Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2, F

Percentage Accurate: 77.4% → 99.3%

Time: 9.8s

Alternatives: 8

Speedup: 7.7×

Specification

Math

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

FPCore

(FPCore (x y) :precision binary64 (/ (exp (* x (log (/ x (+ x y))))) x))

C

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

Fortran

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

Java

public static double code(double x, double y) {
	return Math.exp((x * Math.log((x / (x + y))))) / x;
}

Python

def code(x, y):
	return math.exp((x * math.log((x / (x + y))))) / x

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 77.4% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x y) :precision binary64 (/ (exp (* x (log (/ x (+ x y))))) x))

C

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

Fortran

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

Java

public static double code(double x, double y) {
	return Math.exp((x * Math.log((x / (x + y))))) / x;
}

Python

def code(x, y):
	return math.exp((x * math.log((x / (x + y))))) / x

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{e^{-y}}{x}\\ \mathbf{if}\;x \leq -19000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 2.3 \cdot 10^{-5}:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (exp (- y)) x)))
   (if (<= x -19000000.0) t_0 (if (<= x 2.3e-5) (/ 1.0 x) t_0))))

C

double code(double x, double y) {
	double t_0 = exp(-y) / x;
	double tmp;
	if (x <= -19000000.0) {
		tmp = t_0;
	} else if (x <= 2.3e-5) {
		tmp = 1.0 / x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

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 = exp(-y) / x
    if (x <= (-19000000.0d0)) then
        tmp = t_0
    else if (x <= 2.3d-5) then
        tmp = 1.0d0 / x
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = Math.exp(-y) / x;
	double tmp;
	if (x <= -19000000.0) {
		tmp = t_0;
	} else if (x <= 2.3e-5) {
		tmp = 1.0 / x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.exp(-y) / x
	tmp = 0
	if x <= -19000000.0:
		tmp = t_0
	elif x <= 2.3e-5:
		tmp = 1.0 / x
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(exp(Float64(-y)) / x)
	tmp = 0.0
	if (x <= -19000000.0)
		tmp = t_0;
	elseif (x <= 2.3e-5)
		tmp = Float64(1.0 / x);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = exp(-y) / x;
	tmp = 0.0;
	if (x <= -19000000.0)
		tmp = t_0;
	elseif (x <= 2.3e-5)
		tmp = 1.0 / x;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Exp[(-y)], $MachinePrecision] / x), $MachinePrecision]}, If[LessEqual[x, -19000000.0], t$95$0, If[LessEqual[x, 2.3e-5], N[(1.0 / x), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.9e7 or 2.3e-5 < x

   1. Initial program 72.3%

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

   3. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

         \[\leadsto \frac{e^{\color{blue}{\mathsf{neg}\left(y\right)}}}{x} \]

      2. lower-neg.f64 100.0

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

   5. Applied rewrites 100.0%

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

   if -1.9e7 < x < 2.3e-5

   1. Initial program 84.9%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 99.2%

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

Alternative 2: 77.0% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -7.5 \cdot 10^{-5}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\frac{\frac{\frac{\frac{\mathsf{fma}\left(x, x, x\right)}{2}}{x}}{x}}{x}, y, \frac{-1}{x}\right), y, \frac{1}{x}\right)\\ \mathbf{elif}\;y \leq 150:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{elif}\;y \leq 2.3 \cdot 10^{+181}:\\ \;\;\;\;\frac{\frac{x}{y}}{x \cdot x} \cdot y\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -7.5e-5)
   (fma (fma (/ (/ (/ (/ (fma x x x) 2.0) x) x) x) y (/ -1.0 x)) y (/ 1.0 x))
   (if (<= y 150.0)
     (/ 1.0 x)
     (if (<= y 2.3e+181) (* (/ (/ x y) (* x x)) y) (/ 1.0 x)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -7.5e-5) {
		tmp = fma(fma(((((fma(x, x, x) / 2.0) / x) / x) / x), y, (-1.0 / x)), y, (1.0 / x));
	} else if (y <= 150.0) {
		tmp = 1.0 / x;
	} else if (y <= 2.3e+181) {
		tmp = ((x / y) / (x * x)) * y;
	} else {
		tmp = 1.0 / x;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -7.5e-5)
		tmp = fma(fma(Float64(Float64(Float64(Float64(fma(x, x, x) / 2.0) / x) / x) / x), y, Float64(-1.0 / x)), y, Float64(1.0 / x));
	elseif (y <= 150.0)
		tmp = Float64(1.0 / x);
	elseif (y <= 2.3e+181)
		tmp = Float64(Float64(Float64(x / y) / Float64(x * x)) * y);
	else
		tmp = Float64(1.0 / x);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, -7.5e-5], N[(N[(N[(N[(N[(N[(N[(x * x + x), $MachinePrecision] / 2.0), $MachinePrecision] / x), $MachinePrecision] / x), $MachinePrecision] / x), $MachinePrecision] * y + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision] * y + N[(1.0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 150.0], N[(1.0 / x), $MachinePrecision], If[LessEqual[y, 2.3e+181], N[(N[(N[(x / y), $MachinePrecision] / N[(x * x), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision], N[(1.0 / x), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < -7.49999999999999934e-5

   1. Initial program 50.3%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

   5. Applied rewrites 23.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{\frac{0.5}{x}}{x} + \frac{0.5}{x}, y, \frac{-1}{x}\right), y, \frac{1}{x}\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 8.1%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(x, x, \left(x \cdot 2\right) \cdot 0.5\right)}{\left(x \cdot 2\right) \cdot \left(x \cdot x\right)}, y, \frac{-1}{x}\right), y, \frac{1}{x}\right) \]
   8. Step-by-step derivation

   9. Applied rewrites 48.3%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{\frac{\frac{\frac{\mathsf{fma}\left(x, x, x\right)}{2}}{x}}{x}}{x}, y, \frac{-1}{x}\right), y, \frac{1}{x}\right) \]

   if -7.49999999999999934e-5 < y < 150 or 2.2999999999999999e181 < y

   1. Initial program 94.2%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 95.0%

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

   if 150 < y < 2.2999999999999999e181

   1. Initial program 40.2%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 3.1

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

   5. Applied rewrites 3.1%

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

   7. Applied rewrites 27.9%

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

   8. Taylor expanded in y around 0

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

   10. Applied rewrites 38.1%

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

   12. Applied rewrites 57.6%

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

Alternative 3: 76.5% accurate, 4.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -6.2 \cdot 10^{+196}:\\ \;\;\;\;\frac{\frac{\left(-y\right) \cdot x}{x}}{x}\\ \mathbf{elif}\;y \leq 150:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{elif}\;y \leq 2.3 \cdot 10^{+181}:\\ \;\;\;\;\frac{\frac{x}{y}}{x \cdot x} \cdot y\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -6.2e+196)
   (/ (/ (* (- y) x) x) x)
   (if (<= y 150.0)
     (/ 1.0 x)
     (if (<= y 2.3e+181) (* (/ (/ x y) (* x x)) y) (/ 1.0 x)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -6.2e+196) {
		tmp = ((-y * x) / x) / x;
	} else if (y <= 150.0) {
		tmp = 1.0 / x;
	} else if (y <= 2.3e+181) {
		tmp = ((x / y) / (x * x)) * y;
	} else {
		tmp = 1.0 / x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-6.2d+196)) then
        tmp = ((-y * x) / x) / x
    else if (y <= 150.0d0) then
        tmp = 1.0d0 / x
    else if (y <= 2.3d+181) then
        tmp = ((x / y) / (x * x)) * y
    else
        tmp = 1.0d0 / x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -6.2e+196) {
		tmp = ((-y * x) / x) / x;
	} else if (y <= 150.0) {
		tmp = 1.0 / x;
	} else if (y <= 2.3e+181) {
		tmp = ((x / y) / (x * x)) * y;
	} else {
		tmp = 1.0 / x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -6.2e+196:
		tmp = ((-y * x) / x) / x
	elif y <= 150.0:
		tmp = 1.0 / x
	elif y <= 2.3e+181:
		tmp = ((x / y) / (x * x)) * y
	else:
		tmp = 1.0 / x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -6.2e+196)
		tmp = Float64(Float64(Float64(Float64(-y) * x) / x) / x);
	elseif (y <= 150.0)
		tmp = Float64(1.0 / x);
	elseif (y <= 2.3e+181)
		tmp = Float64(Float64(Float64(x / y) / Float64(x * x)) * y);
	else
		tmp = Float64(1.0 / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -6.2e+196)
		tmp = ((-y * x) / x) / x;
	elseif (y <= 150.0)
		tmp = 1.0 / x;
	elseif (y <= 2.3e+181)
		tmp = ((x / y) / (x * x)) * y;
	else
		tmp = 1.0 / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -6.2e+196], N[(N[(N[((-y) * x), $MachinePrecision] / x), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[y, 150.0], N[(1.0 / x), $MachinePrecision], If[LessEqual[y, 2.3e+181], N[(N[(N[(x / y), $MachinePrecision] / N[(x * x), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision], N[(1.0 / x), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < -6.2000000000000002e196

   1. Initial program 78.5%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 5.1

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

   5. Applied rewrites 5.1%

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

   7. Applied rewrites 78.6%

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

   8. Taylor expanded in y around inf

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

   10. Applied rewrites 78.6%

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

   if -6.2000000000000002e196 < y < 150 or 2.2999999999999999e181 < y

   1. Initial program 85.9%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 86.0%

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

   if 150 < y < 2.2999999999999999e181

   1. Initial program 40.2%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 3.1

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

   5. Applied rewrites 3.1%

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

   7. Applied rewrites 27.9%

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

   8. Taylor expanded in y around 0

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

   10. Applied rewrites 38.1%

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

   12. Applied rewrites 57.6%

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

4. Final simplification 80.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -6.2 \cdot 10^{+196}:\\ \;\;\;\;\frac{\frac{\left(-y\right) \cdot x}{x}}{x}\\ \mathbf{elif}\;y \leq 150:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{elif}\;y \leq 2.3 \cdot 10^{+181}:\\ \;\;\;\;\frac{\frac{x}{y}}{x \cdot x} \cdot y\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 79.0% accurate, 6.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -19000000:\\ \;\;\;\;\frac{\frac{x - y \cdot x}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -19000000.0) (/ (/ (- x (* y x)) x) x) (/ 1.0 x)))

C

double code(double x, double y) {
	double tmp;
	if (x <= -19000000.0) {
		tmp = ((x - (y * x)) / x) / x;
	} else {
		tmp = 1.0 / x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-19000000.0d0)) then
        tmp = ((x - (y * x)) / x) / x
    else
        tmp = 1.0d0 / x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -19000000.0) {
		tmp = ((x - (y * x)) / x) / x;
	} else {
		tmp = 1.0 / x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -19000000.0:
		tmp = ((x - (y * x)) / x) / x
	else:
		tmp = 1.0 / x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -19000000.0)
		tmp = Float64(Float64(Float64(x - Float64(y * x)) / x) / x);
	else
		tmp = Float64(1.0 / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -19000000.0)
		tmp = ((x - (y * x)) / x) / x;
	else
		tmp = 1.0 / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -19000000.0], N[(N[(N[(x - N[(y * x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] / x), $MachinePrecision], N[(1.0 / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.9e7

   1. Initial program 67.1%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 44.9

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

   5. Applied rewrites 44.9%

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

   7. Applied rewrites 62.3%

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

   if -1.9e7 < x

   1. Initial program 80.5%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 80.6%

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

Alternative 5: 78.8% accurate, 7.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -19000000:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.5, y, -1\right), y, 1\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -19000000.0) (/ (fma (fma 0.5 y -1.0) y 1.0) x) (/ 1.0 x)))

C

double code(double x, double y) {
	double tmp;
	if (x <= -19000000.0) {
		tmp = fma(fma(0.5, y, -1.0), y, 1.0) / x;
	} else {
		tmp = 1.0 / x;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -19000000.0)
		tmp = Float64(fma(fma(0.5, y, -1.0), y, 1.0) / x);
	else
		tmp = Float64(1.0 / x);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, -19000000.0], N[(N[(N[(0.5 * y + -1.0), $MachinePrecision] * y + 1.0), $MachinePrecision] / x), $MachinePrecision], N[(1.0 / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.9e7

   1. Initial program 67.1%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

   5. Applied rewrites 57.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{\frac{0.5}{x}}{x} + \frac{0.5}{x}, y, \frac{-1}{x}\right), y, \frac{1}{x}\right)} \]

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 61.1%

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(0.5, y, -1\right), y, 1\right)}{\color{blue}{x}} \]

   if -1.9e7 < x

   1. Initial program 80.5%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 80.6%

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

Alternative 6: 75.0% accurate, 8.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -9.1 \cdot 10^{+195}:\\ \;\;\;\;\frac{\left(y \cdot y\right) \cdot 0.5}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -9.1e+195) (/ (* (* y y) 0.5) x) (/ 1.0 x)))

C

double code(double x, double y) {
	double tmp;
	if (y <= -9.1e+195) {
		tmp = ((y * y) * 0.5) / x;
	} else {
		tmp = 1.0 / x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-9.1d+195)) then
        tmp = ((y * y) * 0.5d0) / x
    else
        tmp = 1.0d0 / x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -9.1e+195) {
		tmp = ((y * y) * 0.5) / x;
	} else {
		tmp = 1.0 / x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -9.1e+195:
		tmp = ((y * y) * 0.5) / x
	else:
		tmp = 1.0 / x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -9.1e+195)
		tmp = Float64(Float64(Float64(y * y) * 0.5) / x);
	else
		tmp = Float64(1.0 / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -9.1e+195)
		tmp = ((y * y) * 0.5) / x;
	else
		tmp = 1.0 / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -9.1e+195], N[(N[(N[(y * y), $MachinePrecision] * 0.5), $MachinePrecision] / x), $MachinePrecision], N[(1.0 / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -9.0999999999999998e195

   1. Initial program 78.5%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

   5. Applied rewrites 62.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{\frac{0.5}{x}}{x} + \frac{0.5}{x}, y, \frac{-1}{x}\right), y, \frac{1}{x}\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 16.8%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(x, x, \left(x \cdot 2\right) \cdot 0.5\right)}{\left(x \cdot 2\right) \cdot \left(x \cdot x\right)}, y, \frac{-1}{x}\right), y, \frac{1}{x}\right) \]

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 78.6%

       \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(0.5, y, -1\right), y, 1\right)}{\color{blue}{x}} \]

   11. Taylor expanded in y around inf

       \[\leadsto \frac{\frac{1}{2} \cdot {y}^{2}}{x} \]
   12. Step-by-step derivation

   13. Applied rewrites 78.6%

       \[\leadsto \frac{\left(y \cdot y\right) \cdot 0.5}{x} \]

   if -9.0999999999999998e195 < y

   1. Initial program 77.3%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 74.9%

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

Alternative 7: 73.8% accurate, 8.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3.6 \cdot 10^{+207}:\\ \;\;\;\;\left(\frac{y}{x} \cdot y\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -3.6e+207) (* (* (/ y x) y) 0.5) (/ 1.0 x)))

C

double code(double x, double y) {
	double tmp;
	if (y <= -3.6e+207) {
		tmp = ((y / x) * y) * 0.5;
	} else {
		tmp = 1.0 / x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-3.6d+207)) then
        tmp = ((y / x) * y) * 0.5d0
    else
        tmp = 1.0d0 / x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -3.6e+207) {
		tmp = ((y / x) * y) * 0.5;
	} else {
		tmp = 1.0 / x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -3.6e+207:
		tmp = ((y / x) * y) * 0.5
	else:
		tmp = 1.0 / x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -3.6e+207)
		tmp = Float64(Float64(Float64(y / x) * y) * 0.5);
	else
		tmp = Float64(1.0 / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -3.6e+207)
		tmp = ((y / x) * y) * 0.5;
	else
		tmp = 1.0 / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -3.6e+207], N[(N[(N[(y / x), $MachinePrecision] * y), $MachinePrecision] * 0.5), $MachinePrecision], N[(1.0 / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -3.60000000000000014e207

   1. Initial program 78.5%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

   5. Applied rewrites 62.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{\frac{0.5}{x}}{x} + \frac{0.5}{x}, y, \frac{-1}{x}\right), y, \frac{1}{x}\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 16.8%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(x, x, \left(x \cdot 2\right) \cdot 0.5\right)}{\left(x \cdot 2\right) \cdot \left(x \cdot x\right)}, y, \frac{-1}{x}\right), y, \frac{1}{x}\right) \]

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 78.6%

       \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(0.5, y, -1\right), y, 1\right)}{\color{blue}{x}} \]

   11. Taylor expanded in y around inf

       \[\leadsto \frac{1}{2} \cdot \frac{{y}^{2}}{\color{blue}{x}} \]
   12. Step-by-step derivation

   13. Applied rewrites 64.1%

       \[\leadsto \left(y \cdot \frac{y}{x}\right) \cdot 0.5 \]

   if -3.60000000000000014e207 < y

   1. Initial program 77.3%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 74.9%

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

4. Final simplification 74.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.6 \cdot 10^{+207}:\\ \;\;\;\;\left(\frac{y}{x} \cdot y\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 74.8% accurate, 19.3× speedup

Math

\[\begin{array}{l} \\ \frac{1}{x} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.3%

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

3. Taylor expanded in y around 0

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

5. Applied rewrites 72.1%

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

Developer Target 1: 77.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{e^{\frac{-1}{y}}}{x}\\ t_1 := \frac{{\left(\frac{x}{y + x}\right)}^{x}}{x}\\ \mathbf{if}\;y < -3.7311844206647956 \cdot 10^{+94}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y < 2.817959242728288 \cdot 10^{+37}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y < 2.347387415166998 \cdot 10^{+178}:\\ \;\;\;\;\log \left(e^{t\_1}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (exp (/ -1.0 y)) x)) (t_1 (/ (pow (/ x (+ y x)) x) x)))
   (if (< y -3.7311844206647956e+94)
     t_0
     (if (< y 2.817959242728288e+37)
       t_1
       (if (< y 2.347387415166998e+178) (log (exp t_1)) t_0)))))

C

double code(double x, double y) {
	double t_0 = exp((-1.0 / y)) / x;
	double t_1 = pow((x / (y + x)), x) / x;
	double tmp;
	if (y < -3.7311844206647956e+94) {
		tmp = t_0;
	} else if (y < 2.817959242728288e+37) {
		tmp = t_1;
	} else if (y < 2.347387415166998e+178) {
		tmp = log(exp(t_1));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

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 = exp(((-1.0d0) / y)) / x
    t_1 = ((x / (y + x)) ** x) / x
    if (y < (-3.7311844206647956d+94)) then
        tmp = t_0
    else if (y < 2.817959242728288d+37) then
        tmp = t_1
    else if (y < 2.347387415166998d+178) then
        tmp = log(exp(t_1))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = Math.exp((-1.0 / y)) / x;
	double t_1 = Math.pow((x / (y + x)), x) / x;
	double tmp;
	if (y < -3.7311844206647956e+94) {
		tmp = t_0;
	} else if (y < 2.817959242728288e+37) {
		tmp = t_1;
	} else if (y < 2.347387415166998e+178) {
		tmp = Math.log(Math.exp(t_1));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.exp((-1.0 / y)) / x
	t_1 = math.pow((x / (y + x)), x) / x
	tmp = 0
	if y < -3.7311844206647956e+94:
		tmp = t_0
	elif y < 2.817959242728288e+37:
		tmp = t_1
	elif y < 2.347387415166998e+178:
		tmp = math.log(math.exp(t_1))
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(exp(Float64(-1.0 / y)) / x)
	t_1 = Float64((Float64(x / Float64(y + x)) ^ x) / x)
	tmp = 0.0
	if (y < -3.7311844206647956e+94)
		tmp = t_0;
	elseif (y < 2.817959242728288e+37)
		tmp = t_1;
	elseif (y < 2.347387415166998e+178)
		tmp = log(exp(t_1));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = exp((-1.0 / y)) / x;
	t_1 = ((x / (y + x)) ^ x) / x;
	tmp = 0.0;
	if (y < -3.7311844206647956e+94)
		tmp = t_0;
	elseif (y < 2.817959242728288e+37)
		tmp = t_1;
	elseif (y < 2.347387415166998e+178)
		tmp = log(exp(t_1));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Exp[N[(-1.0 / y), $MachinePrecision]], $MachinePrecision] / x), $MachinePrecision]}, Block[{t$95$1 = N[(N[Power[N[(x / N[(y + x), $MachinePrecision]), $MachinePrecision], x], $MachinePrecision] / x), $MachinePrecision]}, If[Less[y, -3.7311844206647956e+94], t$95$0, If[Less[y, 2.817959242728288e+37], t$95$1, If[Less[y, 2.347387415166998e+178], N[Log[N[Exp[t$95$1], $MachinePrecision]], $MachinePrecision], t$95$0]]]]]

Reproduce

herbie shell --seed 2024243 
(FPCore (x y)
  :name "Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2, F"
  :precision binary64

  :alt
  (! :herbie-platform default (if (< y -37311844206647956000000000000000000000000000000000000000000000000000000000000000000000000000000) (/ (exp (/ -1 y)) x) (if (< y 28179592427282880000000000000000000000) (/ (pow (/ x (+ y x)) x) x) (if (< y 23473874151669980000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000) (log (exp (/ (pow (/ x (+ y x)) x) x))) (/ (exp (/ -1 y)) x)))))

  (/ (exp (* x (log (/ x (+ x y))))) x))