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

Percentage Accurate: 69.2% → 99.8%

Time: 9.3s

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: 69.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 70.4%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. times-frac N/A

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

   5. associate-*l/ N/A

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

   6. lift-*.f64 N/A

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

   7. times-frac N/A

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

   8. lower-*.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. lift-+.f64 N/A

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

   11. +-commutative N/A

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

   12. lower-+.f64 N/A

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

   13. lower-/.f64 N/A

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

   14. lower-/.f64 99.8

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

   15. lift-+.f64 N/A

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

   16. +-commutative N/A

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

   17. lower-+.f64 99.8

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

   18. lift-+.f64 N/A

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

   19. +-commutative N/A

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

   20. lower-+.f64 99.8

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

   21. lift-+.f64 N/A

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

   22. +-commutative N/A

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

   23. lower-+.f64 99.8

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

4. Applied rewrites 99.8%

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

Alternative 2: 95.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_0 := 1 + \left(y + x\right)\\ \mathbf{if}\;y \leq 5.5 \cdot 10^{+99}:\\ \;\;\;\;\frac{\frac{y}{y + x} \cdot x}{t\_0 \cdot \left(y + x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} \cdot \frac{\frac{y}{t\_0}}{y + x}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ 1.0 (+ y x))))
   (if (<= y 5.5e+99)
     (/ (* (/ y (+ y x)) x) (* t_0 (+ y x)))
     (* (/ x y) (/ (/ y t_0) (+ y x))))))

C

assert(x < y);
double code(double x, double y) {
	double t_0 = 1.0 + (y + x);
	double tmp;
	if (y <= 5.5e+99) {
		tmp = ((y / (y + x)) * x) / (t_0 * (y + x));
	} else {
		tmp = (x / y) * ((y / t_0) / (y + x));
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 1.0d0 + (y + x)
    if (y <= 5.5d+99) then
        tmp = ((y / (y + x)) * x) / (t_0 * (y + x))
    else
        tmp = (x / y) * ((y / t_0) / (y + x))
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double t_0 = 1.0 + (y + x);
	double tmp;
	if (y <= 5.5e+99) {
		tmp = ((y / (y + x)) * x) / (t_0 * (y + x));
	} else {
		tmp = (x / y) * ((y / t_0) / (y + x));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	t_0 = 1.0 + (y + x)
	tmp = 0
	if y <= 5.5e+99:
		tmp = ((y / (y + x)) * x) / (t_0 * (y + x))
	else:
		tmp = (x / y) * ((y / t_0) / (y + x))
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	t_0 = Float64(1.0 + Float64(y + x))
	tmp = 0.0
	if (y <= 5.5e+99)
		tmp = Float64(Float64(Float64(y / Float64(y + x)) * x) / Float64(t_0 * Float64(y + x)));
	else
		tmp = Float64(Float64(x / y) * Float64(Float64(y / t_0) / Float64(y + x)));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	t_0 = 1.0 + (y + x);
	tmp = 0.0;
	if (y <= 5.5e+99)
		tmp = ((y / (y + x)) * x) / (t_0 * (y + x));
	else
		tmp = (x / y) * ((y / t_0) / (y + x));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := Block[{t$95$0 = N[(1.0 + N[(y + x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, 5.5e+99], N[(N[(N[(y / N[(y + x), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision] / N[(t$95$0 * N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] * N[(N[(y / t$95$0), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < 5.5000000000000002e99

   1. Initial program 73.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-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. associate-*r/ N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-+.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 96.3

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

      17. lift-+.f64 N/A

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

      18. +-commutative N/A

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

      19. lower-+.f64 96.3

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

      20. lift-+.f64 N/A

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

      21. +-commutative N/A

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

      22. lower-+.f64 96.3

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

      23. lift-+.f64 N/A

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

      24. +-commutative N/A

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

      25. lower-+.f64 96.3

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

   4. Applied rewrites 96.3%

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

   if 5.5000000000000002e99 < y

   1. Initial program 57.2%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   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 0

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

      1. lower-/.f64 93.9

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

   7. Applied rewrites 93.9%

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

Alternative 3: 88.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_0 := 1 + \left(y + x\right)\\ \mathbf{if}\;y \leq 1.4 \cdot 10^{-143}:\\ \;\;\;\;\frac{\frac{y}{t\_0}}{\mathsf{fma}\left(2, y, x\right)}\\ \mathbf{elif}\;y \leq 8 \cdot 10^{+94}:\\ \;\;\;\;\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{t\_0} \cdot x}{y + x}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ 1.0 (+ y x))))
   (if (<= y 1.4e-143)
     (/ (/ y t_0) (fma 2.0 y x))
     (if (<= y 8e+94)
       (/ (* x y) (* (* (+ x y) (+ x y)) (+ (+ x y) 1.0)))
       (/ (* (/ 1.0 t_0) x) (+ y x))))))

C

assert(x < y);
double code(double x, double y) {
	double t_0 = 1.0 + (y + x);
	double tmp;
	if (y <= 1.4e-143) {
		tmp = (y / t_0) / fma(2.0, y, x);
	} else if (y <= 8e+94) {
		tmp = (x * y) / (((x + y) * (x + y)) * ((x + y) + 1.0));
	} else {
		tmp = ((1.0 / t_0) * x) / (y + x);
	}
	return tmp;
}

Julia

x, y = sort([x, y])
function code(x, y)
	t_0 = Float64(1.0 + Float64(y + x))
	tmp = 0.0
	if (y <= 1.4e-143)
		tmp = Float64(Float64(y / t_0) / fma(2.0, y, x));
	elseif (y <= 8e+94)
		tmp = Float64(Float64(x * y) / Float64(Float64(Float64(x + y) * Float64(x + y)) * Float64(Float64(x + y) + 1.0)));
	else
		tmp = Float64(Float64(Float64(1.0 / t_0) * x) / Float64(y + x));
	end
	return tmp
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := Block[{t$95$0 = N[(1.0 + N[(y + x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, 1.4e-143], N[(N[(y / t$95$0), $MachinePrecision] / N[(2.0 * y + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 8e+94], N[(N[(x * y), $MachinePrecision] / N[(N[(N[(x + y), $MachinePrecision] * N[(x + y), $MachinePrecision]), $MachinePrecision] * N[(N[(x + y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 / t$95$0), $MachinePrecision] * x), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < 1.3999999999999999e-143

   1. Initial program 67.9%

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

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

      4. lift-/.f64 N/A

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

      5. frac-times N/A

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

      6. lift-/.f64 N/A

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

      7. clear-num N/A

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

      8. div-inv N/A

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

      9. clear-num N/A

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

      10. lift-/.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 98.7

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

   6. Applied rewrites 98.7%

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

   7. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 58.2

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

   9. Applied rewrites 58.2%

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

   if 1.3999999999999999e-143 < y < 8.0000000000000002e94

   1. Initial program 89.1%

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

   if 8.0000000000000002e94 < y

   1. Initial program 59.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. associate-*l/ N/A

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

      6. lift-*.f64 N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-/.f64 99.8

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

      15. lift-+.f64 N/A

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

      16. +-commutative N/A

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

      17. lower-+.f64 99.8

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

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

      20. lower-+.f64 99.8

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

      21. lift-+.f64 N/A

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

      22. +-commutative N/A

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

      23. lower-+.f64 99.8

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

   4. Applied rewrites 99.8%

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

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

      3. lift-/.f64 N/A

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

      4. frac-2neg N/A

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

      5. associate-*r/ N/A

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

      6. lower-/.f64 N/A

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

   6. Applied rewrites 100.0%

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

   7. Taylor expanded in x around 0

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

   9. Applied rewrites 88.7%

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

4. Final simplification 69.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.4 \cdot 10^{-143}:\\ \;\;\;\;\frac{\frac{y}{1 + \left(y + x\right)}}{\mathsf{fma}\left(2, y, x\right)}\\ \mathbf{elif}\;y \leq 8 \cdot 10^{+94}:\\ \;\;\;\;\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{1 + \left(y + x\right)} \cdot x}{y + x}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 88.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 1.4 \cdot 10^{-143}:\\ \;\;\;\;\frac{\frac{y}{1 + \left(y + x\right)}}{\mathsf{fma}\left(2, y, x\right)}\\ \mathbf{elif}\;y \leq 1.6 \cdot 10^{+95}:\\ \;\;\;\;\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-x}{y}}{-\left(y + x\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 1.4e-143)
   (/ (/ y (+ 1.0 (+ y x))) (fma 2.0 y x))
   (if (<= y 1.6e+95)
     (/ (* x y) (* (* (+ x y) (+ x y)) (+ (+ x y) 1.0)))
     (/ (/ (- x) y) (- (+ y x))))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 1.4e-143) {
		tmp = (y / (1.0 + (y + x))) / fma(2.0, y, x);
	} else if (y <= 1.6e+95) {
		tmp = (x * y) / (((x + y) * (x + y)) * ((x + y) + 1.0));
	} else {
		tmp = (-x / y) / -(y + x);
	}
	return tmp;
}

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= 1.4e-143)
		tmp = Float64(Float64(y / Float64(1.0 + Float64(y + x))) / fma(2.0, y, x));
	elseif (y <= 1.6e+95)
		tmp = Float64(Float64(x * y) / Float64(Float64(Float64(x + y) * Float64(x + y)) * Float64(Float64(x + y) + 1.0)));
	else
		tmp = Float64(Float64(Float64(-x) / y) / Float64(-Float64(y + x)));
	end
	return tmp
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[y, 1.4e-143], N[(N[(y / N[(1.0 + N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 * y + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.6e+95], N[(N[(x * y), $MachinePrecision] / N[(N[(N[(x + y), $MachinePrecision] * N[(x + y), $MachinePrecision]), $MachinePrecision] * N[(N[(x + y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[((-x) / y), $MachinePrecision] / (-N[(y + x), $MachinePrecision])), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 1.3999999999999999e-143

   1. Initial program 67.9%

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

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

      4. lift-/.f64 N/A

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

      5. frac-times N/A

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

      6. lift-/.f64 N/A

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

      7. clear-num N/A

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

      8. div-inv N/A

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

      9. clear-num N/A

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

      10. lift-/.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 98.7

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

   6. Applied rewrites 98.7%

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

   7. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 58.2

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

   9. Applied rewrites 58.2%

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

   if 1.3999999999999999e-143 < y < 1.6e95

   1. Initial program 89.1%

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

   if 1.6e95 < y

   1. Initial program 59.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. associate-*l/ N/A

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

      6. lift-*.f64 N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-/.f64 99.8

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

      15. lift-+.f64 N/A

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

      16. +-commutative N/A

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

      17. lower-+.f64 99.8

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

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

      20. lower-+.f64 99.8

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

      21. lift-+.f64 N/A

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

      22. +-commutative N/A

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

      23. lower-+.f64 99.8

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

   4. Applied rewrites 99.8%

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

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

      3. lift-/.f64 N/A

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

      4. frac-2neg N/A

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

      5. associate-*r/ N/A

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

      6. lower-/.f64 N/A

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

   6. Applied rewrites 100.0%

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

   7. Taylor expanded in y around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 88.4

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

   9. Applied rewrites 88.4%

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

Alternative 5: 84.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_0 := 1 + \left(y + x\right)\\ \mathbf{if}\;x \leq -4.8 \cdot 10^{+129}:\\ \;\;\;\;\frac{\frac{y}{t\_0}}{\mathsf{fma}\left(2, y, x\right)}\\ \mathbf{elif}\;x \leq -1.35 \cdot 10^{-80}:\\ \;\;\;\;y \cdot \frac{x}{t\_0 \cdot \left(\mathsf{fma}\left(2, y, x\right) \cdot x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{1 + y}}{y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ 1.0 (+ y x))))
   (if (<= x -4.8e+129)
     (/ (/ y t_0) (fma 2.0 y x))
     (if (<= x -1.35e-80)
       (* y (/ x (* t_0 (* (fma 2.0 y x) x))))
       (/ (/ x (+ 1.0 y)) y)))))

C

assert(x < y);
double code(double x, double y) {
	double t_0 = 1.0 + (y + x);
	double tmp;
	if (x <= -4.8e+129) {
		tmp = (y / t_0) / fma(2.0, y, x);
	} else if (x <= -1.35e-80) {
		tmp = y * (x / (t_0 * (fma(2.0, y, x) * x)));
	} else {
		tmp = (x / (1.0 + y)) / y;
	}
	return tmp;
}

Julia

x, y = sort([x, y])
function code(x, y)
	t_0 = Float64(1.0 + Float64(y + x))
	tmp = 0.0
	if (x <= -4.8e+129)
		tmp = Float64(Float64(y / t_0) / fma(2.0, y, x));
	elseif (x <= -1.35e-80)
		tmp = Float64(y * Float64(x / Float64(t_0 * Float64(fma(2.0, y, x) * x))));
	else
		tmp = Float64(Float64(x / Float64(1.0 + y)) / y);
	end
	return tmp
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := Block[{t$95$0 = N[(1.0 + N[(y + x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -4.8e+129], N[(N[(y / t$95$0), $MachinePrecision] / N[(2.0 * y + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -1.35e-80], N[(y * N[(x / N[(t$95$0 * N[(N[(2.0 * y + x), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(1.0 + y), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -4.7999999999999997e129

   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 100.0

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

      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 100.0

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

      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 100.0

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

      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 100.0

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

   4. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\frac{x}{y + x} \cdot \frac{\frac{y}{1 + \left(y + x\right)}}{y + x}} \]
   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. clear-num N/A

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

      4. lift-/.f64 N/A

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

      5. frac-times N/A

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

      6. lift-/.f64 N/A

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

      7. clear-num N/A

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

      8. div-inv N/A

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

      9. clear-num N/A

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

      10. lift-/.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 97.6

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

   6. Applied rewrites 97.6%

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

   7. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 91.0

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

   9. Applied rewrites 91.0%

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

   if -4.7999999999999997e129 < x < -1.3500000000000001e-80

   1. Initial program 62.1%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. unpow2 N/A

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

      4. distribute-rgt-in N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-fma.f64 49.4

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

   5. Applied rewrites 49.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 57.1

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 57.1

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

      10. lift-+.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lift-+.f64 57.1

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

   7. Applied rewrites 57.1%

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

   if -1.3500000000000001e-80 < x

   1. Initial program 73.1%

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

   3. 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 56.7

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

   5. Applied rewrites 56.7%

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

   7. Applied rewrites 57.7%

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

4. Final simplification 61.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.8 \cdot 10^{+129}:\\ \;\;\;\;\frac{\frac{y}{1 + \left(y + x\right)}}{\mathsf{fma}\left(2, y, x\right)}\\ \mathbf{elif}\;x \leq -1.35 \cdot 10^{-80}:\\ \;\;\;\;y \cdot \frac{x}{\left(1 + \left(y + x\right)\right) \cdot \left(\mathsf{fma}\left(2, y, x\right) \cdot x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{1 + y}}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 96.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_0 := 1 + \left(y + x\right)\\ \mathbf{if}\;x \leq -5 \cdot 10^{+147}:\\ \;\;\;\;\frac{\frac{y}{t\_0}}{\mathsf{fma}\left(2, y, x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{y + x} \cdot x}{t\_0 \cdot \left(y + x\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ 1.0 (+ y x))))
   (if (<= x -5e+147)
     (/ (/ y t_0) (fma 2.0 y x))
     (/ (* (/ y (+ y x)) x) (* t_0 (+ y x))))))

C

assert(x < y);
double code(double x, double y) {
	double t_0 = 1.0 + (y + x);
	double tmp;
	if (x <= -5e+147) {
		tmp = (y / t_0) / fma(2.0, y, x);
	} else {
		tmp = ((y / (y + x)) * x) / (t_0 * (y + x));
	}
	return tmp;
}

Julia

x, y = sort([x, y])
function code(x, y)
	t_0 = Float64(1.0 + Float64(y + x))
	tmp = 0.0
	if (x <= -5e+147)
		tmp = Float64(Float64(y / t_0) / fma(2.0, y, x));
	else
		tmp = Float64(Float64(Float64(y / Float64(y + x)) * x) / Float64(t_0 * Float64(y + x)));
	end
	return tmp
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := Block[{t$95$0 = N[(1.0 + N[(y + x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -5e+147], N[(N[(y / t$95$0), $MachinePrecision] / N[(2.0 * y + x), $MachinePrecision]), $MachinePrecision], N[(N[(N[(y / N[(y + x), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision] / N[(t$95$0 * N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -5.0000000000000002e147

   1. Initial program 68.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. associate-*l/ N/A

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

      6. lift-*.f64 N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-/.f64 99.9

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

      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. clear-num N/A

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

      4. lift-/.f64 N/A

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

      5. frac-times N/A

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

      6. lift-/.f64 N/A

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

      7. clear-num N/A

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

      8. div-inv N/A

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

      9. clear-num N/A

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

      10. lift-/.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 98.5

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

   6. Applied rewrites 98.5%

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

   7. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 91.9

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

   9. Applied rewrites 91.9%

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

   if -5.0000000000000002e147 < x

   1. Initial program 70.6%

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

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-+.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 94.6

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

      17. lift-+.f64 N/A

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

      18. +-commutative N/A

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

      19. lower-+.f64 94.6

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

      20. lift-+.f64 N/A

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

      21. +-commutative N/A

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

      22. lower-+.f64 94.6

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

      23. lift-+.f64 N/A

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

      24. +-commutative N/A

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

      25. lower-+.f64 94.6

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

   4. Applied rewrites 94.6%

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

Alternative 7: 96.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_0 := 1 + \left(y + x\right)\\ \mathbf{if}\;x \leq -5 \cdot 10^{+157}:\\ \;\;\;\;\frac{\frac{y}{t\_0}}{\mathsf{fma}\left(2, y, x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{t\_0 \cdot \left(y + x\right)} \cdot \frac{x}{y + x}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ 1.0 (+ y x))))
   (if (<= x -5e+157)
     (/ (/ y t_0) (fma 2.0 y x))
     (* (/ y (* t_0 (+ y x))) (/ x (+ y x))))))

C

assert(x < y);
double code(double x, double y) {
	double t_0 = 1.0 + (y + x);
	double tmp;
	if (x <= -5e+157) {
		tmp = (y / t_0) / fma(2.0, y, x);
	} else {
		tmp = (y / (t_0 * (y + x))) * (x / (y + x));
	}
	return tmp;
}

Julia

x, y = sort([x, y])
function code(x, y)
	t_0 = Float64(1.0 + Float64(y + x))
	tmp = 0.0
	if (x <= -5e+157)
		tmp = Float64(Float64(y / t_0) / fma(2.0, y, x));
	else
		tmp = Float64(Float64(y / Float64(t_0 * Float64(y + x))) * Float64(x / Float64(y + x)));
	end
	return tmp
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := Block[{t$95$0 = N[(1.0 + N[(y + x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -5e+157], N[(N[(y / t$95$0), $MachinePrecision] / N[(2.0 * y + x), $MachinePrecision]), $MachinePrecision], N[(N[(y / N[(t$95$0 * N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(x / N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -4.99999999999999976e157

   1. Initial program 68.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. associate-*l/ N/A

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

      6. lift-*.f64 N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-/.f64 99.9

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

      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. clear-num N/A

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

      4. lift-/.f64 N/A

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

      5. frac-times N/A

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

      6. lift-/.f64 N/A

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

      7. clear-num N/A

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

      8. div-inv N/A

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

      9. clear-num N/A

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

      10. lift-/.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 98.5

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

   6. Applied rewrites 98.5%

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

   7. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 91.9

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

   9. Applied rewrites 91.9%

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

   if -4.99999999999999976e157 < x

   1. Initial program 70.6%

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

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

      8. times-frac N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 N/A

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

      16. lift-+.f64 N/A

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

      17. +-commutative N/A

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

      18. lower-+.f64 N/A

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

      19. lift-+.f64 N/A

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

      20. +-commutative N/A

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

      21. lower-+.f64 N/A

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

      22. lower-/.f64 94.6

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

      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 94.6%

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

Alternative 8: 99.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \frac{\frac{y}{1 + \left(y + x\right)}}{\mathsf{fma}\left(\frac{y}{x} + 2, y, x\right)} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (/ (/ y (+ 1.0 (+ y x))) (fma (+ (/ y x) 2.0) y x)))

C

assert(x < y);
double code(double x, double y) {
	return (y / (1.0 + (y + x))) / fma(((y / x) + 2.0), y, x);
}

Julia

x, y = sort([x, y])
function code(x, y)
	return Float64(Float64(y / Float64(1.0 + Float64(y + x))) / fma(Float64(Float64(y / x) + 2.0), y, x))
end

Wolfram

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

Derivation

1. Initial program 70.4%

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

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

   3. clear-num N/A

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

   4. lift-/.f64 N/A

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

   5. frac-times N/A

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

   6. lift-/.f64 N/A

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

   7. clear-num N/A

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

   8. div-inv N/A

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

   9. clear-num N/A

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

   10. lift-/.f64 N/A

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

   11. lower-/.f64 N/A

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

   12. lower-*.f64 N/A

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

   13. lower-/.f64 98.8

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

6. Applied rewrites 98.8%

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

7. Taylor expanded in y around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. +-commutative N/A

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

   5. lower-+.f64 N/A

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

   6. lower-/.f64 98.8

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

9. Applied rewrites 98.8%

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

Alternative 9: 82.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 6.9 \cdot 10^{-162}:\\ \;\;\;\;\frac{\frac{y}{1 + \left(y + x\right)}}{\mathsf{fma}\left(2, y, x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{1 + y}}{y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 6.9e-162)
   (/ (/ y (+ 1.0 (+ y x))) (fma 2.0 y x))
   (/ (/ x (+ 1.0 y)) y)))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 6.9e-162) {
		tmp = (y / (1.0 + (y + x))) / fma(2.0, y, x);
	} else {
		tmp = (x / (1.0 + y)) / y;
	}
	return tmp;
}

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= 6.9e-162)
		tmp = Float64(Float64(y / Float64(1.0 + Float64(y + x))) / fma(2.0, y, x));
	else
		tmp = Float64(Float64(x / Float64(1.0 + y)) / y);
	end
	return tmp
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[y, 6.9e-162], N[(N[(y / N[(1.0 + N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 * y + x), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(1.0 + y), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 6.9000000000000004e-162

   1. Initial program 69.2%

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

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

      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. clear-num N/A

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

      4. lift-/.f64 N/A

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

      5. frac-times N/A

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

      6. lift-/.f64 N/A

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

      7. clear-num N/A

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

      8. div-inv N/A

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

      9. clear-num N/A

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

      10. lift-/.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 98.7

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

   6. Applied rewrites 98.7%

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

   7. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 59.0

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

   9. Applied rewrites 59.0%

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

   if 6.9000000000000004e-162 < y

   1. Initial program 72.1%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. distribute-lft-in N/A

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

      4. *-rgt-identity N/A

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

      5. lower-fma.f64 60.6

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

   5. Applied rewrites 60.6%

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

   7. Applied rewrites 63.6%

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

4. Final simplification 60.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 6.9 \cdot 10^{-162}:\\ \;\;\;\;\frac{\frac{y}{1 + \left(y + x\right)}}{\mathsf{fma}\left(2, y, x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{1 + y}}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 81.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 6.9 \cdot 10^{-162}:\\ \;\;\;\;\frac{\frac{y}{-1 - x}}{-\left(y + x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{1 + y}}{y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 6.9e-162) (/ (/ y (- -1.0 x)) (- (+ y x))) (/ (/ x (+ 1.0 y)) y)))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 6.9e-162) {
		tmp = (y / (-1.0 - x)) / -(y + x);
	} else {
		tmp = (x / (1.0 + 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 <= 6.9d-162) then
        tmp = (y / ((-1.0d0) - x)) / -(y + x)
    else
        tmp = (x / (1.0d0 + y)) / y
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (y <= 6.9e-162) {
		tmp = (y / (-1.0 - x)) / -(y + x);
	} else {
		tmp = (x / (1.0 + y)) / y;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if y <= 6.9e-162:
		tmp = (y / (-1.0 - x)) / -(y + x)
	else:
		tmp = (x / (1.0 + y)) / y
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= 6.9e-162)
		tmp = Float64(Float64(y / Float64(-1.0 - x)) / Float64(-Float64(y + x)));
	else
		tmp = Float64(Float64(x / Float64(1.0 + y)) / y);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 6.9e-162)
		tmp = (y / (-1.0 - x)) / -(y + x);
	else
		tmp = (x / (1.0 + 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, 6.9e-162], N[(N[(y / N[(-1.0 - x), $MachinePrecision]), $MachinePrecision] / (-N[(y + x), $MachinePrecision])), $MachinePrecision], N[(N[(x / N[(1.0 + y), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 6.9000000000000004e-162

   1. Initial program 69.2%

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

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

      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. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. frac-2neg N/A

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

      5. associate-*r/ N/A

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

      6. lower-/.f64 N/A

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

   6. Applied rewrites 99.8%

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

   7. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

      4. lower-/.f64 N/A

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

      5. distribute-lft-in N/A

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

      6. metadata-eval N/A

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

      7. mul-1-neg N/A

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

      8. unsub-neg N/A

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

      9. lower--.f64 58.1

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

   9. Applied rewrites 58.1%

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

   if 6.9000000000000004e-162 < y

   1. Initial program 72.1%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. distribute-lft-in N/A

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

      4. *-rgt-identity N/A

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

      5. lower-fma.f64 60.6

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

   5. Applied rewrites 60.6%

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

   7. Applied rewrites 63.6%

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

4. Final simplification 60.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 6.9 \cdot 10^{-162}:\\ \;\;\;\;\frac{\frac{y}{-1 - x}}{-\left(y + x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{1 + y}}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 80.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -1000000000000:\\ \;\;\;\;\frac{\frac{-y}{x}}{-\left(y + x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{1 + y}}{y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -1000000000000.0)
   (/ (/ (- y) x) (- (+ y x)))
   (/ (/ x (+ 1.0 y)) y)))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -1000000000000.0) {
		tmp = (-y / x) / -(y + x);
	} else {
		tmp = (x / (1.0 + 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 (x <= (-1000000000000.0d0)) then
        tmp = (-y / x) / -(y + x)
    else
        tmp = (x / (1.0d0 + y)) / y
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (x <= -1000000000000.0) {
		tmp = (-y / x) / -(y + x);
	} else {
		tmp = (x / (1.0 + y)) / y;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -1000000000000.0:
		tmp = (-y / x) / -(y + x)
	else:
		tmp = (x / (1.0 + y)) / y
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -1000000000000.0)
		tmp = Float64(Float64(Float64(-y) / x) / Float64(-Float64(y + x)));
	else
		tmp = Float64(Float64(x / Float64(1.0 + y)) / y);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1000000000000.0)
		tmp = (-y / x) / -(y + x);
	else
		tmp = (x / (1.0 + 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[x, -1000000000000.0], N[(N[((-y) / x), $MachinePrecision] / (-N[(y + x), $MachinePrecision])), $MachinePrecision], N[(N[(x / N[(1.0 + y), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1e12

   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. 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. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. frac-2neg N/A

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

      5. associate-*r/ N/A

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

      6. lower-/.f64 N/A

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

   6. Applied rewrites 99.8%

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

   7. Taylor expanded in x around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 68.6

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

   9. Applied rewrites 68.6%

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

   if -1e12 < x

   1. Initial program 73.1%

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

   3. 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 58.0

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

   5. Applied rewrites 58.0%

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

   7. Applied rewrites 59.0%

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

4. Final simplification 61.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1000000000000:\\ \;\;\;\;\frac{\frac{-y}{x}}{-\left(y + x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{1 + y}}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 80.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 7 \cdot 10^{-152}:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(x, x, x\right)}\\ \mathbf{elif}\;y \leq 10^{+100}:\\ \;\;\;\;\frac{x}{y \cdot y + y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 7e-152)
   (/ y (fma x x x))
   (if (<= y 1e+100) (/ x (+ (* y y) y)) (/ (/ x y) y))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 7e-152) {
		tmp = y / fma(x, x, x);
	} else if (y <= 1e+100) {
		tmp = x / ((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 <= 7e-152)
		tmp = Float64(y / fma(x, x, x));
	elseif (y <= 1e+100)
		tmp = Float64(x / Float64(Float64(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, 7e-152], N[(y / N[(x * x + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1e+100], N[(x / N[(N[(y * y), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 7.0000000000000002e-152

   1. Initial program 69.2%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. distribute-lft-in N/A

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

      4. *-rgt-identity N/A

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

      5. lower-fma.f64 57.6

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

   5. Applied rewrites 57.6%

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

   if 7.0000000000000002e-152 < y < 1.00000000000000002e100

   1. Initial program 84.8%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. distribute-lft-in N/A

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

      4. *-rgt-identity N/A

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

      5. lower-fma.f64 44.9

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

   5. Applied rewrites 44.9%

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

   7. Applied rewrites 44.9%

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

   if 1.00000000000000002e100 < y

   1. Initial program 57.2%

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

   3. Taylor expanded in y around inf

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

      1. unpow2 N/A

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

      2. associate-/r* N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 87.7

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

   5. Applied rewrites 87.7%

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

4. Final simplification 60.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 7 \cdot 10^{-152}:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(x, x, x\right)}\\ \mathbf{elif}\;y \leq 10^{+100}:\\ \;\;\;\;\frac{x}{y \cdot y + y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 80.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -1000000000000:\\ \;\;\;\;\frac{\frac{y}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{1 + y}}{y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -1000000000000.0) (/ (/ y x) x) (/ (/ x (+ 1.0 y)) y)))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -1000000000000.0) {
		tmp = (y / x) / x;
	} else {
		tmp = (x / (1.0 + 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 (x <= (-1000000000000.0d0)) then
        tmp = (y / x) / x
    else
        tmp = (x / (1.0d0 + y)) / y
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (x <= -1000000000000.0) {
		tmp = (y / x) / x;
	} else {
		tmp = (x / (1.0 + y)) / y;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -1000000000000.0:
		tmp = (y / x) / x
	else:
		tmp = (x / (1.0 + y)) / y
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -1000000000000.0)
		tmp = Float64(Float64(y / x) / x);
	else
		tmp = Float64(Float64(x / Float64(1.0 + y)) / y);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1000000000000.0)
		tmp = (y / x) / x;
	else
		tmp = (x / (1.0 + 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[x, -1000000000000.0], N[(N[(y / x), $MachinePrecision] / x), $MachinePrecision], N[(N[(x / N[(1.0 + y), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1e12

   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 x around inf

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

      1. unpow2 N/A

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

      2. associate-/r* N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 68.0

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

   5. Applied rewrites 68.0%

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

   if -1e12 < x

   1. Initial program 73.1%

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

   3. 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 58.0

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

   5. Applied rewrites 58.0%

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

   7. Applied rewrites 59.0%

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

4. Final simplification 61.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1000000000000:\\ \;\;\;\;\frac{\frac{y}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{1 + y}}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 78.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -1000000000000:\\ \;\;\;\;\frac{\frac{y}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot y + y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -1000000000000.0) (/ (/ y x) x) (/ x (+ (* y y) y))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -1000000000000.0) {
		tmp = (y / x) / x;
	} else {
		tmp = x / ((y * 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 (x <= (-1000000000000.0d0)) then
        tmp = (y / x) / x
    else
        tmp = x / ((y * y) + y)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (x <= -1000000000000.0) {
		tmp = (y / x) / x;
	} else {
		tmp = x / ((y * y) + y);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -1000000000000.0:
		tmp = (y / x) / x
	else:
		tmp = x / ((y * y) + y)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -1000000000000.0)
		tmp = Float64(Float64(y / x) / x);
	else
		tmp = Float64(x / Float64(Float64(y * y) + y));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1000000000000.0)
		tmp = (y / x) / x;
	else
		tmp = x / ((y * 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[x, -1000000000000.0], N[(N[(y / x), $MachinePrecision] / x), $MachinePrecision], N[(x / N[(N[(y * y), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1e12

   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 x around inf

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

      1. unpow2 N/A

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

      2. associate-/r* N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 68.0

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

   5. Applied rewrites 68.0%

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

   if -1e12 < x

   1. Initial program 73.1%

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

   3. 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 58.0

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

   5. Applied rewrites 58.0%

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

   7. Applied rewrites 58.0%

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

4. Final simplification 60.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1000000000000:\\ \;\;\;\;\frac{\frac{y}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot y + y}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 78.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 7 \cdot 10^{-152}:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(x, x, x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot y + y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 7e-152) (/ y (fma x x x)) (/ x (+ (* y y) y))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 7e-152) {
		tmp = y / fma(x, x, x);
	} else {
		tmp = x / ((y * y) + y);
	}
	return tmp;
}

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= 7e-152)
		tmp = Float64(y / fma(x, x, x));
	else
		tmp = Float64(x / Float64(Float64(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, 7e-152], N[(y / N[(x * x + x), $MachinePrecision]), $MachinePrecision], N[(x / N[(N[(y * y), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 7.0000000000000002e-152

   1. Initial program 69.2%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. distribute-lft-in N/A

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

      4. *-rgt-identity N/A

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

      5. lower-fma.f64 57.6

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

   5. Applied rewrites 57.6%

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

   if 7.0000000000000002e-152 < y

   1. Initial program 72.3%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. distribute-lft-in N/A

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

      4. *-rgt-identity N/A

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

      5. lower-fma.f64 61.3

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

   5. Applied rewrites 61.3%

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

   7. Applied rewrites 61.3%

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

4. Final simplification 59.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 7 \cdot 10^{-152}:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(x, x, x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot y + y}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 78.9% accurate, 1.6× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 7 \cdot 10^{-152}:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(x, x, x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(y, y, y\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 7e-152) (/ y (fma x x x)) (/ x (fma y y y))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 7e-152) {
		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 <= 7e-152)
		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, 7e-152], 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 < 7.0000000000000002e-152

   1. Initial program 69.2%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. distribute-lft-in N/A

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

      4. *-rgt-identity N/A

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

      5. lower-fma.f64 57.6

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

   5. Applied rewrites 57.6%

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

   if 7.0000000000000002e-152 < y

   1. Initial program 72.3%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. distribute-lft-in N/A

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

      4. *-rgt-identity N/A

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

      5. lower-fma.f64 61.3

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

   5. Applied rewrites 61.3%

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

4. Final simplification 59.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 7 \cdot 10^{-152}:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(x, x, x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(y, y, y\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 76.1% accurate, 1.6× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -1000000000000:\\ \;\;\;\;\frac{y}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(y, y, y\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -1000000000000.0) (/ y (* x x)) (/ x (fma y y y))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -1000000000000.0) {
		tmp = y / (x * x);
	} else {
		tmp = x / fma(y, y, y);
	}
	return tmp;
}

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -1000000000000.0)
		tmp = Float64(y / Float64(x * x));
	else
		tmp = Float64(x / fma(y, y, y));
	end
	return tmp
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[x, -1000000000000.0], N[(y / N[(x * x), $MachinePrecision]), $MachinePrecision], N[(x / N[(y * y + y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1e12

   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. 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

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 67.0

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

   7. Applied rewrites 67.0%

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

   if -1e12 < x

   1. Initial program 73.1%

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

   3. 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 58.0

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

   5. Applied rewrites 58.0%

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

4. Final simplification 60.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1000000000000:\\ \;\;\;\;\frac{y}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(y, y, y\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 63.7% accurate, 1.7× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 1.55 \cdot 10^{-7}:\\ \;\;\;\;\frac{y}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 1.55e-7) (/ y (* x x)) (/ x (* y y))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 1.55e-7) {
		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 <= 1.55d-7) 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 <= 1.55e-7) {
		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 <= 1.55e-7:
		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 <= 1.55e-7)
		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 <= 1.55e-7)
		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, 1.55e-7], N[(y / N[(x * x), $MachinePrecision]), $MachinePrecision], N[(x / N[(y * y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.55e-7

   1. Initial program 71.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. associate-*l/ N/A

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

      6. lift-*.f64 N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-/.f64 99.8

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

      15. lift-+.f64 N/A

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

      16. +-commutative N/A

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

      17. lower-+.f64 99.8

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

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

      20. lower-+.f64 99.8

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

      21. lift-+.f64 N/A

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

      22. +-commutative N/A

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

      23. lower-+.f64 99.8

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 47.7

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

   7. Applied rewrites 47.7%

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

   if 1.55e-7 < y

   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 99.7

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

      15. lift-+.f64 N/A

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

      16. +-commutative N/A

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

      17. lower-+.f64 99.7

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

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

      20. lower-+.f64 99.7

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

      21. lift-+.f64 N/A

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

      22. +-commutative N/A

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

      23. lower-+.f64 99.7

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 76.2

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

   7. Applied rewrites 76.2%

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

Alternative 19: 36.8% 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 70.4%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. times-frac N/A

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

   5. associate-*l/ N/A

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

   6. lift-*.f64 N/A

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

   7. times-frac N/A

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

   8. lower-*.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. lift-+.f64 N/A

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

   11. +-commutative N/A

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

   12. lower-+.f64 N/A

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

   13. lower-/.f64 N/A

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

   14. lower-/.f64 99.8

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

   15. lift-+.f64 N/A

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

   16. +-commutative N/A

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

   17. lower-+.f64 99.8

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

   18. lift-+.f64 N/A

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

   19. +-commutative N/A

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

   20. lower-+.f64 99.8

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

   21. lift-+.f64 N/A

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

   22. +-commutative N/A

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

   23. lower-+.f64 99.8

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

4. Applied rewrites 99.8%

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

5. Taylor expanded in y around inf

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

   1. lower-/.f64 N/A

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

   2. unpow2 N/A

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

   3. lower-*.f64 38.2

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

7. Applied rewrites 38.2%

   \[\leadsto \color{blue}{\frac{x}{y \cdot y}} \]
8. 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 2024324 
(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))))