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

Percentage Accurate: 68.8% → 99.2%

Time: 12.9s

Alternatives: 21

Speedup: 1.2×

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: 68.8% 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: 99.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Derivation

1. Initial program 66.6%

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

   1. associate-*l* 66.6%

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

   2. times-frac 91.9%

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

   3. associate-+r+ 91.9%

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

4. Applied egg-rr 91.9%

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

   1. clear-num 91.1%

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

   2. un-div-inv 91.1%

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

   3. +-commutative 91.1%

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

   4. *-un-lft-identity 91.1%

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

   5. times-frac 99.0%

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

   6. /-rgt-identity 99.0%

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

   7. +-commutative 99.0%

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

6. Applied egg-rr 99.0%

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

7. Final simplification 99.0%

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

Alternative 2: 95.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_0 := x + \left(y + 1\right)\\ \mathbf{if}\;x \leq -1.32 \cdot 10^{+154}:\\ \;\;\;\;\frac{\frac{y}{x + y}}{t\_0}\\ \mathbf{elif}\;x \leq -2.8 \cdot 10^{-15}:\\ \;\;\;\;\frac{y}{t\_0 \cdot \left(x + y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{x + y}}{\left(x + y\right) \cdot \frac{y + 1}{y}}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ x (+ y 1.0))))
   (if (<= x -1.32e+154)
     (/ (/ y (+ x y)) t_0)
     (if (<= x -2.8e-15)
       (/ y (* t_0 (+ x y)))
       (/ (/ x (+ x y)) (* (+ x y) (/ (+ y 1.0) y)))))))

C

assert(x < y);
double code(double x, double y) {
	double t_0 = x + (y + 1.0);
	double tmp;
	if (x <= -1.32e+154) {
		tmp = (y / (x + y)) / t_0;
	} else if (x <= -2.8e-15) {
		tmp = y / (t_0 * (x + y));
	} else {
		tmp = (x / (x + y)) / ((x + y) * ((y + 1.0) / y));
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x + (y + 1.0d0)
    if (x <= (-1.32d+154)) then
        tmp = (y / (x + y)) / t_0
    else if (x <= (-2.8d-15)) then
        tmp = y / (t_0 * (x + y))
    else
        tmp = (x / (x + y)) / ((x + y) * ((y + 1.0d0) / y))
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double t_0 = x + (y + 1.0);
	double tmp;
	if (x <= -1.32e+154) {
		tmp = (y / (x + y)) / t_0;
	} else if (x <= -2.8e-15) {
		tmp = y / (t_0 * (x + y));
	} else {
		tmp = (x / (x + y)) / ((x + y) * ((y + 1.0) / y));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	t_0 = x + (y + 1.0)
	tmp = 0
	if x <= -1.32e+154:
		tmp = (y / (x + y)) / t_0
	elif x <= -2.8e-15:
		tmp = y / (t_0 * (x + y))
	else:
		tmp = (x / (x + y)) / ((x + y) * ((y + 1.0) / y))
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	t_0 = Float64(x + Float64(y + 1.0))
	tmp = 0.0
	if (x <= -1.32e+154)
		tmp = Float64(Float64(y / Float64(x + y)) / t_0);
	elseif (x <= -2.8e-15)
		tmp = Float64(y / Float64(t_0 * Float64(x + y)));
	else
		tmp = Float64(Float64(x / Float64(x + y)) / Float64(Float64(x + y) * Float64(Float64(y + 1.0) / y)));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	t_0 = x + (y + 1.0);
	tmp = 0.0;
	if (x <= -1.32e+154)
		tmp = (y / (x + y)) / t_0;
	elseif (x <= -2.8e-15)
		tmp = y / (t_0 * (x + y));
	else
		tmp = (x / (x + y)) / ((x + y) * ((y + 1.0) / y));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := Block[{t$95$0 = N[(x + N[(y + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.32e+154], N[(N[(y / N[(x + y), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision], If[LessEqual[x, -2.8e-15], N[(y / N[(t$95$0 * N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(x + y), $MachinePrecision]), $MachinePrecision] / N[(N[(x + y), $MachinePrecision] * N[(N[(y + 1.0), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.31999999999999998e154

   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. Add Preprocessing
   3. Step-by-step derivation

      1. associate-*l* 67.3%

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

      2. times-frac 84.8%

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

      3. associate-+r+ 84.8%

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

   4. Applied egg-rr 84.8%

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

      1. clear-num 84.8%

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

      2. associate-/r* 100.0%

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

      3. frac-times 99.9%

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

      4. *-un-lft-identity 99.9%

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

      5. +-commutative 99.9%

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

      6. +-commutative 99.9%

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

   6. Applied egg-rr 99.9%

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

   7. Taylor expanded in y around 0 99.9%

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

   8. Taylor expanded in y around 0 93.7%

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

   if -1.31999999999999998e154 < x < -2.80000000000000014e-15

   1. Initial program 75.1%

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

      1. associate-*l* 75.1%

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

      2. times-frac 91.9%

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

      3. associate-+r+ 91.9%

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

   4. Applied egg-rr 91.9%

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

   5. Taylor expanded in x around inf 80.4%

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

   if -2.80000000000000014e-15 < x

   1. Initial program 64.7%

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

      1. associate-*l* 64.7%

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

      2. times-frac 93.3%

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

      3. associate-+r+ 93.3%

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

   4. Applied egg-rr 93.3%

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

      1. clear-num 92.4%

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

      2. un-div-inv 92.4%

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

      3. +-commutative 92.4%

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

      4. *-un-lft-identity 92.4%

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

      5. times-frac 98.9%

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

      6. /-rgt-identity 98.9%

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

      7. +-commutative 98.9%

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

   6. Applied egg-rr 98.9%

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

   7. Taylor expanded in x around 0 82.9%

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

      1. +-commutative 82.9%

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

   9. Simplified 82.9%

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

4. Final simplification 84.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.32 \cdot 10^{+154}:\\ \;\;\;\;\frac{\frac{y}{x + y}}{x + \left(y + 1\right)}\\ \mathbf{elif}\;x \leq -2.8 \cdot 10^{-15}:\\ \;\;\;\;\frac{y}{\left(x + \left(y + 1\right)\right) \cdot \left(x + y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{x + y}}{\left(x + y\right) \cdot \frac{y + 1}{y}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 91.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 0.065:\\ \;\;\;\;\frac{y}{x + y} \cdot \frac{x}{\left(x + y\right) \cdot \left(x + 1\right)}\\ \mathbf{elif}\;y \leq 1.65 \cdot 10^{+60}:\\ \;\;\;\;\frac{x \cdot y}{\left(y + 1\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x + y} \cdot \frac{x}{y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 0.065)
   (* (/ y (+ x y)) (/ x (* (+ x y) (+ x 1.0))))
   (if (<= y 1.65e+60)
     (/ (* x y) (* (+ y 1.0) (* (+ x y) (+ x y))))
     (* (/ 1.0 (+ x y)) (/ x y)))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 0.065) {
		tmp = (y / (x + y)) * (x / ((x + y) * (x + 1.0)));
	} else if (y <= 1.65e+60) {
		tmp = (x * y) / ((y + 1.0) * ((x + y) * (x + y)));
	} else {
		tmp = (1.0 / (x + y)) * (x / y);
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 0.065d0) then
        tmp = (y / (x + y)) * (x / ((x + y) * (x + 1.0d0)))
    else if (y <= 1.65d+60) then
        tmp = (x * y) / ((y + 1.0d0) * ((x + y) * (x + y)))
    else
        tmp = (1.0d0 / (x + y)) * (x / y)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (y <= 0.065) {
		tmp = (y / (x + y)) * (x / ((x + y) * (x + 1.0)));
	} else if (y <= 1.65e+60) {
		tmp = (x * y) / ((y + 1.0) * ((x + y) * (x + y)));
	} else {
		tmp = (1.0 / (x + y)) * (x / y);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if y <= 0.065:
		tmp = (y / (x + y)) * (x / ((x + y) * (x + 1.0)))
	elif y <= 1.65e+60:
		tmp = (x * y) / ((y + 1.0) * ((x + y) * (x + y)))
	else:
		tmp = (1.0 / (x + y)) * (x / y)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= 0.065)
		tmp = Float64(Float64(y / Float64(x + y)) * Float64(x / Float64(Float64(x + y) * Float64(x + 1.0))));
	elseif (y <= 1.65e+60)
		tmp = Float64(Float64(x * y) / Float64(Float64(y + 1.0) * Float64(Float64(x + y) * Float64(x + y))));
	else
		tmp = Float64(Float64(1.0 / Float64(x + y)) * Float64(x / y));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 0.065)
		tmp = (y / (x + y)) * (x / ((x + y) * (x + 1.0)));
	elseif (y <= 1.65e+60)
		tmp = (x * y) / ((y + 1.0) * ((x + y) * (x + y)));
	else
		tmp = (1.0 / (x + y)) * (x / y);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[y, 0.065], N[(N[(y / N[(x + y), $MachinePrecision]), $MachinePrecision] * N[(x / N[(N[(x + y), $MachinePrecision] * N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.65e+60], N[(N[(x * y), $MachinePrecision] / N[(N[(y + 1.0), $MachinePrecision] * N[(N[(x + y), $MachinePrecision] * N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[(x + y), $MachinePrecision]), $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 0.065000000000000002

   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. Add Preprocessing

   3. Taylor expanded in x around inf 58.6%

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

      1. *-commutative 58.6%

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

      2. associate-*l* 58.6%

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

      3. times-frac 81.9%

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

      4. +-commutative 81.9%

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

      5. +-commutative 81.9%

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

   5. Applied egg-rr 81.9%

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

   if 0.065000000000000002 < y < 1.6499999999999999e60

   1. Initial program 71.7%

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

   3. Taylor expanded in x around 0 71.5%

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

   if 1.6499999999999999e60 < y

   1. Initial program 63.8%

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

      1. associate-*l* 63.8%

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

      2. times-frac 80.9%

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

      3. associate-+r+ 80.9%

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

   4. Applied egg-rr 80.9%

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

   5. Taylor expanded in y around inf 79.1%

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

      1. clear-num 79.0%

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

      2. +-commutative 79.0%

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

      3. frac-times 78.7%

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

      4. metadata-eval 78.7%

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

      5. +-commutative 78.7%

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

   7. Applied egg-rr 78.7%

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

      1. associate-/r/ 78.7%

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

      2. associate-/r/ 79.1%

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

   9. Simplified 79.1%

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

4. Final simplification 80.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 0.065:\\ \;\;\;\;\frac{y}{x + y} \cdot \frac{x}{\left(x + y\right) \cdot \left(x + 1\right)}\\ \mathbf{elif}\;y \leq 1.65 \cdot 10^{+60}:\\ \;\;\;\;\frac{x \cdot y}{\left(y + 1\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x + y} \cdot \frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 54.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -6.2 \cdot 10^{+232}:\\ \;\;\;\;\frac{x}{y \cdot \left(x + 1\right)}\\ \mathbf{elif}\;x \leq -4.2 \cdot 10^{+29}:\\ \;\;\;\;\frac{\frac{x}{y}}{y}\\ \mathbf{elif}\;x \leq 5.6 \cdot 10^{-59}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} \cdot \frac{1}{y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -6.2e+232)
   (/ x (* y (+ x 1.0)))
   (if (<= x -4.2e+29)
     (/ (/ x y) y)
     (if (<= x 5.6e-59) (/ x (* y (+ y 1.0))) (* (/ x y) (/ 1.0 y))))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -6.2e+232) {
		tmp = x / (y * (x + 1.0));
	} else if (x <= -4.2e+29) {
		tmp = (x / y) / y;
	} else if (x <= 5.6e-59) {
		tmp = x / (y * (y + 1.0));
	} else {
		tmp = (x / y) * (1.0 / y);
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-6.2d+232)) then
        tmp = x / (y * (x + 1.0d0))
    else if (x <= (-4.2d+29)) then
        tmp = (x / y) / y
    else if (x <= 5.6d-59) then
        tmp = x / (y * (y + 1.0d0))
    else
        tmp = (x / y) * (1.0d0 / y)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (x <= -6.2e+232) {
		tmp = x / (y * (x + 1.0));
	} else if (x <= -4.2e+29) {
		tmp = (x / y) / y;
	} else if (x <= 5.6e-59) {
		tmp = x / (y * (y + 1.0));
	} else {
		tmp = (x / y) * (1.0 / y);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -6.2e+232:
		tmp = x / (y * (x + 1.0))
	elif x <= -4.2e+29:
		tmp = (x / y) / y
	elif x <= 5.6e-59:
		tmp = x / (y * (y + 1.0))
	else:
		tmp = (x / y) * (1.0 / y)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -6.2e+232)
		tmp = Float64(x / Float64(y * Float64(x + 1.0)));
	elseif (x <= -4.2e+29)
		tmp = Float64(Float64(x / y) / y);
	elseif (x <= 5.6e-59)
		tmp = Float64(x / Float64(y * Float64(y + 1.0)));
	else
		tmp = Float64(Float64(x / y) * Float64(1.0 / y));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -6.2e+232)
		tmp = x / (y * (x + 1.0));
	elseif (x <= -4.2e+29)
		tmp = (x / y) / y;
	elseif (x <= 5.6e-59)
		tmp = x / (y * (y + 1.0));
	else
		tmp = (x / y) * (1.0 / y);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[x, -6.2e+232], N[(x / N[(y * N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -4.2e+29], N[(N[(x / y), $MachinePrecision] / y), $MachinePrecision], If[LessEqual[x, 5.6e-59], N[(x / N[(y * N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] * N[(1.0 / y), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < -6.19999999999999966e232

   1. Initial program 61.9%

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

   3. Taylor expanded in x around inf 61.9%

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

   4. Taylor expanded in y around inf 31.0%

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

      1. +-commutative 31.0%

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

   6. Simplified 31.0%

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

   if -6.19999999999999966e232 < x < -4.2000000000000003e29

   1. Initial program 73.2%

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

      1. associate-*l* 73.2%

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

      2. times-frac 83.8%

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

      3. associate-+r+ 83.8%

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

   4. Applied egg-rr 83.8%

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

      1. clear-num 82.6%

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

      2. un-div-inv 82.7%

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

      3. +-commutative 82.7%

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

      4. *-un-lft-identity 82.7%

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

      5. times-frac 98.6%

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

      6. /-rgt-identity 98.6%

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

      7. +-commutative 98.6%

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

   6. Applied egg-rr 98.6%

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

   7. Taylor expanded in y around inf 20.5%

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

   8. Taylor expanded in x around 0 19.7%

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

   if -4.2000000000000003e29 < x < 5.59999999999999961e-59

   1. Initial program 72.4%

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

      1. associate-/l* 88.4%

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

      2. associate-+l+ 88.4%

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

   3. Simplified 88.4%

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

   5. Taylor expanded in x around 0 76.1%

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

      1. +-commutative 76.1%

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

   7. Simplified 76.1%

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

   if 5.59999999999999961e-59 < x

   1. Initial program 53.6%

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

      1. associate-*l* 53.6%

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

      2. times-frac 82.6%

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

      3. associate-+r+ 82.6%

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

   4. Applied egg-rr 82.6%

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

   5. Taylor expanded in y around inf 34.3%

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

   6. Taylor expanded in x around 0 33.6%

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

Alternative 5: 95.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_0 := \frac{x}{x + y}\\ \mathbf{if}\;y \leq 1.02 \cdot 10^{+142}:\\ \;\;\;\;t\_0 \cdot \frac{y}{\left(x + \left(y + 1\right)\right) \cdot \left(x + y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0}{\left(x + y\right) \cdot \frac{y + 1}{y}}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ x (+ x y))))
   (if (<= y 1.02e+142)
     (* t_0 (/ y (* (+ x (+ y 1.0)) (+ x y))))
     (/ t_0 (* (+ x y) (/ (+ y 1.0) y))))))

C

assert(x < y);
double code(double x, double y) {
	double t_0 = x / (x + y);
	double tmp;
	if (y <= 1.02e+142) {
		tmp = t_0 * (y / ((x + (y + 1.0)) * (x + y)));
	} else {
		tmp = t_0 / ((x + y) * ((y + 1.0) / y));
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x / (x + y)
    if (y <= 1.02d+142) then
        tmp = t_0 * (y / ((x + (y + 1.0d0)) * (x + y)))
    else
        tmp = t_0 / ((x + y) * ((y + 1.0d0) / y))
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double t_0 = x / (x + y);
	double tmp;
	if (y <= 1.02e+142) {
		tmp = t_0 * (y / ((x + (y + 1.0)) * (x + y)));
	} else {
		tmp = t_0 / ((x + y) * ((y + 1.0) / y));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	t_0 = x / (x + y)
	tmp = 0
	if y <= 1.02e+142:
		tmp = t_0 * (y / ((x + (y + 1.0)) * (x + y)))
	else:
		tmp = t_0 / ((x + y) * ((y + 1.0) / y))
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	t_0 = Float64(x / Float64(x + y))
	tmp = 0.0
	if (y <= 1.02e+142)
		tmp = Float64(t_0 * Float64(y / Float64(Float64(x + Float64(y + 1.0)) * Float64(x + y))));
	else
		tmp = Float64(t_0 / Float64(Float64(x + y) * Float64(Float64(y + 1.0) / y)));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	t_0 = x / (x + y);
	tmp = 0.0;
	if (y <= 1.02e+142)
		tmp = t_0 * (y / ((x + (y + 1.0)) * (x + y)));
	else
		tmp = t_0 / ((x + y) * ((y + 1.0) / y));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := Block[{t$95$0 = N[(x / N[(x + y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, 1.02e+142], N[(t$95$0 * N[(y / N[(N[(x + N[(y + 1.0), $MachinePrecision]), $MachinePrecision] * N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 / N[(N[(x + y), $MachinePrecision] * N[(N[(y + 1.0), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < 1.0199999999999999e142

   1. Initial program 66.3%

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

      1. associate-*l* 66.3%

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

      2. times-frac 93.6%

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

      3. associate-+r+ 93.6%

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

   4. Applied egg-rr 93.6%

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

   if 1.0199999999999999e142 < y

   1. Initial program 68.4%

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

      1. associate-*l* 68.4%

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

      2. times-frac 81.1%

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

      3. associate-+r+ 81.1%

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

   4. Applied egg-rr 81.1%

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

      1. clear-num 81.1%

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

      2. un-div-inv 81.1%

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

      3. +-commutative 81.1%

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

      4. *-un-lft-identity 81.1%

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

      5. times-frac 99.8%

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

      6. /-rgt-identity 99.8%

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

      7. +-commutative 99.8%

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

   6. Applied egg-rr 99.8%

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

   7. Taylor expanded in x around 0 89.9%

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

      1. +-commutative 89.9%

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

   9. Simplified 89.9%

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

4. Final simplification 93.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.02 \cdot 10^{+142}:\\ \;\;\;\;\frac{x}{x + y} \cdot \frac{y}{\left(x + \left(y + 1\right)\right) \cdot \left(x + y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{x + y}}{\left(x + y\right) \cdot \frac{y + 1}{y}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 90.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 6.2 \cdot 10^{+29}:\\ \;\;\;\;\frac{y}{x + y} \cdot \frac{x}{\left(x + y\right) \cdot \left(x + 1\right)}\\ \mathbf{elif}\;y \leq 1.02 \cdot 10^{+142}:\\ \;\;\;\;x \cdot \left(\frac{1}{x + y} \cdot \frac{1}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + y} \cdot \frac{1}{y + 1}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 6.2e+29)
   (* (/ y (+ x y)) (/ x (* (+ x y) (+ x 1.0))))
   (if (<= y 1.02e+142)
     (* x (* (/ 1.0 (+ x y)) (/ 1.0 y)))
     (* (/ x (+ x y)) (/ 1.0 (+ y 1.0))))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 6.2e+29) {
		tmp = (y / (x + y)) * (x / ((x + y) * (x + 1.0)));
	} else if (y <= 1.02e+142) {
		tmp = x * ((1.0 / (x + y)) * (1.0 / y));
	} else {
		tmp = (x / (x + y)) * (1.0 / (y + 1.0));
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 6.2d+29) then
        tmp = (y / (x + y)) * (x / ((x + y) * (x + 1.0d0)))
    else if (y <= 1.02d+142) then
        tmp = x * ((1.0d0 / (x + y)) * (1.0d0 / y))
    else
        tmp = (x / (x + y)) * (1.0d0 / (y + 1.0d0))
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (y <= 6.2e+29) {
		tmp = (y / (x + y)) * (x / ((x + y) * (x + 1.0)));
	} else if (y <= 1.02e+142) {
		tmp = x * ((1.0 / (x + y)) * (1.0 / y));
	} else {
		tmp = (x / (x + y)) * (1.0 / (y + 1.0));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if y <= 6.2e+29:
		tmp = (y / (x + y)) * (x / ((x + y) * (x + 1.0)))
	elif y <= 1.02e+142:
		tmp = x * ((1.0 / (x + y)) * (1.0 / y))
	else:
		tmp = (x / (x + y)) * (1.0 / (y + 1.0))
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= 6.2e+29)
		tmp = Float64(Float64(y / Float64(x + y)) * Float64(x / Float64(Float64(x + y) * Float64(x + 1.0))));
	elseif (y <= 1.02e+142)
		tmp = Float64(x * Float64(Float64(1.0 / Float64(x + y)) * Float64(1.0 / y)));
	else
		tmp = Float64(Float64(x / Float64(x + y)) * Float64(1.0 / Float64(y + 1.0)));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 6.2e+29)
		tmp = (y / (x + y)) * (x / ((x + y) * (x + 1.0)));
	elseif (y <= 1.02e+142)
		tmp = x * ((1.0 / (x + y)) * (1.0 / y));
	else
		tmp = (x / (x + y)) * (1.0 / (y + 1.0));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[y, 6.2e+29], N[(N[(y / N[(x + y), $MachinePrecision]), $MachinePrecision] * N[(x / N[(N[(x + y), $MachinePrecision] * N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.02e+142], N[(x * N[(N[(1.0 / N[(x + y), $MachinePrecision]), $MachinePrecision] * N[(1.0 / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(x + y), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 6.1999999999999998e29

   1. Initial program 67.5%

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

   3. Taylor expanded in x around inf 58.3%

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

      1. *-commutative 58.3%

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

      2. associate-*l* 58.3%

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

      3. times-frac 81.2%

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

      4. +-commutative 81.2%

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

      5. +-commutative 81.2%

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

   5. Applied egg-rr 81.2%

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

   if 6.1999999999999998e29 < y < 1.0199999999999999e142

   1. Initial program 56.1%

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

      1. associate-/l* 77.0%

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

      2. associate-+l+ 77.0%

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

   3. Simplified 77.0%

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

      1. *-un-lft-identity 77.0%

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

      2. associate-+r+ 77.0%

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

      3. associate-*l* 77.0%

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

      4. times-frac 80.6%

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

      5. associate-+r+ 80.6%

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

   6. Applied egg-rr 80.6%

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

   7. Taylor expanded in y around inf 72.2%

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

   if 1.0199999999999999e142 < y

   1. Initial program 68.4%

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

      1. associate-*l* 68.4%

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

      2. times-frac 81.1%

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

      3. associate-+r+ 81.1%

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

   4. Applied egg-rr 81.1%

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

   5. Taylor expanded in x around 0 89.7%

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

      1. +-commutative 89.7%

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

   7. Simplified 89.7%

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

4. Final simplification 81.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 6.2 \cdot 10^{+29}:\\ \;\;\;\;\frac{y}{x + y} \cdot \frac{x}{\left(x + y\right) \cdot \left(x + 1\right)}\\ \mathbf{elif}\;y \leq 1.02 \cdot 10^{+142}:\\ \;\;\;\;x \cdot \left(\frac{1}{x + y} \cdot \frac{1}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + y} \cdot \frac{1}{y + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 87.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_0 := x + \left(y + 1\right)\\ \mathbf{if}\;x \leq -1.04 \cdot 10^{+157}:\\ \;\;\;\;\frac{\frac{y}{x + y}}{t\_0}\\ \mathbf{elif}\;x \leq -5 \cdot 10^{-152}:\\ \;\;\;\;\frac{y}{t\_0 \cdot \left(x + y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{x + y}}{y + 1}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ x (+ y 1.0))))
   (if (<= x -1.04e+157)
     (/ (/ y (+ x y)) t_0)
     (if (<= x -5e-152) (/ y (* t_0 (+ x y))) (/ (/ x (+ x y)) (+ y 1.0))))))

C

assert(x < y);
double code(double x, double y) {
	double t_0 = x + (y + 1.0);
	double tmp;
	if (x <= -1.04e+157) {
		tmp = (y / (x + y)) / t_0;
	} else if (x <= -5e-152) {
		tmp = y / (t_0 * (x + y));
	} else {
		tmp = (x / (x + y)) / (y + 1.0);
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x + (y + 1.0d0)
    if (x <= (-1.04d+157)) then
        tmp = (y / (x + y)) / t_0
    else if (x <= (-5d-152)) then
        tmp = y / (t_0 * (x + y))
    else
        tmp = (x / (x + y)) / (y + 1.0d0)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double t_0 = x + (y + 1.0);
	double tmp;
	if (x <= -1.04e+157) {
		tmp = (y / (x + y)) / t_0;
	} else if (x <= -5e-152) {
		tmp = y / (t_0 * (x + y));
	} else {
		tmp = (x / (x + y)) / (y + 1.0);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	t_0 = x + (y + 1.0)
	tmp = 0
	if x <= -1.04e+157:
		tmp = (y / (x + y)) / t_0
	elif x <= -5e-152:
		tmp = y / (t_0 * (x + y))
	else:
		tmp = (x / (x + y)) / (y + 1.0)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	t_0 = Float64(x + Float64(y + 1.0))
	tmp = 0.0
	if (x <= -1.04e+157)
		tmp = Float64(Float64(y / Float64(x + y)) / t_0);
	elseif (x <= -5e-152)
		tmp = Float64(y / Float64(t_0 * Float64(x + y)));
	else
		tmp = Float64(Float64(x / Float64(x + y)) / Float64(y + 1.0));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	t_0 = x + (y + 1.0);
	tmp = 0.0;
	if (x <= -1.04e+157)
		tmp = (y / (x + y)) / t_0;
	elseif (x <= -5e-152)
		tmp = y / (t_0 * (x + y));
	else
		tmp = (x / (x + y)) / (y + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := Block[{t$95$0 = N[(x + N[(y + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.04e+157], N[(N[(y / N[(x + y), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision], If[LessEqual[x, -5e-152], N[(y / N[(t$95$0 * N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(x + y), $MachinePrecision]), $MachinePrecision] / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.04000000000000007e157

   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. Add Preprocessing
   3. Step-by-step derivation

      1. associate-*l* 67.3%

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

      2. times-frac 84.8%

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

      3. associate-+r+ 84.8%

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

   4. Applied egg-rr 84.8%

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

      1. clear-num 84.8%

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

      2. associate-/r* 100.0%

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

      3. frac-times 99.9%

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

      4. *-un-lft-identity 99.9%

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

      5. +-commutative 99.9%

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

      6. +-commutative 99.9%

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

   6. Applied egg-rr 99.9%

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

   7. Taylor expanded in y around 0 99.9%

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

   8. Taylor expanded in y around 0 93.7%

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

   if -1.04000000000000007e157 < x < -4.9999999999999997e-152

   1. Initial program 76.2%

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

      1. associate-*l* 76.2%

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

      2. times-frac 95.4%

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

      3. associate-+r+ 95.4%

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

   4. Applied egg-rr 95.4%

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

   5. Taylor expanded in x around inf 73.9%

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

   if -4.9999999999999997e-152 < x

   1. Initial program 62.2%

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

      1. associate-*l* 62.2%

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

      2. times-frac 92.1%

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

      3. associate-+r+ 92.1%

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

   4. Applied egg-rr 92.1%

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

      1. clear-num 91.0%

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

      2. un-div-inv 91.0%

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

      3. +-commutative 91.0%

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

      4. *-un-lft-identity 91.0%

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

      5. times-frac 98.8%

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

      6. /-rgt-identity 98.8%

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

      7. +-commutative 98.8%

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

   6. Applied egg-rr 98.8%

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

   7. Taylor expanded in x around 0 61.6%

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

      1. +-commutative 61.6%

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

   9. Simplified 61.6%

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

4. Final simplification 69.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.04 \cdot 10^{+157}:\\ \;\;\;\;\frac{\frac{y}{x + y}}{x + \left(y + 1\right)}\\ \mathbf{elif}\;x \leq -5 \cdot 10^{-152}:\\ \;\;\;\;\frac{y}{\left(x + \left(y + 1\right)\right) \cdot \left(x + y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{x + y}}{y + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 87.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -2.8 \cdot 10^{+154}:\\ \;\;\;\;\frac{\frac{y}{x + y}}{x}\\ \mathbf{elif}\;x \leq -1.9 \cdot 10^{-151}:\\ \;\;\;\;\frac{y}{\left(x + \left(y + 1\right)\right) \cdot \left(x + y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{x + y}}{y + 1}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -2.8e+154)
   (/ (/ y (+ x y)) x)
   (if (<= x -1.9e-151)
     (/ y (* (+ x (+ y 1.0)) (+ x y)))
     (/ (/ x (+ x y)) (+ y 1.0)))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -2.8e+154) {
		tmp = (y / (x + y)) / x;
	} else if (x <= -1.9e-151) {
		tmp = y / ((x + (y + 1.0)) * (x + y));
	} else {
		tmp = (x / (x + y)) / (y + 1.0);
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-2.8d+154)) then
        tmp = (y / (x + y)) / x
    else if (x <= (-1.9d-151)) then
        tmp = y / ((x + (y + 1.0d0)) * (x + y))
    else
        tmp = (x / (x + y)) / (y + 1.0d0)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (x <= -2.8e+154) {
		tmp = (y / (x + y)) / x;
	} else if (x <= -1.9e-151) {
		tmp = y / ((x + (y + 1.0)) * (x + y));
	} else {
		tmp = (x / (x + y)) / (y + 1.0);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -2.8e+154:
		tmp = (y / (x + y)) / x
	elif x <= -1.9e-151:
		tmp = y / ((x + (y + 1.0)) * (x + y))
	else:
		tmp = (x / (x + y)) / (y + 1.0)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -2.8e+154)
		tmp = Float64(Float64(y / Float64(x + y)) / x);
	elseif (x <= -1.9e-151)
		tmp = Float64(y / Float64(Float64(x + Float64(y + 1.0)) * Float64(x + y)));
	else
		tmp = Float64(Float64(x / Float64(x + y)) / Float64(y + 1.0));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -2.8e+154)
		tmp = (y / (x + y)) / x;
	elseif (x <= -1.9e-151)
		tmp = y / ((x + (y + 1.0)) * (x + y));
	else
		tmp = (x / (x + y)) / (y + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[x, -2.8e+154], N[(N[(y / N[(x + y), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[x, -1.9e-151], N[(y / N[(N[(x + N[(y + 1.0), $MachinePrecision]), $MachinePrecision] * N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(x + y), $MachinePrecision]), $MachinePrecision] / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.7999999999999999e154

   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. Add Preprocessing
   3. Step-by-step derivation

      1. associate-*l* 67.3%

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

      2. times-frac 84.8%

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

      3. associate-+r+ 84.8%

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

   4. Applied egg-rr 84.8%

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

      1. clear-num 84.8%

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

      2. associate-/r* 100.0%

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

      3. frac-times 99.9%

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

      4. *-un-lft-identity 99.9%

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

      5. +-commutative 99.9%

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

      6. +-commutative 99.9%

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

   6. Applied egg-rr 99.9%

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

   7. Taylor expanded in x around inf 93.7%

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

   if -2.7999999999999999e154 < x < -1.89999999999999985e-151

   1. Initial program 76.2%

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

      1. associate-*l* 76.2%

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

      2. times-frac 95.4%

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

      3. associate-+r+ 95.4%

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

   4. Applied egg-rr 95.4%

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

   5. Taylor expanded in x around inf 73.9%

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

   if -1.89999999999999985e-151 < x

   1. Initial program 62.2%

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

      1. associate-*l* 62.2%

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

      2. times-frac 92.1%

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

      3. associate-+r+ 92.1%

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

   4. Applied egg-rr 92.1%

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

      1. clear-num 91.0%

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

      2. un-div-inv 91.0%

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

      3. +-commutative 91.0%

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

      4. *-un-lft-identity 91.0%

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

      5. times-frac 98.8%

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

      6. /-rgt-identity 98.8%

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

      7. +-commutative 98.8%

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

   6. Applied egg-rr 98.8%

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

   7. Taylor expanded in x around 0 61.6%

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

      1. +-commutative 61.6%

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

   9. Simplified 61.6%

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

4. Final simplification 69.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.8 \cdot 10^{+154}:\\ \;\;\;\;\frac{\frac{y}{x + y}}{x}\\ \mathbf{elif}\;x \leq -1.9 \cdot 10^{-151}:\\ \;\;\;\;\frac{y}{\left(x + \left(y + 1\right)\right) \cdot \left(x + y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{x + y}}{y + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 95.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -7 \cdot 10^{-28}:\\ \;\;\;\;\frac{\frac{y}{x}}{\left(x + \left(y + 1\right)\right) \cdot \left(1 + \frac{y}{x}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{x + y}}{\left(x + y\right) \cdot \frac{y + 1}{y}}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -7e-28)
   (/ (/ y x) (* (+ x (+ y 1.0)) (+ 1.0 (/ y x))))
   (/ (/ x (+ x y)) (* (+ x y) (/ (+ y 1.0) y)))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -7e-28) {
		tmp = (y / x) / ((x + (y + 1.0)) * (1.0 + (y / x)));
	} else {
		tmp = (x / (x + y)) / ((x + y) * ((y + 1.0) / y));
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-7d-28)) then
        tmp = (y / x) / ((x + (y + 1.0d0)) * (1.0d0 + (y / x)))
    else
        tmp = (x / (x + y)) / ((x + y) * ((y + 1.0d0) / y))
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (x <= -7e-28) {
		tmp = (y / x) / ((x + (y + 1.0)) * (1.0 + (y / x)));
	} else {
		tmp = (x / (x + y)) / ((x + y) * ((y + 1.0) / y));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -7e-28:
		tmp = (y / x) / ((x + (y + 1.0)) * (1.0 + (y / x)))
	else:
		tmp = (x / (x + y)) / ((x + y) * ((y + 1.0) / y))
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -7e-28)
		tmp = Float64(Float64(y / x) / Float64(Float64(x + Float64(y + 1.0)) * Float64(1.0 + Float64(y / x))));
	else
		tmp = Float64(Float64(x / Float64(x + y)) / Float64(Float64(x + y) * Float64(Float64(y + 1.0) / y)));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -7e-28)
		tmp = (y / x) / ((x + (y + 1.0)) * (1.0 + (y / x)));
	else
		tmp = (x / (x + y)) / ((x + y) * ((y + 1.0) / y));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[x, -7e-28], N[(N[(y / x), $MachinePrecision] / N[(N[(x + N[(y + 1.0), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(x + y), $MachinePrecision]), $MachinePrecision] / N[(N[(x + y), $MachinePrecision] * N[(N[(y + 1.0), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -6.9999999999999999e-28

   1. Initial program 72.7%

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

      1. associate-*l* 72.7%

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

      2. times-frac 89.0%

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

      3. associate-+r+ 89.0%

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

   4. Applied egg-rr 89.0%

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

      1. clear-num 89.0%

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

      2. associate-/r* 99.8%

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

      3. frac-times 99.6%

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

      4. *-un-lft-identity 99.6%

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

      5. +-commutative 99.6%

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

      6. +-commutative 99.6%

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

   6. Applied egg-rr 99.6%

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

   7. Taylor expanded in y around 0 99.6%

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

   8. Taylor expanded in y around 0 87.5%

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

   if -6.9999999999999999e-28 < x

   1. Initial program 63.9%

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

      1. associate-*l* 63.9%

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

      2. times-frac 93.1%

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

      3. associate-+r+ 93.1%

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

   4. Applied egg-rr 93.1%

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

      1. clear-num 92.2%

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

      2. un-div-inv 92.2%

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

      3. +-commutative 92.2%

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

      4. *-un-lft-identity 92.2%

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

      5. times-frac 98.9%

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

      6. /-rgt-identity 98.9%

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

      7. +-commutative 98.9%

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

   6. Applied egg-rr 98.9%

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

   7. Taylor expanded in x around 0 82.5%

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

      1. +-commutative 82.5%

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

   9. Simplified 82.5%

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

4. Final simplification 84.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -7 \cdot 10^{-28}:\\ \;\;\;\;\frac{\frac{y}{x}}{\left(x + \left(y + 1\right)\right) \cdot \left(1 + \frac{y}{x}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{x + y}}{\left(x + y\right) \cdot \frac{y + 1}{y}}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 82.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -9 \cdot 10^{-141}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + 1}\\ \mathbf{elif}\;x \leq 5.6 \cdot 10^{-59}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x + y} \cdot \frac{x}{y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -9e-141)
   (/ (/ y x) (+ x 1.0))
   (if (<= x 5.6e-59) (/ x (* y (+ y 1.0))) (* (/ 1.0 (+ x y)) (/ x y)))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -9e-141) {
		tmp = (y / x) / (x + 1.0);
	} else if (x <= 5.6e-59) {
		tmp = x / (y * (y + 1.0));
	} else {
		tmp = (1.0 / (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 <= (-9d-141)) then
        tmp = (y / x) / (x + 1.0d0)
    else if (x <= 5.6d-59) then
        tmp = x / (y * (y + 1.0d0))
    else
        tmp = (1.0d0 / (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 <= -9e-141) {
		tmp = (y / x) / (x + 1.0);
	} else if (x <= 5.6e-59) {
		tmp = x / (y * (y + 1.0));
	} else {
		tmp = (1.0 / (x + y)) * (x / y);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -9e-141:
		tmp = (y / x) / (x + 1.0)
	elif x <= 5.6e-59:
		tmp = x / (y * (y + 1.0))
	else:
		tmp = (1.0 / (x + y)) * (x / y)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -9e-141)
		tmp = Float64(Float64(y / x) / Float64(x + 1.0));
	elseif (x <= 5.6e-59)
		tmp = Float64(x / Float64(y * Float64(y + 1.0)));
	else
		tmp = Float64(Float64(1.0 / 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 <= -9e-141)
		tmp = (y / x) / (x + 1.0);
	elseif (x <= 5.6e-59)
		tmp = x / (y * (y + 1.0));
	else
		tmp = (1.0 / (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, -9e-141], N[(N[(y / x), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 5.6e-59], N[(x / N[(y * N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[(x + y), $MachinePrecision]), $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -9.0000000000000001e-141

   1. Initial program 73.5%

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

      1. associate-/l* 82.6%

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

      2. associate-+l+ 82.6%

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

   3. Simplified 82.6%

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

   5. Taylor expanded in y around 0 65.3%

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

      1. associate-/r* 68.4%

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

      2. +-commutative 68.4%

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

   7. Simplified 68.4%

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

   if -9.0000000000000001e-141 < x < 5.59999999999999961e-59

   1. Initial program 69.0%

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

      1. associate-/l* 86.6%

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

      2. associate-+l+ 86.6%

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

   3. Simplified 86.6%

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

   5. Taylor expanded in x around 0 83.5%

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

      1. +-commutative 83.5%

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

   7. Simplified 83.5%

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

   if 5.59999999999999961e-59 < x

   1. Initial program 53.6%

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

      1. associate-*l* 53.6%

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

      2. times-frac 82.6%

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

      3. associate-+r+ 82.6%

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

   4. Applied egg-rr 82.6%

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

   5. Taylor expanded in y around inf 34.3%

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

      1. clear-num 34.2%

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

      2. +-commutative 34.2%

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

      3. frac-times 34.1%

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

      4. metadata-eval 34.1%

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

      5. +-commutative 34.1%

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

   7. Applied egg-rr 34.1%

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

      1. associate-/r/ 34.0%

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

      2. associate-/r/ 34.2%

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

   9. Simplified 34.2%

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

Alternative 11: 82.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -9 \cdot 10^{-141}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + 1}\\ \mathbf{elif}\;x \leq 5.6 \cdot 10^{-59}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} \cdot \frac{1}{y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -9e-141)
   (/ (/ y x) (+ x 1.0))
   (if (<= x 5.6e-59) (/ x (* y (+ y 1.0))) (* (/ x y) (/ 1.0 y)))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -9e-141) {
		tmp = (y / x) / (x + 1.0);
	} else if (x <= 5.6e-59) {
		tmp = x / (y * (y + 1.0));
	} else {
		tmp = (x / y) * (1.0 / y);
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-9d-141)) then
        tmp = (y / x) / (x + 1.0d0)
    else if (x <= 5.6d-59) then
        tmp = x / (y * (y + 1.0d0))
    else
        tmp = (x / y) * (1.0d0 / y)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (x <= -9e-141) {
		tmp = (y / x) / (x + 1.0);
	} else if (x <= 5.6e-59) {
		tmp = x / (y * (y + 1.0));
	} else {
		tmp = (x / y) * (1.0 / y);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -9e-141:
		tmp = (y / x) / (x + 1.0)
	elif x <= 5.6e-59:
		tmp = x / (y * (y + 1.0))
	else:
		tmp = (x / y) * (1.0 / y)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -9e-141)
		tmp = Float64(Float64(y / x) / Float64(x + 1.0));
	elseif (x <= 5.6e-59)
		tmp = Float64(x / Float64(y * Float64(y + 1.0)));
	else
		tmp = Float64(Float64(x / y) * Float64(1.0 / y));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -9e-141)
		tmp = (y / x) / (x + 1.0);
	elseif (x <= 5.6e-59)
		tmp = x / (y * (y + 1.0));
	else
		tmp = (x / y) * (1.0 / y);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[x, -9e-141], N[(N[(y / x), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 5.6e-59], N[(x / N[(y * N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] * N[(1.0 / y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -9.0000000000000001e-141

   1. Initial program 73.5%

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

      1. associate-/l* 82.6%

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

      2. associate-+l+ 82.6%

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

   3. Simplified 82.6%

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

   5. Taylor expanded in y around 0 65.3%

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

      1. associate-/r* 68.4%

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

      2. +-commutative 68.4%

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

   7. Simplified 68.4%

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

   if -9.0000000000000001e-141 < x < 5.59999999999999961e-59

   1. Initial program 69.0%

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

      1. associate-/l* 86.6%

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

      2. associate-+l+ 86.6%

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

   3. Simplified 86.6%

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

   5. Taylor expanded in x around 0 83.5%

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

      1. +-commutative 83.5%

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

   7. Simplified 83.5%

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

   if 5.59999999999999961e-59 < x

   1. Initial program 53.6%

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

      1. associate-*l* 53.6%

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

      2. times-frac 82.6%

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

      3. associate-+r+ 82.6%

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

   4. Applied egg-rr 82.6%

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

   5. Taylor expanded in y around inf 34.3%

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

   6. Taylor expanded in x around 0 33.6%

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

Alternative 12: 80.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -8.8 \cdot 10^{-141}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{elif}\;x \leq 5.6 \cdot 10^{-59}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} \cdot \frac{1}{y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -8.8e-141)
   (/ y (* x (+ x 1.0)))
   (if (<= x 5.6e-59) (/ x (* y (+ y 1.0))) (* (/ x y) (/ 1.0 y)))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -8.8e-141) {
		tmp = y / (x * (x + 1.0));
	} else if (x <= 5.6e-59) {
		tmp = x / (y * (y + 1.0));
	} else {
		tmp = (x / y) * (1.0 / y);
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-8.8d-141)) then
        tmp = y / (x * (x + 1.0d0))
    else if (x <= 5.6d-59) then
        tmp = x / (y * (y + 1.0d0))
    else
        tmp = (x / y) * (1.0d0 / y)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (x <= -8.8e-141) {
		tmp = y / (x * (x + 1.0));
	} else if (x <= 5.6e-59) {
		tmp = x / (y * (y + 1.0));
	} else {
		tmp = (x / y) * (1.0 / y);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -8.8e-141:
		tmp = y / (x * (x + 1.0))
	elif x <= 5.6e-59:
		tmp = x / (y * (y + 1.0))
	else:
		tmp = (x / y) * (1.0 / y)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -8.8e-141)
		tmp = Float64(y / Float64(x * Float64(x + 1.0)));
	elseif (x <= 5.6e-59)
		tmp = Float64(x / Float64(y * Float64(y + 1.0)));
	else
		tmp = Float64(Float64(x / y) * Float64(1.0 / y));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -8.8e-141)
		tmp = y / (x * (x + 1.0));
	elseif (x <= 5.6e-59)
		tmp = x / (y * (y + 1.0));
	else
		tmp = (x / y) * (1.0 / y);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[x, -8.8e-141], N[(y / N[(x * N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 5.6e-59], N[(x / N[(y * N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] * N[(1.0 / y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -8.80000000000000037e-141

   1. Initial program 73.5%

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

      1. associate-/l* 82.6%

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

      2. associate-+l+ 82.6%

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

   3. Simplified 82.6%

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

   5. Taylor expanded in y around 0 65.3%

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

   if -8.80000000000000037e-141 < x < 5.59999999999999961e-59

   1. Initial program 69.0%

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

      1. associate-/l* 86.6%

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

      2. associate-+l+ 86.6%

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

   3. Simplified 86.6%

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

   5. Taylor expanded in x around 0 83.5%

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

      1. +-commutative 83.5%

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

   7. Simplified 83.5%

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

   if 5.59999999999999961e-59 < x

   1. Initial program 53.6%

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

      1. associate-*l* 53.6%

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

      2. times-frac 82.6%

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

      3. associate-+r+ 82.6%

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

   4. Applied egg-rr 82.6%

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

   5. Taylor expanded in y around inf 34.3%

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

   6. Taylor expanded in x around 0 33.6%

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

4. Final simplification 62.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -8.8 \cdot 10^{-141}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{elif}\;x \leq 5.6 \cdot 10^{-59}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} \cdot \frac{1}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 82.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -9.5 \cdot 10^{-150}:\\ \;\;\;\;\frac{\frac{y}{x + 1}}{x + y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{x + y}}{y + 1}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -9.5e-150) (/ (/ y (+ x 1.0)) (+ x y)) (/ (/ x (+ x y)) (+ y 1.0))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -9.5e-150) {
		tmp = (y / (x + 1.0)) / (x + y);
	} else {
		tmp = (x / (x + y)) / (y + 1.0);
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-9.5d-150)) then
        tmp = (y / (x + 1.0d0)) / (x + y)
    else
        tmp = (x / (x + y)) / (y + 1.0d0)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (x <= -9.5e-150) {
		tmp = (y / (x + 1.0)) / (x + y);
	} else {
		tmp = (x / (x + y)) / (y + 1.0);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -9.5e-150:
		tmp = (y / (x + 1.0)) / (x + y)
	else:
		tmp = (x / (x + y)) / (y + 1.0)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -9.5e-150)
		tmp = Float64(Float64(y / Float64(x + 1.0)) / Float64(x + y));
	else
		tmp = Float64(Float64(x / Float64(x + y)) / Float64(y + 1.0));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -9.5e-150)
		tmp = (y / (x + 1.0)) / (x + y);
	else
		tmp = (x / (x + y)) / (y + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[x, -9.5e-150], N[(N[(y / N[(x + 1.0), $MachinePrecision]), $MachinePrecision] / N[(x + y), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(x + y), $MachinePrecision]), $MachinePrecision] / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -9.50000000000000013e-150

   1. Initial program 73.1%

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

   3. Taylor expanded in x around inf 63.3%

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

   4. Taylor expanded in x around inf 51.3%

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

      1. associate-*l* 51.3%

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

      2. +-commutative 51.3%

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

      3. times-frac 65.1%

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

      4. +-commutative 65.1%

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

   6. Applied egg-rr 65.1%

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

      1. associate-*l/ 65.1%

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

      2. associate-/l* 54.5%

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

      3. times-frac 68.4%

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

      4. *-inverses 68.4%

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

   8. Simplified 68.4%

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

   if -9.50000000000000013e-150 < x

   1. Initial program 62.2%

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

      1. associate-*l* 62.2%

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

      2. times-frac 92.1%

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

      3. associate-+r+ 92.1%

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

   4. Applied egg-rr 92.1%

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

      1. clear-num 91.0%

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

      2. un-div-inv 91.0%

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

      3. +-commutative 91.0%

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

      4. *-un-lft-identity 91.0%

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

      5. times-frac 98.8%

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

      6. /-rgt-identity 98.8%

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

      7. +-commutative 98.8%

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

   6. Applied egg-rr 98.8%

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

   7. Taylor expanded in x around 0 61.6%

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

      1. +-commutative 61.6%

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

   9. Simplified 61.6%

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

4. Final simplification 64.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -9.5 \cdot 10^{-150}:\\ \;\;\;\;\frac{\frac{y}{x + 1}}{x + y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{x + y}}{y + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 82.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -9.5 \cdot 10^{-150}:\\ \;\;\;\;\frac{\frac{y}{x + y}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{x + y}}{y + 1}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -9.5e-150) (/ (/ y (+ x y)) (+ x 1.0)) (/ (/ x (+ x y)) (+ y 1.0))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -9.5e-150) {
		tmp = (y / (x + y)) / (x + 1.0);
	} else {
		tmp = (x / (x + y)) / (y + 1.0);
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-9.5d-150)) then
        tmp = (y / (x + y)) / (x + 1.0d0)
    else
        tmp = (x / (x + y)) / (y + 1.0d0)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (x <= -9.5e-150) {
		tmp = (y / (x + y)) / (x + 1.0);
	} else {
		tmp = (x / (x + y)) / (y + 1.0);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -9.5e-150:
		tmp = (y / (x + y)) / (x + 1.0)
	else:
		tmp = (x / (x + y)) / (y + 1.0)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -9.5e-150)
		tmp = Float64(Float64(y / Float64(x + y)) / Float64(x + 1.0));
	else
		tmp = Float64(Float64(x / Float64(x + y)) / Float64(y + 1.0));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -9.5e-150)
		tmp = (y / (x + y)) / (x + 1.0);
	else
		tmp = (x / (x + y)) / (y + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[x, -9.5e-150], N[(N[(y / N[(x + y), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(x + y), $MachinePrecision]), $MachinePrecision] / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -9.50000000000000013e-150

   1. Initial program 73.1%

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

      1. associate-*l* 73.1%

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

      2. times-frac 91.6%

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

      3. associate-+r+ 91.6%

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

   4. Applied egg-rr 91.6%

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

      1. clear-num 91.6%

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

      2. associate-/r* 99.8%

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

      3. frac-times 99.6%

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

      4. *-un-lft-identity 99.6%

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

      5. +-commutative 99.6%

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

      6. +-commutative 99.6%

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

   6. Applied egg-rr 99.6%

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

   7. Taylor expanded in y around 0 68.4%

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

      1. +-commutative 68.4%

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

   9. Simplified 68.4%

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

   if -9.50000000000000013e-150 < x

   1. Initial program 62.2%

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

      1. associate-*l* 62.2%

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

      2. times-frac 92.1%

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

      3. associate-+r+ 92.1%

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

   4. Applied egg-rr 92.1%

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

      1. clear-num 91.0%

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

      2. un-div-inv 91.0%

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

      3. +-commutative 91.0%

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

      4. *-un-lft-identity 91.0%

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

      5. times-frac 98.8%

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

      6. /-rgt-identity 98.8%

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

      7. +-commutative 98.8%

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

   6. Applied egg-rr 98.8%

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

   7. Taylor expanded in x around 0 61.6%

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

      1. +-commutative 61.6%

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

   9. Simplified 61.6%

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

4. Final simplification 64.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -9.5 \cdot 10^{-150}:\\ \;\;\;\;\frac{\frac{y}{x + y}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{x + y}}{y + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 82.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -9 \cdot 10^{-141}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{x + y}}{y + 1}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -9e-141) (/ (/ y x) (+ x 1.0)) (/ (/ x (+ x y)) (+ y 1.0))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -9e-141) {
		tmp = (y / x) / (x + 1.0);
	} else {
		tmp = (x / (x + y)) / (y + 1.0);
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-9d-141)) then
        tmp = (y / x) / (x + 1.0d0)
    else
        tmp = (x / (x + y)) / (y + 1.0d0)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (x <= -9e-141) {
		tmp = (y / x) / (x + 1.0);
	} else {
		tmp = (x / (x + y)) / (y + 1.0);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -9e-141:
		tmp = (y / x) / (x + 1.0)
	else:
		tmp = (x / (x + y)) / (y + 1.0)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -9e-141)
		tmp = Float64(Float64(y / x) / Float64(x + 1.0));
	else
		tmp = Float64(Float64(x / Float64(x + y)) / Float64(y + 1.0));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -9e-141)
		tmp = (y / x) / (x + 1.0);
	else
		tmp = (x / (x + y)) / (y + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[x, -9e-141], N[(N[(y / x), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(x + y), $MachinePrecision]), $MachinePrecision] / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -9.0000000000000001e-141

   1. Initial program 73.5%

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

      1. associate-/l* 82.6%

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

      2. associate-+l+ 82.6%

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

   3. Simplified 82.6%

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

   5. Taylor expanded in y around 0 65.3%

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

      1. associate-/r* 68.4%

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

      2. +-commutative 68.4%

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

   7. Simplified 68.4%

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

   if -9.0000000000000001e-141 < x

   1. Initial program 62.1%

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

      1. associate-*l* 62.1%

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

      2. times-frac 92.2%

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

      3. associate-+r+ 92.2%

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

   4. Applied egg-rr 92.2%

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

      1. clear-num 91.1%

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

      2. un-div-inv 91.1%

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

      3. +-commutative 91.1%

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

      4. *-un-lft-identity 91.1%

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

      5. times-frac 98.8%

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

      6. /-rgt-identity 98.8%

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

      7. +-commutative 98.8%

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

   6. Applied egg-rr 98.8%

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

   7. Taylor expanded in x around 0 61.5%

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

      1. +-commutative 61.5%

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

   9. Simplified 61.5%

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

4. Final simplification 64.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -9 \cdot 10^{-141}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{x + y}}{y + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 51.8% accurate, 1.4× speedup

Math

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

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 6.0) (/ x (* y (+ x 1.0))) (/ (/ x y) y)))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 6.0) {
		tmp = x / (y * (x + 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 (y <= 6.0d0) then
        tmp = x / (y * (x + 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 (y <= 6.0) {
		tmp = x / (y * (x + 1.0));
	} else {
		tmp = (x / y) / y;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if y <= 6.0:
		tmp = x / (y * (x + 1.0))
	else:
		tmp = (x / y) / y
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= 6.0)
		tmp = Float64(x / Float64(y * Float64(x + 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 (y <= 6.0)
		tmp = x / (y * (x + 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[y, 6.0], N[(x / N[(y * N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 6

   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. Add Preprocessing

   3. Taylor expanded in x around inf 58.6%

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

   4. Taylor expanded in y around inf 32.4%

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

      1. +-commutative 32.4%

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

   6. Simplified 32.4%

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

   if 6 < y

   1. Initial program 65.8%

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

      1. associate-*l* 65.8%

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

      2. times-frac 84.4%

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

      3. associate-+r+ 84.4%

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

   4. Applied egg-rr 84.4%

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

      1. clear-num 84.4%

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

      2. un-div-inv 84.4%

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

      3. +-commutative 84.4%

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

      4. *-un-lft-identity 84.4%

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

      5. times-frac 99.8%

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

      6. /-rgt-identity 99.8%

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

      7. +-commutative 99.8%

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

   6. Applied egg-rr 99.8%

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

   7. Taylor expanded in y around inf 70.3%

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

   8. Taylor expanded in x around 0 69.9%

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

Alternative 17: 49.2% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= 1.0)
		tmp = Float64(x / y);
	else
		tmp = Float64(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 (y <= 1.0)
		tmp = x / y;
	else
		tmp = (x / y) / y;
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if y < 1

   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. associate-/l* 81.9%

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

      2. associate-+l+ 81.9%

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

   3. Simplified 81.9%

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

   5. Taylor expanded in x around 0 41.4%

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

      1. +-commutative 41.4%

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

   7. Simplified 41.4%

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

   8. Taylor expanded in y around 0 24.7%

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

   if 1 < y

   1. Initial program 65.8%

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

      1. associate-*l* 65.8%

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

      2. times-frac 84.4%

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

      3. associate-+r+ 84.4%

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

   4. Applied egg-rr 84.4%

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

      1. clear-num 84.4%

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

      2. un-div-inv 84.4%

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

      3. +-commutative 84.4%

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

      4. *-un-lft-identity 84.4%

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

      5. times-frac 99.8%

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

      6. /-rgt-identity 99.8%

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

      7. +-commutative 99.8%

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

   6. Applied egg-rr 99.8%

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

   7. Taylor expanded in y around inf 70.3%

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

   8. Taylor expanded in x around 0 69.9%

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

Alternative 18: 28.9% accurate, 1.7× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -170000:\\ \;\;\;\;\frac{1}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{y}{x}}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -170000.0) (/ 1.0 (+ x 1.0)) (/ 1.0 (/ y x))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -170000.0) {
		tmp = 1.0 / (x + 1.0);
	} else {
		tmp = 1.0 / (y / x);
	}
	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 <= (-170000.0d0)) then
        tmp = 1.0d0 / (x + 1.0d0)
    else
        tmp = 1.0d0 / (y / x)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (x <= -170000.0) {
		tmp = 1.0 / (x + 1.0);
	} else {
		tmp = 1.0 / (y / x);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -170000.0:
		tmp = 1.0 / (x + 1.0)
	else:
		tmp = 1.0 / (y / x)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -170000.0)
		tmp = Float64(1.0 / Float64(x + 1.0));
	else
		tmp = Float64(1.0 / Float64(y / x));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -170000.0)
		tmp = 1.0 / (x + 1.0);
	else
		tmp = 1.0 / (y / x);
	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, -170000.0], N[(1.0 / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(y / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.7e5

   1. Initial program 70.1%

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

   3. Taylor expanded in x around inf 66.6%

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

   4. Taylor expanded in x around inf 63.8%

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

   5. Taylor expanded in y around inf 6.4%

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

      1. +-commutative 6.4%

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

   7. Simplified 6.4%

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

   if -1.7e5 < x

   1. Initial program 65.3%

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

      1. associate-/l* 82.1%

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

      2. associate-+l+ 82.1%

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

   3. Simplified 82.1%

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

   5. Taylor expanded in x around 0 60.7%

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

      1. +-commutative 60.7%

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

   7. Simplified 60.7%

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

   8. Taylor expanded in y around 0 35.4%

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

      1. clear-num 35.3%

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

      2. inv-pow 35.3%

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

   10. Applied egg-rr 35.3%

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

       1. unpow-1 35.3%

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

   12. Simplified 35.3%

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

Alternative 19: 28.5% accurate, 1.7× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -500000:\\ \;\;\;\;\frac{1}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -500000.0) (/ 1.0 (+ x 1.0)) (/ x y)))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -500000.0) {
		tmp = 1.0 / (x + 1.0);
	} 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 <= (-500000.0d0)) then
        tmp = 1.0d0 / (x + 1.0d0)
    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 <= -500000.0) {
		tmp = 1.0 / (x + 1.0);
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -500000.0:
		tmp = 1.0 / (x + 1.0)
	else:
		tmp = x / y
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -500000.0)
		tmp = Float64(1.0 / Float64(x + 1.0));
	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 <= -500000.0)
		tmp = 1.0 / (x + 1.0);
	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, -500000.0], N[(1.0 / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], N[(x / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -5e5

   1. Initial program 70.1%

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

   3. Taylor expanded in x around inf 66.6%

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

   4. Taylor expanded in x around inf 63.8%

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

   5. Taylor expanded in y around inf 6.4%

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

      1. +-commutative 6.4%

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

   7. Simplified 6.4%

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

   if -5e5 < x

   1. Initial program 65.3%

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

      1. associate-/l* 82.1%

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

      2. associate-+l+ 82.1%

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

   3. Simplified 82.1%

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

   5. Taylor expanded in x around 0 60.7%

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

      1. +-commutative 60.7%

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

   7. Simplified 60.7%

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

   8. Taylor expanded in y around 0 35.4%

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

Alternative 20: 26.9% 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 66.6%

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

   1. associate-/l* 81.1%

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

   2. associate-+l+ 81.1%

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

3. Simplified 81.1%

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

5. Taylor expanded in x around 0 47.9%

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

   1. +-commutative 47.9%

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

7. Simplified 47.9%

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

8. Taylor expanded in y around 0 26.7%

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

Alternative 21: 4.1% accurate, 5.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 66.6%

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

   1. associate-*l* 66.6%

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

   2. times-frac 91.9%

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

   3. associate-+r+ 91.9%

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

4. Applied egg-rr 91.9%

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

5. Taylor expanded in y around inf 39.1%

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

6. Taylor expanded in x around inf 4.4%

   \[\leadsto \color{blue}{\frac{1}{y}} \]
7. Add Preprocessing

Developer Target 1: 99.8% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024137 
(FPCore (x y)
  :name "Numeric.SpecFunctions:incompleteBetaApprox from math-functions-0.1.5.2, A"
  :precision binary64

  :alt
  (! :herbie-platform default (/ (/ (/ x (+ (+ y 1) x)) (+ y x)) (/ 1 (/ y (+ y x)))))

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