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

Percentage Accurate: 69.1% → 99.8%

Time: 16.6s

Alternatives: 19

Speedup: 1.9×

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: 69.1% 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, 1.0× speedup

Math

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

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (* (/ x (+ x y)) (/ (/ y (+ 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(x / Float64(x + y)) * Float64(Float64(y / Float64(x + Float64(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[(x / N[(x + y), $MachinePrecision]), $MachinePrecision] * N[(N[(y / N[(x + N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 67.9%

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

   1. times-frac 89.0%

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

   2. +-commutative 89.0%

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

   3. +-commutative 89.0%

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

   4. associate-/r/ 85.8%

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

   5. +-commutative 85.8%

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

   6. associate-/r/ 85.8%

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

   7. associate-*l/ 83.0%

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

   8. *-commutative 83.0%

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

   9. distribute-rgt1-in 63.3%

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

   10. fma-def 83.0%

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

   11. +-commutative 83.0%

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

   12. +-commutative 83.0%

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

   13. cube-unmult 83.0%

       \[\leadsto \frac{x}{\frac{\mathsf{fma}\left(x + y, x + y, \color{blue}{{\left(y + x\right)}^{3}}\right)}{y}} \]

   14. +-commutative 83.0%

       \[\leadsto \frac{x}{\frac{\mathsf{fma}\left(x + y, x + y, {\color{blue}{\left(x + y\right)}}^{3}\right)}{y}} \]

3. Simplified 83.0%

   \[\leadsto \color{blue}{\frac{x}{\frac{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}{y}}} \]
4. Step-by-step derivation

   1. associate-/l* 67.9%

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

   2. fma-udef 54.2%

      \[\leadsto \frac{x \cdot y}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right) + {\left(x + y\right)}^{3}}} \]

   3. cube-mult 54.2%

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

   4. distribute-rgt1-in 67.9%

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

   5. *-commutative 67.9%

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

   6. times-frac 89.0%

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

   7. associate-*l/ 82.7%

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

   8. times-frac 99.9%

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

   9. associate-+r+ 99.9%

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

5. Applied egg-rr 99.9%

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

6. Final simplification 99.9%

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

Alternative 2: 94.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_0 := \frac{x}{x + y}\\ t_1 := \frac{y}{x + \left(y + 1\right)}\\ \mathbf{if}\;y \leq 5.6 \cdot 10^{-230}:\\ \;\;\;\;\frac{t_1}{x + y \cdot 2}\\ \mathbf{elif}\;y \leq 7.2 \cdot 10^{-164}:\\ \;\;\;\;y \cdot \frac{t_0}{\left(x + y\right) \cdot \left(y + 1\right)}\\ \mathbf{elif}\;y \leq 2.4 \cdot 10^{+161}:\\ \;\;\;\;t_1 \cdot \frac{x}{\left(x + y\right) \cdot \left(x + y\right)}\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \frac{1}{x + y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ x (+ x y))) (t_1 (/ y (+ x (+ y 1.0)))))
   (if (<= y 5.6e-230)
     (/ t_1 (+ x (* y 2.0)))
     (if (<= y 7.2e-164)
       (* y (/ t_0 (* (+ x y) (+ y 1.0))))
       (if (<= y 2.4e+161)
         (* t_1 (/ x (* (+ x y) (+ x y))))
         (* t_0 (/ 1.0 (+ x y))))))))

C

assert(x < y);
double code(double x, double y) {
	double t_0 = x / (x + y);
	double t_1 = y / (x + (y + 1.0));
	double tmp;
	if (y <= 5.6e-230) {
		tmp = t_1 / (x + (y * 2.0));
	} else if (y <= 7.2e-164) {
		tmp = y * (t_0 / ((x + y) * (y + 1.0)));
	} else if (y <= 2.4e+161) {
		tmp = t_1 * (x / ((x + y) * (x + y)));
	} else {
		tmp = t_0 * (1.0 / (x + y));
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = x / (x + y)
    t_1 = y / (x + (y + 1.0d0))
    if (y <= 5.6d-230) then
        tmp = t_1 / (x + (y * 2.0d0))
    else if (y <= 7.2d-164) then
        tmp = y * (t_0 / ((x + y) * (y + 1.0d0)))
    else if (y <= 2.4d+161) then
        tmp = t_1 * (x / ((x + y) * (x + y)))
    else
        tmp = t_0 * (1.0d0 / (x + y))
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double t_0 = x / (x + y);
	double t_1 = y / (x + (y + 1.0));
	double tmp;
	if (y <= 5.6e-230) {
		tmp = t_1 / (x + (y * 2.0));
	} else if (y <= 7.2e-164) {
		tmp = y * (t_0 / ((x + y) * (y + 1.0)));
	} else if (y <= 2.4e+161) {
		tmp = t_1 * (x / ((x + y) * (x + y)));
	} else {
		tmp = t_0 * (1.0 / (x + y));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	t_0 = x / (x + y)
	t_1 = y / (x + (y + 1.0))
	tmp = 0
	if y <= 5.6e-230:
		tmp = t_1 / (x + (y * 2.0))
	elif y <= 7.2e-164:
		tmp = y * (t_0 / ((x + y) * (y + 1.0)))
	elif y <= 2.4e+161:
		tmp = t_1 * (x / ((x + y) * (x + y)))
	else:
		tmp = t_0 * (1.0 / (x + y))
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	t_0 = Float64(x / Float64(x + y))
	t_1 = Float64(y / Float64(x + Float64(y + 1.0)))
	tmp = 0.0
	if (y <= 5.6e-230)
		tmp = Float64(t_1 / Float64(x + Float64(y * 2.0)));
	elseif (y <= 7.2e-164)
		tmp = Float64(y * Float64(t_0 / Float64(Float64(x + y) * Float64(y + 1.0))));
	elseif (y <= 2.4e+161)
		tmp = Float64(t_1 * Float64(x / Float64(Float64(x + y) * Float64(x + y))));
	else
		tmp = Float64(t_0 * Float64(1.0 / Float64(x + y)));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	t_0 = x / (x + y);
	t_1 = y / (x + (y + 1.0));
	tmp = 0.0;
	if (y <= 5.6e-230)
		tmp = t_1 / (x + (y * 2.0));
	elseif (y <= 7.2e-164)
		tmp = y * (t_0 / ((x + y) * (y + 1.0)));
	elseif (y <= 2.4e+161)
		tmp = t_1 * (x / ((x + y) * (x + y)));
	else
		tmp = t_0 * (1.0 / (x + y));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := Block[{t$95$0 = N[(x / N[(x + y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(y / N[(x + N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, 5.6e-230], N[(t$95$1 / N[(x + N[(y * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 7.2e-164], N[(y * N[(t$95$0 / N[(N[(x + y), $MachinePrecision] * N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2.4e+161], N[(t$95$1 * N[(x / N[(N[(x + y), $MachinePrecision] * N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(1.0 / N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if y < 5.6000000000000002e-230

   1. Initial program 65.9%

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

      1. times-frac 87.8%

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

      2. +-commutative 87.8%

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

      3. +-commutative 87.8%

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

      4. associate-/r/ 86.8%

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

      5. +-commutative 86.8%

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

      6. associate-/r/ 86.8%

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

      7. associate-*l/ 83.9%

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

      8. *-commutative 83.9%

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

      9. distribute-rgt1-in 55.7%

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

      10. fma-def 83.9%

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

      11. +-commutative 83.9%

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

      12. +-commutative 83.9%

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

      13. cube-unmult 83.9%

          \[\leadsto \frac{x}{\frac{\mathsf{fma}\left(x + y, x + y, \color{blue}{{\left(y + x\right)}^{3}}\right)}{y}} \]

      14. +-commutative 83.9%

          \[\leadsto \frac{x}{\frac{\mathsf{fma}\left(x + y, x + y, {\color{blue}{\left(x + y\right)}}^{3}\right)}{y}} \]

   3. Simplified 83.9%

      \[\leadsto \color{blue}{\frac{x}{\frac{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}{y}}} \]
   4. Step-by-step derivation

      1. associate-/l* 65.9%

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

      2. fma-udef 46.3%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right) + {\left(x + y\right)}^{3}}} \]

      3. cube-mult 46.3%

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

      4. distribute-rgt1-in 65.9%

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

      5. *-commutative 65.9%

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

      6. times-frac 87.8%

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

      7. associate-*l/ 82.3%

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

      8. times-frac 99.9%

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

      9. associate-+r+ 99.9%

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

   5. Applied egg-rr 99.9%

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

      1. clear-num 99.8%

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

      2. frac-times 99.8%

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

      3. *-un-lft-identity 99.8%

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

   7. Applied egg-rr 99.8%

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

   8. Taylor expanded in x around inf 58.3%

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

      1. *-commutative 58.3%

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

   10. Simplified 58.3%

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

   if 5.6000000000000002e-230 < y < 7.19999999999999988e-164

   1. Initial program 58.0%

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

      1. times-frac 70.1%

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

      2. /-rgt-identity 70.1%

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

      3. /-rgt-identity 70.1%

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

      4. associate-+l+ 70.1%

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

   3. Simplified 70.1%

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

   4. Taylor expanded in x around 0 63.7%

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

      1. +-commutative 63.7%

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

   6. Simplified 63.7%

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

      1. associate-/r* 93.3%

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

      2. clear-num 93.3%

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

      3. frac-times 93.3%

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

      4. *-commutative 93.3%

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

      5. *-un-lft-identity 93.3%

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

   8. Applied egg-rr 93.3%

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

      1. associate-*r/ 93.6%

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

      2. associate-/r/ 93.3%

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

   10. Simplified 93.3%

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

   if 7.19999999999999988e-164 < y < 2.3999999999999999e161

   1. Initial program 72.5%

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

      1. times-frac 96.3%

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

      2. /-rgt-identity 96.3%

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

      3. /-rgt-identity 96.3%

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

      4. associate-+l+ 96.3%

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

   3. Simplified 96.3%

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

   if 2.3999999999999999e161 < y

   1. Initial program 68.4%

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

      1. times-frac 82.9%

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

      2. +-commutative 82.9%

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

      3. +-commutative 82.9%

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

      4. associate-/r/ 82.9%

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

      5. +-commutative 82.9%

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

      6. associate-/r/ 82.9%

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

      7. associate-*l/ 82.9%

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

      8. *-commutative 82.9%

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

      9. distribute-rgt1-in 79.6%

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

      10. fma-def 82.9%

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

      11. +-commutative 82.9%

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

      12. +-commutative 82.9%

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

      13. cube-unmult 82.9%

          \[\leadsto \frac{x}{\frac{\mathsf{fma}\left(x + y, x + y, \color{blue}{{\left(y + x\right)}^{3}}\right)}{y}} \]

      14. +-commutative 82.9%

          \[\leadsto \frac{x}{\frac{\mathsf{fma}\left(x + y, x + y, {\color{blue}{\left(x + y\right)}}^{3}\right)}{y}} \]

   3. Simplified 82.9%

      \[\leadsto \color{blue}{\frac{x}{\frac{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}{y}}} \]
   4. Step-by-step derivation

      1. associate-/l* 68.4%

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

      2. fma-udef 68.4%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right) + {\left(x + y\right)}^{3}}} \]

      3. cube-mult 68.4%

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

      4. distribute-rgt1-in 68.4%

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

      5. *-commutative 68.4%

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

      6. times-frac 82.9%

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

      7. associate-*l/ 82.9%

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

      8. times-frac 99.9%

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

      9. associate-+r+ 99.9%

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

   5. Applied egg-rr 99.9%

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

   6. Taylor expanded in y around inf 88.2%

      \[\leadsto \frac{x}{x + y} \cdot \frac{\color{blue}{1}}{x + y} \]
3. Recombined 4 regimes into one program.

4. Final simplification 76.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 5.6 \cdot 10^{-230}:\\ \;\;\;\;\frac{\frac{y}{x + \left(y + 1\right)}}{x + y \cdot 2}\\ \mathbf{elif}\;y \leq 7.2 \cdot 10^{-164}:\\ \;\;\;\;y \cdot \frac{\frac{x}{x + y}}{\left(x + y\right) \cdot \left(y + 1\right)}\\ \mathbf{elif}\;y \leq 2.4 \cdot 10^{+161}:\\ \;\;\;\;\frac{y}{x + \left(y + 1\right)} \cdot \frac{x}{\left(x + y\right) \cdot \left(x + y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + y} \cdot \frac{1}{x + y}\\ \end{array} \]

Alternative 3: 90.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_0 := \frac{x}{\left(x + y\right) \cdot \left(x + y\right)}\\ \mathbf{if}\;x \leq -1.35 \cdot 10^{+157}:\\ \;\;\;\;\frac{\frac{y}{x + \left(y + 1\right)}}{x + y \cdot 2}\\ \mathbf{elif}\;x \leq -12.2:\\ \;\;\;\;t_0 \cdot \frac{y}{x}\\ \mathbf{elif}\;x \leq -2.85 \cdot 10^{-149}:\\ \;\;\;\;t_0 \cdot \frac{y}{y + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + y} \cdot \frac{1}{y + 1}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ x (* (+ x y) (+ x y)))))
   (if (<= x -1.35e+157)
     (/ (/ y (+ x (+ y 1.0))) (+ x (* y 2.0)))
     (if (<= x -12.2)
       (* t_0 (/ y x))
       (if (<= x -2.85e-149)
         (* t_0 (/ y (+ y 1.0)))
         (* (/ x (+ x y)) (/ 1.0 (+ y 1.0))))))))

C

assert(x < y);
double code(double x, double y) {
	double t_0 = x / ((x + y) * (x + y));
	double tmp;
	if (x <= -1.35e+157) {
		tmp = (y / (x + (y + 1.0))) / (x + (y * 2.0));
	} else if (x <= -12.2) {
		tmp = t_0 * (y / x);
	} else if (x <= -2.85e-149) {
		tmp = t_0 * (y / (y + 1.0));
	} else {
		tmp = (x / (x + y)) * (1.0 / (y + 1.0));
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x / ((x + y) * (x + y))
    if (x <= (-1.35d+157)) then
        tmp = (y / (x + (y + 1.0d0))) / (x + (y * 2.0d0))
    else if (x <= (-12.2d0)) then
        tmp = t_0 * (y / x)
    else if (x <= (-2.85d-149)) then
        tmp = t_0 * (y / (y + 1.0d0))
    else
        tmp = (x / (x + y)) * (1.0d0 / (y + 1.0d0))
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double t_0 = x / ((x + y) * (x + y));
	double tmp;
	if (x <= -1.35e+157) {
		tmp = (y / (x + (y + 1.0))) / (x + (y * 2.0));
	} else if (x <= -12.2) {
		tmp = t_0 * (y / x);
	} else if (x <= -2.85e-149) {
		tmp = t_0 * (y / (y + 1.0));
	} else {
		tmp = (x / (x + y)) * (1.0 / (y + 1.0));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	t_0 = x / ((x + y) * (x + y))
	tmp = 0
	if x <= -1.35e+157:
		tmp = (y / (x + (y + 1.0))) / (x + (y * 2.0))
	elif x <= -12.2:
		tmp = t_0 * (y / x)
	elif x <= -2.85e-149:
		tmp = t_0 * (y / (y + 1.0))
	else:
		tmp = (x / (x + y)) * (1.0 / (y + 1.0))
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	t_0 = Float64(x / Float64(Float64(x + y) * Float64(x + y)))
	tmp = 0.0
	if (x <= -1.35e+157)
		tmp = Float64(Float64(y / Float64(x + Float64(y + 1.0))) / Float64(x + Float64(y * 2.0)));
	elseif (x <= -12.2)
		tmp = Float64(t_0 * Float64(y / x));
	elseif (x <= -2.85e-149)
		tmp = Float64(t_0 * Float64(y / Float64(y + 1.0)));
	else
		tmp = Float64(Float64(x / Float64(x + y)) * Float64(1.0 / Float64(y + 1.0)));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	t_0 = x / ((x + y) * (x + y));
	tmp = 0.0;
	if (x <= -1.35e+157)
		tmp = (y / (x + (y + 1.0))) / (x + (y * 2.0));
	elseif (x <= -12.2)
		tmp = t_0 * (y / x);
	elseif (x <= -2.85e-149)
		tmp = t_0 * (y / (y + 1.0));
	else
		tmp = (x / (x + y)) * (1.0 / (y + 1.0));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := Block[{t$95$0 = N[(x / N[(N[(x + y), $MachinePrecision] * N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.35e+157], N[(N[(y / N[(x + N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x + N[(y * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -12.2], N[(t$95$0 * N[(y / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -2.85e-149], N[(t$95$0 * N[(y / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(x + y), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -1.35e157

   1. Initial program 62.2%

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

      1. times-frac 89.0%

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

      2. +-commutative 89.0%

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

      3. +-commutative 89.0%

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

      4. associate-/r/ 89.0%

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

      5. +-commutative 89.0%

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

      6. associate-/r/ 89.0%

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

      7. associate-*l/ 89.0%

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

      8. *-commutative 89.0%

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

      9. distribute-rgt1-in 2.8%

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

      10. fma-def 89.0%

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

      11. +-commutative 89.0%

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

      12. +-commutative 89.0%

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

      13. cube-unmult 89.0%

          \[\leadsto \frac{x}{\frac{\mathsf{fma}\left(x + y, x + y, \color{blue}{{\left(y + x\right)}^{3}}\right)}{y}} \]

      14. +-commutative 89.0%

          \[\leadsto \frac{x}{\frac{\mathsf{fma}\left(x + y, x + y, {\color{blue}{\left(x + y\right)}}^{3}\right)}{y}} \]

   3. Simplified 89.0%

      \[\leadsto \color{blue}{\frac{x}{\frac{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}{y}}} \]
   4. Step-by-step derivation

      1. associate-/l* 62.2%

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

      2. fma-udef 0.0%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right) + {\left(x + y\right)}^{3}}} \]

      3. cube-mult 0.0%

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

      4. distribute-rgt1-in 62.2%

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

      5. *-commutative 62.2%

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

      6. times-frac 89.0%

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

      7. associate-*l/ 89.0%

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

      8. times-frac 100.0%

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

      9. associate-+r+ 100.0%

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

   5. Applied egg-rr 100.0%

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

      1. clear-num 100.0%

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

      2. frac-times 99.9%

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

      3. *-un-lft-identity 99.9%

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

   7. Applied egg-rr 99.9%

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

   8. Taylor expanded in x around inf 89.2%

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

      1. *-commutative 89.2%

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

   10. Simplified 89.2%

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

   if -1.35e157 < x < -12.199999999999999

   1. Initial program 60.5%

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

      1. times-frac 90.8%

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

      2. /-rgt-identity 90.8%

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

      3. /-rgt-identity 90.8%

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

      4. associate-+l+ 90.8%

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

   3. Simplified 90.8%

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

   4. Taylor expanded in x around inf 80.4%

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

   if -12.199999999999999 < x < -2.8499999999999999e-149

   1. Initial program 95.5%

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

      1. times-frac 99.7%

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

      2. /-rgt-identity 99.7%

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

      3. /-rgt-identity 99.7%

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

      4. associate-+l+ 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in x around 0 99.1%

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

      1. +-commutative 99.1%

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

   6. Simplified 99.1%

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

   if -2.8499999999999999e-149 < x

   1. Initial program 66.8%

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

      1. times-frac 87.0%

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

      2. +-commutative 87.0%

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

      3. +-commutative 87.0%

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

      4. associate-/r/ 84.2%

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

      5. +-commutative 84.2%

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

      6. associate-/r/ 84.3%

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

      7. associate-*l/ 82.1%

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

      8. *-commutative 82.1%

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

      9. distribute-rgt1-in 75.1%

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

      10. fma-def 82.1%

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

      11. +-commutative 82.1%

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

      12. +-commutative 82.1%

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

      13. cube-unmult 82.1%

          \[\leadsto \frac{x}{\frac{\mathsf{fma}\left(x + y, x + y, \color{blue}{{\left(y + x\right)}^{3}}\right)}{y}} \]

      14. +-commutative 82.1%

          \[\leadsto \frac{x}{\frac{\mathsf{fma}\left(x + y, x + y, {\color{blue}{\left(x + y\right)}}^{3}\right)}{y}} \]

   3. Simplified 82.1%

      \[\leadsto \color{blue}{\frac{x}{\frac{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}{y}}} \]
   4. Step-by-step derivation

      1. associate-/l* 66.8%

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

      2. fma-udef 61.2%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right) + {\left(x + y\right)}^{3}}} \]

      3. cube-mult 61.1%

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

      4. distribute-rgt1-in 66.8%

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

      5. *-commutative 66.8%

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

      6. times-frac 87.0%

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

      7. associate-*l/ 77.4%

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

      8. times-frac 99.9%

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

      9. associate-+r+ 99.9%

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

   5. Applied egg-rr 99.9%

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

   6. Taylor expanded in x around 0 59.0%

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

      1. +-commutative 59.0%

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

   8. Simplified 59.0%

      \[\leadsto \frac{x}{x + y} \cdot \color{blue}{\frac{1}{y + 1}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 70.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.35 \cdot 10^{+157}:\\ \;\;\;\;\frac{\frac{y}{x + \left(y + 1\right)}}{x + y \cdot 2}\\ \mathbf{elif}\;x \leq -12.2:\\ \;\;\;\;\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{x}\\ \mathbf{elif}\;x \leq -2.85 \cdot 10^{-149}:\\ \;\;\;\;\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{y + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + y} \cdot \frac{1}{y + 1}\\ \end{array} \]

Alternative 4: 92.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_0 := \frac{x}{x + y}\\ \mathbf{if}\;x \leq -1.28 \cdot 10^{+157}:\\ \;\;\;\;\frac{\frac{y}{x + \left(y + 1\right)}}{x + y \cdot 2}\\ \mathbf{elif}\;x \leq -750:\\ \;\;\;\;\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{x}\\ \mathbf{elif}\;x \leq -4.3 \cdot 10^{-221}:\\ \;\;\;\;y \cdot \frac{t_0}{\left(x + y\right) \cdot \left(y + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \frac{1}{y + 1}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ x (+ x y))))
   (if (<= x -1.28e+157)
     (/ (/ y (+ x (+ y 1.0))) (+ x (* y 2.0)))
     (if (<= x -750.0)
       (* (/ x (* (+ x y) (+ x y))) (/ y x))
       (if (<= x -4.3e-221)
         (* y (/ t_0 (* (+ x y) (+ y 1.0))))
         (* t_0 (/ 1.0 (+ y 1.0))))))))

C

assert(x < y);
double code(double x, double y) {
	double t_0 = x / (x + y);
	double tmp;
	if (x <= -1.28e+157) {
		tmp = (y / (x + (y + 1.0))) / (x + (y * 2.0));
	} else if (x <= -750.0) {
		tmp = (x / ((x + y) * (x + y))) * (y / x);
	} else if (x <= -4.3e-221) {
		tmp = y * (t_0 / ((x + y) * (y + 1.0)));
	} else {
		tmp = t_0 * (1.0 / (y + 1.0));
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x / (x + y)
    if (x <= (-1.28d+157)) then
        tmp = (y / (x + (y + 1.0d0))) / (x + (y * 2.0d0))
    else if (x <= (-750.0d0)) then
        tmp = (x / ((x + y) * (x + y))) * (y / x)
    else if (x <= (-4.3d-221)) then
        tmp = y * (t_0 / ((x + y) * (y + 1.0d0)))
    else
        tmp = t_0 * (1.0d0 / (y + 1.0d0))
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double t_0 = x / (x + y);
	double tmp;
	if (x <= -1.28e+157) {
		tmp = (y / (x + (y + 1.0))) / (x + (y * 2.0));
	} else if (x <= -750.0) {
		tmp = (x / ((x + y) * (x + y))) * (y / x);
	} else if (x <= -4.3e-221) {
		tmp = y * (t_0 / ((x + y) * (y + 1.0)));
	} else {
		tmp = t_0 * (1.0 / (y + 1.0));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	t_0 = x / (x + y)
	tmp = 0
	if x <= -1.28e+157:
		tmp = (y / (x + (y + 1.0))) / (x + (y * 2.0))
	elif x <= -750.0:
		tmp = (x / ((x + y) * (x + y))) * (y / x)
	elif x <= -4.3e-221:
		tmp = y * (t_0 / ((x + y) * (y + 1.0)))
	else:
		tmp = t_0 * (1.0 / (y + 1.0))
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	t_0 = Float64(x / Float64(x + y))
	tmp = 0.0
	if (x <= -1.28e+157)
		tmp = Float64(Float64(y / Float64(x + Float64(y + 1.0))) / Float64(x + Float64(y * 2.0)));
	elseif (x <= -750.0)
		tmp = Float64(Float64(x / Float64(Float64(x + y) * Float64(x + y))) * Float64(y / x));
	elseif (x <= -4.3e-221)
		tmp = Float64(y * Float64(t_0 / Float64(Float64(x + y) * Float64(y + 1.0))));
	else
		tmp = Float64(t_0 * Float64(1.0 / Float64(y + 1.0)));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	t_0 = x / (x + y);
	tmp = 0.0;
	if (x <= -1.28e+157)
		tmp = (y / (x + (y + 1.0))) / (x + (y * 2.0));
	elseif (x <= -750.0)
		tmp = (x / ((x + y) * (x + y))) * (y / x);
	elseif (x <= -4.3e-221)
		tmp = y * (t_0 / ((x + y) * (y + 1.0)));
	else
		tmp = t_0 * (1.0 / (y + 1.0));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := Block[{t$95$0 = N[(x / N[(x + y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.28e+157], N[(N[(y / N[(x + N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x + N[(y * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -750.0], N[(N[(x / N[(N[(x + y), $MachinePrecision] * N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(y / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -4.3e-221], N[(y * N[(t$95$0 / N[(N[(x + y), $MachinePrecision] * N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(1.0 / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -1.28000000000000001e157

   1. Initial program 62.2%

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

      1. times-frac 89.0%

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

      2. +-commutative 89.0%

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

      3. +-commutative 89.0%

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

      4. associate-/r/ 89.0%

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

      5. +-commutative 89.0%

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

      6. associate-/r/ 89.0%

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

      7. associate-*l/ 89.0%

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

      8. *-commutative 89.0%

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

      9. distribute-rgt1-in 2.8%

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

      10. fma-def 89.0%

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

      11. +-commutative 89.0%

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

      12. +-commutative 89.0%

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

      13. cube-unmult 89.0%

          \[\leadsto \frac{x}{\frac{\mathsf{fma}\left(x + y, x + y, \color{blue}{{\left(y + x\right)}^{3}}\right)}{y}} \]

      14. +-commutative 89.0%

          \[\leadsto \frac{x}{\frac{\mathsf{fma}\left(x + y, x + y, {\color{blue}{\left(x + y\right)}}^{3}\right)}{y}} \]

   3. Simplified 89.0%

      \[\leadsto \color{blue}{\frac{x}{\frac{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}{y}}} \]
   4. Step-by-step derivation

      1. associate-/l* 62.2%

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

      2. fma-udef 0.0%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right) + {\left(x + y\right)}^{3}}} \]

      3. cube-mult 0.0%

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

      4. distribute-rgt1-in 62.2%

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

      5. *-commutative 62.2%

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

      6. times-frac 89.0%

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

      7. associate-*l/ 89.0%

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

      8. times-frac 100.0%

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

      9. associate-+r+ 100.0%

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

   5. Applied egg-rr 100.0%

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

      1. clear-num 100.0%

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

      2. frac-times 99.9%

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

      3. *-un-lft-identity 99.9%

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

   7. Applied egg-rr 99.9%

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

   8. Taylor expanded in x around inf 89.2%

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

      1. *-commutative 89.2%

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

   10. Simplified 89.2%

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

   if -1.28000000000000001e157 < x < -750

   1. Initial program 60.5%

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

      1. times-frac 90.8%

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

      2. /-rgt-identity 90.8%

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

      3. /-rgt-identity 90.8%

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

      4. associate-+l+ 90.8%

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

   3. Simplified 90.8%

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

   4. Taylor expanded in x around inf 80.4%

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

   if -750 < x < -4.2999999999999998e-221

   1. Initial program 96.5%

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

      1. times-frac 99.6%

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

      2. /-rgt-identity 99.6%

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

      3. /-rgt-identity 99.6%

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

      4. associate-+l+ 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in x around 0 99.2%

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

      1. +-commutative 99.2%

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

   6. Simplified 99.2%

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

      1. associate-/r* 99.1%

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

      2. clear-num 99.0%

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

      3. frac-times 99.1%

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

      4. *-commutative 99.1%

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

      5. *-un-lft-identity 99.1%

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

   8. Applied egg-rr 99.1%

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

      1. associate-*r/ 99.2%

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

      2. associate-/r/ 97.9%

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

   10. Simplified 97.9%

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

   if -4.2999999999999998e-221 < x

   1. Initial program 65.3%

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

      1. times-frac 86.4%

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

      2. +-commutative 86.4%

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

      3. +-commutative 86.4%

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

      4. associate-/r/ 83.5%

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

      5. +-commutative 83.5%

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

      6. associate-/r/ 83.5%

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

      7. associate-*l/ 81.3%

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

      8. *-commutative 81.3%

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

      9. distribute-rgt1-in 74.6%

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

      10. fma-def 81.3%

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

      11. +-commutative 81.3%

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

      12. +-commutative 81.3%

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

      13. cube-unmult 81.3%

          \[\leadsto \frac{x}{\frac{\mathsf{fma}\left(x + y, x + y, \color{blue}{{\left(y + x\right)}^{3}}\right)}{y}} \]

      14. +-commutative 81.3%

          \[\leadsto \frac{x}{\frac{\mathsf{fma}\left(x + y, x + y, {\color{blue}{\left(x + y\right)}}^{3}\right)}{y}} \]

   3. Simplified 81.3%

      \[\leadsto \color{blue}{\frac{x}{\frac{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}{y}}} \]
   4. Step-by-step derivation

      1. associate-/l* 65.3%

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

      2. fma-udef 60.0%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right) + {\left(x + y\right)}^{3}}} \]

      3. cube-mult 60.0%

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

      4. distribute-rgt1-in 65.3%

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

      5. *-commutative 65.3%

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

      6. times-frac 86.4%

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

      7. associate-*l/ 76.4%

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

      8. times-frac 99.9%

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

      9. associate-+r+ 99.9%

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

   5. Applied egg-rr 99.9%

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

   6. Taylor expanded in x around 0 57.1%

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

      1. +-commutative 57.1%

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

   8. Simplified 57.1%

      \[\leadsto \frac{x}{x + y} \cdot \color{blue}{\frac{1}{y + 1}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 69.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.28 \cdot 10^{+157}:\\ \;\;\;\;\frac{\frac{y}{x + \left(y + 1\right)}}{x + y \cdot 2}\\ \mathbf{elif}\;x \leq -750:\\ \;\;\;\;\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{x}\\ \mathbf{elif}\;x \leq -4.3 \cdot 10^{-221}:\\ \;\;\;\;y \cdot \frac{\frac{x}{x + y}}{\left(x + y\right) \cdot \left(y + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + y} \cdot \frac{1}{y + 1}\\ \end{array} \]

Alternative 5: 83.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_0 := \frac{\frac{y}{x + \left(y + 1\right)}}{x + y \cdot 2}\\ \mathbf{if}\;x \leq -1.28 \cdot 10^{+157}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq -720000:\\ \;\;\;\;\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{x}\\ \mathbf{elif}\;x \leq -2.4 \cdot 10^{-54}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y} \cdot \frac{x}{y + 1}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (/ y (+ x (+ y 1.0))) (+ x (* y 2.0)))))
   (if (<= x -1.28e+157)
     t_0
     (if (<= x -720000.0)
       (* (/ x (* (+ x y) (+ x y))) (/ y x))
       (if (<= x -2.4e-54) t_0 (* (/ 1.0 y) (/ x (+ y 1.0))))))))

C

assert(x < y);
double code(double x, double y) {
	double t_0 = (y / (x + (y + 1.0))) / (x + (y * 2.0));
	double tmp;
	if (x <= -1.28e+157) {
		tmp = t_0;
	} else if (x <= -720000.0) {
		tmp = (x / ((x + y) * (x + y))) * (y / x);
	} else if (x <= -2.4e-54) {
		tmp = t_0;
	} else {
		tmp = (1.0 / y) * (x / (y + 1.0));
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (y / (x + (y + 1.0d0))) / (x + (y * 2.0d0))
    if (x <= (-1.28d+157)) then
        tmp = t_0
    else if (x <= (-720000.0d0)) then
        tmp = (x / ((x + y) * (x + y))) * (y / x)
    else if (x <= (-2.4d-54)) then
        tmp = t_0
    else
        tmp = (1.0d0 / y) * (x / (y + 1.0d0))
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double t_0 = (y / (x + (y + 1.0))) / (x + (y * 2.0));
	double tmp;
	if (x <= -1.28e+157) {
		tmp = t_0;
	} else if (x <= -720000.0) {
		tmp = (x / ((x + y) * (x + y))) * (y / x);
	} else if (x <= -2.4e-54) {
		tmp = t_0;
	} else {
		tmp = (1.0 / y) * (x / (y + 1.0));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	t_0 = (y / (x + (y + 1.0))) / (x + (y * 2.0))
	tmp = 0
	if x <= -1.28e+157:
		tmp = t_0
	elif x <= -720000.0:
		tmp = (x / ((x + y) * (x + y))) * (y / x)
	elif x <= -2.4e-54:
		tmp = t_0
	else:
		tmp = (1.0 / y) * (x / (y + 1.0))
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	t_0 = Float64(Float64(y / Float64(x + Float64(y + 1.0))) / Float64(x + Float64(y * 2.0)))
	tmp = 0.0
	if (x <= -1.28e+157)
		tmp = t_0;
	elseif (x <= -720000.0)
		tmp = Float64(Float64(x / Float64(Float64(x + y) * Float64(x + y))) * Float64(y / x));
	elseif (x <= -2.4e-54)
		tmp = t_0;
	else
		tmp = Float64(Float64(1.0 / y) * Float64(x / Float64(y + 1.0)));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	t_0 = (y / (x + (y + 1.0))) / (x + (y * 2.0));
	tmp = 0.0;
	if (x <= -1.28e+157)
		tmp = t_0;
	elseif (x <= -720000.0)
		tmp = (x / ((x + y) * (x + y))) * (y / x);
	elseif (x <= -2.4e-54)
		tmp = t_0;
	else
		tmp = (1.0 / y) * (x / (y + 1.0));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := Block[{t$95$0 = N[(N[(y / N[(x + N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x + N[(y * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.28e+157], t$95$0, If[LessEqual[x, -720000.0], N[(N[(x / N[(N[(x + y), $MachinePrecision] * N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(y / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -2.4e-54], t$95$0, N[(N[(1.0 / y), $MachinePrecision] * N[(x / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.28000000000000001e157 or -7.2e5 < x < -2.40000000000000013e-54

   1. Initial program 69.3%

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

      1. times-frac 91.1%

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

      2. +-commutative 91.1%

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

      3. +-commutative 91.1%

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

      4. associate-/r/ 90.5%

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

      5. +-commutative 90.5%

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

      6. associate-/r/ 90.5%

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

      7. associate-*l/ 90.5%

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

      8. *-commutative 90.5%

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

      9. distribute-rgt1-in 18.3%

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

      10. fma-def 90.5%

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

      11. +-commutative 90.5%

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

      12. +-commutative 90.5%

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

      13. cube-unmult 90.5%

          \[\leadsto \frac{x}{\frac{\mathsf{fma}\left(x + y, x + y, \color{blue}{{\left(y + x\right)}^{3}}\right)}{y}} \]

      14. +-commutative 90.5%

          \[\leadsto \frac{x}{\frac{\mathsf{fma}\left(x + y, x + y, {\color{blue}{\left(x + y\right)}}^{3}\right)}{y}} \]

   3. Simplified 90.5%

      \[\leadsto \color{blue}{\frac{x}{\frac{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}{y}}} \]
   4. Step-by-step derivation

      1. associate-/l* 69.3%

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

      2. fma-udef 16.6%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right) + {\left(x + y\right)}^{3}}} \]

      3. cube-mult 16.6%

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

      4. distribute-rgt1-in 69.3%

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

      5. *-commutative 69.3%

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

      6. times-frac 91.1%

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

      7. associate-*l/ 91.1%

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

      8. times-frac 100.0%

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

      9. associate-+r+ 100.0%

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

   5. Applied egg-rr 100.0%

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

      1. clear-num 100.0%

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

      2. frac-times 99.9%

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

      3. *-un-lft-identity 99.9%

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

   7. Applied egg-rr 99.9%

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

   8. Taylor expanded in x around inf 86.9%

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

      1. *-commutative 86.9%

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

   10. Simplified 86.9%

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

   if -1.28000000000000001e157 < x < -7.2e5

   1. Initial program 57.2%

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

      1. times-frac 90.0%

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

      2. /-rgt-identity 90.0%

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

      3. /-rgt-identity 90.0%

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

      4. associate-+l+ 90.0%

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

   3. Simplified 90.0%

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

   4. Taylor expanded in x around inf 82.4%

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

   if -2.40000000000000013e-54 < x

   1. Initial program 69.7%

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

      1. times-frac 88.4%

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

      2. /-rgt-identity 88.4%

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

      3. /-rgt-identity 88.4%

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

      4. associate-+l+ 88.4%

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

   3. Simplified 88.4%

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

   4. Taylor expanded in x around 0 58.8%

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

      1. +-commutative 58.8%

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

   6. Simplified 58.8%

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

      1. *-un-lft-identity 58.8%

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

      2. times-frac 59.6%

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

   8. Applied egg-rr 59.6%

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

4. Final simplification 67.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.28 \cdot 10^{+157}:\\ \;\;\;\;\frac{\frac{y}{x + \left(y + 1\right)}}{x + y \cdot 2}\\ \mathbf{elif}\;x \leq -720000:\\ \;\;\;\;\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{x}\\ \mathbf{elif}\;x \leq -2.4 \cdot 10^{-54}:\\ \;\;\;\;\frac{\frac{y}{x + \left(y + 1\right)}}{x + y \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y} \cdot \frac{x}{y + 1}\\ \end{array} \]

Alternative 6: 83.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_0 := \frac{\frac{y}{x + \left(y + 1\right)}}{x}\\ \mathbf{if}\;x \leq -2.4 \cdot 10^{+142}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq -5000000:\\ \;\;\;\;\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{x}\\ \mathbf{elif}\;x \leq -1.6 \cdot 10^{-51}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y} \cdot \frac{x}{y + 1}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (/ y (+ x (+ y 1.0))) x)))
   (if (<= x -2.4e+142)
     t_0
     (if (<= x -5000000.0)
       (* (/ x (* (+ x y) (+ x y))) (/ y x))
       (if (<= x -1.6e-51) t_0 (* (/ 1.0 y) (/ x (+ y 1.0))))))))

C

assert(x < y);
double code(double x, double y) {
	double t_0 = (y / (x + (y + 1.0))) / x;
	double tmp;
	if (x <= -2.4e+142) {
		tmp = t_0;
	} else if (x <= -5000000.0) {
		tmp = (x / ((x + y) * (x + y))) * (y / x);
	} else if (x <= -1.6e-51) {
		tmp = t_0;
	} else {
		tmp = (1.0 / y) * (x / (y + 1.0));
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (y / (x + (y + 1.0d0))) / x
    if (x <= (-2.4d+142)) then
        tmp = t_0
    else if (x <= (-5000000.0d0)) then
        tmp = (x / ((x + y) * (x + y))) * (y / x)
    else if (x <= (-1.6d-51)) then
        tmp = t_0
    else
        tmp = (1.0d0 / y) * (x / (y + 1.0d0))
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double t_0 = (y / (x + (y + 1.0))) / x;
	double tmp;
	if (x <= -2.4e+142) {
		tmp = t_0;
	} else if (x <= -5000000.0) {
		tmp = (x / ((x + y) * (x + y))) * (y / x);
	} else if (x <= -1.6e-51) {
		tmp = t_0;
	} else {
		tmp = (1.0 / y) * (x / (y + 1.0));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	t_0 = (y / (x + (y + 1.0))) / x
	tmp = 0
	if x <= -2.4e+142:
		tmp = t_0
	elif x <= -5000000.0:
		tmp = (x / ((x + y) * (x + y))) * (y / x)
	elif x <= -1.6e-51:
		tmp = t_0
	else:
		tmp = (1.0 / y) * (x / (y + 1.0))
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	t_0 = Float64(Float64(y / Float64(x + Float64(y + 1.0))) / x)
	tmp = 0.0
	if (x <= -2.4e+142)
		tmp = t_0;
	elseif (x <= -5000000.0)
		tmp = Float64(Float64(x / Float64(Float64(x + y) * Float64(x + y))) * Float64(y / x));
	elseif (x <= -1.6e-51)
		tmp = t_0;
	else
		tmp = Float64(Float64(1.0 / y) * Float64(x / Float64(y + 1.0)));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	t_0 = (y / (x + (y + 1.0))) / x;
	tmp = 0.0;
	if (x <= -2.4e+142)
		tmp = t_0;
	elseif (x <= -5000000.0)
		tmp = (x / ((x + y) * (x + y))) * (y / x);
	elseif (x <= -1.6e-51)
		tmp = t_0;
	else
		tmp = (1.0 / y) * (x / (y + 1.0));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := Block[{t$95$0 = N[(N[(y / N[(x + N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]}, If[LessEqual[x, -2.4e+142], t$95$0, If[LessEqual[x, -5000000.0], N[(N[(x / N[(N[(x + y), $MachinePrecision] * N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(y / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -1.6e-51], t$95$0, N[(N[(1.0 / y), $MachinePrecision] * N[(x / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.3999999999999999e142 or -5e6 < x < -1.6e-51

   1. Initial program 70.5%

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

      1. times-frac 92.0%

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

      2. /-rgt-identity 92.0%

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

      3. /-rgt-identity 92.0%

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

      4. associate-+l+ 92.0%

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

   3. Simplified 92.0%

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

   4. Taylor expanded in x around inf 87.8%

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

      1. associate-*l/ 87.9%

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

      2. *-un-lft-identity 87.9%

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

   6. Applied egg-rr 87.9%

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

   if -2.3999999999999999e142 < x < -5e6

   1. Initial program 53.3%

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

      1. times-frac 88.5%

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

      2. /-rgt-identity 88.5%

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

      3. /-rgt-identity 88.5%

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

      4. associate-+l+ 88.5%

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

   3. Simplified 88.5%

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

   4. Taylor expanded in x around inf 79.6%

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

   if -1.6e-51 < x

   1. Initial program 69.7%

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

      1. times-frac 88.4%

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

      2. /-rgt-identity 88.4%

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

      3. /-rgt-identity 88.4%

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

      4. associate-+l+ 88.4%

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

   3. Simplified 88.4%

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

   4. Taylor expanded in x around 0 58.8%

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

      1. +-commutative 58.8%

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

   6. Simplified 58.8%

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

      1. *-un-lft-identity 58.8%

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

      2. times-frac 59.6%

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

   8. Applied egg-rr 59.6%

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

4. Final simplification 67.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.4 \cdot 10^{+142}:\\ \;\;\;\;\frac{\frac{y}{x + \left(y + 1\right)}}{x}\\ \mathbf{elif}\;x \leq -5000000:\\ \;\;\;\;\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{x}\\ \mathbf{elif}\;x \leq -1.6 \cdot 10^{-51}:\\ \;\;\;\;\frac{\frac{y}{x + \left(y + 1\right)}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y} \cdot \frac{x}{y + 1}\\ \end{array} \]

Alternative 7: 83.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -1.28 \cdot 10^{+157}:\\ \;\;\;\;\frac{x}{x + y} \cdot \frac{\frac{y}{x}}{x + y}\\ \mathbf{elif}\;x \leq -6000000:\\ \;\;\;\;\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{x}\\ \mathbf{elif}\;x \leq -1.45 \cdot 10^{-50}:\\ \;\;\;\;\frac{\frac{y}{x + \left(y + 1\right)}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y} \cdot \frac{x}{y + 1}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -1.28e+157)
   (* (/ x (+ x y)) (/ (/ y x) (+ x y)))
   (if (<= x -6000000.0)
     (* (/ x (* (+ x y) (+ x y))) (/ y x))
     (if (<= x -1.45e-50)
       (/ (/ y (+ x (+ y 1.0))) x)
       (* (/ 1.0 y) (/ x (+ y 1.0)))))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -1.28e+157) {
		tmp = (x / (x + y)) * ((y / x) / (x + y));
	} else if (x <= -6000000.0) {
		tmp = (x / ((x + y) * (x + y))) * (y / x);
	} else if (x <= -1.45e-50) {
		tmp = (y / (x + (y + 1.0))) / x;
	} else {
		tmp = (1.0 / y) * (x / (y + 1.0));
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-1.28d+157)) then
        tmp = (x / (x + y)) * ((y / x) / (x + y))
    else if (x <= (-6000000.0d0)) then
        tmp = (x / ((x + y) * (x + y))) * (y / x)
    else if (x <= (-1.45d-50)) then
        tmp = (y / (x + (y + 1.0d0))) / x
    else
        tmp = (1.0d0 / y) * (x / (y + 1.0d0))
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (x <= -1.28e+157) {
		tmp = (x / (x + y)) * ((y / x) / (x + y));
	} else if (x <= -6000000.0) {
		tmp = (x / ((x + y) * (x + y))) * (y / x);
	} else if (x <= -1.45e-50) {
		tmp = (y / (x + (y + 1.0))) / x;
	} else {
		tmp = (1.0 / y) * (x / (y + 1.0));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -1.28e+157:
		tmp = (x / (x + y)) * ((y / x) / (x + y))
	elif x <= -6000000.0:
		tmp = (x / ((x + y) * (x + y))) * (y / x)
	elif x <= -1.45e-50:
		tmp = (y / (x + (y + 1.0))) / x
	else:
		tmp = (1.0 / y) * (x / (y + 1.0))
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -1.28e+157)
		tmp = Float64(Float64(x / Float64(x + y)) * Float64(Float64(y / x) / Float64(x + y)));
	elseif (x <= -6000000.0)
		tmp = Float64(Float64(x / Float64(Float64(x + y) * Float64(x + y))) * Float64(y / x));
	elseif (x <= -1.45e-50)
		tmp = Float64(Float64(y / Float64(x + Float64(y + 1.0))) / x);
	else
		tmp = Float64(Float64(1.0 / y) * Float64(x / Float64(y + 1.0)));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1.28e+157)
		tmp = (x / (x + y)) * ((y / x) / (x + y));
	elseif (x <= -6000000.0)
		tmp = (x / ((x + y) * (x + y))) * (y / x);
	elseif (x <= -1.45e-50)
		tmp = (y / (x + (y + 1.0))) / x;
	else
		tmp = (1.0 / y) * (x / (y + 1.0));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[x, -1.28e+157], N[(N[(x / N[(x + y), $MachinePrecision]), $MachinePrecision] * N[(N[(y / x), $MachinePrecision] / N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -6000000.0], N[(N[(x / N[(N[(x + y), $MachinePrecision] * N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(y / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -1.45e-50], N[(N[(y / N[(x + N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], N[(N[(1.0 / y), $MachinePrecision] * N[(x / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < -1.28000000000000001e157

   1. Initial program 62.2%

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

      1. times-frac 89.0%

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

      2. +-commutative 89.0%

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

      3. +-commutative 89.0%

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

      4. associate-/r/ 89.0%

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

      5. +-commutative 89.0%

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

      6. associate-/r/ 89.0%

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

      7. associate-*l/ 89.0%

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

      8. *-commutative 89.0%

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

      9. distribute-rgt1-in 2.8%

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

      10. fma-def 89.0%

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

      11. +-commutative 89.0%

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

      12. +-commutative 89.0%

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

      13. cube-unmult 89.0%

          \[\leadsto \frac{x}{\frac{\mathsf{fma}\left(x + y, x + y, \color{blue}{{\left(y + x\right)}^{3}}\right)}{y}} \]

      14. +-commutative 89.0%

          \[\leadsto \frac{x}{\frac{\mathsf{fma}\left(x + y, x + y, {\color{blue}{\left(x + y\right)}}^{3}\right)}{y}} \]

   3. Simplified 89.0%

      \[\leadsto \color{blue}{\frac{x}{\frac{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}{y}}} \]
   4. Step-by-step derivation

      1. associate-/l* 62.2%

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

      2. fma-udef 0.0%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right) + {\left(x + y\right)}^{3}}} \]

      3. cube-mult 0.0%

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

      4. distribute-rgt1-in 62.2%

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

      5. *-commutative 62.2%

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

      6. times-frac 89.0%

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

      7. associate-*l/ 89.0%

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

      8. times-frac 100.0%

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

      9. associate-+r+ 100.0%

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

   5. Applied egg-rr 100.0%

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

   6. Taylor expanded in x around inf 89.2%

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

   if -1.28000000000000001e157 < x < -6e6

   1. Initial program 57.2%

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

      1. times-frac 90.0%

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

      2. /-rgt-identity 90.0%

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

      3. /-rgt-identity 90.0%

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

      4. associate-+l+ 90.0%

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

   3. Simplified 90.0%

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

   4. Taylor expanded in x around inf 82.4%

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

   if -6e6 < x < -1.45000000000000004e-50

   1. Initial program 99.6%

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

      1. times-frac 99.6%

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

      2. /-rgt-identity 99.6%

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

      3. /-rgt-identity 99.6%

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

      4. associate-+l+ 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in x around inf 75.5%

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

      1. associate-*l/ 75.7%

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

      2. *-un-lft-identity 75.7%

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

   6. Applied egg-rr 75.7%

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

   if -1.45000000000000004e-50 < x

   1. Initial program 69.7%

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

      1. times-frac 88.4%

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

      2. /-rgt-identity 88.4%

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

      3. /-rgt-identity 88.4%

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

      4. associate-+l+ 88.4%

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

   3. Simplified 88.4%

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

   4. Taylor expanded in x around 0 58.8%

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

      1. +-commutative 58.8%

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

   6. Simplified 58.8%

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

      1. *-un-lft-identity 58.8%

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

      2. times-frac 59.6%

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

   8. Applied egg-rr 59.6%

      \[\leadsto \color{blue}{\frac{1}{y} \cdot \frac{x}{y + 1}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 67.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.28 \cdot 10^{+157}:\\ \;\;\;\;\frac{x}{x + y} \cdot \frac{\frac{y}{x}}{x + y}\\ \mathbf{elif}\;x \leq -6000000:\\ \;\;\;\;\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{x}\\ \mathbf{elif}\;x \leq -1.45 \cdot 10^{-50}:\\ \;\;\;\;\frac{\frac{y}{x + \left(y + 1\right)}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y} \cdot \frac{x}{y + 1}\\ \end{array} \]

Alternative 8: 79.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{\frac{y}{x}}{x}\\ \mathbf{elif}\;x \leq -5.2 \cdot 10^{-50}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -1.0)
   (/ (/ y x) x)
   (if (<= x -5.2e-50) (/ y x) (/ x (* y (+ y 1.0))))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -1.0) {
		tmp = (y / x) / x;
	} else if (x <= -5.2e-50) {
		tmp = y / x;
	} else {
		tmp = x / (y * (y + 1.0));
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-1.0d0)) then
        tmp = (y / x) / x
    else if (x <= (-5.2d-50)) then
        tmp = y / x
    else
        tmp = x / (y * (y + 1.0d0))
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (x <= -1.0) {
		tmp = (y / x) / x;
	} else if (x <= -5.2e-50) {
		tmp = y / x;
	} else {
		tmp = x / (y * (y + 1.0));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -1.0:
		tmp = (y / x) / x
	elif x <= -5.2e-50:
		tmp = y / x
	else:
		tmp = x / (y * (y + 1.0))
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -1.0)
		tmp = Float64(Float64(y / x) / x);
	elseif (x <= -5.2e-50)
		tmp = Float64(y / x);
	else
		tmp = Float64(x / Float64(y * Float64(y + 1.0)));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1.0)
		tmp = (y / x) / x;
	elseif (x <= -5.2e-50)
		tmp = y / x;
	else
		tmp = x / (y * (y + 1.0));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[x, -1.0], N[(N[(y / x), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[x, -5.2e-50], N[(y / x), $MachinePrecision], N[(x / N[(y * N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1

   1. Initial program 61.3%

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

      1. times-frac 90.0%

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

      2. /-rgt-identity 90.0%

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

      3. /-rgt-identity 90.0%

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

      4. associate-+l+ 90.0%

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

   3. Simplified 90.0%

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

   4. Taylor expanded in x around inf 75.6%

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

      1. associate-*l/ 75.6%

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

      2. *-un-lft-identity 75.6%

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

   6. Applied egg-rr 75.6%

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

   7. Taylor expanded in x around inf 73.1%

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

   if -1 < x < -5.2000000000000003e-50

   1. Initial program 99.4%

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

      1. times-frac 99.7%

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

      2. /-rgt-identity 99.7%

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

      3. /-rgt-identity 99.7%

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

      4. associate-+l+ 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in x around 0 96.8%

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

      1. +-commutative 96.8%

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

   6. Simplified 96.8%

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

   7. Taylor expanded in y around 0 57.8%

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

   if -5.2000000000000003e-50 < x

   1. Initial program 69.7%

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

      1. times-frac 88.4%

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

      2. /-rgt-identity 88.4%

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

      3. /-rgt-identity 88.4%

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

      4. associate-+l+ 88.4%

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

   3. Simplified 88.4%

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

   4. Taylor expanded in x around 0 58.8%

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

      1. +-commutative 58.8%

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

   6. Simplified 58.8%

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

4. Final simplification 62.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{\frac{y}{x}}{x}\\ \mathbf{elif}\;x \leq -5.2 \cdot 10^{-50}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \end{array} \]

Alternative 9: 80.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{+59}:\\ \;\;\;\;\frac{\frac{y}{x}}{x}\\ \mathbf{elif}\;x \leq -5.2 \cdot 10^{-50}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -2e+59)
   (/ (/ y x) x)
   (if (<= x -5.2e-50) (/ y (* x (+ x 1.0))) (/ x (* y (+ y 1.0))))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -2e+59) {
		tmp = (y / x) / x;
	} else if (x <= -5.2e-50) {
		tmp = y / (x * (x + 1.0));
	} else {
		tmp = x / (y * (y + 1.0));
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-2d+59)) then
        tmp = (y / x) / x
    else if (x <= (-5.2d-50)) then
        tmp = y / (x * (x + 1.0d0))
    else
        tmp = x / (y * (y + 1.0d0))
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (x <= -2e+59) {
		tmp = (y / x) / x;
	} else if (x <= -5.2e-50) {
		tmp = y / (x * (x + 1.0));
	} else {
		tmp = x / (y * (y + 1.0));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -2e+59:
		tmp = (y / x) / x
	elif x <= -5.2e-50:
		tmp = y / (x * (x + 1.0))
	else:
		tmp = x / (y * (y + 1.0))
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -2e+59)
		tmp = Float64(Float64(y / x) / x);
	elseif (x <= -5.2e-50)
		tmp = Float64(y / Float64(x * Float64(x + 1.0)));
	else
		tmp = Float64(x / Float64(y * Float64(y + 1.0)));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -2e+59)
		tmp = (y / x) / x;
	elseif (x <= -5.2e-50)
		tmp = y / (x * (x + 1.0));
	else
		tmp = x / (y * (y + 1.0));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[x, -2e+59], N[(N[(y / x), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[x, -5.2e-50], N[(y / N[(x * N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x / N[(y * N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.99999999999999994e59

   1. Initial program 60.1%

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

      1. times-frac 89.5%

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

      2. /-rgt-identity 89.5%

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

      3. /-rgt-identity 89.5%

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

      4. associate-+l+ 89.5%

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

   3. Simplified 89.5%

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

   4. Taylor expanded in x around inf 80.2%

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

      1. associate-*l/ 80.3%

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

      2. *-un-lft-identity 80.3%

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

   6. Applied egg-rr 80.3%

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

   7. Taylor expanded in x around inf 79.9%

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

   if -1.99999999999999994e59 < x < -5.2000000000000003e-50

   1. Initial program 76.8%

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

      1. times-frac 94.5%

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

      2. /-rgt-identity 94.5%

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

      3. /-rgt-identity 94.5%

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

      4. associate-+l+ 94.5%

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

   3. Simplified 94.5%

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

   4. Taylor expanded in y around 0 53.7%

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

   if -5.2000000000000003e-50 < x

   1. Initial program 69.7%

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

      1. times-frac 88.4%

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

      2. /-rgt-identity 88.4%

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

      3. /-rgt-identity 88.4%

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

      4. associate-+l+ 88.4%

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

   3. Simplified 88.4%

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

   4. Taylor expanded in x around 0 58.8%

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

      1. +-commutative 58.8%

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

   6. Simplified 58.8%

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

4. Final simplification 63.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{+59}:\\ \;\;\;\;\frac{\frac{y}{x}}{x}\\ \mathbf{elif}\;x \leq -5.2 \cdot 10^{-50}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \end{array} \]

Alternative 10: 81.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{+65}:\\ \;\;\;\;\frac{\frac{y}{x}}{x}\\ \mathbf{elif}\;x \leq -1.8 \cdot 10^{-50}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + 1}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -1e+65)
   (/ (/ y x) x)
   (if (<= x -1.8e-50) (/ y (* x (+ x 1.0))) (/ (/ x y) (+ y 1.0)))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -1e+65) {
		tmp = (y / x) / x;
	} else if (x <= -1.8e-50) {
		tmp = y / (x * (x + 1.0));
	} else {
		tmp = (x / y) / (y + 1.0);
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-1d+65)) then
        tmp = (y / x) / x
    else if (x <= (-1.8d-50)) then
        tmp = y / (x * (x + 1.0d0))
    else
        tmp = (x / y) / (y + 1.0d0)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (x <= -1e+65) {
		tmp = (y / x) / x;
	} else if (x <= -1.8e-50) {
		tmp = y / (x * (x + 1.0));
	} else {
		tmp = (x / y) / (y + 1.0);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -1e+65:
		tmp = (y / x) / x
	elif x <= -1.8e-50:
		tmp = y / (x * (x + 1.0))
	else:
		tmp = (x / y) / (y + 1.0)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -1e+65)
		tmp = Float64(Float64(y / x) / x);
	elseif (x <= -1.8e-50)
		tmp = Float64(y / Float64(x * Float64(x + 1.0)));
	else
		tmp = Float64(Float64(x / y) / Float64(y + 1.0));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1e+65)
		tmp = (y / x) / x;
	elseif (x <= -1.8e-50)
		tmp = y / (x * (x + 1.0));
	else
		tmp = (x / y) / (y + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[x, -1e+65], N[(N[(y / x), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[x, -1.8e-50], N[(y / N[(x * N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -9.9999999999999999e64

   1. Initial program 60.1%

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

      1. times-frac 89.5%

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

      2. /-rgt-identity 89.5%

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

      3. /-rgt-identity 89.5%

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

      4. associate-+l+ 89.5%

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

   3. Simplified 89.5%

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

   4. Taylor expanded in x around inf 80.2%

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

      1. associate-*l/ 80.3%

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

      2. *-un-lft-identity 80.3%

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

   6. Applied egg-rr 80.3%

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

   7. Taylor expanded in x around inf 79.9%

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

   if -9.9999999999999999e64 < x < -1.7999999999999999e-50

   1. Initial program 76.8%

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

      1. times-frac 94.5%

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

      2. /-rgt-identity 94.5%

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

      3. /-rgt-identity 94.5%

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

      4. associate-+l+ 94.5%

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

   3. Simplified 94.5%

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

   4. Taylor expanded in y around 0 53.7%

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

   if -1.7999999999999999e-50 < x

   1. Initial program 69.7%

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

      1. times-frac 88.4%

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

      2. +-commutative 88.4%

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

      3. +-commutative 88.4%

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

      4. associate-/r/ 85.9%

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

      5. +-commutative 85.9%

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

      6. associate-/r/ 85.9%

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

      7. associate-*l/ 84.0%

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

      8. *-commutative 84.0%

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

      9. distribute-rgt1-in 76.1%

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

      10. fma-def 84.0%

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

      11. +-commutative 84.0%

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

      12. +-commutative 84.0%

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

      13. cube-unmult 84.0%

          \[\leadsto \frac{x}{\frac{\mathsf{fma}\left(x + y, x + y, \color{blue}{{\left(y + x\right)}^{3}}\right)}{y}} \]

      14. +-commutative 84.0%

          \[\leadsto \frac{x}{\frac{\mathsf{fma}\left(x + y, x + y, {\color{blue}{\left(x + y\right)}}^{3}\right)}{y}} \]

   3. Simplified 84.0%

      \[\leadsto \color{blue}{\frac{x}{\frac{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}{y}}} \]
   4. Step-by-step derivation

      1. associate-/l* 69.8%

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

      2. fma-udef 63.0%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right) + {\left(x + y\right)}^{3}}} \]

      3. cube-mult 63.0%

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

      4. distribute-rgt1-in 69.7%

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

      5. *-commutative 69.7%

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

      6. times-frac 88.4%

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

      7. associate-*l/ 79.2%

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

      8. times-frac 99.9%

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

      9. associate-+r+ 99.9%

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

   5. Applied egg-rr 99.9%

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

      1. clear-num 99.8%

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

      2. associate-*l/ 99.8%

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

      3. *-un-lft-identity 99.8%

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

      4. associate-/l/ 95.5%

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

   7. Applied egg-rr 95.5%

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

   8. Taylor expanded in x around 0 58.8%

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

      1. +-commutative 58.8%

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

      2. associate-/r* 59.6%

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

   10. Simplified 59.6%

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

4. Final simplification 64.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{+65}:\\ \;\;\;\;\frac{\frac{y}{x}}{x}\\ \mathbf{elif}\;x \leq -1.8 \cdot 10^{-50}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + 1}\\ \end{array} \]

Alternative 11: 81.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -8.5 \cdot 10^{-52}:\\ \;\;\;\;\frac{\frac{y}{x + 1}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y} \cdot \frac{x}{y + 1}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -8.5e-52) (/ (/ y (+ x 1.0)) x) (* (/ 1.0 y) (/ x (+ y 1.0)))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -8.5e-52) {
		tmp = (y / (x + 1.0)) / x;
	} else {
		tmp = (1.0 / y) * (x / (y + 1.0));
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-8.5d-52)) then
        tmp = (y / (x + 1.0d0)) / x
    else
        tmp = (1.0d0 / y) * (x / (y + 1.0d0))
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (x <= -8.5e-52) {
		tmp = (y / (x + 1.0)) / x;
	} else {
		tmp = (1.0 / y) * (x / (y + 1.0));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -8.5e-52:
		tmp = (y / (x + 1.0)) / x
	else:
		tmp = (1.0 / y) * (x / (y + 1.0))
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -8.5e-52)
		tmp = Float64(Float64(y / Float64(x + 1.0)) / x);
	else
		tmp = Float64(Float64(1.0 / y) * Float64(x / Float64(y + 1.0)));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -8.5e-52)
		tmp = (y / (x + 1.0)) / x;
	else
		tmp = (1.0 / y) * (x / (y + 1.0));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[x, -8.5e-52], N[(N[(y / N[(x + 1.0), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], N[(N[(1.0 / y), $MachinePrecision] * N[(x / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -8.50000000000000006e-52

   1. Initial program 63.7%

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

      1. times-frac 90.6%

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

      2. /-rgt-identity 90.6%

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

      3. /-rgt-identity 90.6%

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

      4. associate-+l+ 90.6%

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

   3. Simplified 90.6%

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

   4. Taylor expanded in x around inf 74.6%

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

      1. associate-*l/ 74.7%

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

      2. *-un-lft-identity 74.7%

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

   6. Applied egg-rr 74.7%

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

   7. Taylor expanded in y around 0 74.2%

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

      1. +-commutative 74.2%

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

   9. Simplified 74.2%

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

   if -8.50000000000000006e-52 < x

   1. Initial program 69.7%

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

      1. times-frac 88.4%

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

      2. /-rgt-identity 88.4%

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

      3. /-rgt-identity 88.4%

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

      4. associate-+l+ 88.4%

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

   3. Simplified 88.4%

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

   4. Taylor expanded in x around 0 58.8%

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

      1. +-commutative 58.8%

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

   6. Simplified 58.8%

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

      1. *-un-lft-identity 58.8%

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

      2. times-frac 59.6%

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

   8. Applied egg-rr 59.6%

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

4. Final simplification 64.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -8.5 \cdot 10^{-52}:\\ \;\;\;\;\frac{\frac{y}{x + 1}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y} \cdot \frac{x}{y + 1}\\ \end{array} \]

Alternative 12: 81.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -5.2 \cdot 10^{-50}:\\ \;\;\;\;\frac{\frac{y}{x + \left(y + 1\right)}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y} \cdot \frac{x}{y + 1}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -5.2e-50)
   (/ (/ y (+ x (+ y 1.0))) x)
   (* (/ 1.0 y) (/ x (+ y 1.0)))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -5.2e-50) {
		tmp = (y / (x + (y + 1.0))) / x;
	} else {
		tmp = (1.0 / y) * (x / (y + 1.0));
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-5.2d-50)) then
        tmp = (y / (x + (y + 1.0d0))) / x
    else
        tmp = (1.0d0 / y) * (x / (y + 1.0d0))
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (x <= -5.2e-50) {
		tmp = (y / (x + (y + 1.0))) / x;
	} else {
		tmp = (1.0 / y) * (x / (y + 1.0));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -5.2e-50:
		tmp = (y / (x + (y + 1.0))) / x
	else:
		tmp = (1.0 / y) * (x / (y + 1.0))
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -5.2e-50)
		tmp = Float64(Float64(y / Float64(x + Float64(y + 1.0))) / x);
	else
		tmp = Float64(Float64(1.0 / y) * Float64(x / Float64(y + 1.0)));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -5.2e-50)
		tmp = (y / (x + (y + 1.0))) / x;
	else
		tmp = (1.0 / y) * (x / (y + 1.0));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[x, -5.2e-50], N[(N[(y / N[(x + N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], N[(N[(1.0 / y), $MachinePrecision] * N[(x / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -5.2000000000000003e-50

   1. Initial program 63.7%

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

      1. times-frac 90.6%

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

      2. /-rgt-identity 90.6%

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

      3. /-rgt-identity 90.6%

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

      4. associate-+l+ 90.6%

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

   3. Simplified 90.6%

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

   4. Taylor expanded in x around inf 74.6%

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

      1. associate-*l/ 74.7%

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

      2. *-un-lft-identity 74.7%

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

   6. Applied egg-rr 74.7%

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

   if -5.2000000000000003e-50 < x

   1. Initial program 69.7%

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

      1. times-frac 88.4%

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

      2. /-rgt-identity 88.4%

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

      3. /-rgt-identity 88.4%

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

      4. associate-+l+ 88.4%

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

   3. Simplified 88.4%

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

   4. Taylor expanded in x around 0 58.8%

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

      1. +-commutative 58.8%

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

   6. Simplified 58.8%

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

      1. *-un-lft-identity 58.8%

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

      2. times-frac 59.6%

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

   8. Applied egg-rr 59.6%

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

4. Final simplification 64.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.2 \cdot 10^{-50}:\\ \;\;\;\;\frac{\frac{y}{x + \left(y + 1\right)}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y} \cdot \frac{x}{y + 1}\\ \end{array} \]

Alternative 13: 81.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -3.1 \cdot 10^{-50}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + 1}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -3.1e-50) (/ (/ y x) (+ x 1.0)) (/ (/ x y) (+ y 1.0))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -3.1e-50) {
		tmp = (y / x) / (x + 1.0);
	} else {
		tmp = (x / y) / (y + 1.0);
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-3.1d-50)) then
        tmp = (y / x) / (x + 1.0d0)
    else
        tmp = (x / y) / (y + 1.0d0)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (x <= -3.1e-50) {
		tmp = (y / x) / (x + 1.0);
	} else {
		tmp = (x / y) / (y + 1.0);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -3.1e-50:
		tmp = (y / x) / (x + 1.0)
	else:
		tmp = (x / y) / (y + 1.0)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -3.1e-50)
		tmp = Float64(Float64(y / x) / Float64(x + 1.0));
	else
		tmp = Float64(Float64(x / y) / Float64(y + 1.0));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -3.1e-50)
		tmp = (y / x) / (x + 1.0);
	else
		tmp = (x / y) / (y + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[x, -3.1e-50], N[(N[(y / x), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -3.1000000000000002e-50

   1. Initial program 63.7%

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

      1. times-frac 90.6%

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

      2. /-rgt-identity 90.6%

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

      3. /-rgt-identity 90.6%

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

      4. associate-+l+ 90.6%

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

   3. Simplified 90.6%

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

   4. Taylor expanded in y around 0 74.2%

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

      1. associate-/r* 74.1%

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

      2. +-commutative 74.1%

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

   6. Simplified 74.1%

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

   if -3.1000000000000002e-50 < x

   1. Initial program 69.7%

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

      1. times-frac 88.4%

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

      2. +-commutative 88.4%

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

      3. +-commutative 88.4%

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

      4. associate-/r/ 85.9%

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

      5. +-commutative 85.9%

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

      6. associate-/r/ 85.9%

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

      7. associate-*l/ 84.0%

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

      8. *-commutative 84.0%

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

      9. distribute-rgt1-in 76.1%

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

      10. fma-def 84.0%

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

      11. +-commutative 84.0%

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

      12. +-commutative 84.0%

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

      13. cube-unmult 84.0%

          \[\leadsto \frac{x}{\frac{\mathsf{fma}\left(x + y, x + y, \color{blue}{{\left(y + x\right)}^{3}}\right)}{y}} \]

      14. +-commutative 84.0%

          \[\leadsto \frac{x}{\frac{\mathsf{fma}\left(x + y, x + y, {\color{blue}{\left(x + y\right)}}^{3}\right)}{y}} \]

   3. Simplified 84.0%

      \[\leadsto \color{blue}{\frac{x}{\frac{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}{y}}} \]
   4. Step-by-step derivation

      1. associate-/l* 69.8%

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

      2. fma-udef 63.0%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right) + {\left(x + y\right)}^{3}}} \]

      3. cube-mult 63.0%

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

      4. distribute-rgt1-in 69.7%

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

      5. *-commutative 69.7%

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

      6. times-frac 88.4%

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

      7. associate-*l/ 79.2%

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

      8. times-frac 99.9%

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

      9. associate-+r+ 99.9%

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

   5. Applied egg-rr 99.9%

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

      1. clear-num 99.8%

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

      2. associate-*l/ 99.8%

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

      3. *-un-lft-identity 99.8%

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

      4. associate-/l/ 95.5%

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

   7. Applied egg-rr 95.5%

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

   8. Taylor expanded in x around 0 58.8%

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

      1. +-commutative 58.8%

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

      2. associate-/r* 59.6%

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

   10. Simplified 59.6%

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

4. Final simplification 64.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.1 \cdot 10^{-50}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + 1}\\ \end{array} \]

Alternative 14: 81.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -1.35 \cdot 10^{-51}:\\ \;\;\;\;\frac{\frac{y}{x + 1}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + 1}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -1.35e-51) (/ (/ y (+ x 1.0)) x) (/ (/ x y) (+ y 1.0))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -1.35e-51) {
		tmp = (y / (x + 1.0)) / x;
	} else {
		tmp = (x / y) / (y + 1.0);
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-1.35d-51)) then
        tmp = (y / (x + 1.0d0)) / x
    else
        tmp = (x / y) / (y + 1.0d0)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (x <= -1.35e-51) {
		tmp = (y / (x + 1.0)) / x;
	} else {
		tmp = (x / y) / (y + 1.0);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -1.35e-51:
		tmp = (y / (x + 1.0)) / x
	else:
		tmp = (x / y) / (y + 1.0)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -1.35e-51)
		tmp = Float64(Float64(y / Float64(x + 1.0)) / x);
	else
		tmp = Float64(Float64(x / y) / Float64(y + 1.0));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1.35e-51)
		tmp = (y / (x + 1.0)) / x;
	else
		tmp = (x / y) / (y + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[x, -1.35e-51], N[(N[(y / N[(x + 1.0), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.3499999999999999e-51

   1. Initial program 63.7%

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

      1. times-frac 90.6%

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

      2. /-rgt-identity 90.6%

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

      3. /-rgt-identity 90.6%

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

      4. associate-+l+ 90.6%

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

   3. Simplified 90.6%

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

   4. Taylor expanded in x around inf 74.6%

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

      1. associate-*l/ 74.7%

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

      2. *-un-lft-identity 74.7%

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

   6. Applied egg-rr 74.7%

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

   7. Taylor expanded in y around 0 74.2%

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

      1. +-commutative 74.2%

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

   9. Simplified 74.2%

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

   if -1.3499999999999999e-51 < x

   1. Initial program 69.7%

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

      1. times-frac 88.4%

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

      2. +-commutative 88.4%

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

      3. +-commutative 88.4%

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

      4. associate-/r/ 85.9%

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

      5. +-commutative 85.9%

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

      6. associate-/r/ 85.9%

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

      7. associate-*l/ 84.0%

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

      8. *-commutative 84.0%

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

      9. distribute-rgt1-in 76.1%

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

      10. fma-def 84.0%

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

      11. +-commutative 84.0%

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

      12. +-commutative 84.0%

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

      13. cube-unmult 84.0%

          \[\leadsto \frac{x}{\frac{\mathsf{fma}\left(x + y, x + y, \color{blue}{{\left(y + x\right)}^{3}}\right)}{y}} \]

      14. +-commutative 84.0%

          \[\leadsto \frac{x}{\frac{\mathsf{fma}\left(x + y, x + y, {\color{blue}{\left(x + y\right)}}^{3}\right)}{y}} \]

   3. Simplified 84.0%

      \[\leadsto \color{blue}{\frac{x}{\frac{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}{y}}} \]
   4. Step-by-step derivation

      1. associate-/l* 69.8%

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

      2. fma-udef 63.0%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right) + {\left(x + y\right)}^{3}}} \]

      3. cube-mult 63.0%

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

      4. distribute-rgt1-in 69.7%

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

      5. *-commutative 69.7%

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

      6. times-frac 88.4%

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

      7. associate-*l/ 79.2%

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

      8. times-frac 99.9%

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

      9. associate-+r+ 99.9%

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

   5. Applied egg-rr 99.9%

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

      1. clear-num 99.8%

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

      2. associate-*l/ 99.8%

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

      3. *-un-lft-identity 99.8%

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

      4. associate-/l/ 95.5%

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

   7. Applied egg-rr 95.5%

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

   8. Taylor expanded in x around 0 58.8%

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

      1. +-commutative 58.8%

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

      2. associate-/r* 59.6%

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

   10. Simplified 59.6%

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

4. Final simplification 64.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.35 \cdot 10^{-51}:\\ \;\;\;\;\frac{\frac{y}{x + 1}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + 1}\\ \end{array} \]

Alternative 15: 66.2% accurate, 2.4× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{\frac{y}{x}}{x}\\ \mathbf{elif}\;x \leq -3.8 \cdot 10^{-212}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -1.0) (/ (/ y x) x) (if (<= x -3.8e-212) (/ y x) (/ x y))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -1.0) {
		tmp = (y / x) / x;
	} else if (x <= -3.8e-212) {
		tmp = y / x;
	} else {
		tmp = x / y;
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-1.0d0)) then
        tmp = (y / x) / x
    else if (x <= (-3.8d-212)) then
        tmp = y / x
    else
        tmp = x / y
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (x <= -1.0) {
		tmp = (y / x) / x;
	} else if (x <= -3.8e-212) {
		tmp = y / x;
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -1.0:
		tmp = (y / x) / x
	elif x <= -3.8e-212:
		tmp = y / x
	else:
		tmp = x / y
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -1.0)
		tmp = Float64(Float64(y / x) / x);
	elseif (x <= -3.8e-212)
		tmp = Float64(y / x);
	else
		tmp = Float64(x / y);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1.0)
		tmp = (y / x) / x;
	elseif (x <= -3.8e-212)
		tmp = y / x;
	else
		tmp = x / y;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[x, -1.0], N[(N[(y / x), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[x, -3.8e-212], N[(y / x), $MachinePrecision], N[(x / y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1

   1. Initial program 61.3%

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

      1. times-frac 90.0%

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

      2. /-rgt-identity 90.0%

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

      3. /-rgt-identity 90.0%

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

      4. associate-+l+ 90.0%

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

   3. Simplified 90.0%

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

   4. Taylor expanded in x around inf 75.6%

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

      1. associate-*l/ 75.6%

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

      2. *-un-lft-identity 75.6%

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

   6. Applied egg-rr 75.6%

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

   7. Taylor expanded in x around inf 73.1%

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

   if -1 < x < -3.80000000000000022e-212

   1. Initial program 96.5%

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

      1. times-frac 99.6%

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

      2. /-rgt-identity 99.6%

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

      3. /-rgt-identity 99.6%

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

      4. associate-+l+ 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in x around 0 99.2%

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

      1. +-commutative 99.2%

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

   6. Simplified 99.2%

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

   7. Taylor expanded in y around 0 30.0%

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

   if -3.80000000000000022e-212 < x

   1. Initial program 65.3%

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

      1. times-frac 86.4%

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

      2. /-rgt-identity 86.4%

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

      3. /-rgt-identity 86.4%

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

      4. associate-+l+ 86.4%

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

   3. Simplified 86.4%

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

   4. Taylor expanded in x around 0 55.5%

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

      1. +-commutative 55.5%

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

   6. Simplified 55.5%

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

   7. Taylor expanded in y around 0 39.7%

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

4. Final simplification 48.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{\frac{y}{x}}{x}\\ \mathbf{elif}\;x \leq -3.8 \cdot 10^{-212}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]

Alternative 16: 27.7% accurate, 3.4× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -7.2 \cdot 10^{-18}:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y) :precision binary64 (if (<= x -7.2e-18) (/ 1.0 x) (/ x y)))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -7.2e-18) {
		tmp = 1.0 / x;
	} else {
		tmp = x / y;
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-7.2d-18)) then
        tmp = 1.0d0 / x
    else
        tmp = x / y
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (x <= -7.2e-18) {
		tmp = 1.0 / x;
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -7.2e-18:
		tmp = 1.0 / x
	else:
		tmp = x / y
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -7.2e-18)
		tmp = Float64(1.0 / x);
	else
		tmp = Float64(x / y);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -7.2e-18)
		tmp = 1.0 / x;
	else
		tmp = x / y;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[x, -7.2e-18], N[(1.0 / x), $MachinePrecision], N[(x / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -7.20000000000000021e-18

   1. Initial program 62.3%

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

      1. times-frac 90.2%

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

      2. /-rgt-identity 90.2%

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

      3. /-rgt-identity 90.2%

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

      4. associate-+l+ 90.2%

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

   3. Simplified 90.2%

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

   4. Taylor expanded in x around inf 74.9%

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

   5. Taylor expanded in y around inf 5.6%

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

   if -7.20000000000000021e-18 < x

   1. Initial program 70.2%

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

      1. times-frac 88.6%

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

      2. /-rgt-identity 88.6%

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

      3. /-rgt-identity 88.6%

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

      4. associate-+l+ 88.6%

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

   3. Simplified 88.6%

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

   4. Taylor expanded in x around 0 58.5%

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

      1. +-commutative 58.5%

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

   6. Simplified 58.5%

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

   7. Taylor expanded in y around 0 41.3%

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

4. Final simplification 30.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -7.2 \cdot 10^{-18}:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]

Alternative 17: 42.6% accurate, 3.4× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -3.8 \cdot 10^{-212}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y) :precision binary64 (if (<= x -3.8e-212) (/ y x) (/ x y)))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -3.8e-212) {
		tmp = y / x;
	} else {
		tmp = x / y;
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-3.8d-212)) then
        tmp = y / x
    else
        tmp = x / y
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (x <= -3.8e-212) {
		tmp = y / x;
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -3.8e-212:
		tmp = y / x
	else:
		tmp = x / y
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -3.8e-212)
		tmp = Float64(y / x);
	else
		tmp = Float64(x / y);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -3.8e-212)
		tmp = y / x;
	else
		tmp = x / y;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[x, -3.8e-212], N[(y / x), $MachinePrecision], N[(x / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -3.80000000000000022e-212

   1. Initial program 71.8%

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

      1. times-frac 92.8%

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

      2. /-rgt-identity 92.8%

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

      3. /-rgt-identity 92.8%

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

      4. associate-+l+ 92.8%

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

   3. Simplified 92.8%

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

   4. Taylor expanded in x around 0 76.3%

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

      1. +-commutative 76.3%

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

   6. Simplified 76.3%

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

   7. Taylor expanded in y around 0 28.2%

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

   if -3.80000000000000022e-212 < x

   1. Initial program 65.3%

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

      1. times-frac 86.4%

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

      2. /-rgt-identity 86.4%

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

      3. /-rgt-identity 86.4%

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

      4. associate-+l+ 86.4%

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

   3. Simplified 86.4%

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

   4. Taylor expanded in x around 0 55.5%

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

      1. +-commutative 55.5%

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

   6. Simplified 55.5%

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

   7. Taylor expanded in y around 0 39.7%

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

4. Final simplification 35.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.8 \cdot 10^{-212}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]

Alternative 18: 4.3% accurate, 5.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 67.9%

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

   1. times-frac 89.0%

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

   2. /-rgt-identity 89.0%

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

   3. /-rgt-identity 89.0%

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

   4. associate-+l+ 89.0%

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

3. Simplified 89.0%

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

4. Taylor expanded in x around inf 52.6%

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

5. Taylor expanded in y around inf 4.3%

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

6. Final simplification 4.3%

   \[\leadsto \frac{1}{x} \]

Alternative 19: 3.5% accurate, 17.0× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ 1 \end{array} \]

FPCore

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

C

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

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 = 1.0d0
end function

Java

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

Python

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

Julia

x, y = sort([x, y])
function code(x, y)
	return 1.0
end

MATLAB

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

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := 1.0

Derivation

1. Initial program 67.9%

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

   1. times-frac 89.0%

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

   2. +-commutative 89.0%

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

   3. +-commutative 89.0%

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

   4. associate-/r/ 85.8%

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

   5. +-commutative 85.8%

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

   6. associate-/r/ 85.8%

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

   7. associate-*l/ 83.0%

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

   8. *-commutative 83.0%

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

   9. distribute-rgt1-in 63.3%

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

   10. fma-def 83.0%

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

   11. +-commutative 83.0%

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

   12. +-commutative 83.0%

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

   13. cube-unmult 83.0%

       \[\leadsto \frac{x}{\frac{\mathsf{fma}\left(x + y, x + y, \color{blue}{{\left(y + x\right)}^{3}}\right)}{y}} \]

   14. +-commutative 83.0%

       \[\leadsto \frac{x}{\frac{\mathsf{fma}\left(x + y, x + y, {\color{blue}{\left(x + y\right)}}^{3}\right)}{y}} \]

3. Simplified 83.0%

   \[\leadsto \color{blue}{\frac{x}{\frac{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}{y}}} \]
4. Step-by-step derivation

   1. associate-/l* 67.9%

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

   2. fma-udef 54.2%

      \[\leadsto \frac{x \cdot y}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right) + {\left(x + y\right)}^{3}}} \]

   3. cube-mult 54.2%

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

   4. distribute-rgt1-in 67.9%

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

   5. *-commutative 67.9%

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

   6. times-frac 89.0%

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

   7. associate-*l/ 82.7%

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

   8. times-frac 99.9%

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

   9. associate-+r+ 99.9%

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

5. Applied egg-rr 99.9%

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

6. Taylor expanded in x around 0 50.1%

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

   1. +-commutative 50.1%

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

8. Simplified 50.1%

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

9. Taylor expanded in y around 0 3.4%

   \[\leadsto \color{blue}{1} \]

10. Final simplification 3.4%

    \[\leadsto 1 \]

Developer target: 99.8% accurate, 0.9× 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 2023301 
(FPCore (x y)
  :name "Numeric.SpecFunctions:incompleteBetaApprox from math-functions-0.1.5.2, A"
  :precision binary64

  :herbie-target
  (/ (/ (/ x (+ (+ y 1.0) x)) (+ y x)) (/ 1.0 (/ y (+ y x))))

  (/ (* x y) (* (* (+ x y) (+ x y)) (+ (+ x y) 1.0))))