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

Percentage Accurate: 69.0% → 99.8%

Time: 19.4s

Alternatives: 16

Speedup: 1.4×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 69.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 67.0%

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

   1. *-commutative 67.0%

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

   2. associate-*l* 67.0%

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

   3. times-frac 94.3%

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

   4. +-commutative 94.3%

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

   5. +-commutative 94.3%

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

   6. associate-+r+ 94.3%

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

   7. +-commutative 94.3%

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

   8. associate-+l+ 94.3%

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

4. Applied egg-rr 94.3%

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

   1. frac-times 67.0%

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

   2. *-commutative 67.0%

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

   3. +-commutative 67.0%

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

   4. +-commutative 67.0%

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

   5. associate-+r+ 67.0%

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

   6. +-commutative 67.0%

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

6. Applied egg-rr 67.0%

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

   1. *-commutative 67.0%

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

   2. times-frac 94.3%

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

   3. associate-/r* 99.8%

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

8. Simplified 99.8%

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

9. Final simplification 99.8%

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

Alternative 2: 92.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_0 := x + \left(y + 1\right)\\ \mathbf{if}\;x \leq -1.45 \cdot 10^{+30}:\\ \;\;\;\;\frac{\frac{x}{x + y}}{t\_0} \cdot \frac{y}{x}\\ \mathbf{elif}\;x \leq -7.4 \cdot 10^{-181}:\\ \;\;\;\;x \cdot \frac{y}{t\_0 \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + 1}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ x (+ y 1.0))))
   (if (<= x -1.45e+30)
     (* (/ (/ x (+ x y)) t_0) (/ y x))
     (if (<= x -7.4e-181)
       (* x (/ y (* t_0 (* (+ x y) (+ x y)))))
       (/ (/ 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 <= -1.45e+30) {
		tmp = ((x / (x + y)) / t_0) * (y / x);
	} else if (x <= -7.4e-181) {
		tmp = x * (y / (t_0 * ((x + y) * (x + y))));
	} else {
		tmp = (x / y) / (y + 1.0);
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x + (y + 1.0d0)
    if (x <= (-1.45d+30)) then
        tmp = ((x / (x + y)) / t_0) * (y / x)
    else if (x <= (-7.4d-181)) then
        tmp = x * (y / (t_0 * ((x + y) * (x + y))))
    else
        tmp = (x / y) / (y + 1.0d0)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double t_0 = x + (y + 1.0);
	double tmp;
	if (x <= -1.45e+30) {
		tmp = ((x / (x + y)) / t_0) * (y / x);
	} else if (x <= -7.4e-181) {
		tmp = x * (y / (t_0 * ((x + y) * (x + y))));
	} else {
		tmp = (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 <= -1.45e+30:
		tmp = ((x / (x + y)) / t_0) * (y / x)
	elif x <= -7.4e-181:
		tmp = x * (y / (t_0 * ((x + y) * (x + y))))
	else:
		tmp = (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 <= -1.45e+30)
		tmp = Float64(Float64(Float64(x / Float64(x + y)) / t_0) * Float64(y / x));
	elseif (x <= -7.4e-181)
		tmp = Float64(x * Float64(y / Float64(t_0 * Float64(Float64(x + y) * Float64(x + y)))));
	else
		tmp = Float64(Float64(x / y) / Float64(y + 1.0));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	t_0 = x + (y + 1.0);
	tmp = 0.0;
	if (x <= -1.45e+30)
		tmp = ((x / (x + y)) / t_0) * (y / x);
	elseif (x <= -7.4e-181)
		tmp = x * (y / (t_0 * ((x + y) * (x + y))));
	else
		tmp = (x / y) / (y + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := Block[{t$95$0 = N[(x + N[(y + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.45e+30], N[(N[(N[(x / N[(x + y), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] * N[(y / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -7.4e-181], N[(x * N[(y / N[(t$95$0 * N[(N[(x + y), $MachinePrecision] * N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.4499999999999999e30

   1. Initial program 55.3%

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

      1. *-commutative 55.3%

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

      2. associate-*l* 55.3%

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

      3. times-frac 85.3%

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

      4. +-commutative 85.3%

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

      5. +-commutative 85.3%

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

      6. associate-+r+ 85.3%

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

      7. +-commutative 85.3%

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

      8. associate-+l+ 85.3%

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

   4. Applied egg-rr 85.3%

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

      1. frac-times 55.3%

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

      2. *-commutative 55.3%

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

      3. +-commutative 55.3%

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

      4. +-commutative 55.3%

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

      5. associate-+r+ 55.3%

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

      6. +-commutative 55.3%

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

   6. Applied egg-rr 55.3%

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

      1. *-commutative 55.3%

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

      2. times-frac 85.3%

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

      3. associate-/r* 99.7%

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

   8. Simplified 99.7%

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

   9. Taylor expanded in y around 0 84.4%

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

   if -1.4499999999999999e30 < x < -7.39999999999999968e-181

   1. Initial program 76.6%

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

      1. associate-/l* 96.5%

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

      2. associate-+l+ 96.5%

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

   3. Simplified 96.5%

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

   if -7.39999999999999968e-181 < x

   1. Initial program 68.5%

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

      1. associate-/l* 84.5%

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

      2. associate-+l+ 84.5%

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

   3. Simplified 84.5%

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

   5. Taylor expanded in x around 0 63.1%

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

      1. associate-/r* 64.5%

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

      2. +-commutative 64.5%

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

   7. Simplified 64.5%

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

4. Final simplification 72.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.45 \cdot 10^{+30}:\\ \;\;\;\;\frac{\frac{x}{x + y}}{x + \left(y + 1\right)} \cdot \frac{y}{x}\\ \mathbf{elif}\;x \leq -7.4 \cdot 10^{-181}:\\ \;\;\;\;x \cdot \frac{y}{\left(x + \left(y + 1\right)\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 95.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -1.4 \cdot 10^{+49}:\\ \;\;\;\;\frac{\frac{x}{x + y}}{x + \left(y + 1\right)} \cdot \frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{\frac{y}{\left(x + y\right) \cdot \left(y + \left(x + 1\right)\right)}}{x + y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -1.4e+49)
   (* (/ (/ x (+ x y)) (+ x (+ y 1.0))) (/ y x))
   (* x (/ (/ y (* (+ x y) (+ y (+ x 1.0)))) (+ x y)))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -1.4e+49) {
		tmp = ((x / (x + y)) / (x + (y + 1.0))) * (y / x);
	} else {
		tmp = x * ((y / ((x + y) * (y + (x + 1.0)))) / (x + y));
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-1.4d+49)) then
        tmp = ((x / (x + y)) / (x + (y + 1.0d0))) * (y / x)
    else
        tmp = x * ((y / ((x + y) * (y + (x + 1.0d0)))) / (x + y))
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (x <= -1.4e+49) {
		tmp = ((x / (x + y)) / (x + (y + 1.0))) * (y / x);
	} else {
		tmp = x * ((y / ((x + y) * (y + (x + 1.0)))) / (x + y));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -1.4e+49:
		tmp = ((x / (x + y)) / (x + (y + 1.0))) * (y / x)
	else:
		tmp = x * ((y / ((x + y) * (y + (x + 1.0)))) / (x + y))
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -1.4e+49)
		tmp = Float64(Float64(Float64(x / Float64(x + y)) / Float64(x + Float64(y + 1.0))) * Float64(y / x));
	else
		tmp = Float64(x * Float64(Float64(y / Float64(Float64(x + y) * Float64(y + Float64(x + 1.0)))) / Float64(x + y)));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1.4e+49)
		tmp = ((x / (x + y)) / (x + (y + 1.0))) * (y / x);
	else
		tmp = x * ((y / ((x + y) * (y + (x + 1.0)))) / (x + y));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[x, -1.4e+49], N[(N[(N[(x / N[(x + y), $MachinePrecision]), $MachinePrecision] / N[(x + N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(y / x), $MachinePrecision]), $MachinePrecision], N[(x * N[(N[(y / N[(N[(x + y), $MachinePrecision] * N[(y + N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.3999999999999999e49

   1. Initial program 54.5%

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

      1. *-commutative 54.5%

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

      2. associate-*l* 54.5%

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

      3. times-frac 85.7%

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

      4. +-commutative 85.7%

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

      5. +-commutative 85.7%

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

      6. associate-+r+ 85.7%

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

      7. +-commutative 85.7%

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

      8. associate-+l+ 85.7%

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

   4. Applied egg-rr 85.7%

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

      1. frac-times 54.5%

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

      2. *-commutative 54.5%

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

      3. +-commutative 54.5%

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

      4. +-commutative 54.5%

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

      5. associate-+r+ 54.5%

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

      6. +-commutative 54.5%

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

   6. Applied egg-rr 54.5%

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

      1. *-commutative 54.5%

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

      2. times-frac 85.7%

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

      3. associate-/r* 99.7%

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

   8. Simplified 99.7%

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

   9. Taylor expanded in y around 0 86.9%

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

   if -1.3999999999999999e49 < x

   1. Initial program 69.7%

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

      1. associate-/l* 86.0%

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

      2. associate-+l+ 86.0%

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

   3. Simplified 86.0%

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

      1. *-un-lft-identity 86.0%

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

      2. associate-+r+ 86.0%

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

      3. associate-*l* 86.0%

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

      4. times-frac 95.6%

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

      5. +-commutative 95.6%

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

      6. +-commutative 95.6%

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

      7. associate-+r+ 95.6%

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

      8. +-commutative 95.6%

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

      9. associate-+l+ 95.6%

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

   6. Applied egg-rr 95.6%

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

      1. associate-*l/ 95.7%

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

      2. *-lft-identity 95.7%

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

      3. +-commutative 95.7%

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

   8. Simplified 95.7%

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

4. Final simplification 94.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.4 \cdot 10^{+49}:\\ \;\;\;\;\frac{\frac{x}{x + y}}{x + \left(y + 1\right)} \cdot \frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{\frac{y}{\left(x + y\right) \cdot \left(y + \left(x + 1\right)\right)}}{x + y}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 96.5% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < -1.35000000000000003e154

   1. Initial program 52.0%

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

      1. *-commutative 52.0%

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

      2. associate-*l* 52.0%

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

      3. times-frac 87.2%

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

      4. +-commutative 87.2%

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

      5. +-commutative 87.2%

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

      6. associate-+r+ 87.2%

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

      7. +-commutative 87.2%

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

      8. associate-+l+ 87.2%

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

   4. Applied egg-rr 87.2%

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

      1. frac-times 52.0%

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

      2. *-commutative 52.0%

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

      3. +-commutative 52.0%

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

      4. +-commutative 52.0%

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

      5. associate-+r+ 52.0%

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

      6. +-commutative 52.0%

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

   6. Applied egg-rr 52.0%

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

      1. *-commutative 52.0%

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

      2. times-frac 87.2%

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

      3. associate-/r* 99.9%

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

   8. Simplified 99.9%

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

   9. Taylor expanded in y around 0 92.4%

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

   if -1.35000000000000003e154 < x

   1. Initial program 68.9%

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

      1. *-commutative 68.9%

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

      2. associate-*l* 68.9%

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

      3. times-frac 95.3%

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

      4. +-commutative 95.3%

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

      5. +-commutative 95.3%

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

      6. associate-+r+ 95.3%

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

      7. +-commutative 95.3%

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

      8. associate-+l+ 95.3%

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

   4. Applied egg-rr 95.3%

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

4. Final simplification 94.9%

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

Alternative 5: 84.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -1.75 \cdot 10^{-154}:\\ \;\;\;\;\frac{y}{x} \cdot \frac{x}{\left(x + y\right) \cdot \left(y + \left(x + 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + 1}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -1.75e-154)
   (* (/ y x) (/ x (* (+ x y) (+ y (+ x 1.0)))))
   (/ (/ x y) (+ y 1.0))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -1.75e-154) {
		tmp = (y / x) * (x / ((x + y) * (y + (x + 1.0))));
	} else {
		tmp = (x / y) / (y + 1.0);
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-1.75d-154)) then
        tmp = (y / x) * (x / ((x + y) * (y + (x + 1.0d0))))
    else
        tmp = (x / y) / (y + 1.0d0)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (x <= -1.75e-154) {
		tmp = (y / x) * (x / ((x + y) * (y + (x + 1.0))));
	} else {
		tmp = (x / y) / (y + 1.0);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -1.75e-154:
		tmp = (y / x) * (x / ((x + y) * (y + (x + 1.0))))
	else:
		tmp = (x / y) / (y + 1.0)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -1.75e-154)
		tmp = Float64(Float64(y / x) * Float64(x / Float64(Float64(x + y) * Float64(y + Float64(x + 1.0)))));
	else
		tmp = Float64(Float64(x / y) / Float64(y + 1.0));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1.75e-154)
		tmp = (y / x) * (x / ((x + y) * (y + (x + 1.0))));
	else
		tmp = (x / y) / (y + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[x, -1.75e-154], N[(N[(y / x), $MachinePrecision] * N[(x / N[(N[(x + y), $MachinePrecision] * N[(y + N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.75e-154

   1. Initial program 62.7%

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

      1. *-commutative 62.7%

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

      2. associate-*l* 62.7%

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

      3. times-frac 90.9%

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

      4. +-commutative 90.9%

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

      5. +-commutative 90.9%

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

      6. associate-+r+ 90.9%

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

      7. +-commutative 90.9%

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

      8. associate-+l+ 90.9%

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

   4. Applied egg-rr 90.9%

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

   5. Taylor expanded in y around 0 73.7%

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

   if -1.75e-154 < x

   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* 84.8%

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

      2. associate-+l+ 84.8%

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

   3. Simplified 84.8%

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

   5. Taylor expanded in x around 0 63.7%

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

      1. associate-/r* 65.1%

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

      2. +-commutative 65.1%

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

   7. Simplified 65.1%

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

4. Final simplification 67.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.75 \cdot 10^{-154}:\\ \;\;\;\;\frac{y}{x} \cdot \frac{x}{\left(x + y\right) \cdot \left(y + \left(x + 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 87.2% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-1.6d-154)) then
        tmp = ((x / (x + y)) / (x + (y + 1.0d0))) * (y / x)
    else
        tmp = (x / y) / (y + 1.0d0)
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1.6e-154)
		tmp = ((x / (x + y)) / (x + (y + 1.0))) * (y / x);
	else
		tmp = (x / y) / (y + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[x, -1.6e-154], N[(N[(N[(x / N[(x + y), $MachinePrecision]), $MachinePrecision] / N[(x + N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(y / x), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.60000000000000002e-154

   1. Initial program 62.7%

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

      1. *-commutative 62.7%

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

      2. associate-*l* 62.7%

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

      3. times-frac 90.9%

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

      4. +-commutative 90.9%

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

      5. +-commutative 90.9%

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

      6. associate-+r+ 90.9%

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

      7. +-commutative 90.9%

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

      8. associate-+l+ 90.9%

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

   4. Applied egg-rr 90.9%

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

      1. frac-times 62.7%

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

      2. *-commutative 62.7%

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

      3. +-commutative 62.7%

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

      4. +-commutative 62.7%

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

      5. associate-+r+ 62.7%

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

      6. +-commutative 62.7%

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

   6. Applied egg-rr 62.7%

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

      1. *-commutative 62.7%

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

      2. times-frac 90.9%

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

      3. associate-/r* 99.7%

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

   8. Simplified 99.7%

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

   9. Taylor expanded in y around 0 75.5%

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

   if -1.60000000000000002e-154 < x

   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* 84.8%

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

      2. associate-+l+ 84.8%

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

   3. Simplified 84.8%

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

   5. Taylor expanded in x around 0 63.7%

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

      1. associate-/r* 65.1%

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

      2. +-commutative 65.1%

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

   7. Simplified 65.1%

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

4. Final simplification 68.5%

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

Alternative 7: 82.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -1.75 \cdot 10^{-154}:\\ \;\;\;\;\frac{y}{x + y} \cdot \frac{1}{x + \left(y + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + 1}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -1.75e-154)
   (* (/ y (+ x y)) (/ 1.0 (+ x (+ y 1.0))))
   (/ (/ x y) (+ y 1.0))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -1.75e-154) {
		tmp = (y / (x + y)) * (1.0 / (x + (y + 1.0)));
	} else {
		tmp = (x / y) / (y + 1.0);
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-1.75d-154)) then
        tmp = (y / (x + y)) * (1.0d0 / (x + (y + 1.0d0)))
    else
        tmp = (x / y) / (y + 1.0d0)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (x <= -1.75e-154) {
		tmp = (y / (x + y)) * (1.0 / (x + (y + 1.0)));
	} else {
		tmp = (x / y) / (y + 1.0);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -1.75e-154:
		tmp = (y / (x + y)) * (1.0 / (x + (y + 1.0)))
	else:
		tmp = (x / y) / (y + 1.0)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -1.75e-154)
		tmp = Float64(Float64(y / Float64(x + y)) * Float64(1.0 / Float64(x + Float64(y + 1.0))));
	else
		tmp = Float64(Float64(x / y) / Float64(y + 1.0));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1.75e-154)
		tmp = (y / (x + y)) * (1.0 / (x + (y + 1.0)));
	else
		tmp = (x / y) / (y + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[x, -1.75e-154], N[(N[(y / N[(x + y), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(x + N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.75e-154

   1. Initial program 62.7%

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

      1. *-commutative 62.7%

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

      2. associate-*l* 62.7%

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

      3. times-frac 90.9%

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

      4. +-commutative 90.9%

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

      5. +-commutative 90.9%

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

      6. associate-+r+ 90.9%

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

      7. +-commutative 90.9%

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

      8. associate-+l+ 90.9%

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

   4. Applied egg-rr 90.9%

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

      1. frac-times 62.7%

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

      2. *-commutative 62.7%

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

      3. +-commutative 62.7%

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

      4. +-commutative 62.7%

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

      5. associate-+r+ 62.7%

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

      6. +-commutative 62.7%

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

   6. Applied egg-rr 62.7%

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

      1. *-commutative 62.7%

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

      2. times-frac 90.9%

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

      3. associate-/r* 99.7%

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

   8. Simplified 99.7%

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

   9. Taylor expanded in x around inf 66.6%

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

   if -1.75e-154 < x

   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* 84.8%

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

      2. associate-+l+ 84.8%

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

   3. Simplified 84.8%

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

   5. Taylor expanded in x around 0 63.7%

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

      1. associate-/r* 65.1%

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

      2. +-commutative 65.1%

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

   7. Simplified 65.1%

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

4. Final simplification 65.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.75 \cdot 10^{-154}:\\ \;\;\;\;\frac{y}{x + y} \cdot \frac{1}{x + \left(y + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 68.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{y}{x} \cdot \frac{1}{x}\\ \mathbf{elif}\;x \leq -1.75 \cdot 10^{-154}:\\ \;\;\;\;y \cdot \frac{1}{x + y}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{y}{x}}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -1.0)
   (* (/ y x) (/ 1.0 x))
   (if (<= x -1.75e-154) (* y (/ 1.0 (+ x y))) (/ 1.0 (/ y x)))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -1.0) {
		tmp = (y / x) * (1.0 / x);
	} else if (x <= -1.75e-154) {
		tmp = y * (1.0 / (x + y));
	} 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 <= (-1.0d0)) then
        tmp = (y / x) * (1.0d0 / x)
    else if (x <= (-1.75d-154)) then
        tmp = y * (1.0d0 / (x + y))
    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 <= -1.0) {
		tmp = (y / x) * (1.0 / x);
	} else if (x <= -1.75e-154) {
		tmp = y * (1.0 / (x + y));
	} else {
		tmp = 1.0 / (y / x);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -1.0:
		tmp = (y / x) * (1.0 / x)
	elif x <= -1.75e-154:
		tmp = y * (1.0 / (x + y))
	else:
		tmp = 1.0 / (y / x)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -1.0)
		tmp = Float64(Float64(y / x) * Float64(1.0 / x));
	elseif (x <= -1.75e-154)
		tmp = Float64(y * Float64(1.0 / Float64(x + y)));
	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 <= -1.0)
		tmp = (y / x) * (1.0 / x);
	elseif (x <= -1.75e-154)
		tmp = y * (1.0 / (x + y));
	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, -1.0], N[(N[(y / x), $MachinePrecision] * N[(1.0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -1.75e-154], N[(y * N[(1.0 / N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(y / x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1

   1. Initial program 56.0%

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

      1. *-commutative 56.0%

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

      2. associate-*l* 56.0%

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

      3. times-frac 86.1%

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

      4. +-commutative 86.1%

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

      5. +-commutative 86.1%

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

      6. associate-+r+ 86.1%

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

      7. +-commutative 86.1%

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

      8. associate-+l+ 86.1%

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

   4. Applied egg-rr 86.1%

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

   5. Taylor expanded in x around inf 74.2%

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

   6. Taylor expanded in y around 0 73.8%

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

   if -1 < x < -1.75e-154

   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. *-un-lft-identity 75.1%

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

      2. associate-*l* 75.2%

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

      3. times-frac 75.5%

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

      4. +-commutative 75.5%

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

      5. *-commutative 75.5%

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

      6. +-commutative 75.5%

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

      7. associate-+r+ 75.5%

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

      8. +-commutative 75.5%

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

      9. associate-+l+ 75.5%

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

   4. Applied egg-rr 75.5%

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

   5. Taylor expanded in y around 0 25.8%

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

      1. +-commutative 25.8%

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

   7. Simplified 25.8%

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

   8. Taylor expanded in x around 0 46.2%

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

   if -1.75e-154 < x

   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* 84.8%

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

      2. associate-+l+ 84.8%

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

   3. Simplified 84.8%

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

   5. Taylor expanded in x around 0 63.6%

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

      1. associate-/r* 64.1%

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

      2. +-commutative 64.1%

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

   7. Simplified 64.1%

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

   8. Taylor expanded in y around 0 39.9%

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

      1. un-div-inv 39.9%

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

      2. clear-num 40.5%

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

   10. Applied egg-rr 40.5%

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

4. Final simplification 48.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{y}{x} \cdot \frac{1}{x}\\ \mathbf{elif}\;x \leq -1.75 \cdot 10^{-154}:\\ \;\;\;\;y \cdot \frac{1}{x + y}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{y}{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 80.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{y}{x} \cdot \frac{1}{x}\\ \mathbf{elif}\;x \leq -1.75 \cdot 10^{-154}:\\ \;\;\;\;y \cdot \frac{1}{x + y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -1.0)
   (* (/ y x) (/ 1.0 x))
   (if (<= x -1.75e-154) (* y (/ 1.0 (+ x y))) (/ x (* y (+ y 1.0))))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -1.0) {
		tmp = (y / x) * (1.0 / x);
	} else if (x <= -1.75e-154) {
		tmp = y * (1.0 / (x + y));
	} else {
		tmp = x / (y * (y + 1.0));
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-1.0d0)) then
        tmp = (y / x) * (1.0d0 / x)
    else if (x <= (-1.75d-154)) then
        tmp = y * (1.0d0 / (x + y))
    else
        tmp = x / (y * (y + 1.0d0))
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (x <= -1.0) {
		tmp = (y / x) * (1.0 / x);
	} else if (x <= -1.75e-154) {
		tmp = y * (1.0 / (x + y));
	} else {
		tmp = x / (y * (y + 1.0));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -1.0:
		tmp = (y / x) * (1.0 / x)
	elif x <= -1.75e-154:
		tmp = y * (1.0 / (x + y))
	else:
		tmp = x / (y * (y + 1.0))
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -1.0)
		tmp = Float64(Float64(y / x) * Float64(1.0 / x));
	elseif (x <= -1.75e-154)
		tmp = Float64(y * Float64(1.0 / Float64(x + y)));
	else
		tmp = Float64(x / Float64(y * Float64(y + 1.0)));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1.0)
		tmp = (y / x) * (1.0 / x);
	elseif (x <= -1.75e-154)
		tmp = y * (1.0 / (x + y));
	else
		tmp = x / (y * (y + 1.0));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[x, -1.0], N[(N[(y / x), $MachinePrecision] * N[(1.0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -1.75e-154], N[(y * N[(1.0 / N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x / N[(y * N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1

   1. Initial program 56.0%

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

      1. *-commutative 56.0%

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

      2. associate-*l* 56.0%

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

      3. times-frac 86.1%

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

      4. +-commutative 86.1%

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

      5. +-commutative 86.1%

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

      6. associate-+r+ 86.1%

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

      7. +-commutative 86.1%

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

      8. associate-+l+ 86.1%

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

   4. Applied egg-rr 86.1%

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

   5. Taylor expanded in x around inf 74.2%

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

   6. Taylor expanded in y around 0 73.8%

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

   if -1 < x < -1.75e-154

   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. *-un-lft-identity 75.1%

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

      2. associate-*l* 75.2%

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

      3. times-frac 75.5%

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

      4. +-commutative 75.5%

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

      5. *-commutative 75.5%

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

      6. +-commutative 75.5%

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

      7. associate-+r+ 75.5%

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

      8. +-commutative 75.5%

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

      9. associate-+l+ 75.5%

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

   4. Applied egg-rr 75.5%

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

   5. Taylor expanded in y around 0 25.8%

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

      1. +-commutative 25.8%

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

   7. Simplified 25.8%

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

   8. Taylor expanded in x around 0 46.2%

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

   if -1.75e-154 < x

   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* 84.8%

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

      2. associate-+l+ 84.8%

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

   3. Simplified 84.8%

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

   5. Taylor expanded in x around 0 63.7%

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

4. Final simplification 63.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{y}{x} \cdot \frac{1}{x}\\ \mathbf{elif}\;x \leq -1.75 \cdot 10^{-154}:\\ \;\;\;\;y \cdot \frac{1}{x + y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 44.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 3.3 \cdot 10^{-190}:\\ \;\;\;\;y \cdot \frac{1}{x + y}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{y}{x}}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 3.3e-190) (* y (/ 1.0 (+ x y))) (/ 1.0 (/ y x))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 3.3e-190) {
		tmp = y * (1.0 / (x + y));
	} 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 (y <= 3.3d-190) then
        tmp = y * (1.0d0 / (x + y))
    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 (y <= 3.3e-190) {
		tmp = y * (1.0 / (x + y));
	} else {
		tmp = 1.0 / (y / x);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if y <= 3.3e-190:
		tmp = y * (1.0 / (x + y))
	else:
		tmp = 1.0 / (y / x)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= 3.3e-190)
		tmp = Float64(y * Float64(1.0 / Float64(x + y)));
	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 (y <= 3.3e-190)
		tmp = y * (1.0 / (x + y));
	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[y, 3.3e-190], N[(y * N[(1.0 / N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(y / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 3.30000000000000019e-190

   1. Initial program 64.8%

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

      1. *-un-lft-identity 64.8%

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

      2. associate-*l* 64.8%

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

      3. times-frac 68.4%

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

      4. +-commutative 68.4%

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

      5. *-commutative 68.4%

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

      6. +-commutative 68.4%

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

      7. associate-+r+ 68.4%

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

      8. +-commutative 68.4%

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

      9. associate-+l+ 68.4%

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

   4. Applied egg-rr 68.4%

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

   5. Taylor expanded in y around 0 37.9%

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

      1. +-commutative 37.9%

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

   7. Simplified 37.9%

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

   8. Taylor expanded in x around 0 33.3%

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

   if 3.30000000000000019e-190 < y

   1. Initial program 69.9%

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

      1. associate-/l* 84.8%

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

      2. associate-+l+ 84.8%

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

   3. Simplified 84.8%

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

   5. Taylor expanded in x around 0 62.2%

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

      1. associate-/r* 63.5%

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

      2. +-commutative 63.5%

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

   7. Simplified 63.5%

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

   8. Taylor expanded in y around 0 34.5%

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

      1. un-div-inv 34.5%

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

      2. clear-num 35.5%

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

   10. Applied egg-rr 35.5%

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

4. Final simplification 34.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 3.3 \cdot 10^{-190}:\\ \;\;\;\;y \cdot \frac{1}{x + y}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{y}{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 79.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -1.75 \cdot 10^{-154}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -1.75e-154) (/ y (* x (+ x 1.0))) (/ x (* y (+ y 1.0)))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -1.75e-154) {
		tmp = y / (x * (x + 1.0));
	} else {
		tmp = x / (y * (y + 1.0));
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-1.75d-154)) then
        tmp = y / (x * (x + 1.0d0))
    else
        tmp = x / (y * (y + 1.0d0))
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (x <= -1.75e-154) {
		tmp = y / (x * (x + 1.0));
	} else {
		tmp = x / (y * (y + 1.0));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -1.75e-154:
		tmp = y / (x * (x + 1.0))
	else:
		tmp = x / (y * (y + 1.0))
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -1.75e-154)
		tmp = Float64(y / Float64(x * Float64(x + 1.0)));
	else
		tmp = Float64(x / Float64(y * Float64(y + 1.0)));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1.75e-154)
		tmp = y / (x * (x + 1.0));
	else
		tmp = x / (y * (y + 1.0));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[x, -1.75e-154], N[(y / N[(x * N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x / N[(y * N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.75e-154

   1. Initial program 62.7%

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

      1. associate-/l* 82.8%

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

      2. associate-+l+ 82.8%

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

   3. Simplified 82.8%

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

   5. Taylor expanded in y around 0 67.5%

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

      1. +-commutative 67.5%

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

   7. Simplified 67.5%

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

   if -1.75e-154 < x

   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* 84.8%

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

      2. associate-+l+ 84.8%

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

   3. Simplified 84.8%

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

   5. Taylor expanded in x around 0 63.7%

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

4. Final simplification 64.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.75 \cdot 10^{-154}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 80.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -1.75 \cdot 10^{-154}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + 1}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -1.75e-154) (/ y (* x (+ x 1.0))) (/ (/ x y) (+ y 1.0))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -1.75e-154) {
		tmp = y / (x * (x + 1.0));
	} else {
		tmp = (x / y) / (y + 1.0);
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-1.75d-154)) then
        tmp = y / (x * (x + 1.0d0))
    else
        tmp = (x / y) / (y + 1.0d0)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (x <= -1.75e-154) {
		tmp = y / (x * (x + 1.0));
	} else {
		tmp = (x / y) / (y + 1.0);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -1.75e-154:
		tmp = y / (x * (x + 1.0))
	else:
		tmp = (x / y) / (y + 1.0)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -1.75e-154)
		tmp = Float64(y / Float64(x * Float64(x + 1.0)));
	else
		tmp = Float64(Float64(x / y) / Float64(y + 1.0));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1.75e-154)
		tmp = y / (x * (x + 1.0));
	else
		tmp = (x / y) / (y + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[x, -1.75e-154], N[(y / N[(x * N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.75e-154

   1. Initial program 62.7%

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

      1. associate-/l* 82.8%

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

      2. associate-+l+ 82.8%

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

   3. Simplified 82.8%

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

   5. Taylor expanded in y around 0 67.5%

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

      1. +-commutative 67.5%

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

   7. Simplified 67.5%

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

   if -1.75e-154 < x

   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* 84.8%

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

      2. associate-+l+ 84.8%

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

   3. Simplified 84.8%

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

   5. Taylor expanded in x around 0 63.7%

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

      1. associate-/r* 65.1%

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

      2. +-commutative 65.1%

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

   7. Simplified 65.1%

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

4. Final simplification 65.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.75 \cdot 10^{-154}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 82.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -1.75 \cdot 10^{-154}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + 1}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -1.75e-154) (/ (/ y x) (+ x 1.0)) (/ (/ x y) (+ y 1.0))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -1.75e-154) {
		tmp = (y / x) / (x + 1.0);
	} else {
		tmp = (x / y) / (y + 1.0);
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-1.75d-154)) then
        tmp = (y / x) / (x + 1.0d0)
    else
        tmp = (x / y) / (y + 1.0d0)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (x <= -1.75e-154) {
		tmp = (y / x) / (x + 1.0);
	} else {
		tmp = (x / y) / (y + 1.0);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -1.75e-154:
		tmp = (y / x) / (x + 1.0)
	else:
		tmp = (x / y) / (y + 1.0)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -1.75e-154)
		tmp = Float64(Float64(y / x) / Float64(x + 1.0));
	else
		tmp = Float64(Float64(x / y) / Float64(y + 1.0));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1.75e-154)
		tmp = (y / x) / (x + 1.0);
	else
		tmp = (x / y) / (y + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[x, -1.75e-154], N[(N[(y / x), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.75e-154

   1. Initial program 62.7%

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

      1. *-commutative 62.7%

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

      2. associate-*l* 62.7%

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

      3. times-frac 90.9%

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

      4. +-commutative 90.9%

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

      5. +-commutative 90.9%

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

      6. associate-+r+ 90.9%

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

      7. +-commutative 90.9%

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

      8. associate-+l+ 90.9%

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

   4. Applied egg-rr 90.9%

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

      1. frac-times 62.7%

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

      2. *-commutative 62.7%

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

      3. +-commutative 62.7%

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

      4. +-commutative 62.7%

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

      5. associate-+r+ 62.7%

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

      6. +-commutative 62.7%

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

   6. Applied egg-rr 62.7%

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

      1. *-commutative 62.7%

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

      2. times-frac 90.9%

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

      3. associate-/r* 99.7%

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

   8. Simplified 99.7%

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

      1. log1p-expm1-u 99.7%

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

   10. Applied egg-rr 99.7%

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

   11. Taylor expanded in y around 0 67.5%

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

       1. associate-/r* 65.9%

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

       2. +-commutative 65.9%

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

   13. Simplified 65.9%

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

   if -1.75e-154 < x

   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* 84.8%

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

      2. associate-+l+ 84.8%

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

   3. Simplified 84.8%

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

   5. Taylor expanded in x around 0 63.7%

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

      1. associate-/r* 65.1%

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

      2. +-commutative 65.1%

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

   7. Simplified 65.1%

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

4. Final simplification 65.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.75 \cdot 10^{-154}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 26.7% accurate, 3.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 67.0%

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

   1. associate-/l* 84.1%

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

   2. associate-+l+ 84.1%

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

3. Simplified 84.1%

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

5. Taylor expanded in x around 0 53.9%

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

   1. associate-/r* 54.6%

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

   2. +-commutative 54.6%

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

7. Simplified 54.6%

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

8. Taylor expanded in y around 0 29.2%

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

   1. un-div-inv 29.2%

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

   2. clear-num 29.6%

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

10. Applied egg-rr 29.6%

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

11. Final simplification 29.6%

    \[\leadsto \frac{1}{\frac{y}{x}} \]
12. Add Preprocessing

Alternative 15: 4.0% 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 67.0%

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

   1. *-commutative 67.0%

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

   2. associate-*l* 67.0%

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

   3. times-frac 94.3%

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

   4. +-commutative 94.3%

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

   5. +-commutative 94.3%

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

   6. associate-+r+ 94.3%

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

   7. +-commutative 94.3%

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

   8. associate-+l+ 94.3%

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

4. Applied egg-rr 94.3%

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

   1. frac-times 67.0%

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

   2. *-commutative 67.0%

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

   3. +-commutative 67.0%

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

   4. +-commutative 67.0%

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

   5. associate-+r+ 67.0%

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

   6. +-commutative 67.0%

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

6. Applied egg-rr 67.0%

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

   1. *-commutative 67.0%

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

   2. times-frac 94.3%

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

   3. associate-/r* 99.8%

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

8. Simplified 99.8%

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

9. Taylor expanded in y around 0 55.0%

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

10. Taylor expanded in y around inf 4.1%

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

11. Final simplification 4.1%

    \[\leadsto \frac{1}{y} \]
12. Add Preprocessing

Alternative 16: 26.2% accurate, 5.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 67.0%

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

   1. associate-/l* 84.1%

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

   2. associate-+l+ 84.1%

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

3. Simplified 84.1%

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

5. Taylor expanded in x around 0 54.0%

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

6. Taylor expanded in y around 0 29.2%

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

7. Final simplification 29.2%

   \[\leadsto \frac{x}{y} \]
8. Add Preprocessing

Developer target: 99.8% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024067 
(FPCore (x y)
  :name "Numeric.SpecFunctions:incompleteBetaApprox from math-functions-0.1.5.2, A"
  :precision binary64

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

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