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

Percentage Accurate: 69.8% → 99.8%

Time: 9.0s

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: 69.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 67.1%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. 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

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

   4. +-commutative N/A

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

   5. associate-*r/ N/A

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

   6. lift-/.f64 N/A

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

   7. associate-*r/ N/A

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

   8. *-commutative N/A

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

   9. associate-/r* N/A

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

   10. *-commutative N/A

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

   11. times-frac N/A

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

   12. lower-*.f64 N/A

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

   13. lower-/.f64 N/A

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

   14. +-commutative N/A

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

   15. lift-+.f64 N/A

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

   16. lower-/.f64 99.8

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

   17. lift-+.f64 N/A

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

   18. lift-+.f64 N/A

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

   19. +-commutative N/A

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

6. Applied rewrites 99.8%

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

7. Final simplification 99.8%

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

Alternative 2: 95.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_0 := \left(1 + x\right) + y\\ \mathbf{if}\;y \leq -1.32 \cdot 10^{-256}:\\ \;\;\;\;1 \cdot \frac{\frac{y}{1 + \left(x + y\right)}}{x + y}\\ \mathbf{elif}\;y \leq 1.5 \cdot 10^{+146}:\\ \;\;\;\;\frac{\frac{y}{t\_0 \cdot \left(x + y\right)}}{x + y} \cdot x\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \frac{\frac{x}{x + y}}{t\_0}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ (+ 1.0 x) y)))
   (if (<= y -1.32e-256)
     (* 1.0 (/ (/ y (+ 1.0 (+ x y))) (+ x y)))
     (if (<= y 1.5e+146)
       (* (/ (/ y (* t_0 (+ x y))) (+ x y)) x)
       (* 1.0 (/ (/ x (+ x y)) t_0))))))

C

assert(x < y);
double code(double x, double y) {
	double t_0 = (1.0 + x) + y;
	double tmp;
	if (y <= -1.32e-256) {
		tmp = 1.0 * ((y / (1.0 + (x + y))) / (x + y));
	} else if (y <= 1.5e+146) {
		tmp = ((y / (t_0 * (x + y))) / (x + y)) * x;
	} else {
		tmp = 1.0 * ((x / (x + y)) / t_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) :: t_0
    real(8) :: tmp
    t_0 = (1.0d0 + x) + y
    if (y <= (-1.32d-256)) then
        tmp = 1.0d0 * ((y / (1.0d0 + (x + y))) / (x + y))
    else if (y <= 1.5d+146) then
        tmp = ((y / (t_0 * (x + y))) / (x + y)) * x
    else
        tmp = 1.0d0 * ((x / (x + y)) / t_0)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double t_0 = (1.0 + x) + y;
	double tmp;
	if (y <= -1.32e-256) {
		tmp = 1.0 * ((y / (1.0 + (x + y))) / (x + y));
	} else if (y <= 1.5e+146) {
		tmp = ((y / (t_0 * (x + y))) / (x + y)) * x;
	} else {
		tmp = 1.0 * ((x / (x + y)) / t_0);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	t_0 = (1.0 + x) + y
	tmp = 0
	if y <= -1.32e-256:
		tmp = 1.0 * ((y / (1.0 + (x + y))) / (x + y))
	elif y <= 1.5e+146:
		tmp = ((y / (t_0 * (x + y))) / (x + y)) * x
	else:
		tmp = 1.0 * ((x / (x + y)) / t_0)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	t_0 = Float64(Float64(1.0 + x) + y)
	tmp = 0.0
	if (y <= -1.32e-256)
		tmp = Float64(1.0 * Float64(Float64(y / Float64(1.0 + Float64(x + y))) / Float64(x + y)));
	elseif (y <= 1.5e+146)
		tmp = Float64(Float64(Float64(y / Float64(t_0 * Float64(x + y))) / Float64(x + y)) * x);
	else
		tmp = Float64(1.0 * Float64(Float64(x / Float64(x + y)) / t_0));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	t_0 = (1.0 + x) + y;
	tmp = 0.0;
	if (y <= -1.32e-256)
		tmp = 1.0 * ((y / (1.0 + (x + y))) / (x + y));
	elseif (y <= 1.5e+146)
		tmp = ((y / (t_0 * (x + y))) / (x + y)) * x;
	else
		tmp = 1.0 * ((x / (x + y)) / t_0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := Block[{t$95$0 = N[(N[(1.0 + x), $MachinePrecision] + y), $MachinePrecision]}, If[LessEqual[y, -1.32e-256], N[(1.0 * N[(N[(y / N[(1.0 + N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.5e+146], N[(N[(N[(y / N[(t$95$0 * N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x + y), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision], N[(1.0 * N[(N[(x / N[(x + y), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.32e-256

   1. Initial program 66.2%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   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 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 50.7%

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

   if -1.32e-256 < y < 1.50000000000000001e146

   1. Initial program 68.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. 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 99.9

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

      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 99.9

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

      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 99.9

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

      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 99.9

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

   4. Applied rewrites 99.9%

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

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

      3. associate-*l/ N/A

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

      4. lift-/.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. associate-/r* N/A

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

      7. lift-*.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. associate-/l* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

   6. Applied rewrites 94.8%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lower-/.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lift-+.f64 94.8

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

   8. Applied rewrites 94.8%

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

   if 1.50000000000000001e146 < y

   1. Initial program 63.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. 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

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

      4. +-commutative N/A

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

      5. associate-*r/ N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*r/ N/A

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

      8. *-commutative N/A

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

      9. associate-/r* N/A

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

      10. *-commutative N/A

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

      11. times-frac N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. +-commutative N/A

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

      15. lift-+.f64 N/A

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

      16. lower-/.f64 99.9

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

      17. lift-+.f64 N/A

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

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

   6. Applied rewrites 99.9%

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

   7. Taylor expanded in x around 0

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

   9. Applied rewrites 99.9%

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

4. Final simplification 75.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.32 \cdot 10^{-256}:\\ \;\;\;\;1 \cdot \frac{\frac{y}{1 + \left(x + y\right)}}{x + y}\\ \mathbf{elif}\;y \leq 1.5 \cdot 10^{+146}:\\ \;\;\;\;\frac{\frac{y}{\left(\left(1 + x\right) + y\right) \cdot \left(x + y\right)}}{x + y} \cdot x\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \frac{\frac{x}{x + y}}{\left(1 + x\right) + y}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 93.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_0 := 1 + \left(x + y\right)\\ \mathbf{if}\;x \leq -1.35 \cdot 10^{+170}:\\ \;\;\;\;1 \cdot \frac{\frac{y}{t\_0}}{x + y}\\ \mathbf{elif}\;x \leq -8 \cdot 10^{-8}:\\ \;\;\;\;1 \cdot \frac{y}{t\_0 \cdot \left(x + y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\left(1 + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{x + y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ 1.0 (+ x y))))
   (if (<= x -1.35e+170)
     (* 1.0 (/ (/ y t_0) (+ x y)))
     (if (<= x -8e-8)
       (* 1.0 (/ y (* t_0 (+ x y))))
       (* (/ x (* (+ 1.0 y) (+ x y))) (/ y (+ x y)))))))

C

assert(x < y);
double code(double x, double y) {
	double t_0 = 1.0 + (x + y);
	double tmp;
	if (x <= -1.35e+170) {
		tmp = 1.0 * ((y / t_0) / (x + y));
	} else if (x <= -8e-8) {
		tmp = 1.0 * (y / (t_0 * (x + y)));
	} else {
		tmp = (x / ((1.0 + y) * (x + y))) * (y / (x + y));
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 1.0d0 + (x + y)
    if (x <= (-1.35d+170)) then
        tmp = 1.0d0 * ((y / t_0) / (x + y))
    else if (x <= (-8d-8)) then
        tmp = 1.0d0 * (y / (t_0 * (x + y)))
    else
        tmp = (x / ((1.0d0 + y) * (x + y))) * (y / (x + y))
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double t_0 = 1.0 + (x + y);
	double tmp;
	if (x <= -1.35e+170) {
		tmp = 1.0 * ((y / t_0) / (x + y));
	} else if (x <= -8e-8) {
		tmp = 1.0 * (y / (t_0 * (x + y)));
	} else {
		tmp = (x / ((1.0 + y) * (x + y))) * (y / (x + y));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	t_0 = 1.0 + (x + y)
	tmp = 0
	if x <= -1.35e+170:
		tmp = 1.0 * ((y / t_0) / (x + y))
	elif x <= -8e-8:
		tmp = 1.0 * (y / (t_0 * (x + y)))
	else:
		tmp = (x / ((1.0 + y) * (x + y))) * (y / (x + y))
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	t_0 = Float64(1.0 + Float64(x + y))
	tmp = 0.0
	if (x <= -1.35e+170)
		tmp = Float64(1.0 * Float64(Float64(y / t_0) / Float64(x + y)));
	elseif (x <= -8e-8)
		tmp = Float64(1.0 * Float64(y / Float64(t_0 * Float64(x + y))));
	else
		tmp = Float64(Float64(x / Float64(Float64(1.0 + y) * Float64(x + y))) * Float64(y / Float64(x + y)));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	t_0 = 1.0 + (x + y);
	tmp = 0.0;
	if (x <= -1.35e+170)
		tmp = 1.0 * ((y / t_0) / (x + y));
	elseif (x <= -8e-8)
		tmp = 1.0 * (y / (t_0 * (x + y)));
	else
		tmp = (x / ((1.0 + y) * (x + y))) * (y / (x + y));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := Block[{t$95$0 = N[(1.0 + N[(x + y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.35e+170], N[(1.0 * N[(N[(y / t$95$0), $MachinePrecision] / N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -8e-8], N[(1.0 * N[(y / N[(t$95$0 * N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(N[(1.0 + y), $MachinePrecision] * N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(y / N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.3500000000000001e170

   1. Initial program 68.8%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   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 99.9

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

      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 99.9

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

      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 99.9

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

      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 99.9

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

   4. Applied rewrites 99.9%

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

   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 86.0%

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

   if -1.3500000000000001e170 < x < -8.0000000000000002e-8

   1. Initial program 62.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

      8. times-frac N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 N/A

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

      16. lift-+.f64 N/A

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

      17. +-commutative N/A

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

      18. lower-+.f64 N/A

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

      19. lift-+.f64 N/A

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

      20. +-commutative N/A

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

      21. lower-+.f64 N/A

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

      22. lower-/.f64 97.3

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

      23. lift-+.f64 N/A

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

      24. +-commutative N/A

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

   4. Applied rewrites 97.3%

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

   5. Taylor expanded in x around inf

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

   7. Applied rewrites 87.1%

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

   if -8.0000000000000002e-8 < x

   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 0

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

      1. lower-+.f64 61.8

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

   5. Applied rewrites 61.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*l* N/A

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

      9. times-frac N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. +-commutative N/A

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

      13. lift-+.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. +-commutative N/A

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

      18. lift-+.f64 84.5

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

   7. Applied rewrites 84.5%

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

4. Final simplification 85.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.35 \cdot 10^{+170}:\\ \;\;\;\;1 \cdot \frac{\frac{y}{1 + \left(x + y\right)}}{x + y}\\ \mathbf{elif}\;x \leq -8 \cdot 10^{-8}:\\ \;\;\;\;1 \cdot \frac{y}{\left(1 + \left(x + y\right)\right) \cdot \left(x + y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\left(1 + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{x + y}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 93.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_0 := 1 + \left(x + y\right)\\ \mathbf{if}\;x \leq -1.35 \cdot 10^{+170}:\\ \;\;\;\;1 \cdot \frac{\frac{y}{t\_0}}{x + y}\\ \mathbf{elif}\;x \leq -8 \cdot 10^{-8}:\\ \;\;\;\;1 \cdot \frac{y}{t\_0 \cdot \left(x + y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{\left(1 + y\right) \cdot \left(x + y\right)}}{x + y} \cdot x\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ 1.0 (+ x y))))
   (if (<= x -1.35e+170)
     (* 1.0 (/ (/ y t_0) (+ x y)))
     (if (<= x -8e-8)
       (* 1.0 (/ y (* t_0 (+ x y))))
       (* (/ (/ y (* (+ 1.0 y) (+ x y))) (+ x y)) x)))))

C

assert(x < y);
double code(double x, double y) {
	double t_0 = 1.0 + (x + y);
	double tmp;
	if (x <= -1.35e+170) {
		tmp = 1.0 * ((y / t_0) / (x + y));
	} else if (x <= -8e-8) {
		tmp = 1.0 * (y / (t_0 * (x + y)));
	} else {
		tmp = ((y / ((1.0 + y) * (x + y))) / (x + 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 + (x + y)
    if (x <= (-1.35d+170)) then
        tmp = 1.0d0 * ((y / t_0) / (x + y))
    else if (x <= (-8d-8)) then
        tmp = 1.0d0 * (y / (t_0 * (x + y)))
    else
        tmp = ((y / ((1.0d0 + y) * (x + y))) / (x + 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 + (x + y);
	double tmp;
	if (x <= -1.35e+170) {
		tmp = 1.0 * ((y / t_0) / (x + y));
	} else if (x <= -8e-8) {
		tmp = 1.0 * (y / (t_0 * (x + y)));
	} else {
		tmp = ((y / ((1.0 + y) * (x + y))) / (x + y)) * x;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	t_0 = 1.0 + (x + y)
	tmp = 0
	if x <= -1.35e+170:
		tmp = 1.0 * ((y / t_0) / (x + y))
	elif x <= -8e-8:
		tmp = 1.0 * (y / (t_0 * (x + y)))
	else:
		tmp = ((y / ((1.0 + y) * (x + y))) / (x + y)) * x
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	t_0 = Float64(1.0 + Float64(x + y))
	tmp = 0.0
	if (x <= -1.35e+170)
		tmp = Float64(1.0 * Float64(Float64(y / t_0) / Float64(x + y)));
	elseif (x <= -8e-8)
		tmp = Float64(1.0 * Float64(y / Float64(t_0 * Float64(x + y))));
	else
		tmp = Float64(Float64(Float64(y / Float64(Float64(1.0 + y) * Float64(x + y))) / Float64(x + y)) * x);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	t_0 = 1.0 + (x + y);
	tmp = 0.0;
	if (x <= -1.35e+170)
		tmp = 1.0 * ((y / t_0) / (x + y));
	elseif (x <= -8e-8)
		tmp = 1.0 * (y / (t_0 * (x + y)));
	else
		tmp = ((y / ((1.0 + y) * (x + y))) / (x + y)) * x;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := Block[{t$95$0 = N[(1.0 + N[(x + y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.35e+170], N[(1.0 * N[(N[(y / t$95$0), $MachinePrecision] / N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -8e-8], N[(1.0 * N[(y / N[(t$95$0 * N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(y / N[(N[(1.0 + y), $MachinePrecision] * N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x + y), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.3500000000000001e170

   1. Initial program 68.8%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   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 99.9

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

      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 99.9

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

      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 99.9

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

      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 99.9

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

   4. Applied rewrites 99.9%

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

   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 86.0%

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

   if -1.3500000000000001e170 < x < -8.0000000000000002e-8

   1. Initial program 62.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

      8. times-frac N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 N/A

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

      16. lift-+.f64 N/A

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

      17. +-commutative N/A

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

      18. lower-+.f64 N/A

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

      19. lift-+.f64 N/A

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

      20. +-commutative N/A

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

      21. lower-+.f64 N/A

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

      22. lower-/.f64 97.3

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

      23. lift-+.f64 N/A

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

      24. +-commutative N/A

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

   4. Applied rewrites 97.3%

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

   5. Taylor expanded in x around inf

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

   7. Applied rewrites 87.1%

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

   if -8.0000000000000002e-8 < x

   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 99.9

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

      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 99.9

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

      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 99.9

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

      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 99.9

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

   4. Applied rewrites 99.9%

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

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

      3. associate-*l/ N/A

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

      4. lift-/.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. associate-/r* N/A

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

      7. lift-*.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. associate-/l* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

   6. Applied rewrites 93.9%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lower-/.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lift-+.f64 93.7

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

   8. Applied rewrites 93.7%

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

   9. Taylor expanded in x around 0

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

       1. lower-+.f64 88.3

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

   11. Applied rewrites 88.3%

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

4. Final simplification 87.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.35 \cdot 10^{+170}:\\ \;\;\;\;1 \cdot \frac{\frac{y}{1 + \left(x + y\right)}}{x + y}\\ \mathbf{elif}\;x \leq -8 \cdot 10^{-8}:\\ \;\;\;\;1 \cdot \frac{y}{\left(1 + \left(x + y\right)\right) \cdot \left(x + y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{\left(1 + y\right) \cdot \left(x + y\right)}}{x + y} \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 95.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 1.5 \cdot 10^{+146}:\\ \;\;\;\;\frac{x}{\left(1 + \left(x + y\right)\right) \cdot \left(x + y\right)} \cdot \frac{y}{x + y}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \frac{\frac{x}{x + y}}{\left(1 + x\right) + y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 1.5e+146)
   (* (/ x (* (+ 1.0 (+ x y)) (+ x y))) (/ y (+ x y)))
   (* 1.0 (/ (/ x (+ x y)) (+ (+ 1.0 x) y)))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 1.5e+146) {
		tmp = (x / ((1.0 + (x + y)) * (x + y))) * (y / (x + y));
	} else {
		tmp = 1.0 * ((x / (x + y)) / ((1.0 + x) + y));
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 1.5d+146) then
        tmp = (x / ((1.0d0 + (x + y)) * (x + y))) * (y / (x + y))
    else
        tmp = 1.0d0 * ((x / (x + y)) / ((1.0d0 + x) + y))
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (y <= 1.5e+146) {
		tmp = (x / ((1.0 + (x + y)) * (x + y))) * (y / (x + y));
	} else {
		tmp = 1.0 * ((x / (x + y)) / ((1.0 + x) + y));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if y <= 1.5e+146:
		tmp = (x / ((1.0 + (x + y)) * (x + y))) * (y / (x + y))
	else:
		tmp = 1.0 * ((x / (x + y)) / ((1.0 + x) + y))
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= 1.5e+146)
		tmp = Float64(Float64(x / Float64(Float64(1.0 + Float64(x + y)) * Float64(x + y))) * Float64(y / Float64(x + y)));
	else
		tmp = Float64(1.0 * Float64(Float64(x / Float64(x + y)) / Float64(Float64(1.0 + x) + y)));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 1.5e+146)
		tmp = (x / ((1.0 + (x + y)) * (x + y))) * (y / (x + y));
	else
		tmp = 1.0 * ((x / (x + y)) / ((1.0 + x) + y));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[y, 1.5e+146], N[(N[(x / N[(N[(1.0 + N[(x + y), $MachinePrecision]), $MachinePrecision] * N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(y / N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 * N[(N[(x / N[(x + y), $MachinePrecision]), $MachinePrecision] / N[(N[(1.0 + x), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.50000000000000001e146

   1. Initial program 67.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 95.6

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

      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 95.6

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

      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 95.6

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

      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 95.6

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

   4. Applied rewrites 95.6%

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

   if 1.50000000000000001e146 < y

   1. Initial program 63.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. 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

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

      4. +-commutative N/A

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

      5. associate-*r/ N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*r/ N/A

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

      8. *-commutative N/A

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

      9. associate-/r* N/A

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

      10. *-commutative N/A

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

      11. times-frac N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. +-commutative N/A

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

      15. lift-+.f64 N/A

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

      16. lower-/.f64 99.9

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

      17. lift-+.f64 N/A

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

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

   6. Applied rewrites 99.9%

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

   7. Taylor expanded in x around 0

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

   9. Applied rewrites 99.9%

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

4. Final simplification 96.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.5 \cdot 10^{+146}:\\ \;\;\;\;\frac{x}{\left(1 + \left(x + y\right)\right) \cdot \left(x + y\right)} \cdot \frac{y}{x + y}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \frac{\frac{x}{x + y}}{\left(1 + x\right) + y}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 95.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_0 := \frac{x}{x + y}\\ \mathbf{if}\;y \leq 1.5 \cdot 10^{+146}:\\ \;\;\;\;\frac{y}{\left(1 + \left(x + y\right)\right) \cdot \left(x + y\right)} \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \frac{t\_0}{\left(1 + x\right) + y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ x (+ x y))))
   (if (<= y 1.5e+146)
     (* (/ y (* (+ 1.0 (+ x y)) (+ x y))) t_0)
     (* 1.0 (/ t_0 (+ (+ 1.0 x) y))))))

C

assert(x < y);
double code(double x, double y) {
	double t_0 = x / (x + y);
	double tmp;
	if (y <= 1.5e+146) {
		tmp = (y / ((1.0 + (x + y)) * (x + y))) * t_0;
	} else {
		tmp = 1.0 * (t_0 / ((1.0 + x) + y));
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x / (x + y)
    if (y <= 1.5d+146) then
        tmp = (y / ((1.0d0 + (x + y)) * (x + y))) * t_0
    else
        tmp = 1.0d0 * (t_0 / ((1.0d0 + x) + y))
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double t_0 = x / (x + y);
	double tmp;
	if (y <= 1.5e+146) {
		tmp = (y / ((1.0 + (x + y)) * (x + y))) * t_0;
	} else {
		tmp = 1.0 * (t_0 / ((1.0 + x) + y));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	t_0 = x / (x + y)
	tmp = 0
	if y <= 1.5e+146:
		tmp = (y / ((1.0 + (x + y)) * (x + y))) * t_0
	else:
		tmp = 1.0 * (t_0 / ((1.0 + x) + y))
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	t_0 = Float64(x / Float64(x + y))
	tmp = 0.0
	if (y <= 1.5e+146)
		tmp = Float64(Float64(y / Float64(Float64(1.0 + Float64(x + y)) * Float64(x + y))) * t_0);
	else
		tmp = Float64(1.0 * Float64(t_0 / Float64(Float64(1.0 + x) + y)));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	t_0 = x / (x + y);
	tmp = 0.0;
	if (y <= 1.5e+146)
		tmp = (y / ((1.0 + (x + y)) * (x + y))) * t_0;
	else
		tmp = 1.0 * (t_0 / ((1.0 + x) + y));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := Block[{t$95$0 = N[(x / N[(x + y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, 1.5e+146], N[(N[(y / N[(N[(1.0 + N[(x + y), $MachinePrecision]), $MachinePrecision] * N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision], N[(1.0 * N[(t$95$0 / N[(N[(1.0 + x), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < 1.50000000000000001e146

   1. Initial program 67.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

      8. times-frac N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 N/A

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

      16. lift-+.f64 N/A

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

      17. +-commutative N/A

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

      18. lower-+.f64 N/A

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

      19. lift-+.f64 N/A

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

      20. +-commutative N/A

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

      21. lower-+.f64 N/A

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

      22. lower-/.f64 95.6

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

      23. lift-+.f64 N/A

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

      24. +-commutative N/A

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

   4. Applied rewrites 95.6%

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

   if 1.50000000000000001e146 < y

   1. Initial program 63.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. 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

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

      4. +-commutative N/A

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

      5. associate-*r/ N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*r/ N/A

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

      8. *-commutative N/A

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

      9. associate-/r* N/A

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

      10. *-commutative N/A

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

      11. times-frac N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. +-commutative N/A

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

      15. lift-+.f64 N/A

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

      16. lower-/.f64 99.9

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

      17. lift-+.f64 N/A

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

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

   6. Applied rewrites 99.9%

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

   7. Taylor expanded in x around 0

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

   9. Applied rewrites 99.9%

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

4. Final simplification 96.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.5 \cdot 10^{+146}:\\ \;\;\;\;\frac{y}{\left(1 + \left(x + y\right)\right) \cdot \left(x + y\right)} \cdot \frac{x}{x + y}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \frac{\frac{x}{x + y}}{\left(1 + x\right) + y}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 99.8% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 67.1%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. 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. Final simplification 99.8%

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

Alternative 8: 87.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_0 := \left(1 + x\right) + y\\ \mathbf{if}\;y \leq 1.15 \cdot 10^{-139}:\\ \;\;\;\;1 \cdot \frac{\frac{y}{1 + \left(x + y\right)}}{x + y}\\ \mathbf{elif}\;y \leq 1.5 \cdot 10^{+146}:\\ \;\;\;\;\frac{1 \cdot x}{t\_0 \cdot \left(x + y\right)}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \frac{\frac{x}{x + y}}{t\_0}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ (+ 1.0 x) y)))
   (if (<= y 1.15e-139)
     (* 1.0 (/ (/ y (+ 1.0 (+ x y))) (+ x y)))
     (if (<= y 1.5e+146)
       (/ (* 1.0 x) (* t_0 (+ x y)))
       (* 1.0 (/ (/ x (+ x y)) t_0))))))

C

assert(x < y);
double code(double x, double y) {
	double t_0 = (1.0 + x) + y;
	double tmp;
	if (y <= 1.15e-139) {
		tmp = 1.0 * ((y / (1.0 + (x + y))) / (x + y));
	} else if (y <= 1.5e+146) {
		tmp = (1.0 * x) / (t_0 * (x + y));
	} else {
		tmp = 1.0 * ((x / (x + y)) / t_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) :: t_0
    real(8) :: tmp
    t_0 = (1.0d0 + x) + y
    if (y <= 1.15d-139) then
        tmp = 1.0d0 * ((y / (1.0d0 + (x + y))) / (x + y))
    else if (y <= 1.5d+146) then
        tmp = (1.0d0 * x) / (t_0 * (x + y))
    else
        tmp = 1.0d0 * ((x / (x + y)) / t_0)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double t_0 = (1.0 + x) + y;
	double tmp;
	if (y <= 1.15e-139) {
		tmp = 1.0 * ((y / (1.0 + (x + y))) / (x + y));
	} else if (y <= 1.5e+146) {
		tmp = (1.0 * x) / (t_0 * (x + y));
	} else {
		tmp = 1.0 * ((x / (x + y)) / t_0);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	t_0 = (1.0 + x) + y
	tmp = 0
	if y <= 1.15e-139:
		tmp = 1.0 * ((y / (1.0 + (x + y))) / (x + y))
	elif y <= 1.5e+146:
		tmp = (1.0 * x) / (t_0 * (x + y))
	else:
		tmp = 1.0 * ((x / (x + y)) / t_0)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	t_0 = Float64(Float64(1.0 + x) + y)
	tmp = 0.0
	if (y <= 1.15e-139)
		tmp = Float64(1.0 * Float64(Float64(y / Float64(1.0 + Float64(x + y))) / Float64(x + y)));
	elseif (y <= 1.5e+146)
		tmp = Float64(Float64(1.0 * x) / Float64(t_0 * Float64(x + y)));
	else
		tmp = Float64(1.0 * Float64(Float64(x / Float64(x + y)) / t_0));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	t_0 = (1.0 + x) + y;
	tmp = 0.0;
	if (y <= 1.15e-139)
		tmp = 1.0 * ((y / (1.0 + (x + y))) / (x + y));
	elseif (y <= 1.5e+146)
		tmp = (1.0 * x) / (t_0 * (x + y));
	else
		tmp = 1.0 * ((x / (x + y)) / t_0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := Block[{t$95$0 = N[(N[(1.0 + x), $MachinePrecision] + y), $MachinePrecision]}, If[LessEqual[y, 1.15e-139], N[(1.0 * N[(N[(y / N[(1.0 + N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.5e+146], N[(N[(1.0 * x), $MachinePrecision] / N[(t$95$0 * N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 * N[(N[(x / N[(x + y), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < 1.15000000000000006e-139

   1. Initial program 67.3%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   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 99.9

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

      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 99.9

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

      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 99.9

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

      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 99.9

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

   4. Applied rewrites 99.9%

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

   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 65.0%

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

   if 1.15000000000000006e-139 < y < 1.50000000000000001e146

   1. Initial program 68.1%

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

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

      4. +-commutative N/A

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

      5. associate-*r/ N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*r/ N/A

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

      8. *-commutative N/A

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

      9. associate-/r* N/A

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

      10. *-commutative N/A

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

      11. times-frac N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. +-commutative N/A

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

      15. lift-+.f64 N/A

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

      16. lower-/.f64 99.8

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

      17. lift-+.f64 N/A

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

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

   6. Applied rewrites 99.8%

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

   7. Taylor expanded in x around 0

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

   9. Applied rewrites 59.6%

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

       1. lift-*.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. lift-/.f64 N/A

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

       4. associate-/l/ N/A

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

       5. lift-+.f64 N/A

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

       6. lift-+.f64 N/A

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

       7. associate-+l+ N/A

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

       8. +-commutative N/A

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

       9. lift-+.f64 N/A

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

       10. lift-+.f64 N/A

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

       11. *-commutative N/A

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

       12. lift-*.f64 N/A

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

       13. associate-*r/ N/A

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

       14. lower-/.f64 N/A

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

       15. lower-*.f64 77.8

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

   11. Applied rewrites 77.8%

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

   if 1.50000000000000001e146 < y

   1. Initial program 63.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. 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

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

      4. +-commutative N/A

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

      5. associate-*r/ N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*r/ N/A

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

      8. *-commutative N/A

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

      9. associate-/r* N/A

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

      10. *-commutative N/A

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

      11. times-frac N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. +-commutative N/A

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

      15. lift-+.f64 N/A

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

      16. lower-/.f64 99.9

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

      17. lift-+.f64 N/A

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

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

   6. Applied rewrites 99.9%

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

   7. Taylor expanded in x around 0

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

   9. Applied rewrites 99.9%

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

4. Final simplification 71.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.15 \cdot 10^{-139}:\\ \;\;\;\;1 \cdot \frac{\frac{y}{1 + \left(x + y\right)}}{x + y}\\ \mathbf{elif}\;y \leq 1.5 \cdot 10^{+146}:\\ \;\;\;\;\frac{1 \cdot x}{\left(\left(1 + x\right) + y\right) \cdot \left(x + y\right)}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \frac{\frac{x}{x + y}}{\left(1 + x\right) + y}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 87.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 1.15 \cdot 10^{-139}:\\ \;\;\;\;1 \cdot \frac{\frac{y}{1 + \left(x + y\right)}}{x + y}\\ \mathbf{elif}\;y \leq 1.5 \cdot 10^{+146}:\\ \;\;\;\;\frac{1 \cdot x}{\left(\left(1 + x\right) + y\right) \cdot \left(x + y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 1.15e-139)
   (* 1.0 (/ (/ y (+ 1.0 (+ x y))) (+ x y)))
   (if (<= y 1.5e+146)
     (/ (* 1.0 x) (* (+ (+ 1.0 x) y) (+ x y)))
     (/ (/ x y) y))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 1.15e-139) {
		tmp = 1.0 * ((y / (1.0 + (x + y))) / (x + y));
	} else if (y <= 1.5e+146) {
		tmp = (1.0 * x) / (((1.0 + x) + y) * (x + y));
	} 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 <= 1.15d-139) then
        tmp = 1.0d0 * ((y / (1.0d0 + (x + y))) / (x + y))
    else if (y <= 1.5d+146) then
        tmp = (1.0d0 * x) / (((1.0d0 + x) + y) * (x + y))
    else
        tmp = (x / y) / y
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (y <= 1.15e-139) {
		tmp = 1.0 * ((y / (1.0 + (x + y))) / (x + y));
	} else if (y <= 1.5e+146) {
		tmp = (1.0 * x) / (((1.0 + x) + y) * (x + y));
	} else {
		tmp = (x / y) / y;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if y <= 1.15e-139:
		tmp = 1.0 * ((y / (1.0 + (x + y))) / (x + y))
	elif y <= 1.5e+146:
		tmp = (1.0 * x) / (((1.0 + x) + y) * (x + y))
	else:
		tmp = (x / y) / y
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= 1.15e-139)
		tmp = Float64(1.0 * Float64(Float64(y / Float64(1.0 + Float64(x + y))) / Float64(x + y)));
	elseif (y <= 1.5e+146)
		tmp = Float64(Float64(1.0 * x) / Float64(Float64(Float64(1.0 + x) + y) * Float64(x + y)));
	else
		tmp = Float64(Float64(x / y) / y);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 1.15e-139)
		tmp = 1.0 * ((y / (1.0 + (x + y))) / (x + y));
	elseif (y <= 1.5e+146)
		tmp = (1.0 * x) / (((1.0 + x) + y) * (x + y));
	else
		tmp = (x / y) / y;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[y, 1.15e-139], N[(1.0 * N[(N[(y / N[(1.0 + N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.5e+146], N[(N[(1.0 * x), $MachinePrecision] / N[(N[(N[(1.0 + x), $MachinePrecision] + y), $MachinePrecision] * N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 1.15000000000000006e-139

   1. Initial program 67.3%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   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 99.9

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

      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 99.9

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

      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 99.9

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

      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 99.9

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

   4. Applied rewrites 99.9%

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

   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 65.0%

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

   if 1.15000000000000006e-139 < y < 1.50000000000000001e146

   1. Initial program 68.1%

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

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

      4. +-commutative N/A

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

      5. associate-*r/ N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*r/ N/A

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

      8. *-commutative N/A

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

      9. associate-/r* N/A

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

      10. *-commutative N/A

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

      11. times-frac N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. +-commutative N/A

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

      15. lift-+.f64 N/A

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

      16. lower-/.f64 99.8

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

      17. lift-+.f64 N/A

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

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

   6. Applied rewrites 99.8%

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

   7. Taylor expanded in x around 0

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

   9. Applied rewrites 59.6%

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

       1. lift-*.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. lift-/.f64 N/A

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

       4. associate-/l/ N/A

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

       5. lift-+.f64 N/A

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

       6. lift-+.f64 N/A

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

       7. associate-+l+ N/A

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

       8. +-commutative N/A

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

       9. lift-+.f64 N/A

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

       10. lift-+.f64 N/A

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

       11. *-commutative N/A

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

       12. lift-*.f64 N/A

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

       13. associate-*r/ N/A

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

       14. lower-/.f64 N/A

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

       15. lower-*.f64 77.8

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

   11. Applied rewrites 77.8%

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

   if 1.50000000000000001e146 < y

   1. Initial program 63.8%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 84.6

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

   5. Applied rewrites 84.6%

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

   7. Applied rewrites 99.9%

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

4. Final simplification 71.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.15 \cdot 10^{-139}:\\ \;\;\;\;1 \cdot \frac{\frac{y}{1 + \left(x + y\right)}}{x + y}\\ \mathbf{elif}\;y \leq 1.5 \cdot 10^{+146}:\\ \;\;\;\;\frac{1 \cdot x}{\left(\left(1 + x\right) + y\right) \cdot \left(x + y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 87.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 1.15 \cdot 10^{-139}:\\ \;\;\;\;\frac{1}{1 + x} \cdot \frac{y}{x + y}\\ \mathbf{elif}\;y \leq 1.5 \cdot 10^{+146}:\\ \;\;\;\;\frac{1 \cdot x}{\left(\left(1 + x\right) + y\right) \cdot \left(x + y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 1.15e-139)
   (* (/ 1.0 (+ 1.0 x)) (/ y (+ x y)))
   (if (<= y 1.5e+146)
     (/ (* 1.0 x) (* (+ (+ 1.0 x) y) (+ x y)))
     (/ (/ x y) y))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 1.15e-139) {
		tmp = (1.0 / (1.0 + x)) * (y / (x + y));
	} else if (y <= 1.5e+146) {
		tmp = (1.0 * x) / (((1.0 + x) + y) * (x + y));
	} 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 <= 1.15d-139) then
        tmp = (1.0d0 / (1.0d0 + x)) * (y / (x + y))
    else if (y <= 1.5d+146) then
        tmp = (1.0d0 * x) / (((1.0d0 + x) + y) * (x + y))
    else
        tmp = (x / y) / y
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (y <= 1.15e-139) {
		tmp = (1.0 / (1.0 + x)) * (y / (x + y));
	} else if (y <= 1.5e+146) {
		tmp = (1.0 * x) / (((1.0 + x) + y) * (x + y));
	} else {
		tmp = (x / y) / y;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if y <= 1.15e-139:
		tmp = (1.0 / (1.0 + x)) * (y / (x + y))
	elif y <= 1.5e+146:
		tmp = (1.0 * x) / (((1.0 + x) + y) * (x + y))
	else:
		tmp = (x / y) / y
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= 1.15e-139)
		tmp = Float64(Float64(1.0 / Float64(1.0 + x)) * Float64(y / Float64(x + y)));
	elseif (y <= 1.5e+146)
		tmp = Float64(Float64(1.0 * x) / Float64(Float64(Float64(1.0 + x) + y) * Float64(x + y)));
	else
		tmp = Float64(Float64(x / y) / y);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 1.15e-139)
		tmp = (1.0 / (1.0 + x)) * (y / (x + y));
	elseif (y <= 1.5e+146)
		tmp = (1.0 * x) / (((1.0 + x) + y) * (x + y));
	else
		tmp = (x / y) / y;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[y, 1.15e-139], N[(N[(1.0 / N[(1.0 + x), $MachinePrecision]), $MachinePrecision] * N[(y / N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.5e+146], N[(N[(1.0 * x), $MachinePrecision] / N[(N[(N[(1.0 + x), $MachinePrecision] + y), $MachinePrecision] * N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 1.15000000000000006e-139

   1. Initial program 67.3%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   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 99.9

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

      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 99.9

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

      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 99.9

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

      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 99.9

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

   4. Applied rewrites 99.9%

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

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

      4. +-commutative N/A

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

      5. associate-*r/ N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*r/ N/A

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

      8. *-commutative N/A

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

      9. associate-/r* N/A

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

      10. *-commutative N/A

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

      11. times-frac N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. +-commutative N/A

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

      15. lift-+.f64 N/A

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

      16. lower-/.f64 99.8

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

      17. lift-+.f64 N/A

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

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

   6. Applied rewrites 99.8%

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

   7. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 64.6

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

   9. Applied rewrites 64.6%

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

   if 1.15000000000000006e-139 < y < 1.50000000000000001e146

   1. Initial program 68.1%

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

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

      4. +-commutative N/A

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

      5. associate-*r/ N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*r/ N/A

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

      8. *-commutative N/A

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

      9. associate-/r* N/A

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

      10. *-commutative N/A

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

      11. times-frac N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. +-commutative N/A

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

      15. lift-+.f64 N/A

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

      16. lower-/.f64 99.8

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

      17. lift-+.f64 N/A

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

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

   6. Applied rewrites 99.8%

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

   7. Taylor expanded in x around 0

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

   9. Applied rewrites 59.6%

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

       1. lift-*.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. lift-/.f64 N/A

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

       4. associate-/l/ N/A

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

       5. lift-+.f64 N/A

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

       6. lift-+.f64 N/A

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

       7. associate-+l+ N/A

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

       8. +-commutative N/A

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

       9. lift-+.f64 N/A

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

       10. lift-+.f64 N/A

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

       11. *-commutative N/A

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

       12. lift-*.f64 N/A

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

       13. associate-*r/ N/A

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

       14. lower-/.f64 N/A

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

       15. lower-*.f64 77.8

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

   11. Applied rewrites 77.8%

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

   if 1.50000000000000001e146 < y

   1. Initial program 63.8%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 84.6

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

   5. Applied rewrites 84.6%

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

   7. Applied rewrites 99.9%

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

4. Final simplification 71.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.15 \cdot 10^{-139}:\\ \;\;\;\;\frac{1}{1 + x} \cdot \frac{y}{x + y}\\ \mathbf{elif}\;y \leq 1.5 \cdot 10^{+146}:\\ \;\;\;\;\frac{1 \cdot x}{\left(\left(1 + x\right) + y\right) \cdot \left(x + y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 86.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 1.15 \cdot 10^{-139}:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(x, x, x\right)}\\ \mathbf{elif}\;y \leq 1.5 \cdot 10^{+146}:\\ \;\;\;\;\frac{1 \cdot x}{\left(\left(1 + x\right) + y\right) \cdot \left(x + y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 1.15e-139)
   (/ y (fma x x x))
   (if (<= y 1.5e+146)
     (/ (* 1.0 x) (* (+ (+ 1.0 x) y) (+ x y)))
     (/ (/ x y) y))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 1.15e-139) {
		tmp = y / fma(x, x, x);
	} else if (y <= 1.5e+146) {
		tmp = (1.0 * x) / (((1.0 + x) + y) * (x + y));
	} else {
		tmp = (x / y) / y;
	}
	return tmp;
}

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= 1.15e-139)
		tmp = Float64(y / fma(x, x, x));
	elseif (y <= 1.5e+146)
		tmp = Float64(Float64(1.0 * x) / Float64(Float64(Float64(1.0 + x) + y) * Float64(x + y)));
	else
		tmp = Float64(Float64(x / y) / y);
	end
	return tmp
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[y, 1.15e-139], N[(y / N[(x * x + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.5e+146], N[(N[(1.0 * x), $MachinePrecision] / N[(N[(N[(1.0 + x), $MachinePrecision] + y), $MachinePrecision] * N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 1.15000000000000006e-139

   1. Initial program 67.3%

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

   3. Taylor expanded in y around 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 62.8

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

   5. Applied rewrites 62.8%

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

   if 1.15000000000000006e-139 < y < 1.50000000000000001e146

   1. Initial program 68.1%

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

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

      4. +-commutative N/A

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

      5. associate-*r/ N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*r/ N/A

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

      8. *-commutative N/A

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

      9. associate-/r* N/A

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

      10. *-commutative N/A

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

      11. times-frac N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. +-commutative N/A

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

      15. lift-+.f64 N/A

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

      16. lower-/.f64 99.8

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

      17. lift-+.f64 N/A

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

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

   6. Applied rewrites 99.8%

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

   7. Taylor expanded in x around 0

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

   9. Applied rewrites 59.6%

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

       1. lift-*.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. lift-/.f64 N/A

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

       4. associate-/l/ N/A

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

       5. lift-+.f64 N/A

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

       6. lift-+.f64 N/A

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

       7. associate-+l+ N/A

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

       8. +-commutative N/A

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

       9. lift-+.f64 N/A

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

       10. lift-+.f64 N/A

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

       11. *-commutative N/A

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

       12. lift-*.f64 N/A

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

       13. associate-*r/ N/A

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

       14. lower-/.f64 N/A

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

       15. lower-*.f64 77.8

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

   11. Applied rewrites 77.8%

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

   if 1.50000000000000001e146 < y

   1. Initial program 63.8%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 84.6

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

   5. Applied rewrites 84.6%

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

   7. Applied rewrites 99.9%

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

Alternative 12: 85.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -1.35 \cdot 10^{+170}:\\ \;\;\;\;\frac{1}{x} \cdot \frac{y}{x + y}\\ \mathbf{elif}\;x \leq -2.05 \cdot 10^{-170}:\\ \;\;\;\;1 \cdot \frac{y}{\left(1 + \left(x + y\right)\right) \cdot \left(x + y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(y, y, y\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -1.35e+170)
   (* (/ 1.0 x) (/ y (+ x y)))
   (if (<= x -2.05e-170)
     (* 1.0 (/ y (* (+ 1.0 (+ x y)) (+ x y))))
     (/ x (fma y y y)))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -1.35e+170) {
		tmp = (1.0 / x) * (y / (x + y));
	} else if (x <= -2.05e-170) {
		tmp = 1.0 * (y / ((1.0 + (x + y)) * (x + y)));
	} else {
		tmp = x / fma(y, y, y);
	}
	return tmp;
}

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -1.35e+170)
		tmp = Float64(Float64(1.0 / x) * Float64(y / Float64(x + y)));
	elseif (x <= -2.05e-170)
		tmp = Float64(1.0 * Float64(y / Float64(Float64(1.0 + Float64(x + y)) * Float64(x + y))));
	else
		tmp = Float64(x / fma(y, y, y));
	end
	return tmp
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[x, -1.35e+170], N[(N[(1.0 / x), $MachinePrecision] * N[(y / N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -2.05e-170], N[(1.0 * N[(y / N[(N[(1.0 + N[(x + y), $MachinePrecision]), $MachinePrecision] * N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x / N[(y * y + y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.3500000000000001e170

   1. Initial program 68.8%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   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 99.9

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

      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 99.9

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

      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 99.9

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

      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 99.9

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

   4. Applied rewrites 99.9%

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

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

      4. +-commutative N/A

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

      5. associate-*r/ N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*r/ N/A

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

      8. *-commutative N/A

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

      9. associate-/r* N/A

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

      10. *-commutative N/A

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

      11. times-frac N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. +-commutative N/A

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

      15. lift-+.f64 N/A

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

      16. lower-/.f64 99.9

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

      17. lift-+.f64 N/A

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

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

   6. Applied rewrites 99.9%

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

   7. Taylor expanded in x around inf

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

      1. lower-/.f64 85.5

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

   9. Applied rewrites 85.5%

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

   if -1.3500000000000001e170 < x < -2.04999999999999983e-170

   1. Initial program 72.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

      8. times-frac N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 N/A

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

      16. lift-+.f64 N/A

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

      17. +-commutative N/A

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

      18. lower-+.f64 N/A

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

      19. lift-+.f64 N/A

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

      20. +-commutative N/A

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

      21. lower-+.f64 N/A

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

      22. lower-/.f64 98.6

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

      23. lift-+.f64 N/A

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

      24. +-commutative N/A

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

   4. Applied rewrites 98.6%

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

   5. Taylor expanded in x around inf

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

   7. Applied rewrites 76.6%

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

   if -2.04999999999999983e-170 < x

   1. Initial program 64.5%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   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 55.6

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

   5. Applied rewrites 55.6%

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

4. Final simplification 65.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.35 \cdot 10^{+170}:\\ \;\;\;\;\frac{1}{x} \cdot \frac{y}{x + y}\\ \mathbf{elif}\;x \leq -2.05 \cdot 10^{-170}:\\ \;\;\;\;1 \cdot \frac{y}{\left(1 + \left(x + y\right)\right) \cdot \left(x + y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(y, y, y\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 85.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -1.35 \cdot 10^{+170}:\\ \;\;\;\;\frac{\frac{y}{x}}{x}\\ \mathbf{elif}\;x \leq -2.05 \cdot 10^{-170}:\\ \;\;\;\;1 \cdot \frac{y}{\left(1 + \left(x + y\right)\right) \cdot \left(x + y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(y, y, y\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -1.35e+170)
   (/ (/ y x) x)
   (if (<= x -2.05e-170)
     (* 1.0 (/ y (* (+ 1.0 (+ x y)) (+ x y))))
     (/ x (fma y y y)))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -1.35e+170) {
		tmp = (y / x) / x;
	} else if (x <= -2.05e-170) {
		tmp = 1.0 * (y / ((1.0 + (x + y)) * (x + y)));
	} else {
		tmp = x / fma(y, y, y);
	}
	return tmp;
}

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -1.35e+170)
		tmp = Float64(Float64(y / x) / x);
	elseif (x <= -2.05e-170)
		tmp = Float64(1.0 * Float64(y / Float64(Float64(1.0 + Float64(x + y)) * Float64(x + y))));
	else
		tmp = Float64(x / fma(y, y, y));
	end
	return tmp
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[x, -1.35e+170], N[(N[(y / x), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[x, -2.05e-170], N[(1.0 * N[(y / N[(N[(1.0 + N[(x + y), $MachinePrecision]), $MachinePrecision] * N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x / N[(y * y + y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.3500000000000001e170

   1. Initial program 68.8%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 85.5

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

   5. Applied rewrites 85.5%

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

   7. Applied rewrites 85.4%

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

   if -1.3500000000000001e170 < x < -2.04999999999999983e-170

   1. Initial program 72.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

      8. times-frac N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 N/A

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

      16. lift-+.f64 N/A

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

      17. +-commutative N/A

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

      18. lower-+.f64 N/A

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

      19. lift-+.f64 N/A

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

      20. +-commutative N/A

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

      21. lower-+.f64 N/A

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

      22. lower-/.f64 98.6

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

      23. lift-+.f64 N/A

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

      24. +-commutative N/A

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

   4. Applied rewrites 98.6%

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

   5. Taylor expanded in x around inf

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

   7. Applied rewrites 76.6%

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

   if -2.04999999999999983e-170 < x

   1. Initial program 64.5%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   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 55.6

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

   5. Applied rewrites 55.6%

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

4. Final simplification 65.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.35 \cdot 10^{+170}:\\ \;\;\;\;\frac{\frac{y}{x}}{x}\\ \mathbf{elif}\;x \leq -2.05 \cdot 10^{-170}:\\ \;\;\;\;1 \cdot \frac{y}{\left(1 + \left(x + y\right)\right) \cdot \left(x + y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(y, y, y\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 85.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -1.35 \cdot 10^{+170}:\\ \;\;\;\;\frac{\frac{y}{x}}{x}\\ \mathbf{elif}\;x \leq -2.05 \cdot 10^{-170}:\\ \;\;\;\;\frac{1}{\left(\left(1 + x\right) + y\right) \cdot \left(x + y\right)} \cdot y\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(y, y, y\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -1.35e+170)
   (/ (/ y x) x)
   (if (<= x -2.05e-170)
     (* (/ 1.0 (* (+ (+ 1.0 x) y) (+ x y))) y)
     (/ x (fma y y y)))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -1.35e+170) {
		tmp = (y / x) / x;
	} else if (x <= -2.05e-170) {
		tmp = (1.0 / (((1.0 + x) + y) * (x + y))) * y;
	} else {
		tmp = x / fma(y, y, y);
	}
	return tmp;
}

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -1.35e+170)
		tmp = Float64(Float64(y / x) / x);
	elseif (x <= -2.05e-170)
		tmp = Float64(Float64(1.0 / Float64(Float64(Float64(1.0 + x) + y) * Float64(x + y))) * y);
	else
		tmp = Float64(x / fma(y, y, y));
	end
	return tmp
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[x, -1.35e+170], N[(N[(y / x), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[x, -2.05e-170], N[(N[(1.0 / N[(N[(N[(1.0 + x), $MachinePrecision] + y), $MachinePrecision] * N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision], N[(x / N[(y * y + y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.3500000000000001e170

   1. Initial program 68.8%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 85.5

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

   5. Applied rewrites 85.5%

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

   7. Applied rewrites 85.4%

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

   if -1.3500000000000001e170 < x < -2.04999999999999983e-170

   1. Initial program 72.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

      8. times-frac N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 N/A

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

      16. lift-+.f64 N/A

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

      17. +-commutative N/A

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

      18. lower-+.f64 N/A

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

      19. lift-+.f64 N/A

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

      20. +-commutative N/A

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

      21. lower-+.f64 N/A

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

      22. lower-/.f64 98.6

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

      23. lift-+.f64 N/A

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

      24. +-commutative N/A

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

   4. Applied rewrites 98.6%

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

   5. Taylor expanded in x around inf

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

   7. Applied rewrites 76.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 76.5

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

      7. lift-+.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. associate-+l+ N/A

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

      11. lift-+.f64 N/A

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

      12. lift-+.f64 76.5

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

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lift-+.f64 76.5

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

   9. Applied rewrites 76.5%

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

   if -2.04999999999999983e-170 < x

   1. Initial program 64.5%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   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 55.6

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

   5. Applied rewrites 55.6%

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

4. Final simplification 65.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.35 \cdot 10^{+170}:\\ \;\;\;\;\frac{\frac{y}{x}}{x}\\ \mathbf{elif}\;x \leq -2.05 \cdot 10^{-170}:\\ \;\;\;\;\frac{1}{\left(\left(1 + x\right) + y\right) \cdot \left(x + y\right)} \cdot y\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(y, y, y\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 81.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 1.15 \cdot 10^{-139}:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(x, x, x\right)}\\ \mathbf{elif}\;y \leq 2 \cdot 10^{+42}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(y, y, y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 1.15e-139)
   (/ y (fma x x x))
   (if (<= y 2e+42) (/ x (fma y y y)) (/ (/ x y) y))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 1.15e-139) {
		tmp = y / fma(x, x, x);
	} else if (y <= 2e+42) {
		tmp = x / fma(y, y, y);
	} else {
		tmp = (x / y) / y;
	}
	return tmp;
}

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= 1.15e-139)
		tmp = Float64(y / fma(x, x, x));
	elseif (y <= 2e+42)
		tmp = Float64(x / fma(y, y, y));
	else
		tmp = Float64(Float64(x / y) / y);
	end
	return tmp
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[y, 1.15e-139], N[(y / N[(x * x + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2e+42], N[(x / N[(y * y + y), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 1.15000000000000006e-139

   1. Initial program 67.3%

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

   3. Taylor expanded in y around 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 62.8

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

   5. Applied rewrites 62.8%

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

   if 1.15000000000000006e-139 < y < 2.00000000000000009e42

   1. Initial program 77.4%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      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 53.4

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

   5. Applied rewrites 53.4%

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

   if 2.00000000000000009e42 < y

   1. Initial program 59.7%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 76.3

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

   5. Applied rewrites 76.3%

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

   7. Applied rewrites 85.0%

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

Alternative 16: 78.9% accurate, 1.6× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 1.15 \cdot 10^{-139}:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(x, x, x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(y, y, y\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 1.15e-139) (/ y (fma x x x)) (/ x (fma y y y))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 1.15e-139) {
		tmp = y / fma(x, x, x);
	} else {
		tmp = x / fma(y, y, y);
	}
	return tmp;
}

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= 1.15e-139)
		tmp = Float64(y / fma(x, x, x));
	else
		tmp = Float64(x / fma(y, y, y));
	end
	return tmp
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[y, 1.15e-139], N[(y / N[(x * x + x), $MachinePrecision]), $MachinePrecision], N[(x / N[(y * y + y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.15000000000000006e-139

   1. Initial program 67.3%

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

   3. Taylor expanded in y around 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 62.8

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

   5. Applied rewrites 62.8%

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

   if 1.15000000000000006e-139 < y

   1. Initial program 66.6%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      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 67.3

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

   5. Applied rewrites 67.3%

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

Alternative 17: 76.9% accurate, 1.6× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -1.1 \cdot 10^{+14}:\\ \;\;\;\;\frac{y}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(y, y, y\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -1.1e+14) (/ y (* x x)) (/ x (fma y y y))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -1.1e+14) {
		tmp = y / (x * x);
	} else {
		tmp = x / fma(y, y, y);
	}
	return tmp;
}

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -1.1e+14)
		tmp = Float64(y / Float64(x * x));
	else
		tmp = Float64(x / fma(y, y, y));
	end
	return tmp
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[x, -1.1e+14], N[(y / N[(x * x), $MachinePrecision]), $MachinePrecision], N[(x / N[(y * y + y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.1e14

   1. Initial program 64.5%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 78.4

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

   5. Applied rewrites 78.4%

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

   if -1.1e14 < x

   1. Initial program 68.0%

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

   3. Taylor expanded in x around 0

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

      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 55.2

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

   5. Applied rewrites 55.2%

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

Alternative 18: 64.9% accurate, 1.7× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 3 \cdot 10^{+20}:\\ \;\;\;\;\frac{y}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 3e+20) (/ y (* x x)) (/ x (* y y))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 3e+20) {
		tmp = y / (x * x);
	} else {
		tmp = x / (y * y);
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 3d+20) then
        tmp = y / (x * x)
    else
        tmp = x / (y * y)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (y <= 3e+20) {
		tmp = y / (x * x);
	} else {
		tmp = x / (y * y);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if y <= 3e+20:
		tmp = y / (x * x)
	else:
		tmp = x / (y * y)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= 3e+20)
		tmp = Float64(y / Float64(x * x));
	else
		tmp = Float64(x / Float64(y * y));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 3e+20)
		tmp = y / (x * x);
	else
		tmp = x / (y * y);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[y, 3e+20], N[(y / N[(x * x), $MachinePrecision]), $MachinePrecision], N[(x / N[(y * y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 3e20

   1. Initial program 68.3%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 41.2

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

   5. Applied rewrites 41.2%

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

   if 3e20 < y

   1. Initial program 63.1%

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

   3. Taylor expanded in y around inf

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

      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 77.8

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

   5. Applied rewrites 77.8%

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

Alternative 19: 37.7% 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 67.1%

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

3. Taylor expanded in y around inf

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

   1. lower-/.f64 N/A

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

   2. unpow2 N/A

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

   3. lower-*.f64 34.8

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

5. Applied rewrites 34.8%

   \[\leadsto \color{blue}{\frac{x}{y \cdot y}} \]
6. Add Preprocessing

Developer Target 1: 99.8% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024298 
(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))))