) / (((x + y) * (x + y)) * (1.0 + y))
	else:
		tmp = (x / (y + x)) * (1.0 / y)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= 4.6e-129)
		tmp = Float64(Float64(y / Float64(x - -1.0)) / Float64(y + x));
	elseif (y <= 9.4e+86)
		tmp = Float64(Float64(x * y) / Float64(Float64(Float64(x + y) * Float64(x + y)) * Float64(1.0 + y)));
	else
		tmp = Float64(Float64(x / Float64(y + x)) * Float64(1.0 / y));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 4.6e-129)
		tmp = (y / (x - -1.0)) / (y + x);
	elseif (y <= 9.4e+86)
		tmp = (x * y) / (((x + y) * (x + y)) * (1.0 + y));
	else
		tmp = (x / (y + x)) * (1.0 / y);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[y, 4.6e-129], N[(N[(y / N[(x - -1.0), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 9.4e+86], N[(N[(x * y), $MachinePrecision] / N[(N[(N[(x + y), $MachinePrecision] * N[(x + y), $MachinePrecision]), $MachinePrecision] * N[(1.0 + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(y + x), $MachinePrecision]), $MachinePrecision] * N[(1.0 / y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 4.5999999999999999e-129

   1. Initial program 70.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      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

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

      5. associate-*l* N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 95.7

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

      15. lift-+.f64 N/A

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

      16. metadata-eval N/A

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

      17. fp-cancel-sign-sub-inv N/A

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

      18. metadata-eval N/A

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

      19. metadata-eval N/A

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

      20. metadata-eval N/A

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

      21. lower--.f64 N/A

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

      22. lift-+.f64 N/A

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

      23. +-commutative N/A

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

      24. lower-+.f64 N/A

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

      25. metadata-eval 95.7

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

      26. lift-+.f64 N/A

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

   4. Applied rewrites 95.7%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/r* N/A

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. lift-+.f64 N/A

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

      9. associate-*l/ N/A

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

      10. lower-/.f64 N/A

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

   6. Applied rewrites 99.8%

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

   7. Taylor expanded in y around 0

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

   9. Applied rewrites 55.3%

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

   if 4.5999999999999999e-129 < y < 9.4000000000000004e86

   1. Initial program 89.2%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 79.8%

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

   if 9.4000000000000004e86 < y

   1. Initial program 49.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      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

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

      5. associate-*l* N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 79.9

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

      15. lift-+.f64 N/A

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

      16. metadata-eval N/A

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

      17. fp-cancel-sign-sub-inv N/A

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

      18. metadata-eval N/A

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

      19. metadata-eval N/A

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

      20. metadata-eval N/A

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

      21. lower--.f64 N/A

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

      22. lift-+.f64 N/A

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

      23. +-commutative N/A

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

      24. lower-+.f64 N/A

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

      25. metadata-eval 79.9

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

      26. lift-+.f64 N/A

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

   4. Applied rewrites 79.9%

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

   5. Taylor expanded in y around inf

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

   7. Applied rewrites 81.5%

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

Alternative 8: 82.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 1.5 \cdot 10^{-87}:\\ \;\;\;\;\frac{\frac{y}{x - -1}}{y + x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y + x} \cdot \frac{1}{1 + y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 1.5e-87)
   (/ (/ y (- x -1.0)) (+ y x))
   (* (/ x (+ y x)) (/ 1.0 (+ 1.0 y)))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 1.5e-87) {
		tmp = (y / (x - -1.0)) / (y + x);
	} else {
		tmp = (x / (y + x)) * (1.0 / (1.0 + y));
	}
	return tmp;
}

Fortran

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

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

real(8) function code(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 1.5d-87) then
        tmp = (y / (x - (-1.0d0))) / (y + x)
    else
        tmp = (x / (y + x)) * (1.0d0 / (1.0d0 + y))
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (y <= 1.5e-87) {
		tmp = (y / (x - -1.0)) / (y + x);
	} else {
		tmp = (x / (y + x)) * (1.0 / (1.0 + y));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if y <= 1.5e-87:
		tmp = (y / (x - -1.0)) / (y + x)
	else:
		tmp = (x / (y + x)) * (1.0 / (1.0 + y))
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= 1.5e-87)
		tmp = Float64(Float64(y / Float64(x - -1.0)) / Float64(y + x));
	else
		tmp = Float64(Float64(x / Float64(y + x)) * Float64(1.0 / Float64(1.0 + y)));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 1.5e-87)
		tmp = (y / (x - -1.0)) / (y + x);
	else
		tmp = (x / (y + x)) * (1.0 / (1.0 + y));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[y, 1.5e-87], N[(N[(y / N[(x - -1.0), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(y + x), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(1.0 + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.50000000000000008e-87

   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. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      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

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

      5. associate-*l* N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 96.0

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

      15. lift-+.f64 N/A

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

      16. metadata-eval N/A

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

      17. fp-cancel-sign-sub-inv N/A

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

      18. metadata-eval N/A

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

      19. metadata-eval N/A

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

      20. metadata-eval N/A

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

      21. lower--.f64 N/A

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

      22. lift-+.f64 N/A

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

      23. +-commutative N/A

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

      24. lower-+.f64 N/A

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

      25. metadata-eval 96.0

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

      26. lift-+.f64 N/A

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

   4. Applied rewrites 96.0%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/r* N/A

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. lift-+.f64 N/A

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

      9. associate-*l/ N/A

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

      10. lower-/.f64 N/A

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

   6. Applied rewrites 99.8%

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

   7. Taylor expanded in y around 0

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

   9. Applied rewrites 56.3%

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

   if 1.50000000000000008e-87 < y

   1. Initial program 62.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      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

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

      5. associate-*l* N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 87.0

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

      15. lift-+.f64 N/A

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

      16. metadata-eval N/A

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

      17. fp-cancel-sign-sub-inv N/A

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

      18. metadata-eval N/A

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

      19. metadata-eval N/A

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

      20. metadata-eval N/A

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

      21. lower--.f64 N/A

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

      22. lift-+.f64 N/A

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

      23. +-commutative N/A

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

      24. lower-+.f64 N/A

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

      25. metadata-eval 87.0

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

      26. lift-+.f64 N/A

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

   4. Applied rewrites 87.0%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 77.9%

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

Alternative 9: 81.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 1.5 \cdot 10^{-87}:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(x, x, x\right)}\\ \mathbf{elif}\;y \leq 5 \cdot 10^{+61}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(y, y, y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 1.5e-87)
   (/ y (fma x x x))
   (if (<= y 5e+61) (/ x (fma y y y)) (/ (/ x y) y))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 1.5e-87) {
		tmp = y / fma(x, x, x);
	} else if (y <= 5e+61) {
		tmp = x / fma(y, y, y);
	} else {
		tmp = (x / y) / y;
	}
	return tmp;
}

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= 1.5e-87)
		tmp = Float64(y / fma(x, x, x));
	elseif (y <= 5e+61)
		tmp = Float64(x / fma(y, y, y));
	else
		tmp = Float64(Float64(x / 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[y, 1.5e-87], N[(y / N[(x * x + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 5e+61], N[(x / N[(y * y + y), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 1.50000000000000008e-87

   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 y around 0

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

   5. Applied rewrites 54.6%

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

   if 1.50000000000000008e-87 < y < 5.00000000000000018e61

   1. Initial program 88.4%

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

   5. Applied rewrites 70.2%

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

   if 5.00000000000000018e61 < y

   1. Initial program 51.5%

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

   3. Taylor expanded in y around inf

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

   5. Applied rewrites 76.3%

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

   7. Applied rewrites 80.7%

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

Alternative 10: 82.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 1.5 \cdot 10^{-87}:\\ \;\;\;\;\frac{\frac{y}{x - -1}}{y + x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{1 + y}}{y + x}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 1.5e-87) (/ (/ y (- x -1.0)) (+ y x)) (/ (/ x (+ 1.0 y)) (+ y x))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 1.5e-87) {
		tmp = (y / (x - -1.0)) / (y + x);
	} else {
		tmp = (x / (1.0 + y)) / (y + x);
	}
	return tmp;
}

Fortran

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

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

real(8) function code(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 1.5d-87) then
        tmp = (y / (x - (-1.0d0))) / (y + x)
    else
        tmp = (x / (1.0d0 + y)) / (y + x)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (y <= 1.5e-87) {
		tmp = (y / (x - -1.0)) / (y + x);
	} else {
		tmp = (x / (1.0 + y)) / (y + x);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if y <= 1.5e-87:
		tmp = (y / (x - -1.0)) / (y + x)
	else:
		tmp = (x / (1.0 + y)) / (y + x)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= 1.5e-87)
		tmp = Float64(Float64(y / Float64(x - -1.0)) / Float64(y + x));
	else
		tmp = Float64(Float64(x / Float64(1.0 + y)) / Float64(y + x));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 1.5e-87)
		tmp = (y / (x - -1.0)) / (y + x);
	else
		tmp = (x / (1.0 + y)) / (y + x);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[y, 1.5e-87], N[(N[(y / N[(x - -1.0), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(1.0 + y), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.50000000000000008e-87

   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. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      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

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

      5. associate-*l* N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 96.0

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

      15. lift-+.f64 N/A

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

      16. metadata-eval N/A

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

      17. fp-cancel-sign-sub-inv N/A

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

      18. metadata-eval N/A

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

      19. metadata-eval N/A

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

      20. metadata-eval N/A

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

      21. lower--.f64 N/A

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

      22. lift-+.f64 N/A

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

      23. +-commutative N/A

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

      24. lower-+.f64 N/A

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

      25. metadata-eval 96.0

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

      26. lift-+.f64 N/A

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

   4. Applied rewrites 96.0%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/r* N/A

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. lift-+.f64 N/A

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

      9. associate-*l/ N/A

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

      10. lower-/.f64 N/A

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

   6. Applied rewrites 99.8%

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

   7. Taylor expanded in y around 0

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

   9. Applied rewrites 56.3%

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

   if 1.50000000000000008e-87 < y

   1. Initial program 62.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      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

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

      5. associate-*l* N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 87.0

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

      15. lift-+.f64 N/A

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

      16. metadata-eval N/A

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

      17. fp-cancel-sign-sub-inv N/A

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

      18. metadata-eval N/A

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

      19. metadata-eval N/A

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

      20. metadata-eval N/A

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

      21. lower--.f64 N/A

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

      22. lift-+.f64 N/A

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

      23. +-commutative N/A

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

      24. lower-+.f64 N/A

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

      25. metadata-eval 87.0

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

      26. lift-+.f64 N/A

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

   4. Applied rewrites 87.0%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/r* N/A

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. lift-+.f64 N/A

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

      9. associate-*l/ N/A

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

      10. lower-/.f64 N/A

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

   6. Applied rewrites 99.8%

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

   7. Taylor expanded in x around 0

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

   9. Applied rewrites 77.9%

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

Alternative 11: 81.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 1.5 \cdot 10^{-87}:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(x, x, x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{1 + y}}{y + x}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 1.5e-87) (/ y (fma x x x)) (/ (/ x (+ 1.0 y)) (+ y x))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 1.5e-87) {
		tmp = y / fma(x, x, x);
	} else {
		tmp = (x / (1.0 + y)) / (y + x);
	}
	return tmp;
}

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= 1.5e-87)
		tmp = Float64(y / fma(x, x, x));
	else
		tmp = Float64(Float64(x / Float64(1.0 + y)) / Float64(y + x));
	end
	return tmp
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[y, 1.5e-87], N[(y / N[(x * x + x), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(1.0 + y), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.50000000000000008e-87

   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 y around 0

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

   5. Applied rewrites 54.6%

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

   if 1.50000000000000008e-87 < y

   1. Initial program 62.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      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

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

      5. associate-*l* N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 87.0

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

      15. lift-+.f64 N/A

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

      16. metadata-eval N/A

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

      17. fp-cancel-sign-sub-inv N/A

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

      18. metadata-eval N/A

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

      19. metadata-eval N/A

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

      20. metadata-eval N/A

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

      21. lower--.f64 N/A

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

      22. lift-+.f64 N/A

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

      23. +-commutative N/A

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

      24. lower-+.f64 N/A

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

      25. metadata-eval 87.0

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

      26. lift-+.f64 N/A

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

   4. Applied rewrites 87.0%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/r* N/A

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. lift-+.f64 N/A

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

      9. associate-*l/ N/A

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

      10. lower-/.f64 N/A

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

   6. Applied rewrites 99.8%

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

   7. Taylor expanded in x around 0

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

   9. Applied rewrites 77.9%

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

Alternative 12: 81.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 1.5 \cdot 10^{-87}:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(x, x, x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y - -1}}{y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 1.5e-87) (/ y (fma x x x)) (/ (/ x (- y -1.0)) y)))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 1.5e-87) {
		tmp = y / fma(x, x, x);
	} else {
		tmp = (x / (y - -1.0)) / y;
	}
	return tmp;
}

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= 1.5e-87)
		tmp = Float64(y / fma(x, x, x));
	else
		tmp = Float64(Float64(x / Float64(y - -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[y, 1.5e-87], N[(y / N[(x * x + x), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(y - -1.0), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.50000000000000008e-87

   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 y around 0

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

   5. Applied rewrites 54.6%

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

   if 1.50000000000000008e-87 < y

   1. Initial program 62.9%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 74.4%

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

   7. Applied rewrites 77.5%

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

Alternative 13: 65.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 1.02 \cdot 10^{-87}:\\ \;\;\;\;\frac{y}{x \cdot x}\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 1.02e-87) (/ y (* x x)) (if (<= y 1.0) (/ x y) (/ x (* y y)))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 1.02e-87) {
		tmp = y / (x * x);
	} else if (y <= 1.0) {
		tmp = x / y;
	} else {
		tmp = x / (y * y);
	}
	return tmp;
}

Fortran

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

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

real(8) function code(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 1.02d-87) then
        tmp = y / (x * x)
    else if (y <= 1.0d0) then
        tmp = x / y
    else
        tmp = x / (y * y)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (y <= 1.02e-87) {
		tmp = y / (x * x);
	} else if (y <= 1.0) {
		tmp = x / y;
	} else {
		tmp = x / (y * y);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if y <= 1.02e-87:
		tmp = y / (x * x)
	elif y <= 1.0:
		tmp = x / y
	else:
		tmp = x / (y * y)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= 1.02e-87)
		tmp = Float64(y / Float64(x * x));
	elseif (y <= 1.0)
		tmp = Float64(x / y);
	else
		tmp = Float64(x / Float64(y * y));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 1.02e-87)
		tmp = y / (x * x);
	elseif (y <= 1.0)
		tmp = x / y;
	else
		tmp = x / (y * y);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[y, 1.02e-87], N[(y / N[(x * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.0], N[(x / y), $MachinePrecision], N[(x / N[(y * y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 1.02000000000000009e-87

   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 inf

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

   5. Applied rewrites 38.0%

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

   if 1.02000000000000009e-87 < y < 1

   1. Initial program 89.9%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 66.3%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 60.7%

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

   if 1 < y

   1. Initial program 54.5%

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

   3. Taylor expanded in y around inf

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

   5. Applied rewrites 76.9%

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

Alternative 14: 79.2% accurate, 1.6× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 1.5 \cdot 10^{-87}:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(x, x, x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(y, y, y\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 1.5e-87) (/ y (fma x x x)) (/ x (fma y y y))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 1.5e-87) {
		tmp = y / fma(x, x, x);
	} else {
		tmp = x / fma(y, y, y);
	}
	return tmp;
}

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= 1.5e-87)
		tmp = Float64(y / fma(x, x, x));
	else
		tmp = Float64(x / fma(y, 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[y, 1.5e-87], N[(y / N[(x * x + x), $MachinePrecision]), $MachinePrecision], N[(x / N[(y * y + y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.50000000000000008e-87

   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 y around 0

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

   5. Applied rewrites 54.6%

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

   if 1.50000000000000008e-87 < y

   1. Initial program 62.9%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 74.4%

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

Alternative 15: 76.2% accurate, 1.6× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -1400000:\\ \;\;\;\;\frac{y}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(y, y, y\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -1400000.0) (/ y (* x x)) (/ x (fma y y y))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -1400000.0) {
		tmp = y / (x * x);
	} else {
		tmp = x / fma(y, y, y);
	}
	return tmp;
}

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -1400000.0)
		tmp = Float64(y / Float64(x * x));
	else
		tmp = Float64(x / fma(y, 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, -1400000.0], N[(y / N[(x * x), $MachinePrecision]), $MachinePrecision], N[(x / N[(y * y + y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.4e6

   1. Initial program 64.9%

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

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 71.3%

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

   if -1.4e6 < x

   1. Initial program 70.1%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 60.9%

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

Alternative 16: 47.1% 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 73.7%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 45.1%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 29.0%

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

   if 1 < y

   1. Initial program 54.5%

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

   3. Taylor expanded in y around inf

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

   5. Applied rewrites 76.9%

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

Alternative 17: 26.7% 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 68.9%

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

3. Taylor expanded in x around 0

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

5. Applied rewrites 53.0%

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

6. Taylor expanded in y around 0

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

8. Applied rewrites 27.9%

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

Developer Target 1: 99.8% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2025019 
(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))))