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

Percentage Accurate: 68.6% → 99.4%

Time: 13.3s

Alternatives: 17

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: 68.6% 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: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 70.6%

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

   1. *-commutative 70.6%

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

   2. associate-*l* 70.7%

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

   3. times-frac 93.3%

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

   4. +-commutative 93.3%

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

   5. +-commutative 93.3%

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

   6. associate-+r+ 93.3%

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

   7. +-commutative 93.3%

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

   8. associate-+l+ 93.3%

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

4. Applied egg-rr 93.3%

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

   1. clear-num 93.3%

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

   2. associate-/r* 99.8%

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

   3. +-commutative 99.8%

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

   4. frac-times 99.3%

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

   5. *-un-lft-identity 99.3%

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

   6. +-commutative 99.3%

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

   7. +-commutative 99.3%

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

6. Applied egg-rr 99.3%

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

Alternative 2: 97.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_0 := y + \left(x + 1\right)\\ t_1 := \frac{x + y}{y} \cdot t\_0\\ \mathbf{if}\;y \leq -3.3 \cdot 10^{-92}:\\ \;\;\;\;\frac{1}{t\_1}\\ \mathbf{elif}\;y \leq 1.5 \cdot 10^{+147}:\\ \;\;\;\;\frac{y}{x + y} \cdot \frac{x}{\left(x + y\right) \cdot t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{t\_1}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ y (+ x 1.0))) (t_1 (* (/ (+ x y) y) t_0)))
   (if (<= y -3.3e-92)
     (/ 1.0 t_1)
     (if (<= y 1.5e+147)
       (* (/ y (+ x y)) (/ x (* (+ x y) t_0)))
       (/ (/ x y) t_1)))))

C

assert(x < y);
double code(double x, double y) {
	double t_0 = y + (x + 1.0);
	double t_1 = ((x + y) / y) * t_0;
	double tmp;
	if (y <= -3.3e-92) {
		tmp = 1.0 / t_1;
	} else if (y <= 1.5e+147) {
		tmp = (y / (x + y)) * (x / ((x + y) * t_0));
	} else {
		tmp = (x / y) / t_1;
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = y + (x + 1.0d0)
    t_1 = ((x + y) / y) * t_0
    if (y <= (-3.3d-92)) then
        tmp = 1.0d0 / t_1
    else if (y <= 1.5d+147) then
        tmp = (y / (x + y)) * (x / ((x + y) * t_0))
    else
        tmp = (x / y) / t_1
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double t_0 = y + (x + 1.0);
	double t_1 = ((x + y) / y) * t_0;
	double tmp;
	if (y <= -3.3e-92) {
		tmp = 1.0 / t_1;
	} else if (y <= 1.5e+147) {
		tmp = (y / (x + y)) * (x / ((x + y) * t_0));
	} else {
		tmp = (x / y) / t_1;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	t_0 = y + (x + 1.0)
	t_1 = ((x + y) / y) * t_0
	tmp = 0
	if y <= -3.3e-92:
		tmp = 1.0 / t_1
	elif y <= 1.5e+147:
		tmp = (y / (x + y)) * (x / ((x + y) * t_0))
	else:
		tmp = (x / y) / t_1
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	t_0 = Float64(y + Float64(x + 1.0))
	t_1 = Float64(Float64(Float64(x + y) / y) * t_0)
	tmp = 0.0
	if (y <= -3.3e-92)
		tmp = Float64(1.0 / t_1);
	elseif (y <= 1.5e+147)
		tmp = Float64(Float64(y / Float64(x + y)) * Float64(x / Float64(Float64(x + y) * t_0)));
	else
		tmp = Float64(Float64(x / y) / t_1);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	t_0 = y + (x + 1.0);
	t_1 = ((x + y) / y) * t_0;
	tmp = 0.0;
	if (y <= -3.3e-92)
		tmp = 1.0 / t_1;
	elseif (y <= 1.5e+147)
		tmp = (y / (x + y)) * (x / ((x + y) * t_0));
	else
		tmp = (x / y) / t_1;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := Block[{t$95$0 = N[(y + N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(x + y), $MachinePrecision] / y), $MachinePrecision] * t$95$0), $MachinePrecision]}, If[LessEqual[y, -3.3e-92], N[(1.0 / t$95$1), $MachinePrecision], If[LessEqual[y, 1.5e+147], N[(N[(y / N[(x + y), $MachinePrecision]), $MachinePrecision] * N[(x / N[(N[(x + y), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / t$95$1), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -3.29999999999999998e-92

   1. Initial program 59.8%

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

      1. *-commutative 59.8%

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

      2. associate-*l* 59.8%

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

      3. times-frac 86.3%

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

      4. +-commutative 86.3%

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

      5. +-commutative 86.3%

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

      6. associate-+r+ 86.3%

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

      7. +-commutative 86.3%

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

      8. associate-+l+ 86.3%

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

   4. Applied egg-rr 86.3%

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

      1. clear-num 86.3%

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

      2. associate-/r* 99.7%

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

      3. +-commutative 99.7%

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

      4. frac-times 98.3%

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

      5. *-un-lft-identity 98.3%

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

      6. +-commutative 98.3%

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

      7. +-commutative 98.3%

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

   6. Applied egg-rr 98.3%

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

   7. Taylor expanded in x around inf 36.9%

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

   if -3.29999999999999998e-92 < y < 1.49999999999999997e147

   1. Initial program 78.4%

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

      1. *-commutative 78.4%

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

      2. associate-*l* 78.4%

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

      3. times-frac 99.2%

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

      4. +-commutative 99.2%

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

      5. +-commutative 99.2%

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

      6. associate-+r+ 99.2%

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

      7. +-commutative 99.2%

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

      8. associate-+l+ 99.2%

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

   4. Applied egg-rr 99.2%

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

   if 1.49999999999999997e147 < y

   1. Initial program 63.5%

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

      1. *-commutative 63.5%

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

      2. associate-*l* 63.5%

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

      3. times-frac 83.8%

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

      4. +-commutative 83.8%

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

      5. +-commutative 83.8%

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

      6. associate-+r+ 83.8%

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

      7. +-commutative 83.8%

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

      8. associate-+l+ 83.8%

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

   4. Applied egg-rr 83.8%

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

      1. clear-num 83.8%

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

      2. associate-/r* 100.0%

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

      3. +-commutative 100.0%

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

      4. frac-times 99.5%

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

      5. *-un-lft-identity 99.5%

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

      6. +-commutative 99.5%

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

      7. +-commutative 99.5%

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

   6. Applied egg-rr 99.5%

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

   7. Taylor expanded in x around 0 92.8%

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

4. Final simplification 77.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.3 \cdot 10^{-92}:\\ \;\;\;\;\frac{1}{\frac{x + y}{y} \cdot \left(y + \left(x + 1\right)\right)}\\ \mathbf{elif}\;y \leq 1.5 \cdot 10^{+147}:\\ \;\;\;\;\frac{y}{x + y} \cdot \frac{x}{\left(x + y\right) \cdot \left(y + \left(x + 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{\frac{x + y}{y} \cdot \left(y + \left(x + 1\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 92.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_0 := y + \left(x + 1\right)\\ \mathbf{if}\;x \leq -1.2 \cdot 10^{+32}:\\ \;\;\;\;\frac{\frac{\frac{x}{t\_0}}{x + y}}{\frac{x}{y}}\\ \mathbf{elif}\;x \leq -5 \cdot 10^{-165}:\\ \;\;\;\;x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(x + \left(y + 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{\frac{x + y}{y} \cdot t\_0}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ y (+ x 1.0))))
   (if (<= x -1.2e+32)
     (/ (/ (/ x t_0) (+ x y)) (/ x y))
     (if (<= x -5e-165)
       (* x (/ y (* (* (+ x y) (+ x y)) (+ x (+ y 1.0)))))
       (/ (/ x y) (* (/ (+ x y) y) t_0))))))

C

assert(x < y);
double code(double x, double y) {
	double t_0 = y + (x + 1.0);
	double tmp;
	if (x <= -1.2e+32) {
		tmp = ((x / t_0) / (x + y)) / (x / y);
	} else if (x <= -5e-165) {
		tmp = x * (y / (((x + y) * (x + y)) * (x + (y + 1.0))));
	} else {
		tmp = (x / y) / (((x + y) / y) * t_0);
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = y + (x + 1.0d0)
    if (x <= (-1.2d+32)) then
        tmp = ((x / t_0) / (x + y)) / (x / y)
    else if (x <= (-5d-165)) then
        tmp = x * (y / (((x + y) * (x + y)) * (x + (y + 1.0d0))))
    else
        tmp = (x / y) / (((x + y) / y) * t_0)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double t_0 = y + (x + 1.0);
	double tmp;
	if (x <= -1.2e+32) {
		tmp = ((x / t_0) / (x + y)) / (x / y);
	} else if (x <= -5e-165) {
		tmp = x * (y / (((x + y) * (x + y)) * (x + (y + 1.0))));
	} else {
		tmp = (x / y) / (((x + y) / y) * t_0);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	t_0 = y + (x + 1.0)
	tmp = 0
	if x <= -1.2e+32:
		tmp = ((x / t_0) / (x + y)) / (x / y)
	elif x <= -5e-165:
		tmp = x * (y / (((x + y) * (x + y)) * (x + (y + 1.0))))
	else:
		tmp = (x / y) / (((x + y) / y) * t_0)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	t_0 = Float64(y + Float64(x + 1.0))
	tmp = 0.0
	if (x <= -1.2e+32)
		tmp = Float64(Float64(Float64(x / t_0) / Float64(x + y)) / Float64(x / y));
	elseif (x <= -5e-165)
		tmp = Float64(x * Float64(y / Float64(Float64(Float64(x + y) * Float64(x + y)) * Float64(x + Float64(y + 1.0)))));
	else
		tmp = Float64(Float64(x / y) / Float64(Float64(Float64(x + y) / y) * t_0));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	t_0 = y + (x + 1.0);
	tmp = 0.0;
	if (x <= -1.2e+32)
		tmp = ((x / t_0) / (x + y)) / (x / y);
	elseif (x <= -5e-165)
		tmp = x * (y / (((x + y) * (x + y)) * (x + (y + 1.0))));
	else
		tmp = (x / y) / (((x + y) / y) * t_0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := Block[{t$95$0 = N[(y + N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.2e+32], N[(N[(N[(x / t$95$0), $MachinePrecision] / N[(x + y), $MachinePrecision]), $MachinePrecision] / N[(x / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -5e-165], N[(x * N[(y / N[(N[(N[(x + y), $MachinePrecision] * N[(x + y), $MachinePrecision]), $MachinePrecision] * N[(x + N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / N[(N[(N[(x + y), $MachinePrecision] / y), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.19999999999999996e32

   1. Initial program 65.0%

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

      1. *-commutative 65.0%

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

      2. associate-*l* 65.0%

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

      3. times-frac 88.1%

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

      4. +-commutative 88.1%

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

      5. +-commutative 88.1%

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

      6. associate-+r+ 88.1%

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

      7. +-commutative 88.1%

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

      8. associate-+l+ 88.1%

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

   4. Applied egg-rr 88.1%

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

   5. Taylor expanded in y around 0 87.7%

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

      1. clear-num 87.7%

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

      2. frac-times 82.9%

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

      3. *-un-lft-identity 82.9%

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

      4. *-commutative 82.9%

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

      5. +-commutative 82.9%

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

      6. +-commutative 82.9%

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

   7. Applied egg-rr 82.9%

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

      1. associate-/l/ 87.7%

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

      2. associate-/r* 88.7%

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

   9. Simplified 88.7%

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

   if -1.19999999999999996e32 < x < -4.99999999999999981e-165

   1. Initial program 77.7%

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

      1. associate-/l* 93.3%

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

      2. associate-+l+ 93.3%

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

   3. Simplified 93.3%

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

   if -4.99999999999999981e-165 < x

   1. Initial program 71.0%

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

      1. *-commutative 71.0%

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

      2. associate-*l* 71.0%

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

      3. times-frac 93.7%

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

      4. +-commutative 93.7%

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

      5. +-commutative 93.7%

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

      6. associate-+r+ 93.7%

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

      7. +-commutative 93.7%

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

      8. associate-+l+ 93.7%

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

   4. Applied egg-rr 93.7%

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

      1. clear-num 93.6%

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

      2. associate-/r* 99.8%

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

      3. +-commutative 99.8%

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

      4. frac-times 99.0%

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

      5. *-un-lft-identity 99.0%

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

      6. +-commutative 99.0%

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

      7. +-commutative 99.0%

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

   6. Applied egg-rr 99.0%

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

   7. Taylor expanded in x around 0 56.4%

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

Alternative 4: 73.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_0 := \frac{\frac{x}{y}}{y}\\ \mathbf{if}\;x \leq -7.2 \cdot 10^{-6}:\\ \;\;\;\;\frac{y}{x} \cdot \frac{1}{x}\\ \mathbf{elif}\;x \leq -9.2 \cdot 10^{-44}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq -2.5 \cdot 10^{-91}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{elif}\;x \leq 8.5 \cdot 10^{-129}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (/ x y) y)))
   (if (<= x -7.2e-6)
     (* (/ y x) (/ 1.0 x))
     (if (<= x -9.2e-44)
       t_0
       (if (<= x -2.5e-91) (/ y x) (if (<= x 8.5e-129) (/ x y) t_0))))))

C

assert(x < y);
double code(double x, double y) {
	double t_0 = (x / y) / y;
	double tmp;
	if (x <= -7.2e-6) {
		tmp = (y / x) * (1.0 / x);
	} else if (x <= -9.2e-44) {
		tmp = t_0;
	} else if (x <= -2.5e-91) {
		tmp = y / x;
	} else if (x <= 8.5e-129) {
		tmp = x / y;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (x / y) / y
    if (x <= (-7.2d-6)) then
        tmp = (y / x) * (1.0d0 / x)
    else if (x <= (-9.2d-44)) then
        tmp = t_0
    else if (x <= (-2.5d-91)) then
        tmp = y / x
    else if (x <= 8.5d-129) then
        tmp = x / y
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double t_0 = (x / y) / y;
	double tmp;
	if (x <= -7.2e-6) {
		tmp = (y / x) * (1.0 / x);
	} else if (x <= -9.2e-44) {
		tmp = t_0;
	} else if (x <= -2.5e-91) {
		tmp = y / x;
	} else if (x <= 8.5e-129) {
		tmp = x / y;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	t_0 = (x / y) / y
	tmp = 0
	if x <= -7.2e-6:
		tmp = (y / x) * (1.0 / x)
	elif x <= -9.2e-44:
		tmp = t_0
	elif x <= -2.5e-91:
		tmp = y / x
	elif x <= 8.5e-129:
		tmp = x / y
	else:
		tmp = t_0
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	t_0 = Float64(Float64(x / y) / y)
	tmp = 0.0
	if (x <= -7.2e-6)
		tmp = Float64(Float64(y / x) * Float64(1.0 / x));
	elseif (x <= -9.2e-44)
		tmp = t_0;
	elseif (x <= -2.5e-91)
		tmp = Float64(y / x);
	elseif (x <= 8.5e-129)
		tmp = Float64(x / y);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	t_0 = (x / y) / y;
	tmp = 0.0;
	if (x <= -7.2e-6)
		tmp = (y / x) * (1.0 / x);
	elseif (x <= -9.2e-44)
		tmp = t_0;
	elseif (x <= -2.5e-91)
		tmp = y / x;
	elseif (x <= 8.5e-129)
		tmp = x / y;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := Block[{t$95$0 = N[(N[(x / y), $MachinePrecision] / y), $MachinePrecision]}, If[LessEqual[x, -7.2e-6], N[(N[(y / x), $MachinePrecision] * N[(1.0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -9.2e-44], t$95$0, If[LessEqual[x, -2.5e-91], N[(y / x), $MachinePrecision], If[LessEqual[x, 8.5e-129], N[(x / y), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -7.19999999999999967e-6

   1. Initial program 65.6%

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

      1. *-commutative 65.6%

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

      2. associate-*l* 65.7%

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

      3. times-frac 89.3%

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

      4. +-commutative 89.3%

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

      5. +-commutative 89.3%

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

      6. associate-+r+ 89.3%

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

      7. +-commutative 89.3%

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

      8. associate-+l+ 89.3%

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

   4. Applied egg-rr 89.3%

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

   5. Taylor expanded in y around 0 87.6%

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

   6. Taylor expanded in x around inf 77.0%

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

   if -7.19999999999999967e-6 < x < -9.19999999999999992e-44 or 8.49999999999999937e-129 < x

   1. Initial program 76.1%

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

      1. *-commutative 76.1%

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

      2. associate-*l* 76.1%

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

      3. times-frac 91.6%

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

      4. +-commutative 91.6%

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

      5. +-commutative 91.6%

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

      6. associate-+r+ 91.6%

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

      7. +-commutative 91.6%

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

      8. associate-+l+ 91.6%

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

   4. Applied egg-rr 91.6%

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

      1. clear-num 91.5%

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

      2. associate-/r* 99.7%

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

      3. +-commutative 99.7%

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

      4. frac-times 98.6%

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

      5. *-un-lft-identity 98.6%

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

      6. +-commutative 98.6%

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

      7. +-commutative 98.6%

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

   6. Applied egg-rr 98.6%

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

   7. Taylor expanded in x around 0 49.1%

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

   8. Taylor expanded in y around inf 33.7%

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

   if -9.19999999999999992e-44 < x < -2.49999999999999999e-91

   1. Initial program 72.8%

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

      1. associate-/l* 92.7%

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

      2. associate-+l+ 92.7%

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

   3. Simplified 92.7%

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

   5. Taylor expanded in y around 0 65.0%

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

   6. Taylor expanded in x around 0 65.0%

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

   if -2.49999999999999999e-91 < x < 8.49999999999999937e-129

   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. Step-by-step derivation

      1. associate-/l* 72.0%

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

      2. associate-+l+ 72.0%

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

   3. Simplified 72.0%

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

   5. Taylor expanded in x around 0 78.1%

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

      1. +-commutative 78.1%

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

   7. Simplified 78.1%

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

   8. Taylor expanded in y around 0 61.3%

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

Alternative 5: 85.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_0 := y + \left(x + 1\right)\\ \mathbf{if}\;x \leq -8 \cdot 10^{+192}:\\ \;\;\;\;\frac{1}{\frac{x + y}{y} \cdot t\_0}\\ \mathbf{elif}\;x \leq -1.52 \cdot 10^{-90}:\\ \;\;\;\;\frac{x}{\left(x + y\right) \cdot t\_0} \cdot \frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + 1}}{x + y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ y (+ x 1.0))))
   (if (<= x -8e+192)
     (/ 1.0 (* (/ (+ x y) y) t_0))
     (if (<= x -1.52e-90)
       (* (/ x (* (+ x y) t_0)) (/ y x))
       (/ (/ x (+ y 1.0)) (+ x y))))))

C

assert(x < y);
double code(double x, double y) {
	double t_0 = y + (x + 1.0);
	double tmp;
	if (x <= -8e+192) {
		tmp = 1.0 / (((x + y) / y) * t_0);
	} else if (x <= -1.52e-90) {
		tmp = (x / ((x + y) * t_0)) * (y / x);
	} else {
		tmp = (x / (y + 1.0)) / (x + y);
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = y + (x + 1.0d0)
    if (x <= (-8d+192)) then
        tmp = 1.0d0 / (((x + y) / y) * t_0)
    else if (x <= (-1.52d-90)) then
        tmp = (x / ((x + y) * t_0)) * (y / x)
    else
        tmp = (x / (y + 1.0d0)) / (x + y)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double t_0 = y + (x + 1.0);
	double tmp;
	if (x <= -8e+192) {
		tmp = 1.0 / (((x + y) / y) * t_0);
	} else if (x <= -1.52e-90) {
		tmp = (x / ((x + y) * t_0)) * (y / x);
	} else {
		tmp = (x / (y + 1.0)) / (x + y);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	t_0 = y + (x + 1.0)
	tmp = 0
	if x <= -8e+192:
		tmp = 1.0 / (((x + y) / y) * t_0)
	elif x <= -1.52e-90:
		tmp = (x / ((x + y) * t_0)) * (y / x)
	else:
		tmp = (x / (y + 1.0)) / (x + y)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	t_0 = Float64(y + Float64(x + 1.0))
	tmp = 0.0
	if (x <= -8e+192)
		tmp = Float64(1.0 / Float64(Float64(Float64(x + y) / y) * t_0));
	elseif (x <= -1.52e-90)
		tmp = Float64(Float64(x / Float64(Float64(x + y) * t_0)) * Float64(y / x));
	else
		tmp = Float64(Float64(x / Float64(y + 1.0)) / Float64(x + y));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	t_0 = y + (x + 1.0);
	tmp = 0.0;
	if (x <= -8e+192)
		tmp = 1.0 / (((x + y) / y) * t_0);
	elseif (x <= -1.52e-90)
		tmp = (x / ((x + y) * t_0)) * (y / x);
	else
		tmp = (x / (y + 1.0)) / (x + y);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := Block[{t$95$0 = N[(y + N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -8e+192], N[(1.0 / N[(N[(N[(x + y), $MachinePrecision] / y), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -1.52e-90], N[(N[(x / N[(N[(x + y), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] * N[(y / x), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(y + 1.0), $MachinePrecision]), $MachinePrecision] / N[(x + y), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -8.00000000000000033e192

   1. Initial program 61.7%

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

      1. *-commutative 61.7%

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

      2. associate-*l* 61.7%

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

      3. times-frac 86.4%

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

      4. +-commutative 86.4%

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

      5. +-commutative 86.4%

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

      6. associate-+r+ 86.4%

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

      7. +-commutative 86.4%

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

      8. associate-+l+ 86.4%

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

   4. Applied egg-rr 86.4%

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

      1. clear-num 86.4%

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

      2. associate-/r* 99.9%

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

      3. +-commutative 99.9%

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

      4. frac-times 99.9%

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

      5. *-un-lft-identity 99.9%

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

      6. +-commutative 99.9%

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

      7. +-commutative 99.9%

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

   6. Applied egg-rr 99.9%

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

   7. Taylor expanded in x around inf 85.5%

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

   if -8.00000000000000033e192 < x < -1.51999999999999991e-90

   1. Initial program 73.6%

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

      1. *-commutative 73.6%

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

      2. associate-*l* 73.6%

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

      3. times-frac 95.2%

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

      4. +-commutative 95.2%

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

      5. +-commutative 95.2%

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

      6. associate-+r+ 95.2%

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

      7. +-commutative 95.2%

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

      8. associate-+l+ 95.2%

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

   4. Applied egg-rr 95.2%

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

   5. Taylor expanded in y around 0 81.1%

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

   if -1.51999999999999991e-90 < x

   1. Initial program 71.3%

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

      1. associate-/l* 79.9%

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

      2. associate-+l+ 79.9%

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

   3. Simplified 79.9%

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

      1. *-un-lft-identity 79.9%

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

      2. associate-+r+ 79.9%

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

      3. associate-*l* 79.9%

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

      4. times-frac 91.8%

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

      5. +-commutative 91.8%

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

      6. +-commutative 91.8%

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

      7. associate-+r+ 91.8%

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

      8. +-commutative 91.8%

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

      9. associate-+l+ 91.8%

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

   6. Applied egg-rr 91.8%

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

      1. associate-*l/ 91.8%

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

      2. *-lft-identity 91.8%

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

      3. +-commutative 91.8%

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

   8. Simplified 91.8%

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

   9. Taylor expanded in x around 0 48.5%

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

       1. +-commutative 48.5%

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

   11. Simplified 48.5%

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

       1. clear-num 48.5%

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

       2. un-div-inv 48.5%

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

       3. div-inv 48.5%

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

       4. +-commutative 48.5%

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

       5. clear-num 48.5%

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

       6. /-rgt-identity 48.5%

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

   13. Applied egg-rr 48.5%

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

       1. associate-/l/ 48.2%

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

   15. Simplified 48.2%

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

4. Final simplification 60.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -8 \cdot 10^{+192}:\\ \;\;\;\;\frac{1}{\frac{x + y}{y} \cdot \left(y + \left(x + 1\right)\right)}\\ \mathbf{elif}\;x \leq -1.52 \cdot 10^{-90}:\\ \;\;\;\;\frac{x}{\left(x + y\right) \cdot \left(y + \left(x + 1\right)\right)} \cdot \frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + 1}}{x + y}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 95.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_0 := y + \left(x + 1\right)\\ \mathbf{if}\;x \leq -1.35 \cdot 10^{+32}:\\ \;\;\;\;\frac{\frac{\frac{x}{t\_0}}{x + y}}{\frac{x}{y}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{\frac{y}{\left(x + y\right) \cdot t\_0}}{x + y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ y (+ x 1.0))))
   (if (<= x -1.35e+32)
     (/ (/ (/ x t_0) (+ x y)) (/ x y))
     (* x (/ (/ y (* (+ x y) t_0)) (+ x y))))))

C

assert(x < y);
double code(double x, double y) {
	double t_0 = y + (x + 1.0);
	double tmp;
	if (x <= -1.35e+32) {
		tmp = ((x / t_0) / (x + y)) / (x / y);
	} else {
		tmp = x * ((y / ((x + y) * t_0)) / (x + y));
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = y + (x + 1.0d0)
    if (x <= (-1.35d+32)) then
        tmp = ((x / t_0) / (x + y)) / (x / y)
    else
        tmp = x * ((y / ((x + y) * t_0)) / (x + y))
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double t_0 = y + (x + 1.0);
	double tmp;
	if (x <= -1.35e+32) {
		tmp = ((x / t_0) / (x + y)) / (x / y);
	} else {
		tmp = x * ((y / ((x + y) * t_0)) / (x + y));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	t_0 = y + (x + 1.0)
	tmp = 0
	if x <= -1.35e+32:
		tmp = ((x / t_0) / (x + y)) / (x / y)
	else:
		tmp = x * ((y / ((x + y) * t_0)) / (x + y))
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	t_0 = Float64(y + Float64(x + 1.0))
	tmp = 0.0
	if (x <= -1.35e+32)
		tmp = Float64(Float64(Float64(x / t_0) / Float64(x + y)) / Float64(x / y));
	else
		tmp = Float64(x * Float64(Float64(y / Float64(Float64(x + y) * t_0)) / Float64(x + y)));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	t_0 = y + (x + 1.0);
	tmp = 0.0;
	if (x <= -1.35e+32)
		tmp = ((x / t_0) / (x + y)) / (x / y);
	else
		tmp = x * ((y / ((x + y) * t_0)) / (x + y));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := Block[{t$95$0 = N[(y + N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.35e+32], N[(N[(N[(x / t$95$0), $MachinePrecision] / N[(x + y), $MachinePrecision]), $MachinePrecision] / N[(x / y), $MachinePrecision]), $MachinePrecision], N[(x * N[(N[(y / N[(N[(x + y), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] / N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.35000000000000006e32

   1. Initial program 65.0%

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

      1. *-commutative 65.0%

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

      2. associate-*l* 65.0%

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

      3. times-frac 88.1%

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

      4. +-commutative 88.1%

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

      5. +-commutative 88.1%

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

      6. associate-+r+ 88.1%

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

      7. +-commutative 88.1%

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

      8. associate-+l+ 88.1%

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

   4. Applied egg-rr 88.1%

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

   5. Taylor expanded in y around 0 87.7%

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

      1. clear-num 87.7%

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

      2. frac-times 82.9%

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

      3. *-un-lft-identity 82.9%

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

      4. *-commutative 82.9%

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

      5. +-commutative 82.9%

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

      6. +-commutative 82.9%

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

   7. Applied egg-rr 82.9%

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

      1. associate-/l/ 87.7%

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

      2. associate-/r* 88.7%

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

   9. Simplified 88.7%

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

   if -1.35000000000000006e32 < x

   1. Initial program 72.3%

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

      1. associate-/l* 82.0%

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

      2. associate-+l+ 82.0%

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

   3. Simplified 82.0%

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

      1. *-un-lft-identity 82.0%

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

      2. associate-+r+ 82.0%

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

      3. associate-*l* 82.0%

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

      4. times-frac 93.0%

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

      5. +-commutative 93.0%

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

      6. +-commutative 93.0%

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

      7. associate-+r+ 93.0%

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

      8. +-commutative 93.0%

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

      9. associate-+l+ 93.0%

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

   6. Applied egg-rr 93.0%

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

      1. associate-*l/ 93.1%

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

      2. *-lft-identity 93.1%

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

      3. +-commutative 93.1%

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

   8. Simplified 93.1%

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

4. Final simplification 92.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.35 \cdot 10^{+32}:\\ \;\;\;\;\frac{\frac{\frac{x}{y + \left(x + 1\right)}}{x + y}}{\frac{x}{y}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{\frac{y}{\left(x + y\right) \cdot \left(y + \left(x + 1\right)\right)}}{x + y}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 86.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_0 := y + \left(x + 1\right)\\ \mathbf{if}\;x \leq -1.52 \cdot 10^{-90}:\\ \;\;\;\;\frac{\frac{\frac{x}{t\_0}}{x + y}}{\frac{x}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{\frac{x + y}{y} \cdot t\_0}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ y (+ x 1.0))))
   (if (<= x -1.52e-90)
     (/ (/ (/ x t_0) (+ x y)) (/ x y))
     (/ (/ x y) (* (/ (+ x y) y) t_0)))))

C

assert(x < y);
double code(double x, double y) {
	double t_0 = y + (x + 1.0);
	double tmp;
	if (x <= -1.52e-90) {
		tmp = ((x / t_0) / (x + y)) / (x / y);
	} else {
		tmp = (x / y) / (((x + y) / y) * t_0);
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = y + (x + 1.0d0)
    if (x <= (-1.52d-90)) then
        tmp = ((x / t_0) / (x + y)) / (x / y)
    else
        tmp = (x / y) / (((x + y) / y) * t_0)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double t_0 = y + (x + 1.0);
	double tmp;
	if (x <= -1.52e-90) {
		tmp = ((x / t_0) / (x + y)) / (x / y);
	} else {
		tmp = (x / y) / (((x + y) / y) * t_0);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	t_0 = y + (x + 1.0)
	tmp = 0
	if x <= -1.52e-90:
		tmp = ((x / t_0) / (x + y)) / (x / y)
	else:
		tmp = (x / y) / (((x + y) / y) * t_0)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	t_0 = Float64(y + Float64(x + 1.0))
	tmp = 0.0
	if (x <= -1.52e-90)
		tmp = Float64(Float64(Float64(x / t_0) / Float64(x + y)) / Float64(x / y));
	else
		tmp = Float64(Float64(x / y) / Float64(Float64(Float64(x + y) / y) * t_0));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	t_0 = y + (x + 1.0);
	tmp = 0.0;
	if (x <= -1.52e-90)
		tmp = ((x / t_0) / (x + y)) / (x / y);
	else
		tmp = (x / y) / (((x + y) / y) * t_0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := Block[{t$95$0 = N[(y + N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.52e-90], N[(N[(N[(x / t$95$0), $MachinePrecision] / N[(x + y), $MachinePrecision]), $MachinePrecision] / N[(x / y), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / N[(N[(N[(x + y), $MachinePrecision] / y), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.51999999999999991e-90

   1. Initial program 69.5%

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

      1. *-commutative 69.5%

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

      2. associate-*l* 69.5%

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

      3. times-frac 92.1%

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

      4. +-commutative 92.1%

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

      5. +-commutative 92.1%

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

      6. associate-+r+ 92.1%

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

      7. +-commutative 92.1%

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

      8. associate-+l+ 92.1%

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

   4. Applied egg-rr 92.1%

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

   5. Taylor expanded in y around 0 82.9%

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

      1. clear-num 82.8%

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

      2. frac-times 78.6%

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

      3. *-un-lft-identity 78.6%

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

      4. *-commutative 78.6%

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

      5. +-commutative 78.6%

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

      6. +-commutative 78.6%

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

   7. Applied egg-rr 78.6%

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

      1. associate-/l/ 82.9%

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

      2. associate-/r* 83.5%

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

   9. Simplified 83.5%

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

   if -1.51999999999999991e-90 < x

   1. Initial program 71.3%

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

      1. *-commutative 71.3%

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

      2. associate-*l* 71.3%

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

      3. times-frac 94.0%

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

      4. +-commutative 94.0%

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

      5. +-commutative 94.0%

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

      6. associate-+r+ 94.0%

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

      7. +-commutative 94.0%

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

      8. associate-+l+ 94.0%

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

   4. Applied egg-rr 94.0%

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

      1. clear-num 93.9%

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

      2. associate-/r* 99.8%

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

      3. +-commutative 99.8%

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

      4. frac-times 99.0%

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

      5. *-un-lft-identity 99.0%

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

      6. +-commutative 99.0%

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

      7. +-commutative 99.0%

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

   6. Applied egg-rr 99.0%

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

   7. Taylor expanded in x around 0 57.3%

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

Alternative 8: 85.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 3.4 \cdot 10^{-50}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{\frac{x + y}{y} \cdot \left(y + \left(x + 1\right)\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 3.4e-50)
   (/ (/ y x) (+ x 1.0))
   (/ (/ x y) (* (/ (+ x y) y) (+ y (+ x 1.0))))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 3.4e-50) {
		tmp = (y / x) / (x + 1.0);
	} else {
		tmp = (x / y) / (((x + y) / 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 (y <= 3.4d-50) then
        tmp = (y / x) / (x + 1.0d0)
    else
        tmp = (x / y) / (((x + y) / 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 (y <= 3.4e-50) {
		tmp = (y / x) / (x + 1.0);
	} else {
		tmp = (x / y) / (((x + y) / y) * (y + (x + 1.0)));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if y <= 3.4e-50:
		tmp = (y / x) / (x + 1.0)
	else:
		tmp = (x / y) / (((x + y) / y) * (y + (x + 1.0)))
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= 3.4e-50)
		tmp = Float64(Float64(y / x) / Float64(x + 1.0));
	else
		tmp = Float64(Float64(x / y) / Float64(Float64(Float64(x + y) / 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 (y <= 3.4e-50)
		tmp = (y / x) / (x + 1.0);
	else
		tmp = (x / y) / (((x + y) / 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[y, 3.4e-50], N[(N[(y / x), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / N[(N[(N[(x + y), $MachinePrecision] / y), $MachinePrecision] * N[(y + N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 3.40000000000000014e-50

   1. Initial program 69.5%

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

      1. associate-/l* 80.2%

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

      2. associate-+l+ 80.2%

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

   3. Simplified 80.2%

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

   5. Taylor expanded in y around 0 66.1%

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

      1. associate-/r* 66.6%

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

      2. +-commutative 66.6%

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

   7. Simplified 66.6%

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

   if 3.40000000000000014e-50 < y

   1. Initial program 73.9%

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

      1. *-commutative 73.9%

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

      2. associate-*l* 73.9%

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

      3. times-frac 91.6%

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

      4. +-commutative 91.6%

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

      5. +-commutative 91.6%

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

      6. associate-+r+ 91.6%

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

      7. +-commutative 91.6%

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

      8. associate-+l+ 91.6%

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

   4. Applied egg-rr 91.6%

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

      1. clear-num 91.6%

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

      2. associate-/r* 99.8%

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

      3. +-commutative 99.8%

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

      4. frac-times 99.7%

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

      5. *-un-lft-identity 99.7%

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

      6. +-commutative 99.7%

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

      7. +-commutative 99.7%

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

   6. Applied egg-rr 99.7%

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

   7. Taylor expanded in x around 0 78.9%

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

Alternative 9: 65.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 3.1 \cdot 10^{-229}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 3.1e-229) (/ y x) (if (<= y 1.0) (/ x y) (/ (/ x y) y))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 3.1e-229) {
		tmp = y / x;
	} else if (y <= 1.0) {
		tmp = x / y;
	} else {
		tmp = (x / y) / y;
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 3.1d-229) then
        tmp = y / x
    else if (y <= 1.0d0) then
        tmp = x / y
    else
        tmp = (x / y) / y
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (y <= 3.1e-229) {
		tmp = y / x;
	} else if (y <= 1.0) {
		tmp = x / y;
	} else {
		tmp = (x / y) / y;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if y <= 3.1e-229:
		tmp = y / x
	elif y <= 1.0:
		tmp = x / y
	else:
		tmp = (x / y) / y
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= 3.1e-229)
		tmp = Float64(y / x);
	elseif (y <= 1.0)
		tmp = Float64(x / y);
	else
		tmp = Float64(Float64(x / y) / y);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 3.1e-229)
		tmp = y / x;
	elseif (y <= 1.0)
		tmp = x / y;
	else
		tmp = (x / y) / y;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[y, 3.1e-229], N[(y / x), $MachinePrecision], If[LessEqual[y, 1.0], N[(x / y), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 3.1000000000000001e-229

   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* 80.1%

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

      2. associate-+l+ 80.1%

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

   3. Simplified 80.1%

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

   5. Taylor expanded in y around 0 62.4%

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

   6. Taylor expanded in x around 0 39.8%

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

   if 3.1000000000000001e-229 < y < 1

   1. Initial program 84.9%

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

      1. associate-/l* 85.2%

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

      2. associate-+l+ 85.2%

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

   3. Simplified 85.2%

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

   5. Taylor expanded in x around 0 22.5%

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

      1. +-commutative 22.5%

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

   7. Simplified 22.5%

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

   8. Taylor expanded in y around 0 22.5%

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

   if 1 < y

   1. Initial program 68.6%

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

      1. *-commutative 68.6%

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

      2. associate-*l* 68.6%

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

      3. times-frac 90.0%

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

      4. +-commutative 90.0%

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

      5. +-commutative 90.0%

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

      6. associate-+r+ 90.0%

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

      7. +-commutative 90.0%

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

      8. associate-+l+ 90.0%

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

   4. Applied egg-rr 90.0%

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

      1. clear-num 89.9%

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

      2. associate-/r* 99.9%

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

      3. +-commutative 99.9%

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

      4. frac-times 99.7%

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

      5. *-un-lft-identity 99.7%

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

      6. +-commutative 99.7%

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

      7. +-commutative 99.7%

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

   6. Applied egg-rr 99.7%

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

   7. Taylor expanded in x around 0 85.5%

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

   8. Taylor expanded in y around inf 69.5%

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

Alternative 10: 81.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 6.5 \cdot 10^{-49}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + 1}}{x + y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 6.5e-49) (/ (/ y x) (+ x 1.0)) (/ (/ x (+ y 1.0)) (+ x y))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 6.5e-49) {
		tmp = (y / x) / (x + 1.0);
	} else {
		tmp = (x / (y + 1.0)) / (x + y);
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 6.5d-49) then
        tmp = (y / x) / (x + 1.0d0)
    else
        tmp = (x / (y + 1.0d0)) / (x + y)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (y <= 6.5e-49) {
		tmp = (y / x) / (x + 1.0);
	} else {
		tmp = (x / (y + 1.0)) / (x + y);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if y <= 6.5e-49:
		tmp = (y / x) / (x + 1.0)
	else:
		tmp = (x / (y + 1.0)) / (x + y)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= 6.5e-49)
		tmp = Float64(Float64(y / x) / Float64(x + 1.0));
	else
		tmp = Float64(Float64(x / Float64(y + 1.0)) / Float64(x + y));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 6.5e-49)
		tmp = (y / x) / (x + 1.0);
	else
		tmp = (x / (y + 1.0)) / (x + y);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[y, 6.5e-49], N[(N[(y / x), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(y + 1.0), $MachinePrecision]), $MachinePrecision] / N[(x + y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 6.49999999999999968e-49

   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* 80.3%

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

      2. associate-+l+ 80.3%

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

   3. Simplified 80.3%

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

   5. Taylor expanded in y around 0 66.2%

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

      1. associate-/r* 66.8%

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

      2. +-commutative 66.8%

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

   7. Simplified 66.8%

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

   if 6.49999999999999968e-49 < y

   1. Initial program 73.5%

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

      1. associate-/l* 85.4%

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

      2. associate-+l+ 85.4%

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

   3. Simplified 85.4%

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

      1. *-un-lft-identity 85.4%

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

      2. associate-+r+ 85.4%

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

      3. associate-*l* 85.5%

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

      4. times-frac 89.9%

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

      5. +-commutative 89.9%

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

      6. +-commutative 89.9%

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

      7. associate-+r+ 89.9%

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

      8. +-commutative 89.9%

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

      9. associate-+l+ 89.9%

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

   6. Applied egg-rr 89.9%

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

      1. associate-*l/ 89.9%

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

      2. *-lft-identity 89.9%

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

      3. +-commutative 89.9%

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

   8. Simplified 89.9%

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

   9. Taylor expanded in x around 0 68.6%

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

       1. +-commutative 68.6%

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

   11. Simplified 68.6%

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

       1. clear-num 67.3%

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

       2. un-div-inv 67.3%

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

       3. div-inv 67.2%

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

       4. +-commutative 67.2%

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

       5. clear-num 67.3%

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

       6. /-rgt-identity 67.3%

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

   13. Applied egg-rr 67.3%

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

       1. associate-/l/ 65.3%

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

   15. Simplified 65.3%

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

Alternative 11: 81.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 1.08 \cdot 10^{-47}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + 1}}{y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 1.08e-47) (/ (/ y x) (+ x 1.0)) (/ (/ x (+ y 1.0)) y)))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 1.08e-47) {
		tmp = (y / x) / (x + 1.0);
	} else {
		tmp = (x / (y + 1.0)) / y;
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 1.08d-47) then
        tmp = (y / x) / (x + 1.0d0)
    else
        tmp = (x / (y + 1.0d0)) / y
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (y <= 1.08e-47) {
		tmp = (y / x) / (x + 1.0);
	} else {
		tmp = (x / (y + 1.0)) / y;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if y <= 1.08e-47:
		tmp = (y / x) / (x + 1.0)
	else:
		tmp = (x / (y + 1.0)) / y
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= 1.08e-47)
		tmp = Float64(Float64(y / x) / Float64(x + 1.0));
	else
		tmp = Float64(Float64(x / Float64(y + 1.0)) / y);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 1.08e-47)
		tmp = (y / x) / (x + 1.0);
	else
		tmp = (x / (y + 1.0)) / y;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[y, 1.08e-47], N[(N[(y / x), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(y + 1.0), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.08000000000000005e-47

   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* 80.3%

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

      2. associate-+l+ 80.3%

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

   3. Simplified 80.3%

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

   5. Taylor expanded in y around 0 66.2%

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

      1. associate-/r* 66.8%

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

      2. +-commutative 66.8%

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

   7. Simplified 66.8%

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

   if 1.08000000000000005e-47 < y

   1. Initial program 73.5%

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

      1. associate-/l* 85.4%

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

      2. associate-+l+ 85.4%

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

   3. Simplified 85.4%

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

   5. Taylor expanded in x around 0 62.3%

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

      1. +-commutative 62.3%

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

   7. Simplified 62.3%

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

      1. associate-/r* 64.8%

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

      2. div-inv 64.7%

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

   9. Applied egg-rr 64.7%

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

       1. associate-*l/ 64.7%

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

       2. un-div-inv 64.8%

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

   11. Applied egg-rr 64.8%

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

Alternative 12: 79.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -7.2 \cdot 10^{-6}:\\ \;\;\;\;\frac{y}{x} \cdot \frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + 1}}{y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -7.2e-6) (* (/ y x) (/ 1.0 x)) (/ (/ x (+ y 1.0)) y)))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -7.2e-6) {
		tmp = (y / x) * (1.0 / x);
	} else {
		tmp = (x / (y + 1.0)) / y;
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-7.2d-6)) then
        tmp = (y / x) * (1.0d0 / x)
    else
        tmp = (x / (y + 1.0d0)) / y
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (x <= -7.2e-6) {
		tmp = (y / x) * (1.0 / x);
	} else {
		tmp = (x / (y + 1.0)) / y;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -7.2e-6:
		tmp = (y / x) * (1.0 / x)
	else:
		tmp = (x / (y + 1.0)) / y
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -7.2e-6)
		tmp = Float64(Float64(y / x) * Float64(1.0 / x));
	else
		tmp = Float64(Float64(x / Float64(y + 1.0)) / y);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -7.2e-6)
		tmp = (y / x) * (1.0 / x);
	else
		tmp = (x / (y + 1.0)) / y;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[x, -7.2e-6], N[(N[(y / x), $MachinePrecision] * N[(1.0 / x), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(y + 1.0), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -7.19999999999999967e-6

   1. Initial program 65.6%

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

      1. *-commutative 65.6%

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

      2. associate-*l* 65.7%

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

      3. times-frac 89.3%

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

      4. +-commutative 89.3%

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

      5. +-commutative 89.3%

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

      6. associate-+r+ 89.3%

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

      7. +-commutative 89.3%

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

      8. associate-+l+ 89.3%

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

   4. Applied egg-rr 89.3%

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

   5. Taylor expanded in y around 0 87.6%

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

   6. Taylor expanded in x around inf 77.0%

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

   if -7.19999999999999967e-6 < x

   1. Initial program 72.4%

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

      1. associate-/l* 81.4%

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

      2. associate-+l+ 81.4%

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

   3. Simplified 81.4%

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

   5. Taylor expanded in x around 0 47.6%

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

      1. +-commutative 47.6%

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

   7. Simplified 47.6%

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

      1. associate-/r* 48.8%

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

      2. div-inv 48.7%

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

   9. Applied egg-rr 48.7%

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

       1. associate-*l/ 48.7%

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

       2. un-div-inv 48.8%

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

   11. Applied egg-rr 48.8%

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

Alternative 13: 78.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -7.2 \cdot 10^{-6}:\\ \;\;\;\;\frac{y}{x} \cdot \frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -7.2e-6) (* (/ y x) (/ 1.0 x)) (/ x (* y (+ y 1.0)))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -7.2e-6) {
		tmp = (y / x) * (1.0 / x);
	} else {
		tmp = x / (y * (y + 1.0));
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-7.2d-6)) then
        tmp = (y / x) * (1.0d0 / x)
    else
        tmp = x / (y * (y + 1.0d0))
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (x <= -7.2e-6) {
		tmp = (y / x) * (1.0 / x);
	} else {
		tmp = x / (y * (y + 1.0));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -7.2e-6:
		tmp = (y / x) * (1.0 / x)
	else:
		tmp = x / (y * (y + 1.0))
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -7.2e-6)
		tmp = Float64(Float64(y / x) * Float64(1.0 / x));
	else
		tmp = Float64(x / Float64(y * Float64(y + 1.0)));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -7.2e-6)
		tmp = (y / x) * (1.0 / x);
	else
		tmp = x / (y * (y + 1.0));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[x, -7.2e-6], N[(N[(y / x), $MachinePrecision] * N[(1.0 / x), $MachinePrecision]), $MachinePrecision], N[(x / N[(y * N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -7.19999999999999967e-6

   1. Initial program 65.6%

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

      1. *-commutative 65.6%

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

      2. associate-*l* 65.7%

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

      3. times-frac 89.3%

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

      4. +-commutative 89.3%

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

      5. +-commutative 89.3%

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

      6. associate-+r+ 89.3%

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

      7. +-commutative 89.3%

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

      8. associate-+l+ 89.3%

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

   4. Applied egg-rr 89.3%

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

   5. Taylor expanded in y around 0 87.6%

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

   6. Taylor expanded in x around inf 77.0%

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

   if -7.19999999999999967e-6 < x

   1. Initial program 72.4%

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

      1. associate-/l* 81.4%

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

      2. associate-+l+ 81.4%

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

   3. Simplified 81.4%

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

   5. Taylor expanded in x around 0 47.6%

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

      1. +-commutative 47.6%

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

   7. Simplified 47.6%

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

Alternative 14: 43.5% accurate, 2.1× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -4 \cdot 10^{-93}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y) :precision binary64 (if (<= x -4e-93) (/ y x) (/ x y)))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -4e-93) {
		tmp = y / x;
	} else {
		tmp = x / y;
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-4d-93)) then
        tmp = y / x
    else
        tmp = x / y
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (x <= -4e-93) {
		tmp = y / x;
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -4e-93:
		tmp = y / x
	else:
		tmp = x / y
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -4e-93)
		tmp = Float64(y / x);
	else
		tmp = Float64(x / y);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -4e-93)
		tmp = y / x;
	else
		tmp = x / y;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[x, -4e-93], N[(y / x), $MachinePrecision], N[(x / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -3.9999999999999996e-93

   1. Initial program 69.8%

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

      1. 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 y around 0 70.0%

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

   6. Taylor expanded in x around 0 34.8%

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

   if -3.9999999999999996e-93 < x

   1. Initial program 71.1%

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

      1. associate-/l* 79.8%

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

      2. associate-+l+ 79.8%

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

   3. Simplified 79.8%

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

   5. Taylor expanded in x around 0 46.0%

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

      1. +-commutative 46.0%

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

   7. Simplified 46.0%

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

   8. Taylor expanded in y around 0 25.6%

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

Alternative 15: 28.1% accurate, 2.1× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -7.2 \cdot 10^{-6}:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y) :precision binary64 (if (<= x -7.2e-6) (/ 1.0 x) (/ x y)))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -7.2e-6) {
		tmp = 1.0 / x;
	} else {
		tmp = x / y;
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-7.2d-6)) then
        tmp = 1.0d0 / x
    else
        tmp = x / y
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (x <= -7.2e-6) {
		tmp = 1.0 / x;
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -7.2e-6:
		tmp = 1.0 / x
	else:
		tmp = x / y
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -7.2e-6)
		tmp = Float64(1.0 / x);
	else
		tmp = Float64(x / y);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -7.2e-6)
		tmp = 1.0 / x;
	else
		tmp = x / y;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[x, -7.2e-6], N[(1.0 / x), $MachinePrecision], N[(x / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -7.19999999999999967e-6

   1. Initial program 65.6%

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

      1. *-commutative 65.6%

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

      2. associate-*l* 65.7%

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

      3. times-frac 89.3%

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

      4. +-commutative 89.3%

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

      5. +-commutative 89.3%

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

      6. associate-+r+ 89.3%

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

      7. +-commutative 89.3%

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

      8. associate-+l+ 89.3%

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

   4. Applied egg-rr 89.3%

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

      1. clear-num 89.2%

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

      2. associate-/r* 99.8%

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

      3. +-commutative 99.8%

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

      4. frac-times 99.8%

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

      5. *-un-lft-identity 99.8%

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

      6. +-commutative 99.8%

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

      7. +-commutative 99.8%

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

   6. Applied egg-rr 99.8%

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

   7. Taylor expanded in x around 0 49.8%

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

   8. Taylor expanded in x around inf 5.9%

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

   if -7.19999999999999967e-6 < x

   1. Initial program 72.4%

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

      1. associate-/l* 81.4%

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

      2. associate-+l+ 81.4%

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

   3. Simplified 81.4%

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

   5. Taylor expanded in x around 0 47.6%

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

      1. +-commutative 47.6%

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

   7. Simplified 47.6%

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

   8. Taylor expanded in y around 0 23.4%

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

Alternative 16: 4.3% accurate, 5.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 70.6%

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

   1. *-commutative 70.6%

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

   2. associate-*l* 70.7%

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

   3. times-frac 93.3%

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

   4. +-commutative 93.3%

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

   5. +-commutative 93.3%

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

   6. associate-+r+ 93.3%

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

   7. +-commutative 93.3%

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

   8. associate-+l+ 93.3%

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

4. Applied egg-rr 93.3%

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

   1. clear-num 93.3%

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

   2. associate-/r* 99.8%

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

   3. +-commutative 99.8%

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

   4. frac-times 99.3%

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

   5. *-un-lft-identity 99.3%

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

   6. +-commutative 99.3%

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

   7. +-commutative 99.3%

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

6. Applied egg-rr 99.3%

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

7. Taylor expanded in x around 0 55.2%

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

8. Taylor expanded in x around inf 4.5%

   \[\leadsto \color{blue}{\frac{1}{x}} \]
9. Add Preprocessing

Alternative 17: 3.5% accurate, 17.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 70.6%

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

   1. associate-/l* 81.6%

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

   2. associate-+l+ 81.6%

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

3. Simplified 81.6%

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

   1. *-un-lft-identity 81.6%

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

   2. associate-+r+ 81.6%

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

   3. associate-*l* 81.6%

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

   4. times-frac 90.8%

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

   5. +-commutative 90.8%

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

   6. +-commutative 90.8%

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

   7. associate-+r+ 90.8%

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

   8. +-commutative 90.8%

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

   9. associate-+l+ 90.8%

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

6. Applied egg-rr 90.8%

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

   1. associate-*l/ 90.8%

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

   2. *-lft-identity 90.8%

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

   3. +-commutative 90.8%

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

8. Simplified 90.8%

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

9. Taylor expanded in x around 0 43.5%

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

    1. +-commutative 43.5%

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

11. Simplified 43.5%

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

12. Taylor expanded in y around 0 3.4%

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

13. Taylor expanded in x around 0 3.4%

    \[\leadsto \color{blue}{1} \]
14. 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 2024085 
(FPCore (x y)
  :name "Numeric.SpecFunctions:incompleteBetaApprox from math-functions-0.1.5.2, A"
  :precision binary64

  :alt
  (/ (/ (/ x (+ (+ y 1.0) x)) (+ y x)) (/ 1.0 (/ y (+ y x))))

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