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

Percentage Accurate: 69.4% → 99.7%

Time: 14.0s

Alternatives: 21

Speedup: 1.4×

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: 69.4% 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: 99.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

assert(x < y);
double code(double x, double y) {
	return ((x / (x + y)) / (x + y)) / ((y + (x + 1.0)) / 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)) / (x + y)) / ((y + (x + 1.0d0)) / y)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 68.3%

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

   1. associate-/l* 82.0%

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

   2. associate-+l+ 82.0%

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

3. Simplified 82.0%

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

   1. clear-num 82.0%

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

   2. associate-+r+ 82.0%

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

   3. *-commutative 82.0%

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

   4. distribute-rgt1-in 62.5%

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

   5. cube-mult 62.5%

      \[\leadsto x \cdot \frac{1}{\frac{\left(x + y\right) \cdot \left(x + y\right) + \color{blue}{{\left(x + y\right)}^{3}}}{y}} \]

   6. un-div-inv 62.5%

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

   7. cube-mult 62.5%

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

   8. distribute-rgt1-in 82.1%

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

   9. *-commutative 82.1%

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

   10. associate-/l* 83.9%

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

   11. pow2 83.8%

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

   12. +-commutative 83.8%

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

6. Applied egg-rr 83.8%

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

   1. associate-/r* 86.0%

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

   2. +-commutative 86.0%

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

8. Simplified 86.0%

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

   1. *-un-lft-identity 86.0%

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

   2. unpow2 86.0%

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

   3. times-frac 99.7%

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

10. Applied egg-rr 99.7%

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

    1. associate-*l/ 99.7%

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

    2. *-lft-identity 99.7%

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

12. Simplified 99.7%

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

13. Final simplification 99.7%

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

Alternative 2: 95.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_0 := \frac{y}{x + y}\\ \mathbf{if}\;x \leq -1.4 \cdot 10^{+154}:\\ \;\;\;\;\frac{t\_0}{x + \left(y + 1\right)}\\ \mathbf{elif}\;x \leq -5.2 \cdot 10^{-14}:\\ \;\;\;\;\frac{y}{\left(x + y\right) \cdot \left(y + \left(x + 1\right)\right)}\\ \mathbf{elif}\;x \leq 2.9 \cdot 10^{-71}:\\ \;\;\;\;\frac{x \cdot t\_0}{\left(x + y\right) \cdot \left(y + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y} \cdot \frac{x}{y + 1}\\ \end{array} \end{array} \]

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))))
   (if (<= x -1.4e+154)
     (/ t_0 (+ x (+ y 1.0)))
     (if (<= x -5.2e-14)
       (/ y (* (+ x y) (+ y (+ x 1.0))))
       (if (<= x 2.9e-71)
         (/ (* x t_0) (* (+ x y) (+ y 1.0)))
         (* (/ 1.0 y) (/ x (+ y 1.0))))))))

C

assert(x < y);
double code(double x, double y) {
	double t_0 = y / (x + y);
	double tmp;
	if (x <= -1.4e+154) {
		tmp = t_0 / (x + (y + 1.0));
	} else if (x <= -5.2e-14) {
		tmp = y / ((x + y) * (y + (x + 1.0)));
	} else if (x <= 2.9e-71) {
		tmp = (x * t_0) / ((x + y) * (y + 1.0));
	} 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)
    if (x <= (-1.4d+154)) then
        tmp = t_0 / (x + (y + 1.0d0))
    else if (x <= (-5.2d-14)) then
        tmp = y / ((x + y) * (y + (x + 1.0d0)))
    else if (x <= 2.9d-71) then
        tmp = (x * t_0) / ((x + y) * (y + 1.0d0))
    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);
	double tmp;
	if (x <= -1.4e+154) {
		tmp = t_0 / (x + (y + 1.0));
	} else if (x <= -5.2e-14) {
		tmp = y / ((x + y) * (y + (x + 1.0)));
	} else if (x <= 2.9e-71) {
		tmp = (x * t_0) / ((x + y) * (y + 1.0));
	} 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)
	tmp = 0
	if x <= -1.4e+154:
		tmp = t_0 / (x + (y + 1.0))
	elif x <= -5.2e-14:
		tmp = y / ((x + y) * (y + (x + 1.0)))
	elif x <= 2.9e-71:
		tmp = (x * t_0) / ((x + y) * (y + 1.0))
	else:
		tmp = (1.0 / y) * (x / (y + 1.0))
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	t_0 = Float64(y / Float64(x + y))
	tmp = 0.0
	if (x <= -1.4e+154)
		tmp = Float64(t_0 / Float64(x + Float64(y + 1.0)));
	elseif (x <= -5.2e-14)
		tmp = Float64(y / Float64(Float64(x + y) * Float64(y + Float64(x + 1.0))));
	elseif (x <= 2.9e-71)
		tmp = Float64(Float64(x * t_0) / Float64(Float64(x + y) * Float64(y + 1.0)));
	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);
	tmp = 0.0;
	if (x <= -1.4e+154)
		tmp = t_0 / (x + (y + 1.0));
	elseif (x <= -5.2e-14)
		tmp = y / ((x + y) * (y + (x + 1.0)));
	elseif (x <= 2.9e-71)
		tmp = (x * t_0) / ((x + y) * (y + 1.0));
	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[(y / N[(x + y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.4e+154], N[(t$95$0 / N[(x + N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -5.2e-14], N[(y / N[(N[(x + y), $MachinePrecision] * N[(y + N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 2.9e-71], N[(N[(x * t$95$0), $MachinePrecision] / N[(N[(x + y), $MachinePrecision] * N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $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 < -1.4e154

   1. Initial program 70.4%

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

      1. associate-*l* 70.4%

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

      2. times-frac 87.8%

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

      3. +-commutative 87.8%

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

      4. +-commutative 87.8%

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

      5. associate-+r+ 87.8%

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

      6. +-commutative 87.8%

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

      7. associate-+l+ 87.8%

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

   4. Applied egg-rr 87.8%

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

   5. Taylor expanded in x around inf 87.8%

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

      1. *-un-lft-identity 87.8%

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

      2. associate-/r* 91.0%

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

      3. +-commutative 91.0%

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

      4. +-commutative 91.0%

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

      5. associate-+r+ 91.0%

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

      6. +-commutative 91.0%

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

      7. associate-+l+ 91.0%

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

   7. Applied egg-rr 91.0%

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

   if -1.4e154 < x < -5.19999999999999993e-14

   1. Initial program 71.4%

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

      1. associate-*l* 71.4%

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

      2. times-frac 90.3%

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

      3. +-commutative 90.3%

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

      4. +-commutative 90.3%

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

      5. associate-+r+ 90.3%

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

      6. +-commutative 90.3%

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

      7. associate-+l+ 90.3%

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

   4. Applied egg-rr 90.3%

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

   5. Taylor expanded in x around inf 83.6%

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

   if -5.19999999999999993e-14 < x < 2.8999999999999999e-71

   1. Initial program 69.4%

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

      1. associate-*l* 69.3%

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

      2. times-frac 99.8%

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

      3. +-commutative 99.8%

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

      4. +-commutative 99.8%

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

      5. associate-+r+ 99.8%

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

      6. +-commutative 99.8%

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

      7. associate-+l+ 99.8%

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

   4. Applied egg-rr 99.8%

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

      1. *-commutative 99.8%

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

      2. associate-/r* 99.8%

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

      3. associate-+r+ 99.8%

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

      4. +-commutative 99.8%

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

      5. frac-times 99.9%

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

      6. *-commutative 99.9%

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

      7. +-commutative 99.9%

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

      8. associate-+r+ 99.9%

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

      9. distribute-lft-in 99.9%

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

      10. +-commutative 99.9%

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

      11. distribute-lft-in 99.9%

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

   6. Applied egg-rr 99.9%

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

   7. Taylor expanded in x around 0 99.9%

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

   if 2.8999999999999999e-71 < x

   1. Initial program 64.1%

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

      1. associate-/l* 78.1%

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

      2. associate-+l+ 78.1%

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

   3. Simplified 78.1%

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

   5. Taylor expanded in x around 0 28.6%

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

      1. +-commutative 28.6%

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

   7. Simplified 28.6%

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

      1. *-un-lft-identity 28.6%

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

      2. times-frac 33.5%

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

   9. Applied egg-rr 33.5%

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

4. Final simplification 79.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.4 \cdot 10^{+154}:\\ \;\;\;\;\frac{\frac{y}{x + y}}{x + \left(y + 1\right)}\\ \mathbf{elif}\;x \leq -5.2 \cdot 10^{-14}:\\ \;\;\;\;\frac{y}{\left(x + y\right) \cdot \left(y + \left(x + 1\right)\right)}\\ \mathbf{elif}\;x \leq 2.9 \cdot 10^{-71}:\\ \;\;\;\;\frac{x \cdot \frac{y}{x + y}}{\left(x + y\right) \cdot \left(y + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y} \cdot \frac{x}{y + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 95.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -2.2 \cdot 10^{+154}:\\ \;\;\;\;\frac{\frac{y}{x + y}}{x + \left(y + 1\right)}\\ \mathbf{elif}\;x \leq -4.2 \cdot 10^{-14}:\\ \;\;\;\;\frac{y}{\left(x + y\right) \cdot \left(y + \left(x + 1\right)\right)}\\ \mathbf{elif}\;x \leq 3.2 \cdot 10^{-111}:\\ \;\;\;\;\frac{x}{x + y} \cdot \frac{y}{\left(x + y\right) \cdot \left(y + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y} \cdot \frac{x}{y + 1}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -2.2e+154)
   (/ (/ y (+ x y)) (+ x (+ y 1.0)))
   (if (<= x -4.2e-14)
     (/ y (* (+ x y) (+ y (+ x 1.0))))
     (if (<= x 3.2e-111)
       (* (/ x (+ x y)) (/ y (* (+ x y) (+ y 1.0))))
       (* (/ 1.0 y) (/ x (+ y 1.0)))))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -2.2e+154) {
		tmp = (y / (x + y)) / (x + (y + 1.0));
	} else if (x <= -4.2e-14) {
		tmp = y / ((x + y) * (y + (x + 1.0)));
	} else if (x <= 3.2e-111) {
		tmp = (x / (x + y)) * (y / ((x + y) * (y + 1.0)));
	} 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) :: tmp
    if (x <= (-2.2d+154)) then
        tmp = (y / (x + y)) / (x + (y + 1.0d0))
    else if (x <= (-4.2d-14)) then
        tmp = y / ((x + y) * (y + (x + 1.0d0)))
    else if (x <= 3.2d-111) then
        tmp = (x / (x + y)) * (y / ((x + y) * (y + 1.0d0)))
    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 <= -2.2e+154) {
		tmp = (y / (x + y)) / (x + (y + 1.0));
	} else if (x <= -4.2e-14) {
		tmp = y / ((x + y) * (y + (x + 1.0)));
	} else if (x <= 3.2e-111) {
		tmp = (x / (x + y)) * (y / ((x + y) * (y + 1.0)));
	} 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 <= -2.2e+154:
		tmp = (y / (x + y)) / (x + (y + 1.0))
	elif x <= -4.2e-14:
		tmp = y / ((x + y) * (y + (x + 1.0)))
	elif x <= 3.2e-111:
		tmp = (x / (x + y)) * (y / ((x + y) * (y + 1.0)))
	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 <= -2.2e+154)
		tmp = Float64(Float64(y / Float64(x + y)) / Float64(x + Float64(y + 1.0)));
	elseif (x <= -4.2e-14)
		tmp = Float64(y / Float64(Float64(x + y) * Float64(y + Float64(x + 1.0))));
	elseif (x <= 3.2e-111)
		tmp = Float64(Float64(x / Float64(x + y)) * Float64(y / Float64(Float64(x + y) * Float64(y + 1.0))));
	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)
	tmp = 0.0;
	if (x <= -2.2e+154)
		tmp = (y / (x + y)) / (x + (y + 1.0));
	elseif (x <= -4.2e-14)
		tmp = y / ((x + y) * (y + (x + 1.0)));
	elseif (x <= 3.2e-111)
		tmp = (x / (x + y)) * (y / ((x + y) * (y + 1.0)));
	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_] := If[LessEqual[x, -2.2e+154], N[(N[(y / N[(x + y), $MachinePrecision]), $MachinePrecision] / N[(x + N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -4.2e-14], N[(y / N[(N[(x + y), $MachinePrecision] * N[(y + N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 3.2e-111], N[(N[(x / N[(x + y), $MachinePrecision]), $MachinePrecision] * N[(y / N[(N[(x + y), $MachinePrecision] * N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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 < -2.2000000000000001e154

   1. Initial program 70.4%

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

      1. associate-*l* 70.4%

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

      2. times-frac 87.8%

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

      3. +-commutative 87.8%

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

      4. +-commutative 87.8%

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

      5. associate-+r+ 87.8%

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

      6. +-commutative 87.8%

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

      7. associate-+l+ 87.8%

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

   4. Applied egg-rr 87.8%

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

   5. Taylor expanded in x around inf 87.8%

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

      1. *-un-lft-identity 87.8%

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

      2. associate-/r* 91.0%

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

      3. +-commutative 91.0%

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

      4. +-commutative 91.0%

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

      5. associate-+r+ 91.0%

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

      6. +-commutative 91.0%

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

      7. associate-+l+ 91.0%

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

   7. Applied egg-rr 91.0%

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

   if -2.2000000000000001e154 < x < -4.1999999999999998e-14

   1. Initial program 71.4%

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

      1. associate-*l* 71.4%

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

      2. times-frac 90.3%

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

      3. +-commutative 90.3%

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

      4. +-commutative 90.3%

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

      5. associate-+r+ 90.3%

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

      6. +-commutative 90.3%

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

      7. associate-+l+ 90.3%

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

   4. Applied egg-rr 90.3%

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

   5. Taylor expanded in x around inf 83.6%

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

   if -4.1999999999999998e-14 < x < 3.1999999999999998e-111

   1. Initial program 68.7%

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

      1. associate-*l* 68.7%

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

      2. times-frac 99.8%

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

      3. +-commutative 99.8%

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

      4. +-commutative 99.8%

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

      5. associate-+r+ 99.8%

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

      6. +-commutative 99.8%

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

      7. associate-+l+ 99.8%

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

   4. Applied egg-rr 99.8%

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

   5. Taylor expanded in x around 0 99.8%

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

      1. +-commutative 99.8%

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

   7. Simplified 99.8%

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

   if 3.1999999999999998e-111 < x

   1. Initial program 65.6%

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

      1. associate-/l* 79.2%

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

      2. associate-+l+ 79.2%

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

   3. Simplified 79.2%

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

   5. Taylor expanded in x around 0 31.1%

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

      1. +-commutative 31.1%

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

   7. Simplified 31.1%

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

      1. *-un-lft-identity 31.1%

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

      2. times-frac 35.4%

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

   9. Applied egg-rr 35.4%

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

4. Final simplification 77.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.2 \cdot 10^{+154}:\\ \;\;\;\;\frac{\frac{y}{x + y}}{x + \left(y + 1\right)}\\ \mathbf{elif}\;x \leq -4.2 \cdot 10^{-14}:\\ \;\;\;\;\frac{y}{\left(x + y\right) \cdot \left(y + \left(x + 1\right)\right)}\\ \mathbf{elif}\;x \leq 3.2 \cdot 10^{-111}:\\ \;\;\;\;\frac{x}{x + y} \cdot \frac{y}{\left(x + y\right) \cdot \left(y + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y} \cdot \frac{x}{y + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 93.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_0 := x + \left(y + 1\right)\\ \mathbf{if}\;x \leq -1.6 \cdot 10^{+58}:\\ \;\;\;\;\frac{\frac{y}{x + y}}{t\_0}\\ \mathbf{elif}\;x \leq -1.95 \cdot 10^{-59}:\\ \;\;\;\;x \cdot \frac{y}{t\_0 \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{x}{x + y}}{x + y}}{\frac{y + 1}{y}}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ x (+ y 1.0))))
   (if (<= x -1.6e+58)
     (/ (/ y (+ x y)) t_0)
     (if (<= x -1.95e-59)
       (* x (/ y (* t_0 (* (+ x y) (+ x y)))))
       (/ (/ (/ x (+ x y)) (+ x y)) (/ (+ y 1.0) y))))))

C

assert(x < y);
double code(double x, double y) {
	double t_0 = x + (y + 1.0);
	double tmp;
	if (x <= -1.6e+58) {
		tmp = (y / (x + y)) / t_0;
	} else if (x <= -1.95e-59) {
		tmp = x * (y / (t_0 * ((x + y) * (x + y))));
	} else {
		tmp = ((x / (x + y)) / (x + y)) / ((y + 1.0) / y);
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x + (y + 1.0d0)
    if (x <= (-1.6d+58)) then
        tmp = (y / (x + y)) / t_0
    else if (x <= (-1.95d-59)) then
        tmp = x * (y / (t_0 * ((x + y) * (x + y))))
    else
        tmp = ((x / (x + y)) / (x + y)) / ((y + 1.0d0) / y)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double t_0 = x + (y + 1.0);
	double tmp;
	if (x <= -1.6e+58) {
		tmp = (y / (x + y)) / t_0;
	} else if (x <= -1.95e-59) {
		tmp = x * (y / (t_0 * ((x + y) * (x + y))));
	} else {
		tmp = ((x / (x + y)) / (x + y)) / ((y + 1.0) / y);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	t_0 = x + (y + 1.0)
	tmp = 0
	if x <= -1.6e+58:
		tmp = (y / (x + y)) / t_0
	elif x <= -1.95e-59:
		tmp = x * (y / (t_0 * ((x + y) * (x + y))))
	else:
		tmp = ((x / (x + y)) / (x + y)) / ((y + 1.0) / y)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	t_0 = Float64(x + Float64(y + 1.0))
	tmp = 0.0
	if (x <= -1.6e+58)
		tmp = Float64(Float64(y / Float64(x + y)) / t_0);
	elseif (x <= -1.95e-59)
		tmp = Float64(x * Float64(y / Float64(t_0 * Float64(Float64(x + y) * Float64(x + y)))));
	else
		tmp = Float64(Float64(Float64(x / Float64(x + y)) / Float64(x + y)) / Float64(Float64(y + 1.0) / y));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	t_0 = x + (y + 1.0);
	tmp = 0.0;
	if (x <= -1.6e+58)
		tmp = (y / (x + y)) / t_0;
	elseif (x <= -1.95e-59)
		tmp = x * (y / (t_0 * ((x + y) * (x + y))));
	else
		tmp = ((x / (x + y)) / (x + y)) / ((y + 1.0) / 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[(y + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.6e+58], N[(N[(y / N[(x + y), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision], If[LessEqual[x, -1.95e-59], N[(x * N[(y / N[(t$95$0 * N[(N[(x + y), $MachinePrecision] * N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x / N[(x + y), $MachinePrecision]), $MachinePrecision] / N[(x + y), $MachinePrecision]), $MachinePrecision] / N[(N[(y + 1.0), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.60000000000000008e58

   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. Add Preprocessing
   3. Step-by-step derivation

      1. associate-*l* 65.9%

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

      2. times-frac 86.3%

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

      3. +-commutative 86.3%

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

      4. +-commutative 86.3%

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

      5. associate-+r+ 86.3%

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

      6. +-commutative 86.3%

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

      7. associate-+l+ 86.3%

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

   4. Applied egg-rr 86.3%

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

   5. Taylor expanded in x around inf 84.5%

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

      1. *-un-lft-identity 84.5%

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

      2. associate-/r* 82.1%

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

      3. +-commutative 82.1%

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

      4. +-commutative 82.1%

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

      5. associate-+r+ 82.1%

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

      6. +-commutative 82.1%

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

      7. associate-+l+ 82.1%

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

   7. Applied egg-rr 82.1%

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

   if -1.60000000000000008e58 < x < -1.95000000000000009e-59

   1. Initial program 94.6%

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

      1. associate-/l* 94.6%

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

      2. associate-+l+ 94.6%

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

   3. Simplified 94.6%

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

   if -1.95000000000000009e-59 < x

   1. Initial program 66.4%

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

      1. associate-/l* 81.7%

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

      2. associate-+l+ 81.7%

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

   3. Simplified 81.7%

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

      1. clear-num 81.7%

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

      2. associate-+r+ 81.7%

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

      3. *-commutative 81.7%

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

      4. distribute-rgt1-in 70.5%

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

      5. cube-mult 70.5%

         \[\leadsto x \cdot \frac{1}{\frac{\left(x + y\right) \cdot \left(x + y\right) + \color{blue}{{\left(x + y\right)}^{3}}}{y}} \]

      6. un-div-inv 70.6%

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

      7. cube-mult 70.6%

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

      8. distribute-rgt1-in 81.8%

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

      9. *-commutative 81.8%

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

      10. associate-/l* 83.7%

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

      11. pow2 83.7%

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

      12. +-commutative 83.7%

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

   6. Applied egg-rr 83.7%

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

      1. associate-/r* 84.6%

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

      2. +-commutative 84.6%

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

   8. Simplified 84.6%

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

      1. *-un-lft-identity 84.6%

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

      2. unpow2 84.7%

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

      3. times-frac 99.7%

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

   10. Applied egg-rr 99.7%

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

       1. associate-*l/ 99.7%

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

       2. *-lft-identity 99.7%

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

   12. Simplified 99.7%

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

   13. Taylor expanded in x around 0 84.2%

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

4. Final simplification 84.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.6 \cdot 10^{+58}:\\ \;\;\;\;\frac{\frac{y}{x + y}}{x + \left(y + 1\right)}\\ \mathbf{elif}\;x \leq -1.95 \cdot 10^{-59}:\\ \;\;\;\;x \cdot \frac{y}{\left(x + \left(y + 1\right)\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{x}{x + y}}{x + y}}{\frac{y + 1}{y}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 90.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 3.35 \cdot 10^{-163}:\\ \;\;\;\;\frac{\frac{y}{x + 1}}{x}\\ \mathbf{elif}\;y \leq 6.6 \cdot 10^{-5}:\\ \;\;\;\;x \cdot \frac{y}{\left(x + 1\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}\\ \mathbf{elif}\;y \leq 5.7 \cdot 10^{+109}:\\ \;\;\;\;\frac{x}{\left(x + y\right) \cdot \left(y + \left(x + 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + y} \cdot \frac{1}{y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 3.35e-163)
   (/ (/ y (+ x 1.0)) x)
   (if (<= y 6.6e-5)
     (* x (/ y (* (+ x 1.0) (* (+ x y) (+ x y)))))
     (if (<= y 5.7e+109)
       (/ x (* (+ x y) (+ y (+ x 1.0))))
       (* (/ x (+ x y)) (/ 1.0 y))))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 3.35e-163) {
		tmp = (y / (x + 1.0)) / x;
	} else if (y <= 6.6e-5) {
		tmp = x * (y / ((x + 1.0) * ((x + y) * (x + y))));
	} else if (y <= 5.7e+109) {
		tmp = x / ((x + y) * (y + (x + 1.0)));
	} else {
		tmp = (x / (x + y)) * (1.0 / y);
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 3.35d-163) then
        tmp = (y / (x + 1.0d0)) / x
    else if (y <= 6.6d-5) then
        tmp = x * (y / ((x + 1.0d0) * ((x + y) * (x + y))))
    else if (y <= 5.7d+109) then
        tmp = x / ((x + y) * (y + (x + 1.0d0)))
    else
        tmp = (x / (x + y)) * (1.0d0 / y)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (y <= 3.35e-163) {
		tmp = (y / (x + 1.0)) / x;
	} else if (y <= 6.6e-5) {
		tmp = x * (y / ((x + 1.0) * ((x + y) * (x + y))));
	} else if (y <= 5.7e+109) {
		tmp = x / ((x + y) * (y + (x + 1.0)));
	} else {
		tmp = (x / (x + y)) * (1.0 / y);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if y <= 3.35e-163:
		tmp = (y / (x + 1.0)) / x
	elif y <= 6.6e-5:
		tmp = x * (y / ((x + 1.0) * ((x + y) * (x + y))))
	elif y <= 5.7e+109:
		tmp = x / ((x + y) * (y + (x + 1.0)))
	else:
		tmp = (x / (x + y)) * (1.0 / y)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= 3.35e-163)
		tmp = Float64(Float64(y / Float64(x + 1.0)) / x);
	elseif (y <= 6.6e-5)
		tmp = Float64(x * Float64(y / Float64(Float64(x + 1.0) * Float64(Float64(x + y) * Float64(x + y)))));
	elseif (y <= 5.7e+109)
		tmp = Float64(x / Float64(Float64(x + y) * Float64(y + Float64(x + 1.0))));
	else
		tmp = Float64(Float64(x / Float64(x + y)) * Float64(1.0 / y));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 3.35e-163)
		tmp = (y / (x + 1.0)) / x;
	elseif (y <= 6.6e-5)
		tmp = x * (y / ((x + 1.0) * ((x + y) * (x + y))));
	elseif (y <= 5.7e+109)
		tmp = x / ((x + y) * (y + (x + 1.0)));
	else
		tmp = (x / (x + y)) * (1.0 / y);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[y, 3.35e-163], N[(N[(y / N[(x + 1.0), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[y, 6.6e-5], N[(x * N[(y / N[(N[(x + 1.0), $MachinePrecision] * N[(N[(x + y), $MachinePrecision] * N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 5.7e+109], N[(x / N[(N[(x + y), $MachinePrecision] * N[(y + N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(x + y), $MachinePrecision]), $MachinePrecision] * N[(1.0 / y), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if y < 3.3500000000000001e-163

   1. Initial program 65.0%

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

      1. associate-/l* 80.1%

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

      2. associate-+l+ 80.1%

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

   3. Simplified 80.1%

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

      1. clear-num 80.1%

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

      2. associate-+r+ 80.1%

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

      3. *-commutative 80.1%

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

      4. distribute-rgt1-in 53.7%

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

      5. cube-mult 53.6%

         \[\leadsto x \cdot \frac{1}{\frac{\left(x + y\right) \cdot \left(x + y\right) + \color{blue}{{\left(x + y\right)}^{3}}}{y}} \]

      6. un-div-inv 53.7%

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

      7. cube-mult 53.7%

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

      8. distribute-rgt1-in 80.2%

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

      9. *-commutative 80.2%

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

      10. associate-/l* 81.3%

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

      11. pow2 81.3%

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

      12. +-commutative 81.3%

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

   6. Applied egg-rr 81.3%

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

      1. associate-/r* 83.5%

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

      2. +-commutative 83.5%

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

   8. Simplified 83.5%

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

      1. *-un-lft-identity 83.5%

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

      2. unpow2 83.5%

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

      3. times-frac 99.7%

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

   10. Applied egg-rr 99.7%

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

       1. associate-*l/ 99.7%

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

       2. *-lft-identity 99.7%

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

   12. Simplified 99.7%

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

   13. Taylor expanded in y around 0 51.3%

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

       1. +-commutative 51.3%

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

   15. Simplified 51.3%

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

       1. *-un-lft-identity 51.3%

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

       2. times-frac 52.7%

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

   17. Applied egg-rr 52.7%

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

       1. associate-*l/ 52.8%

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

       2. *-lft-identity 52.8%

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

       3. +-commutative 52.8%

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

   19. Simplified 52.8%

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

   if 3.3500000000000001e-163 < y < 6.6000000000000005e-5

   1. Initial program 87.2%

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

      1. associate-/l* 93.1%

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

      2. associate-+l+ 93.1%

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

   3. Simplified 93.1%

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

   5. Taylor expanded in y around 0 92.0%

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

      1. +-commutative 92.0%

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

   7. Simplified 92.0%

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

   if 6.6000000000000005e-5 < y < 5.7000000000000002e109

   1. Initial program 75.8%

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

      1. associate-*l* 75.8%

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

      2. times-frac 89.7%

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

      3. +-commutative 89.7%

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

      4. +-commutative 89.7%

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

      5. associate-+r+ 89.7%

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

      6. +-commutative 89.7%

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

      7. associate-+l+ 89.7%

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

   4. Applied egg-rr 89.7%

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

      1. *-commutative 89.7%

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

      2. associate-/r* 99.6%

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

      3. associate-+r+ 99.6%

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

      4. +-commutative 99.6%

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

      5. frac-times 90.1%

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

      6. *-commutative 90.1%

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

      7. +-commutative 90.1%

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

      8. associate-+r+ 90.1%

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

      9. distribute-lft-in 85.4%

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

      10. +-commutative 85.4%

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

      11. distribute-lft-in 90.1%

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

   6. Applied egg-rr 90.1%

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

   7. Taylor expanded in y around inf 90.1%

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

   if 5.7000000000000002e109 < y

   1. Initial program 59.2%

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

      1. associate-*l* 59.2%

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

      2. times-frac 82.2%

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

      3. +-commutative 82.2%

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

      4. +-commutative 82.2%

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

      5. associate-+r+ 82.2%

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

      6. +-commutative 82.2%

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

      7. associate-+l+ 82.2%

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

   4. Applied egg-rr 82.2%

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

   5. Taylor expanded in y around inf 89.4%

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

4. Final simplification 66.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 3.35 \cdot 10^{-163}:\\ \;\;\;\;\frac{\frac{y}{x + 1}}{x}\\ \mathbf{elif}\;y \leq 6.6 \cdot 10^{-5}:\\ \;\;\;\;x \cdot \frac{y}{\left(x + 1\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}\\ \mathbf{elif}\;y \leq 5.7 \cdot 10^{+109}:\\ \;\;\;\;\frac{x}{\left(x + y\right) \cdot \left(y + \left(x + 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + y} \cdot \frac{1}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 94.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -2.1 \cdot 10^{+154}:\\ \;\;\;\;\frac{\frac{y}{x + y}}{x + \left(y + 1\right)}\\ \mathbf{elif}\;x \leq -1.6 \cdot 10^{-30}:\\ \;\;\;\;\frac{y}{\left(x + y\right) \cdot \left(y + \left(x + 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{x}{x + y}}{x + y}}{\frac{y + 1}{y}}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -2.1e+154)
   (/ (/ y (+ x y)) (+ x (+ y 1.0)))
   (if (<= x -1.6e-30)
     (/ y (* (+ x y) (+ y (+ x 1.0))))
     (/ (/ (/ x (+ x y)) (+ x y)) (/ (+ y 1.0) y)))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -2.1e+154) {
		tmp = (y / (x + y)) / (x + (y + 1.0));
	} else if (x <= -1.6e-30) {
		tmp = y / ((x + y) * (y + (x + 1.0)));
	} else {
		tmp = ((x / (x + y)) / (x + y)) / ((y + 1.0) / 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 <= (-2.1d+154)) then
        tmp = (y / (x + y)) / (x + (y + 1.0d0))
    else if (x <= (-1.6d-30)) then
        tmp = y / ((x + y) * (y + (x + 1.0d0)))
    else
        tmp = ((x / (x + y)) / (x + y)) / ((y + 1.0d0) / y)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (x <= -2.1e+154) {
		tmp = (y / (x + y)) / (x + (y + 1.0));
	} else if (x <= -1.6e-30) {
		tmp = y / ((x + y) * (y + (x + 1.0)));
	} else {
		tmp = ((x / (x + y)) / (x + y)) / ((y + 1.0) / y);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -2.1e+154:
		tmp = (y / (x + y)) / (x + (y + 1.0))
	elif x <= -1.6e-30:
		tmp = y / ((x + y) * (y + (x + 1.0)))
	else:
		tmp = ((x / (x + y)) / (x + y)) / ((y + 1.0) / y)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -2.1e+154)
		tmp = Float64(Float64(y / Float64(x + y)) / Float64(x + Float64(y + 1.0)));
	elseif (x <= -1.6e-30)
		tmp = Float64(y / Float64(Float64(x + y) * Float64(y + Float64(x + 1.0))));
	else
		tmp = Float64(Float64(Float64(x / Float64(x + y)) / Float64(x + y)) / Float64(Float64(y + 1.0) / y));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -2.1e+154)
		tmp = (y / (x + y)) / (x + (y + 1.0));
	elseif (x <= -1.6e-30)
		tmp = y / ((x + y) * (y + (x + 1.0)));
	else
		tmp = ((x / (x + y)) / (x + y)) / ((y + 1.0) / 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, -2.1e+154], N[(N[(y / N[(x + y), $MachinePrecision]), $MachinePrecision] / N[(x + N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -1.6e-30], N[(y / N[(N[(x + y), $MachinePrecision] * N[(y + N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x / N[(x + y), $MachinePrecision]), $MachinePrecision] / N[(x + y), $MachinePrecision]), $MachinePrecision] / N[(N[(y + 1.0), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.09999999999999994e154

   1. Initial program 70.4%

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

      1. associate-*l* 70.4%

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

      2. times-frac 87.8%

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

      3. +-commutative 87.8%

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

      4. +-commutative 87.8%

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

      5. associate-+r+ 87.8%

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

      6. +-commutative 87.8%

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

      7. associate-+l+ 87.8%

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

   4. Applied egg-rr 87.8%

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

   5. Taylor expanded in x around inf 87.8%

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

      1. *-un-lft-identity 87.8%

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

      2. associate-/r* 91.0%

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

      3. +-commutative 91.0%

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

      4. +-commutative 91.0%

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

      5. associate-+r+ 91.0%

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

      6. +-commutative 91.0%

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

      7. associate-+l+ 91.0%

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

   7. Applied egg-rr 91.0%

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

   if -2.09999999999999994e154 < x < -1.6e-30

   1. Initial program 73.3%

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

      1. associate-*l* 73.3%

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

      2. times-frac 91.0%

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

      3. +-commutative 91.0%

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

      4. +-commutative 91.0%

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

      5. associate-+r+ 91.0%

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

      6. +-commutative 91.0%

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

      7. associate-+l+ 91.0%

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

   4. Applied egg-rr 91.0%

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

   5. Taylor expanded in x around inf 84.7%

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

   if -1.6e-30 < x

   1. Initial program 67.2%

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

      1. associate-/l* 82.1%

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

      2. associate-+l+ 82.1%

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

   3. Simplified 82.1%

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

      1. clear-num 82.1%

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

      2. associate-+r+ 82.1%

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

      3. *-commutative 82.1%

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

      4. distribute-rgt1-in 71.3%

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

      5. cube-mult 71.2%

         \[\leadsto x \cdot \frac{1}{\frac{\left(x + y\right) \cdot \left(x + y\right) + \color{blue}{{\left(x + y\right)}^{3}}}{y}} \]

      6. un-div-inv 71.3%

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

      7. cube-mult 71.3%

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

      8. distribute-rgt1-in 82.2%

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

      9. *-commutative 82.2%

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

      10. associate-/l* 84.1%

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

      11. pow2 84.1%

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

      12. +-commutative 84.1%

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

   6. Applied egg-rr 84.1%

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

      1. associate-/r* 85.0%

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

      2. +-commutative 85.0%

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

   8. Simplified 85.0%

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

      1. *-un-lft-identity 85.0%

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

      2. unpow2 85.0%

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

      3. times-frac 99.7%

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

   10. Applied egg-rr 99.7%

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

       1. associate-*l/ 99.7%

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

       2. *-lft-identity 99.7%

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

   12. Simplified 99.7%

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

   13. Taylor expanded in x around 0 84.5%

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

4. Final simplification 85.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.1 \cdot 10^{+154}:\\ \;\;\;\;\frac{\frac{y}{x + y}}{x + \left(y + 1\right)}\\ \mathbf{elif}\;x \leq -1.6 \cdot 10^{-30}:\\ \;\;\;\;\frac{y}{\left(x + y\right) \cdot \left(y + \left(x + 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{x}{x + y}}{x + y}}{\frac{y + 1}{y}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 95.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 3.8 \cdot 10^{+125}:\\ \;\;\;\;\frac{x \cdot \frac{y}{x + y}}{\left(x + y\right) \cdot \left(y + \left(x + 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{x}{x + y}}{x + y}}{\frac{y + 1}{y}}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 3.8e+125)
   (/ (* x (/ y (+ x y))) (* (+ x y) (+ y (+ x 1.0))))
   (/ (/ (/ x (+ x y)) (+ x y)) (/ (+ y 1.0) y))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 3.8e+125) {
		tmp = (x * (y / (x + y))) / ((x + y) * (y + (x + 1.0)));
	} else {
		tmp = ((x / (x + y)) / (x + y)) / ((y + 1.0) / y);
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 3.8d+125) then
        tmp = (x * (y / (x + y))) / ((x + y) * (y + (x + 1.0d0)))
    else
        tmp = ((x / (x + y)) / (x + y)) / ((y + 1.0d0) / y)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (y <= 3.8e+125) {
		tmp = (x * (y / (x + y))) / ((x + y) * (y + (x + 1.0)));
	} else {
		tmp = ((x / (x + y)) / (x + y)) / ((y + 1.0) / y);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if y <= 3.8e+125:
		tmp = (x * (y / (x + y))) / ((x + y) * (y + (x + 1.0)))
	else:
		tmp = ((x / (x + y)) / (x + y)) / ((y + 1.0) / y)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= 3.8e+125)
		tmp = Float64(Float64(x * Float64(y / Float64(x + y))) / Float64(Float64(x + y) * Float64(y + Float64(x + 1.0))));
	else
		tmp = Float64(Float64(Float64(x / Float64(x + y)) / Float64(x + y)) / Float64(Float64(y + 1.0) / y));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 3.8e+125)
		tmp = (x * (y / (x + y))) / ((x + y) * (y + (x + 1.0)));
	else
		tmp = ((x / (x + y)) / (x + y)) / ((y + 1.0) / y);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[y, 3.8e+125], N[(N[(x * N[(y / N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(x + y), $MachinePrecision] * N[(y + N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x / N[(x + y), $MachinePrecision]), $MachinePrecision] / N[(x + y), $MachinePrecision]), $MachinePrecision] / N[(N[(y + 1.0), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 3.80000000000000002e125

   1. Initial program 69.2%

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

      1. associate-*l* 69.2%

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

      2. times-frac 95.0%

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

      3. +-commutative 95.0%

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

      4. +-commutative 95.0%

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

      5. associate-+r+ 95.0%

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

      6. +-commutative 95.0%

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

      7. associate-+l+ 95.0%

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

   4. Applied egg-rr 95.0%

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

      1. *-commutative 95.0%

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

      2. associate-/r* 99.8%

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

      3. associate-+r+ 99.8%

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

      4. +-commutative 99.8%

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

      5. frac-times 95.0%

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

      6. *-commutative 95.0%

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

      7. +-commutative 95.0%

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

      8. associate-+r+ 95.0%

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

      9. distribute-lft-in 92.6%

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

      10. +-commutative 92.6%

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

      11. distribute-lft-in 95.0%

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

   6. Applied egg-rr 95.0%

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

   if 3.80000000000000002e125 < y

   1. Initial program 61.9%

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

      1. associate-/l* 79.5%

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

      2. associate-+l+ 79.5%

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

   3. Simplified 79.5%

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

      1. clear-num 79.5%

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

      2. associate-+r+ 79.5%

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

      3. *-commutative 79.5%

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

      4. distribute-rgt1-in 79.5%

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

      5. cube-mult 79.5%

         \[\leadsto x \cdot \frac{1}{\frac{\left(x + y\right) \cdot \left(x + y\right) + \color{blue}{{\left(x + y\right)}^{3}}}{y}} \]

      6. un-div-inv 79.5%

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

      7. cube-mult 79.5%

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

      8. distribute-rgt1-in 79.5%

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

      9. *-commutative 79.5%

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

      10. associate-/l* 82.4%

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

      11. pow2 82.4%

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

      12. +-commutative 82.4%

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

   6. Applied egg-rr 82.4%

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

      1. associate-/r* 82.4%

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

      2. +-commutative 82.4%

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

   8. Simplified 82.4%

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

      1. *-un-lft-identity 82.4%

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

      2. unpow2 82.4%

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

      3. times-frac 99.8%

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

   10. Applied egg-rr 99.8%

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

       1. associate-*l/ 100.0%

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

       2. *-lft-identity 100.0%

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

   12. Simplified 100.0%

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

   13. Taylor expanded in x around 0 94.1%

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

4. Final simplification 94.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 3.8 \cdot 10^{+125}:\\ \;\;\;\;\frac{x \cdot \frac{y}{x + y}}{\left(x + y\right) \cdot \left(y + \left(x + 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{x}{x + y}}{x + y}}{\frac{y + 1}{y}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 95.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_0 := \frac{x}{x + y}\\ \mathbf{if}\;y \leq 3.8 \cdot 10^{+125}:\\ \;\;\;\;t\_0 \cdot \frac{y}{\left(x + y\right) \cdot \left(y + \left(x + 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{t\_0}{x + y}}{\frac{y + 1}{y}}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ x (+ x y))))
   (if (<= y 3.8e+125)
     (* t_0 (/ y (* (+ x y) (+ y (+ x 1.0)))))
     (/ (/ t_0 (+ x y)) (/ (+ y 1.0) y)))))

C

assert(x < y);
double code(double x, double y) {
	double t_0 = x / (x + y);
	double tmp;
	if (y <= 3.8e+125) {
		tmp = t_0 * (y / ((x + y) * (y + (x + 1.0))));
	} else {
		tmp = (t_0 / (x + y)) / ((y + 1.0) / y);
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x / (x + y)
    if (y <= 3.8d+125) then
        tmp = t_0 * (y / ((x + y) * (y + (x + 1.0d0))))
    else
        tmp = (t_0 / (x + y)) / ((y + 1.0d0) / y)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double t_0 = x / (x + y);
	double tmp;
	if (y <= 3.8e+125) {
		tmp = t_0 * (y / ((x + y) * (y + (x + 1.0))));
	} else {
		tmp = (t_0 / (x + y)) / ((y + 1.0) / y);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	t_0 = x / (x + y)
	tmp = 0
	if y <= 3.8e+125:
		tmp = t_0 * (y / ((x + y) * (y + (x + 1.0))))
	else:
		tmp = (t_0 / (x + y)) / ((y + 1.0) / y)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	t_0 = Float64(x / Float64(x + y))
	tmp = 0.0
	if (y <= 3.8e+125)
		tmp = Float64(t_0 * Float64(y / Float64(Float64(x + y) * Float64(y + Float64(x + 1.0)))));
	else
		tmp = Float64(Float64(t_0 / Float64(x + y)) / Float64(Float64(y + 1.0) / y));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	t_0 = x / (x + y);
	tmp = 0.0;
	if (y <= 3.8e+125)
		tmp = t_0 * (y / ((x + y) * (y + (x + 1.0))));
	else
		tmp = (t_0 / (x + y)) / ((y + 1.0) / y);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := Block[{t$95$0 = N[(x / N[(x + y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, 3.8e+125], N[(t$95$0 * N[(y / N[(N[(x + y), $MachinePrecision] * N[(y + N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$0 / N[(x + y), $MachinePrecision]), $MachinePrecision] / N[(N[(y + 1.0), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < 3.80000000000000002e125

   1. Initial program 69.2%

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

      1. associate-*l* 69.2%

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

      2. times-frac 95.0%

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

      3. +-commutative 95.0%

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

      4. +-commutative 95.0%

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

      5. associate-+r+ 95.0%

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

      6. +-commutative 95.0%

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

      7. associate-+l+ 95.0%

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

   4. Applied egg-rr 95.0%

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

   if 3.80000000000000002e125 < y

   1. Initial program 61.9%

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

      1. associate-/l* 79.5%

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

      2. associate-+l+ 79.5%

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

   3. Simplified 79.5%

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

      1. clear-num 79.5%

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

      2. associate-+r+ 79.5%

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

      3. *-commutative 79.5%

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

      4. distribute-rgt1-in 79.5%

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

      5. cube-mult 79.5%

         \[\leadsto x \cdot \frac{1}{\frac{\left(x + y\right) \cdot \left(x + y\right) + \color{blue}{{\left(x + y\right)}^{3}}}{y}} \]

      6. un-div-inv 79.5%

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

      7. cube-mult 79.5%

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

      8. distribute-rgt1-in 79.5%

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

      9. *-commutative 79.5%

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

      10. associate-/l* 82.4%

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

      11. pow2 82.4%

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

      12. +-commutative 82.4%

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

   6. Applied egg-rr 82.4%

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

      1. associate-/r* 82.4%

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

      2. +-commutative 82.4%

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

   8. Simplified 82.4%

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

      1. *-un-lft-identity 82.4%

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

      2. unpow2 82.4%

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

      3. times-frac 99.8%

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

   10. Applied egg-rr 99.8%

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

       1. associate-*l/ 100.0%

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

       2. *-lft-identity 100.0%

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

   12. Simplified 100.0%

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

   13. Taylor expanded in x around 0 94.1%

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

4. Final simplification 94.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 3.8 \cdot 10^{+125}:\\ \;\;\;\;\frac{x}{x + y} \cdot \frac{y}{\left(x + y\right) \cdot \left(y + \left(x + 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{x}{x + y}}{x + y}}{\frac{y + 1}{y}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 86.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 3.5 \cdot 10^{-160}:\\ \;\;\;\;\frac{\frac{y}{x + 1}}{x}\\ \mathbf{elif}\;y \leq 2.85 \cdot 10^{+109}:\\ \;\;\;\;\frac{x}{\left(x + y\right) \cdot \left(y + \left(x + 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + y} \cdot \frac{1}{y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 3.5e-160)
   (/ (/ y (+ x 1.0)) x)
   (if (<= y 2.85e+109)
     (/ x (* (+ x y) (+ y (+ x 1.0))))
     (* (/ x (+ x y)) (/ 1.0 y)))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 3.5e-160) {
		tmp = (y / (x + 1.0)) / x;
	} else if (y <= 2.85e+109) {
		tmp = x / ((x + y) * (y + (x + 1.0)));
	} else {
		tmp = (x / (x + y)) * (1.0 / y);
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 3.5d-160) then
        tmp = (y / (x + 1.0d0)) / x
    else if (y <= 2.85d+109) then
        tmp = x / ((x + y) * (y + (x + 1.0d0)))
    else
        tmp = (x / (x + y)) * (1.0d0 / y)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (y <= 3.5e-160) {
		tmp = (y / (x + 1.0)) / x;
	} else if (y <= 2.85e+109) {
		tmp = x / ((x + y) * (y + (x + 1.0)));
	} else {
		tmp = (x / (x + y)) * (1.0 / y);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if y <= 3.5e-160:
		tmp = (y / (x + 1.0)) / x
	elif y <= 2.85e+109:
		tmp = x / ((x + y) * (y + (x + 1.0)))
	else:
		tmp = (x / (x + y)) * (1.0 / y)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= 3.5e-160)
		tmp = Float64(Float64(y / Float64(x + 1.0)) / x);
	elseif (y <= 2.85e+109)
		tmp = Float64(x / Float64(Float64(x + y) * Float64(y + Float64(x + 1.0))));
	else
		tmp = Float64(Float64(x / Float64(x + y)) * Float64(1.0 / y));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 3.5e-160)
		tmp = (y / (x + 1.0)) / x;
	elseif (y <= 2.85e+109)
		tmp = x / ((x + y) * (y + (x + 1.0)));
	else
		tmp = (x / (x + y)) * (1.0 / y);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[y, 3.5e-160], N[(N[(y / N[(x + 1.0), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[y, 2.85e+109], N[(x / N[(N[(x + y), $MachinePrecision] * N[(y + N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(x + y), $MachinePrecision]), $MachinePrecision] * N[(1.0 / y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 3.5000000000000003e-160

   1. Initial program 65.2%

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

      1. associate-/l* 80.2%

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

      2. associate-+l+ 80.2%

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

   3. Simplified 80.2%

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

      1. clear-num 80.2%

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

      2. associate-+r+ 80.2%

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

      3. *-commutative 80.2%

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

      4. distribute-rgt1-in 53.9%

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

      5. cube-mult 53.9%

         \[\leadsto x \cdot \frac{1}{\frac{\left(x + y\right) \cdot \left(x + y\right) + \color{blue}{{\left(x + y\right)}^{3}}}{y}} \]

      6. un-div-inv 54.0%

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

      7. cube-mult 54.0%

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

      8. distribute-rgt1-in 80.3%

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

      9. *-commutative 80.3%

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

      10. associate-/l* 81.4%

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

      11. pow2 81.4%

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

      12. +-commutative 81.4%

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

   6. Applied egg-rr 81.4%

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

      1. associate-/r* 83.6%

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

      2. +-commutative 83.6%

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

   8. Simplified 83.6%

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

      1. *-un-lft-identity 83.6%

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

      2. unpow2 83.6%

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

      3. times-frac 99.7%

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

   10. Applied egg-rr 99.7%

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

       1. associate-*l/ 99.7%

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

       2. *-lft-identity 99.7%

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

   12. Simplified 99.7%

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

   13. Taylor expanded in y around 0 51.6%

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

       1. +-commutative 51.6%

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

   15. Simplified 51.6%

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

       1. *-un-lft-identity 51.6%

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

       2. times-frac 53.0%

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

   17. Applied egg-rr 53.0%

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

       1. associate-*l/ 53.1%

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

       2. *-lft-identity 53.1%

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

       3. +-commutative 53.1%

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

   19. Simplified 53.1%

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

   if 3.5000000000000003e-160 < y < 2.8500000000000001e109

   1. Initial program 82.8%

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

      1. associate-*l* 82.9%

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

      2. times-frac 96.2%

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

      3. +-commutative 96.2%

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

      4. +-commutative 96.2%

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

      5. associate-+r+ 96.2%

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

      6. +-commutative 96.2%

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

      7. associate-+l+ 96.2%

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

   4. Applied egg-rr 96.2%

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

      1. *-commutative 96.2%

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

      2. associate-/r* 99.8%

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

      3. associate-+r+ 99.8%

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

      4. +-commutative 99.8%

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

      5. frac-times 96.2%

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

      6. *-commutative 96.2%

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

      7. +-commutative 96.2%

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

      8. associate-+r+ 96.2%

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

      9. distribute-lft-in 94.5%

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

      10. +-commutative 94.5%

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

      11. distribute-lft-in 96.2%

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

   6. Applied egg-rr 96.2%

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

   7. Taylor expanded in y around inf 75.5%

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

   if 2.8500000000000001e109 < y

   1. Initial program 59.2%

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

      1. associate-*l* 59.2%

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

      2. times-frac 82.2%

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

      3. +-commutative 82.2%

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

      4. +-commutative 82.2%

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

      5. associate-+r+ 82.2%

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

      6. +-commutative 82.2%

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

      7. associate-+l+ 82.2%

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

   4. Applied egg-rr 82.2%

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

   5. Taylor expanded in y around inf 89.4%

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

4. Final simplification 63.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 3.5 \cdot 10^{-160}:\\ \;\;\;\;\frac{\frac{y}{x + 1}}{x}\\ \mathbf{elif}\;y \leq 2.85 \cdot 10^{+109}:\\ \;\;\;\;\frac{x}{\left(x + y\right) \cdot \left(y + \left(x + 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + y} \cdot \frac{1}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 80.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -4.7 \cdot 10^{+139}:\\ \;\;\;\;\frac{\frac{y}{x}}{x}\\ \mathbf{elif}\;x \leq -2.8 \cdot 10^{-153}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -4.7e+139)
   (/ (/ y x) x)
   (if (<= x -2.8e-153) (/ y (* x (+ x 1.0))) (/ x (* y (+ y 1.0))))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -4.7e+139) {
		tmp = (y / x) / x;
	} else if (x <= -2.8e-153) {
		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 <= (-4.7d+139)) then
        tmp = (y / x) / x
    else if (x <= (-2.8d-153)) 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 <= -4.7e+139) {
		tmp = (y / x) / x;
	} else if (x <= -2.8e-153) {
		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 <= -4.7e+139:
		tmp = (y / x) / x
	elif x <= -2.8e-153:
		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 <= -4.7e+139)
		tmp = Float64(Float64(y / x) / x);
	elseif (x <= -2.8e-153)
		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 <= -4.7e+139)
		tmp = (y / x) / x;
	elseif (x <= -2.8e-153)
		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, -4.7e+139], N[(N[(y / x), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[x, -2.8e-153], 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 < -4.7000000000000001e139

   1. Initial program 71.4%

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

      1. associate-/l* 88.2%

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

      2. associate-+l+ 88.2%

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

   3. Simplified 88.2%

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

   5. Taylor expanded in y around 0 88.2%

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

      1. associate-/r* 90.8%

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

      2. +-commutative 90.8%

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

   7. Simplified 90.8%

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

   8. Taylor expanded in x around inf 90.8%

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

   if -4.7000000000000001e139 < x < -2.8000000000000001e-153

   1. Initial program 78.1%

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

      1. associate-/l* 86.2%

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

      2. associate-+l+ 86.2%

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

   3. Simplified 86.2%

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

   5. Taylor expanded in y around 0 53.2%

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

   if -2.8000000000000001e-153 < x

   1. Initial program 64.7%

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

      1. associate-/l* 79.6%

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

      2. associate-+l+ 79.6%

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

   3. Simplified 79.6%

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

   5. Taylor expanded in x around 0 61.0%

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

      1. +-commutative 61.0%

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

   7. Simplified 61.0%

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

4. Final simplification 63.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.7 \cdot 10^{+139}:\\ \;\;\;\;\frac{\frac{y}{x}}{x}\\ \mathbf{elif}\;x \leq -2.8 \cdot 10^{-153}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 78.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{\frac{y}{x}}{x}\\ \mathbf{elif}\;x \leq -2.8 \cdot 10^{-153}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \end{array} \end{array} \]

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 -2.8e-153) (/ 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 <= -2.8e-153) {
		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 <= (-2.8d-153)) 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 <= -2.8e-153) {
		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 <= -2.8e-153:
		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 <= -2.8e-153)
		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 <= -2.8e-153)
		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, -2.8e-153], 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 70.4%

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

      1. associate-/l* 80.6%

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

      2. associate-+l+ 80.6%

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

   3. Simplified 80.6%

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

   5. Taylor expanded in y around 0 77.9%

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

      1. associate-/r* 79.3%

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

      2. +-commutative 79.3%

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

   7. Simplified 79.3%

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

   8. Taylor expanded in x around inf 79.2%

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

   if -1 < x < -2.8000000000000001e-153

   1. Initial program 86.1%

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

      1. associate-/l* 99.6%

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

      2. associate-+l+ 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in y around 0 42.5%

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

      1. associate-/r* 42.5%

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

      2. +-commutative 42.5%

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

   7. Simplified 42.5%

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

   8. Taylor expanded in x around 0 42.0%

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

   if -2.8000000000000001e-153 < x

   1. Initial program 64.7%

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

      1. associate-/l* 79.6%

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

      2. associate-+l+ 79.6%

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

   3. Simplified 79.6%

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

   5. Taylor expanded in x around 0 61.0%

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

      1. +-commutative 61.0%

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

   7. Simplified 61.0%

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

Alternative 12: 81.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -2.8 \cdot 10^{-153}:\\ \;\;\;\;\frac{\frac{y}{x + 1}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + y} \cdot \frac{1}{y + 1}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -2.8e-153)
   (/ (/ y (+ x 1.0)) x)
   (* (/ x (+ x y)) (/ 1.0 (+ y 1.0)))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -2.8e-153) {
		tmp = (y / (x + 1.0)) / x;
	} 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) :: tmp
    if (x <= (-2.8d-153)) then
        tmp = (y / (x + 1.0d0)) / x
    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 tmp;
	if (x <= -2.8e-153) {
		tmp = (y / (x + 1.0)) / x;
	} else {
		tmp = (x / (x + y)) * (1.0 / (y + 1.0));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -2.8e-153:
		tmp = (y / (x + 1.0)) / x
	else:
		tmp = (x / (x + y)) * (1.0 / (y + 1.0))
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -2.8e-153)
		tmp = Float64(Float64(y / Float64(x + 1.0)) / x);
	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)
	tmp = 0.0;
	if (x <= -2.8e-153)
		tmp = (y / (x + 1.0)) / x;
	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_] := If[LessEqual[x, -2.8e-153], N[(N[(y / N[(x + 1.0), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], N[(N[(x / N[(x + y), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2.8000000000000001e-153

   1. Initial program 75.6%

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

      1. associate-/l* 86.9%

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

      2. associate-+l+ 86.9%

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

   3. Simplified 86.9%

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

      1. clear-num 86.9%

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

      2. associate-+r+ 86.9%

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

      3. *-commutative 86.9%

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

      4. distribute-rgt1-in 47.0%

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

      5. cube-mult 47.0%

         \[\leadsto x \cdot \frac{1}{\frac{\left(x + y\right) \cdot \left(x + y\right) + \color{blue}{{\left(x + y\right)}^{3}}}{y}} \]

      6. un-div-inv 47.1%

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

      7. cube-mult 47.1%

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

      8. distribute-rgt1-in 86.9%

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

      9. *-commutative 86.9%

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

      10. associate-/l* 88.0%

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

      11. pow2 88.0%

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

      12. +-commutative 88.0%

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

   6. Applied egg-rr 88.0%

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

      1. associate-/r* 92.3%

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

      2. +-commutative 92.3%

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

   8. Simplified 92.3%

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

      1. *-un-lft-identity 92.3%

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

      2. unpow2 92.3%

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

      3. times-frac 99.6%

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

   10. Applied egg-rr 99.6%

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

       1. associate-*l/ 99.7%

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

       2. *-lft-identity 99.7%

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

   12. Simplified 99.7%

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

   13. Taylor expanded in y around 0 66.1%

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

       1. +-commutative 66.1%

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

   15. Simplified 66.1%

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

       1. *-un-lft-identity 66.1%

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

       2. times-frac 66.9%

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

   17. Applied egg-rr 66.9%

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

       1. associate-*l/ 67.0%

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

       2. *-lft-identity 67.0%

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

       3. +-commutative 67.0%

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

   19. Simplified 67.0%

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

   if -2.8000000000000001e-153 < x

   1. Initial program 64.7%

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

      1. associate-*l* 64.7%

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

      2. times-frac 94.0%

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

      3. +-commutative 94.0%

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

      4. +-commutative 94.0%

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

      5. associate-+r+ 94.0%

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

      6. +-commutative 94.0%

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

      7. associate-+l+ 94.0%

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

   4. Applied egg-rr 94.0%

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

   5. Taylor expanded in x around 0 63.4%

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

      1. +-commutative 63.4%

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

   7. Simplified 63.4%

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

4. Final simplification 64.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.8 \cdot 10^{-153}:\\ \;\;\;\;\frac{\frac{y}{x + 1}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + y} \cdot \frac{1}{y + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 81.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -2.8 \cdot 10^{-153}:\\ \;\;\;\;\frac{\frac{y}{x + 1}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y} \cdot \frac{x}{y + 1}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -2.8e-153) (/ (/ 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 <= -2.8e-153) {
		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 <= (-2.8d-153)) 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 <= -2.8e-153) {
		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 <= -2.8e-153:
		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 <= -2.8e-153)
		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 <= -2.8e-153)
		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, -2.8e-153], 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 < -2.8000000000000001e-153

   1. Initial program 75.6%

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

      1. associate-/l* 86.9%

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

      2. associate-+l+ 86.9%

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

   3. Simplified 86.9%

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

      1. clear-num 86.9%

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

      2. associate-+r+ 86.9%

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

      3. *-commutative 86.9%

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

      4. distribute-rgt1-in 47.0%

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

      5. cube-mult 47.0%

         \[\leadsto x \cdot \frac{1}{\frac{\left(x + y\right) \cdot \left(x + y\right) + \color{blue}{{\left(x + y\right)}^{3}}}{y}} \]

      6. un-div-inv 47.1%

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

      7. cube-mult 47.1%

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

      8. distribute-rgt1-in 86.9%

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

      9. *-commutative 86.9%

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

      10. associate-/l* 88.0%

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

      11. pow2 88.0%

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

      12. +-commutative 88.0%

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

   6. Applied egg-rr 88.0%

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

      1. associate-/r* 92.3%

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

      2. +-commutative 92.3%

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

   8. Simplified 92.3%

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

      1. *-un-lft-identity 92.3%

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

      2. unpow2 92.3%

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

      3. times-frac 99.6%

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

   10. Applied egg-rr 99.6%

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

       1. associate-*l/ 99.7%

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

       2. *-lft-identity 99.7%

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

   12. Simplified 99.7%

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

   13. Taylor expanded in y around 0 66.1%

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

       1. +-commutative 66.1%

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

   15. Simplified 66.1%

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

       1. *-un-lft-identity 66.1%

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

       2. times-frac 66.9%

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

   17. Applied egg-rr 66.9%

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

       1. associate-*l/ 67.0%

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

       2. *-lft-identity 67.0%

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

       3. +-commutative 67.0%

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

   19. Simplified 67.0%

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

   if -2.8000000000000001e-153 < x

   1. Initial program 64.7%

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

      1. associate-/l* 79.6%

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

      2. associate-+l+ 79.6%

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

   3. Simplified 79.6%

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

   5. Taylor expanded in x around 0 61.0%

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

      1. +-commutative 61.0%

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

   7. Simplified 61.0%

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

      1. *-un-lft-identity 61.0%

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

      2. times-frac 62.9%

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

   9. Applied egg-rr 62.9%

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

4. Final simplification 64.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.8 \cdot 10^{-153}:\\ \;\;\;\;\frac{\frac{y}{x + 1}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y} \cdot \frac{x}{y + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 80.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -2.8 \cdot 10^{-153}:\\ \;\;\;\;\frac{\frac{y}{x + 1}}{x}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{\frac{1}{y}}{y + 1}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -2.8e-153) (/ (/ y (+ x 1.0)) x) (* x (/ (/ 1.0 y) (+ y 1.0)))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -2.8e-153) {
		tmp = (y / (x + 1.0)) / x;
	} else {
		tmp = x * ((1.0 / 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 <= (-2.8d-153)) then
        tmp = (y / (x + 1.0d0)) / x
    else
        tmp = x * ((1.0d0 / 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 <= -2.8e-153) {
		tmp = (y / (x + 1.0)) / x;
	} else {
		tmp = x * ((1.0 / y) / (y + 1.0));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -2.8e-153:
		tmp = (y / (x + 1.0)) / x
	else:
		tmp = x * ((1.0 / y) / (y + 1.0))
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -2.8e-153)
		tmp = Float64(Float64(y / Float64(x + 1.0)) / x);
	else
		tmp = Float64(x * Float64(Float64(1.0 / 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 <= -2.8e-153)
		tmp = (y / (x + 1.0)) / x;
	else
		tmp = x * ((1.0 / 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, -2.8e-153], N[(N[(y / N[(x + 1.0), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], N[(x * N[(N[(1.0 / y), $MachinePrecision] / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2.8000000000000001e-153

   1. Initial program 75.6%

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

      1. associate-/l* 86.9%

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

      2. associate-+l+ 86.9%

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

   3. Simplified 86.9%

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

      1. clear-num 86.9%

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

      2. associate-+r+ 86.9%

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

      3. *-commutative 86.9%

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

      4. distribute-rgt1-in 47.0%

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

      5. cube-mult 47.0%

         \[\leadsto x \cdot \frac{1}{\frac{\left(x + y\right) \cdot \left(x + y\right) + \color{blue}{{\left(x + y\right)}^{3}}}{y}} \]

      6. un-div-inv 47.1%

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

      7. cube-mult 47.1%

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

      8. distribute-rgt1-in 86.9%

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

      9. *-commutative 86.9%

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

      10. associate-/l* 88.0%

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

      11. pow2 88.0%

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

      12. +-commutative 88.0%

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

   6. Applied egg-rr 88.0%

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

      1. associate-/r* 92.3%

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

      2. +-commutative 92.3%

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

   8. Simplified 92.3%

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

      1. *-un-lft-identity 92.3%

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

      2. unpow2 92.3%

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

      3. times-frac 99.6%

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

   10. Applied egg-rr 99.6%

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

       1. associate-*l/ 99.7%

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

       2. *-lft-identity 99.7%

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

   12. Simplified 99.7%

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

   13. Taylor expanded in y around 0 66.1%

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

       1. +-commutative 66.1%

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

   15. Simplified 66.1%

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

       1. *-un-lft-identity 66.1%

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

       2. times-frac 66.9%

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

   17. Applied egg-rr 66.9%

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

       1. associate-*l/ 67.0%

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

       2. *-lft-identity 67.0%

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

       3. +-commutative 67.0%

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

   19. Simplified 67.0%

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

   if -2.8000000000000001e-153 < x

   1. Initial program 64.7%

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

      1. associate-/l* 79.6%

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

      2. associate-+l+ 79.6%

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

   3. Simplified 79.6%

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

   5. Taylor expanded in x around 0 60.9%

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

      1. associate-/r* 61.4%

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

      2. +-commutative 61.4%

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

   7. Simplified 61.4%

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

4. Final simplification 63.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.8 \cdot 10^{-153}:\\ \;\;\;\;\frac{\frac{y}{x + 1}}{x}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{\frac{1}{y}}{y + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 67.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{\frac{y}{x}}{x}\\ \mathbf{elif}\;x \leq -2.7 \cdot 10^{-153}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

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 -2.7e-153) (/ 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 <= -2.7e-153) {
		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 <= (-2.7d-153)) 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 <= -2.7e-153) {
		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 <= -2.7e-153:
		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 <= -2.7e-153)
		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 <= -2.7e-153)
		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, -2.7e-153], N[(y / x), $MachinePrecision], N[(x / y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1

   1. Initial program 70.4%

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

      1. associate-/l* 80.6%

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

      2. associate-+l+ 80.6%

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

   3. Simplified 80.6%

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

   5. Taylor expanded in y around 0 77.9%

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

      1. associate-/r* 79.3%

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

      2. +-commutative 79.3%

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

   7. Simplified 79.3%

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

   8. Taylor expanded in x around inf 79.2%

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

   if -1 < x < -2.70000000000000009e-153

   1. Initial program 86.1%

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

      1. associate-/l* 99.6%

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

      2. associate-+l+ 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in y around 0 42.5%

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

      1. associate-/r* 42.5%

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

      2. +-commutative 42.5%

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

   7. Simplified 42.5%

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

   8. Taylor expanded in x around 0 42.0%

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

   if -2.70000000000000009e-153 < x

   1. Initial program 64.7%

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

      1. associate-/l* 79.6%

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

      2. associate-+l+ 79.6%

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

   3. Simplified 79.6%

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

   5. Taylor expanded in x around 0 61.0%

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

      1. +-commutative 61.0%

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

   7. Simplified 61.0%

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

   8. Taylor expanded in y around 0 43.4%

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

Alternative 16: 65.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{y}{x \cdot x}\\ \mathbf{elif}\;x \leq -1.05 \cdot 10^{-153}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

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 -1.05e-153) (/ 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 <= -1.05e-153) {
		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 <= (-1.05d-153)) 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 <= -1.05e-153) {
		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 <= -1.05e-153:
		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(y / Float64(x * x));
	elseif (x <= -1.05e-153)
		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 <= -1.05e-153)
		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[(y / N[(x * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -1.05e-153], N[(y / x), $MachinePrecision], N[(x / y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1

   1. Initial program 70.4%

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

      1. associate-/l* 80.6%

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

      2. associate-+l+ 80.6%

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

   3. Simplified 80.6%

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

      1. clear-num 80.5%

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

      2. associate-+r+ 80.5%

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

      3. *-commutative 80.5%

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

      4. distribute-rgt1-in 29.7%

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

      5. cube-mult 29.7%

         \[\leadsto x \cdot \frac{1}{\frac{\left(x + y\right) \cdot \left(x + y\right) + \color{blue}{{\left(x + y\right)}^{3}}}{y}} \]

      6. un-div-inv 29.7%

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

      7. cube-mult 29.7%

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

      8. distribute-rgt1-in 80.5%

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

      9. *-commutative 80.5%

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

      10. associate-/l* 82.2%

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

      11. pow2 82.2%

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

      12. +-commutative 82.2%

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

   6. Applied egg-rr 82.2%

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

      1. associate-/r* 88.7%

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

      2. +-commutative 88.7%

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

   8. Simplified 88.7%

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

      1. *-un-lft-identity 88.7%

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

      2. unpow2 88.7%

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

      3. times-frac 99.6%

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

   10. Applied egg-rr 99.6%

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

       1. associate-*l/ 99.7%

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

       2. *-lft-identity 99.7%

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

   12. Simplified 99.7%

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

   13. Taylor expanded in y around 0 77.9%

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

       1. +-commutative 77.9%

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

   15. Simplified 77.9%

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

   16. Taylor expanded in x around inf 77.7%

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

   if -1 < x < -1.05000000000000002e-153

   1. Initial program 86.1%

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

      1. associate-/l* 99.6%

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

      2. associate-+l+ 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in y around 0 42.5%

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

      1. associate-/r* 42.5%

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

      2. +-commutative 42.5%

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

   7. Simplified 42.5%

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

   8. Taylor expanded in x around 0 42.0%

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

   if -1.05000000000000002e-153 < x

   1. Initial program 64.7%

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

      1. associate-/l* 79.6%

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

      2. associate-+l+ 79.6%

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

   3. Simplified 79.6%

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

   5. Taylor expanded in x around 0 61.0%

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

      1. +-commutative 61.0%

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

   7. Simplified 61.0%

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

   8. Taylor expanded in y around 0 43.4%

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

Alternative 17: 80.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -2.8 \cdot 10^{-153}:\\ \;\;\;\;\frac{\frac{y}{x + 1}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -2.8e-153) (/ (/ y (+ x 1.0)) x) (/ x (* y (+ y 1.0)))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -2.8e-153) {
		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 <= (-2.8d-153)) 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 <= -2.8e-153) {
		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 <= -2.8e-153:
		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 <= -2.8e-153)
		tmp = Float64(Float64(y / Float64(x + 1.0)) / 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 <= -2.8e-153)
		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, -2.8e-153], N[(N[(y / N[(x + 1.0), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], N[(x / N[(y * N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2.8000000000000001e-153

   1. Initial program 75.6%

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

      1. associate-/l* 86.9%

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

      2. associate-+l+ 86.9%

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

   3. Simplified 86.9%

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

      1. clear-num 86.9%

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

      2. associate-+r+ 86.9%

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

      3. *-commutative 86.9%

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

      4. distribute-rgt1-in 47.0%

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

      5. cube-mult 47.0%

         \[\leadsto x \cdot \frac{1}{\frac{\left(x + y\right) \cdot \left(x + y\right) + \color{blue}{{\left(x + y\right)}^{3}}}{y}} \]

      6. un-div-inv 47.1%

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

      7. cube-mult 47.1%

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

      8. distribute-rgt1-in 86.9%

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

      9. *-commutative 86.9%

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

      10. associate-/l* 88.0%

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

      11. pow2 88.0%

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

      12. +-commutative 88.0%

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

   6. Applied egg-rr 88.0%

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

      1. associate-/r* 92.3%

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

      2. +-commutative 92.3%

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

   8. Simplified 92.3%

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

      1. *-un-lft-identity 92.3%

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

      2. unpow2 92.3%

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

      3. times-frac 99.6%

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

   10. Applied egg-rr 99.6%

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

       1. associate-*l/ 99.7%

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

       2. *-lft-identity 99.7%

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

   12. Simplified 99.7%

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

   13. Taylor expanded in y around 0 66.1%

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

       1. +-commutative 66.1%

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

   15. Simplified 66.1%

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

       1. *-un-lft-identity 66.1%

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

       2. times-frac 66.9%

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

   17. Applied egg-rr 66.9%

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

       1. associate-*l/ 67.0%

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

       2. *-lft-identity 67.0%

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

       3. +-commutative 67.0%

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

   19. Simplified 67.0%

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

   if -2.8000000000000001e-153 < x

   1. Initial program 64.7%

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

      1. associate-/l* 79.6%

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

      2. associate-+l+ 79.6%

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

   3. Simplified 79.6%

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

   5. Taylor expanded in x around 0 61.0%

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

      1. +-commutative 61.0%

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

   7. Simplified 61.0%

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

4. Final simplification 63.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.8 \cdot 10^{-153}:\\ \;\;\;\;\frac{\frac{y}{x + 1}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 80.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -2.8 \cdot 10^{-153}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -2.8e-153) (/ (/ y x) (+ x 1.0)) (/ x (* y (+ y 1.0)))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -2.8e-153) {
		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 <= (-2.8d-153)) 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 <= -2.8e-153) {
		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 <= -2.8e-153:
		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 <= -2.8e-153)
		tmp = Float64(Float64(y / 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 <= -2.8e-153)
		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, -2.8e-153], N[(N[(y / x), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], N[(x / N[(y * N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2.8000000000000001e-153

   1. Initial program 75.6%

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

      1. associate-/l* 86.9%

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

      2. associate-+l+ 86.9%

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

   3. Simplified 86.9%

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

   5. Taylor expanded in y around 0 66.1%

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

      1. associate-/r* 67.1%

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

      2. +-commutative 67.1%

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

   7. Simplified 67.1%

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

   if -2.8000000000000001e-153 < x

   1. Initial program 64.7%

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

      1. associate-/l* 79.6%

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

      2. associate-+l+ 79.6%

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

   3. Simplified 79.6%

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

   5. Taylor expanded in x around 0 61.0%

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

      1. +-commutative 61.0%

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

   7. Simplified 61.0%

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

Alternative 19: 44.5% accurate, 2.1× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -2.4 \cdot 10^{-153}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y) :precision binary64 (if (<= x -2.4e-153) (/ y x) (/ x y)))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -2.4e-153) {
		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 <= (-2.4d-153)) 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 <= -2.4e-153) {
		tmp = y / x;
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -2.4e-153:
		tmp = y / x
	else:
		tmp = x / y
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -2.4e-153)
		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 <= -2.4e-153)
		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, -2.4e-153], N[(y / x), $MachinePrecision], N[(x / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2.4000000000000002e-153

   1. Initial program 75.6%

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

      1. associate-/l* 86.9%

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

      2. associate-+l+ 86.9%

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

   3. Simplified 86.9%

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

   5. Taylor expanded in y around 0 66.1%

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

      1. associate-/r* 67.1%

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

      2. +-commutative 67.1%

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

   7. Simplified 67.1%

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

   8. Taylor expanded in x around 0 36.2%

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

   if -2.4000000000000002e-153 < x

   1. Initial program 64.7%

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

      1. associate-/l* 79.6%

         \[\leadsto \color{blue}{x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      2. associate-+l+ 79.6%

         \[\leadsto x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \color{blue}{\left(x + \left(y + 1\right)\right)}} \]

   3. Simplified 79.6%

      \[\leadsto \color{blue}{x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(x + \left(y + 1\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 61.0%

      \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]
   6. Step-by-step derivation

      1. +-commutative 61.0%

         \[\leadsto \frac{x}{y \cdot \color{blue}{\left(y + 1\right)}} \]

   7. Simplified 61.0%

      \[\leadsto \color{blue}{\frac{x}{y \cdot \left(y + 1\right)}} \]

   8. Taylor expanded in y around 0 43.4%

      \[\leadsto \frac{x}{\color{blue}{y}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 20: 28.1% accurate, 2.1× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -2.45 \cdot 10^{-35}:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y) :precision binary64 (if (<= x -2.45e-35) (/ 1.0 x) (/ x y)))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -2.45e-35) {
		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 <= (-2.45d-35)) 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 <= -2.45e-35) {
		tmp = 1.0 / x;
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -2.45e-35:
		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 <= -2.45e-35)
		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 <= -2.45e-35)
		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, -2.45e-35], N[(1.0 / x), $MachinePrecision], N[(x / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2.4500000000000002e-35

   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. associate-/l* 81.5%

         \[\leadsto \color{blue}{x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      2. associate-+l+ 81.5%

         \[\leadsto x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \color{blue}{\left(x + \left(y + 1\right)\right)}} \]

   3. Simplified 81.5%

      \[\leadsto \color{blue}{x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(x + \left(y + 1\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 54.4%

      \[\leadsto x \cdot \frac{y}{\left(\left(x + y\right) \cdot \color{blue}{y}\right) \cdot \left(x + \left(y + 1\right)\right)} \]

   6. Taylor expanded in x around inf 5.9%

      \[\leadsto \color{blue}{\frac{1}{x}} \]

   if -2.4500000000000002e-35 < x

   1. Initial program 67.2%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/l* 82.1%

         \[\leadsto \color{blue}{x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      2. associate-+l+ 82.1%

         \[\leadsto x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \color{blue}{\left(x + \left(y + 1\right)\right)}} \]

   3. Simplified 82.1%

      \[\leadsto \color{blue}{x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(x + \left(y + 1\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 61.5%

      \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]
   6. Step-by-step derivation

      1. +-commutative 61.5%

         \[\leadsto \frac{x}{y \cdot \color{blue}{\left(y + 1\right)}} \]

   7. Simplified 61.5%

      \[\leadsto \color{blue}{\frac{x}{y \cdot \left(y + 1\right)}} \]

   8. Taylor expanded in y around 0 42.5%

      \[\leadsto \frac{x}{\color{blue}{y}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 21: 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 68.3%

   \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
2. Step-by-step derivation

   1. associate-/l* 82.0%

      \[\leadsto \color{blue}{x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

   2. associate-+l+ 82.0%

      \[\leadsto x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \color{blue}{\left(x + \left(y + 1\right)\right)}} \]

3. Simplified 82.0%

   \[\leadsto \color{blue}{x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(x + \left(y + 1\right)\right)}} \]
4. Add Preprocessing

5. Taylor expanded in x around 0 63.3%

   \[\leadsto x \cdot \frac{y}{\left(\left(x + y\right) \cdot \color{blue}{y}\right) \cdot \left(x + \left(y + 1\right)\right)} \]

6. Taylor expanded in x around inf 3.9%

   \[\leadsto \color{blue}{\frac{1}{x}} \]
7. Add Preprocessing

Developer Target 1: 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 2024135 
(FPCore (x y)
  :name "Numeric.SpecFunctions:incompleteBetaApprox from math-functions-0.1.5.2, A"
  :precision binary64

  :alt
  (! :herbie-platform default (/ (/ (/ x (+ (+ y 1) x)) (+ y x)) (/ 1 (/ y (+ y x)))))

  (/ (* x y) (* (* (+ x y) (+ x y)) (+ (+ x y) 1.0))))