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

Percentage Accurate: 68.4% → 99.8%

Time: 5.9s

Alternatives: 19

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: 68.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 0.8× speedup

Math

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

FPCore

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

C

assert(x < y);
double code(double x, double y) {
	return (((x / ((y + x) - -1.0)) / (y + x)) * y) / (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 + x) - (-1.0d0))) / (y + x)) * y) / (x + y)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 62.0%

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

   1. lift-/.f64 N/A

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

   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. *-commutative N/A

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

   4. lift-*.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. associate-*l* N/A

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

   7. times-frac N/A

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

   8. lower-*.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. lift-+.f64 N/A

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

   11. +-commutative N/A

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

   12. lower-+.f64 N/A

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

   13. lower-/.f64 N/A

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

   14. *-commutative N/A

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

   15. lower-*.f64 91.8

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

4. Applied rewrites 91.8%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-*l/ N/A

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

   4. lift-+.f64 N/A

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

   5. +-commutative N/A

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

   6. lower-/.f64 N/A

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

6. Applied rewrites 67.8%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*l/ N/A

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

   4. lift-*.f64 N/A

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

   5. associate-/l/ N/A

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

   6. lift-/.f64 N/A

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

   7. lift-/.f64 N/A

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

   8. lower-*.f64 99.8

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

8. Applied rewrites 99.8%

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

Alternative 2: 97.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -1.75 \cdot 10^{+154}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(y, \frac{\mathsf{fma}\left(-2, y, -1\right)}{x}, y\right)}{x}}{x + y}\\ \mathbf{elif}\;x \leq 5.2 \cdot 10^{+45}:\\ \;\;\;\;\frac{y}{y + x} \cdot \frac{x}{\left(\left(y + x\right) - -1\right) \cdot \left(y + x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{x}{y}}{y + x} \cdot y}{x + y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -1.75e+154)
   (/ (/ (fma y (/ (fma -2.0 y -1.0) x) y) x) (+ x y))
   (if (<= x 5.2e+45)
     (* (/ y (+ y x)) (/ x (* (- (+ y x) -1.0) (+ y x))))
     (/ (* (/ (/ x y) (+ y x)) y) (+ x y)))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -1.75e+154) {
		tmp = (fma(y, (fma(-2.0, y, -1.0) / x), y) / x) / (x + y);
	} else if (x <= 5.2e+45) {
		tmp = (y / (y + x)) * (x / (((y + x) - -1.0) * (y + x)));
	} else {
		tmp = (((x / y) / (y + x)) * y) / (x + y);
	}
	return tmp;
}

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -1.75e+154)
		tmp = Float64(Float64(fma(y, Float64(fma(-2.0, y, -1.0) / x), y) / x) / Float64(x + y));
	elseif (x <= 5.2e+45)
		tmp = Float64(Float64(y / Float64(y + x)) * Float64(x / Float64(Float64(Float64(y + x) - -1.0) * Float64(y + x))));
	else
		tmp = Float64(Float64(Float64(Float64(x / y) / Float64(y + x)) * y) / Float64(x + y));
	end
	return tmp
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[x, -1.75e+154], N[(N[(N[(y * N[(N[(-2.0 * y + -1.0), $MachinePrecision] / x), $MachinePrecision] + y), $MachinePrecision] / x), $MachinePrecision] / N[(x + y), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 5.2e+45], N[(N[(y / N[(y + x), $MachinePrecision]), $MachinePrecision] * N[(x / N[(N[(N[(y + x), $MachinePrecision] - -1.0), $MachinePrecision] * N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(x / y), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision] / N[(x + y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.7500000000000001e154

   1. Initial program 61.1%

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

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 79.4

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

   4. Applied rewrites 79.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-/.f64 N/A

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

   6. Applied rewrites 61.1%

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

   7. Taylor expanded in x around inf

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

   9. Applied rewrites 91.5%

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

   if -1.7500000000000001e154 < x < 5.20000000000000014e45

   1. Initial program 68.4%

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

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 99.0

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

   4. Applied rewrites 99.0%

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

   if 5.20000000000000014e45 < x

   1. Initial program 38.4%

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

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 73.7

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

   4. Applied rewrites 73.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-/.f64 N/A

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

   6. Applied rewrites 54.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l/ N/A

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

      6. lift-/.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lower-*.f64 99.5

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

   8. Applied rewrites 99.5%

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

   9. Taylor expanded in y around inf

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

   11. Applied rewrites 32.6%

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

Alternative 3: 97.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -1.75 \cdot 10^{+154}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\frac{y}{x} \cdot 3, -y, y\right)}{x}}{x}\\ \mathbf{elif}\;x \leq 5.2 \cdot 10^{+45}:\\ \;\;\;\;\frac{y}{y + x} \cdot \frac{x}{\left(\left(y + x\right) - -1\right) \cdot \left(y + x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{x}{y}}{y + x} \cdot y}{x + y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -1.75e+154)
   (/ (/ (fma (* (/ y x) 3.0) (- y) y) x) x)
   (if (<= x 5.2e+45)
     (* (/ y (+ y x)) (/ x (* (- (+ y x) -1.0) (+ y x))))
     (/ (* (/ (/ x y) (+ y x)) y) (+ x y)))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -1.75e+154) {
		tmp = (fma(((y / x) * 3.0), -y, y) / x) / x;
	} else if (x <= 5.2e+45) {
		tmp = (y / (y + x)) * (x / (((y + x) - -1.0) * (y + x)));
	} else {
		tmp = (((x / y) / (y + x)) * y) / (x + y);
	}
	return tmp;
}

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -1.75e+154)
		tmp = Float64(Float64(fma(Float64(Float64(y / x) * 3.0), Float64(-y), y) / x) / x);
	elseif (x <= 5.2e+45)
		tmp = Float64(Float64(y / Float64(y + x)) * Float64(x / Float64(Float64(Float64(y + x) - -1.0) * Float64(y + x))));
	else
		tmp = Float64(Float64(Float64(Float64(x / y) / Float64(y + x)) * y) / Float64(x + y));
	end
	return tmp
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[x, -1.75e+154], N[(N[(N[(N[(N[(y / x), $MachinePrecision] * 3.0), $MachinePrecision] * (-y) + y), $MachinePrecision] / x), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[x, 5.2e+45], N[(N[(y / N[(y + x), $MachinePrecision]), $MachinePrecision] * N[(x / N[(N[(N[(y + x), $MachinePrecision] - -1.0), $MachinePrecision] * N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(x / y), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision] / N[(x + y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.7500000000000001e154

   1. Initial program 61.1%

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

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 74.8%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 74.8%

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

   10. Applied rewrites 91.5%

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

   if -1.7500000000000001e154 < x < 5.20000000000000014e45

   1. Initial program 68.4%

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

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 99.0

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

   4. Applied rewrites 99.0%

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

   if 5.20000000000000014e45 < x

   1. Initial program 38.4%

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

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 73.7

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

   4. Applied rewrites 73.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-/.f64 N/A

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

   6. Applied rewrites 54.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l/ N/A

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

      6. lift-/.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lower-*.f64 99.5

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

   8. Applied rewrites 99.5%

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

   9. Taylor expanded in y around inf

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

   11. Applied rewrites 32.6%

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

Alternative 4: 97.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -1.75 \cdot 10^{+154}:\\ \;\;\;\;\frac{\frac{1}{y + x} \cdot y}{x + y}\\ \mathbf{elif}\;x \leq 5.2 \cdot 10^{+45}:\\ \;\;\;\;\frac{y}{y + x} \cdot \frac{x}{\left(\left(y + x\right) - -1\right) \cdot \left(y + x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{x}{y}}{y + x} \cdot y}{x + y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -1.75e+154)
   (/ (* (/ 1.0 (+ y x)) y) (+ x y))
   (if (<= x 5.2e+45)
     (* (/ y (+ y x)) (/ x (* (- (+ y x) -1.0) (+ y x))))
     (/ (* (/ (/ x y) (+ y x)) y) (+ x y)))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -1.75e+154) {
		tmp = ((1.0 / (y + x)) * y) / (x + y);
	} else if (x <= 5.2e+45) {
		tmp = (y / (y + x)) * (x / (((y + x) - -1.0) * (y + x)));
	} else {
		tmp = (((x / y) / (y + x)) * y) / (x + y);
	}
	return tmp;
}

Fortran

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

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

real(8) function code(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-1.75d+154)) then
        tmp = ((1.0d0 / (y + x)) * y) / (x + y)
    else if (x <= 5.2d+45) then
        tmp = (y / (y + x)) * (x / (((y + x) - (-1.0d0)) * (y + x)))
    else
        tmp = (((x / y) / (y + x)) * y) / (x + y)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (x <= -1.75e+154) {
		tmp = ((1.0 / (y + x)) * y) / (x + y);
	} else if (x <= 5.2e+45) {
		tmp = (y / (y + x)) * (x / (((y + x) - -1.0) * (y + x)));
	} else {
		tmp = (((x / y) / (y + x)) * y) / (x + y);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -1.75e+154:
		tmp = ((1.0 / (y + x)) * y) / (x + y)
	elif x <= 5.2e+45:
		tmp = (y / (y + x)) * (x / (((y + x) - -1.0) * (y + x)))
	else:
		tmp = (((x / y) / (y + x)) * y) / (x + y)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -1.75e+154)
		tmp = Float64(Float64(Float64(1.0 / Float64(y + x)) * y) / Float64(x + y));
	elseif (x <= 5.2e+45)
		tmp = Float64(Float64(y / Float64(y + x)) * Float64(x / Float64(Float64(Float64(y + x) - -1.0) * Float64(y + x))));
	else
		tmp = Float64(Float64(Float64(Float64(x / y) / Float64(y + x)) * y) / Float64(x + y));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1.75e+154)
		tmp = ((1.0 / (y + x)) * y) / (x + y);
	elseif (x <= 5.2e+45)
		tmp = (y / (y + x)) * (x / (((y + x) - -1.0) * (y + x)));
	else
		tmp = (((x / y) / (y + x)) * y) / (x + y);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[x, -1.75e+154], N[(N[(N[(1.0 / N[(y + x), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision] / N[(x + y), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 5.2e+45], N[(N[(y / N[(y + x), $MachinePrecision]), $MachinePrecision] * N[(x / N[(N[(N[(y + x), $MachinePrecision] - -1.0), $MachinePrecision] * N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(x / y), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision] / N[(x + y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.7500000000000001e154

   1. Initial program 61.1%

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

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 79.4

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

   4. Applied rewrites 79.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-/.f64 N/A

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

   6. Applied rewrites 61.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l/ N/A

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

      6. lift-/.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lower-*.f64 99.9

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

   8. Applied rewrites 99.9%

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

   9. Taylor expanded in x around inf

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

   11. Applied rewrites 92.1%

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

   if -1.7500000000000001e154 < x < 5.20000000000000014e45

   1. Initial program 68.4%

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

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 99.0

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

   4. Applied rewrites 99.0%

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

   if 5.20000000000000014e45 < x

   1. Initial program 38.4%

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

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 73.7

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

   4. Applied rewrites 73.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-/.f64 N/A

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

   6. Applied rewrites 54.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l/ N/A

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

      6. lift-/.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lower-*.f64 99.5

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

   8. Applied rewrites 99.5%

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

   9. Taylor expanded in y around inf

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

   11. Applied rewrites 32.6%

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

Alternative 5: 97.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -1.75 \cdot 10^{+154}:\\ \;\;\;\;\frac{\frac{1}{y + x} \cdot y}{x + y}\\ \mathbf{elif}\;x \leq 5.2 \cdot 10^{+45}:\\ \;\;\;\;\frac{y}{y + x} \cdot \frac{x}{\left(\left(y + x\right) - -1\right) \cdot \left(y + x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{x + y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -1.75e+154)
   (/ (* (/ 1.0 (+ y x)) y) (+ x y))
   (if (<= x 5.2e+45)
     (* (/ y (+ y x)) (/ x (* (- (+ y x) -1.0) (+ y x))))
     (/ (/ x y) (+ x y)))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -1.75e+154) {
		tmp = ((1.0 / (y + x)) * y) / (x + y);
	} else if (x <= 5.2e+45) {
		tmp = (y / (y + x)) * (x / (((y + x) - -1.0) * (y + x)));
	} else {
		tmp = (x / y) / (x + y);
	}
	return tmp;
}

Fortran

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

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

real(8) function code(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-1.75d+154)) then
        tmp = ((1.0d0 / (y + x)) * y) / (x + y)
    else if (x <= 5.2d+45) then
        tmp = (y / (y + x)) * (x / (((y + x) - (-1.0d0)) * (y + x)))
    else
        tmp = (x / y) / (x + y)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (x <= -1.75e+154) {
		tmp = ((1.0 / (y + x)) * y) / (x + y);
	} else if (x <= 5.2e+45) {
		tmp = (y / (y + x)) * (x / (((y + x) - -1.0) * (y + x)));
	} else {
		tmp = (x / y) / (x + y);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -1.75e+154:
		tmp = ((1.0 / (y + x)) * y) / (x + y)
	elif x <= 5.2e+45:
		tmp = (y / (y + x)) * (x / (((y + x) - -1.0) * (y + x)))
	else:
		tmp = (x / y) / (x + y)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -1.75e+154)
		tmp = Float64(Float64(Float64(1.0 / Float64(y + x)) * y) / Float64(x + y));
	elseif (x <= 5.2e+45)
		tmp = Float64(Float64(y / Float64(y + x)) * Float64(x / Float64(Float64(Float64(y + x) - -1.0) * Float64(y + x))));
	else
		tmp = Float64(Float64(x / y) / Float64(x + y));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1.75e+154)
		tmp = ((1.0 / (y + x)) * y) / (x + y);
	elseif (x <= 5.2e+45)
		tmp = (y / (y + x)) * (x / (((y + x) - -1.0) * (y + x)));
	else
		tmp = (x / y) / (x + y);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[x, -1.75e+154], N[(N[(N[(1.0 / N[(y + x), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision] / N[(x + y), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 5.2e+45], N[(N[(y / N[(y + x), $MachinePrecision]), $MachinePrecision] * N[(x / N[(N[(N[(y + x), $MachinePrecision] - -1.0), $MachinePrecision] * N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / N[(x + y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.7500000000000001e154

   1. Initial program 61.1%

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

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 79.4

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

   4. Applied rewrites 79.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-/.f64 N/A

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

   6. Applied rewrites 61.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l/ N/A

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

      6. lift-/.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lower-*.f64 99.9

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

   8. Applied rewrites 99.9%

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

   9. Taylor expanded in x around inf

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

   11. Applied rewrites 92.1%

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

   if -1.7500000000000001e154 < x < 5.20000000000000014e45

   1. Initial program 68.4%

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

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 99.0

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

   4. Applied rewrites 99.0%

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

   if 5.20000000000000014e45 < x

   1. Initial program 38.4%

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

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 73.7

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

   4. Applied rewrites 73.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-/.f64 N/A

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

   6. Applied rewrites 54.8%

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

   7. Taylor expanded in y around inf

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

   9. Applied rewrites 31.9%

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

Alternative 6: 92.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -3100000000000:\\ \;\;\;\;\frac{\frac{y}{x - -1}}{x + y}\\ \mathbf{elif}\;x \leq 5.1 \cdot 10^{+45}:\\ \;\;\;\;\frac{y}{y + x} \cdot \frac{x}{\left(y - -1\right) \cdot \left(y + x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{x + y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -3100000000000.0)
   (/ (/ y (- x -1.0)) (+ x y))
   (if (<= x 5.1e+45)
     (* (/ y (+ y x)) (/ x (* (- y -1.0) (+ y x))))
     (/ (/ x y) (+ x y)))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -3100000000000.0) {
		tmp = (y / (x - -1.0)) / (x + y);
	} else if (x <= 5.1e+45) {
		tmp = (y / (y + x)) * (x / ((y - -1.0) * (y + x)));
	} else {
		tmp = (x / y) / (x + y);
	}
	return tmp;
}

Fortran

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

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

real(8) function code(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-3100000000000.0d0)) then
        tmp = (y / (x - (-1.0d0))) / (x + y)
    else if (x <= 5.1d+45) then
        tmp = (y / (y + x)) * (x / ((y - (-1.0d0)) * (y + x)))
    else
        tmp = (x / y) / (x + y)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (x <= -3100000000000.0) {
		tmp = (y / (x - -1.0)) / (x + y);
	} else if (x <= 5.1e+45) {
		tmp = (y / (y + x)) * (x / ((y - -1.0) * (y + x)));
	} else {
		tmp = (x / y) / (x + y);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -3100000000000.0:
		tmp = (y / (x - -1.0)) / (x + y)
	elif x <= 5.1e+45:
		tmp = (y / (y + x)) * (x / ((y - -1.0) * (y + x)))
	else:
		tmp = (x / y) / (x + y)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -3100000000000.0)
		tmp = Float64(Float64(y / Float64(x - -1.0)) / Float64(x + y));
	elseif (x <= 5.1e+45)
		tmp = Float64(Float64(y / Float64(y + x)) * Float64(x / Float64(Float64(y - -1.0) * Float64(y + x))));
	else
		tmp = Float64(Float64(x / y) / Float64(x + y));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -3100000000000.0)
		tmp = (y / (x - -1.0)) / (x + y);
	elseif (x <= 5.1e+45)
		tmp = (y / (y + x)) * (x / ((y - -1.0) * (y + x)));
	else
		tmp = (x / y) / (x + y);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[x, -3100000000000.0], N[(N[(y / N[(x - -1.0), $MachinePrecision]), $MachinePrecision] / N[(x + y), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 5.1e+45], 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[(x / y), $MachinePrecision] / N[(x + y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -3.1e12

   1. Initial program 64.9%

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

      1. lift-/.f64 N/A

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

      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. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 86.4

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

   4. Applied rewrites 86.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-/.f64 N/A

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

   6. Applied rewrites 70.8%

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

   7. Taylor expanded in y around 0

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

   9. Applied rewrites 80.7%

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

   if -3.1e12 < x < 5.0999999999999997e45

   1. Initial program 68.2%

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

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 95.9%

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

   if 5.0999999999999997e45 < x

   1. Initial program 38.4%

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

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 73.7

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

   4. Applied rewrites 73.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-/.f64 N/A

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

   6. Applied rewrites 54.8%

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

   7. Taylor expanded in y around inf

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

   9. Applied rewrites 31.9%

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

Alternative 7: 87.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -1.6 \cdot 10^{+18}:\\ \;\;\;\;\frac{\frac{1}{x} \cdot y}{x + y}\\ \mathbf{elif}\;x \leq -3.8 \cdot 10^{-149}:\\ \;\;\;\;\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y - -1}}{x + y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -1.6e+18)
   (/ (* (/ 1.0 x) y) (+ x y))
   (if (<= x -3.8e-149)
     (/ (* x y) (* (* (+ x y) (+ x y)) (+ (+ x y) 1.0)))
     (/ (/ x (- y -1.0)) (+ x y)))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -1.6e+18) {
		tmp = ((1.0 / x) * y) / (x + y);
	} else if (x <= -3.8e-149) {
		tmp = (x * y) / (((x + y) * (x + y)) * ((x + y) + 1.0));
	} else {
		tmp = (x / (y - -1.0)) / (x + y);
	}
	return tmp;
}

Fortran

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

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

real(8) function code(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-1.6d+18)) then
        tmp = ((1.0d0 / x) * y) / (x + y)
    else if (x <= (-3.8d-149)) then
        tmp = (x * y) / (((x + y) * (x + y)) * ((x + y) + 1.0d0))
    else
        tmp = (x / (y - (-1.0d0))) / (x + y)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (x <= -1.6e+18) {
		tmp = ((1.0 / x) * y) / (x + y);
	} else if (x <= -3.8e-149) {
		tmp = (x * y) / (((x + y) * (x + y)) * ((x + y) + 1.0));
	} else {
		tmp = (x / (y - -1.0)) / (x + y);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -1.6e+18:
		tmp = ((1.0 / x) * y) / (x + y)
	elif x <= -3.8e-149:
		tmp = (x * y) / (((x + y) * (x + y)) * ((x + y) + 1.0))
	else:
		tmp = (x / (y - -1.0)) / (x + y)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -1.6e+18)
		tmp = Float64(Float64(Float64(1.0 / x) * y) / Float64(x + y));
	elseif (x <= -3.8e-149)
		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 - -1.0)) / Float64(x + y));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1.6e+18)
		tmp = ((1.0 / x) * y) / (x + y);
	elseif (x <= -3.8e-149)
		tmp = (x * y) / (((x + y) * (x + y)) * ((x + y) + 1.0));
	else
		tmp = (x / (y - -1.0)) / (x + y);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[x, -1.6e+18], N[(N[(N[(1.0 / x), $MachinePrecision] * y), $MachinePrecision] / N[(x + y), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -3.8e-149], 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 - -1.0), $MachinePrecision]), $MachinePrecision] / N[(x + y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.6e18

   1. Initial program 64.9%

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

      1. lift-/.f64 N/A

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

      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. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 86.4

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

   4. Applied rewrites 86.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-/.f64 N/A

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

   6. Applied rewrites 70.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l/ N/A

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

      6. lift-/.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lower-*.f64 99.9

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

   8. Applied rewrites 99.9%

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

   9. Taylor expanded in x around inf

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

   11. Applied rewrites 80.8%

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

   if -1.6e18 < x < -3.80000000000000005e-149

   1. Initial program 83.6%

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

   if -3.80000000000000005e-149 < x

   1. Initial program 57.1%

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

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 92.5

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

   4. Applied rewrites 92.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-/.f64 N/A

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

   6. Applied rewrites 63.3%

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

   7. Taylor expanded in x around 0

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

   9. Applied rewrites 60.2%

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

Alternative 8: 99.8% accurate, 0.8× speedup

Math

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

FPCore

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

C

assert(x < y);
double code(double x, double y) {
	return ((x / (x + y)) / ((x + y) - -1.0)) * (y / (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 / (x + y)) / ((x + y) - (-1.0d0))) * (y / (x + y))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 62.0%

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

   1. lift-/.f64 N/A

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

   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. *-commutative N/A

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

   4. lift-*.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. associate-*l* N/A

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

   7. times-frac N/A

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

   8. lower-*.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. lift-+.f64 N/A

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

   11. +-commutative N/A

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

   12. lower-+.f64 N/A

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

   13. lower-/.f64 N/A

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

   14. *-commutative N/A

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

   15. lower-*.f64 91.8

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

4. Applied rewrites 91.8%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 91.8

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

   4. lift-/.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-+.f64 N/A

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

   7. +-commutative N/A

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

   8. *-commutative N/A

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

   9. associate-/r* N/A

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

   10. lower-/.f64 N/A

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

   11. lower-/.f64 N/A

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

   12. lower-+.f64 99.8

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

   13. lift-+.f64 N/A

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

   14. +-commutative N/A

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

   15. lower-+.f64 99.8

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

   16. lift-+.f64 N/A

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

   17. +-commutative N/A

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

   18. lower-+.f64 99.8

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

6. Applied rewrites 99.8%

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

Alternative 9: 80.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -2.7 \cdot 10^{+14}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + y}\\ \mathbf{elif}\;x \leq 3.8 \cdot 10^{+32}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(y, y, y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{x + y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -2.7e+14)
   (/ (/ y x) (+ x y))
   (if (<= x 3.8e+32) (/ x (fma y y y)) (/ (/ x y) (+ x y)))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -2.7e+14) {
		tmp = (y / x) / (x + y);
	} else if (x <= 3.8e+32) {
		tmp = x / fma(y, y, y);
	} else {
		tmp = (x / y) / (x + y);
	}
	return tmp;
}

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -2.7e+14)
		tmp = Float64(Float64(y / x) / Float64(x + y));
	elseif (x <= 3.8e+32)
		tmp = Float64(x / fma(y, y, y));
	else
		tmp = Float64(Float64(x / y) / Float64(x + y));
	end
	return tmp
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[x, -2.7e+14], N[(N[(y / x), $MachinePrecision] / N[(x + y), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 3.8e+32], N[(x / N[(y * y + y), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / N[(x + y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.7e14

   1. Initial program 64.9%

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

      1. lift-/.f64 N/A

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

      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. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 86.4

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

   4. Applied rewrites 86.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-/.f64 N/A

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

   6. Applied rewrites 70.8%

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

   7. Taylor expanded in x around inf

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

   9. Applied rewrites 80.7%

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

   if -2.7e14 < x < 3.8000000000000003e32

   1. Initial program 67.8%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 72.0%

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

   if 3.8000000000000003e32 < x

   1. Initial program 41.0%

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

      1. lift-/.f64 N/A

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

      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. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 74.7

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

   4. Applied rewrites 74.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-/.f64 N/A

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

   6. Applied rewrites 56.7%

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

   7. Taylor expanded in y around inf

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

   9. Applied rewrites 30.7%

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

Alternative 10: 80.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -2.7 \cdot 10^{+14}:\\ \;\;\;\;\frac{\frac{y}{x}}{x}\\ \mathbf{elif}\;x \leq 3.8 \cdot 10^{+32}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(y, y, y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{x + y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -2.7e+14)
   (/ (/ y x) x)
   (if (<= x 3.8e+32) (/ x (fma y y y)) (/ (/ x y) (+ x y)))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -2.7e+14) {
		tmp = (y / x) / x;
	} else if (x <= 3.8e+32) {
		tmp = x / fma(y, y, y);
	} else {
		tmp = (x / y) / (x + y);
	}
	return tmp;
}

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -2.7e+14)
		tmp = Float64(Float64(y / x) / x);
	elseif (x <= 3.8e+32)
		tmp = Float64(x / fma(y, y, y));
	else
		tmp = Float64(Float64(x / y) / Float64(x + y));
	end
	return tmp
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[x, -2.7e+14], N[(N[(y / x), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[x, 3.8e+32], N[(x / N[(y * y + y), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / N[(x + y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.7e14

   1. Initial program 64.9%

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

      1. lift-/.f64 N/A

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

      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. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 86.4

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

   4. Applied rewrites 86.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. frac-2neg N/A

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

      5. lift-neg.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. associate-*r/ N/A

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

      9. lower-/.f64 N/A

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

   6. Applied rewrites 99.9%

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

   7. Taylor expanded in x around inf

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

   9. Applied rewrites 73.6%

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

   11. Applied rewrites 80.3%

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

   if -2.7e14 < x < 3.8000000000000003e32

   1. Initial program 67.8%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 72.0%

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

   if 3.8000000000000003e32 < x

   1. Initial program 41.0%

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

      1. lift-/.f64 N/A

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

      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. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 74.7

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

   4. Applied rewrites 74.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-/.f64 N/A

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

   6. Applied rewrites 56.7%

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

   7. Taylor expanded in y around inf

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

   9. Applied rewrites 30.7%

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

Alternative 11: 69.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -2.7 \cdot 10^{+14}:\\ \;\;\;\;\frac{y}{x \cdot x}\\ \mathbf{elif}\;x \leq -3.7 \cdot 10^{-193} \lor \neg \left(x \leq 4 \cdot 10^{-217}\right):\\ \;\;\;\;\frac{x}{y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -2.7e+14)
   (/ y (* x x))
   (if (or (<= x -3.7e-193) (not (<= x 4e-217))) (/ x (* y y)) (/ x y))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -2.7e+14) {
		tmp = y / (x * x);
	} else if ((x <= -3.7e-193) || !(x <= 4e-217)) {
		tmp = x / (y * y);
	} else {
		tmp = x / y;
	}
	return tmp;
}

Fortran

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

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

real(8) function code(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-2.7d+14)) then
        tmp = y / (x * x)
    else if ((x <= (-3.7d-193)) .or. (.not. (x <= 4d-217))) then
        tmp = x / (y * y)
    else
        tmp = x / y
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (x <= -2.7e+14) {
		tmp = y / (x * x);
	} else if ((x <= -3.7e-193) || !(x <= 4e-217)) {
		tmp = x / (y * y);
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -2.7e+14:
		tmp = y / (x * x)
	elif (x <= -3.7e-193) or not (x <= 4e-217):
		tmp = x / (y * y)
	else:
		tmp = x / y
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -2.7e+14)
		tmp = Float64(y / Float64(x * x));
	elseif ((x <= -3.7e-193) || !(x <= 4e-217))
		tmp = Float64(x / Float64(y * y));
	else
		tmp = Float64(x / y);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -2.7e+14)
		tmp = y / (x * x);
	elseif ((x <= -3.7e-193) || ~((x <= 4e-217)))
		tmp = x / (y * y);
	else
		tmp = x / y;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[x, -2.7e+14], N[(y / N[(x * x), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[x, -3.7e-193], N[Not[LessEqual[x, 4e-217]], $MachinePrecision]], N[(x / N[(y * y), $MachinePrecision]), $MachinePrecision], N[(x / y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.7e14

   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 73.6%

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

   if -2.7e14 < x < -3.7000000000000002e-193 or 4.00000000000000033e-217 < x

   1. Initial program 66.8%

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

   3. Taylor expanded in y around inf

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

   5. Applied rewrites 43.7%

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

   if -3.7000000000000002e-193 < x < 4.00000000000000033e-217

   1. Initial program 43.0%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 93.0%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 85.2%

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

4. Final simplification 58.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.7 \cdot 10^{+14}:\\ \;\;\;\;\frac{y}{x \cdot x}\\ \mathbf{elif}\;x \leq -3.7 \cdot 10^{-193} \lor \neg \left(x \leq 4 \cdot 10^{-217}\right):\\ \;\;\;\;\frac{x}{y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 80.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -2.7 \cdot 10^{+14}:\\ \;\;\;\;\frac{\frac{y}{x}}{x}\\ \mathbf{elif}\;x \leq 1.95 \cdot 10^{+43}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(y, y, y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -2.7e+14)
   (/ (/ y x) x)
   (if (<= x 1.95e+43) (/ x (fma y y y)) (/ (/ x y) y))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -2.7e+14) {
		tmp = (y / x) / x;
	} else if (x <= 1.95e+43) {
		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 (x <= -2.7e+14)
		tmp = Float64(Float64(y / x) / x);
	elseif (x <= 1.95e+43)
		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[x, -2.7e+14], N[(N[(y / x), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[x, 1.95e+43], N[(x / N[(y * y + y), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.7e14

   1. Initial program 64.9%

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

      1. lift-/.f64 N/A

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

      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. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 86.4

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

   4. Applied rewrites 86.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. frac-2neg N/A

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

      5. lift-neg.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. associate-*r/ N/A

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

      9. lower-/.f64 N/A

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

   6. Applied rewrites 99.9%

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

   7. Taylor expanded in x around inf

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

   9. Applied rewrites 73.6%

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

   11. Applied rewrites 80.3%

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

   if -2.7e14 < x < 1.95e43

   1. Initial program 68.0%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 71.5%

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

   if 1.95e43 < x

   1. Initial program 39.7%

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

   3. Taylor expanded in y around inf

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

   5. Applied rewrites 27.7%

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

   7. Applied rewrites 30.7%

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

Alternative 13: 78.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -9 \cdot 10^{-188}:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(x, x, x\right)}\\ \mathbf{elif}\;x \leq 1.95 \cdot 10^{+43}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(y, y, y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -9e-188)
   (/ y (fma x x x))
   (if (<= x 1.95e+43) (/ x (fma y y y)) (/ (/ x y) y))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -9e-188) {
		tmp = y / fma(x, x, x);
	} else if (x <= 1.95e+43) {
		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 (x <= -9e-188)
		tmp = Float64(y / fma(x, x, x));
	elseif (x <= 1.95e+43)
		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[x, -9e-188], N[(y / N[(x * x + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.95e+43], N[(x / N[(y * y + y), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -8.99999999999999986e-188

   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 y around 0

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

   5. Applied rewrites 58.9%

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

   if -8.99999999999999986e-188 < x < 1.95e43

   1. Initial program 64.2%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 72.4%

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

   if 1.95e43 < x

   1. Initial program 39.7%

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

   3. Taylor expanded in y around inf

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

   5. Applied rewrites 27.7%

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

   7. Applied rewrites 30.7%

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

Alternative 14: 80.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -9 \cdot 10^{-188}:\\ \;\;\;\;\frac{\frac{y}{x - -1}}{x + y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y - -1}}{x + y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -9e-188) (/ (/ y (- x -1.0)) (+ x y)) (/ (/ x (- y -1.0)) (+ x y))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -9e-188) {
		tmp = (y / (x - -1.0)) / (x + y);
	} else {
		tmp = (x / (y - -1.0)) / (x + y);
	}
	return tmp;
}

Fortran

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

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

real(8) function code(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-9d-188)) then
        tmp = (y / (x - (-1.0d0))) / (x + y)
    else
        tmp = (x / (y - (-1.0d0))) / (x + y)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (x <= -9e-188) {
		tmp = (y / (x - -1.0)) / (x + y);
	} else {
		tmp = (x / (y - -1.0)) / (x + y);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -9e-188:
		tmp = (y / (x - -1.0)) / (x + y)
	else:
		tmp = (x / (y - -1.0)) / (x + y)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -9e-188)
		tmp = Float64(Float64(y / Float64(x - -1.0)) / Float64(x + y));
	else
		tmp = Float64(Float64(x / Float64(y - -1.0)) / Float64(x + y));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -9e-188)
		tmp = (y / (x - -1.0)) / (x + y);
	else
		tmp = (x / (y - -1.0)) / (x + y);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[x, -9e-188], N[(N[(y / N[(x - -1.0), $MachinePrecision]), $MachinePrecision] / N[(x + y), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(y - -1.0), $MachinePrecision]), $MachinePrecision] / N[(x + y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -8.99999999999999986e-188

   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. 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. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 91.3

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

   4. Applied rewrites 91.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-/.f64 N/A

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

   6. Applied rewrites 74.8%

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

   7. Taylor expanded in y around 0

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

   9. Applied rewrites 63.5%

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

   if -8.99999999999999986e-188 < x

   1. Initial program 56.8%

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

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 92.1

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

   4. Applied rewrites 92.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-/.f64 N/A

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

   6. Applied rewrites 63.3%

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

   7. Taylor expanded in x around 0

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

   9. Applied rewrites 60.0%

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

Alternative 15: 80.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -2.75 \cdot 10^{+14}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y - -1}}{x + y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -2.75e+14) (/ (/ y x) (+ x y)) (/ (/ x (- y -1.0)) (+ x y))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -2.75e+14) {
		tmp = (y / x) / (x + y);
	} else {
		tmp = (x / (y - -1.0)) / (x + y);
	}
	return tmp;
}

Fortran

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

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

real(8) function code(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-2.75d+14)) then
        tmp = (y / x) / (x + y)
    else
        tmp = (x / (y - (-1.0d0))) / (x + y)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (x <= -2.75e+14) {
		tmp = (y / x) / (x + y);
	} else {
		tmp = (x / (y - -1.0)) / (x + y);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -2.75e+14:
		tmp = (y / x) / (x + y)
	else:
		tmp = (x / (y - -1.0)) / (x + y)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -2.75e+14)
		tmp = Float64(Float64(y / x) / Float64(x + y));
	else
		tmp = Float64(Float64(x / Float64(y - -1.0)) / Float64(x + y));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -2.75e+14)
		tmp = (y / x) / (x + y);
	else
		tmp = (x / (y - -1.0)) / (x + y);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[x, -2.75e+14], N[(N[(y / x), $MachinePrecision] / N[(x + y), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(y - -1.0), $MachinePrecision]), $MachinePrecision] / N[(x + y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2.75e14

   1. Initial program 64.9%

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

      1. lift-/.f64 N/A

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

      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. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 86.4

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

   4. Applied rewrites 86.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-/.f64 N/A

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

   6. Applied rewrites 70.8%

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

   7. Taylor expanded in x around inf

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

   9. Applied rewrites 80.7%

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

   if -2.75e14 < x

   1. Initial program 61.1%

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

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 93.6

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

   4. Applied rewrites 93.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-/.f64 N/A

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

   6. Applied rewrites 66.8%

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

   7. Taylor expanded in x around 0

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

   9. Applied rewrites 61.7%

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

Alternative 16: 77.2% accurate, 1.6× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -9 \cdot 10^{-188}:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(x, x, x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(y, y, y\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -9e-188) (/ y (fma x x x)) (/ x (fma y y y))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -9e-188) {
		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 (x <= -9e-188)
		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[x, -9e-188], 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 x < -8.99999999999999986e-188

   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 y around 0

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

   5. Applied rewrites 58.9%

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

   if -8.99999999999999986e-188 < x

   1. Initial program 56.8%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 58.9%

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

Alternative 17: 76.9% accurate, 1.6× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -2.7 \cdot 10^{+14}:\\ \;\;\;\;\frac{y}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(y, y, y\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -2.7e+14) (/ y (* x x)) (/ x (fma y y y))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -2.7e+14) {
		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 <= -2.7e+14)
		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, -2.7e+14], 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 < -2.7e14

   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 73.6%

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

   if -2.7e14 < x

   1. Initial program 61.1%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 60.8%

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

Alternative 18: 47.5% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if y < 1

   1. Initial program 63.4%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   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 44.1%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 26.6%

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

   if 1 < y

   1. Initial program 58.1%

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

   3. Taylor expanded in y around inf

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

   5. Applied rewrites 67.9%

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

Alternative 19: 26.3% 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 62.0%

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

3. Taylor expanded in x around 0

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

5. Applied rewrites 50.3%

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

6. Taylor expanded in y around 0

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

8. Applied rewrites 25.3%

   \[\leadsto \frac{x}{y} \]
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 2025026 
(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))))