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

Percentage Accurate: 69.2% → 99.8%

Time: 10.8s

Alternatives: 20

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 69.2% 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

real(8) function code(x, y)
    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

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

FPCore

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

C

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

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = ((x / (x + y)) * (y / ((x + y) - (-1.0d0)))) / (x + y)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Derivation

1. Initial program 73.3%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. times-frac N/A

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

   5. *-commutative N/A

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

   6. lift-*.f64 N/A

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

   7. associate-/r* N/A

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

   8. associate-*r/ N/A

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

   9. lower-/.f64 N/A

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

4. Applied rewrites 99.9%

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

5. Final simplification 99.9%

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

Alternative 2: 88.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{+162}:\\ \;\;\;\;\frac{1}{x + y} \cdot \frac{y}{x + y}\\ \mathbf{elif}\;x \leq -9.5 \cdot 10^{+39}:\\ \;\;\;\;\frac{1 \cdot y}{\left(x + y\right) \cdot \left(x + y\right)}\\ \mathbf{elif}\;x \leq -1.35 \cdot 10^{-141}:\\ \;\;\;\;\frac{x \cdot y}{\left(\left(\left(x + y\right) - -1\right) \cdot \left(x + y\right)\right) \cdot \left(x + y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{1 + y}}{x + y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -5e+162)
   (* (/ 1.0 (+ x y)) (/ y (+ x y)))
   (if (<= x -9.5e+39)
     (/ (* 1.0 y) (* (+ x y) (+ x y)))
     (if (<= x -1.35e-141)
       (/ (* x y) (* (* (- (+ x y) -1.0) (+ x y)) (+ x y)))
       (/ (/ x (+ 1.0 y)) (+ x y))))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -5e+162) {
		tmp = (1.0 / (x + y)) * (y / (x + y));
	} else if (x <= -9.5e+39) {
		tmp = (1.0 * y) / ((x + y) * (x + y));
	} else if (x <= -1.35e-141) {
		tmp = (x * y) / ((((x + y) - -1.0) * (x + y)) * (x + y));
	} else {
		tmp = (x / (1.0 + y)) / (x + y);
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-5d+162)) then
        tmp = (1.0d0 / (x + y)) * (y / (x + y))
    else if (x <= (-9.5d+39)) then
        tmp = (1.0d0 * y) / ((x + y) * (x + y))
    else if (x <= (-1.35d-141)) then
        tmp = (x * y) / ((((x + y) - (-1.0d0)) * (x + y)) * (x + y))
    else
        tmp = (x / (1.0d0 + 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 <= -5e+162) {
		tmp = (1.0 / (x + y)) * (y / (x + y));
	} else if (x <= -9.5e+39) {
		tmp = (1.0 * y) / ((x + y) * (x + y));
	} else if (x <= -1.35e-141) {
		tmp = (x * y) / ((((x + y) - -1.0) * (x + y)) * (x + y));
	} else {
		tmp = (x / (1.0 + y)) / (x + y);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -5e+162:
		tmp = (1.0 / (x + y)) * (y / (x + y))
	elif x <= -9.5e+39:
		tmp = (1.0 * y) / ((x + y) * (x + y))
	elif x <= -1.35e-141:
		tmp = (x * y) / ((((x + y) - -1.0) * (x + y)) * (x + y))
	else:
		tmp = (x / (1.0 + y)) / (x + y)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -5e+162)
		tmp = Float64(Float64(1.0 / Float64(x + y)) * Float64(y / Float64(x + y)));
	elseif (x <= -9.5e+39)
		tmp = Float64(Float64(1.0 * y) / Float64(Float64(x + y) * Float64(x + y)));
	elseif (x <= -1.35e-141)
		tmp = Float64(Float64(x * y) / Float64(Float64(Float64(Float64(x + y) - -1.0) * Float64(x + y)) * Float64(x + y)));
	else
		tmp = Float64(Float64(x / Float64(1.0 + 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 <= -5e+162)
		tmp = (1.0 / (x + y)) * (y / (x + y));
	elseif (x <= -9.5e+39)
		tmp = (1.0 * y) / ((x + y) * (x + y));
	elseif (x <= -1.35e-141)
		tmp = (x * y) / ((((x + y) - -1.0) * (x + y)) * (x + y));
	else
		tmp = (x / (1.0 + 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, -5e+162], N[(N[(1.0 / N[(x + y), $MachinePrecision]), $MachinePrecision] * N[(y / N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -9.5e+39], N[(N[(1.0 * y), $MachinePrecision] / N[(N[(x + y), $MachinePrecision] * N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -1.35e-141], N[(N[(x * y), $MachinePrecision] / N[(N[(N[(N[(x + y), $MachinePrecision] - -1.0), $MachinePrecision] * N[(x + y), $MachinePrecision]), $MachinePrecision] * N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(1.0 + y), $MachinePrecision]), $MachinePrecision] / N[(x + y), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < -4.9999999999999997e162

   1. Initial program 52.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/r* N/A

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

      8. associate-*r/ N/A

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

      9. lower-/.f64 N/A

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

   4. Applied rewrites 99.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. frac-times N/A

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

      6. *-commutative N/A

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

      7. times-frac N/A

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

      8. lift-/.f64 N/A

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

      9. associate-*l/ N/A

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

      10. associate-/r* N/A

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

      11. lift-*.f64 N/A

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

      12. lift-/.f64 N/A

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

      13. lower-*.f64 68.0

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

   6. Applied rewrites 99.9%

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

   7. Taylor expanded in x around inf

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

   9. Applied rewrites 87.8%

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

   if -4.9999999999999997e162 < x < -9.50000000000000011e39

   1. Initial program 74.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/r* N/A

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

      8. associate-*r/ N/A

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

      9. lower-/.f64 N/A

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

   4. Applied rewrites 99.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. frac-times N/A

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

      6. *-commutative N/A

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

      7. times-frac N/A

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

      8. lift-/.f64 N/A

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

      9. associate-*l/ N/A

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

      10. associate-/r* N/A

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

      11. lift-*.f64 N/A

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

      12. lift-/.f64 N/A

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

      13. lower-*.f64 95.8

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

   6. Applied rewrites 99.8%

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

   7. Taylor expanded in x around inf

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

   9. Applied rewrites 83.0%

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

       1. lift-*.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. frac-2neg N/A

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

       4. lift-+.f64 N/A

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

       5. +-commutative N/A

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

       6. lift-+.f64 N/A

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

       7. lift-/.f64 N/A

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

       8. lift-+.f64 N/A

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

       9. +-commutative N/A

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

       10. lift-+.f64 N/A

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

       11. frac-times N/A

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

       12. lower-/.f64 N/A

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

       13. lower-*.f64 N/A

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

       14. lower-neg.f64 N/A

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

       15. lower-*.f64 N/A

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

       16. lower-neg.f64 95.0

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

       17. lift-+.f64 N/A

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

       18. +-commutative N/A

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

       19. lift-+.f64 95.0

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

       20. lift-+.f64 N/A

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

       21. +-commutative N/A

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

       22. lift-+.f64 95.0

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

   11. Applied rewrites 95.0%

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

   if -9.50000000000000011e39 < x < -1.3500000000000001e-141

   1. Initial program 88.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 88.5

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 88.5

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

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 88.5

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

      16. lift-+.f64 N/A

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

      17. +-commutative N/A

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

      18. lower-+.f64 88.5

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

   4. Applied rewrites 88.5%

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

   if -1.3500000000000001e-141 < x

   1. Initial program 73.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/r* N/A

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

      8. associate-*r/ N/A

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

      9. lower-/.f64 N/A

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 61.7

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

   7. Applied rewrites 61.7%

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

4. Final simplification 71.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{+162}:\\ \;\;\;\;\frac{1}{x + y} \cdot \frac{y}{x + y}\\ \mathbf{elif}\;x \leq -9.5 \cdot 10^{+39}:\\ \;\;\;\;\frac{1 \cdot y}{\left(x + y\right) \cdot \left(x + y\right)}\\ \mathbf{elif}\;x \leq -1.35 \cdot 10^{-141}:\\ \;\;\;\;\frac{x \cdot y}{\left(\left(\left(x + y\right) - -1\right) \cdot \left(x + y\right)\right) \cdot \left(x + y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{1 + y}}{x + y}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 88.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_0 := \left(x + y\right) \cdot \left(x + y\right)\\ \mathbf{if}\;x \leq -5 \cdot 10^{+162}:\\ \;\;\;\;\frac{1}{x + y} \cdot \frac{y}{x + y}\\ \mathbf{elif}\;x \leq -9.5 \cdot 10^{+39}:\\ \;\;\;\;\frac{1 \cdot y}{t\_0}\\ \mathbf{elif}\;x \leq -1.35 \cdot 10^{-141}:\\ \;\;\;\;\frac{x \cdot y}{t\_0 \cdot \left(\left(x + y\right) - -1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{1 + y}}{x + y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (+ x y) (+ x y))))
   (if (<= x -5e+162)
     (* (/ 1.0 (+ x y)) (/ y (+ x y)))
     (if (<= x -9.5e+39)
       (/ (* 1.0 y) t_0)
       (if (<= x -1.35e-141)
         (/ (* x y) (* t_0 (- (+ x y) -1.0)))
         (/ (/ x (+ 1.0 y)) (+ x y)))))))

C

assert(x < y);
double code(double x, double y) {
	double t_0 = (x + y) * (x + y);
	double tmp;
	if (x <= -5e+162) {
		tmp = (1.0 / (x + y)) * (y / (x + y));
	} else if (x <= -9.5e+39) {
		tmp = (1.0 * y) / t_0;
	} else if (x <= -1.35e-141) {
		tmp = (x * y) / (t_0 * ((x + y) - -1.0));
	} else {
		tmp = (x / (1.0 + y)) / (x + y);
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (x + y) * (x + y)
    if (x <= (-5d+162)) then
        tmp = (1.0d0 / (x + y)) * (y / (x + y))
    else if (x <= (-9.5d+39)) then
        tmp = (1.0d0 * y) / t_0
    else if (x <= (-1.35d-141)) then
        tmp = (x * y) / (t_0 * ((x + y) - (-1.0d0)))
    else
        tmp = (x / (1.0d0 + y)) / (x + y)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double t_0 = (x + y) * (x + y);
	double tmp;
	if (x <= -5e+162) {
		tmp = (1.0 / (x + y)) * (y / (x + y));
	} else if (x <= -9.5e+39) {
		tmp = (1.0 * y) / t_0;
	} else if (x <= -1.35e-141) {
		tmp = (x * y) / (t_0 * ((x + y) - -1.0));
	} else {
		tmp = (x / (1.0 + y)) / (x + y);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	t_0 = (x + y) * (x + y)
	tmp = 0
	if x <= -5e+162:
		tmp = (1.0 / (x + y)) * (y / (x + y))
	elif x <= -9.5e+39:
		tmp = (1.0 * y) / t_0
	elif x <= -1.35e-141:
		tmp = (x * y) / (t_0 * ((x + y) - -1.0))
	else:
		tmp = (x / (1.0 + y)) / (x + y)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	t_0 = Float64(Float64(x + y) * Float64(x + y))
	tmp = 0.0
	if (x <= -5e+162)
		tmp = Float64(Float64(1.0 / Float64(x + y)) * Float64(y / Float64(x + y)));
	elseif (x <= -9.5e+39)
		tmp = Float64(Float64(1.0 * y) / t_0);
	elseif (x <= -1.35e-141)
		tmp = Float64(Float64(x * y) / Float64(t_0 * Float64(Float64(x + y) - -1.0)));
	else
		tmp = Float64(Float64(x / Float64(1.0 + y)) / Float64(x + y));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	t_0 = (x + y) * (x + y);
	tmp = 0.0;
	if (x <= -5e+162)
		tmp = (1.0 / (x + y)) * (y / (x + y));
	elseif (x <= -9.5e+39)
		tmp = (1.0 * y) / t_0;
	elseif (x <= -1.35e-141)
		tmp = (x * y) / (t_0 * ((x + y) - -1.0));
	else
		tmp = (x / (1.0 + 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_] := Block[{t$95$0 = N[(N[(x + y), $MachinePrecision] * N[(x + y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -5e+162], N[(N[(1.0 / N[(x + y), $MachinePrecision]), $MachinePrecision] * N[(y / N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -9.5e+39], N[(N[(1.0 * y), $MachinePrecision] / t$95$0), $MachinePrecision], If[LessEqual[x, -1.35e-141], N[(N[(x * y), $MachinePrecision] / N[(t$95$0 * N[(N[(x + y), $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(1.0 + y), $MachinePrecision]), $MachinePrecision] / N[(x + y), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -4.9999999999999997e162

   1. Initial program 52.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/r* N/A

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

      8. associate-*r/ N/A

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

      9. lower-/.f64 N/A

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

   4. Applied rewrites 99.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. frac-times N/A

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

      6. *-commutative N/A

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

      7. times-frac N/A

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

      8. lift-/.f64 N/A

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

      9. associate-*l/ N/A

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

      10. associate-/r* N/A

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

      11. lift-*.f64 N/A

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

      12. lift-/.f64 N/A

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

      13. lower-*.f64 68.0

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

   6. Applied rewrites 99.9%

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

   7. Taylor expanded in x around inf

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

   9. Applied rewrites 87.8%

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

   if -4.9999999999999997e162 < x < -9.50000000000000011e39

   1. Initial program 74.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/r* N/A

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

      8. associate-*r/ N/A

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

      9. lower-/.f64 N/A

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

   4. Applied rewrites 99.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. frac-times N/A

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

      6. *-commutative N/A

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

      7. times-frac N/A

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

      8. lift-/.f64 N/A

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

      9. associate-*l/ N/A

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

      10. associate-/r* N/A

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

      11. lift-*.f64 N/A

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

      12. lift-/.f64 N/A

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

      13. lower-*.f64 95.8

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

   6. Applied rewrites 99.8%

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

   7. Taylor expanded in x around inf

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

   9. Applied rewrites 83.0%

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

       1. lift-*.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. frac-2neg N/A

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

       4. lift-+.f64 N/A

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

       5. +-commutative N/A

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

       6. lift-+.f64 N/A

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

       7. lift-/.f64 N/A

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

       8. lift-+.f64 N/A

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

       9. +-commutative N/A

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

       10. lift-+.f64 N/A

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

       11. frac-times N/A

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

       12. lower-/.f64 N/A

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

       13. lower-*.f64 N/A

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

       14. lower-neg.f64 N/A

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

       15. lower-*.f64 N/A

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

       16. lower-neg.f64 95.0

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

       17. lift-+.f64 N/A

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

       18. +-commutative N/A

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

       19. lift-+.f64 95.0

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

       20. lift-+.f64 N/A

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

       21. +-commutative N/A

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

       22. lift-+.f64 95.0

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

   11. Applied rewrites 95.0%

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

   if -9.50000000000000011e39 < x < -1.3500000000000001e-141

   1. Initial program 88.5%

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

   if -1.3500000000000001e-141 < x

   1. Initial program 73.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/r* N/A

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

      8. associate-*r/ N/A

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

      9. lower-/.f64 N/A

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 61.7

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

   7. Applied rewrites 61.7%

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

4. Final simplification 71.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{+162}:\\ \;\;\;\;\frac{1}{x + y} \cdot \frac{y}{x + y}\\ \mathbf{elif}\;x \leq -9.5 \cdot 10^{+39}:\\ \;\;\;\;\frac{1 \cdot y}{\left(x + y\right) \cdot \left(x + y\right)}\\ \mathbf{elif}\;x \leq -1.35 \cdot 10^{-141}:\\ \;\;\;\;\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) - -1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{1 + y}}{x + y}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 95.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{+162}:\\ \;\;\;\;\frac{\frac{y}{x + y}}{\mathsf{fma}\left(2 + \frac{1}{x}, y, 1 + x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\left(\left(x + y\right) - -1\right) \cdot \left(x + y\right)} \cdot \frac{x}{x + y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -5e+162)
   (/ (/ y (+ x y)) (fma (+ 2.0 (/ 1.0 x)) y (+ 1.0 x)))
   (* (/ y (* (- (+ x y) -1.0) (+ x y))) (/ x (+ x y)))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -5e+162) {
		tmp = (y / (x + y)) / fma((2.0 + (1.0 / x)), y, (1.0 + x));
	} else {
		tmp = (y / (((x + y) - -1.0) * (x + y))) * (x / (x + y));
	}
	return tmp;
}

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -5e+162)
		tmp = Float64(Float64(y / Float64(x + y)) / fma(Float64(2.0 + Float64(1.0 / x)), y, Float64(1.0 + x)));
	else
		tmp = Float64(Float64(y / Float64(Float64(Float64(x + y) - -1.0) * Float64(x + y))) * Float64(x / 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, -5e+162], N[(N[(y / N[(x + y), $MachinePrecision]), $MachinePrecision] / N[(N[(2.0 + N[(1.0 / x), $MachinePrecision]), $MachinePrecision] * y + N[(1.0 + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(y / N[(N[(N[(x + y), $MachinePrecision] - -1.0), $MachinePrecision] * N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(x / N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -4.9999999999999997e162

   1. Initial program 52.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/r* N/A

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

      8. associate-*r/ N/A

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

      9. lower-/.f64 N/A

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

   4. Applied rewrites 99.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. frac-times N/A

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

      6. *-commutative N/A

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

      7. times-frac N/A

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

      8. lift-/.f64 N/A

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

      9. associate-*l/ N/A

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

      10. associate-/r* N/A

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

      11. lift-*.f64 N/A

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

      12. lift-/.f64 N/A

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

      13. lower-*.f64 68.0

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

   6. Applied rewrites 99.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. frac-times N/A

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. *-commutative N/A

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

      10. frac-times N/A

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

      11. lift-/.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. lift-/.f64 N/A

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

      14. lift-+.f64 N/A

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

      15. lift-+.f64 N/A

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

      16. +-commutative N/A

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

      17. +-commutative N/A

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

      18. associate-/r* N/A

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

      19. clear-num N/A

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

   8. Applied rewrites 99.7%

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

   9. Taylor expanded in y around 0

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

       1. +-commutative N/A

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

       2. +-commutative N/A

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

       3. associate-+l+ N/A

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

       4. *-commutative N/A

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

       5. +-commutative N/A

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

       6. lower-fma.f64 N/A

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

       7. +-commutative N/A

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

       8. lower-+.f64 N/A

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

       9. lower-/.f64 N/A

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

       10. lower-+.f64 90.0

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

   11. Applied rewrites 90.0%

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

   if -4.9999999999999997e162 < x

   1. Initial program 75.9%

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

      1. lift-/.f64 N/A

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

      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. *-commutative N/A

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

      8. times-frac N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 N/A

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

      16. lift-+.f64 N/A

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

      17. +-commutative N/A

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

      18. lower-+.f64 N/A

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

      19. lift-+.f64 N/A

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

      20. +-commutative N/A

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

      21. lower-+.f64 N/A

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

      22. lower-/.f64 95.9

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

      23. lift-+.f64 N/A

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

      24. +-commutative N/A

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

   4. Applied rewrites 95.9%

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

4. Final simplification 95.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{+162}:\\ \;\;\;\;\frac{\frac{y}{x + y}}{\mathsf{fma}\left(2 + \frac{1}{x}, y, 1 + x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\left(\left(x + y\right) - -1\right) \cdot \left(x + y\right)} \cdot \frac{x}{x + y}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 95.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_0 := \left(x + y\right) - -1\\ \mathbf{if}\;y \leq -3.7 \cdot 10^{-208}:\\ \;\;\;\;\frac{1 \cdot \frac{y}{t\_0}}{x + y}\\ \mathbf{elif}\;y \leq 0.41:\\ \;\;\;\;\frac{x}{\left(1 + x\right) \cdot \left(x + y\right)} \cdot \frac{y}{x + y}\\ \mathbf{elif}\;y \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\frac{x}{t\_0 \cdot \left(x + y\right)} \cdot 1\\ \mathbf{else}:\\ \;\;\;\;\frac{1 \cdot \frac{x}{x + y}}{x + y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (+ x y) -1.0)))
   (if (<= y -3.7e-208)
     (/ (* 1.0 (/ y t_0)) (+ x y))
     (if (<= y 0.41)
       (* (/ x (* (+ 1.0 x) (+ x y))) (/ y (+ x y)))
       (if (<= y 1.35e+154)
         (* (/ x (* t_0 (+ x y))) 1.0)
         (/ (* 1.0 (/ x (+ x y))) (+ x y)))))))

C

assert(x < y);
double code(double x, double y) {
	double t_0 = (x + y) - -1.0;
	double tmp;
	if (y <= -3.7e-208) {
		tmp = (1.0 * (y / t_0)) / (x + y);
	} else if (y <= 0.41) {
		tmp = (x / ((1.0 + x) * (x + y))) * (y / (x + y));
	} else if (y <= 1.35e+154) {
		tmp = (x / (t_0 * (x + y))) * 1.0;
	} else {
		tmp = (1.0 * (x / (x + y))) / (x + y);
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (x + y) - (-1.0d0)
    if (y <= (-3.7d-208)) then
        tmp = (1.0d0 * (y / t_0)) / (x + y)
    else if (y <= 0.41d0) then
        tmp = (x / ((1.0d0 + x) * (x + y))) * (y / (x + y))
    else if (y <= 1.35d+154) then
        tmp = (x / (t_0 * (x + y))) * 1.0d0
    else
        tmp = (1.0d0 * (x / (x + y))) / (x + y)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double t_0 = (x + y) - -1.0;
	double tmp;
	if (y <= -3.7e-208) {
		tmp = (1.0 * (y / t_0)) / (x + y);
	} else if (y <= 0.41) {
		tmp = (x / ((1.0 + x) * (x + y))) * (y / (x + y));
	} else if (y <= 1.35e+154) {
		tmp = (x / (t_0 * (x + y))) * 1.0;
	} else {
		tmp = (1.0 * (x / (x + y))) / (x + y);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	t_0 = (x + y) - -1.0
	tmp = 0
	if y <= -3.7e-208:
		tmp = (1.0 * (y / t_0)) / (x + y)
	elif y <= 0.41:
		tmp = (x / ((1.0 + x) * (x + y))) * (y / (x + y))
	elif y <= 1.35e+154:
		tmp = (x / (t_0 * (x + y))) * 1.0
	else:
		tmp = (1.0 * (x / (x + y))) / (x + y)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	t_0 = Float64(Float64(x + y) - -1.0)
	tmp = 0.0
	if (y <= -3.7e-208)
		tmp = Float64(Float64(1.0 * Float64(y / t_0)) / Float64(x + y));
	elseif (y <= 0.41)
		tmp = Float64(Float64(x / Float64(Float64(1.0 + x) * Float64(x + y))) * Float64(y / Float64(x + y)));
	elseif (y <= 1.35e+154)
		tmp = Float64(Float64(x / Float64(t_0 * Float64(x + y))) * 1.0);
	else
		tmp = Float64(Float64(1.0 * Float64(x / Float64(x + y))) / Float64(x + y));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	t_0 = (x + y) - -1.0;
	tmp = 0.0;
	if (y <= -3.7e-208)
		tmp = (1.0 * (y / t_0)) / (x + y);
	elseif (y <= 0.41)
		tmp = (x / ((1.0 + x) * (x + y))) * (y / (x + y));
	elseif (y <= 1.35e+154)
		tmp = (x / (t_0 * (x + y))) * 1.0;
	else
		tmp = (1.0 * (x / (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_] := Block[{t$95$0 = N[(N[(x + y), $MachinePrecision] - -1.0), $MachinePrecision]}, If[LessEqual[y, -3.7e-208], N[(N[(1.0 * N[(y / t$95$0), $MachinePrecision]), $MachinePrecision] / N[(x + y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 0.41], N[(N[(x / N[(N[(1.0 + x), $MachinePrecision] * N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(y / N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.35e+154], N[(N[(x / N[(t$95$0 * N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision], N[(N[(1.0 * N[(x / N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x + y), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if y < -3.7000000000000002e-208

   1. Initial program 75.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/r* N/A

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

      8. associate-*r/ N/A

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

      9. lower-/.f64 N/A

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in y around 0

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

   7. Applied rewrites 43.0%

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

   if -3.7000000000000002e-208 < y < 0.409999999999999976

   1. Initial program 77.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-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

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

      16. lift-+.f64 N/A

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

      17. +-commutative N/A

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

      18. lower-+.f64 99.9

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

      19. lift-+.f64 N/A

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

      20. +-commutative N/A

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

      21. lower-+.f64 99.9

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

      22. lift-+.f64 N/A

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

      23. +-commutative N/A

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

      24. lower-+.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 99.6

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

   7. Applied rewrites 99.6%

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

   if 0.409999999999999976 < y < 1.35000000000000003e154

   1. Initial program 67.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-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 89.2

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

      16. lift-+.f64 N/A

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

      17. +-commutative N/A

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

      18. lower-+.f64 89.2

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

      19. lift-+.f64 N/A

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

      20. +-commutative N/A

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

      21. lower-+.f64 89.2

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

      22. lift-+.f64 N/A

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

      23. +-commutative N/A

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

      24. lower-+.f64 89.2

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

   4. Applied rewrites 89.2%

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

   5. Taylor expanded in y around inf

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

   7. Applied rewrites 83.1%

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

   if 1.35000000000000003e154 < y

   1. Initial program 58.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/r* N/A

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

      8. associate-*r/ N/A

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

      9. lower-/.f64 N/A

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in y around inf

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

   7. Applied rewrites 87.1%

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

4. Final simplification 70.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.7 \cdot 10^{-208}:\\ \;\;\;\;\frac{1 \cdot \frac{y}{\left(x + y\right) - -1}}{x + y}\\ \mathbf{elif}\;y \leq 0.41:\\ \;\;\;\;\frac{x}{\left(1 + x\right) \cdot \left(x + y\right)} \cdot \frac{y}{x + y}\\ \mathbf{elif}\;y \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\frac{x}{\left(\left(x + y\right) - -1\right) \cdot \left(x + y\right)} \cdot 1\\ \mathbf{else}:\\ \;\;\;\;\frac{1 \cdot \frac{x}{x + y}}{x + y}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 96.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_0 := \frac{x}{x + y}\\ \mathbf{if}\;y \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\frac{y}{\left(\left(x + y\right) - -1\right) \cdot \left(x + y\right)} \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1 \cdot t\_0}{x + y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ x (+ x y))))
   (if (<= y 1.35e+154)
     (* (/ y (* (- (+ x y) -1.0) (+ x y))) t_0)
     (/ (* 1.0 t_0) (+ x y)))))

C

assert(x < y);
double code(double x, double y) {
	double t_0 = x / (x + y);
	double tmp;
	if (y <= 1.35e+154) {
		tmp = (y / (((x + y) - -1.0) * (x + y))) * t_0;
	} else {
		tmp = (1.0 * t_0) / (x + y);
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x / (x + y)
    if (y <= 1.35d+154) then
        tmp = (y / (((x + y) - (-1.0d0)) * (x + y))) * t_0
    else
        tmp = (1.0d0 * t_0) / (x + y)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double t_0 = x / (x + y);
	double tmp;
	if (y <= 1.35e+154) {
		tmp = (y / (((x + y) - -1.0) * (x + y))) * t_0;
	} else {
		tmp = (1.0 * t_0) / (x + y);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	t_0 = x / (x + y)
	tmp = 0
	if y <= 1.35e+154:
		tmp = (y / (((x + y) - -1.0) * (x + y))) * t_0
	else:
		tmp = (1.0 * t_0) / (x + y)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	t_0 = Float64(x / Float64(x + y))
	tmp = 0.0
	if (y <= 1.35e+154)
		tmp = Float64(Float64(y / Float64(Float64(Float64(x + y) - -1.0) * Float64(x + y))) * t_0);
	else
		tmp = Float64(Float64(1.0 * t_0) / Float64(x + y));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	t_0 = x / (x + y);
	tmp = 0.0;
	if (y <= 1.35e+154)
		tmp = (y / (((x + y) - -1.0) * (x + y))) * t_0;
	else
		tmp = (1.0 * t_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_] := Block[{t$95$0 = N[(x / N[(x + y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, 1.35e+154], N[(N[(y / N[(N[(N[(x + y), $MachinePrecision] - -1.0), $MachinePrecision] * N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision], N[(N[(1.0 * t$95$0), $MachinePrecision] / N[(x + y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < 1.35000000000000003e154

   1. Initial program 75.1%

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

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

      8. times-frac N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 N/A

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

      16. lift-+.f64 N/A

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

      17. +-commutative N/A

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

      18. lower-+.f64 N/A

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

      19. lift-+.f64 N/A

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

      20. +-commutative N/A

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

      21. lower-+.f64 N/A

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

      22. lower-/.f64 94.2

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

      23. lift-+.f64 N/A

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

      24. +-commutative N/A

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

   4. Applied rewrites 94.2%

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

   if 1.35000000000000003e154 < y

   1. Initial program 58.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/r* N/A

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

      8. associate-*r/ N/A

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

      9. lower-/.f64 N/A

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in y around inf

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

   7. Applied rewrites 87.1%

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

4. Final simplification 93.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\frac{y}{\left(\left(x + y\right) - -1\right) \cdot \left(x + y\right)} \cdot \frac{x}{x + y}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 \cdot \frac{x}{x + y}}{x + y}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 99.8% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = (y / (x + y)) * ((x / ((x + y) - (-1.0d0))) / (x + y))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 73.3%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. times-frac N/A

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

   5. *-commutative N/A

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

   6. lift-*.f64 N/A

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

   7. associate-/r* N/A

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

   8. associate-*r/ N/A

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

   9. lower-/.f64 N/A

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

4. Applied rewrites 99.9%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. lift-/.f64 N/A

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

   5. frac-times N/A

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

   6. *-commutative N/A

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

   7. times-frac N/A

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

   8. lift-/.f64 N/A

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

   9. associate-*l/ N/A

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

   10. associate-/r* N/A

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

   11. lift-*.f64 N/A

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

   12. lift-/.f64 N/A

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

   13. lower-*.f64 92.7

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

6. Applied rewrites 99.8%

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

7. Final simplification 99.8%

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

Alternative 8: 84.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq -1.5 \cdot 10^{-16}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + y}\\ \mathbf{elif}\;y \leq 1.95 \cdot 10^{-140}:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(x, x, x\right)}\\ \mathbf{elif}\;y \leq 1.12 \cdot 10^{-26}:\\ \;\;\;\;1 \cdot \frac{x}{\left(1 + x\right) \cdot \left(x + y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{1 + y}}{x + y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y -1.5e-16)
   (/ (/ y x) (+ x y))
   (if (<= y 1.95e-140)
     (/ y (fma x x x))
     (if (<= y 1.12e-26)
       (* 1.0 (/ x (* (+ 1.0 x) (+ x y))))
       (/ (/ x (+ 1.0 y)) (+ x y))))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= -1.5e-16) {
		tmp = (y / x) / (x + y);
	} else if (y <= 1.95e-140) {
		tmp = y / fma(x, x, x);
	} else if (y <= 1.12e-26) {
		tmp = 1.0 * (x / ((1.0 + x) * (x + y)));
	} else {
		tmp = (x / (1.0 + y)) / (x + y);
	}
	return tmp;
}

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= -1.5e-16)
		tmp = Float64(Float64(y / x) / Float64(x + y));
	elseif (y <= 1.95e-140)
		tmp = Float64(y / fma(x, x, x));
	elseif (y <= 1.12e-26)
		tmp = Float64(1.0 * Float64(x / Float64(Float64(1.0 + x) * Float64(x + y))));
	else
		tmp = Float64(Float64(x / Float64(1.0 + 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[y, -1.5e-16], N[(N[(y / x), $MachinePrecision] / N[(x + y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.95e-140], N[(y / N[(x * x + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.12e-26], N[(1.0 * N[(x / N[(N[(1.0 + x), $MachinePrecision] * N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(1.0 + y), $MachinePrecision]), $MachinePrecision] / N[(x + y), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if y < -1.49999999999999997e-16

   1. Initial program 70.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/r* N/A

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

      8. associate-*r/ N/A

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

      9. lower-/.f64 N/A

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in x around inf

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

      1. lower-/.f64 28.7

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

   7. Applied rewrites 28.7%

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

   if -1.49999999999999997e-16 < y < 1.9500000000000001e-140

   1. Initial program 80.0%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. distribute-lft-in N/A

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

      4. *-rgt-identity N/A

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

      5. lower-fma.f64 85.2

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

   5. Applied rewrites 85.2%

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

   if 1.9500000000000001e-140 < y < 1.12e-26

   1. Initial program 79.9%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   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 99.8

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

      16. lift-+.f64 N/A

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

      17. +-commutative N/A

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

      18. lower-+.f64 99.8

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

      19. lift-+.f64 N/A

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

      20. +-commutative N/A

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

      21. lower-+.f64 99.8

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

      22. lift-+.f64 N/A

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

      23. +-commutative N/A

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

      24. lower-+.f64 99.8

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 99.8

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

   7. Applied rewrites 99.8%

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

   8. Taylor expanded in y around inf

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

   10. Applied rewrites 60.5%

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

   if 1.12e-26 < y

   1. Initial program 64.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/r* N/A

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

      8. associate-*r/ N/A

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

      9. lower-/.f64 N/A

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 80.0

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

   7. Applied rewrites 80.0%

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

4. Final simplification 63.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.5 \cdot 10^{-16}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + y}\\ \mathbf{elif}\;y \leq 1.95 \cdot 10^{-140}:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(x, x, x\right)}\\ \mathbf{elif}\;y \leq 1.12 \cdot 10^{-26}:\\ \;\;\;\;1 \cdot \frac{x}{\left(1 + x\right) \cdot \left(x + y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{1 + y}}{x + y}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 84.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq -1.5 \cdot 10^{-16}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + y}\\ \mathbf{elif}\;y \leq 1.95 \cdot 10^{-140}:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(x, x, x\right)}\\ \mathbf{elif}\;y \leq 0.34:\\ \;\;\;\;1 \cdot \frac{x}{\left(1 + x\right) \cdot \left(x + y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{x + y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y -1.5e-16)
   (/ (/ y x) (+ x y))
   (if (<= y 1.95e-140)
     (/ y (fma x x x))
     (if (<= y 0.34)
       (* 1.0 (/ x (* (+ 1.0 x) (+ x y))))
       (/ (/ x y) (+ x y))))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= -1.5e-16) {
		tmp = (y / x) / (x + y);
	} else if (y <= 1.95e-140) {
		tmp = y / fma(x, x, x);
	} else if (y <= 0.34) {
		tmp = 1.0 * (x / ((1.0 + x) * (x + y)));
	} else {
		tmp = (x / y) / (x + y);
	}
	return tmp;
}

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= -1.5e-16)
		tmp = Float64(Float64(y / x) / Float64(x + y));
	elseif (y <= 1.95e-140)
		tmp = Float64(y / fma(x, x, x));
	elseif (y <= 0.34)
		tmp = Float64(1.0 * Float64(x / Float64(Float64(1.0 + x) * Float64(x + 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[y, -1.5e-16], N[(N[(y / x), $MachinePrecision] / N[(x + y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.95e-140], N[(y / N[(x * x + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 0.34], N[(1.0 * N[(x / N[(N[(1.0 + x), $MachinePrecision] * N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / N[(x + y), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if y < -1.49999999999999997e-16

   1. Initial program 70.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/r* N/A

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

      8. associate-*r/ N/A

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

      9. lower-/.f64 N/A

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in x around inf

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

      1. lower-/.f64 28.7

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

   7. Applied rewrites 28.7%

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

   if -1.49999999999999997e-16 < y < 1.9500000000000001e-140

   1. Initial program 80.0%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. distribute-lft-in N/A

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

      4. *-rgt-identity N/A

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

      5. lower-fma.f64 85.2

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

   5. Applied rewrites 85.2%

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

   if 1.9500000000000001e-140 < y < 0.340000000000000024

   1. Initial program 80.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-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 99.8

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

      16. lift-+.f64 N/A

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

      17. +-commutative N/A

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

      18. lower-+.f64 99.8

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

      19. lift-+.f64 N/A

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

      20. +-commutative N/A

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

      21. lower-+.f64 99.8

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

      22. lift-+.f64 N/A

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

      23. +-commutative N/A

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

      24. lower-+.f64 99.8

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 99.8

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

   7. Applied rewrites 99.8%

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

   8. Taylor expanded in y around inf

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

   10. Applied rewrites 62.1%

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

   if 0.340000000000000024 < y

   1. Initial program 63.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/r* N/A

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

      8. associate-*r/ N/A

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

      9. lower-/.f64 N/A

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in y around inf

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

      1. lower-/.f64 78.8

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

   7. Applied rewrites 78.8%

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

4. Final simplification 63.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.5 \cdot 10^{-16}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + y}\\ \mathbf{elif}\;y \leq 1.95 \cdot 10^{-140}:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(x, x, x\right)}\\ \mathbf{elif}\;y \leq 0.34:\\ \;\;\;\;1 \cdot \frac{x}{\left(1 + x\right) \cdot \left(x + y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{x + y}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 87.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_0 := \left(x + y\right) - -1\\ \mathbf{if}\;y \leq 1.95 \cdot 10^{-140}:\\ \;\;\;\;\frac{1 \cdot \frac{y}{t\_0}}{x + y}\\ \mathbf{elif}\;y \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\frac{x}{t\_0 \cdot \left(x + y\right)} \cdot 1\\ \mathbf{else}:\\ \;\;\;\;\frac{1 \cdot \frac{x}{x + y}}{x + y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (+ x y) -1.0)))
   (if (<= y 1.95e-140)
     (/ (* 1.0 (/ y t_0)) (+ x y))
     (if (<= y 1.35e+154)
       (* (/ x (* t_0 (+ x y))) 1.0)
       (/ (* 1.0 (/ x (+ x y))) (+ x y))))))

C

assert(x < y);
double code(double x, double y) {
	double t_0 = (x + y) - -1.0;
	double tmp;
	if (y <= 1.95e-140) {
		tmp = (1.0 * (y / t_0)) / (x + y);
	} else if (y <= 1.35e+154) {
		tmp = (x / (t_0 * (x + y))) * 1.0;
	} else {
		tmp = (1.0 * (x / (x + y))) / (x + y);
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (x + y) - (-1.0d0)
    if (y <= 1.95d-140) then
        tmp = (1.0d0 * (y / t_0)) / (x + y)
    else if (y <= 1.35d+154) then
        tmp = (x / (t_0 * (x + y))) * 1.0d0
    else
        tmp = (1.0d0 * (x / (x + y))) / (x + y)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double t_0 = (x + y) - -1.0;
	double tmp;
	if (y <= 1.95e-140) {
		tmp = (1.0 * (y / t_0)) / (x + y);
	} else if (y <= 1.35e+154) {
		tmp = (x / (t_0 * (x + y))) * 1.0;
	} else {
		tmp = (1.0 * (x / (x + y))) / (x + y);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	t_0 = (x + y) - -1.0
	tmp = 0
	if y <= 1.95e-140:
		tmp = (1.0 * (y / t_0)) / (x + y)
	elif y <= 1.35e+154:
		tmp = (x / (t_0 * (x + y))) * 1.0
	else:
		tmp = (1.0 * (x / (x + y))) / (x + y)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	t_0 = Float64(Float64(x + y) - -1.0)
	tmp = 0.0
	if (y <= 1.95e-140)
		tmp = Float64(Float64(1.0 * Float64(y / t_0)) / Float64(x + y));
	elseif (y <= 1.35e+154)
		tmp = Float64(Float64(x / Float64(t_0 * Float64(x + y))) * 1.0);
	else
		tmp = Float64(Float64(1.0 * Float64(x / Float64(x + y))) / Float64(x + y));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	t_0 = (x + y) - -1.0;
	tmp = 0.0;
	if (y <= 1.95e-140)
		tmp = (1.0 * (y / t_0)) / (x + y);
	elseif (y <= 1.35e+154)
		tmp = (x / (t_0 * (x + y))) * 1.0;
	else
		tmp = (1.0 * (x / (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_] := Block[{t$95$0 = N[(N[(x + y), $MachinePrecision] - -1.0), $MachinePrecision]}, If[LessEqual[y, 1.95e-140], N[(N[(1.0 * N[(y / t$95$0), $MachinePrecision]), $MachinePrecision] / N[(x + y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.35e+154], N[(N[(x / N[(t$95$0 * N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision], N[(N[(1.0 * N[(x / N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x + y), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < 1.9500000000000001e-140

   1. Initial program 75.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/r* N/A

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

      8. associate-*r/ N/A

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

      9. lower-/.f64 N/A

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in y around 0

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

   7. Applied rewrites 59.5%

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

   if 1.9500000000000001e-140 < y < 1.35000000000000003e154

   1. Initial program 74.1%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   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 94.1

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

      16. lift-+.f64 N/A

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

      17. +-commutative N/A

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

      18. lower-+.f64 94.1

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

      19. lift-+.f64 N/A

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

      20. +-commutative N/A

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

      21. lower-+.f64 94.1

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

      22. lift-+.f64 N/A

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

      23. +-commutative N/A

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

      24. lower-+.f64 94.1

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

   4. Applied rewrites 94.1%

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

   5. Taylor expanded in y around inf

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

   7. Applied rewrites 72.3%

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

   if 1.35000000000000003e154 < y

   1. Initial program 58.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/r* N/A

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

      8. associate-*r/ N/A

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

      9. lower-/.f64 N/A

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in y around inf

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

   7. Applied rewrites 87.1%

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

4. Final simplification 65.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.95 \cdot 10^{-140}:\\ \;\;\;\;\frac{1 \cdot \frac{y}{\left(x + y\right) - -1}}{x + y}\\ \mathbf{elif}\;y \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\frac{x}{\left(\left(x + y\right) - -1\right) \cdot \left(x + y\right)} \cdot 1\\ \mathbf{else}:\\ \;\;\;\;\frac{1 \cdot \frac{x}{x + y}}{x + y}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 87.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 1.95 \cdot 10^{-140}:\\ \;\;\;\;\frac{\frac{y}{1 + x}}{x + y}\\ \mathbf{elif}\;y \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\frac{x}{\left(\left(x + y\right) - -1\right) \cdot \left(x + y\right)} \cdot 1\\ \mathbf{else}:\\ \;\;\;\;\frac{1 \cdot \frac{x}{x + y}}{x + y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 1.95e-140)
   (/ (/ y (+ 1.0 x)) (+ x y))
   (if (<= y 1.35e+154)
     (* (/ x (* (- (+ x y) -1.0) (+ x y))) 1.0)
     (/ (* 1.0 (/ x (+ x y))) (+ x y)))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 1.95e-140) {
		tmp = (y / (1.0 + x)) / (x + y);
	} else if (y <= 1.35e+154) {
		tmp = (x / (((x + y) - -1.0) * (x + y))) * 1.0;
	} else {
		tmp = (1.0 * (x / (x + y))) / (x + y);
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 1.95d-140) then
        tmp = (y / (1.0d0 + x)) / (x + y)
    else if (y <= 1.35d+154) then
        tmp = (x / (((x + y) - (-1.0d0)) * (x + y))) * 1.0d0
    else
        tmp = (1.0d0 * (x / (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 (y <= 1.95e-140) {
		tmp = (y / (1.0 + x)) / (x + y);
	} else if (y <= 1.35e+154) {
		tmp = (x / (((x + y) - -1.0) * (x + y))) * 1.0;
	} else {
		tmp = (1.0 * (x / (x + y))) / (x + y);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if y <= 1.95e-140:
		tmp = (y / (1.0 + x)) / (x + y)
	elif y <= 1.35e+154:
		tmp = (x / (((x + y) - -1.0) * (x + y))) * 1.0
	else:
		tmp = (1.0 * (x / (x + y))) / (x + y)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= 1.95e-140)
		tmp = Float64(Float64(y / Float64(1.0 + x)) / Float64(x + y));
	elseif (y <= 1.35e+154)
		tmp = Float64(Float64(x / Float64(Float64(Float64(x + y) - -1.0) * Float64(x + y))) * 1.0);
	else
		tmp = Float64(Float64(1.0 * Float64(x / 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 (y <= 1.95e-140)
		tmp = (y / (1.0 + x)) / (x + y);
	elseif (y <= 1.35e+154)
		tmp = (x / (((x + y) - -1.0) * (x + y))) * 1.0;
	else
		tmp = (1.0 * (x / (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[y, 1.95e-140], N[(N[(y / N[(1.0 + x), $MachinePrecision]), $MachinePrecision] / N[(x + y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.35e+154], N[(N[(x / N[(N[(N[(x + y), $MachinePrecision] - -1.0), $MachinePrecision] * N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision], N[(N[(1.0 * N[(x / N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x + y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 1.9500000000000001e-140

   1. Initial program 75.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/r* N/A

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

      8. associate-*r/ N/A

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

      9. lower-/.f64 N/A

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 58.7

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

   7. Applied rewrites 58.7%

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

   if 1.9500000000000001e-140 < y < 1.35000000000000003e154

   1. Initial program 74.1%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   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 94.1

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

      16. lift-+.f64 N/A

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

      17. +-commutative N/A

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

      18. lower-+.f64 94.1

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

      19. lift-+.f64 N/A

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

      20. +-commutative N/A

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

      21. lower-+.f64 94.1

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

      22. lift-+.f64 N/A

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

      23. +-commutative N/A

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

      24. lower-+.f64 94.1

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

   4. Applied rewrites 94.1%

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

   5. Taylor expanded in y around inf

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

   7. Applied rewrites 72.3%

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

   if 1.35000000000000003e154 < y

   1. Initial program 58.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/r* N/A

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

      8. associate-*r/ N/A

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

      9. lower-/.f64 N/A

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in y around inf

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

   7. Applied rewrites 87.1%

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

4. Final simplification 64.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.95 \cdot 10^{-140}:\\ \;\;\;\;\frac{\frac{y}{1 + x}}{x + y}\\ \mathbf{elif}\;y \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\frac{x}{\left(\left(x + y\right) - -1\right) \cdot \left(x + y\right)} \cdot 1\\ \mathbf{else}:\\ \;\;\;\;\frac{1 \cdot \frac{x}{x + y}}{x + y}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 87.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 1.95 \cdot 10^{-140}:\\ \;\;\;\;\frac{\frac{y}{1 + x}}{x + y}\\ \mathbf{elif}\;y \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\frac{x}{\left(\left(x + y\right) - -1\right) \cdot \left(x + y\right)} \cdot 1\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{x + y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 1.95e-140)
   (/ (/ y (+ 1.0 x)) (+ x y))
   (if (<= y 1.35e+154)
     (* (/ x (* (- (+ x y) -1.0) (+ x y))) 1.0)
     (/ (/ x y) (+ x y)))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 1.95e-140) {
		tmp = (y / (1.0 + x)) / (x + y);
	} else if (y <= 1.35e+154) {
		tmp = (x / (((x + y) - -1.0) * (x + y))) * 1.0;
	} else {
		tmp = (x / y) / (x + y);
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 1.95d-140) then
        tmp = (y / (1.0d0 + x)) / (x + y)
    else if (y <= 1.35d+154) then
        tmp = (x / (((x + y) - (-1.0d0)) * (x + y))) * 1.0d0
    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 (y <= 1.95e-140) {
		tmp = (y / (1.0 + x)) / (x + y);
	} else if (y <= 1.35e+154) {
		tmp = (x / (((x + y) - -1.0) * (x + y))) * 1.0;
	} else {
		tmp = (x / y) / (x + y);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if y <= 1.95e-140:
		tmp = (y / (1.0 + x)) / (x + y)
	elif y <= 1.35e+154:
		tmp = (x / (((x + y) - -1.0) * (x + y))) * 1.0
	else:
		tmp = (x / y) / (x + y)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= 1.95e-140)
		tmp = Float64(Float64(y / Float64(1.0 + x)) / Float64(x + y));
	elseif (y <= 1.35e+154)
		tmp = Float64(Float64(x / Float64(Float64(Float64(x + y) - -1.0) * Float64(x + y))) * 1.0);
	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 (y <= 1.95e-140)
		tmp = (y / (1.0 + x)) / (x + y);
	elseif (y <= 1.35e+154)
		tmp = (x / (((x + y) - -1.0) * (x + y))) * 1.0;
	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[y, 1.95e-140], N[(N[(y / N[(1.0 + x), $MachinePrecision]), $MachinePrecision] / N[(x + y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.35e+154], N[(N[(x / N[(N[(N[(x + y), $MachinePrecision] - -1.0), $MachinePrecision] * N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / N[(x + y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 1.9500000000000001e-140

   1. Initial program 75.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/r* N/A

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

      8. associate-*r/ N/A

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

      9. lower-/.f64 N/A

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 58.7

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

   7. Applied rewrites 58.7%

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

   if 1.9500000000000001e-140 < y < 1.35000000000000003e154

   1. Initial program 74.1%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   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 94.1

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

      16. lift-+.f64 N/A

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

      17. +-commutative N/A

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

      18. lower-+.f64 94.1

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

      19. lift-+.f64 N/A

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

      20. +-commutative N/A

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

      21. lower-+.f64 94.1

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

      22. lift-+.f64 N/A

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

      23. +-commutative N/A

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

      24. lower-+.f64 94.1

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

   4. Applied rewrites 94.1%

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

   5. Taylor expanded in y around inf

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

   7. Applied rewrites 72.3%

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

   if 1.35000000000000003e154 < y

   1. Initial program 58.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/r* N/A

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

      8. associate-*r/ N/A

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

      9. lower-/.f64 N/A

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in y around inf

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

      1. lower-/.f64 87.1

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

   7. Applied rewrites 87.1%

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

4. Final simplification 64.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.95 \cdot 10^{-140}:\\ \;\;\;\;\frac{\frac{y}{1 + x}}{x + y}\\ \mathbf{elif}\;y \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\frac{x}{\left(\left(x + y\right) - -1\right) \cdot \left(x + y\right)} \cdot 1\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{x + y}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 84.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 1.95 \cdot 10^{-140}:\\ \;\;\;\;\frac{\frac{y}{1 + x}}{x + y}\\ \mathbf{elif}\;y \leq 1.12 \cdot 10^{-26}:\\ \;\;\;\;1 \cdot \frac{x}{\left(1 + x\right) \cdot \left(x + y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{1 + y}}{x + y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 1.95e-140)
   (/ (/ y (+ 1.0 x)) (+ x y))
   (if (<= y 1.12e-26)
     (* 1.0 (/ x (* (+ 1.0 x) (+ x y))))
     (/ (/ x (+ 1.0 y)) (+ x y)))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 1.95e-140) {
		tmp = (y / (1.0 + x)) / (x + y);
	} else if (y <= 1.12e-26) {
		tmp = 1.0 * (x / ((1.0 + x) * (x + y)));
	} else {
		tmp = (x / (1.0 + y)) / (x + y);
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 1.95d-140) then
        tmp = (y / (1.0d0 + x)) / (x + y)
    else if (y <= 1.12d-26) then
        tmp = 1.0d0 * (x / ((1.0d0 + x) * (x + y)))
    else
        tmp = (x / (1.0d0 + y)) / (x + y)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (y <= 1.95e-140) {
		tmp = (y / (1.0 + x)) / (x + y);
	} else if (y <= 1.12e-26) {
		tmp = 1.0 * (x / ((1.0 + x) * (x + y)));
	} else {
		tmp = (x / (1.0 + y)) / (x + y);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if y <= 1.95e-140:
		tmp = (y / (1.0 + x)) / (x + y)
	elif y <= 1.12e-26:
		tmp = 1.0 * (x / ((1.0 + x) * (x + y)))
	else:
		tmp = (x / (1.0 + y)) / (x + y)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= 1.95e-140)
		tmp = Float64(Float64(y / Float64(1.0 + x)) / Float64(x + y));
	elseif (y <= 1.12e-26)
		tmp = Float64(1.0 * Float64(x / Float64(Float64(1.0 + x) * Float64(x + y))));
	else
		tmp = Float64(Float64(x / Float64(1.0 + 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 (y <= 1.95e-140)
		tmp = (y / (1.0 + x)) / (x + y);
	elseif (y <= 1.12e-26)
		tmp = 1.0 * (x / ((1.0 + x) * (x + y)));
	else
		tmp = (x / (1.0 + 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[y, 1.95e-140], N[(N[(y / N[(1.0 + x), $MachinePrecision]), $MachinePrecision] / N[(x + y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.12e-26], N[(1.0 * N[(x / N[(N[(1.0 + x), $MachinePrecision] * N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(1.0 + y), $MachinePrecision]), $MachinePrecision] / N[(x + y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 1.9500000000000001e-140

   1. Initial program 75.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/r* N/A

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

      8. associate-*r/ N/A

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

      9. lower-/.f64 N/A

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 58.7

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

   7. Applied rewrites 58.7%

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

   if 1.9500000000000001e-140 < y < 1.12e-26

   1. Initial program 79.9%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   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 99.8

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

      16. lift-+.f64 N/A

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

      17. +-commutative N/A

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

      18. lower-+.f64 99.8

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

      19. lift-+.f64 N/A

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

      20. +-commutative N/A

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

      21. lower-+.f64 99.8

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

      22. lift-+.f64 N/A

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

      23. +-commutative N/A

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

      24. lower-+.f64 99.8

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 99.8

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

   7. Applied rewrites 99.8%

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

   8. Taylor expanded in y around inf

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

   10. Applied rewrites 60.5%

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

   if 1.12e-26 < y

   1. Initial program 64.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/r* N/A

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

      8. associate-*r/ N/A

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

      9. lower-/.f64 N/A

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 80.0

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

   7. Applied rewrites 80.0%

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

4. Final simplification 64.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.95 \cdot 10^{-140}:\\ \;\;\;\;\frac{\frac{y}{1 + x}}{x + y}\\ \mathbf{elif}\;y \leq 1.12 \cdot 10^{-26}:\\ \;\;\;\;1 \cdot \frac{x}{\left(1 + x\right) \cdot \left(x + y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{1 + y}}{x + y}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 79.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -4.8 \cdot 10^{+142}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + y}\\ \mathbf{elif}\;x \leq -3.74 \cdot 10^{-75}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(x, x, x\right) \cdot x} \cdot y\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(y, y, y\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -4.8e+142)
   (/ (/ y x) (+ x y))
   (if (<= x -3.74e-75) (* (/ x (* (fma x x x) x)) y) (/ x (fma y y y)))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -4.8e+142) {
		tmp = (y / x) / (x + y);
	} else if (x <= -3.74e-75) {
		tmp = (x / (fma(x, x, x) * x)) * y;
	} else {
		tmp = x / fma(y, y, y);
	}
	return tmp;
}

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -4.8e+142)
		tmp = Float64(Float64(y / x) / Float64(x + y));
	elseif (x <= -3.74e-75)
		tmp = Float64(Float64(x / Float64(fma(x, x, x) * x)) * y);
	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, -4.8e+142], N[(N[(y / x), $MachinePrecision] / N[(x + y), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -3.74e-75], N[(N[(x / N[(N[(x * x + x), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision], N[(x / N[(y * y + y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -4.7999999999999998e142

   1. Initial program 55.0%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   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. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/r* N/A

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

      8. associate-*r/ N/A

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

      9. lower-/.f64 N/A

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in x around inf

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

      1. lower-/.f64 87.2

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

   7. Applied rewrites 87.2%

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

   if -4.7999999999999998e142 < x < -3.74e-75

   1. Initial program 84.1%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. +-commutative N/A

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

      7. distribute-lft-in N/A

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

      8. *-rgt-identity N/A

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

      9. lower-fma.f64 62.6

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

   5. Applied rewrites 62.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 65.2

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

   7. Applied rewrites 65.2%

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

   if -3.74e-75 < x

   1. Initial program 74.3%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. distribute-lft-in N/A

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

      4. *-rgt-identity N/A

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

      5. lower-fma.f64 62.1

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

   5. Applied rewrites 62.1%

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

4. Final simplification 66.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.8 \cdot 10^{+142}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + y}\\ \mathbf{elif}\;x \leq -3.74 \cdot 10^{-75}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(x, x, x\right) \cdot x} \cdot y\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(y, y, y\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 80.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{+162}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + y}\\ \mathbf{elif}\;x \leq -3.74 \cdot 10^{-75}:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(x, x, x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(y, y, y\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -5e+162)
   (/ (/ y x) (+ x y))
   (if (<= x -3.74e-75) (/ y (fma x x x)) (/ x (fma y y y)))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -5e+162) {
		tmp = (y / x) / (x + y);
	} else if (x <= -3.74e-75) {
		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 <= -5e+162)
		tmp = Float64(Float64(y / x) / Float64(x + y));
	elseif (x <= -3.74e-75)
		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, -5e+162], N[(N[(y / x), $MachinePrecision] / N[(x + y), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -3.74e-75], N[(y / N[(x * x + x), $MachinePrecision]), $MachinePrecision], N[(x / N[(y * y + y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -4.9999999999999997e162

   1. Initial program 52.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/r* N/A

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

      8. associate-*r/ N/A

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

      9. lower-/.f64 N/A

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in x around inf

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

      1. lower-/.f64 87.8

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

   7. Applied rewrites 87.8%

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

   if -4.9999999999999997e162 < x < -3.74e-75

   1. Initial program 82.1%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. distribute-lft-in N/A

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

      4. *-rgt-identity N/A

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

      5. lower-fma.f64 67.5

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

   5. Applied rewrites 67.5%

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

   if -3.74e-75 < x

   1. Initial program 74.3%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. distribute-lft-in N/A

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

      4. *-rgt-identity N/A

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

      5. lower-fma.f64 62.1

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

   5. Applied rewrites 62.1%

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

4. Final simplification 66.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{+162}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + y}\\ \mathbf{elif}\;x \leq -3.74 \cdot 10^{-75}:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(x, x, x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(y, y, y\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 80.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{+162}:\\ \;\;\;\;\frac{\frac{y}{x}}{x}\\ \mathbf{elif}\;x \leq -3.74 \cdot 10^{-75}:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(x, x, x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(y, y, y\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -5e+162)
   (/ (/ y x) x)
   (if (<= x -3.74e-75) (/ y (fma x x x)) (/ x (fma y y y)))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -5e+162) {
		tmp = (y / x) / x;
	} else if (x <= -3.74e-75) {
		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 <= -5e+162)
		tmp = Float64(Float64(y / x) / x);
	elseif (x <= -3.74e-75)
		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, -5e+162], N[(N[(y / x), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[x, -3.74e-75], N[(y / N[(x * x + x), $MachinePrecision]), $MachinePrecision], N[(x / N[(y * y + y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -4.9999999999999997e162

   1. Initial program 52.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/r* N/A

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

      8. associate-*r/ N/A

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

      9. lower-/.f64 N/A

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 68.0

         \[\leadsto \frac{y}{\color{blue}{x \cdot x}} \]

   7. Applied rewrites 68.0%

      \[\leadsto \color{blue}{\frac{y}{x \cdot x}} \]
   8. Step-by-step derivation

   9. Applied rewrites 87.5%

      \[\leadsto \frac{\frac{y}{x}}{\color{blue}{x}} \]

   if -4.9999999999999997e162 < x < -3.74e-75

   1. Initial program 82.1%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{y}{x \cdot \left(1 + x\right)}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{y}{x \cdot \left(1 + x\right)}} \]

      2. +-commutative N/A

         \[\leadsto \frac{y}{x \cdot \color{blue}{\left(x + 1\right)}} \]

      3. distribute-lft-in N/A

         \[\leadsto \frac{y}{\color{blue}{x \cdot x + x \cdot 1}} \]

      4. *-rgt-identity N/A

         \[\leadsto \frac{y}{x \cdot x + \color{blue}{x}} \]

      5. lower-fma.f64 67.5

         \[\leadsto \frac{y}{\color{blue}{\mathsf{fma}\left(x, x, x\right)}} \]

   5. Applied rewrites 67.5%

      \[\leadsto \color{blue}{\frac{y}{\mathsf{fma}\left(x, x, x\right)}} \]

   if -3.74e-75 < x

   1. Initial program 74.3%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]

      2. +-commutative N/A

         \[\leadsto \frac{x}{y \cdot \color{blue}{\left(y + 1\right)}} \]

      3. distribute-lft-in N/A

         \[\leadsto \frac{x}{\color{blue}{y \cdot y + y \cdot 1}} \]

      4. *-rgt-identity N/A

         \[\leadsto \frac{x}{y \cdot y + \color{blue}{y}} \]

      5. lower-fma.f64 62.1

         \[\leadsto \frac{x}{\color{blue}{\mathsf{fma}\left(y, y, y\right)}} \]

   5. Applied rewrites 62.1%

      \[\leadsto \color{blue}{\frac{x}{\mathsf{fma}\left(y, y, y\right)}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 17: 78.5% accurate, 1.6× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -3.74 \cdot 10^{-75}:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(x, x, x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(y, y, y\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -3.74e-75) (/ y (fma x x x)) (/ x (fma y y y))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -3.74e-75) {
		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 <= -3.74e-75)
		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, -3.74e-75], 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 < -3.74e-75

   1. Initial program 70.9%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{y}{x \cdot \left(1 + x\right)}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{y}{x \cdot \left(1 + x\right)}} \]

      2. +-commutative N/A

         \[\leadsto \frac{y}{x \cdot \color{blue}{\left(x + 1\right)}} \]

      3. distribute-lft-in N/A

         \[\leadsto \frac{y}{\color{blue}{x \cdot x + x \cdot 1}} \]

      4. *-rgt-identity N/A

         \[\leadsto \frac{y}{x \cdot x + \color{blue}{x}} \]

      5. lower-fma.f64 67.7

         \[\leadsto \frac{y}{\color{blue}{\mathsf{fma}\left(x, x, x\right)}} \]

   5. Applied rewrites 67.7%

      \[\leadsto \color{blue}{\frac{y}{\mathsf{fma}\left(x, x, x\right)}} \]

   if -3.74e-75 < x

   1. Initial program 74.3%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]

      2. +-commutative N/A

         \[\leadsto \frac{x}{y \cdot \color{blue}{\left(y + 1\right)}} \]

      3. distribute-lft-in N/A

         \[\leadsto \frac{x}{\color{blue}{y \cdot y + y \cdot 1}} \]

      4. *-rgt-identity N/A

         \[\leadsto \frac{x}{y \cdot y + \color{blue}{y}} \]

      5. lower-fma.f64 62.1

         \[\leadsto \frac{x}{\color{blue}{\mathsf{fma}\left(y, y, y\right)}} \]

   5. Applied rewrites 62.1%

      \[\leadsto \color{blue}{\frac{x}{\mathsf{fma}\left(y, y, y\right)}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 18: 75.4% accurate, 1.6× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -4 \cdot 10^{+27}:\\ \;\;\;\;\frac{y}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(y, y, y\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -4e+27) (/ y (* x x)) (/ x (fma y y y))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -4e+27) {
		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 <= -4e+27)
		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, -4e+27], 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 < -4.0000000000000001e27

   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 inf

      \[\leadsto \color{blue}{\frac{y}{{x}^{2}}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{y}{{x}^{2}}} \]

      2. unpow2 N/A

         \[\leadsto \frac{y}{\color{blue}{x \cdot x}} \]

      3. lower-*.f64 73.5

         \[\leadsto \frac{y}{\color{blue}{x \cdot x}} \]

   5. Applied rewrites 73.5%

      \[\leadsto \color{blue}{\frac{y}{x \cdot x}} \]

   if -4.0000000000000001e27 < x

   1. Initial program 75.9%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]

      2. +-commutative N/A

         \[\leadsto \frac{x}{y \cdot \color{blue}{\left(y + 1\right)}} \]

      3. distribute-lft-in N/A

         \[\leadsto \frac{x}{\color{blue}{y \cdot y + y \cdot 1}} \]

      4. *-rgt-identity N/A

         \[\leadsto \frac{x}{y \cdot y + \color{blue}{y}} \]

      5. lower-fma.f64 60.2

         \[\leadsto \frac{x}{\color{blue}{\mathsf{fma}\left(y, y, y\right)}} \]

   5. Applied rewrites 60.2%

      \[\leadsto \color{blue}{\frac{x}{\mathsf{fma}\left(y, y, y\right)}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 19: 63.7% accurate, 1.7× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 3 \cdot 10^{+22}:\\ \;\;\;\;\frac{y}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 3e+22) (/ y (* x x)) (/ x (* y y))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 3e+22) {
		tmp = y / (x * x);
	} else {
		tmp = x / (y * y);
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 3d+22) then
        tmp = y / (x * x)
    else
        tmp = x / (y * y)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (y <= 3e+22) {
		tmp = y / (x * x);
	} else {
		tmp = x / (y * y);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if y <= 3e+22:
		tmp = y / (x * x)
	else:
		tmp = x / (y * y)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= 3e+22)
		tmp = Float64(y / Float64(x * x));
	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 <= 3e+22)
		tmp = y / (x * x);
	else
		tmp = x / (y * y);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[y, 3e+22], N[(y / N[(x * x), $MachinePrecision]), $MachinePrecision], N[(x / N[(y * y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 3e22

   1. Initial program 76.5%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{y}{{x}^{2}}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{y}{{x}^{2}}} \]

      2. unpow2 N/A

         \[\leadsto \frac{y}{\color{blue}{x \cdot x}} \]

      3. lower-*.f64 37.9

         \[\leadsto \frac{y}{\color{blue}{x \cdot x}} \]

   5. Applied rewrites 37.9%

      \[\leadsto \color{blue}{\frac{y}{x \cdot x}} \]

   if 3e22 < y

   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 y around inf

      \[\leadsto \color{blue}{\frac{x}{{y}^{2}}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{{y}^{2}}} \]

      2. unpow2 N/A

         \[\leadsto \frac{x}{\color{blue}{y \cdot y}} \]

      3. lower-*.f64 76.9

         \[\leadsto \frac{x}{\color{blue}{y \cdot y}} \]

   5. Applied rewrites 76.9%

      \[\leadsto \color{blue}{\frac{x}{y \cdot y}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 20: 37.2% accurate, 2.3× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \frac{x}{y \cdot y} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y) :precision binary64 (/ x (* y y)))

C

assert(x < y);
double code(double x, double y) {
	return x / (y * y);
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = x / (y * y)
end function

Java

assert x < y;
public static double code(double x, double y) {
	return x / (y * y);
}

Python

[x, y] = sort([x, y])
def code(x, y):
	return x / (y * y)

Julia

x, y = sort([x, y])
function code(x, y)
	return Float64(x / Float64(y * y))
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp = code(x, y)
	tmp = x / (y * y);
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := N[(x / N[(y * y), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 73.3%

   \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
2. Add Preprocessing

3. Taylor expanded in y around inf

   \[\leadsto \color{blue}{\frac{x}{{y}^{2}}} \]
4. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{x}{{y}^{2}}} \]

   2. unpow2 N/A

      \[\leadsto \frac{x}{\color{blue}{y \cdot y}} \]

   3. lower-*.f64 39.6

      \[\leadsto \frac{x}{\color{blue}{y \cdot y}} \]

5. Applied rewrites 39.6%

   \[\leadsto \color{blue}{\frac{x}{y \cdot y}} \]
6. 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

real(8) function code(x, y)
    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 2024263 
(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))))