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

Percentage Accurate: 69.5% → 99.8%

Time: 8.1s

Alternatives: 19

Speedup: 1.6×

Specification

Math

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

FPCore

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

C

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

Fortran

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: 69.5% 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

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

FPCore

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

C

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

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 + x)) / ((y + x) + 1.0d0)) * (y / (y + x))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 68.2%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. 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. associate-*l/ N/A

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

   6. lift-*.f64 N/A

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

   7. times-frac N/A

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

   8. lower-*.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. lift-+.f64 N/A

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

   11. +-commutative N/A

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

   12. lower-+.f64 N/A

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

   13. lower-/.f64 N/A

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

   14. lower-/.f64 99.8

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

   15. lift-+.f64 N/A

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

   16. +-commutative N/A

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

   17. lower-+.f64 99.8

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

   18. lift-+.f64 N/A

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

   19. +-commutative N/A

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

   20. lower-+.f64 99.8

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

   21. lift-+.f64 N/A

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

   22. +-commutative N/A

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

   23. lower-+.f64 99.8

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

4. Applied rewrites 99.8%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. associate-/l/ N/A

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

   5. associate-*r/ N/A

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

   6. lift-+.f64 N/A

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

   7. lift-+.f64 N/A

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

   8. +-commutative N/A

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

   9. +-commutative N/A

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

   10. lift-+.f64 N/A

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

   11. lift-+.f64 N/A

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

   12. lift-+.f64 N/A

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

   13. +-commutative N/A

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

   14. times-frac N/A

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

   15. lower-*.f64 N/A

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

   16. lower-/.f64 N/A

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

   17. lift-+.f64 N/A

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

   18. +-commutative N/A

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

   19. lift-+.f64 N/A

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

   20. lower-/.f64 N/A

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

   21. +-commutative N/A

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

6. Applied rewrites 99.8%

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

Alternative 2: 84.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -6.8 \cdot 10^{+192}:\\ \;\;\;\;1 \cdot \frac{\frac{y}{1 + \left(y + x\right)}}{y + x}\\ \mathbf{elif}\;x \leq -4.7 \cdot 10^{-71}:\\ \;\;\;\;\frac{1 \cdot y}{\left(\left(x + y\right) + 1\right) \cdot \left(x + y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y + x} \cdot {\left(1 + y\right)}^{-1}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -6.8e+192)
   (* 1.0 (/ (/ y (+ 1.0 (+ y x))) (+ y x)))
   (if (<= x -4.7e-71)
     (/ (* 1.0 y) (* (+ (+ x y) 1.0) (+ x y)))
     (* (/ x (+ y x)) (pow (+ 1.0 y) -1.0)))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -6.8e+192) {
		tmp = 1.0 * ((y / (1.0 + (y + x))) / (y + x));
	} else if (x <= -4.7e-71) {
		tmp = (1.0 * y) / (((x + y) + 1.0) * (x + y));
	} else {
		tmp = (x / (y + x)) * pow((1.0 + y), -1.0);
	}
	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 <= (-6.8d+192)) then
        tmp = 1.0d0 * ((y / (1.0d0 + (y + x))) / (y + x))
    else if (x <= (-4.7d-71)) then
        tmp = (1.0d0 * y) / (((x + y) + 1.0d0) * (x + y))
    else
        tmp = (x / (y + x)) * ((1.0d0 + y) ** (-1.0d0))
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (x <= -6.8e+192) {
		tmp = 1.0 * ((y / (1.0 + (y + x))) / (y + x));
	} else if (x <= -4.7e-71) {
		tmp = (1.0 * y) / (((x + y) + 1.0) * (x + y));
	} else {
		tmp = (x / (y + x)) * Math.pow((1.0 + y), -1.0);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -6.8e+192:
		tmp = 1.0 * ((y / (1.0 + (y + x))) / (y + x))
	elif x <= -4.7e-71:
		tmp = (1.0 * y) / (((x + y) + 1.0) * (x + y))
	else:
		tmp = (x / (y + x)) * math.pow((1.0 + y), -1.0)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -6.8e+192)
		tmp = Float64(1.0 * Float64(Float64(y / Float64(1.0 + Float64(y + x))) / Float64(y + x)));
	elseif (x <= -4.7e-71)
		tmp = Float64(Float64(1.0 * y) / Float64(Float64(Float64(x + y) + 1.0) * Float64(x + y)));
	else
		tmp = Float64(Float64(x / Float64(y + x)) * (Float64(1.0 + y) ^ -1.0));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -6.8e+192)
		tmp = 1.0 * ((y / (1.0 + (y + x))) / (y + x));
	elseif (x <= -4.7e-71)
		tmp = (1.0 * y) / (((x + y) + 1.0) * (x + y));
	else
		tmp = (x / (y + x)) * ((1.0 + y) ^ -1.0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[x, -6.8e+192], N[(1.0 * N[(N[(y / N[(1.0 + N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -4.7e-71], N[(N[(1.0 * y), $MachinePrecision] / N[(N[(N[(x + y), $MachinePrecision] + 1.0), $MachinePrecision] * N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(y + x), $MachinePrecision]), $MachinePrecision] * N[Power[N[(1.0 + y), $MachinePrecision], -1.0], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -6.79999999999999992e192

   1. Initial program 35.5%

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

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

      6. lift-*.f64 N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-/.f64 99.8

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

      15. lift-+.f64 N/A

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

      16. +-commutative N/A

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

      17. lower-+.f64 99.8

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

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

      20. lower-+.f64 99.8

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

      21. lift-+.f64 N/A

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

      22. +-commutative N/A

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

      23. lower-+.f64 99.8

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in x around inf

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

   7. Applied rewrites 92.2%

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

   if -6.79999999999999992e192 < x < -4.69999999999999996e-71

   1. Initial program 73.5%

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

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

      6. lift-*.f64 N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-/.f64 99.7

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

      15. lift-+.f64 N/A

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

      16. +-commutative N/A

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

      17. lower-+.f64 99.7

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

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

      20. lower-+.f64 99.7

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

      21. lift-+.f64 N/A

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

      22. +-commutative N/A

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

      23. lower-+.f64 99.7

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in x around inf

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

   7. Applied rewrites 56.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-/l/ N/A

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

      5. associate-*r/ N/A

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

      6. lift-+.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. +-commutative N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. lower-/.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. lower-+.f64 N/A

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

      18. lower-+.f64 N/A

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

      19. lower-+.f64 76.7

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

   9. Applied rewrites 76.7%

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

   if -4.69999999999999996e-71 < x

   1. Initial program 70.1%

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

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

      6. lift-*.f64 N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-/.f64 99.8

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

      15. lift-+.f64 N/A

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

      16. +-commutative N/A

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

      17. lower-+.f64 99.8

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

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

      20. lower-+.f64 99.8

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

      21. lift-+.f64 N/A

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

      22. +-commutative N/A

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

      23. lower-+.f64 99.8

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 61.5

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

   7. Applied rewrites 61.5%

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

4. Final simplification 67.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -6.8 \cdot 10^{+192}:\\ \;\;\;\;1 \cdot \frac{\frac{y}{1 + \left(y + x\right)}}{y + x}\\ \mathbf{elif}\;x \leq -4.7 \cdot 10^{-71}:\\ \;\;\;\;\frac{1 \cdot y}{\left(\left(x + y\right) + 1\right) \cdot \left(x + y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y + x} \cdot {\left(1 + y\right)}^{-1}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 84.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -6.8 \cdot 10^{+192}:\\ \;\;\;\;\frac{\frac{-y}{x}}{-\left(y + x\right)}\\ \mathbf{elif}\;x \leq -4.7 \cdot 10^{-71}:\\ \;\;\;\;\frac{1 \cdot y}{\left(\left(x + y\right) + 1\right) \cdot \left(x + y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y + x} \cdot {\left(1 + y\right)}^{-1}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -6.8e+192)
   (/ (/ (- y) x) (- (+ y x)))
   (if (<= x -4.7e-71)
     (/ (* 1.0 y) (* (+ (+ x y) 1.0) (+ x y)))
     (* (/ x (+ y x)) (pow (+ 1.0 y) -1.0)))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -6.8e+192) {
		tmp = (-y / x) / -(y + x);
	} else if (x <= -4.7e-71) {
		tmp = (1.0 * y) / (((x + y) + 1.0) * (x + y));
	} else {
		tmp = (x / (y + x)) * pow((1.0 + y), -1.0);
	}
	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 <= (-6.8d+192)) then
        tmp = (-y / x) / -(y + x)
    else if (x <= (-4.7d-71)) then
        tmp = (1.0d0 * y) / (((x + y) + 1.0d0) * (x + y))
    else
        tmp = (x / (y + x)) * ((1.0d0 + y) ** (-1.0d0))
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (x <= -6.8e+192) {
		tmp = (-y / x) / -(y + x);
	} else if (x <= -4.7e-71) {
		tmp = (1.0 * y) / (((x + y) + 1.0) * (x + y));
	} else {
		tmp = (x / (y + x)) * Math.pow((1.0 + y), -1.0);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -6.8e+192:
		tmp = (-y / x) / -(y + x)
	elif x <= -4.7e-71:
		tmp = (1.0 * y) / (((x + y) + 1.0) * (x + y))
	else:
		tmp = (x / (y + x)) * math.pow((1.0 + y), -1.0)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -6.8e+192)
		tmp = Float64(Float64(Float64(-y) / x) / Float64(-Float64(y + x)));
	elseif (x <= -4.7e-71)
		tmp = Float64(Float64(1.0 * y) / Float64(Float64(Float64(x + y) + 1.0) * Float64(x + y)));
	else
		tmp = Float64(Float64(x / Float64(y + x)) * (Float64(1.0 + y) ^ -1.0));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -6.8e+192)
		tmp = (-y / x) / -(y + x);
	elseif (x <= -4.7e-71)
		tmp = (1.0 * y) / (((x + y) + 1.0) * (x + y));
	else
		tmp = (x / (y + x)) * ((1.0 + y) ^ -1.0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[x, -6.8e+192], N[(N[((-y) / x), $MachinePrecision] / (-N[(y + x), $MachinePrecision])), $MachinePrecision], If[LessEqual[x, -4.7e-71], N[(N[(1.0 * y), $MachinePrecision] / N[(N[(N[(x + y), $MachinePrecision] + 1.0), $MachinePrecision] * N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(y + x), $MachinePrecision]), $MachinePrecision] * N[Power[N[(1.0 + y), $MachinePrecision], -1.0], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -6.79999999999999992e192

   1. Initial program 35.5%

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

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

      6. lift-*.f64 N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-/.f64 99.8

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

      15. lift-+.f64 N/A

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

      16. +-commutative N/A

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

      17. lower-+.f64 99.8

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

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

      20. lower-+.f64 99.8

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

      21. lift-+.f64 N/A

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

      22. +-commutative N/A

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

      23. lower-+.f64 99.8

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

   4. Applied rewrites 99.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-/l/ N/A

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

      5. associate-*r/ N/A

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

      6. lift-+.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. +-commutative N/A

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

      10. lift-+.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. +-commutative N/A

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

      14. times-frac N/A

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

      15. lower-*.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. lift-+.f64 N/A

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

      18. +-commutative N/A

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

      19. lift-+.f64 N/A

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

      20. lower-/.f64 N/A

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

      21. +-commutative N/A

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

   6. Applied rewrites 99.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. frac-times N/A

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

      5. associate-*r/ N/A

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

      6. lift-/.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-+.f64 N/A

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

      10. associate-/l/ N/A

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

      11. frac-times N/A

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

      12. sqr-neg-rev N/A

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

      13. associate-/r* N/A

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

      14. lower-/.f64 N/A

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

   8. Applied rewrites 99.7%

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

   9. Taylor expanded in x around inf

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

       1. associate-*r/ N/A

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

       2. lower-/.f64 N/A

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

       3. mul-1-neg N/A

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

       4. lower-neg.f64 92.0

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

   11. Applied rewrites 92.0%

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

   if -6.79999999999999992e192 < x < -4.69999999999999996e-71

   1. Initial program 73.5%

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

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

      6. lift-*.f64 N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-/.f64 99.7

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

      15. lift-+.f64 N/A

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

      16. +-commutative N/A

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

      17. lower-+.f64 99.7

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

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

      20. lower-+.f64 99.7

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

      21. lift-+.f64 N/A

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

      22. +-commutative N/A

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

      23. lower-+.f64 99.7

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in x around inf

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

   7. Applied rewrites 56.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-/l/ N/A

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

      5. associate-*r/ N/A

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

      6. lift-+.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. +-commutative N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. lower-/.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. lower-+.f64 N/A

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

      18. lower-+.f64 N/A

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

      19. lower-+.f64 76.7

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

   9. Applied rewrites 76.7%

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

   if -4.69999999999999996e-71 < x

   1. Initial program 70.1%

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

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

      6. lift-*.f64 N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-/.f64 99.8

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

      15. lift-+.f64 N/A

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

      16. +-commutative N/A

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

      17. lower-+.f64 99.8

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

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

      20. lower-+.f64 99.8

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

      21. lift-+.f64 N/A

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

      22. +-commutative N/A

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

      23. lower-+.f64 99.8

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 61.5

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

   7. Applied rewrites 61.5%

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

4. Final simplification 67.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -6.8 \cdot 10^{+192}:\\ \;\;\;\;\frac{\frac{-y}{x}}{-\left(y + x\right)}\\ \mathbf{elif}\;x \leq -4.7 \cdot 10^{-71}:\\ \;\;\;\;\frac{1 \cdot y}{\left(\left(x + y\right) + 1\right) \cdot \left(x + y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y + x} \cdot {\left(1 + y\right)}^{-1}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 97.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq -2.3 \cdot 10^{-57}:\\ \;\;\;\;\frac{\frac{y - y \cdot \frac{\mathsf{fma}\left(3, y, 1\right)}{x}}{x}}{x}\\ \mathbf{elif}\;y \leq 3.7 \cdot 10^{+139}:\\ \;\;\;\;\frac{\frac{y}{y + x} \cdot x}{\left(1 + \left(y + x\right)\right) \cdot \left(y + x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y} \cdot \frac{y - \mathsf{fma}\left(3, x, 1\right)}{y}}{y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y -2.3e-57)
   (/ (/ (- y (* y (/ (fma 3.0 y 1.0) x))) x) x)
   (if (<= y 3.7e+139)
     (/ (* (/ y (+ y x)) x) (* (+ 1.0 (+ y x)) (+ y x)))
     (/ (* (/ x y) (/ (- y (fma 3.0 x 1.0)) y)) y))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= -2.3e-57) {
		tmp = ((y - (y * (fma(3.0, y, 1.0) / x))) / x) / x;
	} else if (y <= 3.7e+139) {
		tmp = ((y / (y + x)) * x) / ((1.0 + (y + x)) * (y + x));
	} else {
		tmp = ((x / y) * ((y - fma(3.0, x, 1.0)) / y)) / y;
	}
	return tmp;
}

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= -2.3e-57)
		tmp = Float64(Float64(Float64(y - Float64(y * Float64(fma(3.0, y, 1.0) / x))) / x) / x);
	elseif (y <= 3.7e+139)
		tmp = Float64(Float64(Float64(y / Float64(y + x)) * x) / Float64(Float64(1.0 + Float64(y + x)) * Float64(y + x)));
	else
		tmp = Float64(Float64(Float64(x / y) * Float64(Float64(y - fma(3.0, x, 1.0)) / y)) / y);
	end
	return tmp
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[y, -2.3e-57], N[(N[(N[(y - N[(y * N[(N[(3.0 * y + 1.0), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[y, 3.7e+139], N[(N[(N[(y / N[(y + x), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision] / N[(N[(1.0 + N[(y + x), $MachinePrecision]), $MachinePrecision] * N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x / y), $MachinePrecision] * N[(N[(y - N[(3.0 * x + 1.0), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.3e-57

   1. Initial program 67.6%

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

   3. Taylor expanded in x around inf

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

      1. unpow2 N/A

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

      2. associate-/r* N/A

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

      3. lower-/.f64 N/A

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

   5. Applied rewrites 31.3%

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

   if -2.3e-57 < y < 3.69999999999999992e139

   1. Initial program 75.8%

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

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

      6. lift-*.f64 N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-/.f64 99.8

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

      15. lift-+.f64 N/A

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

      16. +-commutative N/A

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

      17. lower-+.f64 99.8

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

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

      20. lower-+.f64 99.8

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

      21. lift-+.f64 N/A

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

      22. +-commutative N/A

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

      23. lower-+.f64 99.8

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

   4. Applied rewrites 99.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-/l/ N/A

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

      5. associate-*r/ N/A

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

      6. lift-+.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. +-commutative N/A

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

      10. lift-+.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. +-commutative N/A

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

      14. times-frac N/A

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

      15. lower-*.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. lift-+.f64 N/A

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

      18. +-commutative N/A

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

      19. lift-+.f64 N/A

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

      20. lower-/.f64 N/A

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

      21. +-commutative N/A

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

   6. Applied rewrites 99.8%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-/l/ N/A

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

      6. associate-*r/ N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 99.2

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 99.2

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

   8. Applied rewrites 99.2%

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

   if 3.69999999999999992e139 < y

   1. Initial program 45.9%

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

   3. Taylor expanded in y around inf

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

      1. unpow2 N/A

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

      2. associate-/r* N/A

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

      3. lower-/.f64 N/A

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

   5. Applied rewrites 81.1%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 81.2%

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

Alternative 5: 97.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_0 := 1 + \left(y + x\right)\\ \mathbf{if}\;y \leq -2.3 \cdot 10^{-57}:\\ \;\;\;\;1 \cdot \frac{\frac{y}{t\_0}}{y + x}\\ \mathbf{elif}\;y \leq 3.7 \cdot 10^{+139}:\\ \;\;\;\;\frac{\frac{y}{y + x} \cdot x}{t\_0 \cdot \left(y + x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y} \cdot \frac{y - \mathsf{fma}\left(3, x, 1\right)}{y}}{y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ 1.0 (+ y x))))
   (if (<= y -2.3e-57)
     (* 1.0 (/ (/ y t_0) (+ y x)))
     (if (<= y 3.7e+139)
       (/ (* (/ y (+ y x)) x) (* t_0 (+ y x)))
       (/ (* (/ x y) (/ (- y (fma 3.0 x 1.0)) y)) y)))))

C

assert(x < y);
double code(double x, double y) {
	double t_0 = 1.0 + (y + x);
	double tmp;
	if (y <= -2.3e-57) {
		tmp = 1.0 * ((y / t_0) / (y + x));
	} else if (y <= 3.7e+139) {
		tmp = ((y / (y + x)) * x) / (t_0 * (y + x));
	} else {
		tmp = ((x / y) * ((y - fma(3.0, x, 1.0)) / y)) / y;
	}
	return tmp;
}

Julia

x, y = sort([x, y])
function code(x, y)
	t_0 = Float64(1.0 + Float64(y + x))
	tmp = 0.0
	if (y <= -2.3e-57)
		tmp = Float64(1.0 * Float64(Float64(y / t_0) / Float64(y + x)));
	elseif (y <= 3.7e+139)
		tmp = Float64(Float64(Float64(y / Float64(y + x)) * x) / Float64(t_0 * Float64(y + x)));
	else
		tmp = Float64(Float64(Float64(x / y) * Float64(Float64(y - fma(3.0, x, 1.0)) / y)) / y);
	end
	return tmp
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := Block[{t$95$0 = N[(1.0 + N[(y + x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -2.3e-57], N[(1.0 * N[(N[(y / t$95$0), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 3.7e+139], N[(N[(N[(y / N[(y + x), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision] / N[(t$95$0 * N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x / y), $MachinePrecision] * N[(N[(y - N[(3.0 * x + 1.0), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.3e-57

   1. Initial program 67.6%

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

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

      6. lift-*.f64 N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-/.f64 99.8

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

      15. lift-+.f64 N/A

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

      16. +-commutative N/A

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

      17. lower-+.f64 99.8

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

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

      20. lower-+.f64 99.8

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

      21. lift-+.f64 N/A

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

      22. +-commutative N/A

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

      23. lower-+.f64 99.8

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in x around inf

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

   7. Applied rewrites 33.2%

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

   if -2.3e-57 < y < 3.69999999999999992e139

   1. Initial program 75.8%

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

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

      6. lift-*.f64 N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-/.f64 99.8

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

      15. lift-+.f64 N/A

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

      16. +-commutative N/A

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

      17. lower-+.f64 99.8

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

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

      20. lower-+.f64 99.8

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

      21. lift-+.f64 N/A

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

      22. +-commutative N/A

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

      23. lower-+.f64 99.8

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

   4. Applied rewrites 99.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-/l/ N/A

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

      5. associate-*r/ N/A

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

      6. lift-+.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. +-commutative N/A

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

      10. lift-+.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. +-commutative N/A

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

      14. times-frac N/A

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

      15. lower-*.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. lift-+.f64 N/A

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

      18. +-commutative N/A

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

      19. lift-+.f64 N/A

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

      20. lower-/.f64 N/A

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

      21. +-commutative N/A

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

   6. Applied rewrites 99.8%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-/l/ N/A

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

      6. associate-*r/ N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 99.2

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 99.2

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

   8. Applied rewrites 99.2%

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

   if 3.69999999999999992e139 < y

   1. Initial program 45.9%

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

   3. Taylor expanded in y around inf

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

      1. unpow2 N/A

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

      2. associate-/r* N/A

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

      3. lower-/.f64 N/A

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

   5. Applied rewrites 81.1%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 81.2%

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

Alternative 6: 96.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_0 := 1 + \left(y + x\right)\\ \mathbf{if}\;y \leq -2.3 \cdot 10^{-57}:\\ \;\;\;\;1 \cdot \frac{\frac{y}{t\_0}}{y + x}\\ \mathbf{elif}\;y \leq 3.7 \cdot 10^{+139}:\\ \;\;\;\;\frac{\frac{y}{y + x} \cdot x}{t\_0 \cdot \left(y + x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-x}{y}}{-\left(y + x\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ 1.0 (+ y x))))
   (if (<= y -2.3e-57)
     (* 1.0 (/ (/ y t_0) (+ y x)))
     (if (<= y 3.7e+139)
       (/ (* (/ y (+ y x)) x) (* t_0 (+ y x)))
       (/ (/ (- x) y) (- (+ y x)))))))

C

assert(x < y);
double code(double x, double y) {
	double t_0 = 1.0 + (y + x);
	double tmp;
	if (y <= -2.3e-57) {
		tmp = 1.0 * ((y / t_0) / (y + x));
	} else if (y <= 3.7e+139) {
		tmp = ((y / (y + x)) * x) / (t_0 * (y + x));
	} else {
		tmp = (-x / y) / -(y + x);
	}
	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 = 1.0d0 + (y + x)
    if (y <= (-2.3d-57)) then
        tmp = 1.0d0 * ((y / t_0) / (y + x))
    else if (y <= 3.7d+139) then
        tmp = ((y / (y + x)) * x) / (t_0 * (y + x))
    else
        tmp = (-x / y) / -(y + x)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double t_0 = 1.0 + (y + x);
	double tmp;
	if (y <= -2.3e-57) {
		tmp = 1.0 * ((y / t_0) / (y + x));
	} else if (y <= 3.7e+139) {
		tmp = ((y / (y + x)) * x) / (t_0 * (y + x));
	} else {
		tmp = (-x / y) / -(y + x);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	t_0 = 1.0 + (y + x)
	tmp = 0
	if y <= -2.3e-57:
		tmp = 1.0 * ((y / t_0) / (y + x))
	elif y <= 3.7e+139:
		tmp = ((y / (y + x)) * x) / (t_0 * (y + x))
	else:
		tmp = (-x / y) / -(y + x)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	t_0 = Float64(1.0 + Float64(y + x))
	tmp = 0.0
	if (y <= -2.3e-57)
		tmp = Float64(1.0 * Float64(Float64(y / t_0) / Float64(y + x)));
	elseif (y <= 3.7e+139)
		tmp = Float64(Float64(Float64(y / Float64(y + x)) * x) / Float64(t_0 * Float64(y + x)));
	else
		tmp = Float64(Float64(Float64(-x) / y) / Float64(-Float64(y + x)));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	t_0 = 1.0 + (y + x);
	tmp = 0.0;
	if (y <= -2.3e-57)
		tmp = 1.0 * ((y / t_0) / (y + x));
	elseif (y <= 3.7e+139)
		tmp = ((y / (y + x)) * x) / (t_0 * (y + x));
	else
		tmp = (-x / y) / -(y + x);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := Block[{t$95$0 = N[(1.0 + N[(y + x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -2.3e-57], N[(1.0 * N[(N[(y / t$95$0), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 3.7e+139], N[(N[(N[(y / N[(y + x), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision] / N[(t$95$0 * N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[((-x) / y), $MachinePrecision] / (-N[(y + x), $MachinePrecision])), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.3e-57

   1. Initial program 67.6%

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

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

      6. lift-*.f64 N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-/.f64 99.8

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

      15. lift-+.f64 N/A

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

      16. +-commutative N/A

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

      17. lower-+.f64 99.8

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

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

      20. lower-+.f64 99.8

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

      21. lift-+.f64 N/A

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

      22. +-commutative N/A

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

      23. lower-+.f64 99.8

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in x around inf

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

   7. Applied rewrites 33.2%

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

   if -2.3e-57 < y < 3.69999999999999992e139

   1. Initial program 75.8%

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

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

      6. lift-*.f64 N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-/.f64 99.8

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

      15. lift-+.f64 N/A

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

      16. +-commutative N/A

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

      17. lower-+.f64 99.8

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

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

      20. lower-+.f64 99.8

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

      21. lift-+.f64 N/A

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

      22. +-commutative N/A

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

      23. lower-+.f64 99.8

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

   4. Applied rewrites 99.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-/l/ N/A

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

      5. associate-*r/ N/A

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

      6. lift-+.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. +-commutative N/A

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

      10. lift-+.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. +-commutative N/A

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

      14. times-frac N/A

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

      15. lower-*.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. lift-+.f64 N/A

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

      18. +-commutative N/A

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

      19. lift-+.f64 N/A

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

      20. lower-/.f64 N/A

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

      21. +-commutative N/A

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

   6. Applied rewrites 99.8%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-/l/ N/A

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

      6. associate-*r/ N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 99.2

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 99.2

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

   8. Applied rewrites 99.2%

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

   if 3.69999999999999992e139 < y

   1. Initial program 45.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. 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. associate-*l/ N/A

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

      6. lift-*.f64 N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-/.f64 99.8

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

      15. lift-+.f64 N/A

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

      16. +-commutative N/A

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

      17. lower-+.f64 99.8

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

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

      20. lower-+.f64 99.8

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

      21. lift-+.f64 N/A

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

      22. +-commutative N/A

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

      23. lower-+.f64 99.8

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

   4. Applied rewrites 99.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-/l/ N/A

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

      5. associate-*r/ N/A

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

      6. lift-+.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. +-commutative N/A

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

      10. lift-+.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. +-commutative N/A

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

      14. times-frac N/A

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

      15. lower-*.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. lift-+.f64 N/A

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

      18. +-commutative N/A

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

      19. lift-+.f64 N/A

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

      20. lower-/.f64 N/A

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

      21. +-commutative N/A

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

   6. Applied rewrites 99.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. frac-times N/A

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

      5. associate-*r/ N/A

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

      6. lift-/.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-+.f64 N/A

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

      10. associate-/l/ N/A

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

      11. frac-times N/A

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

      12. sqr-neg-rev N/A

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

      13. associate-/r* N/A

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

      14. lower-/.f64 N/A

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

   8. Applied rewrites 99.8%

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

   9. Taylor expanded in y around inf

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

       1. associate-*r/ N/A

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

       2. lower-/.f64 N/A

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

       3. mul-1-neg N/A

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

       4. lower-neg.f64 81.8

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

   11. Applied rewrites 81.8%

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

Alternative 7: 96.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_0 := 1 + \left(y + x\right)\\ \mathbf{if}\;y \leq -6.8 \cdot 10^{-50}:\\ \;\;\;\;1 \cdot \frac{\frac{y}{t\_0}}{y + x}\\ \mathbf{elif}\;y \leq 3.7 \cdot 10^{+139}:\\ \;\;\;\;\frac{y}{y + x} \cdot \frac{x}{t\_0 \cdot \left(y + x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-x}{y}}{-\left(y + x\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ 1.0 (+ y x))))
   (if (<= y -6.8e-50)
     (* 1.0 (/ (/ y t_0) (+ y x)))
     (if (<= y 3.7e+139)
       (* (/ y (+ y x)) (/ x (* t_0 (+ y x))))
       (/ (/ (- x) y) (- (+ y x)))))))

C

assert(x < y);
double code(double x, double y) {
	double t_0 = 1.0 + (y + x);
	double tmp;
	if (y <= -6.8e-50) {
		tmp = 1.0 * ((y / t_0) / (y + x));
	} else if (y <= 3.7e+139) {
		tmp = (y / (y + x)) * (x / (t_0 * (y + x)));
	} else {
		tmp = (-x / y) / -(y + x);
	}
	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 = 1.0d0 + (y + x)
    if (y <= (-6.8d-50)) then
        tmp = 1.0d0 * ((y / t_0) / (y + x))
    else if (y <= 3.7d+139) then
        tmp = (y / (y + x)) * (x / (t_0 * (y + x)))
    else
        tmp = (-x / y) / -(y + x)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double t_0 = 1.0 + (y + x);
	double tmp;
	if (y <= -6.8e-50) {
		tmp = 1.0 * ((y / t_0) / (y + x));
	} else if (y <= 3.7e+139) {
		tmp = (y / (y + x)) * (x / (t_0 * (y + x)));
	} else {
		tmp = (-x / y) / -(y + x);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	t_0 = 1.0 + (y + x)
	tmp = 0
	if y <= -6.8e-50:
		tmp = 1.0 * ((y / t_0) / (y + x))
	elif y <= 3.7e+139:
		tmp = (y / (y + x)) * (x / (t_0 * (y + x)))
	else:
		tmp = (-x / y) / -(y + x)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	t_0 = Float64(1.0 + Float64(y + x))
	tmp = 0.0
	if (y <= -6.8e-50)
		tmp = Float64(1.0 * Float64(Float64(y / t_0) / Float64(y + x)));
	elseif (y <= 3.7e+139)
		tmp = Float64(Float64(y / Float64(y + x)) * Float64(x / Float64(t_0 * Float64(y + x))));
	else
		tmp = Float64(Float64(Float64(-x) / y) / Float64(-Float64(y + x)));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	t_0 = 1.0 + (y + x);
	tmp = 0.0;
	if (y <= -6.8e-50)
		tmp = 1.0 * ((y / t_0) / (y + x));
	elseif (y <= 3.7e+139)
		tmp = (y / (y + x)) * (x / (t_0 * (y + x)));
	else
		tmp = (-x / y) / -(y + x);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := Block[{t$95$0 = N[(1.0 + N[(y + x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -6.8e-50], N[(1.0 * N[(N[(y / t$95$0), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 3.7e+139], N[(N[(y / N[(y + x), $MachinePrecision]), $MachinePrecision] * N[(x / N[(t$95$0 * N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[((-x) / y), $MachinePrecision] / (-N[(y + x), $MachinePrecision])), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < -6.80000000000000029e-50

   1. Initial program 66.4%

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

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

      6. lift-*.f64 N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-/.f64 99.8

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

      15. lift-+.f64 N/A

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

      16. +-commutative N/A

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

      17. lower-+.f64 99.8

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

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

      20. lower-+.f64 99.8

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

      21. lift-+.f64 N/A

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

      22. +-commutative N/A

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

      23. lower-+.f64 99.8

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in x around inf

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

   7. Applied rewrites 33.1%

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

   if -6.80000000000000029e-50 < y < 3.69999999999999992e139

   1. Initial program 76.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 99.1

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

      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 99.1

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

      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 99.1

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

      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 99.1

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

   4. Applied rewrites 99.1%

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

   if 3.69999999999999992e139 < y

   1. Initial program 45.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. 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. associate-*l/ N/A

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

      6. lift-*.f64 N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-/.f64 99.8

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

      15. lift-+.f64 N/A

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

      16. +-commutative N/A

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

      17. lower-+.f64 99.8

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

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

      20. lower-+.f64 99.8

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

      21. lift-+.f64 N/A

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

      22. +-commutative N/A

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

      23. lower-+.f64 99.8

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

   4. Applied rewrites 99.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-/l/ N/A

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

      5. associate-*r/ N/A

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

      6. lift-+.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. +-commutative N/A

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

      10. lift-+.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. +-commutative N/A

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

      14. times-frac N/A

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

      15. lower-*.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. lift-+.f64 N/A

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

      18. +-commutative N/A

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

      19. lift-+.f64 N/A

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

      20. lower-/.f64 N/A

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

      21. +-commutative N/A

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

   6. Applied rewrites 99.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. frac-times N/A

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

      5. associate-*r/ N/A

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

      6. lift-/.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-+.f64 N/A

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

      10. associate-/l/ N/A

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

      11. frac-times N/A

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

      12. sqr-neg-rev N/A

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

      13. associate-/r* N/A

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

      14. lower-/.f64 N/A

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

   8. Applied rewrites 99.8%

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

   9. Taylor expanded in y around inf

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

       1. associate-*r/ N/A

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

       2. lower-/.f64 N/A

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

       3. mul-1-neg N/A

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

       4. lower-neg.f64 81.8

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

   11. Applied rewrites 81.8%

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

Alternative 8: 88.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 2.65 \cdot 10^{-145}:\\ \;\;\;\;1 \cdot \frac{\frac{y}{1 + \left(y + x\right)}}{y + x}\\ \mathbf{elif}\;y \leq 5.5 \cdot 10^{+68}:\\ \;\;\;\;\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + x}}{\left(y + x\right) + 1} \cdot 1\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 2.65e-145)
   (* 1.0 (/ (/ y (+ 1.0 (+ y x))) (+ y x)))
   (if (<= y 5.5e+68)
     (/ (* x y) (* (* (+ x y) (+ x y)) (+ (+ x y) 1.0)))
     (* (/ (/ x (+ y x)) (+ (+ y x) 1.0)) 1.0))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 2.65e-145) {
		tmp = 1.0 * ((y / (1.0 + (y + x))) / (y + x));
	} else if (y <= 5.5e+68) {
		tmp = (x * y) / (((x + y) * (x + y)) * ((x + y) + 1.0));
	} else {
		tmp = ((x / (y + x)) / ((y + x) + 1.0)) * 1.0;
	}
	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 <= 2.65d-145) then
        tmp = 1.0d0 * ((y / (1.0d0 + (y + x))) / (y + x))
    else if (y <= 5.5d+68) then
        tmp = (x * y) / (((x + y) * (x + y)) * ((x + y) + 1.0d0))
    else
        tmp = ((x / (y + x)) / ((y + x) + 1.0d0)) * 1.0d0
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (y <= 2.65e-145) {
		tmp = 1.0 * ((y / (1.0 + (y + x))) / (y + x));
	} else if (y <= 5.5e+68) {
		tmp = (x * y) / (((x + y) * (x + y)) * ((x + y) + 1.0));
	} else {
		tmp = ((x / (y + x)) / ((y + x) + 1.0)) * 1.0;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if y <= 2.65e-145:
		tmp = 1.0 * ((y / (1.0 + (y + x))) / (y + x))
	elif y <= 5.5e+68:
		tmp = (x * y) / (((x + y) * (x + y)) * ((x + y) + 1.0))
	else:
		tmp = ((x / (y + x)) / ((y + x) + 1.0)) * 1.0
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= 2.65e-145)
		tmp = Float64(1.0 * Float64(Float64(y / Float64(1.0 + Float64(y + x))) / Float64(y + x)));
	elseif (y <= 5.5e+68)
		tmp = Float64(Float64(x * y) / Float64(Float64(Float64(x + y) * Float64(x + y)) * Float64(Float64(x + y) + 1.0)));
	else
		tmp = Float64(Float64(Float64(x / Float64(y + x)) / Float64(Float64(y + x) + 1.0)) * 1.0);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 2.65e-145)
		tmp = 1.0 * ((y / (1.0 + (y + x))) / (y + x));
	elseif (y <= 5.5e+68)
		tmp = (x * y) / (((x + y) * (x + y)) * ((x + y) + 1.0));
	else
		tmp = ((x / (y + x)) / ((y + x) + 1.0)) * 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[y, 2.65e-145], N[(1.0 * N[(N[(y / N[(1.0 + N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 5.5e+68], N[(N[(x * y), $MachinePrecision] / N[(N[(N[(x + y), $MachinePrecision] * N[(x + y), $MachinePrecision]), $MachinePrecision] * N[(N[(x + y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x / N[(y + x), $MachinePrecision]), $MachinePrecision] / N[(N[(y + x), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 2.65e-145

   1. Initial program 68.0%

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

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

      6. lift-*.f64 N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-/.f64 99.8

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

      15. lift-+.f64 N/A

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

      16. +-commutative N/A

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

      17. lower-+.f64 99.8

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

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

      20. lower-+.f64 99.8

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

      21. lift-+.f64 N/A

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

      22. +-commutative N/A

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

      23. lower-+.f64 99.8

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in x around inf

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

   7. Applied rewrites 56.5%

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

   if 2.65e-145 < y < 5.5000000000000004e68

   1. Initial program 83.9%

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

   if 5.5000000000000004e68 < y

   1. Initial program 53.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. 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. associate-*l/ N/A

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

      6. lift-*.f64 N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-/.f64 99.7

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

      15. lift-+.f64 N/A

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

      16. +-commutative N/A

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

      17. lower-+.f64 99.7

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

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

      20. lower-+.f64 99.7

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

      21. lift-+.f64 N/A

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

      22. +-commutative N/A

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

      23. lower-+.f64 99.7

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

   4. Applied rewrites 99.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-/l/ N/A

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

      5. associate-*r/ N/A

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

      6. lift-+.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. +-commutative N/A

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

      10. lift-+.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. +-commutative N/A

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

      14. times-frac N/A

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

      15. lower-*.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. lift-+.f64 N/A

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

      18. +-commutative N/A

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

      19. lift-+.f64 N/A

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

      20. lower-/.f64 N/A

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

      21. +-commutative N/A

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

   6. Applied rewrites 99.8%

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

   7. Taylor expanded in x around 0

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

   9. Applied rewrites 84.3%

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

Alternative 9: 99.8% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

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 + x)) * ((y / (1.0d0 + (y + x))) / (y + x))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 68.2%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. 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. associate-*l/ N/A

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

   6. lift-*.f64 N/A

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

   7. times-frac N/A

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

   8. lower-*.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. lift-+.f64 N/A

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

   11. +-commutative N/A

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

   12. lower-+.f64 N/A

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

   13. lower-/.f64 N/A

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

   14. lower-/.f64 99.8

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

   15. lift-+.f64 N/A

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

   16. +-commutative N/A

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

   17. lower-+.f64 99.8

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

   18. lift-+.f64 N/A

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

   19. +-commutative N/A

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

   20. lower-+.f64 99.8

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

   21. lift-+.f64 N/A

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

   22. +-commutative N/A

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

   23. lower-+.f64 99.8

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

4. Applied rewrites 99.8%

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

Alternative 10: 84.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -6.8 \cdot 10^{+192}:\\ \;\;\;\;1 \cdot \frac{\frac{y}{1 + \left(y + x\right)}}{y + x}\\ \mathbf{elif}\;x \leq -4.7 \cdot 10^{-71}:\\ \;\;\;\;\frac{1 \cdot y}{\left(\left(x + y\right) + 1\right) \cdot \left(x + y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + x}}{\left(y + x\right) + 1} \cdot 1\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -6.8e+192)
   (* 1.0 (/ (/ y (+ 1.0 (+ y x))) (+ y x)))
   (if (<= x -4.7e-71)
     (/ (* 1.0 y) (* (+ (+ x y) 1.0) (+ x y)))
     (* (/ (/ x (+ y x)) (+ (+ y x) 1.0)) 1.0))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -6.8e+192) {
		tmp = 1.0 * ((y / (1.0 + (y + x))) / (y + x));
	} else if (x <= -4.7e-71) {
		tmp = (1.0 * y) / (((x + y) + 1.0) * (x + y));
	} else {
		tmp = ((x / (y + x)) / ((y + x) + 1.0)) * 1.0;
	}
	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 <= (-6.8d+192)) then
        tmp = 1.0d0 * ((y / (1.0d0 + (y + x))) / (y + x))
    else if (x <= (-4.7d-71)) then
        tmp = (1.0d0 * y) / (((x + y) + 1.0d0) * (x + y))
    else
        tmp = ((x / (y + x)) / ((y + x) + 1.0d0)) * 1.0d0
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (x <= -6.8e+192) {
		tmp = 1.0 * ((y / (1.0 + (y + x))) / (y + x));
	} else if (x <= -4.7e-71) {
		tmp = (1.0 * y) / (((x + y) + 1.0) * (x + y));
	} else {
		tmp = ((x / (y + x)) / ((y + x) + 1.0)) * 1.0;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -6.8e+192:
		tmp = 1.0 * ((y / (1.0 + (y + x))) / (y + x))
	elif x <= -4.7e-71:
		tmp = (1.0 * y) / (((x + y) + 1.0) * (x + y))
	else:
		tmp = ((x / (y + x)) / ((y + x) + 1.0)) * 1.0
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -6.8e+192)
		tmp = Float64(1.0 * Float64(Float64(y / Float64(1.0 + Float64(y + x))) / Float64(y + x)));
	elseif (x <= -4.7e-71)
		tmp = Float64(Float64(1.0 * y) / Float64(Float64(Float64(x + y) + 1.0) * Float64(x + y)));
	else
		tmp = Float64(Float64(Float64(x / Float64(y + x)) / Float64(Float64(y + x) + 1.0)) * 1.0);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -6.8e+192)
		tmp = 1.0 * ((y / (1.0 + (y + x))) / (y + x));
	elseif (x <= -4.7e-71)
		tmp = (1.0 * y) / (((x + y) + 1.0) * (x + y));
	else
		tmp = ((x / (y + x)) / ((y + x) + 1.0)) * 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[x, -6.8e+192], N[(1.0 * N[(N[(y / N[(1.0 + N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -4.7e-71], N[(N[(1.0 * y), $MachinePrecision] / N[(N[(N[(x + y), $MachinePrecision] + 1.0), $MachinePrecision] * N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x / N[(y + x), $MachinePrecision]), $MachinePrecision] / N[(N[(y + x), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -6.79999999999999992e192

   1. Initial program 35.5%

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

         \[\leadsto \color{blue}{\frac{x \cdot \frac{y}{\left(x + y\right) + 1}}{\left(x + y\right) \cdot \left(x + y\right)}} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot \frac{y}{\left(x + y\right) + 1}}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right)}} \]

      7. times-frac N/A

         \[\leadsto \color{blue}{\frac{x}{x + y} \cdot \frac{\frac{y}{\left(x + y\right) + 1}}{x + y}} \]

      8. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{x + y} \cdot \frac{\frac{y}{\left(x + y\right) + 1}}{x + y}} \]

      9. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{x + y}} \cdot \frac{\frac{y}{\left(x + y\right) + 1}}{x + y} \]

      10. lift-+.f64 N/A

          \[\leadsto \frac{x}{\color{blue}{x + y}} \cdot \frac{\frac{y}{\left(x + y\right) + 1}}{x + y} \]

      11. +-commutative N/A

          \[\leadsto \frac{x}{\color{blue}{y + x}} \cdot \frac{\frac{y}{\left(x + y\right) + 1}}{x + y} \]

      12. lower-+.f64 N/A

          \[\leadsto \frac{x}{\color{blue}{y + x}} \cdot \frac{\frac{y}{\left(x + y\right) + 1}}{x + y} \]

      13. lower-/.f64 N/A

          \[\leadsto \frac{x}{y + x} \cdot \color{blue}{\frac{\frac{y}{\left(x + y\right) + 1}}{x + y}} \]

      14. lower-/.f64 99.8

          \[\leadsto \frac{x}{y + x} \cdot \frac{\color{blue}{\frac{y}{\left(x + y\right) + 1}}}{x + y} \]

      15. lift-+.f64 N/A

          \[\leadsto \frac{x}{y + x} \cdot \frac{\frac{y}{\color{blue}{\left(x + y\right) + 1}}}{x + y} \]

      16. +-commutative N/A

          \[\leadsto \frac{x}{y + x} \cdot \frac{\frac{y}{\color{blue}{1 + \left(x + y\right)}}}{x + y} \]

      17. lower-+.f64 99.8

          \[\leadsto \frac{x}{y + x} \cdot \frac{\frac{y}{\color{blue}{1 + \left(x + y\right)}}}{x + y} \]

      18. lift-+.f64 N/A

          \[\leadsto \frac{x}{y + x} \cdot \frac{\frac{y}{1 + \color{blue}{\left(x + y\right)}}}{x + y} \]

      19. +-commutative N/A

          \[\leadsto \frac{x}{y + x} \cdot \frac{\frac{y}{1 + \color{blue}{\left(y + x\right)}}}{x + y} \]

      20. lower-+.f64 99.8

          \[\leadsto \frac{x}{y + x} \cdot \frac{\frac{y}{1 + \color{blue}{\left(y + x\right)}}}{x + y} \]

      21. lift-+.f64 N/A

          \[\leadsto \frac{x}{y + x} \cdot \frac{\frac{y}{1 + \left(y + x\right)}}{\color{blue}{x + y}} \]

      22. +-commutative N/A

          \[\leadsto \frac{x}{y + x} \cdot \frac{\frac{y}{1 + \left(y + x\right)}}{\color{blue}{y + x}} \]

      23. lower-+.f64 99.8

          \[\leadsto \frac{x}{y + x} \cdot \frac{\frac{y}{1 + \left(y + x\right)}}{\color{blue}{y + x}} \]

   4. Applied rewrites 99.8%

      \[\leadsto \color{blue}{\frac{x}{y + x} \cdot \frac{\frac{y}{1 + \left(y + x\right)}}{y + x}} \]

   5. Taylor expanded in x around inf

      \[\leadsto \color{blue}{1} \cdot \frac{\frac{y}{1 + \left(y + x\right)}}{y + x} \]
   6. Step-by-step derivation

   7. Applied rewrites 92.2%

      \[\leadsto \color{blue}{1} \cdot \frac{\frac{y}{1 + \left(y + x\right)}}{y + x} \]

   if -6.79999999999999992e192 < x < -4.69999999999999996e-71

   1. Initial program 73.5%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   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. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{x \cdot \frac{y}{\left(x + y\right) + 1}}{\left(x + y\right) \cdot \left(x + y\right)}} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot \frac{y}{\left(x + y\right) + 1}}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right)}} \]

      7. times-frac N/A

         \[\leadsto \color{blue}{\frac{x}{x + y} \cdot \frac{\frac{y}{\left(x + y\right) + 1}}{x + y}} \]

      8. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{x + y} \cdot \frac{\frac{y}{\left(x + y\right) + 1}}{x + y}} \]

      9. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{x + y}} \cdot \frac{\frac{y}{\left(x + y\right) + 1}}{x + y} \]

      10. lift-+.f64 N/A

          \[\leadsto \frac{x}{\color{blue}{x + y}} \cdot \frac{\frac{y}{\left(x + y\right) + 1}}{x + y} \]

      11. +-commutative N/A

          \[\leadsto \frac{x}{\color{blue}{y + x}} \cdot \frac{\frac{y}{\left(x + y\right) + 1}}{x + y} \]

      12. lower-+.f64 N/A

          \[\leadsto \frac{x}{\color{blue}{y + x}} \cdot \frac{\frac{y}{\left(x + y\right) + 1}}{x + y} \]

      13. lower-/.f64 N/A

          \[\leadsto \frac{x}{y + x} \cdot \color{blue}{\frac{\frac{y}{\left(x + y\right) + 1}}{x + y}} \]

      14. lower-/.f64 99.7

          \[\leadsto \frac{x}{y + x} \cdot \frac{\color{blue}{\frac{y}{\left(x + y\right) + 1}}}{x + y} \]

      15. lift-+.f64 N/A

          \[\leadsto \frac{x}{y + x} \cdot \frac{\frac{y}{\color{blue}{\left(x + y\right) + 1}}}{x + y} \]

      16. +-commutative N/A

          \[\leadsto \frac{x}{y + x} \cdot \frac{\frac{y}{\color{blue}{1 + \left(x + y\right)}}}{x + y} \]

      17. lower-+.f64 99.7

          \[\leadsto \frac{x}{y + x} \cdot \frac{\frac{y}{\color{blue}{1 + \left(x + y\right)}}}{x + y} \]

      18. lift-+.f64 N/A

          \[\leadsto \frac{x}{y + x} \cdot \frac{\frac{y}{1 + \color{blue}{\left(x + y\right)}}}{x + y} \]

      19. +-commutative N/A

          \[\leadsto \frac{x}{y + x} \cdot \frac{\frac{y}{1 + \color{blue}{\left(y + x\right)}}}{x + y} \]

      20. lower-+.f64 99.7

          \[\leadsto \frac{x}{y + x} \cdot \frac{\frac{y}{1 + \color{blue}{\left(y + x\right)}}}{x + y} \]

      21. lift-+.f64 N/A

          \[\leadsto \frac{x}{y + x} \cdot \frac{\frac{y}{1 + \left(y + x\right)}}{\color{blue}{x + y}} \]

      22. +-commutative N/A

          \[\leadsto \frac{x}{y + x} \cdot \frac{\frac{y}{1 + \left(y + x\right)}}{\color{blue}{y + x}} \]

      23. lower-+.f64 99.7

          \[\leadsto \frac{x}{y + x} \cdot \frac{\frac{y}{1 + \left(y + x\right)}}{\color{blue}{y + x}} \]

   4. Applied rewrites 99.7%

      \[\leadsto \color{blue}{\frac{x}{y + x} \cdot \frac{\frac{y}{1 + \left(y + x\right)}}{y + x}} \]

   5. Taylor expanded in x around inf

      \[\leadsto \color{blue}{1} \cdot \frac{\frac{y}{1 + \left(y + x\right)}}{y + x} \]
   6. Step-by-step derivation

   7. Applied rewrites 56.6%

      \[\leadsto \color{blue}{1} \cdot \frac{\frac{y}{1 + \left(y + x\right)}}{y + x} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{1 \cdot \frac{\frac{y}{1 + \left(y + x\right)}}{y + x}} \]

      2. lift-/.f64 N/A

         \[\leadsto 1 \cdot \color{blue}{\frac{\frac{y}{1 + \left(y + x\right)}}{y + x}} \]

      3. lift-/.f64 N/A

         \[\leadsto 1 \cdot \frac{\color{blue}{\frac{y}{1 + \left(y + x\right)}}}{y + x} \]

      4. associate-/l/ N/A

         \[\leadsto 1 \cdot \color{blue}{\frac{y}{\left(1 + \left(y + x\right)\right) \cdot \left(y + x\right)}} \]

      5. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{1 \cdot y}{\left(1 + \left(y + x\right)\right) \cdot \left(y + x\right)}} \]

      6. lift-+.f64 N/A

         \[\leadsto \frac{1 \cdot y}{\color{blue}{\left(1 + \left(y + x\right)\right)} \cdot \left(y + x\right)} \]

      7. lift-+.f64 N/A

         \[\leadsto \frac{1 \cdot y}{\left(1 + \color{blue}{\left(y + x\right)}\right) \cdot \left(y + x\right)} \]

      8. +-commutative N/A

         \[\leadsto \frac{1 \cdot y}{\left(1 + \color{blue}{\left(x + y\right)}\right) \cdot \left(y + x\right)} \]

      9. +-commutative N/A

         \[\leadsto \frac{1 \cdot y}{\color{blue}{\left(\left(x + y\right) + 1\right)} \cdot \left(y + x\right)} \]

      10. lift-+.f64 N/A

          \[\leadsto \frac{1 \cdot y}{\left(\left(x + y\right) + 1\right) \cdot \color{blue}{\left(y + x\right)}} \]

      11. +-commutative N/A

          \[\leadsto \frac{1 \cdot y}{\left(\left(x + y\right) + 1\right) \cdot \color{blue}{\left(x + y\right)}} \]

      12. *-commutative N/A

          \[\leadsto \frac{1 \cdot y}{\color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      13. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{1 \cdot y}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{\color{blue}{1 \cdot y}}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      15. *-commutative N/A

          \[\leadsto \frac{1 \cdot y}{\color{blue}{\left(\left(x + y\right) + 1\right) \cdot \left(x + y\right)}} \]

      16. lower-*.f64 N/A

          \[\leadsto \frac{1 \cdot y}{\color{blue}{\left(\left(x + y\right) + 1\right) \cdot \left(x + y\right)}} \]

      17. lower-+.f64 N/A

          \[\leadsto \frac{1 \cdot y}{\color{blue}{\left(\left(x + y\right) + 1\right)} \cdot \left(x + y\right)} \]

      18. lower-+.f64 N/A

          \[\leadsto \frac{1 \cdot y}{\left(\color{blue}{\left(x + y\right)} + 1\right) \cdot \left(x + y\right)} \]

      19. lower-+.f64 76.7

          \[\leadsto \frac{1 \cdot y}{\left(\left(x + y\right) + 1\right) \cdot \color{blue}{\left(x + y\right)}} \]

   9. Applied rewrites 76.7%

      \[\leadsto \color{blue}{\frac{1 \cdot y}{\left(\left(x + y\right) + 1\right) \cdot \left(x + y\right)}} \]

   if -4.69999999999999996e-71 < x

   1. Initial program 70.1%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing
   3. 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. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{x \cdot \frac{y}{\left(x + y\right) + 1}}{\left(x + y\right) \cdot \left(x + y\right)}} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot \frac{y}{\left(x + y\right) + 1}}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right)}} \]

      7. times-frac N/A

         \[\leadsto \color{blue}{\frac{x}{x + y} \cdot \frac{\frac{y}{\left(x + y\right) + 1}}{x + y}} \]

      8. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{x + y} \cdot \frac{\frac{y}{\left(x + y\right) + 1}}{x + y}} \]

      9. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{x + y}} \cdot \frac{\frac{y}{\left(x + y\right) + 1}}{x + y} \]

      10. lift-+.f64 N/A

          \[\leadsto \frac{x}{\color{blue}{x + y}} \cdot \frac{\frac{y}{\left(x + y\right) + 1}}{x + y} \]

      11. +-commutative N/A

          \[\leadsto \frac{x}{\color{blue}{y + x}} \cdot \frac{\frac{y}{\left(x + y\right) + 1}}{x + y} \]

      12. lower-+.f64 N/A

          \[\leadsto \frac{x}{\color{blue}{y + x}} \cdot \frac{\frac{y}{\left(x + y\right) + 1}}{x + y} \]

      13. lower-/.f64 N/A

          \[\leadsto \frac{x}{y + x} \cdot \color{blue}{\frac{\frac{y}{\left(x + y\right) + 1}}{x + y}} \]

      14. lower-/.f64 99.8

          \[\leadsto \frac{x}{y + x} \cdot \frac{\color{blue}{\frac{y}{\left(x + y\right) + 1}}}{x + y} \]

      15. lift-+.f64 N/A

          \[\leadsto \frac{x}{y + x} \cdot \frac{\frac{y}{\color{blue}{\left(x + y\right) + 1}}}{x + y} \]

      16. +-commutative N/A

          \[\leadsto \frac{x}{y + x} \cdot \frac{\frac{y}{\color{blue}{1 + \left(x + y\right)}}}{x + y} \]

      17. lower-+.f64 99.8

          \[\leadsto \frac{x}{y + x} \cdot \frac{\frac{y}{\color{blue}{1 + \left(x + y\right)}}}{x + y} \]

      18. lift-+.f64 N/A

          \[\leadsto \frac{x}{y + x} \cdot \frac{\frac{y}{1 + \color{blue}{\left(x + y\right)}}}{x + y} \]

      19. +-commutative N/A

          \[\leadsto \frac{x}{y + x} \cdot \frac{\frac{y}{1 + \color{blue}{\left(y + x\right)}}}{x + y} \]

      20. lower-+.f64 99.8

          \[\leadsto \frac{x}{y + x} \cdot \frac{\frac{y}{1 + \color{blue}{\left(y + x\right)}}}{x + y} \]

      21. lift-+.f64 N/A

          \[\leadsto \frac{x}{y + x} \cdot \frac{\frac{y}{1 + \left(y + x\right)}}{\color{blue}{x + y}} \]

      22. +-commutative N/A

          \[\leadsto \frac{x}{y + x} \cdot \frac{\frac{y}{1 + \left(y + x\right)}}{\color{blue}{y + x}} \]

      23. lower-+.f64 99.8

          \[\leadsto \frac{x}{y + x} \cdot \frac{\frac{y}{1 + \left(y + x\right)}}{\color{blue}{y + x}} \]

   4. Applied rewrites 99.8%

      \[\leadsto \color{blue}{\frac{x}{y + x} \cdot \frac{\frac{y}{1 + \left(y + x\right)}}{y + x}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{y + x} \cdot \frac{\frac{y}{1 + \left(y + x\right)}}{y + x}} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{x}{y + x} \cdot \color{blue}{\frac{\frac{y}{1 + \left(y + x\right)}}{y + x}} \]

      3. lift-/.f64 N/A

         \[\leadsto \frac{x}{y + x} \cdot \frac{\color{blue}{\frac{y}{1 + \left(y + x\right)}}}{y + x} \]

      4. associate-/l/ N/A

         \[\leadsto \frac{x}{y + x} \cdot \color{blue}{\frac{y}{\left(1 + \left(y + x\right)\right) \cdot \left(y + x\right)}} \]

      5. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{\frac{x}{y + x} \cdot y}{\left(1 + \left(y + x\right)\right) \cdot \left(y + x\right)}} \]

      6. lift-+.f64 N/A

         \[\leadsto \frac{\frac{x}{y + x} \cdot y}{\color{blue}{\left(1 + \left(y + x\right)\right)} \cdot \left(y + x\right)} \]

      7. lift-+.f64 N/A

         \[\leadsto \frac{\frac{x}{y + x} \cdot y}{\left(1 + \color{blue}{\left(y + x\right)}\right) \cdot \left(y + x\right)} \]

      8. +-commutative N/A

         \[\leadsto \frac{\frac{x}{y + x} \cdot y}{\left(1 + \color{blue}{\left(x + y\right)}\right) \cdot \left(y + x\right)} \]

      9. +-commutative N/A

         \[\leadsto \frac{\frac{x}{y + x} \cdot y}{\color{blue}{\left(\left(x + y\right) + 1\right)} \cdot \left(y + x\right)} \]

      10. lift-+.f64 N/A

          \[\leadsto \frac{\frac{x}{y + x} \cdot y}{\left(\color{blue}{\left(x + y\right)} + 1\right) \cdot \left(y + x\right)} \]

      11. lift-+.f64 N/A

          \[\leadsto \frac{\frac{x}{y + x} \cdot y}{\color{blue}{\left(\left(x + y\right) + 1\right)} \cdot \left(y + x\right)} \]

      12. lift-+.f64 N/A

          \[\leadsto \frac{\frac{x}{y + x} \cdot y}{\left(\left(x + y\right) + 1\right) \cdot \color{blue}{\left(y + x\right)}} \]

      13. +-commutative N/A

          \[\leadsto \frac{\frac{x}{y + x} \cdot y}{\left(\left(x + y\right) + 1\right) \cdot \color{blue}{\left(x + y\right)}} \]

      14. times-frac N/A

          \[\leadsto \color{blue}{\frac{\frac{x}{y + x}}{\left(x + y\right) + 1} \cdot \frac{y}{x + y}} \]

      15. lower-*.f64 N/A

          \[\leadsto \color{blue}{\frac{\frac{x}{y + x}}{\left(x + y\right) + 1} \cdot \frac{y}{x + y}} \]

      16. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{\frac{x}{y + x}}{\left(x + y\right) + 1}} \cdot \frac{y}{x + y} \]

      17. lift-+.f64 N/A

          \[\leadsto \frac{\frac{x}{y + x}}{\color{blue}{\left(x + y\right)} + 1} \cdot \frac{y}{x + y} \]

      18. +-commutative N/A

          \[\leadsto \frac{\frac{x}{y + x}}{\color{blue}{\left(y + x\right)} + 1} \cdot \frac{y}{x + y} \]

      19. lift-+.f64 N/A

          \[\leadsto \frac{\frac{x}{y + x}}{\color{blue}{\left(y + x\right)} + 1} \cdot \frac{y}{x + y} \]

      20. lower-/.f64 N/A

          \[\leadsto \frac{\frac{x}{y + x}}{\left(y + x\right) + 1} \cdot \color{blue}{\frac{y}{x + y}} \]

      21. +-commutative N/A

          \[\leadsto \frac{\frac{x}{y + x}}{\left(y + x\right) + 1} \cdot \frac{y}{\color{blue}{y + x}} \]

   6. Applied rewrites 99.8%

      \[\leadsto \color{blue}{\frac{\frac{x}{y + x}}{\left(y + x\right) + 1} \cdot \frac{y}{y + x}} \]

   7. Taylor expanded in x around 0

      \[\leadsto \frac{\frac{x}{y + x}}{\left(y + x\right) + 1} \cdot \color{blue}{1} \]
   8. Step-by-step derivation

   9. Applied rewrites 61.9%

      \[\leadsto \frac{\frac{x}{y + x}}{\left(y + x\right) + 1} \cdot \color{blue}{1} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 11: 84.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_0 := -\left(y + x\right)\\ \mathbf{if}\;x \leq -6.8 \cdot 10^{+192}:\\ \;\;\;\;\frac{\frac{-y}{x}}{t\_0}\\ \mathbf{elif}\;x \leq -1.2 \cdot 10^{-56}:\\ \;\;\;\;\frac{1 \cdot y}{\left(\left(x + y\right) + 1\right) \cdot \left(x + y\right)}\\ \mathbf{elif}\;x \leq 8.2 \cdot 10^{-6}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(y, y, y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-x}{y}}{t\_0}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (+ y x))))
   (if (<= x -6.8e+192)
     (/ (/ (- y) x) t_0)
     (if (<= x -1.2e-56)
       (/ (* 1.0 y) (* (+ (+ x y) 1.0) (+ x y)))
       (if (<= x 8.2e-6) (/ x (fma y y y)) (/ (/ (- x) y) t_0))))))

C

assert(x < y);
double code(double x, double y) {
	double t_0 = -(y + x);
	double tmp;
	if (x <= -6.8e+192) {
		tmp = (-y / x) / t_0;
	} else if (x <= -1.2e-56) {
		tmp = (1.0 * y) / (((x + y) + 1.0) * (x + y));
	} else if (x <= 8.2e-6) {
		tmp = x / fma(y, y, y);
	} else {
		tmp = (-x / y) / t_0;
	}
	return tmp;
}

Julia

x, y = sort([x, y])
function code(x, y)
	t_0 = Float64(-Float64(y + x))
	tmp = 0.0
	if (x <= -6.8e+192)
		tmp = Float64(Float64(Float64(-y) / x) / t_0);
	elseif (x <= -1.2e-56)
		tmp = Float64(Float64(1.0 * y) / Float64(Float64(Float64(x + y) + 1.0) * Float64(x + y)));
	elseif (x <= 8.2e-6)
		tmp = Float64(x / fma(y, y, y));
	else
		tmp = Float64(Float64(Float64(-x) / y) / t_0);
	end
	return tmp
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := Block[{t$95$0 = (-N[(y + x), $MachinePrecision])}, If[LessEqual[x, -6.8e+192], N[(N[((-y) / x), $MachinePrecision] / t$95$0), $MachinePrecision], If[LessEqual[x, -1.2e-56], N[(N[(1.0 * y), $MachinePrecision] / N[(N[(N[(x + y), $MachinePrecision] + 1.0), $MachinePrecision] * N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 8.2e-6], N[(x / N[(y * y + y), $MachinePrecision]), $MachinePrecision], N[(N[((-x) / y), $MachinePrecision] / t$95$0), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -6.79999999999999992e192

   1. Initial program 35.5%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   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. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{x \cdot \frac{y}{\left(x + y\right) + 1}}{\left(x + y\right) \cdot \left(x + y\right)}} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot \frac{y}{\left(x + y\right) + 1}}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right)}} \]

      7. times-frac N/A

         \[\leadsto \color{blue}{\frac{x}{x + y} \cdot \frac{\frac{y}{\left(x + y\right) + 1}}{x + y}} \]

      8. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{x + y} \cdot \frac{\frac{y}{\left(x + y\right) + 1}}{x + y}} \]

      9. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{x + y}} \cdot \frac{\frac{y}{\left(x + y\right) + 1}}{x + y} \]

      10. lift-+.f64 N/A

          \[\leadsto \frac{x}{\color{blue}{x + y}} \cdot \frac{\frac{y}{\left(x + y\right) + 1}}{x + y} \]

      11. +-commutative N/A

          \[\leadsto \frac{x}{\color{blue}{y + x}} \cdot \frac{\frac{y}{\left(x + y\right) + 1}}{x + y} \]

      12. lower-+.f64 N/A

          \[\leadsto \frac{x}{\color{blue}{y + x}} \cdot \frac{\frac{y}{\left(x + y\right) + 1}}{x + y} \]

      13. lower-/.f64 N/A

          \[\leadsto \frac{x}{y + x} \cdot \color{blue}{\frac{\frac{y}{\left(x + y\right) + 1}}{x + y}} \]

      14. lower-/.f64 99.8

          \[\leadsto \frac{x}{y + x} \cdot \frac{\color{blue}{\frac{y}{\left(x + y\right) + 1}}}{x + y} \]

      15. lift-+.f64 N/A

          \[\leadsto \frac{x}{y + x} \cdot \frac{\frac{y}{\color{blue}{\left(x + y\right) + 1}}}{x + y} \]

      16. +-commutative N/A

          \[\leadsto \frac{x}{y + x} \cdot \frac{\frac{y}{\color{blue}{1 + \left(x + y\right)}}}{x + y} \]

      17. lower-+.f64 99.8

          \[\leadsto \frac{x}{y + x} \cdot \frac{\frac{y}{\color{blue}{1 + \left(x + y\right)}}}{x + y} \]

      18. lift-+.f64 N/A

          \[\leadsto \frac{x}{y + x} \cdot \frac{\frac{y}{1 + \color{blue}{\left(x + y\right)}}}{x + y} \]

      19. +-commutative N/A

          \[\leadsto \frac{x}{y + x} \cdot \frac{\frac{y}{1 + \color{blue}{\left(y + x\right)}}}{x + y} \]

      20. lower-+.f64 99.8

          \[\leadsto \frac{x}{y + x} \cdot \frac{\frac{y}{1 + \color{blue}{\left(y + x\right)}}}{x + y} \]

      21. lift-+.f64 N/A

          \[\leadsto \frac{x}{y + x} \cdot \frac{\frac{y}{1 + \left(y + x\right)}}{\color{blue}{x + y}} \]

      22. +-commutative N/A

          \[\leadsto \frac{x}{y + x} \cdot \frac{\frac{y}{1 + \left(y + x\right)}}{\color{blue}{y + x}} \]

      23. lower-+.f64 99.8

          \[\leadsto \frac{x}{y + x} \cdot \frac{\frac{y}{1 + \left(y + x\right)}}{\color{blue}{y + x}} \]

   4. Applied rewrites 99.8%

      \[\leadsto \color{blue}{\frac{x}{y + x} \cdot \frac{\frac{y}{1 + \left(y + x\right)}}{y + x}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{y + x} \cdot \frac{\frac{y}{1 + \left(y + x\right)}}{y + x}} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{x}{y + x} \cdot \color{blue}{\frac{\frac{y}{1 + \left(y + x\right)}}{y + x}} \]

      3. lift-/.f64 N/A

         \[\leadsto \frac{x}{y + x} \cdot \frac{\color{blue}{\frac{y}{1 + \left(y + x\right)}}}{y + x} \]

      4. associate-/l/ N/A

         \[\leadsto \frac{x}{y + x} \cdot \color{blue}{\frac{y}{\left(1 + \left(y + x\right)\right) \cdot \left(y + x\right)}} \]

      5. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{\frac{x}{y + x} \cdot y}{\left(1 + \left(y + x\right)\right) \cdot \left(y + x\right)}} \]

      6. lift-+.f64 N/A

         \[\leadsto \frac{\frac{x}{y + x} \cdot y}{\color{blue}{\left(1 + \left(y + x\right)\right)} \cdot \left(y + x\right)} \]

      7. lift-+.f64 N/A

         \[\leadsto \frac{\frac{x}{y + x} \cdot y}{\left(1 + \color{blue}{\left(y + x\right)}\right) \cdot \left(y + x\right)} \]

      8. +-commutative N/A

         \[\leadsto \frac{\frac{x}{y + x} \cdot y}{\left(1 + \color{blue}{\left(x + y\right)}\right) \cdot \left(y + x\right)} \]

      9. +-commutative N/A

         \[\leadsto \frac{\frac{x}{y + x} \cdot y}{\color{blue}{\left(\left(x + y\right) + 1\right)} \cdot \left(y + x\right)} \]

      10. lift-+.f64 N/A

          \[\leadsto \frac{\frac{x}{y + x} \cdot y}{\left(\color{blue}{\left(x + y\right)} + 1\right) \cdot \left(y + x\right)} \]

      11. lift-+.f64 N/A

          \[\leadsto \frac{\frac{x}{y + x} \cdot y}{\color{blue}{\left(\left(x + y\right) + 1\right)} \cdot \left(y + x\right)} \]

      12. lift-+.f64 N/A

          \[\leadsto \frac{\frac{x}{y + x} \cdot y}{\left(\left(x + y\right) + 1\right) \cdot \color{blue}{\left(y + x\right)}} \]

      13. +-commutative N/A

          \[\leadsto \frac{\frac{x}{y + x} \cdot y}{\left(\left(x + y\right) + 1\right) \cdot \color{blue}{\left(x + y\right)}} \]

      14. times-frac N/A

          \[\leadsto \color{blue}{\frac{\frac{x}{y + x}}{\left(x + y\right) + 1} \cdot \frac{y}{x + y}} \]

      15. lower-*.f64 N/A

          \[\leadsto \color{blue}{\frac{\frac{x}{y + x}}{\left(x + y\right) + 1} \cdot \frac{y}{x + y}} \]

      16. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{\frac{x}{y + x}}{\left(x + y\right) + 1}} \cdot \frac{y}{x + y} \]

      17. lift-+.f64 N/A

          \[\leadsto \frac{\frac{x}{y + x}}{\color{blue}{\left(x + y\right)} + 1} \cdot \frac{y}{x + y} \]

      18. +-commutative N/A

          \[\leadsto \frac{\frac{x}{y + x}}{\color{blue}{\left(y + x\right)} + 1} \cdot \frac{y}{x + y} \]

      19. lift-+.f64 N/A

          \[\leadsto \frac{\frac{x}{y + x}}{\color{blue}{\left(y + x\right)} + 1} \cdot \frac{y}{x + y} \]

      20. lower-/.f64 N/A

          \[\leadsto \frac{\frac{x}{y + x}}{\left(y + x\right) + 1} \cdot \color{blue}{\frac{y}{x + y}} \]

      21. +-commutative N/A

          \[\leadsto \frac{\frac{x}{y + x}}{\left(y + x\right) + 1} \cdot \frac{y}{\color{blue}{y + x}} \]

   6. Applied rewrites 99.7%

      \[\leadsto \color{blue}{\frac{\frac{x}{y + x}}{\left(y + x\right) + 1} \cdot \frac{y}{y + x}} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{x}{y + x}}{\left(y + x\right) + 1} \cdot \frac{y}{y + x}} \]

      2. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{x}{y + x}}{\left(y + x\right) + 1}} \cdot \frac{y}{y + x} \]

      3. lift-/.f64 N/A

         \[\leadsto \frac{\frac{x}{y + x}}{\left(y + x\right) + 1} \cdot \color{blue}{\frac{y}{y + x}} \]

      4. frac-times N/A

         \[\leadsto \color{blue}{\frac{\frac{x}{y + x} \cdot y}{\left(\left(y + x\right) + 1\right) \cdot \left(y + x\right)}} \]

      5. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{x}{y + x} \cdot \frac{y}{\left(\left(y + x\right) + 1\right) \cdot \left(y + x\right)}} \]

      6. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{y + x}} \cdot \frac{y}{\left(\left(y + x\right) + 1\right) \cdot \left(y + x\right)} \]

      7. lift-+.f64 N/A

         \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\color{blue}{\left(\left(y + x\right) + 1\right)} \cdot \left(y + x\right)} \]

      8. +-commutative N/A

         \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\color{blue}{\left(1 + \left(y + x\right)\right)} \cdot \left(y + x\right)} \]

      9. lift-+.f64 N/A

         \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\left(1 + \color{blue}{\left(y + x\right)}\right) \cdot \left(y + x\right)} \]

      10. associate-/l/ N/A

          \[\leadsto \frac{x}{y + x} \cdot \color{blue}{\frac{\frac{y}{1 + \left(y + x\right)}}{y + x}} \]

      11. frac-times N/A

          \[\leadsto \color{blue}{\frac{x \cdot \frac{y}{1 + \left(y + x\right)}}{\left(y + x\right) \cdot \left(y + x\right)}} \]

      12. sqr-neg-rev N/A

          \[\leadsto \frac{x \cdot \frac{y}{1 + \left(y + x\right)}}{\color{blue}{\left(\mathsf{neg}\left(\left(y + x\right)\right)\right) \cdot \left(\mathsf{neg}\left(\left(y + x\right)\right)\right)}} \]

      13. associate-/r* N/A

          \[\leadsto \color{blue}{\frac{\frac{x \cdot \frac{y}{1 + \left(y + x\right)}}{\mathsf{neg}\left(\left(y + x\right)\right)}}{\mathsf{neg}\left(\left(y + x\right)\right)}} \]

      14. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{\frac{x \cdot \frac{y}{1 + \left(y + x\right)}}{\mathsf{neg}\left(\left(y + x\right)\right)}}{\mathsf{neg}\left(\left(y + x\right)\right)}} \]

   8. Applied rewrites 99.7%

      \[\leadsto \color{blue}{\frac{\frac{\frac{y}{1 + \left(y + x\right)} \cdot x}{-\left(y + x\right)}}{-\left(y + x\right)}} \]

   9. Taylor expanded in x around inf

      \[\leadsto \frac{\color{blue}{-1 \cdot \frac{y}{x}}}{-\left(y + x\right)} \]
   10. Step-by-step derivation

       1. associate-*r/ N/A

          \[\leadsto \frac{\color{blue}{\frac{-1 \cdot y}{x}}}{-\left(y + x\right)} \]

       2. lower-/.f64 N/A

          \[\leadsto \frac{\color{blue}{\frac{-1 \cdot y}{x}}}{-\left(y + x\right)} \]

       3. mul-1-neg N/A

          \[\leadsto \frac{\frac{\color{blue}{\mathsf{neg}\left(y\right)}}{x}}{-\left(y + x\right)} \]

       4. lower-neg.f64 92.0

          \[\leadsto \frac{\frac{\color{blue}{-y}}{x}}{-\left(y + x\right)} \]

   11. Applied rewrites 92.0%

       \[\leadsto \frac{\color{blue}{\frac{-y}{x}}}{-\left(y + x\right)} \]

   if -6.79999999999999992e192 < x < -1.2e-56

   1. Initial program 73.8%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   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. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{x \cdot \frac{y}{\left(x + y\right) + 1}}{\left(x + y\right) \cdot \left(x + y\right)}} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot \frac{y}{\left(x + y\right) + 1}}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right)}} \]

      7. times-frac N/A

         \[\leadsto \color{blue}{\frac{x}{x + y} \cdot \frac{\frac{y}{\left(x + y\right) + 1}}{x + y}} \]

      8. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{x + y} \cdot \frac{\frac{y}{\left(x + y\right) + 1}}{x + y}} \]

      9. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{x + y}} \cdot \frac{\frac{y}{\left(x + y\right) + 1}}{x + y} \]

      10. lift-+.f64 N/A

          \[\leadsto \frac{x}{\color{blue}{x + y}} \cdot \frac{\frac{y}{\left(x + y\right) + 1}}{x + y} \]

      11. +-commutative N/A

          \[\leadsto \frac{x}{\color{blue}{y + x}} \cdot \frac{\frac{y}{\left(x + y\right) + 1}}{x + y} \]

      12. lower-+.f64 N/A

          \[\leadsto \frac{x}{\color{blue}{y + x}} \cdot \frac{\frac{y}{\left(x + y\right) + 1}}{x + y} \]

      13. lower-/.f64 N/A

          \[\leadsto \frac{x}{y + x} \cdot \color{blue}{\frac{\frac{y}{\left(x + y\right) + 1}}{x + y}} \]

      14. lower-/.f64 99.7

          \[\leadsto \frac{x}{y + x} \cdot \frac{\color{blue}{\frac{y}{\left(x + y\right) + 1}}}{x + y} \]

      15. lift-+.f64 N/A

          \[\leadsto \frac{x}{y + x} \cdot \frac{\frac{y}{\color{blue}{\left(x + y\right) + 1}}}{x + y} \]

      16. +-commutative N/A

          \[\leadsto \frac{x}{y + x} \cdot \frac{\frac{y}{\color{blue}{1 + \left(x + y\right)}}}{x + y} \]

      17. lower-+.f64 99.7

          \[\leadsto \frac{x}{y + x} \cdot \frac{\frac{y}{\color{blue}{1 + \left(x + y\right)}}}{x + y} \]

      18. lift-+.f64 N/A

          \[\leadsto \frac{x}{y + x} \cdot \frac{\frac{y}{1 + \color{blue}{\left(x + y\right)}}}{x + y} \]

      19. +-commutative N/A

          \[\leadsto \frac{x}{y + x} \cdot \frac{\frac{y}{1 + \color{blue}{\left(y + x\right)}}}{x + y} \]

      20. lower-+.f64 99.7

          \[\leadsto \frac{x}{y + x} \cdot \frac{\frac{y}{1 + \color{blue}{\left(y + x\right)}}}{x + y} \]

      21. lift-+.f64 N/A

          \[\leadsto \frac{x}{y + x} \cdot \frac{\frac{y}{1 + \left(y + x\right)}}{\color{blue}{x + y}} \]

      22. +-commutative N/A

          \[\leadsto \frac{x}{y + x} \cdot \frac{\frac{y}{1 + \left(y + x\right)}}{\color{blue}{y + x}} \]

      23. lower-+.f64 99.7

          \[\leadsto \frac{x}{y + x} \cdot \frac{\frac{y}{1 + \left(y + x\right)}}{\color{blue}{y + x}} \]

   4. Applied rewrites 99.7%

      \[\leadsto \color{blue}{\frac{x}{y + x} \cdot \frac{\frac{y}{1 + \left(y + x\right)}}{y + x}} \]

   5. Taylor expanded in x around inf

      \[\leadsto \color{blue}{1} \cdot \frac{\frac{y}{1 + \left(y + x\right)}}{y + x} \]
   6. Step-by-step derivation

   7. Applied rewrites 56.0%

      \[\leadsto \color{blue}{1} \cdot \frac{\frac{y}{1 + \left(y + x\right)}}{y + x} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{1 \cdot \frac{\frac{y}{1 + \left(y + x\right)}}{y + x}} \]

      2. lift-/.f64 N/A

         \[\leadsto 1 \cdot \color{blue}{\frac{\frac{y}{1 + \left(y + x\right)}}{y + x}} \]

      3. lift-/.f64 N/A

         \[\leadsto 1 \cdot \frac{\color{blue}{\frac{y}{1 + \left(y + x\right)}}}{y + x} \]

      4. associate-/l/ N/A

         \[\leadsto 1 \cdot \color{blue}{\frac{y}{\left(1 + \left(y + x\right)\right) \cdot \left(y + x\right)}} \]

      5. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{1 \cdot y}{\left(1 + \left(y + x\right)\right) \cdot \left(y + x\right)}} \]

      6. lift-+.f64 N/A

         \[\leadsto \frac{1 \cdot y}{\color{blue}{\left(1 + \left(y + x\right)\right)} \cdot \left(y + x\right)} \]

      7. lift-+.f64 N/A

         \[\leadsto \frac{1 \cdot y}{\left(1 + \color{blue}{\left(y + x\right)}\right) \cdot \left(y + x\right)} \]

      8. +-commutative N/A

         \[\leadsto \frac{1 \cdot y}{\left(1 + \color{blue}{\left(x + y\right)}\right) \cdot \left(y + x\right)} \]

      9. +-commutative N/A

         \[\leadsto \frac{1 \cdot y}{\color{blue}{\left(\left(x + y\right) + 1\right)} \cdot \left(y + x\right)} \]

      10. lift-+.f64 N/A

          \[\leadsto \frac{1 \cdot y}{\left(\left(x + y\right) + 1\right) \cdot \color{blue}{\left(y + x\right)}} \]

      11. +-commutative N/A

          \[\leadsto \frac{1 \cdot y}{\left(\left(x + y\right) + 1\right) \cdot \color{blue}{\left(x + y\right)}} \]

      12. *-commutative N/A

          \[\leadsto \frac{1 \cdot y}{\color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      13. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{1 \cdot y}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{\color{blue}{1 \cdot y}}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      15. *-commutative N/A

          \[\leadsto \frac{1 \cdot y}{\color{blue}{\left(\left(x + y\right) + 1\right) \cdot \left(x + y\right)}} \]

      16. lower-*.f64 N/A

          \[\leadsto \frac{1 \cdot y}{\color{blue}{\left(\left(x + y\right) + 1\right) \cdot \left(x + y\right)}} \]

      17. lower-+.f64 N/A

          \[\leadsto \frac{1 \cdot y}{\color{blue}{\left(\left(x + y\right) + 1\right)} \cdot \left(x + y\right)} \]

      18. lower-+.f64 N/A

          \[\leadsto \frac{1 \cdot y}{\left(\color{blue}{\left(x + y\right)} + 1\right) \cdot \left(x + y\right)} \]

      19. lower-+.f64 77.2

          \[\leadsto \frac{1 \cdot y}{\left(\left(x + y\right) + 1\right) \cdot \color{blue}{\left(x + y\right)}} \]

   9. Applied rewrites 77.2%

      \[\leadsto \color{blue}{\frac{1 \cdot y}{\left(\left(x + y\right) + 1\right) \cdot \left(x + y\right)}} \]

   if -1.2e-56 < x < 8.1999999999999994e-6

   1. Initial program 78.5%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]

      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 80.5

         \[\leadsto \frac{x}{\color{blue}{\mathsf{fma}\left(y, y, y\right)}} \]

   5. Applied rewrites 80.5%

      \[\leadsto \color{blue}{\frac{x}{\mathsf{fma}\left(y, y, y\right)}} \]

   if 8.1999999999999994e-6 < x

   1. Initial program 56.0%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   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. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{x \cdot \frac{y}{\left(x + y\right) + 1}}{\left(x + y\right) \cdot \left(x + y\right)}} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot \frac{y}{\left(x + y\right) + 1}}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right)}} \]

      7. times-frac N/A

         \[\leadsto \color{blue}{\frac{x}{x + y} \cdot \frac{\frac{y}{\left(x + y\right) + 1}}{x + y}} \]

      8. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{x + y} \cdot \frac{\frac{y}{\left(x + y\right) + 1}}{x + y}} \]

      9. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{x + y}} \cdot \frac{\frac{y}{\left(x + y\right) + 1}}{x + y} \]

      10. lift-+.f64 N/A

          \[\leadsto \frac{x}{\color{blue}{x + y}} \cdot \frac{\frac{y}{\left(x + y\right) + 1}}{x + y} \]

      11. +-commutative N/A

          \[\leadsto \frac{x}{\color{blue}{y + x}} \cdot \frac{\frac{y}{\left(x + y\right) + 1}}{x + y} \]

      12. lower-+.f64 N/A

          \[\leadsto \frac{x}{\color{blue}{y + x}} \cdot \frac{\frac{y}{\left(x + y\right) + 1}}{x + y} \]

      13. lower-/.f64 N/A

          \[\leadsto \frac{x}{y + x} \cdot \color{blue}{\frac{\frac{y}{\left(x + y\right) + 1}}{x + y}} \]

      14. lower-/.f64 99.7

          \[\leadsto \frac{x}{y + x} \cdot \frac{\color{blue}{\frac{y}{\left(x + y\right) + 1}}}{x + y} \]

      15. lift-+.f64 N/A

          \[\leadsto \frac{x}{y + x} \cdot \frac{\frac{y}{\color{blue}{\left(x + y\right) + 1}}}{x + y} \]

      16. +-commutative N/A

          \[\leadsto \frac{x}{y + x} \cdot \frac{\frac{y}{\color{blue}{1 + \left(x + y\right)}}}{x + y} \]

      17. lower-+.f64 99.7

          \[\leadsto \frac{x}{y + x} \cdot \frac{\frac{y}{\color{blue}{1 + \left(x + y\right)}}}{x + y} \]

      18. lift-+.f64 N/A

          \[\leadsto \frac{x}{y + x} \cdot \frac{\frac{y}{1 + \color{blue}{\left(x + y\right)}}}{x + y} \]

      19. +-commutative N/A

          \[\leadsto \frac{x}{y + x} \cdot \frac{\frac{y}{1 + \color{blue}{\left(y + x\right)}}}{x + y} \]

      20. lower-+.f64 99.7

          \[\leadsto \frac{x}{y + x} \cdot \frac{\frac{y}{1 + \color{blue}{\left(y + x\right)}}}{x + y} \]

      21. lift-+.f64 N/A

          \[\leadsto \frac{x}{y + x} \cdot \frac{\frac{y}{1 + \left(y + x\right)}}{\color{blue}{x + y}} \]

      22. +-commutative N/A

          \[\leadsto \frac{x}{y + x} \cdot \frac{\frac{y}{1 + \left(y + x\right)}}{\color{blue}{y + x}} \]

      23. lower-+.f64 99.7

          \[\leadsto \frac{x}{y + x} \cdot \frac{\frac{y}{1 + \left(y + x\right)}}{\color{blue}{y + x}} \]

   4. Applied rewrites 99.7%

      \[\leadsto \color{blue}{\frac{x}{y + x} \cdot \frac{\frac{y}{1 + \left(y + x\right)}}{y + x}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{y + x} \cdot \frac{\frac{y}{1 + \left(y + x\right)}}{y + x}} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{x}{y + x} \cdot \color{blue}{\frac{\frac{y}{1 + \left(y + x\right)}}{y + x}} \]

      3. lift-/.f64 N/A

         \[\leadsto \frac{x}{y + x} \cdot \frac{\color{blue}{\frac{y}{1 + \left(y + x\right)}}}{y + x} \]

      4. associate-/l/ N/A

         \[\leadsto \frac{x}{y + x} \cdot \color{blue}{\frac{y}{\left(1 + \left(y + x\right)\right) \cdot \left(y + x\right)}} \]

      5. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{\frac{x}{y + x} \cdot y}{\left(1 + \left(y + x\right)\right) \cdot \left(y + x\right)}} \]

      6. lift-+.f64 N/A

         \[\leadsto \frac{\frac{x}{y + x} \cdot y}{\color{blue}{\left(1 + \left(y + x\right)\right)} \cdot \left(y + x\right)} \]

      7. lift-+.f64 N/A

         \[\leadsto \frac{\frac{x}{y + x} \cdot y}{\left(1 + \color{blue}{\left(y + x\right)}\right) \cdot \left(y + x\right)} \]

      8. +-commutative N/A

         \[\leadsto \frac{\frac{x}{y + x} \cdot y}{\left(1 + \color{blue}{\left(x + y\right)}\right) \cdot \left(y + x\right)} \]

      9. +-commutative N/A

         \[\leadsto \frac{\frac{x}{y + x} \cdot y}{\color{blue}{\left(\left(x + y\right) + 1\right)} \cdot \left(y + x\right)} \]

      10. lift-+.f64 N/A

          \[\leadsto \frac{\frac{x}{y + x} \cdot y}{\left(\color{blue}{\left(x + y\right)} + 1\right) \cdot \left(y + x\right)} \]

      11. lift-+.f64 N/A

          \[\leadsto \frac{\frac{x}{y + x} \cdot y}{\color{blue}{\left(\left(x + y\right) + 1\right)} \cdot \left(y + x\right)} \]

      12. lift-+.f64 N/A

          \[\leadsto \frac{\frac{x}{y + x} \cdot y}{\left(\left(x + y\right) + 1\right) \cdot \color{blue}{\left(y + x\right)}} \]

      13. +-commutative N/A

          \[\leadsto \frac{\frac{x}{y + x} \cdot y}{\left(\left(x + y\right) + 1\right) \cdot \color{blue}{\left(x + y\right)}} \]

      14. times-frac N/A

          \[\leadsto \color{blue}{\frac{\frac{x}{y + x}}{\left(x + y\right) + 1} \cdot \frac{y}{x + y}} \]

      15. lower-*.f64 N/A

          \[\leadsto \color{blue}{\frac{\frac{x}{y + x}}{\left(x + y\right) + 1} \cdot \frac{y}{x + y}} \]

      16. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{\frac{x}{y + x}}{\left(x + y\right) + 1}} \cdot \frac{y}{x + y} \]

      17. lift-+.f64 N/A

          \[\leadsto \frac{\frac{x}{y + x}}{\color{blue}{\left(x + y\right)} + 1} \cdot \frac{y}{x + y} \]

      18. +-commutative N/A

          \[\leadsto \frac{\frac{x}{y + x}}{\color{blue}{\left(y + x\right)} + 1} \cdot \frac{y}{x + y} \]

      19. lift-+.f64 N/A

          \[\leadsto \frac{\frac{x}{y + x}}{\color{blue}{\left(y + x\right)} + 1} \cdot \frac{y}{x + y} \]

      20. lower-/.f64 N/A

          \[\leadsto \frac{\frac{x}{y + x}}{\left(y + x\right) + 1} \cdot \color{blue}{\frac{y}{x + y}} \]

      21. +-commutative N/A

          \[\leadsto \frac{\frac{x}{y + x}}{\left(y + x\right) + 1} \cdot \frac{y}{\color{blue}{y + x}} \]

   6. Applied rewrites 99.6%

      \[\leadsto \color{blue}{\frac{\frac{x}{y + x}}{\left(y + x\right) + 1} \cdot \frac{y}{y + x}} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{x}{y + x}}{\left(y + x\right) + 1} \cdot \frac{y}{y + x}} \]

      2. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{x}{y + x}}{\left(y + x\right) + 1}} \cdot \frac{y}{y + x} \]

      3. lift-/.f64 N/A

         \[\leadsto \frac{\frac{x}{y + x}}{\left(y + x\right) + 1} \cdot \color{blue}{\frac{y}{y + x}} \]

      4. frac-times N/A

         \[\leadsto \color{blue}{\frac{\frac{x}{y + x} \cdot y}{\left(\left(y + x\right) + 1\right) \cdot \left(y + x\right)}} \]

      5. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{x}{y + x} \cdot \frac{y}{\left(\left(y + x\right) + 1\right) \cdot \left(y + x\right)}} \]

      6. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{y + x}} \cdot \frac{y}{\left(\left(y + x\right) + 1\right) \cdot \left(y + x\right)} \]

      7. lift-+.f64 N/A

         \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\color{blue}{\left(\left(y + x\right) + 1\right)} \cdot \left(y + x\right)} \]

      8. +-commutative N/A

         \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\color{blue}{\left(1 + \left(y + x\right)\right)} \cdot \left(y + x\right)} \]

      9. lift-+.f64 N/A

         \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\left(1 + \color{blue}{\left(y + x\right)}\right) \cdot \left(y + x\right)} \]

      10. associate-/l/ N/A

          \[\leadsto \frac{x}{y + x} \cdot \color{blue}{\frac{\frac{y}{1 + \left(y + x\right)}}{y + x}} \]

      11. frac-times N/A

          \[\leadsto \color{blue}{\frac{x \cdot \frac{y}{1 + \left(y + x\right)}}{\left(y + x\right) \cdot \left(y + x\right)}} \]

      12. sqr-neg-rev N/A

          \[\leadsto \frac{x \cdot \frac{y}{1 + \left(y + x\right)}}{\color{blue}{\left(\mathsf{neg}\left(\left(y + x\right)\right)\right) \cdot \left(\mathsf{neg}\left(\left(y + x\right)\right)\right)}} \]

      13. associate-/r* N/A

          \[\leadsto \color{blue}{\frac{\frac{x \cdot \frac{y}{1 + \left(y + x\right)}}{\mathsf{neg}\left(\left(y + x\right)\right)}}{\mathsf{neg}\left(\left(y + x\right)\right)}} \]

      14. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{\frac{x \cdot \frac{y}{1 + \left(y + x\right)}}{\mathsf{neg}\left(\left(y + x\right)\right)}}{\mathsf{neg}\left(\left(y + x\right)\right)}} \]

   8. Applied rewrites 99.7%

      \[\leadsto \color{blue}{\frac{\frac{\frac{y}{1 + \left(y + x\right)} \cdot x}{-\left(y + x\right)}}{-\left(y + x\right)}} \]

   9. Taylor expanded in y around inf

      \[\leadsto \frac{\color{blue}{-1 \cdot \frac{x}{y}}}{-\left(y + x\right)} \]
   10. Step-by-step derivation

       1. associate-*r/ N/A

          \[\leadsto \frac{\color{blue}{\frac{-1 \cdot x}{y}}}{-\left(y + x\right)} \]

       2. lower-/.f64 N/A

          \[\leadsto \frac{\color{blue}{\frac{-1 \cdot x}{y}}}{-\left(y + x\right)} \]

       3. mul-1-neg N/A

          \[\leadsto \frac{\frac{\color{blue}{\mathsf{neg}\left(x\right)}}{y}}{-\left(y + x\right)} \]

       4. lower-neg.f64 28.5

          \[\leadsto \frac{\frac{\color{blue}{-x}}{y}}{-\left(y + x\right)} \]

   11. Applied rewrites 28.5%

       \[\leadsto \frac{\color{blue}{\frac{-x}{y}}}{-\left(y + x\right)} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 12: 80.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_0 := -\left(y + x\right)\\ \mathbf{if}\;y \leq -7 \cdot 10^{-6}:\\ \;\;\;\;\frac{\frac{-y}{x}}{t\_0}\\ \mathbf{elif}\;y \leq 1350000:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(x, x, x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-x}{y}}{t\_0}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (+ y x))))
   (if (<= y -7e-6)
     (/ (/ (- y) x) t_0)
     (if (<= y 1350000.0) (/ y (fma x x x)) (/ (/ (- x) y) t_0)))))

C

assert(x < y);
double code(double x, double y) {
	double t_0 = -(y + x);
	double tmp;
	if (y <= -7e-6) {
		tmp = (-y / x) / t_0;
	} else if (y <= 1350000.0) {
		tmp = y / fma(x, x, x);
	} else {
		tmp = (-x / y) / t_0;
	}
	return tmp;
}

Julia

x, y = sort([x, y])
function code(x, y)
	t_0 = Float64(-Float64(y + x))
	tmp = 0.0
	if (y <= -7e-6)
		tmp = Float64(Float64(Float64(-y) / x) / t_0);
	elseif (y <= 1350000.0)
		tmp = Float64(y / fma(x, x, x));
	else
		tmp = Float64(Float64(Float64(-x) / y) / t_0);
	end
	return tmp
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := Block[{t$95$0 = (-N[(y + x), $MachinePrecision])}, If[LessEqual[y, -7e-6], N[(N[((-y) / x), $MachinePrecision] / t$95$0), $MachinePrecision], If[LessEqual[y, 1350000.0], N[(y / N[(x * x + x), $MachinePrecision]), $MachinePrecision], N[(N[((-x) / y), $MachinePrecision] / t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < -6.99999999999999989e-6

   1. Initial program 63.7%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot y}}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      4. 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. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{x \cdot \frac{y}{\left(x + y\right) + 1}}{\left(x + y\right) \cdot \left(x + y\right)}} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot \frac{y}{\left(x + y\right) + 1}}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right)}} \]

      7. times-frac N/A

         \[\leadsto \color{blue}{\frac{x}{x + y} \cdot \frac{\frac{y}{\left(x + y\right) + 1}}{x + y}} \]

      8. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{x + y} \cdot \frac{\frac{y}{\left(x + y\right) + 1}}{x + y}} \]

      9. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{x + y}} \cdot \frac{\frac{y}{\left(x + y\right) + 1}}{x + y} \]

      10. lift-+.f64 N/A

          \[\leadsto \frac{x}{\color{blue}{x + y}} \cdot \frac{\frac{y}{\left(x + y\right) + 1}}{x + y} \]

      11. +-commutative N/A

          \[\leadsto \frac{x}{\color{blue}{y + x}} \cdot \frac{\frac{y}{\left(x + y\right) + 1}}{x + y} \]

      12. lower-+.f64 N/A

          \[\leadsto \frac{x}{\color{blue}{y + x}} \cdot \frac{\frac{y}{\left(x + y\right) + 1}}{x + y} \]

      13. lower-/.f64 N/A

          \[\leadsto \frac{x}{y + x} \cdot \color{blue}{\frac{\frac{y}{\left(x + y\right) + 1}}{x + y}} \]

      14. lower-/.f64 99.8

          \[\leadsto \frac{x}{y + x} \cdot \frac{\color{blue}{\frac{y}{\left(x + y\right) + 1}}}{x + y} \]

      15. lift-+.f64 N/A

          \[\leadsto \frac{x}{y + x} \cdot \frac{\frac{y}{\color{blue}{\left(x + y\right) + 1}}}{x + y} \]

      16. +-commutative N/A

          \[\leadsto \frac{x}{y + x} \cdot \frac{\frac{y}{\color{blue}{1 + \left(x + y\right)}}}{x + y} \]

      17. lower-+.f64 99.8

          \[\leadsto \frac{x}{y + x} \cdot \frac{\frac{y}{\color{blue}{1 + \left(x + y\right)}}}{x + y} \]

      18. lift-+.f64 N/A

          \[\leadsto \frac{x}{y + x} \cdot \frac{\frac{y}{1 + \color{blue}{\left(x + y\right)}}}{x + y} \]

      19. +-commutative N/A

          \[\leadsto \frac{x}{y + x} \cdot \frac{\frac{y}{1 + \color{blue}{\left(y + x\right)}}}{x + y} \]

      20. lower-+.f64 99.8

          \[\leadsto \frac{x}{y + x} \cdot \frac{\frac{y}{1 + \color{blue}{\left(y + x\right)}}}{x + y} \]

      21. lift-+.f64 N/A

          \[\leadsto \frac{x}{y + x} \cdot \frac{\frac{y}{1 + \left(y + x\right)}}{\color{blue}{x + y}} \]

      22. +-commutative N/A

          \[\leadsto \frac{x}{y + x} \cdot \frac{\frac{y}{1 + \left(y + x\right)}}{\color{blue}{y + x}} \]

      23. lower-+.f64 99.8

          \[\leadsto \frac{x}{y + x} \cdot \frac{\frac{y}{1 + \left(y + x\right)}}{\color{blue}{y + x}} \]

   4. Applied rewrites 99.8%

      \[\leadsto \color{blue}{\frac{x}{y + x} \cdot \frac{\frac{y}{1 + \left(y + x\right)}}{y + x}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{y + x} \cdot \frac{\frac{y}{1 + \left(y + x\right)}}{y + x}} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{x}{y + x} \cdot \color{blue}{\frac{\frac{y}{1 + \left(y + x\right)}}{y + x}} \]

      3. lift-/.f64 N/A

         \[\leadsto \frac{x}{y + x} \cdot \frac{\color{blue}{\frac{y}{1 + \left(y + x\right)}}}{y + x} \]

      4. associate-/l/ N/A

         \[\leadsto \frac{x}{y + x} \cdot \color{blue}{\frac{y}{\left(1 + \left(y + x\right)\right) \cdot \left(y + x\right)}} \]

      5. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{\frac{x}{y + x} \cdot y}{\left(1 + \left(y + x\right)\right) \cdot \left(y + x\right)}} \]

      6. lift-+.f64 N/A

         \[\leadsto \frac{\frac{x}{y + x} \cdot y}{\color{blue}{\left(1 + \left(y + x\right)\right)} \cdot \left(y + x\right)} \]

      7. lift-+.f64 N/A

         \[\leadsto \frac{\frac{x}{y + x} \cdot y}{\left(1 + \color{blue}{\left(y + x\right)}\right) \cdot \left(y + x\right)} \]

      8. +-commutative N/A

         \[\leadsto \frac{\frac{x}{y + x} \cdot y}{\left(1 + \color{blue}{\left(x + y\right)}\right) \cdot \left(y + x\right)} \]

      9. +-commutative N/A

         \[\leadsto \frac{\frac{x}{y + x} \cdot y}{\color{blue}{\left(\left(x + y\right) + 1\right)} \cdot \left(y + x\right)} \]

      10. lift-+.f64 N/A

          \[\leadsto \frac{\frac{x}{y + x} \cdot y}{\left(\color{blue}{\left(x + y\right)} + 1\right) \cdot \left(y + x\right)} \]

      11. lift-+.f64 N/A

          \[\leadsto \frac{\frac{x}{y + x} \cdot y}{\color{blue}{\left(\left(x + y\right) + 1\right)} \cdot \left(y + x\right)} \]

      12. lift-+.f64 N/A

          \[\leadsto \frac{\frac{x}{y + x} \cdot y}{\left(\left(x + y\right) + 1\right) \cdot \color{blue}{\left(y + x\right)}} \]

      13. +-commutative N/A

          \[\leadsto \frac{\frac{x}{y + x} \cdot y}{\left(\left(x + y\right) + 1\right) \cdot \color{blue}{\left(x + y\right)}} \]

      14. times-frac N/A

          \[\leadsto \color{blue}{\frac{\frac{x}{y + x}}{\left(x + y\right) + 1} \cdot \frac{y}{x + y}} \]

      15. lower-*.f64 N/A

          \[\leadsto \color{blue}{\frac{\frac{x}{y + x}}{\left(x + y\right) + 1} \cdot \frac{y}{x + y}} \]

      16. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{\frac{x}{y + x}}{\left(x + y\right) + 1}} \cdot \frac{y}{x + y} \]

      17. lift-+.f64 N/A

          \[\leadsto \frac{\frac{x}{y + x}}{\color{blue}{\left(x + y\right)} + 1} \cdot \frac{y}{x + y} \]

      18. +-commutative N/A

          \[\leadsto \frac{\frac{x}{y + x}}{\color{blue}{\left(y + x\right)} + 1} \cdot \frac{y}{x + y} \]

      19. lift-+.f64 N/A

          \[\leadsto \frac{\frac{x}{y + x}}{\color{blue}{\left(y + x\right)} + 1} \cdot \frac{y}{x + y} \]

      20. lower-/.f64 N/A

          \[\leadsto \frac{\frac{x}{y + x}}{\left(y + x\right) + 1} \cdot \color{blue}{\frac{y}{x + y}} \]

      21. +-commutative N/A

          \[\leadsto \frac{\frac{x}{y + x}}{\left(y + x\right) + 1} \cdot \frac{y}{\color{blue}{y + x}} \]

   6. Applied rewrites 99.7%

      \[\leadsto \color{blue}{\frac{\frac{x}{y + x}}{\left(y + x\right) + 1} \cdot \frac{y}{y + x}} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{x}{y + x}}{\left(y + x\right) + 1} \cdot \frac{y}{y + x}} \]

      2. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{x}{y + x}}{\left(y + x\right) + 1}} \cdot \frac{y}{y + x} \]

      3. lift-/.f64 N/A

         \[\leadsto \frac{\frac{x}{y + x}}{\left(y + x\right) + 1} \cdot \color{blue}{\frac{y}{y + x}} \]

      4. frac-times N/A

         \[\leadsto \color{blue}{\frac{\frac{x}{y + x} \cdot y}{\left(\left(y + x\right) + 1\right) \cdot \left(y + x\right)}} \]

      5. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{x}{y + x} \cdot \frac{y}{\left(\left(y + x\right) + 1\right) \cdot \left(y + x\right)}} \]

      6. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{y + x}} \cdot \frac{y}{\left(\left(y + x\right) + 1\right) \cdot \left(y + x\right)} \]

      7. lift-+.f64 N/A

         \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\color{blue}{\left(\left(y + x\right) + 1\right)} \cdot \left(y + x\right)} \]

      8. +-commutative N/A

         \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\color{blue}{\left(1 + \left(y + x\right)\right)} \cdot \left(y + x\right)} \]

      9. lift-+.f64 N/A

         \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\left(1 + \color{blue}{\left(y + x\right)}\right) \cdot \left(y + x\right)} \]

      10. associate-/l/ N/A

          \[\leadsto \frac{x}{y + x} \cdot \color{blue}{\frac{\frac{y}{1 + \left(y + x\right)}}{y + x}} \]

      11. frac-times N/A

          \[\leadsto \color{blue}{\frac{x \cdot \frac{y}{1 + \left(y + x\right)}}{\left(y + x\right) \cdot \left(y + x\right)}} \]

      12. sqr-neg-rev N/A

          \[\leadsto \frac{x \cdot \frac{y}{1 + \left(y + x\right)}}{\color{blue}{\left(\mathsf{neg}\left(\left(y + x\right)\right)\right) \cdot \left(\mathsf{neg}\left(\left(y + x\right)\right)\right)}} \]

      13. associate-/r* N/A

          \[\leadsto \color{blue}{\frac{\frac{x \cdot \frac{y}{1 + \left(y + x\right)}}{\mathsf{neg}\left(\left(y + x\right)\right)}}{\mathsf{neg}\left(\left(y + x\right)\right)}} \]

      14. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{\frac{x \cdot \frac{y}{1 + \left(y + x\right)}}{\mathsf{neg}\left(\left(y + x\right)\right)}}{\mathsf{neg}\left(\left(y + x\right)\right)}} \]

   8. Applied rewrites 99.8%

      \[\leadsto \color{blue}{\frac{\frac{\frac{y}{1 + \left(y + x\right)} \cdot x}{-\left(y + x\right)}}{-\left(y + x\right)}} \]

   9. Taylor expanded in x around inf

      \[\leadsto \frac{\color{blue}{-1 \cdot \frac{y}{x}}}{-\left(y + x\right)} \]
   10. Step-by-step derivation

       1. associate-*r/ N/A

          \[\leadsto \frac{\color{blue}{\frac{-1 \cdot y}{x}}}{-\left(y + x\right)} \]

       2. lower-/.f64 N/A

          \[\leadsto \frac{\color{blue}{\frac{-1 \cdot y}{x}}}{-\left(y + x\right)} \]

       3. mul-1-neg N/A

          \[\leadsto \frac{\frac{\color{blue}{\mathsf{neg}\left(y\right)}}{x}}{-\left(y + x\right)} \]

       4. lower-neg.f64 30.7

          \[\leadsto \frac{\frac{\color{blue}{-y}}{x}}{-\left(y + x\right)} \]

   11. Applied rewrites 30.7%

       \[\leadsto \frac{\color{blue}{\frac{-y}{x}}}{-\left(y + x\right)} \]

   if -6.99999999999999989e-6 < y < 1.35e6

   1. Initial program 75.0%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   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 75.7

         \[\leadsto \frac{y}{\color{blue}{\mathsf{fma}\left(x, x, x\right)}} \]

   5. Applied rewrites 75.7%

      \[\leadsto \color{blue}{\frac{y}{\mathsf{fma}\left(x, x, x\right)}} \]

   if 1.35e6 < y

   1. Initial program 62.3%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   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. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{x \cdot \frac{y}{\left(x + y\right) + 1}}{\left(x + y\right) \cdot \left(x + y\right)}} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot \frac{y}{\left(x + y\right) + 1}}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right)}} \]

      7. times-frac N/A

         \[\leadsto \color{blue}{\frac{x}{x + y} \cdot \frac{\frac{y}{\left(x + y\right) + 1}}{x + y}} \]

      8. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{x + y} \cdot \frac{\frac{y}{\left(x + y\right) + 1}}{x + y}} \]

      9. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{x + y}} \cdot \frac{\frac{y}{\left(x + y\right) + 1}}{x + y} \]

      10. lift-+.f64 N/A

          \[\leadsto \frac{x}{\color{blue}{x + y}} \cdot \frac{\frac{y}{\left(x + y\right) + 1}}{x + y} \]

      11. +-commutative N/A

          \[\leadsto \frac{x}{\color{blue}{y + x}} \cdot \frac{\frac{y}{\left(x + y\right) + 1}}{x + y} \]

      12. lower-+.f64 N/A

          \[\leadsto \frac{x}{\color{blue}{y + x}} \cdot \frac{\frac{y}{\left(x + y\right) + 1}}{x + y} \]

      13. lower-/.f64 N/A

          \[\leadsto \frac{x}{y + x} \cdot \color{blue}{\frac{\frac{y}{\left(x + y\right) + 1}}{x + y}} \]

      14. lower-/.f64 99.7

          \[\leadsto \frac{x}{y + x} \cdot \frac{\color{blue}{\frac{y}{\left(x + y\right) + 1}}}{x + y} \]

      15. lift-+.f64 N/A

          \[\leadsto \frac{x}{y + x} \cdot \frac{\frac{y}{\color{blue}{\left(x + y\right) + 1}}}{x + y} \]

      16. +-commutative N/A

          \[\leadsto \frac{x}{y + x} \cdot \frac{\frac{y}{\color{blue}{1 + \left(x + y\right)}}}{x + y} \]

      17. lower-+.f64 99.7

          \[\leadsto \frac{x}{y + x} \cdot \frac{\frac{y}{\color{blue}{1 + \left(x + y\right)}}}{x + y} \]

      18. lift-+.f64 N/A

          \[\leadsto \frac{x}{y + x} \cdot \frac{\frac{y}{1 + \color{blue}{\left(x + y\right)}}}{x + y} \]

      19. +-commutative N/A

          \[\leadsto \frac{x}{y + x} \cdot \frac{\frac{y}{1 + \color{blue}{\left(y + x\right)}}}{x + y} \]

      20. lower-+.f64 99.7

          \[\leadsto \frac{x}{y + x} \cdot \frac{\frac{y}{1 + \color{blue}{\left(y + x\right)}}}{x + y} \]

      21. lift-+.f64 N/A

          \[\leadsto \frac{x}{y + x} \cdot \frac{\frac{y}{1 + \left(y + x\right)}}{\color{blue}{x + y}} \]

      22. +-commutative N/A

          \[\leadsto \frac{x}{y + x} \cdot \frac{\frac{y}{1 + \left(y + x\right)}}{\color{blue}{y + x}} \]

      23. lower-+.f64 99.7

          \[\leadsto \frac{x}{y + x} \cdot \frac{\frac{y}{1 + \left(y + x\right)}}{\color{blue}{y + x}} \]

   4. Applied rewrites 99.7%

      \[\leadsto \color{blue}{\frac{x}{y + x} \cdot \frac{\frac{y}{1 + \left(y + x\right)}}{y + x}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{y + x} \cdot \frac{\frac{y}{1 + \left(y + x\right)}}{y + x}} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{x}{y + x} \cdot \color{blue}{\frac{\frac{y}{1 + \left(y + x\right)}}{y + x}} \]

      3. lift-/.f64 N/A

         \[\leadsto \frac{x}{y + x} \cdot \frac{\color{blue}{\frac{y}{1 + \left(y + x\right)}}}{y + x} \]

      4. associate-/l/ N/A

         \[\leadsto \frac{x}{y + x} \cdot \color{blue}{\frac{y}{\left(1 + \left(y + x\right)\right) \cdot \left(y + x\right)}} \]

      5. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{\frac{x}{y + x} \cdot y}{\left(1 + \left(y + x\right)\right) \cdot \left(y + x\right)}} \]

      6. lift-+.f64 N/A

         \[\leadsto \frac{\frac{x}{y + x} \cdot y}{\color{blue}{\left(1 + \left(y + x\right)\right)} \cdot \left(y + x\right)} \]

      7. lift-+.f64 N/A

         \[\leadsto \frac{\frac{x}{y + x} \cdot y}{\left(1 + \color{blue}{\left(y + x\right)}\right) \cdot \left(y + x\right)} \]

      8. +-commutative N/A

         \[\leadsto \frac{\frac{x}{y + x} \cdot y}{\left(1 + \color{blue}{\left(x + y\right)}\right) \cdot \left(y + x\right)} \]

      9. +-commutative N/A

         \[\leadsto \frac{\frac{x}{y + x} \cdot y}{\color{blue}{\left(\left(x + y\right) + 1\right)} \cdot \left(y + x\right)} \]

      10. lift-+.f64 N/A

          \[\leadsto \frac{\frac{x}{y + x} \cdot y}{\left(\color{blue}{\left(x + y\right)} + 1\right) \cdot \left(y + x\right)} \]

      11. lift-+.f64 N/A

          \[\leadsto \frac{\frac{x}{y + x} \cdot y}{\color{blue}{\left(\left(x + y\right) + 1\right)} \cdot \left(y + x\right)} \]

      12. lift-+.f64 N/A

          \[\leadsto \frac{\frac{x}{y + x} \cdot y}{\left(\left(x + y\right) + 1\right) \cdot \color{blue}{\left(y + x\right)}} \]

      13. +-commutative N/A

          \[\leadsto \frac{\frac{x}{y + x} \cdot y}{\left(\left(x + y\right) + 1\right) \cdot \color{blue}{\left(x + y\right)}} \]

      14. times-frac N/A

          \[\leadsto \color{blue}{\frac{\frac{x}{y + x}}{\left(x + y\right) + 1} \cdot \frac{y}{x + y}} \]

      15. lower-*.f64 N/A

          \[\leadsto \color{blue}{\frac{\frac{x}{y + x}}{\left(x + y\right) + 1} \cdot \frac{y}{x + y}} \]

      16. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{\frac{x}{y + x}}{\left(x + y\right) + 1}} \cdot \frac{y}{x + y} \]

      17. lift-+.f64 N/A

          \[\leadsto \frac{\frac{x}{y + x}}{\color{blue}{\left(x + y\right)} + 1} \cdot \frac{y}{x + y} \]

      18. +-commutative N/A

          \[\leadsto \frac{\frac{x}{y + x}}{\color{blue}{\left(y + x\right)} + 1} \cdot \frac{y}{x + y} \]

      19. lift-+.f64 N/A

          \[\leadsto \frac{\frac{x}{y + x}}{\color{blue}{\left(y + x\right)} + 1} \cdot \frac{y}{x + y} \]

      20. lower-/.f64 N/A

          \[\leadsto \frac{\frac{x}{y + x}}{\left(y + x\right) + 1} \cdot \color{blue}{\frac{y}{x + y}} \]

      21. +-commutative N/A

          \[\leadsto \frac{\frac{x}{y + x}}{\left(y + x\right) + 1} \cdot \frac{y}{\color{blue}{y + x}} \]

   6. Applied rewrites 99.8%

      \[\leadsto \color{blue}{\frac{\frac{x}{y + x}}{\left(y + x\right) + 1} \cdot \frac{y}{y + x}} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{x}{y + x}}{\left(y + x\right) + 1} \cdot \frac{y}{y + x}} \]

      2. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{x}{y + x}}{\left(y + x\right) + 1}} \cdot \frac{y}{y + x} \]

      3. lift-/.f64 N/A

         \[\leadsto \frac{\frac{x}{y + x}}{\left(y + x\right) + 1} \cdot \color{blue}{\frac{y}{y + x}} \]

      4. frac-times N/A

         \[\leadsto \color{blue}{\frac{\frac{x}{y + x} \cdot y}{\left(\left(y + x\right) + 1\right) \cdot \left(y + x\right)}} \]

      5. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{x}{y + x} \cdot \frac{y}{\left(\left(y + x\right) + 1\right) \cdot \left(y + x\right)}} \]

      6. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{y + x}} \cdot \frac{y}{\left(\left(y + x\right) + 1\right) \cdot \left(y + x\right)} \]

      7. lift-+.f64 N/A

         \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\color{blue}{\left(\left(y + x\right) + 1\right)} \cdot \left(y + x\right)} \]

      8. +-commutative N/A

         \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\color{blue}{\left(1 + \left(y + x\right)\right)} \cdot \left(y + x\right)} \]

      9. lift-+.f64 N/A

         \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\left(1 + \color{blue}{\left(y + x\right)}\right) \cdot \left(y + x\right)} \]

      10. associate-/l/ N/A

          \[\leadsto \frac{x}{y + x} \cdot \color{blue}{\frac{\frac{y}{1 + \left(y + x\right)}}{y + x}} \]

      11. frac-times N/A

          \[\leadsto \color{blue}{\frac{x \cdot \frac{y}{1 + \left(y + x\right)}}{\left(y + x\right) \cdot \left(y + x\right)}} \]

      12. sqr-neg-rev N/A

          \[\leadsto \frac{x \cdot \frac{y}{1 + \left(y + x\right)}}{\color{blue}{\left(\mathsf{neg}\left(\left(y + x\right)\right)\right) \cdot \left(\mathsf{neg}\left(\left(y + x\right)\right)\right)}} \]

      13. associate-/r* N/A

          \[\leadsto \color{blue}{\frac{\frac{x \cdot \frac{y}{1 + \left(y + x\right)}}{\mathsf{neg}\left(\left(y + x\right)\right)}}{\mathsf{neg}\left(\left(y + x\right)\right)}} \]

      14. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{\frac{x \cdot \frac{y}{1 + \left(y + x\right)}}{\mathsf{neg}\left(\left(y + x\right)\right)}}{\mathsf{neg}\left(\left(y + x\right)\right)}} \]

   8. Applied rewrites 99.8%

      \[\leadsto \color{blue}{\frac{\frac{\frac{y}{1 + \left(y + x\right)} \cdot x}{-\left(y + x\right)}}{-\left(y + x\right)}} \]

   9. Taylor expanded in y around inf

      \[\leadsto \frac{\color{blue}{-1 \cdot \frac{x}{y}}}{-\left(y + x\right)} \]
   10. Step-by-step derivation

       1. associate-*r/ N/A

          \[\leadsto \frac{\color{blue}{\frac{-1 \cdot x}{y}}}{-\left(y + x\right)} \]

       2. lower-/.f64 N/A

          \[\leadsto \frac{\color{blue}{\frac{-1 \cdot x}{y}}}{-\left(y + x\right)} \]

       3. mul-1-neg N/A

          \[\leadsto \frac{\frac{\color{blue}{\mathsf{neg}\left(x\right)}}{y}}{-\left(y + x\right)} \]

       4. lower-neg.f64 79.2

          \[\leadsto \frac{\frac{\color{blue}{-x}}{y}}{-\left(y + x\right)} \]

   11. Applied rewrites 79.2%

       \[\leadsto \frac{\color{blue}{\frac{-x}{y}}}{-\left(y + x\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 13: 80.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq -7 \cdot 10^{-6}:\\ \;\;\;\;\frac{\frac{y}{x}}{x}\\ \mathbf{elif}\;y \leq 1350000:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(x, x, x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-x}{y}}{-\left(y + x\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y -7e-6)
   (/ (/ y x) x)
   (if (<= y 1350000.0) (/ y (fma x x x)) (/ (/ (- x) y) (- (+ y x))))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= -7e-6) {
		tmp = (y / x) / x;
	} else if (y <= 1350000.0) {
		tmp = y / fma(x, x, x);
	} else {
		tmp = (-x / y) / -(y + x);
	}
	return tmp;
}

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= -7e-6)
		tmp = Float64(Float64(y / x) / x);
	elseif (y <= 1350000.0)
		tmp = Float64(y / fma(x, x, x));
	else
		tmp = Float64(Float64(Float64(-x) / y) / Float64(-Float64(y + x)));
	end
	return tmp
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[y, -7e-6], N[(N[(y / x), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[y, 1350000.0], N[(y / N[(x * x + x), $MachinePrecision]), $MachinePrecision], N[(N[((-x) / y), $MachinePrecision] / (-N[(y + x), $MachinePrecision])), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -6.99999999999999989e-6

   1. Initial program 63.7%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{y}{{x}^{2}}} \]
   4. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \frac{y}{\color{blue}{x \cdot x}} \]

      2. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{y}{x}}{x}} \]

      3. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{y}{x}}{x}} \]

      4. lower-/.f64 30.1

         \[\leadsto \frac{\color{blue}{\frac{y}{x}}}{x} \]

   5. Applied rewrites 30.1%

      \[\leadsto \color{blue}{\frac{\frac{y}{x}}{x}} \]

   if -6.99999999999999989e-6 < y < 1.35e6

   1. Initial program 75.0%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   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 75.7

         \[\leadsto \frac{y}{\color{blue}{\mathsf{fma}\left(x, x, x\right)}} \]

   5. Applied rewrites 75.7%

      \[\leadsto \color{blue}{\frac{y}{\mathsf{fma}\left(x, x, x\right)}} \]

   if 1.35e6 < y

   1. Initial program 62.3%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   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. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{x \cdot \frac{y}{\left(x + y\right) + 1}}{\left(x + y\right) \cdot \left(x + y\right)}} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot \frac{y}{\left(x + y\right) + 1}}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right)}} \]

      7. times-frac N/A

         \[\leadsto \color{blue}{\frac{x}{x + y} \cdot \frac{\frac{y}{\left(x + y\right) + 1}}{x + y}} \]

      8. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{x + y} \cdot \frac{\frac{y}{\left(x + y\right) + 1}}{x + y}} \]

      9. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{x + y}} \cdot \frac{\frac{y}{\left(x + y\right) + 1}}{x + y} \]

      10. lift-+.f64 N/A

          \[\leadsto \frac{x}{\color{blue}{x + y}} \cdot \frac{\frac{y}{\left(x + y\right) + 1}}{x + y} \]

      11. +-commutative N/A

          \[\leadsto \frac{x}{\color{blue}{y + x}} \cdot \frac{\frac{y}{\left(x + y\right) + 1}}{x + y} \]

      12. lower-+.f64 N/A

          \[\leadsto \frac{x}{\color{blue}{y + x}} \cdot \frac{\frac{y}{\left(x + y\right) + 1}}{x + y} \]

      13. lower-/.f64 N/A

          \[\leadsto \frac{x}{y + x} \cdot \color{blue}{\frac{\frac{y}{\left(x + y\right) + 1}}{x + y}} \]

      14. lower-/.f64 99.7

          \[\leadsto \frac{x}{y + x} \cdot \frac{\color{blue}{\frac{y}{\left(x + y\right) + 1}}}{x + y} \]

      15. lift-+.f64 N/A

          \[\leadsto \frac{x}{y + x} \cdot \frac{\frac{y}{\color{blue}{\left(x + y\right) + 1}}}{x + y} \]

      16. +-commutative N/A

          \[\leadsto \frac{x}{y + x} \cdot \frac{\frac{y}{\color{blue}{1 + \left(x + y\right)}}}{x + y} \]

      17. lower-+.f64 99.7

          \[\leadsto \frac{x}{y + x} \cdot \frac{\frac{y}{\color{blue}{1 + \left(x + y\right)}}}{x + y} \]

      18. lift-+.f64 N/A

          \[\leadsto \frac{x}{y + x} \cdot \frac{\frac{y}{1 + \color{blue}{\left(x + y\right)}}}{x + y} \]

      19. +-commutative N/A

          \[\leadsto \frac{x}{y + x} \cdot \frac{\frac{y}{1 + \color{blue}{\left(y + x\right)}}}{x + y} \]

      20. lower-+.f64 99.7

          \[\leadsto \frac{x}{y + x} \cdot \frac{\frac{y}{1 + \color{blue}{\left(y + x\right)}}}{x + y} \]

      21. lift-+.f64 N/A

          \[\leadsto \frac{x}{y + x} \cdot \frac{\frac{y}{1 + \left(y + x\right)}}{\color{blue}{x + y}} \]

      22. +-commutative N/A

          \[\leadsto \frac{x}{y + x} \cdot \frac{\frac{y}{1 + \left(y + x\right)}}{\color{blue}{y + x}} \]

      23. lower-+.f64 99.7

          \[\leadsto \frac{x}{y + x} \cdot \frac{\frac{y}{1 + \left(y + x\right)}}{\color{blue}{y + x}} \]

   4. Applied rewrites 99.7%

      \[\leadsto \color{blue}{\frac{x}{y + x} \cdot \frac{\frac{y}{1 + \left(y + x\right)}}{y + x}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{y + x} \cdot \frac{\frac{y}{1 + \left(y + x\right)}}{y + x}} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{x}{y + x} \cdot \color{blue}{\frac{\frac{y}{1 + \left(y + x\right)}}{y + x}} \]

      3. lift-/.f64 N/A

         \[\leadsto \frac{x}{y + x} \cdot \frac{\color{blue}{\frac{y}{1 + \left(y + x\right)}}}{y + x} \]

      4. associate-/l/ N/A

         \[\leadsto \frac{x}{y + x} \cdot \color{blue}{\frac{y}{\left(1 + \left(y + x\right)\right) \cdot \left(y + x\right)}} \]

      5. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{\frac{x}{y + x} \cdot y}{\left(1 + \left(y + x\right)\right) \cdot \left(y + x\right)}} \]

      6. lift-+.f64 N/A

         \[\leadsto \frac{\frac{x}{y + x} \cdot y}{\color{blue}{\left(1 + \left(y + x\right)\right)} \cdot \left(y + x\right)} \]

      7. lift-+.f64 N/A

         \[\leadsto \frac{\frac{x}{y + x} \cdot y}{\left(1 + \color{blue}{\left(y + x\right)}\right) \cdot \left(y + x\right)} \]

      8. +-commutative N/A

         \[\leadsto \frac{\frac{x}{y + x} \cdot y}{\left(1 + \color{blue}{\left(x + y\right)}\right) \cdot \left(y + x\right)} \]

      9. +-commutative N/A

         \[\leadsto \frac{\frac{x}{y + x} \cdot y}{\color{blue}{\left(\left(x + y\right) + 1\right)} \cdot \left(y + x\right)} \]

      10. lift-+.f64 N/A

          \[\leadsto \frac{\frac{x}{y + x} \cdot y}{\left(\color{blue}{\left(x + y\right)} + 1\right) \cdot \left(y + x\right)} \]

      11. lift-+.f64 N/A

          \[\leadsto \frac{\frac{x}{y + x} \cdot y}{\color{blue}{\left(\left(x + y\right) + 1\right)} \cdot \left(y + x\right)} \]

      12. lift-+.f64 N/A

          \[\leadsto \frac{\frac{x}{y + x} \cdot y}{\left(\left(x + y\right) + 1\right) \cdot \color{blue}{\left(y + x\right)}} \]

      13. +-commutative N/A

          \[\leadsto \frac{\frac{x}{y + x} \cdot y}{\left(\left(x + y\right) + 1\right) \cdot \color{blue}{\left(x + y\right)}} \]

      14. times-frac N/A

          \[\leadsto \color{blue}{\frac{\frac{x}{y + x}}{\left(x + y\right) + 1} \cdot \frac{y}{x + y}} \]

      15. lower-*.f64 N/A

          \[\leadsto \color{blue}{\frac{\frac{x}{y + x}}{\left(x + y\right) + 1} \cdot \frac{y}{x + y}} \]

      16. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{\frac{x}{y + x}}{\left(x + y\right) + 1}} \cdot \frac{y}{x + y} \]

      17. lift-+.f64 N/A

          \[\leadsto \frac{\frac{x}{y + x}}{\color{blue}{\left(x + y\right)} + 1} \cdot \frac{y}{x + y} \]

      18. +-commutative N/A

          \[\leadsto \frac{\frac{x}{y + x}}{\color{blue}{\left(y + x\right)} + 1} \cdot \frac{y}{x + y} \]

      19. lift-+.f64 N/A

          \[\leadsto \frac{\frac{x}{y + x}}{\color{blue}{\left(y + x\right)} + 1} \cdot \frac{y}{x + y} \]

      20. lower-/.f64 N/A

          \[\leadsto \frac{\frac{x}{y + x}}{\left(y + x\right) + 1} \cdot \color{blue}{\frac{y}{x + y}} \]

      21. +-commutative N/A

          \[\leadsto \frac{\frac{x}{y + x}}{\left(y + x\right) + 1} \cdot \frac{y}{\color{blue}{y + x}} \]

   6. Applied rewrites 99.8%

      \[\leadsto \color{blue}{\frac{\frac{x}{y + x}}{\left(y + x\right) + 1} \cdot \frac{y}{y + x}} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{x}{y + x}}{\left(y + x\right) + 1} \cdot \frac{y}{y + x}} \]

      2. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{x}{y + x}}{\left(y + x\right) + 1}} \cdot \frac{y}{y + x} \]

      3. lift-/.f64 N/A

         \[\leadsto \frac{\frac{x}{y + x}}{\left(y + x\right) + 1} \cdot \color{blue}{\frac{y}{y + x}} \]

      4. frac-times N/A

         \[\leadsto \color{blue}{\frac{\frac{x}{y + x} \cdot y}{\left(\left(y + x\right) + 1\right) \cdot \left(y + x\right)}} \]

      5. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{x}{y + x} \cdot \frac{y}{\left(\left(y + x\right) + 1\right) \cdot \left(y + x\right)}} \]

      6. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{y + x}} \cdot \frac{y}{\left(\left(y + x\right) + 1\right) \cdot \left(y + x\right)} \]

      7. lift-+.f64 N/A

         \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\color{blue}{\left(\left(y + x\right) + 1\right)} \cdot \left(y + x\right)} \]

      8. +-commutative N/A

         \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\color{blue}{\left(1 + \left(y + x\right)\right)} \cdot \left(y + x\right)} \]

      9. lift-+.f64 N/A

         \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\left(1 + \color{blue}{\left(y + x\right)}\right) \cdot \left(y + x\right)} \]

      10. associate-/l/ N/A

          \[\leadsto \frac{x}{y + x} \cdot \color{blue}{\frac{\frac{y}{1 + \left(y + x\right)}}{y + x}} \]

      11. frac-times N/A

          \[\leadsto \color{blue}{\frac{x \cdot \frac{y}{1 + \left(y + x\right)}}{\left(y + x\right) \cdot \left(y + x\right)}} \]

      12. sqr-neg-rev N/A

          \[\leadsto \frac{x \cdot \frac{y}{1 + \left(y + x\right)}}{\color{blue}{\left(\mathsf{neg}\left(\left(y + x\right)\right)\right) \cdot \left(\mathsf{neg}\left(\left(y + x\right)\right)\right)}} \]

      13. associate-/r* N/A

          \[\leadsto \color{blue}{\frac{\frac{x \cdot \frac{y}{1 + \left(y + x\right)}}{\mathsf{neg}\left(\left(y + x\right)\right)}}{\mathsf{neg}\left(\left(y + x\right)\right)}} \]

      14. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{\frac{x \cdot \frac{y}{1 + \left(y + x\right)}}{\mathsf{neg}\left(\left(y + x\right)\right)}}{\mathsf{neg}\left(\left(y + x\right)\right)}} \]

   8. Applied rewrites 99.8%

      \[\leadsto \color{blue}{\frac{\frac{\frac{y}{1 + \left(y + x\right)} \cdot x}{-\left(y + x\right)}}{-\left(y + x\right)}} \]

   9. Taylor expanded in y around inf

      \[\leadsto \frac{\color{blue}{-1 \cdot \frac{x}{y}}}{-\left(y + x\right)} \]
   10. Step-by-step derivation

       1. associate-*r/ N/A

          \[\leadsto \frac{\color{blue}{\frac{-1 \cdot x}{y}}}{-\left(y + x\right)} \]

       2. lower-/.f64 N/A

          \[\leadsto \frac{\color{blue}{\frac{-1 \cdot x}{y}}}{-\left(y + x\right)} \]

       3. mul-1-neg N/A

          \[\leadsto \frac{\frac{\color{blue}{\mathsf{neg}\left(x\right)}}{y}}{-\left(y + x\right)} \]

       4. lower-neg.f64 79.2

          \[\leadsto \frac{\frac{\color{blue}{-x}}{y}}{-\left(y + x\right)} \]

   11. Applied rewrites 79.2%

       \[\leadsto \frac{\color{blue}{\frac{-x}{y}}}{-\left(y + x\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 14: 80.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq -7 \cdot 10^{-6}:\\ \;\;\;\;\frac{\frac{y}{x}}{x}\\ \mathbf{elif}\;y \leq 1350000:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(x, x, x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y -7e-6)
   (/ (/ y x) x)
   (if (<= y 1350000.0) (/ y (fma x x x)) (/ (/ x y) y))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= -7e-6) {
		tmp = (y / x) / x;
	} else if (y <= 1350000.0) {
		tmp = y / fma(x, x, x);
	} else {
		tmp = (x / y) / y;
	}
	return tmp;
}

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= -7e-6)
		tmp = Float64(Float64(y / x) / x);
	elseif (y <= 1350000.0)
		tmp = Float64(y / fma(x, x, x));
	else
		tmp = Float64(Float64(x / y) / y);
	end
	return tmp
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[y, -7e-6], N[(N[(y / x), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[y, 1350000.0], N[(y / N[(x * x + x), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -6.99999999999999989e-6

   1. Initial program 63.7%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{y}{{x}^{2}}} \]
   4. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \frac{y}{\color{blue}{x \cdot x}} \]

      2. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{y}{x}}{x}} \]

      3. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{y}{x}}{x}} \]

      4. lower-/.f64 30.1

         \[\leadsto \frac{\color{blue}{\frac{y}{x}}}{x} \]

   5. Applied rewrites 30.1%

      \[\leadsto \color{blue}{\frac{\frac{y}{x}}{x}} \]

   if -6.99999999999999989e-6 < y < 1.35e6

   1. Initial program 75.0%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   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 75.7

         \[\leadsto \frac{y}{\color{blue}{\mathsf{fma}\left(x, x, x\right)}} \]

   5. Applied rewrites 75.7%

      \[\leadsto \color{blue}{\frac{y}{\mathsf{fma}\left(x, x, x\right)}} \]

   if 1.35e6 < y

   1. Initial program 62.3%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\frac{x}{{y}^{2}}} \]
   4. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \frac{x}{\color{blue}{y \cdot y}} \]

      2. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{x}{y}}{y}} \]

      3. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{x}{y}}{y}} \]

      4. lower-/.f64 78.8

         \[\leadsto \frac{\color{blue}{\frac{x}{y}}}{y} \]

   5. Applied rewrites 78.8%

      \[\leadsto \color{blue}{\frac{\frac{x}{y}}{y}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 15: 79.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 1350000:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(x, x, x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 1350000.0) (/ y (fma x x x)) (/ (/ x y) y)))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 1350000.0) {
		tmp = y / fma(x, x, x);
	} else {
		tmp = (x / y) / y;
	}
	return tmp;
}

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= 1350000.0)
		tmp = Float64(y / fma(x, x, x));
	else
		tmp = Float64(Float64(x / y) / y);
	end
	return tmp
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[y, 1350000.0], N[(y / N[(x * x + x), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.35e6

   1. Initial program 70.4%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{y}{x \cdot \left(1 + x\right)}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{y}{x \cdot \left(1 + x\right)}} \]

      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 51.8

         \[\leadsto \frac{y}{\color{blue}{\mathsf{fma}\left(x, x, x\right)}} \]

   5. Applied rewrites 51.8%

      \[\leadsto \color{blue}{\frac{y}{\mathsf{fma}\left(x, x, x\right)}} \]

   if 1.35e6 < y

   1. Initial program 62.3%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\frac{x}{{y}^{2}}} \]
   4. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \frac{x}{\color{blue}{y \cdot y}} \]

      2. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{x}{y}}{y}} \]

      3. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{x}{y}}{y}} \]

      4. lower-/.f64 78.8

         \[\leadsto \frac{\color{blue}{\frac{x}{y}}}{y} \]

   5. Applied rewrites 78.8%

      \[\leadsto \color{blue}{\frac{\frac{x}{y}}{y}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 16: 78.4% accurate, 1.6× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -9 \cdot 10^{-47}:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(x, x, x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(y, y, y\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -9e-47) (/ y (fma x x x)) (/ x (fma y y y))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -9e-47) {
		tmp = y / fma(x, x, x);
	} else {
		tmp = x / fma(y, y, y);
	}
	return tmp;
}

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -9e-47)
		tmp = Float64(y / fma(x, x, x));
	else
		tmp = Float64(x / fma(y, y, y));
	end
	return tmp
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[x, -9e-47], 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 < -9e-47

   1. Initial program 63.2%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{y}{x \cdot \left(1 + x\right)}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{y}{x \cdot \left(1 + x\right)}} \]

      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 58.5

         \[\leadsto \frac{y}{\color{blue}{\mathsf{fma}\left(x, x, x\right)}} \]

   5. Applied rewrites 58.5%

      \[\leadsto \color{blue}{\frac{y}{\mathsf{fma}\left(x, x, x\right)}} \]

   if -9e-47 < x

   1. Initial program 70.2%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]

      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 60.7

         \[\leadsto \frac{x}{\color{blue}{\mathsf{fma}\left(y, y, y\right)}} \]

   5. Applied rewrites 60.7%

      \[\leadsto \color{blue}{\frac{x}{\mathsf{fma}\left(y, y, y\right)}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 17: 76.6% accurate, 1.6× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -1100000000:\\ \;\;\;\;\frac{y}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(y, y, y\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -1100000000.0) (/ y (* x x)) (/ x (fma y y y))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -1100000000.0) {
		tmp = y / (x * x);
	} else {
		tmp = x / fma(y, y, y);
	}
	return tmp;
}

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -1100000000.0)
		tmp = Float64(y / Float64(x * x));
	else
		tmp = Float64(x / fma(y, y, y));
	end
	return tmp
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[x, -1100000000.0], N[(y / N[(x * x), $MachinePrecision]), $MachinePrecision], N[(x / N[(y * y + y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.1e9

   1. Initial program 55.3%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{y}{{x}^{2}}} \]
   4. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \frac{y}{\color{blue}{x \cdot x}} \]

      2. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{y}{x}}{x}} \]

      3. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{y}{x}}{x}} \]

      4. lower-/.f64 78.2

         \[\leadsto \frac{\color{blue}{\frac{y}{x}}}{x} \]

   5. Applied rewrites 78.2%

      \[\leadsto \color{blue}{\frac{\frac{y}{x}}{x}} \]
   6. Step-by-step derivation

   7. Applied rewrites 66.7%

      \[\leadsto \frac{y}{\color{blue}{x \cdot x}} \]

   if -1.1e9 < x

   1. Initial program 71.8%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]

      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 61.6

         \[\leadsto \frac{x}{\color{blue}{\mathsf{fma}\left(y, y, y\right)}} \]

   5. Applied rewrites 61.6%

      \[\leadsto \color{blue}{\frac{x}{\mathsf{fma}\left(y, y, y\right)}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 18: 64.7% accurate, 1.7× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 1350000:\\ \;\;\;\;\frac{y}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 1350000.0) (/ y (* x x)) (/ x (* y y))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 1350000.0) {
		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 <= 1350000.0d0) 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 <= 1350000.0) {
		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 <= 1350000.0:
		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 <= 1350000.0)
		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 <= 1350000.0)
		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, 1350000.0], N[(y / N[(x * x), $MachinePrecision]), $MachinePrecision], N[(x / N[(y * y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.35e6

   1. Initial program 70.4%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{y}{{x}^{2}}} \]
   4. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \frac{y}{\color{blue}{x \cdot x}} \]

      2. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{y}{x}}{x}} \]

      3. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{y}{x}}{x}} \]

      4. lower-/.f64 43.3

         \[\leadsto \frac{\color{blue}{\frac{y}{x}}}{x} \]

   5. Applied rewrites 43.3%

      \[\leadsto \color{blue}{\frac{\frac{y}{x}}{x}} \]
   6. Step-by-step derivation

   7. Applied rewrites 37.9%

      \[\leadsto \frac{y}{\color{blue}{x \cdot x}} \]

   if 1.35e6 < y

   1. Initial program 62.3%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   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. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{x \cdot \frac{y}{\left(x + y\right) + 1}}{\left(x + y\right) \cdot \left(x + y\right)}} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot \frac{y}{\left(x + y\right) + 1}}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right)}} \]

      7. times-frac N/A

         \[\leadsto \color{blue}{\frac{x}{x + y} \cdot \frac{\frac{y}{\left(x + y\right) + 1}}{x + y}} \]

      8. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{x + y} \cdot \frac{\frac{y}{\left(x + y\right) + 1}}{x + y}} \]

      9. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{x + y}} \cdot \frac{\frac{y}{\left(x + y\right) + 1}}{x + y} \]

      10. lift-+.f64 N/A

          \[\leadsto \frac{x}{\color{blue}{x + y}} \cdot \frac{\frac{y}{\left(x + y\right) + 1}}{x + y} \]

      11. +-commutative N/A

          \[\leadsto \frac{x}{\color{blue}{y + x}} \cdot \frac{\frac{y}{\left(x + y\right) + 1}}{x + y} \]

      12. lower-+.f64 N/A

          \[\leadsto \frac{x}{\color{blue}{y + x}} \cdot \frac{\frac{y}{\left(x + y\right) + 1}}{x + y} \]

      13. lower-/.f64 N/A

          \[\leadsto \frac{x}{y + x} \cdot \color{blue}{\frac{\frac{y}{\left(x + y\right) + 1}}{x + y}} \]

      14. lower-/.f64 99.7

          \[\leadsto \frac{x}{y + x} \cdot \frac{\color{blue}{\frac{y}{\left(x + y\right) + 1}}}{x + y} \]

      15. lift-+.f64 N/A

          \[\leadsto \frac{x}{y + x} \cdot \frac{\frac{y}{\color{blue}{\left(x + y\right) + 1}}}{x + y} \]

      16. +-commutative N/A

          \[\leadsto \frac{x}{y + x} \cdot \frac{\frac{y}{\color{blue}{1 + \left(x + y\right)}}}{x + y} \]

      17. lower-+.f64 99.7

          \[\leadsto \frac{x}{y + x} \cdot \frac{\frac{y}{\color{blue}{1 + \left(x + y\right)}}}{x + y} \]

      18. lift-+.f64 N/A

          \[\leadsto \frac{x}{y + x} \cdot \frac{\frac{y}{1 + \color{blue}{\left(x + y\right)}}}{x + y} \]

      19. +-commutative N/A

          \[\leadsto \frac{x}{y + x} \cdot \frac{\frac{y}{1 + \color{blue}{\left(y + x\right)}}}{x + y} \]

      20. lower-+.f64 99.7

          \[\leadsto \frac{x}{y + x} \cdot \frac{\frac{y}{1 + \color{blue}{\left(y + x\right)}}}{x + y} \]

      21. lift-+.f64 N/A

          \[\leadsto \frac{x}{y + x} \cdot \frac{\frac{y}{1 + \left(y + x\right)}}{\color{blue}{x + y}} \]

      22. +-commutative N/A

          \[\leadsto \frac{x}{y + x} \cdot \frac{\frac{y}{1 + \left(y + x\right)}}{\color{blue}{y + x}} \]

      23. lower-+.f64 99.7

          \[\leadsto \frac{x}{y + x} \cdot \frac{\frac{y}{1 + \left(y + x\right)}}{\color{blue}{y + x}} \]

   4. Applied rewrites 99.7%

      \[\leadsto \color{blue}{\frac{x}{y + x} \cdot \frac{\frac{y}{1 + \left(y + x\right)}}{y + x}} \]

   5. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\frac{x}{{y}^{2}}} \]
   6. 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 73.2

         \[\leadsto \frac{x}{\color{blue}{y \cdot y}} \]

   7. Applied rewrites 73.2%

      \[\leadsto \color{blue}{\frac{x}{y \cdot y}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 47.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1350000:\\ \;\;\;\;\frac{y}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot y}\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 37.6% 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 68.2%

   \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \color{blue}{\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{\color{blue}{x \cdot y}}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

   3. 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. associate-*l/ N/A

      \[\leadsto \color{blue}{\frac{x \cdot \frac{y}{\left(x + y\right) + 1}}{\left(x + y\right) \cdot \left(x + y\right)}} \]

   6. lift-*.f64 N/A

      \[\leadsto \frac{x \cdot \frac{y}{\left(x + y\right) + 1}}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right)}} \]

   7. times-frac N/A

      \[\leadsto \color{blue}{\frac{x}{x + y} \cdot \frac{\frac{y}{\left(x + y\right) + 1}}{x + y}} \]

   8. lower-*.f64 N/A

      \[\leadsto \color{blue}{\frac{x}{x + y} \cdot \frac{\frac{y}{\left(x + y\right) + 1}}{x + y}} \]

   9. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{x}{x + y}} \cdot \frac{\frac{y}{\left(x + y\right) + 1}}{x + y} \]

   10. lift-+.f64 N/A

       \[\leadsto \frac{x}{\color{blue}{x + y}} \cdot \frac{\frac{y}{\left(x + y\right) + 1}}{x + y} \]

   11. +-commutative N/A

       \[\leadsto \frac{x}{\color{blue}{y + x}} \cdot \frac{\frac{y}{\left(x + y\right) + 1}}{x + y} \]

   12. lower-+.f64 N/A

       \[\leadsto \frac{x}{\color{blue}{y + x}} \cdot \frac{\frac{y}{\left(x + y\right) + 1}}{x + y} \]

   13. lower-/.f64 N/A

       \[\leadsto \frac{x}{y + x} \cdot \color{blue}{\frac{\frac{y}{\left(x + y\right) + 1}}{x + y}} \]

   14. lower-/.f64 99.8

       \[\leadsto \frac{x}{y + x} \cdot \frac{\color{blue}{\frac{y}{\left(x + y\right) + 1}}}{x + y} \]

   15. lift-+.f64 N/A

       \[\leadsto \frac{x}{y + x} \cdot \frac{\frac{y}{\color{blue}{\left(x + y\right) + 1}}}{x + y} \]

   16. +-commutative N/A

       \[\leadsto \frac{x}{y + x} \cdot \frac{\frac{y}{\color{blue}{1 + \left(x + y\right)}}}{x + y} \]

   17. lower-+.f64 99.8

       \[\leadsto \frac{x}{y + x} \cdot \frac{\frac{y}{\color{blue}{1 + \left(x + y\right)}}}{x + y} \]

   18. lift-+.f64 N/A

       \[\leadsto \frac{x}{y + x} \cdot \frac{\frac{y}{1 + \color{blue}{\left(x + y\right)}}}{x + y} \]

   19. +-commutative N/A

       \[\leadsto \frac{x}{y + x} \cdot \frac{\frac{y}{1 + \color{blue}{\left(y + x\right)}}}{x + y} \]

   20. lower-+.f64 99.8

       \[\leadsto \frac{x}{y + x} \cdot \frac{\frac{y}{1 + \color{blue}{\left(y + x\right)}}}{x + y} \]

   21. lift-+.f64 N/A

       \[\leadsto \frac{x}{y + x} \cdot \frac{\frac{y}{1 + \left(y + x\right)}}{\color{blue}{x + y}} \]

   22. +-commutative N/A

       \[\leadsto \frac{x}{y + x} \cdot \frac{\frac{y}{1 + \left(y + x\right)}}{\color{blue}{y + x}} \]

   23. lower-+.f64 99.8

       \[\leadsto \frac{x}{y + x} \cdot \frac{\frac{y}{1 + \left(y + x\right)}}{\color{blue}{y + x}} \]

4. Applied rewrites 99.8%

   \[\leadsto \color{blue}{\frac{x}{y + x} \cdot \frac{\frac{y}{1 + \left(y + x\right)}}{y + x}} \]

5. Taylor expanded in y around inf

   \[\leadsto \color{blue}{\frac{x}{{y}^{2}}} \]
6. 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 41.9

      \[\leadsto \frac{x}{\color{blue}{y \cdot y}} \]

7. Applied rewrites 41.9%

   \[\leadsto \color{blue}{\frac{x}{y \cdot y}} \]

8. Final simplification 41.9%

   \[\leadsto \frac{x}{y \cdot y} \]
9. Add Preprocessing

Developer Target 1: 99.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{\frac{x}{\left(y + 1\right) + x}}{y + x}}{\frac{1}{\frac{y}{y + x}}} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (/ (/ (/ x (+ (+ y 1.0) x)) (+ y x)) (/ 1.0 (/ y (+ y x)))))

C

double code(double x, double y) {
	return ((x / ((y + 1.0) + x)) / (y + x)) / (1.0 / (y / (y + x)));
}

Fortran

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 2024329 
(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))))