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

Percentage Accurate: 69.0% → 99.8%

Time: 10.1s

Alternatives: 19

Speedup: 1.6×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 69.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 73.9%

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

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

   6. lift-*.f64 N/A

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

   7. associate-/r* N/A

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

   8. associate-*r/ N/A

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

   9. lower-/.f64 N/A

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

4. Applied rewrites 99.9%

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

5. Final simplification 99.9%

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

Alternative 2: 87.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_0 := \left(x + y\right) + 1\\ \mathbf{if}\;y \leq 2.1 \cdot 10^{-130}:\\ \;\;\;\;\frac{1 \cdot \frac{y}{t\_0}}{x + y}\\ \mathbf{elif}\;y \leq 2.9 \cdot 10^{+79}:\\ \;\;\;\;\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 \cdot \frac{x}{x + y}}{x + y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ (+ x y) 1.0)))
   (if (<= y 2.1e-130)
     (/ (* 1.0 (/ y t_0)) (+ x y))
     (if (<= y 2.9e+79)
       (/ (* x y) (* (* (+ x y) (+ x y)) t_0))
       (/ (* 1.0 (/ x (+ x y))) (+ x y))))))

C

assert(x < y);
double code(double x, double y) {
	double t_0 = (x + y) + 1.0;
	double tmp;
	if (y <= 2.1e-130) {
		tmp = (1.0 * (y / t_0)) / (x + y);
	} else if (y <= 2.9e+79) {
		tmp = (x * y) / (((x + y) * (x + y)) * t_0);
	} else {
		tmp = (1.0 * (x / (x + y))) / (x + y);
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (x + y) + 1.0d0
    if (y <= 2.1d-130) then
        tmp = (1.0d0 * (y / t_0)) / (x + y)
    else if (y <= 2.9d+79) then
        tmp = (x * y) / (((x + y) * (x + y)) * t_0)
    else
        tmp = (1.0d0 * (x / (x + y))) / (x + y)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double t_0 = (x + y) + 1.0;
	double tmp;
	if (y <= 2.1e-130) {
		tmp = (1.0 * (y / t_0)) / (x + y);
	} else if (y <= 2.9e+79) {
		tmp = (x * y) / (((x + y) * (x + y)) * t_0);
	} else {
		tmp = (1.0 * (x / (x + y))) / (x + y);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	t_0 = (x + y) + 1.0
	tmp = 0
	if y <= 2.1e-130:
		tmp = (1.0 * (y / t_0)) / (x + y)
	elif y <= 2.9e+79:
		tmp = (x * y) / (((x + y) * (x + y)) * t_0)
	else:
		tmp = (1.0 * (x / (x + y))) / (x + y)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	t_0 = Float64(Float64(x + y) + 1.0)
	tmp = 0.0
	if (y <= 2.1e-130)
		tmp = Float64(Float64(1.0 * Float64(y / t_0)) / Float64(x + y));
	elseif (y <= 2.9e+79)
		tmp = Float64(Float64(x * y) / Float64(Float64(Float64(x + y) * Float64(x + y)) * t_0));
	else
		tmp = Float64(Float64(1.0 * Float64(x / Float64(x + y))) / Float64(x + y));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	t_0 = (x + y) + 1.0;
	tmp = 0.0;
	if (y <= 2.1e-130)
		tmp = (1.0 * (y / t_0)) / (x + y);
	elseif (y <= 2.9e+79)
		tmp = (x * y) / (((x + y) * (x + y)) * t_0);
	else
		tmp = (1.0 * (x / (x + y))) / (x + y);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := Block[{t$95$0 = N[(N[(x + y), $MachinePrecision] + 1.0), $MachinePrecision]}, If[LessEqual[y, 2.1e-130], N[(N[(1.0 * N[(y / t$95$0), $MachinePrecision]), $MachinePrecision] / N[(x + y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2.9e+79], N[(N[(x * y), $MachinePrecision] / N[(N[(N[(x + y), $MachinePrecision] * N[(x + y), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 * N[(x / N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x + y), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < 2.10000000000000002e-130

   1. Initial program 72.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/r* N/A

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

      8. associate-*r/ N/A

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

      9. lower-/.f64 N/A

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in y around 0

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

   7. Applied rewrites 57.2%

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

   if 2.10000000000000002e-130 < y < 2.89999999999999992e79

   1. Initial program 89.8%

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

   if 2.89999999999999992e79 < y

   1. Initial program 60.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/r* N/A

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

      8. associate-*r/ N/A

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

      9. lower-/.f64 N/A

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in y around inf

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

   7. Applied rewrites 86.0%

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

4. Final simplification 68.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.1 \cdot 10^{-130}:\\ \;\;\;\;\frac{1 \cdot \frac{y}{\left(x + y\right) + 1}}{x + y}\\ \mathbf{elif}\;y \leq 2.9 \cdot 10^{+79}:\\ \;\;\;\;\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 \cdot \frac{x}{x + y}}{x + y}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 95.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_0 := \left(x + y\right) + 1\\ \mathbf{if}\;x \leq -1.95 \cdot 10^{+158}:\\ \;\;\;\;\frac{1}{x + y} \cdot \frac{y}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{x + y} \cdot x}{t\_0 \cdot \left(x + y\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ (+ x y) 1.0)))
   (if (<= x -1.95e+158)
     (* (/ 1.0 (+ x y)) (/ y t_0))
     (/ (* (/ y (+ x y)) x) (* t_0 (+ x y))))))

C

assert(x < y);
double code(double x, double y) {
	double t_0 = (x + y) + 1.0;
	double tmp;
	if (x <= -1.95e+158) {
		tmp = (1.0 / (x + y)) * (y / t_0);
	} else {
		tmp = ((y / (x + y)) * x) / (t_0 * (x + y));
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (x + y) + 1.0d0
    if (x <= (-1.95d+158)) then
        tmp = (1.0d0 / (x + y)) * (y / t_0)
    else
        tmp = ((y / (x + y)) * x) / (t_0 * (x + y))
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double t_0 = (x + y) + 1.0;
	double tmp;
	if (x <= -1.95e+158) {
		tmp = (1.0 / (x + y)) * (y / t_0);
	} else {
		tmp = ((y / (x + y)) * x) / (t_0 * (x + y));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	t_0 = (x + y) + 1.0
	tmp = 0
	if x <= -1.95e+158:
		tmp = (1.0 / (x + y)) * (y / t_0)
	else:
		tmp = ((y / (x + y)) * x) / (t_0 * (x + y))
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	t_0 = Float64(Float64(x + y) + 1.0)
	tmp = 0.0
	if (x <= -1.95e+158)
		tmp = Float64(Float64(1.0 / Float64(x + y)) * Float64(y / t_0));
	else
		tmp = Float64(Float64(Float64(y / Float64(x + y)) * x) / Float64(t_0 * Float64(x + y)));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	t_0 = (x + y) + 1.0;
	tmp = 0.0;
	if (x <= -1.95e+158)
		tmp = (1.0 / (x + y)) * (y / t_0);
	else
		tmp = ((y / (x + y)) * x) / (t_0 * (x + y));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := Block[{t$95$0 = N[(N[(x + y), $MachinePrecision] + 1.0), $MachinePrecision]}, If[LessEqual[x, -1.95e+158], N[(N[(1.0 / N[(x + y), $MachinePrecision]), $MachinePrecision] * N[(y / t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(y / N[(x + y), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision] / N[(t$95$0 * N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.95e158

   1. Initial program 59.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/r* N/A

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

      8. associate-*r/ N/A

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

      9. lower-/.f64 N/A

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in y around 0

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

   7. Applied rewrites 90.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 90.8

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 90.8

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 90.8

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

   9. Applied rewrites 90.8%

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

   if -1.95e158 < x

   1. Initial program 75.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. times-frac N/A

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

      8. associate-*r/ N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-+.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 96.3

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

      17. lift-+.f64 N/A

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

      18. +-commutative N/A

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

      19. lower-+.f64 96.3

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

      20. lift-+.f64 N/A

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

      21. +-commutative N/A

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

      22. lower-+.f64 96.3

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

      23. lift-+.f64 N/A

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

      24. +-commutative N/A

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

      25. lower-+.f64 96.3

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

   4. Applied rewrites 96.3%

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

4. Final simplification 95.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.95 \cdot 10^{+158}:\\ \;\;\;\;\frac{1}{x + y} \cdot \frac{y}{\left(x + y\right) + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{x + y} \cdot x}{\left(\left(x + y\right) + 1\right) \cdot \left(x + y\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 95.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_0 := \left(x + y\right) + 1\\ \mathbf{if}\;x \leq -1.95 \cdot 10^{+158}:\\ \;\;\;\;\frac{1}{x + y} \cdot \frac{y}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{t\_0 \cdot \left(x + y\right)} \cdot \frac{x}{x + y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ (+ x y) 1.0)))
   (if (<= x -1.95e+158)
     (* (/ 1.0 (+ x y)) (/ y t_0))
     (* (/ y (* t_0 (+ x y))) (/ x (+ x y))))))

C

assert(x < y);
double code(double x, double y) {
	double t_0 = (x + y) + 1.0;
	double tmp;
	if (x <= -1.95e+158) {
		tmp = (1.0 / (x + y)) * (y / t_0);
	} else {
		tmp = (y / (t_0 * (x + y))) * (x / (x + y));
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (x + y) + 1.0d0
    if (x <= (-1.95d+158)) then
        tmp = (1.0d0 / (x + y)) * (y / t_0)
    else
        tmp = (y / (t_0 * (x + y))) * (x / (x + y))
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double t_0 = (x + y) + 1.0;
	double tmp;
	if (x <= -1.95e+158) {
		tmp = (1.0 / (x + y)) * (y / t_0);
	} else {
		tmp = (y / (t_0 * (x + y))) * (x / (x + y));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	t_0 = (x + y) + 1.0
	tmp = 0
	if x <= -1.95e+158:
		tmp = (1.0 / (x + y)) * (y / t_0)
	else:
		tmp = (y / (t_0 * (x + y))) * (x / (x + y))
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	t_0 = Float64(Float64(x + y) + 1.0)
	tmp = 0.0
	if (x <= -1.95e+158)
		tmp = Float64(Float64(1.0 / Float64(x + y)) * Float64(y / t_0));
	else
		tmp = Float64(Float64(y / Float64(t_0 * Float64(x + y))) * Float64(x / Float64(x + y)));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	t_0 = (x + y) + 1.0;
	tmp = 0.0;
	if (x <= -1.95e+158)
		tmp = (1.0 / (x + y)) * (y / t_0);
	else
		tmp = (y / (t_0 * (x + y))) * (x / (x + y));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := Block[{t$95$0 = N[(N[(x + y), $MachinePrecision] + 1.0), $MachinePrecision]}, If[LessEqual[x, -1.95e+158], N[(N[(1.0 / N[(x + y), $MachinePrecision]), $MachinePrecision] * N[(y / t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(y / N[(t$95$0 * N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(x / N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.95e158

   1. Initial program 59.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/r* N/A

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

      8. associate-*r/ N/A

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

      9. lower-/.f64 N/A

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in y around 0

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

   7. Applied rewrites 90.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 90.8

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 90.8

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 90.8

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

   9. Applied rewrites 90.8%

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

   if -1.95e158 < x

   1. Initial program 75.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

      8. times-frac N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 N/A

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

      16. lift-+.f64 N/A

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

      17. +-commutative N/A

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

      18. lower-+.f64 N/A

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

      19. lift-+.f64 N/A

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

      20. +-commutative N/A

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

      21. lower-+.f64 N/A

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

      22. lower-/.f64 96.3

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

      23. lift-+.f64 N/A

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

      24. +-commutative N/A

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

   4. Applied rewrites 96.3%

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

4. Final simplification 95.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.95 \cdot 10^{+158}:\\ \;\;\;\;\frac{1}{x + y} \cdot \frac{y}{\left(x + y\right) + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\left(\left(x + y\right) + 1\right) \cdot \left(x + y\right)} \cdot \frac{x}{x + y}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.8% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 73.9%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. times-frac N/A

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

   5. associate-*l/ N/A

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

   6. lift-*.f64 N/A

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

   7. times-frac N/A

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

   8. lower-*.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. lift-+.f64 N/A

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

   11. +-commutative N/A

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

   12. lower-+.f64 N/A

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

   13. lower-/.f64 N/A

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

   14. lower-/.f64 99.8

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

   15. lift-+.f64 N/A

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

   16. +-commutative N/A

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

   17. lower-+.f64 99.8

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

   18. lift-+.f64 N/A

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

   19. +-commutative N/A

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

   20. lower-+.f64 99.8

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

   21. lift-+.f64 N/A

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

   22. +-commutative N/A

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

   23. lower-+.f64 99.8

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

4. Applied rewrites 99.8%

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

5. Final simplification 99.8%

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

Alternative 6: 93.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 1.02 \cdot 10^{-10}:\\ \;\;\;\;\frac{\frac{y}{x + y} \cdot x}{\left(x + 1\right) \cdot \left(x + y\right)}\\ \mathbf{elif}\;y \leq 2.9 \cdot 10^{+146}:\\ \;\;\;\;\frac{1 \cdot x}{\left(\left(x + y\right) + 1\right) \cdot \left(x + y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 \cdot \frac{x}{x + y}}{x + y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 1.02e-10)
   (/ (* (/ y (+ x y)) x) (* (+ x 1.0) (+ x y)))
   (if (<= y 2.9e+146)
     (/ (* 1.0 x) (* (+ (+ x y) 1.0) (+ x y)))
     (/ (* 1.0 (/ x (+ x y))) (+ x y)))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 1.02e-10) {
		tmp = ((y / (x + y)) * x) / ((x + 1.0) * (x + y));
	} else if (y <= 2.9e+146) {
		tmp = (1.0 * x) / (((x + y) + 1.0) * (x + y));
	} else {
		tmp = (1.0 * (x / (x + y))) / (x + y);
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 1.02d-10) then
        tmp = ((y / (x + y)) * x) / ((x + 1.0d0) * (x + y))
    else if (y <= 2.9d+146) then
        tmp = (1.0d0 * x) / (((x + y) + 1.0d0) * (x + y))
    else
        tmp = (1.0d0 * (x / (x + y))) / (x + y)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (y <= 1.02e-10) {
		tmp = ((y / (x + y)) * x) / ((x + 1.0) * (x + y));
	} else if (y <= 2.9e+146) {
		tmp = (1.0 * x) / (((x + y) + 1.0) * (x + y));
	} else {
		tmp = (1.0 * (x / (x + y))) / (x + y);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if y <= 1.02e-10:
		tmp = ((y / (x + y)) * x) / ((x + 1.0) * (x + y))
	elif y <= 2.9e+146:
		tmp = (1.0 * x) / (((x + y) + 1.0) * (x + y))
	else:
		tmp = (1.0 * (x / (x + y))) / (x + y)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= 1.02e-10)
		tmp = Float64(Float64(Float64(y / Float64(x + y)) * x) / Float64(Float64(x + 1.0) * Float64(x + y)));
	elseif (y <= 2.9e+146)
		tmp = Float64(Float64(1.0 * x) / Float64(Float64(Float64(x + y) + 1.0) * Float64(x + y)));
	else
		tmp = Float64(Float64(1.0 * Float64(x / Float64(x + y))) / Float64(x + y));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 1.02e-10)
		tmp = ((y / (x + y)) * x) / ((x + 1.0) * (x + y));
	elseif (y <= 2.9e+146)
		tmp = (1.0 * x) / (((x + y) + 1.0) * (x + y));
	else
		tmp = (1.0 * (x / (x + y))) / (x + y);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[y, 1.02e-10], N[(N[(N[(y / N[(x + y), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision] / N[(N[(x + 1.0), $MachinePrecision] * N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2.9e+146], N[(N[(1.0 * x), $MachinePrecision] / N[(N[(N[(x + y), $MachinePrecision] + 1.0), $MachinePrecision] * N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 * N[(x / N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x + y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 1.01999999999999997e-10

   1. Initial program 76.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. times-frac N/A

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

      8. associate-*r/ N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-+.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 96.4

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

      17. lift-+.f64 N/A

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

      18. +-commutative N/A

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

      19. lower-+.f64 96.4

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

      20. lift-+.f64 N/A

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

      21. +-commutative N/A

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

      22. lower-+.f64 96.4

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

      23. lift-+.f64 N/A

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

      24. +-commutative N/A

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

      25. lower-+.f64 96.4

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

   4. Applied rewrites 96.4%

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

   5. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 86.2

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

   7. Applied rewrites 86.2%

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

   if 1.01999999999999997e-10 < y < 2.8999999999999998e146

   1. Initial program 59.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. times-frac N/A

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

      8. associate-*r/ N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-+.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 92.1

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

      17. lift-+.f64 N/A

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

      18. +-commutative N/A

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

      19. lower-+.f64 92.1

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

      20. lift-+.f64 N/A

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

      21. +-commutative N/A

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

      22. lower-+.f64 92.1

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

      23. lift-+.f64 N/A

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

      24. +-commutative N/A

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

      25. lower-+.f64 92.1

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

   4. Applied rewrites 92.1%

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

   5. Taylor expanded in y around inf

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

   7. Applied rewrites 79.2%

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

   if 2.8999999999999998e146 < y

   1. Initial program 75.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/r* N/A

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

      8. associate-*r/ N/A

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

      9. lower-/.f64 N/A

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in y around inf

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

   7. Applied rewrites 100.0%

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

4. Final simplification 86.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.02 \cdot 10^{-10}:\\ \;\;\;\;\frac{\frac{y}{x + y} \cdot x}{\left(x + 1\right) \cdot \left(x + y\right)}\\ \mathbf{elif}\;y \leq 2.9 \cdot 10^{+146}:\\ \;\;\;\;\frac{1 \cdot x}{\left(\left(x + y\right) + 1\right) \cdot \left(x + y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 \cdot \frac{x}{x + y}}{x + y}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 86.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_0 := \left(x + y\right) + 1\\ \mathbf{if}\;y \leq 1.85 \cdot 10^{-118}:\\ \;\;\;\;\frac{1 \cdot \frac{y}{t\_0}}{x + y}\\ \mathbf{elif}\;y \leq 2.9 \cdot 10^{+146}:\\ \;\;\;\;\frac{1 \cdot x}{t\_0 \cdot \left(x + y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 \cdot \frac{x}{x + y}}{x + y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ (+ x y) 1.0)))
   (if (<= y 1.85e-118)
     (/ (* 1.0 (/ y t_0)) (+ x y))
     (if (<= y 2.9e+146)
       (/ (* 1.0 x) (* t_0 (+ x y)))
       (/ (* 1.0 (/ x (+ x y))) (+ x y))))))

C

assert(x < y);
double code(double x, double y) {
	double t_0 = (x + y) + 1.0;
	double tmp;
	if (y <= 1.85e-118) {
		tmp = (1.0 * (y / t_0)) / (x + y);
	} else if (y <= 2.9e+146) {
		tmp = (1.0 * x) / (t_0 * (x + y));
	} else {
		tmp = (1.0 * (x / (x + y))) / (x + y);
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (x + y) + 1.0d0
    if (y <= 1.85d-118) then
        tmp = (1.0d0 * (y / t_0)) / (x + y)
    else if (y <= 2.9d+146) then
        tmp = (1.0d0 * x) / (t_0 * (x + y))
    else
        tmp = (1.0d0 * (x / (x + y))) / (x + y)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double t_0 = (x + y) + 1.0;
	double tmp;
	if (y <= 1.85e-118) {
		tmp = (1.0 * (y / t_0)) / (x + y);
	} else if (y <= 2.9e+146) {
		tmp = (1.0 * x) / (t_0 * (x + y));
	} else {
		tmp = (1.0 * (x / (x + y))) / (x + y);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	t_0 = (x + y) + 1.0
	tmp = 0
	if y <= 1.85e-118:
		tmp = (1.0 * (y / t_0)) / (x + y)
	elif y <= 2.9e+146:
		tmp = (1.0 * x) / (t_0 * (x + y))
	else:
		tmp = (1.0 * (x / (x + y))) / (x + y)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	t_0 = Float64(Float64(x + y) + 1.0)
	tmp = 0.0
	if (y <= 1.85e-118)
		tmp = Float64(Float64(1.0 * Float64(y / t_0)) / Float64(x + y));
	elseif (y <= 2.9e+146)
		tmp = Float64(Float64(1.0 * x) / Float64(t_0 * Float64(x + y)));
	else
		tmp = Float64(Float64(1.0 * Float64(x / Float64(x + y))) / Float64(x + y));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	t_0 = (x + y) + 1.0;
	tmp = 0.0;
	if (y <= 1.85e-118)
		tmp = (1.0 * (y / t_0)) / (x + y);
	elseif (y <= 2.9e+146)
		tmp = (1.0 * x) / (t_0 * (x + y));
	else
		tmp = (1.0 * (x / (x + y))) / (x + y);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := Block[{t$95$0 = N[(N[(x + y), $MachinePrecision] + 1.0), $MachinePrecision]}, If[LessEqual[y, 1.85e-118], N[(N[(1.0 * N[(y / t$95$0), $MachinePrecision]), $MachinePrecision] / N[(x + y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2.9e+146], N[(N[(1.0 * x), $MachinePrecision] / N[(t$95$0 * N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 * N[(x / N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x + y), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < 1.85000000000000007e-118

   1. Initial program 73.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/r* N/A

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

      8. associate-*r/ N/A

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

      9. lower-/.f64 N/A

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in y around 0

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

   7. Applied rewrites 57.9%

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

   if 1.85000000000000007e-118 < y < 2.8999999999999998e146

   1. Initial program 74.5%

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

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

      13. +-commutative N/A

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

      14. lower-+.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 95.4

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

      17. lift-+.f64 N/A

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

      18. +-commutative N/A

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

      19. lower-+.f64 95.4

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

      20. lift-+.f64 N/A

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

      21. +-commutative N/A

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

      22. lower-+.f64 95.4

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

      23. lift-+.f64 N/A

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

      24. +-commutative N/A

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

      25. lower-+.f64 95.4

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

   4. Applied rewrites 95.4%

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

   5. Taylor expanded in y around inf

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

   7. Applied rewrites 79.1%

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

   if 2.8999999999999998e146 < y

   1. Initial program 75.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/r* N/A

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

      8. associate-*r/ N/A

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

      9. lower-/.f64 N/A

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in y around inf

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

   7. Applied rewrites 100.0%

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

4. Final simplification 68.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.85 \cdot 10^{-118}:\\ \;\;\;\;\frac{1 \cdot \frac{y}{\left(x + y\right) + 1}}{x + y}\\ \mathbf{elif}\;y \leq 2.9 \cdot 10^{+146}:\\ \;\;\;\;\frac{1 \cdot x}{\left(\left(x + y\right) + 1\right) \cdot \left(x + y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 \cdot \frac{x}{x + y}}{x + y}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 86.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 1.85 \cdot 10^{-118}:\\ \;\;\;\;\frac{\frac{y}{x + 1}}{x + y}\\ \mathbf{elif}\;y \leq 2.9 \cdot 10^{+146}:\\ \;\;\;\;\frac{1 \cdot x}{\left(\left(x + y\right) + 1\right) \cdot \left(x + y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 \cdot \frac{x}{x + y}}{x + y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 1.85e-118)
   (/ (/ y (+ x 1.0)) (+ x y))
   (if (<= y 2.9e+146)
     (/ (* 1.0 x) (* (+ (+ x y) 1.0) (+ x y)))
     (/ (* 1.0 (/ x (+ x y))) (+ x y)))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 1.85e-118) {
		tmp = (y / (x + 1.0)) / (x + y);
	} else if (y <= 2.9e+146) {
		tmp = (1.0 * x) / (((x + y) + 1.0) * (x + y));
	} else {
		tmp = (1.0 * (x / (x + y))) / (x + y);
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 1.85d-118) then
        tmp = (y / (x + 1.0d0)) / (x + y)
    else if (y <= 2.9d+146) then
        tmp = (1.0d0 * x) / (((x + y) + 1.0d0) * (x + y))
    else
        tmp = (1.0d0 * (x / (x + y))) / (x + y)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (y <= 1.85e-118) {
		tmp = (y / (x + 1.0)) / (x + y);
	} else if (y <= 2.9e+146) {
		tmp = (1.0 * x) / (((x + y) + 1.0) * (x + y));
	} else {
		tmp = (1.0 * (x / (x + y))) / (x + y);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if y <= 1.85e-118:
		tmp = (y / (x + 1.0)) / (x + y)
	elif y <= 2.9e+146:
		tmp = (1.0 * x) / (((x + y) + 1.0) * (x + y))
	else:
		tmp = (1.0 * (x / (x + y))) / (x + y)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= 1.85e-118)
		tmp = Float64(Float64(y / Float64(x + 1.0)) / Float64(x + y));
	elseif (y <= 2.9e+146)
		tmp = Float64(Float64(1.0 * x) / Float64(Float64(Float64(x + y) + 1.0) * Float64(x + y)));
	else
		tmp = Float64(Float64(1.0 * Float64(x / Float64(x + y))) / Float64(x + y));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 1.85e-118)
		tmp = (y / (x + 1.0)) / (x + y);
	elseif (y <= 2.9e+146)
		tmp = (1.0 * x) / (((x + y) + 1.0) * (x + y));
	else
		tmp = (1.0 * (x / (x + y))) / (x + y);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[y, 1.85e-118], N[(N[(y / N[(x + 1.0), $MachinePrecision]), $MachinePrecision] / N[(x + y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2.9e+146], N[(N[(1.0 * x), $MachinePrecision] / N[(N[(N[(x + y), $MachinePrecision] + 1.0), $MachinePrecision] * N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 * N[(x / N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x + y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 1.85000000000000007e-118

   1. Initial program 73.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/r* N/A

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

      8. associate-*r/ N/A

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

      9. lower-/.f64 N/A

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 57.1

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

   7. Applied rewrites 57.1%

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

   if 1.85000000000000007e-118 < y < 2.8999999999999998e146

   1. Initial program 74.5%

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

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

      13. +-commutative N/A

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

      14. lower-+.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 95.4

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

      17. lift-+.f64 N/A

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

      18. +-commutative N/A

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

      19. lower-+.f64 95.4

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

      20. lift-+.f64 N/A

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

      21. +-commutative N/A

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

      22. lower-+.f64 95.4

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

      23. lift-+.f64 N/A

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

      24. +-commutative N/A

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

      25. lower-+.f64 95.4

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

   4. Applied rewrites 95.4%

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

   5. Taylor expanded in y around inf

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

   7. Applied rewrites 79.1%

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

   if 2.8999999999999998e146 < y

   1. Initial program 75.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/r* N/A

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

      8. associate-*r/ N/A

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

      9. lower-/.f64 N/A

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in y around inf

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

   7. Applied rewrites 100.0%

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

4. Final simplification 67.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.85 \cdot 10^{-118}:\\ \;\;\;\;\frac{\frac{y}{x + 1}}{x + y}\\ \mathbf{elif}\;y \leq 2.9 \cdot 10^{+146}:\\ \;\;\;\;\frac{1 \cdot x}{\left(\left(x + y\right) + 1\right) \cdot \left(x + y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 \cdot \frac{x}{x + y}}{x + y}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 86.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 1.85 \cdot 10^{-118}:\\ \;\;\;\;\frac{\frac{y}{x + 1}}{x + y}\\ \mathbf{elif}\;y \leq 2.9 \cdot 10^{+146}:\\ \;\;\;\;\frac{1 \cdot x}{\left(\left(x + y\right) + 1\right) \cdot \left(x + y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{x + y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 1.85e-118)
   (/ (/ y (+ x 1.0)) (+ x y))
   (if (<= y 2.9e+146)
     (/ (* 1.0 x) (* (+ (+ x y) 1.0) (+ x y)))
     (/ (/ x y) (+ x y)))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 1.85e-118) {
		tmp = (y / (x + 1.0)) / (x + y);
	} else if (y <= 2.9e+146) {
		tmp = (1.0 * x) / (((x + y) + 1.0) * (x + y));
	} else {
		tmp = (x / y) / (x + y);
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 1.85d-118) then
        tmp = (y / (x + 1.0d0)) / (x + y)
    else if (y <= 2.9d+146) then
        tmp = (1.0d0 * x) / (((x + y) + 1.0d0) * (x + y))
    else
        tmp = (x / y) / (x + y)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (y <= 1.85e-118) {
		tmp = (y / (x + 1.0)) / (x + y);
	} else if (y <= 2.9e+146) {
		tmp = (1.0 * x) / (((x + y) + 1.0) * (x + y));
	} else {
		tmp = (x / y) / (x + y);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if y <= 1.85e-118:
		tmp = (y / (x + 1.0)) / (x + y)
	elif y <= 2.9e+146:
		tmp = (1.0 * x) / (((x + y) + 1.0) * (x + y))
	else:
		tmp = (x / y) / (x + y)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= 1.85e-118)
		tmp = Float64(Float64(y / Float64(x + 1.0)) / Float64(x + y));
	elseif (y <= 2.9e+146)
		tmp = Float64(Float64(1.0 * x) / Float64(Float64(Float64(x + y) + 1.0) * Float64(x + y)));
	else
		tmp = Float64(Float64(x / y) / Float64(x + y));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 1.85e-118)
		tmp = (y / (x + 1.0)) / (x + y);
	elseif (y <= 2.9e+146)
		tmp = (1.0 * x) / (((x + y) + 1.0) * (x + y));
	else
		tmp = (x / y) / (x + y);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[y, 1.85e-118], N[(N[(y / N[(x + 1.0), $MachinePrecision]), $MachinePrecision] / N[(x + y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2.9e+146], N[(N[(1.0 * x), $MachinePrecision] / N[(N[(N[(x + y), $MachinePrecision] + 1.0), $MachinePrecision] * N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / N[(x + y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 1.85000000000000007e-118

   1. Initial program 73.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/r* N/A

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

      8. associate-*r/ N/A

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

      9. lower-/.f64 N/A

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 57.1

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

   7. Applied rewrites 57.1%

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

   if 1.85000000000000007e-118 < y < 2.8999999999999998e146

   1. Initial program 74.5%

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

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

      13. +-commutative N/A

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

      14. lower-+.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 95.4

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

      17. lift-+.f64 N/A

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

      18. +-commutative N/A

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

      19. lower-+.f64 95.4

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

      20. lift-+.f64 N/A

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

      21. +-commutative N/A

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

      22. lower-+.f64 95.4

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

      23. lift-+.f64 N/A

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

      24. +-commutative N/A

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

      25. lower-+.f64 95.4

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

   4. Applied rewrites 95.4%

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

   5. Taylor expanded in y around inf

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

   7. Applied rewrites 79.1%

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

   if 2.8999999999999998e146 < y

   1. Initial program 75.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/r* N/A

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

      8. associate-*r/ N/A

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

      9. lower-/.f64 N/A

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in y around inf

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

      1. lower-/.f64 100.0

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

   7. Applied rewrites 100.0%

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

4. Final simplification 67.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.85 \cdot 10^{-118}:\\ \;\;\;\;\frac{\frac{y}{x + 1}}{x + y}\\ \mathbf{elif}\;y \leq 2.9 \cdot 10^{+146}:\\ \;\;\;\;\frac{1 \cdot x}{\left(\left(x + y\right) + 1\right) \cdot \left(x + y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{x + y}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 87.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -1.75 \cdot 10^{+148}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + y}\\ \mathbf{elif}\;x \leq -1.45 \cdot 10^{-117}:\\ \;\;\;\;\frac{1 \cdot y}{\left(\left(x + y\right) + 1\right) \cdot \left(x + y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{1 + y}}{y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -1.75e+148)
   (/ (/ y x) (+ x y))
   (if (<= x -1.45e-117)
     (/ (* 1.0 y) (* (+ (+ x y) 1.0) (+ x y)))
     (/ (/ x (+ 1.0 y)) y))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -1.75e+148) {
		tmp = (y / x) / (x + y);
	} else if (x <= -1.45e-117) {
		tmp = (1.0 * y) / (((x + y) + 1.0) * (x + y));
	} else {
		tmp = (x / (1.0 + y)) / y;
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-1.75d+148)) then
        tmp = (y / x) / (x + y)
    else if (x <= (-1.45d-117)) then
        tmp = (1.0d0 * y) / (((x + y) + 1.0d0) * (x + y))
    else
        tmp = (x / (1.0d0 + y)) / y
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (x <= -1.75e+148) {
		tmp = (y / x) / (x + y);
	} else if (x <= -1.45e-117) {
		tmp = (1.0 * y) / (((x + y) + 1.0) * (x + y));
	} else {
		tmp = (x / (1.0 + y)) / y;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -1.75e+148:
		tmp = (y / x) / (x + y)
	elif x <= -1.45e-117:
		tmp = (1.0 * y) / (((x + y) + 1.0) * (x + y))
	else:
		tmp = (x / (1.0 + y)) / y
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -1.75e+148)
		tmp = Float64(Float64(y / x) / Float64(x + y));
	elseif (x <= -1.45e-117)
		tmp = Float64(Float64(1.0 * y) / Float64(Float64(Float64(x + y) + 1.0) * Float64(x + y)));
	else
		tmp = Float64(Float64(x / Float64(1.0 + y)) / y);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1.75e+148)
		tmp = (y / x) / (x + y);
	elseif (x <= -1.45e-117)
		tmp = (1.0 * y) / (((x + y) + 1.0) * (x + y));
	else
		tmp = (x / (1.0 + y)) / y;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[x, -1.75e+148], N[(N[(y / x), $MachinePrecision] / N[(x + y), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -1.45e-117], N[(N[(1.0 * y), $MachinePrecision] / N[(N[(N[(x + y), $MachinePrecision] + 1.0), $MachinePrecision] * N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(1.0 + y), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.7499999999999999e148

   1. Initial program 58.9%

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

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

      6. lift-*.f64 N/A

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

      7. associate-/r* N/A

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

      8. associate-*r/ N/A

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

      9. lower-/.f64 N/A

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in x around inf

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

      1. lower-/.f64 91.1

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

   7. Applied rewrites 91.1%

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

   if -1.7499999999999999e148 < x < -1.45e-117

   1. Initial program 77.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/r* N/A

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

      8. associate-*r/ N/A

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

      9. lower-/.f64 N/A

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in y around 0

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

   7. Applied rewrites 54.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-/.f64 N/A

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

      5. frac-times N/A

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

      6. lift-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 77.9

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 77.9

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

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 77.9

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

   9. Applied rewrites 77.9%

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

   if -1.45e-117 < x

   1. Initial program 75.4%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-lft-in N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 75.4

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

      12. lift-+.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-+.f64 75.4

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

   4. Applied rewrites 75.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. lift-+.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. +-commutative N/A

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

      9. lift-+.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. associate-*l/ N/A

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

      13. *-commutative N/A

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

      14. lift-fma.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. distribute-lft-out N/A

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

      17. lift-+.f64 N/A

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

      18. associate-/l/ N/A

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

   6. Applied rewrites 99.5%

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

   7. Taylor expanded in x around 0

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

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

   9. Applied rewrites 59.7%

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

   11. Applied rewrites 61.3%

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

4. Final simplification 68.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.75 \cdot 10^{+148}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + y}\\ \mathbf{elif}\;x \leq -1.45 \cdot 10^{-117}:\\ \;\;\;\;\frac{1 \cdot y}{\left(\left(x + y\right) + 1\right) \cdot \left(x + y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{1 + y}}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 69.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_0 := \frac{x}{y \cdot y}\\ \mathbf{if}\;x \leq -2 \cdot 10^{+20}:\\ \;\;\;\;\frac{y}{x \cdot x}\\ \mathbf{elif}\;x \leq -1.35 \cdot 10^{-133}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 1.55 \cdot 10^{-186}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

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 y))))
   (if (<= x -2e+20)
     (/ y (* x x))
     (if (<= x -1.35e-133) t_0 (if (<= x 1.55e-186) (/ x y) t_0)))))

C

assert(x < y);
double code(double x, double y) {
	double t_0 = x / (y * y);
	double tmp;
	if (x <= -2e+20) {
		tmp = y / (x * x);
	} else if (x <= -1.35e-133) {
		tmp = t_0;
	} else if (x <= 1.55e-186) {
		tmp = x / y;
	} else {
		tmp = 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 = x / (y * y)
    if (x <= (-2d+20)) then
        tmp = y / (x * x)
    else if (x <= (-1.35d-133)) then
        tmp = t_0
    else if (x <= 1.55d-186) then
        tmp = x / y
    else
        tmp = 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 * y);
	double tmp;
	if (x <= -2e+20) {
		tmp = y / (x * x);
	} else if (x <= -1.35e-133) {
		tmp = t_0;
	} else if (x <= 1.55e-186) {
		tmp = x / y;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	t_0 = x / (y * y)
	tmp = 0
	if x <= -2e+20:
		tmp = y / (x * x)
	elif x <= -1.35e-133:
		tmp = t_0
	elif x <= 1.55e-186:
		tmp = x / y
	else:
		tmp = t_0
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	t_0 = Float64(x / Float64(y * y))
	tmp = 0.0
	if (x <= -2e+20)
		tmp = Float64(y / Float64(x * x));
	elseif (x <= -1.35e-133)
		tmp = t_0;
	elseif (x <= 1.55e-186)
		tmp = Float64(x / y);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	t_0 = x / (y * y);
	tmp = 0.0;
	if (x <= -2e+20)
		tmp = y / (x * x);
	elseif (x <= -1.35e-133)
		tmp = t_0;
	elseif (x <= 1.55e-186)
		tmp = x / y;
	else
		tmp = 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[(x / N[(y * y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -2e+20], N[(y / N[(x * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -1.35e-133], t$95$0, If[LessEqual[x, 1.55e-186], N[(x / y), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if x < -2e20

   1. Initial program 65.5%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   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 77.9

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

   5. Applied rewrites 77.9%

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

   if -2e20 < x < -1.3499999999999999e-133 or 1.55000000000000005e-186 < x

   1. Initial program 77.0%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 40.5

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

   5. Applied rewrites 40.5%

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

   if -1.3499999999999999e-133 < x < 1.55000000000000005e-186

   1. Initial program 74.0%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-lft-in N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 74.0

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

      12. lift-+.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-+.f64 74.0

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

   4. Applied rewrites 74.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. lift-+.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. +-commutative N/A

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

      9. lift-+.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. associate-*l/ N/A

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

      13. *-commutative N/A

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

      14. lift-fma.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. distribute-lft-out N/A

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

      17. lift-+.f64 N/A

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

      18. associate-/l/ N/A

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

   6. Applied rewrites 99.8%

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

   7. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]
   8. 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 90.9

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

   9. Applied rewrites 90.9%

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

   10. Taylor expanded in y around 0

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

   12. Applied rewrites 80.7%

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

Alternative 12: 82.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -8 \cdot 10^{-41}:\\ \;\;\;\;\frac{\frac{y}{x + 1}}{x + y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{1 + y}}{y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -8e-41) (/ (/ y (+ x 1.0)) (+ x y)) (/ (/ x (+ 1.0 y)) y)))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -8e-41) {
		tmp = (y / (x + 1.0)) / (x + y);
	} else {
		tmp = (x / (1.0 + y)) / y;
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-8d-41)) then
        tmp = (y / (x + 1.0d0)) / (x + y)
    else
        tmp = (x / (1.0d0 + y)) / y
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (x <= -8e-41) {
		tmp = (y / (x + 1.0)) / (x + y);
	} else {
		tmp = (x / (1.0 + y)) / y;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -8e-41:
		tmp = (y / (x + 1.0)) / (x + y)
	else:
		tmp = (x / (1.0 + y)) / y
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -8e-41)
		tmp = Float64(Float64(y / Float64(x + 1.0)) / Float64(x + y));
	else
		tmp = Float64(Float64(x / Float64(1.0 + y)) / y);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -8e-41)
		tmp = (y / (x + 1.0)) / (x + y);
	else
		tmp = (x / (1.0 + y)) / y;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[x, -8e-41], N[(N[(y / N[(x + 1.0), $MachinePrecision]), $MachinePrecision] / N[(x + y), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(1.0 + y), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -8.00000000000000005e-41

   1. Initial program 67.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/r* N/A

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

      8. associate-*r/ N/A

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

      9. lower-/.f64 N/A

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 78.7

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

   7. Applied rewrites 78.7%

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

   if -8.00000000000000005e-41 < x

   1. Initial program 75.9%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-lft-in N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 75.9

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

      12. lift-+.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-+.f64 75.9

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

   4. Applied rewrites 75.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. lift-+.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. +-commutative N/A

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

      9. lift-+.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. associate-*l/ N/A

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

      13. *-commutative N/A

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

      14. lift-fma.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. distribute-lft-out N/A

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

      17. lift-+.f64 N/A

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

      18. associate-/l/ N/A

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

   6. Applied rewrites 99.5%

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

   7. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. distribute-lft-in N/A

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

      4. *-rgt-identity N/A

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

      5. lower-fma.f64 61.2

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

   9. Applied rewrites 61.2%

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

   11. Applied rewrites 62.6%

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

4. Final simplification 66.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -8 \cdot 10^{-41}:\\ \;\;\;\;\frac{\frac{y}{x + 1}}{x + y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{1 + y}}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 79.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{+20}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{1 + y}}{y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -2e+20) (/ (/ y x) (+ x y)) (/ (/ x (+ 1.0 y)) y)))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -2e+20) {
		tmp = (y / x) / (x + y);
	} else {
		tmp = (x / (1.0 + y)) / y;
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-2d+20)) then
        tmp = (y / x) / (x + y)
    else
        tmp = (x / (1.0d0 + y)) / y
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (x <= -2e+20) {
		tmp = (y / x) / (x + y);
	} else {
		tmp = (x / (1.0 + y)) / y;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -2e+20:
		tmp = (y / x) / (x + y)
	else:
		tmp = (x / (1.0 + y)) / y
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -2e+20)
		tmp = Float64(Float64(y / x) / Float64(x + y));
	else
		tmp = Float64(Float64(x / Float64(1.0 + y)) / y);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -2e+20)
		tmp = (y / x) / (x + y);
	else
		tmp = (x / (1.0 + y)) / y;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[x, -2e+20], N[(N[(y / x), $MachinePrecision] / N[(x + y), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(1.0 + y), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2e20

   1. Initial program 65.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/r* N/A

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

      8. associate-*r/ N/A

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

      9. lower-/.f64 N/A

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in x around inf

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

      1. lower-/.f64 83.7

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

   7. Applied rewrites 83.7%

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

   if -2e20 < x

   1. Initial program 76.1%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-lft-in N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 76.1

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

      12. lift-+.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-+.f64 76.1

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

   4. Applied rewrites 76.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. lift-+.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. +-commutative N/A

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

      9. lift-+.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. associate-*l/ N/A

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

      13. *-commutative N/A

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

      14. lift-fma.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. distribute-lft-out N/A

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

      17. lift-+.f64 N/A

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

      18. associate-/l/ N/A

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

   6. Applied rewrites 99.5%

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

   7. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. distribute-lft-in N/A

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

      4. *-rgt-identity N/A

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

      5. lower-fma.f64 60.7

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

   9. Applied rewrites 60.7%

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

   11. Applied rewrites 62.1%

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

4. Final simplification 66.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{+20}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{1 + y}}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 79.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{+20}:\\ \;\;\;\;\frac{\frac{y}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{1 + y}}{y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -2e+20) (/ (/ y x) x) (/ (/ x (+ 1.0 y)) y)))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -2e+20) {
		tmp = (y / x) / x;
	} else {
		tmp = (x / (1.0 + y)) / y;
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-2d+20)) then
        tmp = (y / x) / x
    else
        tmp = (x / (1.0d0 + y)) / y
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (x <= -2e+20) {
		tmp = (y / x) / x;
	} else {
		tmp = (x / (1.0 + y)) / y;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -2e+20:
		tmp = (y / x) / x
	else:
		tmp = (x / (1.0 + y)) / y
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -2e+20)
		tmp = Float64(Float64(y / x) / x);
	else
		tmp = Float64(Float64(x / Float64(1.0 + y)) / y);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -2e+20)
		tmp = (y / x) / x;
	else
		tmp = (x / (1.0 + y)) / y;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[x, -2e+20], N[(N[(y / x), $MachinePrecision] / x), $MachinePrecision], N[(N[(x / N[(1.0 + y), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2e20

   1. Initial program 65.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/r* N/A

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

      8. associate-*r/ N/A

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

      9. lower-/.f64 N/A

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 77.9

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

   7. Applied rewrites 77.9%

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

   9. Applied rewrites 83.4%

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

   if -2e20 < x

   1. Initial program 76.1%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-lft-in N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 76.1

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

      12. lift-+.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-+.f64 76.1

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

   4. Applied rewrites 76.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. lift-+.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. +-commutative N/A

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

      9. lift-+.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. associate-*l/ N/A

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

      13. *-commutative N/A

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

      14. lift-fma.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. distribute-lft-out N/A

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

      17. lift-+.f64 N/A

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

      18. associate-/l/ N/A

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

   6. Applied rewrites 99.5%

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

   7. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. distribute-lft-in N/A

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

      4. *-rgt-identity N/A

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

      5. lower-fma.f64 60.7

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

   9. Applied rewrites 60.7%

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

   11. Applied rewrites 62.1%

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

Alternative 15: 78.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{+20}:\\ \;\;\;\;\frac{\frac{y}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(y, y, y\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -2e+20) (/ (/ y x) x) (/ x (fma y y y))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -2e+20) {
		tmp = (y / x) / x;
	} else {
		tmp = x / fma(y, y, y);
	}
	return tmp;
}

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -2e+20)
		tmp = Float64(Float64(y / x) / x);
	else
		tmp = Float64(x / fma(y, y, y));
	end
	return tmp
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[x, -2e+20], N[(N[(y / x), $MachinePrecision] / x), $MachinePrecision], N[(x / N[(y * y + y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2e20

   1. Initial program 65.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/r* N/A

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

      8. associate-*r/ N/A

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

      9. lower-/.f64 N/A

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 77.9

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

   7. Applied rewrites 77.9%

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

   9. Applied rewrites 83.4%

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

   if -2e20 < x

   1. Initial program 76.1%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. distribute-lft-in N/A

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

      4. *-rgt-identity N/A

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

      5. lower-fma.f64 60.7

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

   5. Applied rewrites 60.7%

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

Alternative 16: 78.4% accurate, 1.6× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -8 \cdot 10^{-41}:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(x, x, x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(y, y, y\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -8e-41) (/ y (fma x x x)) (/ x (fma y y y))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -8e-41) {
		tmp = y / fma(x, x, x);
	} else {
		tmp = x / fma(y, y, y);
	}
	return tmp;
}

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -8e-41)
		tmp = Float64(y / fma(x, x, x));
	else
		tmp = Float64(x / fma(y, y, y));
	end
	return tmp
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[x, -8e-41], N[(y / N[(x * x + x), $MachinePrecision]), $MachinePrecision], N[(x / N[(y * y + y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -8.00000000000000005e-41

   1. Initial program 67.8%

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

   3. Taylor expanded in y around 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 73.7

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

   5. Applied rewrites 73.7%

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

   if -8.00000000000000005e-41 < x

   1. Initial program 75.9%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. distribute-lft-in N/A

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

      4. *-rgt-identity N/A

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

      5. lower-fma.f64 61.2

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

   5. Applied rewrites 61.2%

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

Alternative 17: 76.0% accurate, 1.6× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{+20}:\\ \;\;\;\;\frac{y}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(y, y, y\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -2e+20) (/ y (* x x)) (/ x (fma y y y))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -2e+20) {
		tmp = y / (x * x);
	} else {
		tmp = x / fma(y, y, y);
	}
	return tmp;
}

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -2e+20)
		tmp = Float64(y / Float64(x * x));
	else
		tmp = Float64(x / fma(y, y, y));
	end
	return tmp
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[x, -2e+20], N[(y / N[(x * x), $MachinePrecision]), $MachinePrecision], N[(x / N[(y * y + y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2e20

   1. Initial program 65.5%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   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 77.9

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

   5. Applied rewrites 77.9%

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

   if -2e20 < x

   1. Initial program 76.1%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. distribute-lft-in N/A

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

      4. *-rgt-identity N/A

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

      5. lower-fma.f64 60.7

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

   5. Applied rewrites 60.7%

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

Alternative 18: 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 76.5%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-lft-in N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 76.5

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

      12. lift-+.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-+.f64 76.5

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

   4. Applied rewrites 76.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. lift-+.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. +-commutative N/A

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

      9. lift-+.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. associate-*l/ N/A

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

      13. *-commutative N/A

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

      14. lift-fma.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. distribute-lft-out N/A

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

      17. lift-+.f64 N/A

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

      18. associate-/l/ N/A

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

   6. Applied rewrites 99.3%

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

   7. Taylor expanded in x around 0

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

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

   9. Applied rewrites 44.0%

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

   10. Taylor expanded in y around 0

       \[\leadsto \frac{x}{\color{blue}{y}} \]
   11. Step-by-step derivation

   12. Applied rewrites 28.8%

       \[\leadsto \frac{x}{\color{blue}{y}} \]

   if 1 < y

   1. Initial program 66.1%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\frac{x}{{y}^{2}}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{{y}^{2}}} \]

      2. unpow2 N/A

         \[\leadsto \frac{x}{\color{blue}{y \cdot y}} \]

      3. lower-*.f64 70.5

         \[\leadsto \frac{x}{\color{blue}{y \cdot y}} \]

   5. Applied rewrites 70.5%

      \[\leadsto \color{blue}{\frac{x}{y \cdot y}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 19: 26.5% 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 73.9%

   \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right)} \cdot \left(\left(x + y\right) + 1\right)} \]

   2. lift-+.f64 N/A

      \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \color{blue}{\left(x + y\right)}\right) \cdot \left(\left(x + y\right) + 1\right)} \]

   3. +-commutative N/A

      \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \color{blue}{\left(y + x\right)}\right) \cdot \left(\left(x + y\right) + 1\right)} \]

   4. distribute-lft-in N/A

      \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(x + y\right) \cdot y + \left(x + y\right) \cdot x\right)} \cdot \left(\left(x + y\right) + 1\right)} \]

   5. *-commutative N/A

      \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot y + \color{blue}{x \cdot \left(x + y\right)}\right) \cdot \left(\left(x + y\right) + 1\right)} \]

   6. lower-fma.f64 N/A

      \[\leadsto \frac{x \cdot y}{\color{blue}{\mathsf{fma}\left(x + y, y, x \cdot \left(x + y\right)\right)} \cdot \left(\left(x + y\right) + 1\right)} \]

   7. lift-+.f64 N/A

      \[\leadsto \frac{x \cdot y}{\mathsf{fma}\left(\color{blue}{x + y}, y, x \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

   8. +-commutative N/A

      \[\leadsto \frac{x \cdot y}{\mathsf{fma}\left(\color{blue}{y + x}, y, x \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

   9. lower-+.f64 N/A

      \[\leadsto \frac{x \cdot y}{\mathsf{fma}\left(\color{blue}{y + x}, y, x \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

   10. *-commutative N/A

       \[\leadsto \frac{x \cdot y}{\mathsf{fma}\left(y + x, y, \color{blue}{\left(x + y\right) \cdot x}\right) \cdot \left(\left(x + y\right) + 1\right)} \]

   11. lower-*.f64 73.9

       \[\leadsto \frac{x \cdot y}{\mathsf{fma}\left(y + x, y, \color{blue}{\left(x + y\right) \cdot x}\right) \cdot \left(\left(x + y\right) + 1\right)} \]

   12. lift-+.f64 N/A

       \[\leadsto \frac{x \cdot y}{\mathsf{fma}\left(y + x, y, \color{blue}{\left(x + y\right)} \cdot x\right) \cdot \left(\left(x + y\right) + 1\right)} \]

   13. +-commutative N/A

       \[\leadsto \frac{x \cdot y}{\mathsf{fma}\left(y + x, y, \color{blue}{\left(y + x\right)} \cdot x\right) \cdot \left(\left(x + y\right) + 1\right)} \]

   14. lower-+.f64 73.9

       \[\leadsto \frac{x \cdot y}{\mathsf{fma}\left(y + x, y, \color{blue}{\left(y + x\right)} \cdot x\right) \cdot \left(\left(x + y\right) + 1\right)} \]

4. Applied rewrites 73.9%

   \[\leadsto \frac{x \cdot y}{\color{blue}{\mathsf{fma}\left(y + x, y, \left(y + x\right) \cdot x\right)} \cdot \left(\left(x + y\right) + 1\right)} \]
5. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \color{blue}{\frac{x \cdot y}{\mathsf{fma}\left(y + x, y, \left(y + x\right) \cdot x\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{\color{blue}{x \cdot y}}{\mathsf{fma}\left(y + x, y, \left(y + x\right) \cdot x\right) \cdot \left(\left(x + y\right) + 1\right)} \]

   3. lift-*.f64 N/A

      \[\leadsto \frac{x \cdot y}{\color{blue}{\mathsf{fma}\left(y + x, y, \left(y + x\right) \cdot x\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

   4. times-frac N/A

      \[\leadsto \color{blue}{\frac{x}{\mathsf{fma}\left(y + x, y, \left(y + x\right) \cdot x\right)} \cdot \frac{y}{\left(x + y\right) + 1}} \]

   5. lift-+.f64 N/A

      \[\leadsto \frac{x}{\mathsf{fma}\left(y + x, y, \left(y + x\right) \cdot x\right)} \cdot \frac{y}{\color{blue}{\left(x + y\right) + 1}} \]

   6. lift-+.f64 N/A

      \[\leadsto \frac{x}{\mathsf{fma}\left(y + x, y, \left(y + x\right) \cdot x\right)} \cdot \frac{y}{\color{blue}{\left(x + y\right)} + 1} \]

   7. +-commutative N/A

      \[\leadsto \frac{x}{\mathsf{fma}\left(y + x, y, \left(y + x\right) \cdot x\right)} \cdot \frac{y}{\color{blue}{1 + \left(x + y\right)}} \]

   8. +-commutative N/A

      \[\leadsto \frac{x}{\mathsf{fma}\left(y + x, y, \left(y + x\right) \cdot x\right)} \cdot \frac{y}{1 + \color{blue}{\left(y + x\right)}} \]

   9. lift-+.f64 N/A

      \[\leadsto \frac{x}{\mathsf{fma}\left(y + x, y, \left(y + x\right) \cdot x\right)} \cdot \frac{y}{1 + \color{blue}{\left(y + x\right)}} \]

   10. lift-+.f64 N/A

       \[\leadsto \frac{x}{\mathsf{fma}\left(y + x, y, \left(y + x\right) \cdot x\right)} \cdot \frac{y}{\color{blue}{1 + \left(y + x\right)}} \]

   11. lift-/.f64 N/A

       \[\leadsto \frac{x}{\mathsf{fma}\left(y + x, y, \left(y + x\right) \cdot x\right)} \cdot \color{blue}{\frac{y}{1 + \left(y + x\right)}} \]

   12. associate-*l/ N/A

       \[\leadsto \color{blue}{\frac{x \cdot \frac{y}{1 + \left(y + x\right)}}{\mathsf{fma}\left(y + x, y, \left(y + x\right) \cdot x\right)}} \]

   13. *-commutative N/A

       \[\leadsto \frac{\color{blue}{\frac{y}{1 + \left(y + x\right)} \cdot x}}{\mathsf{fma}\left(y + x, y, \left(y + x\right) \cdot x\right)} \]

   14. lift-fma.f64 N/A

       \[\leadsto \frac{\frac{y}{1 + \left(y + x\right)} \cdot x}{\color{blue}{\left(y + x\right) \cdot y + \left(y + x\right) \cdot x}} \]

   15. lift-*.f64 N/A

       \[\leadsto \frac{\frac{y}{1 + \left(y + x\right)} \cdot x}{\left(y + x\right) \cdot y + \color{blue}{\left(y + x\right) \cdot x}} \]

   16. distribute-lft-out N/A

       \[\leadsto \frac{\frac{y}{1 + \left(y + x\right)} \cdot x}{\color{blue}{\left(y + x\right) \cdot \left(y + x\right)}} \]

   17. lift-+.f64 N/A

       \[\leadsto \frac{\frac{y}{1 + \left(y + x\right)} \cdot x}{\left(y + x\right) \cdot \color{blue}{\left(y + x\right)}} \]

   18. associate-/l/ N/A

       \[\leadsto \color{blue}{\frac{\frac{\frac{y}{1 + \left(y + x\right)} \cdot x}{y + x}}{y + x}} \]

6. Applied rewrites 99.4%

   \[\leadsto \color{blue}{\frac{\frac{x}{x + y}}{\frac{\left(x + y\right) + 1}{y} \cdot \left(x + y\right)}} \]

7. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]
8. 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 50.7

      \[\leadsto \frac{x}{\color{blue}{\mathsf{fma}\left(y, y, y\right)}} \]

9. Applied rewrites 50.7%

   \[\leadsto \color{blue}{\frac{x}{\mathsf{fma}\left(y, y, y\right)}} \]

10. Taylor expanded in y around 0

    \[\leadsto \frac{x}{\color{blue}{y}} \]
11. Step-by-step derivation

12. Applied rewrites 28.7%

    \[\leadsto \frac{x}{\color{blue}{y}} \]
13. 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 2024249 
(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))))