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

Percentage Accurate: 70.0% → 99.8%

Time: 14.5s

Alternatives: 19

Speedup: 1.5×

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: 70.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 67.0%

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

   1. times-frac 87.9%

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

   2. associate-+r+ 87.9%

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

   3. associate-*l/ 81.9%

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

   4. times-frac 99.8%

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

3. Applied egg-rr 99.8%

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

4. Final simplification 99.8%

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

Alternative 2: 97.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_0 := \frac{y}{x + \left(y + 1\right)}\\ \mathbf{if}\;x \leq -1.35 \cdot 10^{+154}:\\ \;\;\;\;\frac{t_0}{x + y}\\ \mathbf{elif}\;x \leq -4 \cdot 10^{-16}:\\ \;\;\;\;t_0 \cdot \frac{x}{\left(x + y\right) \cdot \left(x + y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + y} \cdot \frac{\frac{y}{y + 1}}{x + y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ y (+ x (+ y 1.0)))))
   (if (<= x -1.35e+154)
     (/ t_0 (+ x y))
     (if (<= x -4e-16)
       (* t_0 (/ x (* (+ x y) (+ x y))))
       (* (/ x (+ x y)) (/ (/ y (+ y 1.0)) (+ x y)))))))

C

assert(x < y);
double code(double x, double y) {
	double t_0 = y / (x + (y + 1.0));
	double tmp;
	if (x <= -1.35e+154) {
		tmp = t_0 / (x + y);
	} else if (x <= -4e-16) {
		tmp = t_0 * (x / ((x + y) * (x + y)));
	} else {
		tmp = (x / (x + y)) * ((y / (y + 1.0)) / (x + y));
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = y / (x + (y + 1.0d0))
    if (x <= (-1.35d+154)) then
        tmp = t_0 / (x + y)
    else if (x <= (-4d-16)) then
        tmp = t_0 * (x / ((x + y) * (x + y)))
    else
        tmp = (x / (x + y)) * ((y / (y + 1.0d0)) / (x + y))
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double t_0 = y / (x + (y + 1.0));
	double tmp;
	if (x <= -1.35e+154) {
		tmp = t_0 / (x + y);
	} else if (x <= -4e-16) {
		tmp = t_0 * (x / ((x + y) * (x + y)));
	} else {
		tmp = (x / (x + y)) * ((y / (y + 1.0)) / (x + y));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	t_0 = y / (x + (y + 1.0))
	tmp = 0
	if x <= -1.35e+154:
		tmp = t_0 / (x + y)
	elif x <= -4e-16:
		tmp = t_0 * (x / ((x + y) * (x + y)))
	else:
		tmp = (x / (x + y)) * ((y / (y + 1.0)) / (x + y))
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	t_0 = Float64(y / Float64(x + Float64(y + 1.0)))
	tmp = 0.0
	if (x <= -1.35e+154)
		tmp = Float64(t_0 / Float64(x + y));
	elseif (x <= -4e-16)
		tmp = Float64(t_0 * Float64(x / Float64(Float64(x + y) * Float64(x + y))));
	else
		tmp = Float64(Float64(x / Float64(x + y)) * Float64(Float64(y / Float64(y + 1.0)) / Float64(x + y)));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	t_0 = y / (x + (y + 1.0));
	tmp = 0.0;
	if (x <= -1.35e+154)
		tmp = t_0 / (x + y);
	elseif (x <= -4e-16)
		tmp = t_0 * (x / ((x + y) * (x + y)));
	else
		tmp = (x / (x + y)) * ((y / (y + 1.0)) / (x + y));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := Block[{t$95$0 = N[(y / N[(x + N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.35e+154], N[(t$95$0 / N[(x + y), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -4e-16], N[(t$95$0 * N[(x / N[(N[(x + y), $MachinePrecision] * N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(x + y), $MachinePrecision]), $MachinePrecision] * N[(N[(y / N[(y + 1.0), $MachinePrecision]), $MachinePrecision] / N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.35000000000000003e154

   1. Initial program 50.2%

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

      1. times-frac 75.3%

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

      2. associate-+r+ 75.3%

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

      3. associate-*l/ 75.3%

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

      4. times-frac 99.9%

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

   3. Applied egg-rr 99.9%

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

   4. Taylor expanded in x around inf 82.2%

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

   if -1.35000000000000003e154 < x < -3.9999999999999999e-16

   1. Initial program 70.6%

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

      1. times-frac 91.6%

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

      2. /-rgt-identity 91.6%

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

      3. associate-/l/ 91.6%

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

      4. *-lft-identity 91.6%

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

      5. associate-+l+ 91.6%

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

   3. Simplified 91.6%

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

   if -3.9999999999999999e-16 < x

   1. Initial program 68.8%

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

      1. times-frac 89.1%

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

      2. associate-+r+ 89.1%

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

      3. associate-*l/ 80.5%

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

      4. times-frac 99.8%

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

   3. Applied egg-rr 99.8%

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

   4. Taylor expanded in x around 0 83.8%

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

      1. +-commutative 83.8%

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

   6. Simplified 83.8%

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

4. Final simplification 85.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.35 \cdot 10^{+154}:\\ \;\;\;\;\frac{\frac{y}{x + \left(y + 1\right)}}{x + y}\\ \mathbf{elif}\;x \leq -4 \cdot 10^{-16}:\\ \;\;\;\;\frac{y}{x + \left(y + 1\right)} \cdot \frac{x}{\left(x + y\right) \cdot \left(x + y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + y} \cdot \frac{\frac{y}{y + 1}}{x + y}\\ \end{array} \]

Alternative 3: 88.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -1.36 \cdot 10^{+141}:\\ \;\;\;\;\frac{\frac{y}{x + \left(y + 1\right)}}{x + y}\\ \mathbf{elif}\;x \leq -1.3 \cdot 10^{-145}:\\ \;\;\;\;\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + 1}}{x + y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -1.36e+141)
   (/ (/ y (+ x (+ y 1.0))) (+ x y))
   (if (<= x -1.3e-145)
     (* (/ x (* (+ x y) (+ x y))) (/ y (+ x 1.0)))
     (/ (/ x (+ y 1.0)) (+ x y)))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -1.36e+141) {
		tmp = (y / (x + (y + 1.0))) / (x + y);
	} else if (x <= -1.3e-145) {
		tmp = (x / ((x + y) * (x + y))) * (y / (x + 1.0));
	} else {
		tmp = (x / (y + 1.0)) / (x + y);
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-1.36d+141)) then
        tmp = (y / (x + (y + 1.0d0))) / (x + y)
    else if (x <= (-1.3d-145)) then
        tmp = (x / ((x + y) * (x + y))) * (y / (x + 1.0d0))
    else
        tmp = (x / (y + 1.0d0)) / (x + y)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (x <= -1.36e+141) {
		tmp = (y / (x + (y + 1.0))) / (x + y);
	} else if (x <= -1.3e-145) {
		tmp = (x / ((x + y) * (x + y))) * (y / (x + 1.0));
	} else {
		tmp = (x / (y + 1.0)) / (x + y);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -1.36e+141:
		tmp = (y / (x + (y + 1.0))) / (x + y)
	elif x <= -1.3e-145:
		tmp = (x / ((x + y) * (x + y))) * (y / (x + 1.0))
	else:
		tmp = (x / (y + 1.0)) / (x + y)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -1.36e+141)
		tmp = Float64(Float64(y / Float64(x + Float64(y + 1.0))) / Float64(x + y));
	elseif (x <= -1.3e-145)
		tmp = Float64(Float64(x / Float64(Float64(x + y) * Float64(x + y))) * Float64(y / Float64(x + 1.0)));
	else
		tmp = Float64(Float64(x / Float64(y + 1.0)) / Float64(x + y));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1.36e+141)
		tmp = (y / (x + (y + 1.0))) / (x + y);
	elseif (x <= -1.3e-145)
		tmp = (x / ((x + y) * (x + y))) * (y / (x + 1.0));
	else
		tmp = (x / (y + 1.0)) / (x + y);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[x, -1.36e+141], N[(N[(y / N[(x + N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x + y), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -1.3e-145], N[(N[(x / N[(N[(x + y), $MachinePrecision] * N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(y / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(y + 1.0), $MachinePrecision]), $MachinePrecision] / N[(x + y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.36e141

   1. Initial program 48.0%

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

      1. times-frac 79.3%

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

      2. associate-+r+ 79.3%

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

      3. associate-*l/ 79.4%

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

      4. times-frac 99.9%

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

   3. Applied egg-rr 99.9%

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

   4. Taylor expanded in x around inf 83.7%

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

   if -1.36e141 < x < -1.3e-145

   1. Initial program 77.5%

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

      1. times-frac 94.0%

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

      2. /-rgt-identity 94.0%

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

      3. associate-/l/ 94.0%

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

      4. *-lft-identity 94.0%

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

      5. associate-+l+ 94.0%

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

   3. Simplified 94.0%

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

   4. Taylor expanded in y around 0 79.4%

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

      1. +-commutative 79.4%

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

   6. Simplified 79.4%

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

   if -1.3e-145 < x

   1. Initial program 67.0%

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

      1. times-frac 87.4%

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

      2. associate-+r+ 87.4%

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

      3. associate-*l/ 80.5%

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

      4. times-frac 99.8%

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

   3. Applied egg-rr 99.8%

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

   4. Taylor expanded in x around 0 81.2%

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

      1. +-commutative 81.2%

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

   6. Simplified 81.2%

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

      1. associate-*l/ 81.2%

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

      2. associate-/l/ 81.5%

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

      3. +-commutative 81.5%

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

      4. +-commutative 81.5%

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

      5. +-commutative 81.5%

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

   8. Applied egg-rr 81.5%

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

   9. Taylor expanded in x around 0 62.2%

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

       1. +-commutative 62.2%

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

   11. Simplified 62.2%

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

4. Final simplification 69.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.36 \cdot 10^{+141}:\\ \;\;\;\;\frac{\frac{y}{x + \left(y + 1\right)}}{x + y}\\ \mathbf{elif}\;x \leq -1.3 \cdot 10^{-145}:\\ \;\;\;\;\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + 1}}{x + y}\\ \end{array} \]

Alternative 4: 92.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -1.46 \cdot 10^{+16}:\\ \;\;\;\;\frac{\frac{y}{x + \left(y + 1\right)}}{x + y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + y} \cdot \frac{\frac{y}{y + 1}}{x + y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -1.46e+16)
   (/ (/ y (+ x (+ y 1.0))) (+ x y))
   (* (/ x (+ x y)) (/ (/ y (+ y 1.0)) (+ x y)))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -1.46e+16) {
		tmp = (y / (x + (y + 1.0))) / (x + y);
	} else {
		tmp = (x / (x + y)) * ((y / (y + 1.0)) / (x + y));
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-1.46d+16)) then
        tmp = (y / (x + (y + 1.0d0))) / (x + y)
    else
        tmp = (x / (x + y)) * ((y / (y + 1.0d0)) / (x + y))
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (x <= -1.46e+16) {
		tmp = (y / (x + (y + 1.0))) / (x + y);
	} else {
		tmp = (x / (x + y)) * ((y / (y + 1.0)) / (x + y));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -1.46e+16:
		tmp = (y / (x + (y + 1.0))) / (x + y)
	else:
		tmp = (x / (x + y)) * ((y / (y + 1.0)) / (x + y))
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -1.46e+16)
		tmp = Float64(Float64(y / Float64(x + Float64(y + 1.0))) / Float64(x + y));
	else
		tmp = Float64(Float64(x / Float64(x + y)) * Float64(Float64(y / Float64(y + 1.0)) / Float64(x + y)));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1.46e+16)
		tmp = (y / (x + (y + 1.0))) / (x + y);
	else
		tmp = (x / (x + y)) * ((y / (y + 1.0)) / (x + y));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[x, -1.46e+16], N[(N[(y / N[(x + N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x + y), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(x + y), $MachinePrecision]), $MachinePrecision] * N[(N[(y / N[(y + 1.0), $MachinePrecision]), $MachinePrecision] / N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.46e16

   1. Initial program 58.9%

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

      1. times-frac 83.2%

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

      2. associate-+r+ 83.2%

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

      3. associate-*l/ 83.3%

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

      4. times-frac 99.8%

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

   3. Applied egg-rr 99.8%

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

   4. Taylor expanded in x around inf 75.8%

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

   if -1.46e16 < x

   1. Initial program 69.8%

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

      1. times-frac 89.6%

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

      2. associate-+r+ 89.6%

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

      3. associate-*l/ 81.4%

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

      4. times-frac 99.8%

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

   3. Applied egg-rr 99.8%

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

   4. Taylor expanded in x around 0 84.4%

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

      1. +-commutative 84.4%

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

   6. Simplified 84.4%

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

4. Final simplification 82.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.46 \cdot 10^{+16}:\\ \;\;\;\;\frac{\frac{y}{x + \left(y + 1\right)}}{x + y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + y} \cdot \frac{\frac{y}{y + 1}}{x + y}\\ \end{array} \]

Alternative 5: 79.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{y}{x \cdot x}\\ \mathbf{elif}\;x \leq -3.4 \cdot 10^{-108}:\\ \;\;\;\;\frac{y}{x + y}\\ \mathbf{elif}\;x \leq 2.5 \cdot 10^{-33}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -1.0)
   (/ y (* x x))
   (if (<= x -3.4e-108)
     (/ y (+ x y))
     (if (<= x 2.5e-33) (/ x (* y (+ y 1.0))) (/ (/ x y) y)))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -1.0) {
		tmp = y / (x * x);
	} else if (x <= -3.4e-108) {
		tmp = y / (x + y);
	} else if (x <= 2.5e-33) {
		tmp = x / (y * (y + 1.0));
	} else {
		tmp = (x / y) / y;
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-1.0d0)) then
        tmp = y / (x * x)
    else if (x <= (-3.4d-108)) then
        tmp = y / (x + y)
    else if (x <= 2.5d-33) then
        tmp = x / (y * (y + 1.0d0))
    else
        tmp = (x / y) / y
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (x <= -1.0) {
		tmp = y / (x * x);
	} else if (x <= -3.4e-108) {
		tmp = y / (x + y);
	} else if (x <= 2.5e-33) {
		tmp = x / (y * (y + 1.0));
	} else {
		tmp = (x / y) / y;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -1.0:
		tmp = y / (x * x)
	elif x <= -3.4e-108:
		tmp = y / (x + y)
	elif x <= 2.5e-33:
		tmp = x / (y * (y + 1.0))
	else:
		tmp = (x / y) / 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 <= -3.4e-108)
		tmp = Float64(y / Float64(x + y));
	elseif (x <= 2.5e-33)
		tmp = Float64(x / Float64(y * Float64(y + 1.0)));
	else
		tmp = Float64(Float64(x / y) / y);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1.0)
		tmp = y / (x * x);
	elseif (x <= -3.4e-108)
		tmp = y / (x + y);
	elseif (x <= 2.5e-33)
		tmp = x / (y * (y + 1.0));
	else
		tmp = (x / y) / y;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[x, -1.0], N[(y / N[(x * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -3.4e-108], N[(y / N[(x + y), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 2.5e-33], N[(x / N[(y * N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / y), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < -1

   1. Initial program 60.7%

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

      1. associate-*r/ 71.5%

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

      2. *-commutative 71.5%

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

      3. distribute-rgt1-in 34.6%

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

      4. fma-def 71.5%

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

      5. cube-unmult 71.5%

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

   3. Simplified 71.5%

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

   4. Taylor expanded in x around inf 69.2%

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

      1. unpow2 69.2%

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

   6. Simplified 69.2%

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

   if -1 < x < -3.40000000000000002e-108

   1. Initial program 84.3%

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

      1. times-frac 99.6%

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

      2. associate-+r+ 99.6%

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

      3. associate-*l/ 88.1%

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

      4. times-frac 99.7%

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

   3. Applied egg-rr 99.7%

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

   4. Taylor expanded in x around 0 98.4%

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

      1. +-commutative 98.4%

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

   6. Simplified 98.4%

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

      1. associate-*l/ 98.4%

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

      2. associate-/l/ 98.4%

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

      3. +-commutative 98.4%

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

      4. +-commutative 98.4%

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

      5. +-commutative 98.4%

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

   8. Applied egg-rr 98.4%

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

   9. Taylor expanded in y around 0 52.5%

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

   if -3.40000000000000002e-108 < x < 2.50000000000000014e-33

   1. Initial program 72.6%

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

      1. times-frac 88.6%

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

      2. /-rgt-identity 88.6%

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

      3. associate-/l/ 88.6%

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

      4. *-lft-identity 88.6%

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

      5. associate-+l+ 88.6%

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

   3. Simplified 88.6%

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

   4. Taylor expanded in x around 0 83.8%

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

      1. +-commutative 83.8%

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

   6. Simplified 83.8%

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

   if 2.50000000000000014e-33 < x

   1. Initial program 60.0%

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

      1. associate-*r/ 75.1%

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

      2. *-commutative 75.1%

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

      3. distribute-rgt1-in 69.3%

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

      4. fma-def 75.1%

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

      5. cube-unmult 75.1%

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

   3. Simplified 75.1%

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

   4. Taylor expanded in y around inf 29.7%

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

      1. unpow2 29.7%

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

   6. Simplified 29.7%

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

      1. associate-/r* 33.1%

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

      2. div-inv 33.0%

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

   8. Applied egg-rr 33.0%

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

      1. associate-*r/ 33.1%

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

      2. *-rgt-identity 33.1%

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

   10. Simplified 33.1%

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

4. Final simplification 62.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{y}{x \cdot x}\\ \mathbf{elif}\;x \leq -3.4 \cdot 10^{-108}:\\ \;\;\;\;\frac{y}{x + y}\\ \mathbf{elif}\;x \leq 2.5 \cdot 10^{-33}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y}\\ \end{array} \]

Alternative 6: 82.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -3.4 \cdot 10^{-108}:\\ \;\;\;\;\frac{\frac{y}{x + \left(y + 1\right)}}{x + y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + 1}}{x + y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -3.4e-108)
   (/ (/ y (+ x (+ y 1.0))) (+ x y))
   (/ (/ x (+ y 1.0)) (+ x y))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -3.4e-108) {
		tmp = (y / (x + (y + 1.0))) / (x + y);
	} else {
		tmp = (x / (y + 1.0)) / (x + y);
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-3.4d-108)) then
        tmp = (y / (x + (y + 1.0d0))) / (x + y)
    else
        tmp = (x / (y + 1.0d0)) / (x + y)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (x <= -3.4e-108) {
		tmp = (y / (x + (y + 1.0))) / (x + y);
	} else {
		tmp = (x / (y + 1.0)) / (x + y);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -3.4e-108:
		tmp = (y / (x + (y + 1.0))) / (x + y)
	else:
		tmp = (x / (y + 1.0)) / (x + y)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -3.4e-108)
		tmp = Float64(Float64(y / Float64(x + Float64(y + 1.0))) / Float64(x + y));
	else
		tmp = Float64(Float64(x / Float64(y + 1.0)) / Float64(x + y));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -3.4e-108)
		tmp = (y / (x + (y + 1.0))) / (x + y);
	else
		tmp = (x / (y + 1.0)) / (x + y);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[x, -3.4e-108], N[(N[(y / N[(x + N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x + y), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(y + 1.0), $MachinePrecision]), $MachinePrecision] / N[(x + y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -3.40000000000000002e-108

   1. Initial program 66.9%

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

      1. times-frac 88.1%

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

      2. associate-+r+ 88.1%

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

      3. associate-*l/ 85.1%

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

      4. times-frac 99.8%

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

   3. Applied egg-rr 99.8%

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

   4. Taylor expanded in x around inf 68.1%

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

   if -3.40000000000000002e-108 < x

   1. Initial program 67.0%

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

      1. times-frac 87.8%

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

      2. associate-+r+ 87.8%

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

      3. associate-*l/ 80.0%

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

      4. times-frac 99.8%

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

   3. Applied egg-rr 99.8%

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

   4. Taylor expanded in x around 0 81.9%

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

      1. +-commutative 81.9%

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

   6. Simplified 81.9%

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

      1. associate-*l/ 81.9%

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

      2. associate-/l/ 82.2%

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

      3. +-commutative 82.2%

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

      4. +-commutative 82.2%

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

      5. +-commutative 82.2%

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

   8. Applied egg-rr 82.2%

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

   9. Taylor expanded in x around 0 61.8%

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

       1. +-commutative 61.8%

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

   11. Simplified 61.8%

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

4. Final simplification 64.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.4 \cdot 10^{-108}:\\ \;\;\;\;\frac{\frac{y}{x + \left(y + 1\right)}}{x + y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + 1}}{x + y}\\ \end{array} \]

Alternative 7: 65.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_0 := \frac{x}{y \cdot y}\\ \mathbf{if}\;y \leq -9 \cdot 10^{+80}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 4.4 \cdot 10^{-119}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{elif}\;y \leq 200:\\ \;\;\;\;\frac{x}{y} - x\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ x (* y y))))
   (if (<= y -9e+80)
     t_0
     (if (<= y 4.4e-119) (/ y x) (if (<= y 200.0) (- (/ x y) x) t_0)))))

C

assert(x < y);
double code(double x, double y) {
	double t_0 = x / (y * y);
	double tmp;
	if (y <= -9e+80) {
		tmp = t_0;
	} else if (y <= 4.4e-119) {
		tmp = y / x;
	} else if (y <= 200.0) {
		tmp = (x / y) - x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x / (y * y)
    if (y <= (-9d+80)) then
        tmp = t_0
    else if (y <= 4.4d-119) then
        tmp = y / x
    else if (y <= 200.0d0) then
        tmp = (x / y) - x
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double t_0 = x / (y * y);
	double tmp;
	if (y <= -9e+80) {
		tmp = t_0;
	} else if (y <= 4.4e-119) {
		tmp = y / x;
	} else if (y <= 200.0) {
		tmp = (x / y) - x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	t_0 = x / (y * y)
	tmp = 0
	if y <= -9e+80:
		tmp = t_0
	elif y <= 4.4e-119:
		tmp = y / x
	elif y <= 200.0:
		tmp = (x / y) - x
	else:
		tmp = t_0
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	t_0 = Float64(x / Float64(y * y))
	tmp = 0.0
	if (y <= -9e+80)
		tmp = t_0;
	elseif (y <= 4.4e-119)
		tmp = Float64(y / x);
	elseif (y <= 200.0)
		tmp = Float64(Float64(x / y) - x);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	t_0 = x / (y * y);
	tmp = 0.0;
	if (y <= -9e+80)
		tmp = t_0;
	elseif (y <= 4.4e-119)
		tmp = y / x;
	elseif (y <= 200.0)
		tmp = (x / y) - x;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := Block[{t$95$0 = N[(x / N[(y * y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -9e+80], t$95$0, If[LessEqual[y, 4.4e-119], N[(y / x), $MachinePrecision], If[LessEqual[y, 200.0], N[(N[(x / y), $MachinePrecision] - x), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if y < -9.00000000000000013e80 or 200 < y

   1. Initial program 51.5%

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

      1. associate-*r/ 69.3%

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

      2. *-commutative 69.3%

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

      3. distribute-rgt1-in 47.6%

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

      4. fma-def 69.3%

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

      5. cube-unmult 69.3%

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

   3. Simplified 69.3%

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

   4. Taylor expanded in y around inf 67.8%

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

      1. unpow2 67.8%

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

   6. Simplified 67.8%

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

   if -9.00000000000000013e80 < y < 4.4000000000000001e-119

   1. Initial program 78.7%

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

      1. times-frac 91.0%

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

      2. associate-+r+ 91.0%

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

      3. associate-*l/ 80.3%

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

      4. times-frac 99.8%

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

   3. Applied egg-rr 99.8%

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

   4. Taylor expanded in x around 0 77.4%

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

      1. +-commutative 77.4%

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

   6. Simplified 77.4%

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

   7. Taylor expanded in y around 0 48.3%

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

   if 4.4000000000000001e-119 < y < 200

   1. Initial program 84.0%

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

      1. times-frac 99.5%

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

      2. /-rgt-identity 99.5%

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

      3. associate-/l/ 99.5%

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

      4. *-lft-identity 99.5%

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

      5. associate-+l+ 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in x around 0 43.1%

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

      1. distribute-rgt-in 43.1%

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

      2. *-lft-identity 43.1%

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

   6. Simplified 43.1%

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

   7. Taylor expanded in y around 0 40.9%

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

      1. neg-mul-1 40.9%

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

      2. unsub-neg 40.9%

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

   9. Simplified 40.9%

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

4. Final simplification 56.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -9 \cdot 10^{+80}:\\ \;\;\;\;\frac{x}{y \cdot y}\\ \mathbf{elif}\;y \leq 4.4 \cdot 10^{-119}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{elif}\;y \leq 200:\\ \;\;\;\;\frac{x}{y} - x\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot y}\\ \end{array} \]

Alternative 8: 72.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{y}{x \cdot x}\\ \mathbf{elif}\;x \leq -2.25 \cdot 10^{-122}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{elif}\;x \leq 10^{-100}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -1.0)
   (/ y (* x x))
   (if (<= x -2.25e-122) (/ y x) (if (<= x 1e-100) (/ x y) (/ x (* y y))))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -1.0) {
		tmp = y / (x * x);
	} else if (x <= -2.25e-122) {
		tmp = y / x;
	} else if (x <= 1e-100) {
		tmp = x / y;
	} else {
		tmp = x / (y * y);
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-1.0d0)) then
        tmp = y / (x * x)
    else if (x <= (-2.25d-122)) then
        tmp = y / x
    else if (x <= 1d-100) then
        tmp = x / y
    else
        tmp = x / (y * y)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (x <= -1.0) {
		tmp = y / (x * x);
	} else if (x <= -2.25e-122) {
		tmp = y / x;
	} else if (x <= 1e-100) {
		tmp = x / y;
	} else {
		tmp = x / (y * 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 <= -2.25e-122:
		tmp = y / x
	elif x <= 1e-100:
		tmp = x / y
	else:
		tmp = x / (y * 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 <= -2.25e-122)
		tmp = Float64(y / x);
	elseif (x <= 1e-100)
		tmp = Float64(x / y);
	else
		tmp = Float64(x / Float64(y * y));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1.0)
		tmp = y / (x * x);
	elseif (x <= -2.25e-122)
		tmp = y / x;
	elseif (x <= 1e-100)
		tmp = x / y;
	else
		tmp = x / (y * y);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[x, -1.0], N[(y / N[(x * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -2.25e-122], N[(y / x), $MachinePrecision], If[LessEqual[x, 1e-100], N[(x / y), $MachinePrecision], N[(x / N[(y * y), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < -1

   1. Initial program 60.7%

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

      1. associate-*r/ 71.5%

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

      2. *-commutative 71.5%

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

      3. distribute-rgt1-in 34.6%

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

      4. fma-def 71.5%

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

      5. cube-unmult 71.5%

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

   3. Simplified 71.5%

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

   4. Taylor expanded in x around inf 69.2%

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

      1. unpow2 69.2%

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

   6. Simplified 69.2%

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

   if -1 < x < -2.2499999999999999e-122

   1. Initial program 83.1%

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

      1. times-frac 99.5%

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

      2. associate-+r+ 99.5%

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

      3. associate-*l/ 86.5%

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

      4. times-frac 99.8%

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

   3. Applied egg-rr 99.8%

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

   4. Taylor expanded in x around 0 98.7%

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

      1. +-commutative 98.7%

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

   6. Simplified 98.7%

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

   7. Taylor expanded in y around 0 49.1%

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

   if -2.2499999999999999e-122 < x < 1e-100

   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. Step-by-step derivation

      1. times-frac 85.4%

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

      2. /-rgt-identity 85.4%

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

      3. associate-/l/ 85.4%

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

      4. *-lft-identity 85.4%

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

      5. associate-+l+ 85.4%

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

   3. Simplified 85.4%

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

   4. Taylor expanded in x around 0 84.6%

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

      1. distribute-rgt-in 84.6%

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

      2. *-lft-identity 84.6%

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

   6. Simplified 84.6%

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

   7. Taylor expanded in y around 0 69.2%

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

   if 1e-100 < x

   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-*r/ 77.5%

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

      2. *-commutative 77.5%

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

      3. distribute-rgt1-in 68.2%

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

      4. fma-def 77.5%

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

      5. cube-unmult 77.5%

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

   3. Simplified 77.5%

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

   4. Taylor expanded in y around inf 37.2%

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

      1. unpow2 37.2%

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

   6. Simplified 37.2%

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

4. Final simplification 56.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{y}{x \cdot x}\\ \mathbf{elif}\;x \leq -2.25 \cdot 10^{-122}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{elif}\;x \leq 10^{-100}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot y}\\ \end{array} \]

Alternative 9: 72.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{y}{x \cdot x}\\ \mathbf{elif}\;x \leq -1.86 \cdot 10^{-121}:\\ \;\;\;\;\frac{y}{x + y}\\ \mathbf{elif}\;x \leq 9.2 \cdot 10^{-101}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -1.0)
   (/ y (* x x))
   (if (<= x -1.86e-121)
     (/ y (+ x y))
     (if (<= x 9.2e-101) (/ x y) (/ x (* y y))))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -1.0) {
		tmp = y / (x * x);
	} else if (x <= -1.86e-121) {
		tmp = y / (x + y);
	} else if (x <= 9.2e-101) {
		tmp = x / y;
	} else {
		tmp = x / (y * y);
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-1.0d0)) then
        tmp = y / (x * x)
    else if (x <= (-1.86d-121)) then
        tmp = y / (x + y)
    else if (x <= 9.2d-101) then
        tmp = x / y
    else
        tmp = x / (y * y)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (x <= -1.0) {
		tmp = y / (x * x);
	} else if (x <= -1.86e-121) {
		tmp = y / (x + y);
	} else if (x <= 9.2e-101) {
		tmp = x / y;
	} else {
		tmp = x / (y * 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 <= -1.86e-121:
		tmp = y / (x + y)
	elif x <= 9.2e-101:
		tmp = x / y
	else:
		tmp = x / (y * 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 <= -1.86e-121)
		tmp = Float64(y / Float64(x + y));
	elseif (x <= 9.2e-101)
		tmp = Float64(x / y);
	else
		tmp = Float64(x / Float64(y * y));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1.0)
		tmp = y / (x * x);
	elseif (x <= -1.86e-121)
		tmp = y / (x + y);
	elseif (x <= 9.2e-101)
		tmp = x / y;
	else
		tmp = x / (y * y);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[x, -1.0], N[(y / N[(x * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -1.86e-121], N[(y / N[(x + y), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 9.2e-101], N[(x / y), $MachinePrecision], N[(x / N[(y * y), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < -1

   1. Initial program 60.7%

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

      1. associate-*r/ 71.5%

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

      2. *-commutative 71.5%

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

      3. distribute-rgt1-in 34.6%

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

      4. fma-def 71.5%

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

      5. cube-unmult 71.5%

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

   3. Simplified 71.5%

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

   4. Taylor expanded in x around inf 69.2%

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

      1. unpow2 69.2%

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

   6. Simplified 69.2%

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

   if -1 < x < -1.8599999999999999e-121

   1. Initial program 83.1%

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

      1. times-frac 99.5%

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

      2. associate-+r+ 99.5%

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

      3. associate-*l/ 86.5%

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

      4. times-frac 99.8%

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

   3. Applied egg-rr 99.8%

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

   4. Taylor expanded in x around 0 98.7%

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

      1. +-commutative 98.7%

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

   6. Simplified 98.7%

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

      1. associate-*l/ 98.6%

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

      2. associate-/l/ 98.6%

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

      3. +-commutative 98.6%

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

      4. +-commutative 98.6%

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

      5. +-commutative 98.6%

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

   8. Applied egg-rr 98.6%

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

   9. Taylor expanded in y around 0 49.4%

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

   if -1.8599999999999999e-121 < x < 9.1999999999999998e-101

   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. Step-by-step derivation

      1. times-frac 85.4%

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

      2. /-rgt-identity 85.4%

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

      3. associate-/l/ 85.4%

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

      4. *-lft-identity 85.4%

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

      5. associate-+l+ 85.4%

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

   3. Simplified 85.4%

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

   4. Taylor expanded in x around 0 84.6%

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

      1. distribute-rgt-in 84.6%

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

      2. *-lft-identity 84.6%

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

   6. Simplified 84.6%

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

   7. Taylor expanded in y around 0 69.2%

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

   if 9.1999999999999998e-101 < x

   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-*r/ 77.5%

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

      2. *-commutative 77.5%

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

      3. distribute-rgt1-in 68.2%

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

      4. fma-def 77.5%

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

      5. cube-unmult 77.5%

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

   3. Simplified 77.5%

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

   4. Taylor expanded in y around inf 37.2%

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

      1. unpow2 37.2%

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

   6. Simplified 37.2%

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

4. Final simplification 56.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{y}{x \cdot x}\\ \mathbf{elif}\;x \leq -1.86 \cdot 10^{-121}:\\ \;\;\;\;\frac{y}{x + y}\\ \mathbf{elif}\;x \leq 9.2 \cdot 10^{-101}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot y}\\ \end{array} \]

Alternative 10: 73.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{y}{x \cdot x}\\ \mathbf{elif}\;x \leq -1.85 \cdot 10^{-121}:\\ \;\;\;\;\frac{y}{x + y}\\ \mathbf{elif}\;x \leq 3.5 \cdot 10^{-100}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -1.0)
   (/ y (* x x))
   (if (<= x -1.85e-121)
     (/ y (+ x y))
     (if (<= x 3.5e-100) (/ x y) (/ (/ x y) y)))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -1.0) {
		tmp = y / (x * x);
	} else if (x <= -1.85e-121) {
		tmp = y / (x + y);
	} else if (x <= 3.5e-100) {
		tmp = x / y;
	} else {
		tmp = (x / y) / y;
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-1.0d0)) then
        tmp = y / (x * x)
    else if (x <= (-1.85d-121)) then
        tmp = y / (x + y)
    else if (x <= 3.5d-100) then
        tmp = x / y
    else
        tmp = (x / y) / y
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (x <= -1.0) {
		tmp = y / (x * x);
	} else if (x <= -1.85e-121) {
		tmp = y / (x + y);
	} else if (x <= 3.5e-100) {
		tmp = x / y;
	} else {
		tmp = (x / y) / 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 <= -1.85e-121:
		tmp = y / (x + y)
	elif x <= 3.5e-100:
		tmp = x / y
	else:
		tmp = (x / y) / 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 <= -1.85e-121)
		tmp = Float64(y / Float64(x + y));
	elseif (x <= 3.5e-100)
		tmp = Float64(x / y);
	else
		tmp = Float64(Float64(x / y) / y);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1.0)
		tmp = y / (x * x);
	elseif (x <= -1.85e-121)
		tmp = y / (x + y);
	elseif (x <= 3.5e-100)
		tmp = x / y;
	else
		tmp = (x / y) / y;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[x, -1.0], N[(y / N[(x * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -1.85e-121], N[(y / N[(x + y), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 3.5e-100], N[(x / y), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / y), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < -1

   1. Initial program 60.7%

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

      1. associate-*r/ 71.5%

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

      2. *-commutative 71.5%

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

      3. distribute-rgt1-in 34.6%

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

      4. fma-def 71.5%

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

      5. cube-unmult 71.5%

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

   3. Simplified 71.5%

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

   4. Taylor expanded in x around inf 69.2%

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

      1. unpow2 69.2%

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

   6. Simplified 69.2%

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

   if -1 < x < -1.8500000000000001e-121

   1. Initial program 83.1%

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

      1. times-frac 99.5%

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

      2. associate-+r+ 99.5%

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

      3. associate-*l/ 86.5%

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

      4. times-frac 99.8%

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

   3. Applied egg-rr 99.8%

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

   4. Taylor expanded in x around 0 98.7%

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

      1. +-commutative 98.7%

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

   6. Simplified 98.7%

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

      1. associate-*l/ 98.6%

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

      2. associate-/l/ 98.6%

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

      3. +-commutative 98.6%

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

      4. +-commutative 98.6%

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

      5. +-commutative 98.6%

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

   8. Applied egg-rr 98.6%

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

   9. Taylor expanded in y around 0 49.4%

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

   if -1.8500000000000001e-121 < x < 3.5000000000000001e-100

   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. Step-by-step derivation

      1. times-frac 85.4%

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

      2. /-rgt-identity 85.4%

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

      3. associate-/l/ 85.4%

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

      4. *-lft-identity 85.4%

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

      5. associate-+l+ 85.4%

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

   3. Simplified 85.4%

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

   4. Taylor expanded in x around 0 84.6%

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

      1. distribute-rgt-in 84.6%

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

      2. *-lft-identity 84.6%

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

   6. Simplified 84.6%

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

   7. Taylor expanded in y around 0 69.2%

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

   if 3.5000000000000001e-100 < x

   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-*r/ 77.5%

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

      2. *-commutative 77.5%

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

      3. distribute-rgt1-in 68.2%

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

      4. fma-def 77.5%

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

      5. cube-unmult 77.5%

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

   3. Simplified 77.5%

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

   4. Taylor expanded in y around inf 37.2%

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

      1. unpow2 37.2%

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

   6. Simplified 37.2%

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

      1. associate-/r* 39.9%

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

      2. div-inv 39.9%

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

   8. Applied egg-rr 39.9%

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

      1. associate-*r/ 39.9%

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

      2. *-rgt-identity 39.9%

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

   10. Simplified 39.9%

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

4. Final simplification 57.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{y}{x \cdot x}\\ \mathbf{elif}\;x \leq -1.85 \cdot 10^{-121}:\\ \;\;\;\;\frac{y}{x + y}\\ \mathbf{elif}\;x \leq 3.5 \cdot 10^{-100}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y}\\ \end{array} \]

Alternative 11: 80.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -7.6 \cdot 10^{-110}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{elif}\;x \leq 2.5 \cdot 10^{-33}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -7.6e-110)
   (/ y (* x (+ x 1.0)))
   (if (<= x 2.5e-33) (/ x (* y (+ y 1.0))) (/ (/ x y) y))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -7.6e-110) {
		tmp = y / (x * (x + 1.0));
	} else if (x <= 2.5e-33) {
		tmp = x / (y * (y + 1.0));
	} else {
		tmp = (x / y) / y;
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-7.6d-110)) then
        tmp = y / (x * (x + 1.0d0))
    else if (x <= 2.5d-33) then
        tmp = x / (y * (y + 1.0d0))
    else
        tmp = (x / y) / y
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (x <= -7.6e-110) {
		tmp = y / (x * (x + 1.0));
	} else if (x <= 2.5e-33) {
		tmp = x / (y * (y + 1.0));
	} else {
		tmp = (x / y) / y;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -7.6e-110:
		tmp = y / (x * (x + 1.0))
	elif x <= 2.5e-33:
		tmp = x / (y * (y + 1.0))
	else:
		tmp = (x / y) / y
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -7.6e-110)
		tmp = Float64(y / Float64(x * Float64(x + 1.0)));
	elseif (x <= 2.5e-33)
		tmp = Float64(x / Float64(y * Float64(y + 1.0)));
	else
		tmp = Float64(Float64(x / y) / y);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -7.6e-110)
		tmp = y / (x * (x + 1.0));
	elseif (x <= 2.5e-33)
		tmp = x / (y * (y + 1.0));
	else
		tmp = (x / y) / y;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[x, -7.6e-110], N[(y / N[(x * N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 2.5e-33], N[(x / N[(y * N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -7.5999999999999996e-110

   1. Initial program 67.3%

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

      1. times-frac 88.2%

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

      2. /-rgt-identity 88.2%

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

      3. associate-/l/ 88.2%

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

      4. *-lft-identity 88.2%

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

      5. associate-+l+ 88.2%

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

   3. Simplified 88.2%

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

   4. Taylor expanded in y around 0 64.5%

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

   if -7.5999999999999996e-110 < x < 2.50000000000000014e-33

   1. Initial program 72.3%

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

      1. times-frac 88.5%

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

      2. /-rgt-identity 88.5%

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

      3. associate-/l/ 88.5%

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

      4. *-lft-identity 88.5%

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

      5. associate-+l+ 88.5%

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

   3. Simplified 88.5%

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

   4. Taylor expanded in x around 0 83.6%

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

      1. +-commutative 83.6%

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

   6. Simplified 83.6%

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

   if 2.50000000000000014e-33 < x

   1. Initial program 60.0%

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

      1. associate-*r/ 75.1%

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

      2. *-commutative 75.1%

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

      3. distribute-rgt1-in 69.3%

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

      4. fma-def 75.1%

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

      5. cube-unmult 75.1%

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

   3. Simplified 75.1%

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

   4. Taylor expanded in y around inf 29.7%

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

      1. unpow2 29.7%

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

   6. Simplified 29.7%

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

      1. associate-/r* 33.1%

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

      2. div-inv 33.0%

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

   8. Applied egg-rr 33.0%

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

      1. associate-*r/ 33.1%

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

      2. *-rgt-identity 33.1%

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

   10. Simplified 33.1%

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

4. Final simplification 62.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -7.6 \cdot 10^{-110}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{elif}\;x \leq 2.5 \cdot 10^{-33}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y}\\ \end{array} \]

Alternative 12: 80.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -3.3 \cdot 10^{-108}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{elif}\;x \leq 2.4 \cdot 10^{-33}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{x + y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -3.3e-108)
   (/ y (* x (+ x 1.0)))
   (if (<= x 2.4e-33) (/ x (* y (+ y 1.0))) (/ (/ x y) (+ x y)))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -3.3e-108) {
		tmp = y / (x * (x + 1.0));
	} else if (x <= 2.4e-33) {
		tmp = x / (y * (y + 1.0));
	} else {
		tmp = (x / y) / (x + y);
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-3.3d-108)) then
        tmp = y / (x * (x + 1.0d0))
    else if (x <= 2.4d-33) then
        tmp = x / (y * (y + 1.0d0))
    else
        tmp = (x / y) / (x + y)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (x <= -3.3e-108) {
		tmp = y / (x * (x + 1.0));
	} else if (x <= 2.4e-33) {
		tmp = x / (y * (y + 1.0));
	} else {
		tmp = (x / y) / (x + y);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -3.3e-108:
		tmp = y / (x * (x + 1.0))
	elif x <= 2.4e-33:
		tmp = x / (y * (y + 1.0))
	else:
		tmp = (x / y) / (x + y)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -3.3e-108)
		tmp = Float64(y / Float64(x * Float64(x + 1.0)));
	elseif (x <= 2.4e-33)
		tmp = Float64(x / Float64(y * Float64(y + 1.0)));
	else
		tmp = Float64(Float64(x / y) / Float64(x + y));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -3.3e-108)
		tmp = y / (x * (x + 1.0));
	elseif (x <= 2.4e-33)
		tmp = x / (y * (y + 1.0));
	else
		tmp = (x / y) / (x + y);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[x, -3.3e-108], N[(y / N[(x * N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 2.4e-33], N[(x / N[(y * N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / N[(x + y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -3.3000000000000002e-108

   1. Initial program 66.9%

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

      1. times-frac 88.1%

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

      2. /-rgt-identity 88.1%

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

      3. associate-/l/ 88.1%

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

      4. *-lft-identity 88.1%

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

      5. associate-+l+ 88.1%

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

   3. Simplified 88.1%

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

   4. Taylor expanded in y around 0 65.1%

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

   if -3.3000000000000002e-108 < x < 2.4e-33

   1. Initial program 72.6%

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

      1. times-frac 88.6%

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

      2. /-rgt-identity 88.6%

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

      3. associate-/l/ 88.6%

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

      4. *-lft-identity 88.6%

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

      5. associate-+l+ 88.6%

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

   3. Simplified 88.6%

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

   4. Taylor expanded in x around 0 83.8%

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

      1. +-commutative 83.8%

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

   6. Simplified 83.8%

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

   if 2.4e-33 < x

   1. Initial program 60.0%

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

      1. times-frac 86.9%

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

      2. associate-+r+ 86.9%

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

      3. associate-*l/ 86.9%

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

      4. times-frac 99.8%

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

   3. Applied egg-rr 99.8%

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

   4. Taylor expanded in x around 0 59.1%

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

      1. +-commutative 59.1%

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

   6. Simplified 59.1%

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

      1. associate-*l/ 59.1%

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

      2. associate-/l/ 59.9%

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

      3. +-commutative 59.9%

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

      4. +-commutative 59.9%

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

      5. +-commutative 59.9%

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

   8. Applied egg-rr 59.9%

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

   9. Taylor expanded in y around inf 33.8%

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

4. Final simplification 63.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.3 \cdot 10^{-108}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{elif}\;x \leq 2.4 \cdot 10^{-33}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{x + y}\\ \end{array} \]

Alternative 13: 80.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -3.4 \cdot 10^{-108}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y + 1} \cdot \frac{1}{y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -3.4e-108) (/ y (* x (+ x 1.0))) (* (/ x (+ y 1.0)) (/ 1.0 y))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -3.4e-108) {
		tmp = y / (x * (x + 1.0));
	} else {
		tmp = (x / (y + 1.0)) * (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 <= (-3.4d-108)) then
        tmp = y / (x * (x + 1.0d0))
    else
        tmp = (x / (y + 1.0d0)) * (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 <= -3.4e-108) {
		tmp = y / (x * (x + 1.0));
	} else {
		tmp = (x / (y + 1.0)) * (1.0 / y);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -3.4e-108:
		tmp = y / (x * (x + 1.0))
	else:
		tmp = (x / (y + 1.0)) * (1.0 / y)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -3.4e-108)
		tmp = Float64(y / Float64(x * Float64(x + 1.0)));
	else
		tmp = Float64(Float64(x / Float64(y + 1.0)) * 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 (x <= -3.4e-108)
		tmp = y / (x * (x + 1.0));
	else
		tmp = (x / (y + 1.0)) * (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, -3.4e-108], N[(y / N[(x * N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(y + 1.0), $MachinePrecision]), $MachinePrecision] * N[(1.0 / y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -3.40000000000000002e-108

   1. Initial program 66.9%

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

      1. times-frac 88.1%

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

      2. /-rgt-identity 88.1%

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

      3. associate-/l/ 88.1%

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

      4. *-lft-identity 88.1%

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

      5. associate-+l+ 88.1%

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

   3. Simplified 88.1%

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

   4. Taylor expanded in y around 0 65.1%

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

   if -3.40000000000000002e-108 < x

   1. Initial program 67.0%

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

      1. times-frac 87.8%

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

      2. /-rgt-identity 87.8%

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

      3. associate-/l/ 87.8%

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

      4. *-lft-identity 87.8%

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

      5. associate-+l+ 87.8%

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

   3. Simplified 87.8%

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

   4. Taylor expanded in x around 0 60.0%

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

      1. distribute-rgt-in 60.0%

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

      2. *-lft-identity 60.0%

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

   6. Simplified 60.0%

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

      1. distribute-rgt1-in 60.0%

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

   8. Applied egg-rr 60.0%

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

      1. associate-/r* 61.5%

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

      2. div-inv 61.4%

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

      3. +-commutative 61.4%

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

   10. Applied egg-rr 61.4%

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

4. Final simplification 62.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.4 \cdot 10^{-108}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y + 1} \cdot \frac{1}{y}\\ \end{array} \]

Alternative 14: 81.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -3.4 \cdot 10^{-108}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + \left(y + 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y + 1} \cdot \frac{1}{y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -3.4e-108)
   (/ y (* x (+ x (+ y 1.0))))
   (* (/ x (+ y 1.0)) (/ 1.0 y))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -3.4e-108) {
		tmp = y / (x * (x + (y + 1.0)));
	} else {
		tmp = (x / (y + 1.0)) * (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 <= (-3.4d-108)) then
        tmp = y / (x * (x + (y + 1.0d0)))
    else
        tmp = (x / (y + 1.0d0)) * (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 <= -3.4e-108) {
		tmp = y / (x * (x + (y + 1.0)));
	} else {
		tmp = (x / (y + 1.0)) * (1.0 / y);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -3.4e-108:
		tmp = y / (x * (x + (y + 1.0)))
	else:
		tmp = (x / (y + 1.0)) * (1.0 / y)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -3.4e-108)
		tmp = Float64(y / Float64(x * Float64(x + Float64(y + 1.0))));
	else
		tmp = Float64(Float64(x / Float64(y + 1.0)) * 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 (x <= -3.4e-108)
		tmp = y / (x * (x + (y + 1.0)));
	else
		tmp = (x / (y + 1.0)) * (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, -3.4e-108], N[(y / N[(x * N[(x + N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(y + 1.0), $MachinePrecision]), $MachinePrecision] * N[(1.0 / y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -3.40000000000000002e-108

   1. Initial program 66.9%

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

      1. times-frac 88.1%

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

      2. /-rgt-identity 88.1%

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

      3. associate-/l/ 88.1%

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

      4. *-lft-identity 88.1%

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

      5. associate-+l+ 88.1%

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

   3. Simplified 88.1%

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

   4. Taylor expanded in x around inf 65.3%

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

      1. unpow2 65.3%

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

   6. Simplified 65.3%

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

      1. *-commutative 65.3%

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

      2. associate-/r* 67.3%

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

      3. *-inverses 67.3%

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

      4. frac-times 67.6%

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

      5. *-rgt-identity 67.6%

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

   8. Applied egg-rr 67.6%

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

   if -3.40000000000000002e-108 < x

   1. Initial program 67.0%

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

      1. times-frac 87.8%

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

      2. /-rgt-identity 87.8%

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

      3. associate-/l/ 87.8%

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

      4. *-lft-identity 87.8%

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

      5. associate-+l+ 87.8%

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

   3. Simplified 87.8%

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

   4. Taylor expanded in x around 0 60.0%

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

      1. distribute-rgt-in 60.0%

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

      2. *-lft-identity 60.0%

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

   6. Simplified 60.0%

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

      1. distribute-rgt1-in 60.0%

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

   8. Applied egg-rr 60.0%

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

      1. associate-/r* 61.5%

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

      2. div-inv 61.4%

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

      3. +-commutative 61.4%

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

   10. Applied egg-rr 61.4%

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

4. Final simplification 63.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.4 \cdot 10^{-108}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + \left(y + 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y + 1} \cdot \frac{1}{y}\\ \end{array} \]

Alternative 15: 81.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -7.8 \cdot 10^{-109}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + \left(y + 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + 1}}{x + y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -7.8e-109) (/ y (* x (+ x (+ y 1.0)))) (/ (/ x (+ y 1.0)) (+ x y))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -7.8e-109) {
		tmp = y / (x * (x + (y + 1.0)));
	} else {
		tmp = (x / (y + 1.0)) / (x + y);
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-7.8d-109)) then
        tmp = y / (x * (x + (y + 1.0d0)))
    else
        tmp = (x / (y + 1.0d0)) / (x + y)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (x <= -7.8e-109) {
		tmp = y / (x * (x + (y + 1.0)));
	} else {
		tmp = (x / (y + 1.0)) / (x + y);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -7.8e-109:
		tmp = y / (x * (x + (y + 1.0)))
	else:
		tmp = (x / (y + 1.0)) / (x + y)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -7.8e-109)
		tmp = Float64(y / Float64(x * Float64(x + Float64(y + 1.0))));
	else
		tmp = Float64(Float64(x / Float64(y + 1.0)) / Float64(x + y));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -7.8e-109)
		tmp = y / (x * (x + (y + 1.0)));
	else
		tmp = (x / (y + 1.0)) / (x + y);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[x, -7.8e-109], N[(y / N[(x * N[(x + N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(y + 1.0), $MachinePrecision]), $MachinePrecision] / N[(x + y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -7.80000000000000046e-109

   1. Initial program 66.9%

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

      1. times-frac 88.1%

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

      2. /-rgt-identity 88.1%

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

      3. associate-/l/ 88.1%

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

      4. *-lft-identity 88.1%

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

      5. associate-+l+ 88.1%

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

   3. Simplified 88.1%

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

   4. Taylor expanded in x around inf 65.3%

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

      1. unpow2 65.3%

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

   6. Simplified 65.3%

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

      1. *-commutative 65.3%

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

      2. associate-/r* 67.3%

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

      3. *-inverses 67.3%

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

      4. frac-times 67.6%

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

      5. *-rgt-identity 67.6%

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

   8. Applied egg-rr 67.6%

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

   if -7.80000000000000046e-109 < x

   1. Initial program 67.0%

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

      1. times-frac 87.8%

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

      2. associate-+r+ 87.8%

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

      3. associate-*l/ 80.0%

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

      4. times-frac 99.8%

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

   3. Applied egg-rr 99.8%

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

   4. Taylor expanded in x around 0 81.9%

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

      1. +-commutative 81.9%

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

   6. Simplified 81.9%

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

      1. associate-*l/ 81.9%

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

      2. associate-/l/ 82.2%

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

      3. +-commutative 82.2%

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

      4. +-commutative 82.2%

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

      5. +-commutative 82.2%

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

   8. Applied egg-rr 82.2%

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

   9. Taylor expanded in x around 0 61.8%

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

       1. +-commutative 61.8%

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

   11. Simplified 61.8%

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

4. Final simplification 64.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -7.8 \cdot 10^{-109}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + \left(y + 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + 1}}{x + y}\\ \end{array} \]

Alternative 16: 82.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -3.4 \cdot 10^{-108}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + \left(y + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + 1}}{x + y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -3.4e-108) (/ (/ y x) (+ x (+ y 1.0))) (/ (/ x (+ y 1.0)) (+ x y))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -3.4e-108) {
		tmp = (y / x) / (x + (y + 1.0));
	} else {
		tmp = (x / (y + 1.0)) / (x + y);
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-3.4d-108)) then
        tmp = (y / x) / (x + (y + 1.0d0))
    else
        tmp = (x / (y + 1.0d0)) / (x + y)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (x <= -3.4e-108) {
		tmp = (y / x) / (x + (y + 1.0));
	} else {
		tmp = (x / (y + 1.0)) / (x + y);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -3.4e-108:
		tmp = (y / x) / (x + (y + 1.0))
	else:
		tmp = (x / (y + 1.0)) / (x + y)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -3.4e-108)
		tmp = Float64(Float64(y / x) / Float64(x + Float64(y + 1.0)));
	else
		tmp = Float64(Float64(x / Float64(y + 1.0)) / Float64(x + y));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -3.4e-108)
		tmp = (y / x) / (x + (y + 1.0));
	else
		tmp = (x / (y + 1.0)) / (x + y);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[x, -3.4e-108], N[(N[(y / x), $MachinePrecision] / N[(x + N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(y + 1.0), $MachinePrecision]), $MachinePrecision] / N[(x + y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -3.40000000000000002e-108

   1. Initial program 66.9%

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

      1. times-frac 88.1%

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

      2. /-rgt-identity 88.1%

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

      3. associate-/l/ 88.1%

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

      4. *-lft-identity 88.1%

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

      5. associate-+l+ 88.1%

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

   3. Simplified 88.1%

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

   4. Taylor expanded in x around inf 65.3%

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

      1. unpow2 65.3%

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

   6. Simplified 65.3%

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

      1. expm1-log1p-u 64.9%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{x}{x \cdot x} \cdot \frac{y}{x + \left(y + 1\right)}\right)\right)} \]

      2. expm1-udef 42.2%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{x}{x \cdot x} \cdot \frac{y}{x + \left(y + 1\right)}\right)} - 1} \]

      3. *-commutative 42.2%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{y}{x + \left(y + 1\right)} \cdot \frac{x}{x \cdot x}}\right)} - 1 \]

      4. associate-/r* 42.2%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{y}{x + \left(y + 1\right)} \cdot \color{blue}{\frac{\frac{x}{x}}{x}}\right)} - 1 \]

      5. *-inverses 42.2%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{y}{x + \left(y + 1\right)} \cdot \frac{\color{blue}{1}}{x}\right)} - 1 \]

      6. frac-times 42.2%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{y \cdot 1}{\left(x + \left(y + 1\right)\right) \cdot x}}\right)} - 1 \]

      7. *-rgt-identity 42.2%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{y}}{\left(x + \left(y + 1\right)\right) \cdot x}\right)} - 1 \]

   8. Applied egg-rr 42.2%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{y}{\left(x + \left(y + 1\right)\right) \cdot x}\right)} - 1} \]
   9. Step-by-step derivation

      1. expm1-def 67.2%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{y}{\left(x + \left(y + 1\right)\right) \cdot x}\right)\right)} \]

      2. expm1-log1p 67.6%

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

      3. *-commutative 67.6%

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

      4. associate-/r* 67.4%

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

   10. Simplified 67.4%

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

   if -3.40000000000000002e-108 < x

   1. Initial program 67.0%

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

      1. times-frac 87.8%

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

      2. associate-+r+ 87.8%

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

      3. associate-*l/ 80.0%

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

      4. times-frac 99.8%

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

   3. Applied egg-rr 99.8%

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

   4. Taylor expanded in x around 0 81.9%

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

      1. +-commutative 81.9%

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

   6. Simplified 81.9%

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

      1. associate-*l/ 81.9%

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

      2. associate-/l/ 82.2%

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

      3. +-commutative 82.2%

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

      4. +-commutative 82.2%

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

      5. +-commutative 82.2%

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

   8. Applied egg-rr 82.2%

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

   9. Taylor expanded in x around 0 61.8%

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

       1. +-commutative 61.8%

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

   11. Simplified 61.8%

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

4. Final simplification 63.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.4 \cdot 10^{-108}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + \left(y + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + 1}}{x + y}\\ \end{array} \]

Alternative 17: 44.1% accurate, 3.4× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -8.3 \cdot 10^{-122}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y) :precision binary64 (if (<= x -8.3e-122) (/ y x) (/ x y)))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -8.3e-122) {
		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 <= (-8.3d-122)) 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 <= -8.3e-122) {
		tmp = y / x;
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -8.3e-122:
		tmp = y / x
	else:
		tmp = x / y
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -8.3e-122)
		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 <= -8.3e-122)
		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, -8.3e-122], N[(y / x), $MachinePrecision], N[(x / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -8.3000000000000004e-122

   1. Initial program 67.3%

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

      1. times-frac 88.5%

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

      2. associate-+r+ 88.5%

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

      3. associate-*l/ 84.7%

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

      4. times-frac 99.8%

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

   3. Applied egg-rr 99.8%

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

   4. Taylor expanded in x around 0 64.9%

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

      1. +-commutative 64.9%

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

   6. Simplified 64.9%

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

   7. Taylor expanded in y around 0 30.9%

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

   if -8.3000000000000004e-122 < x

   1. Initial program 66.8%

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

      1. times-frac 87.5%

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

      2. /-rgt-identity 87.5%

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

      3. associate-/l/ 87.5%

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

      4. *-lft-identity 87.5%

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

      5. associate-+l+ 87.5%

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

   3. Simplified 87.5%

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

   4. Taylor expanded in x around 0 59.6%

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

      1. distribute-rgt-in 59.6%

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

      2. *-lft-identity 59.6%

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

   6. Simplified 59.6%

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

   7. Taylor expanded in y around 0 35.2%

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

4. Final simplification 33.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -8.3 \cdot 10^{-122}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]

Alternative 18: 4.3% accurate, 5.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 67.0%

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

   1. times-frac 87.9%

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

   2. associate-+r+ 87.9%

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

   3. associate-*l/ 81.9%

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

   4. times-frac 99.8%

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

3. Applied egg-rr 99.8%

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

4. Taylor expanded in y around inf 39.1%

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

5. Taylor expanded in x around inf 4.1%

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

6. Final simplification 4.1%

   \[\leadsto \frac{1}{x} \]

Alternative 19: 26.3% accurate, 5.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 67.0%

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

   1. times-frac 87.9%

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

   2. /-rgt-identity 87.9%

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

   3. associate-/l/ 87.9%

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

   4. *-lft-identity 87.9%

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

   5. associate-+l+ 87.9%

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

3. Simplified 87.9%

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

4. Taylor expanded in x around 0 48.9%

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

   1. distribute-rgt-in 48.9%

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

   2. *-lft-identity 48.9%

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

6. Simplified 48.9%

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

7. Taylor expanded in y around 0 24.4%

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

8. Final simplification 24.4%

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

Developer target: 99.8% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2023240 
(FPCore (x y)
  :name "Numeric.SpecFunctions:incompleteBetaApprox from math-functions-0.1.5.2, A"
  :precision binary64

  :herbie-target
  (/ (/ (/ x (+ (+ y 1.0) x)) (+ y x)) (/ 1.0 (/ y (+ y x))))

  (/ (* x y) (* (* (+ x y) (+ x y)) (+ (+ x y) 1.0))))