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

Percentage Accurate: 69.6% → 99.8%

Time: 13.3s

Alternatives: 15

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: 69.6% 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

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 66.5%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. times-frac N/A

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

   5. associate-*l/ N/A

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

   6. lift-*.f64 N/A

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

   7. times-frac N/A

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

   8. lower-*.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. lower-/.f64 N/A

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

   11. lower-/.f64 99.8

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

   12. lift-+.f64 N/A

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

   13. lift-+.f64 N/A

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

   14. associate-+l+ N/A

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

   15. +-commutative N/A

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

   16. associate-+l+ N/A

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

   17. lower-+.f64 N/A

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

   18. lower-+.f64 99.8

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

4. Applied rewrites 99.8%

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

5. Final simplification 99.8%

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

Alternative 2: 90.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_0 := \left(x + y\right) \cdot \left(x + y\right)\\ t_1 := y + \left(x + 1\right)\\ \mathbf{if}\;y \leq 10^{-185}:\\ \;\;\;\;\frac{\frac{y}{t\_1}}{x + y} \cdot 1\\ \mathbf{elif}\;y \leq 4 \cdot 10^{+40}:\\ \;\;\;\;y \cdot \frac{x}{t\_1 \cdot t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot 1}{t\_0}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (+ x y) (+ x y))) (t_1 (+ y (+ x 1.0))))
   (if (<= y 1e-185)
     (* (/ (/ y t_1) (+ x y)) 1.0)
     (if (<= y 4e+40) (* y (/ x (* t_1 t_0))) (/ (* x 1.0) t_0)))))

C

assert(x < y);
double code(double x, double y) {
	double t_0 = (x + y) * (x + y);
	double t_1 = y + (x + 1.0);
	double tmp;
	if (y <= 1e-185) {
		tmp = ((y / t_1) / (x + y)) * 1.0;
	} else if (y <= 4e+40) {
		tmp = y * (x / (t_1 * t_0));
	} else {
		tmp = (x * 1.0) / t_0;
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (x + y) * (x + y)
    t_1 = y + (x + 1.0d0)
    if (y <= 1d-185) then
        tmp = ((y / t_1) / (x + y)) * 1.0d0
    else if (y <= 4d+40) then
        tmp = y * (x / (t_1 * t_0))
    else
        tmp = (x * 1.0d0) / t_0
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double t_0 = (x + y) * (x + y);
	double t_1 = y + (x + 1.0);
	double tmp;
	if (y <= 1e-185) {
		tmp = ((y / t_1) / (x + y)) * 1.0;
	} else if (y <= 4e+40) {
		tmp = y * (x / (t_1 * t_0));
	} else {
		tmp = (x * 1.0) / t_0;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	t_0 = (x + y) * (x + y)
	t_1 = y + (x + 1.0)
	tmp = 0
	if y <= 1e-185:
		tmp = ((y / t_1) / (x + y)) * 1.0
	elif y <= 4e+40:
		tmp = y * (x / (t_1 * t_0))
	else:
		tmp = (x * 1.0) / t_0
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	t_0 = Float64(Float64(x + y) * Float64(x + y))
	t_1 = Float64(y + Float64(x + 1.0))
	tmp = 0.0
	if (y <= 1e-185)
		tmp = Float64(Float64(Float64(y / t_1) / Float64(x + y)) * 1.0);
	elseif (y <= 4e+40)
		tmp = Float64(y * Float64(x / Float64(t_1 * t_0)));
	else
		tmp = Float64(Float64(x * 1.0) / t_0);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	t_0 = (x + y) * (x + y);
	t_1 = y + (x + 1.0);
	tmp = 0.0;
	if (y <= 1e-185)
		tmp = ((y / t_1) / (x + y)) * 1.0;
	elseif (y <= 4e+40)
		tmp = y * (x / (t_1 * t_0));
	else
		tmp = (x * 1.0) / t_0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := Block[{t$95$0 = N[(N[(x + y), $MachinePrecision] * N[(x + y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(y + N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, 1e-185], N[(N[(N[(y / t$95$1), $MachinePrecision] / N[(x + y), $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision], If[LessEqual[y, 4e+40], N[(y * N[(x / N[(t$95$1 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * 1.0), $MachinePrecision] / t$95$0), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if y < 9.9999999999999999e-186

   1. Initial program 64.1%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   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. lower-/.f64 N/A

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

      11. lower-/.f64 99.8

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

      12. lift-+.f64 N/A

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

      13. lift-+.f64 N/A

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

      14. associate-+l+ N/A

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

      15. +-commutative N/A

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

      16. associate-+l+ N/A

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

      17. lower-+.f64 N/A

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

      18. lower-+.f64 99.8

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in x around inf

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

   7. Applied rewrites 58.0%

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

   if 9.9999999999999999e-186 < y < 4.00000000000000012e40

   1. Initial program 90.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 97.5

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

      7. lift-+.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. associate-+l+ N/A

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

      10. +-commutative N/A

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

      11. associate-+l+ N/A

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

      12. lower-+.f64 N/A

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

      13. lower-+.f64 97.5

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

   4. Applied rewrites 97.5%

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

   if 4.00000000000000012e40 < y

   1. Initial program 52.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. associate-*l/ N/A

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

      6. lift-*.f64 N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-/.f64 99.8

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

      12. lift-+.f64 N/A

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

      13. lift-+.f64 N/A

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

      14. associate-+l+ N/A

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

      15. +-commutative N/A

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

      16. associate-+l+ N/A

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

      17. lower-+.f64 N/A

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

      18. lower-+.f64 99.8

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in y around inf

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

   7. Applied rewrites 80.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. frac-times N/A

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

      5. lift-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 83.0

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 83.0

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 83.0

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

   9. Applied rewrites 83.0%

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

4. Final simplification 72.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 10^{-185}:\\ \;\;\;\;\frac{\frac{y}{y + \left(x + 1\right)}}{x + y} \cdot 1\\ \mathbf{elif}\;y \leq 4 \cdot 10^{+40}:\\ \;\;\;\;y \cdot \frac{x}{\left(y + \left(x + 1\right)\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot 1}{\left(x + y\right) \cdot \left(x + y\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 96.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_0 := y + \left(x + 1\right)\\ \mathbf{if}\;x \leq -4.6 \cdot 10^{+154}:\\ \;\;\;\;\frac{\frac{y}{t\_0}}{x + y} \cdot 1\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \frac{y}{x + y}}{\left(x + y\right) \cdot t\_0}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ y (+ x 1.0))))
   (if (<= x -4.6e+154)
     (* (/ (/ y t_0) (+ x y)) 1.0)
     (/ (* x (/ y (+ x y))) (* (+ x y) t_0)))))

C

assert(x < y);
double code(double x, double y) {
	double t_0 = y + (x + 1.0);
	double tmp;
	if (x <= -4.6e+154) {
		tmp = ((y / t_0) / (x + y)) * 1.0;
	} else {
		tmp = (x * (y / (x + y))) / ((x + y) * t_0);
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = y + (x + 1.0d0)
    if (x <= (-4.6d+154)) then
        tmp = ((y / t_0) / (x + y)) * 1.0d0
    else
        tmp = (x * (y / (x + y))) / ((x + y) * t_0)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double t_0 = y + (x + 1.0);
	double tmp;
	if (x <= -4.6e+154) {
		tmp = ((y / t_0) / (x + y)) * 1.0;
	} else {
		tmp = (x * (y / (x + y))) / ((x + y) * t_0);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	t_0 = y + (x + 1.0)
	tmp = 0
	if x <= -4.6e+154:
		tmp = ((y / t_0) / (x + y)) * 1.0
	else:
		tmp = (x * (y / (x + y))) / ((x + y) * t_0)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	t_0 = Float64(y + Float64(x + 1.0))
	tmp = 0.0
	if (x <= -4.6e+154)
		tmp = Float64(Float64(Float64(y / t_0) / Float64(x + y)) * 1.0);
	else
		tmp = Float64(Float64(x * Float64(y / Float64(x + y))) / Float64(Float64(x + y) * t_0));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	t_0 = y + (x + 1.0);
	tmp = 0.0;
	if (x <= -4.6e+154)
		tmp = ((y / t_0) / (x + y)) * 1.0;
	else
		tmp = (x * (y / (x + y))) / ((x + y) * t_0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := Block[{t$95$0 = N[(y + N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -4.6e+154], N[(N[(N[(y / t$95$0), $MachinePrecision] / N[(x + y), $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision], N[(N[(x * N[(y / N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(x + y), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -4.6e154

   1. Initial program 41.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. 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. lower-/.f64 N/A

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

      11. lower-/.f64 99.9

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

      12. lift-+.f64 N/A

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

      13. lift-+.f64 N/A

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

      14. associate-+l+ N/A

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

      15. +-commutative N/A

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

      16. associate-+l+ N/A

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

      17. lower-+.f64 N/A

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

      18. lower-+.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in x around inf

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

   7. Applied rewrites 92.4%

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

   if -4.6e154 < x

   1. Initial program 70.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. times-frac N/A

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

      8. 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. lower-*.f64 96.7

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

      13. lift-+.f64 N/A

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

      14. lift-+.f64 N/A

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

      15. associate-+l+ N/A

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

      16. +-commutative N/A

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

      17. associate-+l+ N/A

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

      18. lower-+.f64 N/A

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

      19. lower-+.f64 96.7

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

   4. Applied rewrites 96.7%

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

4. Final simplification 96.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.6 \cdot 10^{+154}:\\ \;\;\;\;\frac{\frac{y}{y + \left(x + 1\right)}}{x + y} \cdot 1\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \frac{y}{x + y}}{\left(x + y\right) \cdot \left(y + \left(x + 1\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 92.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{+149}:\\ \;\;\;\;\frac{\frac{y}{y + \left(x + 1\right)}}{x + y} \cdot 1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\left(x + y\right) \cdot \left(\frac{x + y}{y} \cdot \left(x + \left(y + 1\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -2e+149)
   (* (/ (/ y (+ y (+ x 1.0))) (+ x y)) 1.0)
   (/ x (* (+ x y) (* (/ (+ x y) y) (+ x (+ y 1.0)))))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -2e+149) {
		tmp = ((y / (y + (x + 1.0))) / (x + y)) * 1.0;
	} else {
		tmp = x / ((x + y) * (((x + y) / y) * (x + (y + 1.0))));
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-2d+149)) then
        tmp = ((y / (y + (x + 1.0d0))) / (x + y)) * 1.0d0
    else
        tmp = x / ((x + y) * (((x + y) / y) * (x + (y + 1.0d0))))
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (x <= -2e+149) {
		tmp = ((y / (y + (x + 1.0))) / (x + y)) * 1.0;
	} else {
		tmp = x / ((x + y) * (((x + y) / y) * (x + (y + 1.0))));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -2e+149:
		tmp = ((y / (y + (x + 1.0))) / (x + y)) * 1.0
	else:
		tmp = x / ((x + y) * (((x + y) / y) * (x + (y + 1.0))))
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -2e+149)
		tmp = Float64(Float64(Float64(y / Float64(y + Float64(x + 1.0))) / Float64(x + y)) * 1.0);
	else
		tmp = Float64(x / Float64(Float64(x + y) * Float64(Float64(Float64(x + y) / y) * Float64(x + Float64(y + 1.0)))));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -2e+149)
		tmp = ((y / (y + (x + 1.0))) / (x + y)) * 1.0;
	else
		tmp = x / ((x + y) * (((x + y) / y) * (x + (y + 1.0))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[x, -2e+149], N[(N[(N[(y / N[(y + N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x + y), $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision], N[(x / N[(N[(x + y), $MachinePrecision] * N[(N[(N[(x + y), $MachinePrecision] / y), $MachinePrecision] * N[(x + N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2.0000000000000001e149

   1. Initial program 42.9%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   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. lower-/.f64 N/A

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

      11. lower-/.f64 99.9

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

      12. lift-+.f64 N/A

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

      13. lift-+.f64 N/A

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

      14. associate-+l+ N/A

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

      15. +-commutative N/A

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

      16. associate-+l+ N/A

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

      17. lower-+.f64 N/A

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

      18. lower-+.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in x around inf

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

   7. Applied rewrites 92.6%

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

   if -2.0000000000000001e149 < x

   1. Initial program 69.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. associate-*l/ N/A

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

      6. lift-*.f64 N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-/.f64 99.8

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

      12. lift-+.f64 N/A

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

      13. lift-+.f64 N/A

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

      14. associate-+l+ N/A

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

      15. +-commutative N/A

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

      16. associate-+l+ N/A

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

      17. lower-+.f64 N/A

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

      18. lower-+.f64 99.8

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

   4. Applied rewrites 99.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. frac-times N/A

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

      6. *-commutative N/A

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

      7. *-lft-identity N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. associate-/r/ N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 94.2

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

      14. lift-+.f64 N/A

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

      15. lift-+.f64 N/A

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

      16. associate-+r+ N/A

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

      17. +-commutative N/A

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

      18. lower-+.f64 N/A

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

      19. lower-+.f64 94.2

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

   6. Applied rewrites 94.2%

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

4. Final simplification 94.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{+149}:\\ \;\;\;\;\frac{\frac{y}{y + \left(x + 1\right)}}{x + y} \cdot 1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\left(x + y\right) \cdot \left(\frac{x + y}{y} \cdot \left(x + \left(y + 1\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 93.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{+149}:\\ \;\;\;\;\frac{\frac{y}{y + \left(x + 1\right)}}{x + y} \cdot 1\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{\frac{y}{\left(x + y\right) \cdot \left(x + \left(y + 1\right)\right)}}{x + y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -2e+149)
   (* (/ (/ y (+ y (+ x 1.0))) (+ x y)) 1.0)
   (* x (/ (/ y (* (+ x y) (+ x (+ y 1.0)))) (+ x y)))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -2e+149) {
		tmp = ((y / (y + (x + 1.0))) / (x + y)) * 1.0;
	} else {
		tmp = x * ((y / ((x + y) * (x + (y + 1.0)))) / (x + y));
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-2d+149)) then
        tmp = ((y / (y + (x + 1.0d0))) / (x + y)) * 1.0d0
    else
        tmp = x * ((y / ((x + y) * (x + (y + 1.0d0)))) / (x + y))
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (x <= -2e+149) {
		tmp = ((y / (y + (x + 1.0))) / (x + y)) * 1.0;
	} else {
		tmp = x * ((y / ((x + y) * (x + (y + 1.0)))) / (x + y));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -2e+149:
		tmp = ((y / (y + (x + 1.0))) / (x + y)) * 1.0
	else:
		tmp = x * ((y / ((x + y) * (x + (y + 1.0)))) / (x + y))
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -2e+149)
		tmp = Float64(Float64(Float64(y / Float64(y + Float64(x + 1.0))) / Float64(x + y)) * 1.0);
	else
		tmp = Float64(x * Float64(Float64(y / Float64(Float64(x + y) * Float64(x + Float64(y + 1.0)))) / Float64(x + y)));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -2e+149)
		tmp = ((y / (y + (x + 1.0))) / (x + y)) * 1.0;
	else
		tmp = x * ((y / ((x + y) * (x + (y + 1.0)))) / (x + y));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[x, -2e+149], N[(N[(N[(y / N[(y + N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x + y), $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision], N[(x * N[(N[(y / N[(N[(x + y), $MachinePrecision] * N[(x + N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2.0000000000000001e149

   1. Initial program 42.9%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   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. lower-/.f64 N/A

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

      11. lower-/.f64 99.9

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

      12. lift-+.f64 N/A

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

      13. lift-+.f64 N/A

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

      14. associate-+l+ N/A

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

      15. +-commutative N/A

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

      16. associate-+l+ N/A

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

      17. lower-+.f64 N/A

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

      18. lower-+.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in x around inf

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

   7. Applied rewrites 92.6%

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

   if -2.0000000000000001e149 < x

   1. Initial program 69.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. associate-*l/ N/A

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

      6. lift-*.f64 N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-/.f64 99.8

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

      12. lift-+.f64 N/A

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

      13. lift-+.f64 N/A

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

      14. associate-+l+ N/A

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

      15. +-commutative N/A

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

      16. associate-+l+ N/A

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

      17. lower-+.f64 N/A

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

      18. lower-+.f64 99.8

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

   4. Applied rewrites 99.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 94.6

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

      7. lift-/.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. associate-/l/ N/A

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

      10. lift-*.f64 N/A

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

      11. lower-/.f64 94.6

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

      12. lift-+.f64 N/A

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

      13. lift-+.f64 N/A

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

      14. associate-+r+ N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 N/A

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

      17. lower-+.f64 94.6

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

   6. Applied rewrites 94.6%

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

4. Final simplification 94.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{+149}:\\ \;\;\;\;\frac{\frac{y}{y + \left(x + 1\right)}}{x + y} \cdot 1\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{\frac{y}{\left(x + y\right) \cdot \left(x + \left(y + 1\right)\right)}}{x + y}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 84.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_0 := y + \left(x + 1\right)\\ \mathbf{if}\;y \leq 6 \cdot 10^{-86}:\\ \;\;\;\;\frac{\frac{y}{t\_0}}{x + y} \cdot 1\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot 1}{\left(x + y\right) \cdot t\_0}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ y (+ x 1.0))))
   (if (<= y 6e-86)
     (* (/ (/ y t_0) (+ x y)) 1.0)
     (/ (* x 1.0) (* (+ x y) t_0)))))

C

assert(x < y);
double code(double x, double y) {
	double t_0 = y + (x + 1.0);
	double tmp;
	if (y <= 6e-86) {
		tmp = ((y / t_0) / (x + y)) * 1.0;
	} else {
		tmp = (x * 1.0) / ((x + y) * t_0);
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = y + (x + 1.0d0)
    if (y <= 6d-86) then
        tmp = ((y / t_0) / (x + y)) * 1.0d0
    else
        tmp = (x * 1.0d0) / ((x + y) * t_0)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double t_0 = y + (x + 1.0);
	double tmp;
	if (y <= 6e-86) {
		tmp = ((y / t_0) / (x + y)) * 1.0;
	} else {
		tmp = (x * 1.0) / ((x + y) * t_0);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	t_0 = y + (x + 1.0)
	tmp = 0
	if y <= 6e-86:
		tmp = ((y / t_0) / (x + y)) * 1.0
	else:
		tmp = (x * 1.0) / ((x + y) * t_0)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	t_0 = Float64(y + Float64(x + 1.0))
	tmp = 0.0
	if (y <= 6e-86)
		tmp = Float64(Float64(Float64(y / t_0) / Float64(x + y)) * 1.0);
	else
		tmp = Float64(Float64(x * 1.0) / Float64(Float64(x + y) * t_0));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	t_0 = y + (x + 1.0);
	tmp = 0.0;
	if (y <= 6e-86)
		tmp = ((y / t_0) / (x + y)) * 1.0;
	else
		tmp = (x * 1.0) / ((x + y) * t_0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := Block[{t$95$0 = N[(y + N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, 6e-86], N[(N[(N[(y / t$95$0), $MachinePrecision] / N[(x + y), $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision], N[(N[(x * 1.0), $MachinePrecision] / N[(N[(x + y), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < 6.0000000000000002e-86

   1. Initial program 66.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. associate-*l/ N/A

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

      6. lift-*.f64 N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-/.f64 99.8

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

      12. lift-+.f64 N/A

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

      13. lift-+.f64 N/A

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

      14. associate-+l+ N/A

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

      15. +-commutative N/A

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

      16. associate-+l+ N/A

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

      17. lower-+.f64 N/A

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

      18. lower-+.f64 99.8

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in x around inf

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

   7. Applied rewrites 60.7%

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

   if 6.0000000000000002e-86 < y

   1. Initial program 66.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-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. lower-*.f64 90.4

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

      13. lift-+.f64 N/A

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

      14. lift-+.f64 N/A

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

      15. associate-+l+ N/A

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

      16. +-commutative N/A

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

      17. associate-+l+ N/A

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

      18. lower-+.f64 N/A

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

      19. lower-+.f64 90.4

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

   4. Applied rewrites 90.4%

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

   5. Taylor expanded in y around inf

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

   7. Applied rewrites 83.3%

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

4. Final simplification 69.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 6 \cdot 10^{-86}:\\ \;\;\;\;\frac{\frac{y}{y + \left(x + 1\right)}}{x + y} \cdot 1\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot 1}{\left(x + y\right) \cdot \left(y + \left(x + 1\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 81.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 6 \cdot 10^{-86}:\\ \;\;\;\;\frac{y}{x + x \cdot x}\\ \mathbf{elif}\;y \leq 22000000000:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(y, y, y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot 1}{\left(x + y\right) \cdot \left(x + y\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 6e-86)
   (/ y (+ x (* x x)))
   (if (<= y 22000000000.0)
     (/ x (fma y y y))
     (/ (* x 1.0) (* (+ x y) (+ x y))))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 6e-86) {
		tmp = y / (x + (x * x));
	} else if (y <= 22000000000.0) {
		tmp = x / fma(y, y, y);
	} else {
		tmp = (x * 1.0) / ((x + y) * (x + y));
	}
	return tmp;
}

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= 6e-86)
		tmp = Float64(y / Float64(x + Float64(x * x)));
	elseif (y <= 22000000000.0)
		tmp = Float64(x / fma(y, y, y));
	else
		tmp = Float64(Float64(x * 1.0) / Float64(Float64(x + y) * Float64(x + y)));
	end
	return tmp
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[y, 6e-86], N[(y / N[(x + N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 22000000000.0], N[(x / N[(y * y + y), $MachinePrecision]), $MachinePrecision], N[(N[(x * 1.0), $MachinePrecision] / N[(N[(x + y), $MachinePrecision] * N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 6.0000000000000002e-86

   1. Initial program 66.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. times-frac N/A

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

      8. 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. lower-*.f64 96.2

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

      13. lift-+.f64 N/A

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

      14. lift-+.f64 N/A

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

      15. associate-+l+ N/A

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

      16. +-commutative N/A

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

      17. associate-+l+ N/A

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

      18. lower-+.f64 N/A

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

      19. lower-+.f64 96.2

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

   4. Applied rewrites 96.2%

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

   5. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. unpow2 N/A

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

      5. lower-+.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 57.7

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

   7. Applied rewrites 57.7%

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

   if 6.0000000000000002e-86 < y < 2.2e10

   1. Initial program 92.8%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   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 59.7

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

   5. Applied rewrites 59.7%

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

   if 2.2e10 < y

   1. Initial program 56.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. associate-*l/ N/A

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

      6. lift-*.f64 N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-/.f64 99.8

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

      12. lift-+.f64 N/A

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

      13. lift-+.f64 N/A

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

      14. associate-+l+ N/A

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

      15. +-commutative N/A

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

      16. associate-+l+ N/A

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

      17. lower-+.f64 N/A

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

      18. lower-+.f64 99.8

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in y around inf

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

   7. Applied rewrites 79.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. frac-times N/A

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

      5. lift-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 81.5

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 81.5

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 81.5

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

   9. Applied rewrites 81.5%

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

4. Final simplification 64.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 6 \cdot 10^{-86}:\\ \;\;\;\;\frac{y}{x + x \cdot x}\\ \mathbf{elif}\;y \leq 22000000000:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(y, y, y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot 1}{\left(x + y\right) \cdot \left(x + y\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 83.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 6 \cdot 10^{-86}:\\ \;\;\;\;\frac{y}{x + x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot 1}{\left(x + y\right) \cdot \left(y + \left(x + 1\right)\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 6e-86)
   (/ y (+ x (* x x)))
   (/ (* x 1.0) (* (+ x y) (+ y (+ x 1.0))))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 6e-86) {
		tmp = y / (x + (x * x));
	} else {
		tmp = (x * 1.0) / ((x + y) * (y + (x + 1.0)));
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 6d-86) then
        tmp = y / (x + (x * x))
    else
        tmp = (x * 1.0d0) / ((x + y) * (y + (x + 1.0d0)))
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (y <= 6e-86) {
		tmp = y / (x + (x * x));
	} else {
		tmp = (x * 1.0) / ((x + y) * (y + (x + 1.0)));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if y <= 6e-86:
		tmp = y / (x + (x * x))
	else:
		tmp = (x * 1.0) / ((x + y) * (y + (x + 1.0)))
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= 6e-86)
		tmp = Float64(y / Float64(x + Float64(x * x)));
	else
		tmp = Float64(Float64(x * 1.0) / Float64(Float64(x + y) * Float64(y + Float64(x + 1.0))));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 6e-86)
		tmp = y / (x + (x * x));
	else
		tmp = (x * 1.0) / ((x + y) * (y + (x + 1.0)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[y, 6e-86], N[(y / N[(x + N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * 1.0), $MachinePrecision] / N[(N[(x + y), $MachinePrecision] * N[(y + N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 6.0000000000000002e-86

   1. Initial program 66.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. times-frac N/A

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

      8. 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. lower-*.f64 96.2

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

      13. lift-+.f64 N/A

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

      14. lift-+.f64 N/A

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

      15. associate-+l+ N/A

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

      16. +-commutative N/A

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

      17. associate-+l+ N/A

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

      18. lower-+.f64 N/A

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

      19. lower-+.f64 96.2

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

   4. Applied rewrites 96.2%

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

   5. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. unpow2 N/A

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

      5. lower-+.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 57.7

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

   7. Applied rewrites 57.7%

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

   if 6.0000000000000002e-86 < y

   1. Initial program 66.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-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. lower-*.f64 90.4

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

      13. lift-+.f64 N/A

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

      14. lift-+.f64 N/A

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

      15. associate-+l+ N/A

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

      16. +-commutative N/A

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

      17. associate-+l+ N/A

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

      18. lower-+.f64 N/A

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

      19. lower-+.f64 90.4

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

   4. Applied rewrites 90.4%

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

   5. Taylor expanded in y around inf

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

   7. Applied rewrites 83.3%

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

4. Final simplification 67.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 6 \cdot 10^{-86}:\\ \;\;\;\;\frac{y}{x + x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot 1}{\left(x + y\right) \cdot \left(y + \left(x + 1\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 72.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq -3 \cdot 10^{-164}:\\ \;\;\;\;\frac{y}{x \cdot x}\\ \mathbf{elif}\;y \leq 1.8 \cdot 10^{-123}:\\ \;\;\;\;y \cdot \frac{1}{x}\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y -3e-164)
   (/ y (* x x))
   (if (<= y 1.8e-123) (* y (/ 1.0 x)) (if (<= y 1.0) (/ x y) (/ x (* y y))))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= -3e-164) {
		tmp = y / (x * x);
	} else if (y <= 1.8e-123) {
		tmp = y * (1.0 / x);
	} else if (y <= 1.0) {
		tmp = x / y;
	} else {
		tmp = x / (y * y);
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-3d-164)) then
        tmp = y / (x * x)
    else if (y <= 1.8d-123) then
        tmp = y * (1.0d0 / x)
    else if (y <= 1.0d0) then
        tmp = x / y
    else
        tmp = x / (y * y)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (y <= -3e-164) {
		tmp = y / (x * x);
	} else if (y <= 1.8e-123) {
		tmp = y * (1.0 / x);
	} else if (y <= 1.0) {
		tmp = x / y;
	} else {
		tmp = x / (y * y);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if y <= -3e-164:
		tmp = y / (x * x)
	elif y <= 1.8e-123:
		tmp = y * (1.0 / x)
	elif y <= 1.0:
		tmp = x / y
	else:
		tmp = x / (y * y)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= -3e-164)
		tmp = Float64(y / Float64(x * x));
	elseif (y <= 1.8e-123)
		tmp = Float64(y * Float64(1.0 / x));
	elseif (y <= 1.0)
		tmp = Float64(x / y);
	else
		tmp = Float64(x / Float64(y * y));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -3e-164)
		tmp = y / (x * x);
	elseif (y <= 1.8e-123)
		tmp = y * (1.0 / x);
	elseif (y <= 1.0)
		tmp = x / y;
	else
		tmp = x / (y * y);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[y, -3e-164], N[(y / N[(x * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.8e-123], N[(y * N[(1.0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.0], N[(x / y), $MachinePrecision], N[(x / N[(y * y), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if y < -3.0000000000000001e-164

   1. Initial program 62.6%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 32.5

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

   5. Applied rewrites 32.5%

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

   if -3.0000000000000001e-164 < y < 1.7999999999999998e-123

   1. Initial program 69.4%

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

   3. Taylor expanded in x around inf

      \[\leadsto \frac{x \cdot y}{\color{blue}{{x}^{3}}} \]
   4. Step-by-step derivation

      1. cube-mult N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 54.9

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

   5. Applied rewrites 54.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 55.4

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

   7. Applied rewrites 55.4%

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

   8. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 89.6

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

   10. Applied rewrites 89.6%

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

   11. Taylor expanded in x around 0

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

   13. Applied rewrites 73.5%

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

   if 1.7999999999999998e-123 < y < 1

   1. Initial program 93.8%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   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 51.1

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

   5. Applied rewrites 51.1%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 45.9%

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

   if 1 < y

   1. Initial program 56.7%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 78.0

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

   5. Applied rewrites 78.0%

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

Alternative 10: 73.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq -3 \cdot 10^{-164}:\\ \;\;\;\;\frac{y}{x \cdot x}\\ \mathbf{elif}\;y \leq 1.8 \cdot 10^{-123}:\\ \;\;\;\;y \cdot \frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(y, y, y\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y -3e-164)
   (/ y (* x x))
   (if (<= y 1.8e-123) (* y (/ 1.0 x)) (/ x (fma y y y)))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= -3e-164) {
		tmp = y / (x * x);
	} else if (y <= 1.8e-123) {
		tmp = y * (1.0 / 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 <= -3e-164)
		tmp = Float64(y / Float64(x * x));
	elseif (y <= 1.8e-123)
		tmp = Float64(y * Float64(1.0 / 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, -3e-164], N[(y / N[(x * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.8e-123], N[(y * N[(1.0 / x), $MachinePrecision]), $MachinePrecision], N[(x / N[(y * y + y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -3.0000000000000001e-164

   1. Initial program 62.6%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 32.5

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

   5. Applied rewrites 32.5%

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

   if -3.0000000000000001e-164 < y < 1.7999999999999998e-123

   1. Initial program 69.4%

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

   3. Taylor expanded in x around inf

      \[\leadsto \frac{x \cdot y}{\color{blue}{{x}^{3}}} \]
   4. Step-by-step derivation

      1. cube-mult N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 54.9

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

   5. Applied rewrites 54.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 55.4

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

   7. Applied rewrites 55.4%

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

   8. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 89.6

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

   10. Applied rewrites 89.6%

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

   11. Taylor expanded in x around 0

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

   13. Applied rewrites 73.5%

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

   if 1.7999999999999998e-123 < y

   1. Initial program 68.3%

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

   3. Taylor expanded in x around 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 70.0

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

   5. Applied rewrites 70.0%

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

Alternative 11: 66.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 3 \cdot 10^{-200}:\\ \;\;\;\;\frac{y}{x \cdot x}\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 3e-200) (/ y (* x x)) (if (<= y 1.0) (/ x y) (/ x (* y y)))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 3e-200) {
		tmp = y / (x * x);
	} else if (y <= 1.0) {
		tmp = x / y;
	} else {
		tmp = x / (y * y);
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 3d-200) then
        tmp = y / (x * x)
    else if (y <= 1.0d0) then
        tmp = x / y
    else
        tmp = x / (y * y)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (y <= 3e-200) {
		tmp = y / (x * x);
	} else if (y <= 1.0) {
		tmp = x / y;
	} else {
		tmp = x / (y * y);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if y <= 3e-200:
		tmp = y / (x * x)
	elif y <= 1.0:
		tmp = x / y
	else:
		tmp = x / (y * y)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= 3e-200)
		tmp = Float64(y / Float64(x * x));
	elseif (y <= 1.0)
		tmp = Float64(x / y);
	else
		tmp = Float64(x / Float64(y * y));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 3e-200)
		tmp = y / (x * x);
	elseif (y <= 1.0)
		tmp = x / y;
	else
		tmp = x / (y * y);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[y, 3e-200], N[(y / N[(x * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.0], N[(x / y), $MachinePrecision], N[(x / N[(y * y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 2.99999999999999995e-200

   1. Initial program 64.5%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 41.1

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

   5. Applied rewrites 41.1%

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

   if 2.99999999999999995e-200 < y < 1

   1. Initial program 87.2%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. distribute-lft-in N/A

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

      4. *-rgt-identity N/A

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

      5. lower-fma.f64 45.1

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

   5. Applied rewrites 45.1%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 41.5%

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

   if 1 < y

   1. Initial program 56.7%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 78.0

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

   5. Applied rewrites 78.0%

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

Alternative 12: 79.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 6 \cdot 10^{-86}:\\ \;\;\;\;\frac{y}{x + x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(y, y, y\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 6e-86) (/ y (+ x (* x x))) (/ x (fma y y y))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 6e-86) {
		tmp = y / (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 <= 6e-86)
		tmp = Float64(y / Float64(x + 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[y, 6e-86], N[(y / N[(x + N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x / N[(y * y + y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 6.0000000000000002e-86

   1. Initial program 66.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. times-frac N/A

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

      8. 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. lower-*.f64 96.2

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

      13. lift-+.f64 N/A

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

      14. lift-+.f64 N/A

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

      15. associate-+l+ N/A

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

      16. +-commutative N/A

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

      17. associate-+l+ N/A

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

      18. lower-+.f64 N/A

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

      19. lower-+.f64 96.2

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

   4. Applied rewrites 96.2%

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

   5. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. unpow2 N/A

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

      5. lower-+.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 57.7

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

   7. Applied rewrites 57.7%

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

   if 6.0000000000000002e-86 < y

   1. Initial program 66.4%

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

   3. 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 73.1

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

   5. Applied rewrites 73.1%

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

Alternative 13: 79.7% accurate, 1.6× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 6 \cdot 10^{-86}:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(x, x, x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(y, y, y\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 6e-86) (/ y (fma x x x)) (/ x (fma y y y))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 6e-86) {
		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 <= 6e-86)
		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, 6e-86], 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 < 6.0000000000000002e-86

   1. Initial program 66.5%

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

   3. Taylor expanded in 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 57.7

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

   5. Applied rewrites 57.7%

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

   if 6.0000000000000002e-86 < y

   1. Initial program 66.4%

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

   3. 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 73.1

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

   5. Applied rewrites 73.1%

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

Alternative 14: 46.8% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if y < 1

   1. Initial program 70.3%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   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 44.0

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

   5. Applied rewrites 44.0%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 26.8%

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

   if 1 < y

   1. Initial program 56.7%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 78.0

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

   5. Applied rewrites 78.0%

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

Alternative 15: 26.0% accurate, 3.3× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \frac{x}{y} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 66.5%

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

3. Taylor expanded in x around 0

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

   1. lower-/.f64 N/A

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

   2. +-commutative N/A

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

   3. distribute-lft-in N/A

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

   4. *-rgt-identity N/A

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

   5. lower-fma.f64 53.8

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

5. Applied rewrites 53.8%

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

6. Taylor expanded in y around 0

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

8. Applied rewrites 27.2%

   \[\leadsto \frac{x}{\color{blue}{y}} \]
9. Add Preprocessing

Developer Target 1: 99.8% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024219 
(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))))