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

Percentage Accurate: 78.5% → 78.5%

Time: 2.7s

Alternatives: 7

Speedup: 1.0×

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: 78.5% 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: 78.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \log \left(\frac{x}{x + y}\right)\\ t_1 := \frac{e^{x \cdot t\_0}}{x}\\ \mathbf{if}\;y \leq 1700000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 1.3 \cdot 10^{+209}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (log (/ x (+ x y)))) (t_1 (/ (exp (* x t_0)) x)))
   (if (<= y 1700000.0) t_1 (if (<= y 1.3e+209) t_0 t_1))))

C

double code(double x, double y) {
	double t_0 = log((x / (x + y)));
	double t_1 = exp((x * t_0)) / x;
	double tmp;
	if (y <= 1700000.0) {
		tmp = t_1;
	} else if (y <= 1.3e+209) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	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 = log((x / (x + y)))
    t_1 = exp((x * t_0)) / x
    if (y <= 1700000.0d0) then
        tmp = t_1
    else if (y <= 1.3d+209) then
        tmp = t_0
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = Math.log((x / (x + y)));
	double t_1 = Math.exp((x * t_0)) / x;
	double tmp;
	if (y <= 1700000.0) {
		tmp = t_1;
	} else if (y <= 1.3e+209) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.log((x / (x + y)))
	t_1 = math.exp((x * t_0)) / x
	tmp = 0
	if y <= 1700000.0:
		tmp = t_1
	elif y <= 1.3e+209:
		tmp = t_0
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y)
	t_0 = log(Float64(x / Float64(x + y)))
	t_1 = Float64(exp(Float64(x * t_0)) / x)
	tmp = 0.0
	if (y <= 1700000.0)
		tmp = t_1;
	elseif (y <= 1.3e+209)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = log((x / (x + y)));
	t_1 = exp((x * t_0)) / x;
	tmp = 0.0;
	if (y <= 1700000.0)
		tmp = t_1;
	elseif (y <= 1.3e+209)
		tmp = t_0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[Log[N[(x / N[(x + y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[Exp[N[(x * t$95$0), $MachinePrecision]], $MachinePrecision] / x), $MachinePrecision]}, If[LessEqual[y, 1700000.0], t$95$1, If[LessEqual[y, 1.3e+209], t$95$0, t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if y < 1.7e6 or 1.3e209 < y

   1. Initial program 83.9%

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

   if 1.7e6 < y < 1.3e209

   1. Initial program 31.6%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1 + x \cdot \left(\log x + \left(\log \left(\frac{1}{y}\right) + x \cdot \left(\left(\frac{1}{2} \cdot {\left(\log x + \log \left(\frac{1}{y}\right)\right)}^{2} + x \cdot \left(-1 \cdot \frac{\log x + \log \left(\frac{1}{y}\right)}{y} + \left(\frac{1}{6} \cdot {\left(\log x + \log \left(\frac{1}{y}\right)\right)}^{3} + \frac{1}{2} \cdot \frac{1}{{y}^{2}}\right)\right)\right) - \frac{1}{y}\right)\right)\right)}{x}} \]

   5. Applied rewrites 66.1%

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

Alternative 2: 63.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{x + y}\\ t_1 := \log t\_0\\ \mathbf{if}\;y \leq -850:\\ \;\;\;\;e^{\frac{x}{e^{t\_0}}}\\ \mathbf{elif}\;y \leq 70:\\ \;\;\;\;\frac{t\_0}{x}\\ \mathbf{elif}\;y \leq 4.1 \cdot 10^{+215}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;e^{x \cdot t\_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ x (+ x y))) (t_1 (log t_0)))
   (if (<= y -850.0)
     (exp (/ x (exp t_0)))
     (if (<= y 70.0) (/ t_0 x) (if (<= y 4.1e+215) t_1 (exp (* x t_1)))))))

C

double code(double x, double y) {
	double t_0 = x / (x + y);
	double t_1 = log(t_0);
	double tmp;
	if (y <= -850.0) {
		tmp = exp((x / exp(t_0)));
	} else if (y <= 70.0) {
		tmp = t_0 / x;
	} else if (y <= 4.1e+215) {
		tmp = t_1;
	} else {
		tmp = exp((x * t_1));
	}
	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 = x / (x + y)
    t_1 = log(t_0)
    if (y <= (-850.0d0)) then
        tmp = exp((x / exp(t_0)))
    else if (y <= 70.0d0) then
        tmp = t_0 / x
    else if (y <= 4.1d+215) then
        tmp = t_1
    else
        tmp = exp((x * t_1))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = x / (x + y);
	double t_1 = Math.log(t_0);
	double tmp;
	if (y <= -850.0) {
		tmp = Math.exp((x / Math.exp(t_0)));
	} else if (y <= 70.0) {
		tmp = t_0 / x;
	} else if (y <= 4.1e+215) {
		tmp = t_1;
	} else {
		tmp = Math.exp((x * t_1));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = x / (x + y)
	t_1 = math.log(t_0)
	tmp = 0
	if y <= -850.0:
		tmp = math.exp((x / math.exp(t_0)))
	elif y <= 70.0:
		tmp = t_0 / x
	elif y <= 4.1e+215:
		tmp = t_1
	else:
		tmp = math.exp((x * t_1))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(x / Float64(x + y))
	t_1 = log(t_0)
	tmp = 0.0
	if (y <= -850.0)
		tmp = exp(Float64(x / exp(t_0)));
	elseif (y <= 70.0)
		tmp = Float64(t_0 / x);
	elseif (y <= 4.1e+215)
		tmp = t_1;
	else
		tmp = exp(Float64(x * t_1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = x / (x + y);
	t_1 = log(t_0);
	tmp = 0.0;
	if (y <= -850.0)
		tmp = exp((x / exp(t_0)));
	elseif (y <= 70.0)
		tmp = t_0 / x;
	elseif (y <= 4.1e+215)
		tmp = t_1;
	else
		tmp = exp((x * t_1));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(x / N[(x + y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Log[t$95$0], $MachinePrecision]}, If[LessEqual[y, -850.0], N[Exp[N[(x / N[Exp[t$95$0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[y, 70.0], N[(t$95$0 / x), $MachinePrecision], If[LessEqual[y, 4.1e+215], t$95$1, N[Exp[N[(x * t$95$1), $MachinePrecision]], $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if y < -850

   1. Initial program 47.1%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 3.0%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 1.3%

      \[\leadsto e^{\frac{x}{x + y}} \]

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 19.4%

       \[\leadsto e^{\frac{x}{e^{\frac{x}{x + y}}}} \]

   if -850 < y < 70

   1. Initial program 98.7%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 81.2%

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

   if 70 < y < 4.1000000000000004e215

   1. Initial program 34.4%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1 + x \cdot \left(\log x + \left(\log \left(\frac{1}{y}\right) + x \cdot \left(\left(\frac{1}{2} \cdot {\left(\log x + \log \left(\frac{1}{y}\right)\right)}^{2} + x \cdot \left(-1 \cdot \frac{\log x + \log \left(\frac{1}{y}\right)}{y} + \left(\frac{1}{6} \cdot {\left(\log x + \log \left(\frac{1}{y}\right)\right)}^{3} + \frac{1}{2} \cdot \frac{1}{{y}^{2}}\right)\right)\right) - \frac{1}{y}\right)\right)\right)}{x}} \]

   5. Applied rewrites 64.2%

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

   if 4.1000000000000004e215 < y

   1. Initial program 62.7%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 51.7%

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

Alternative 3: 59.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{x + y}\\ t_1 := \log t\_0\\ \mathbf{if}\;y \leq 70:\\ \;\;\;\;\frac{t\_0}{x}\\ \mathbf{elif}\;y \leq 4.1 \cdot 10^{+215}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;e^{x \cdot t\_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ x (+ x y))) (t_1 (log t_0)))
   (if (<= y 70.0) (/ t_0 x) (if (<= y 4.1e+215) t_1 (exp (* x t_1))))))

C

double code(double x, double y) {
	double t_0 = x / (x + y);
	double t_1 = log(t_0);
	double tmp;
	if (y <= 70.0) {
		tmp = t_0 / x;
	} else if (y <= 4.1e+215) {
		tmp = t_1;
	} else {
		tmp = exp((x * t_1));
	}
	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 = x / (x + y)
    t_1 = log(t_0)
    if (y <= 70.0d0) then
        tmp = t_0 / x
    else if (y <= 4.1d+215) then
        tmp = t_1
    else
        tmp = exp((x * t_1))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = x / (x + y);
	double t_1 = Math.log(t_0);
	double tmp;
	if (y <= 70.0) {
		tmp = t_0 / x;
	} else if (y <= 4.1e+215) {
		tmp = t_1;
	} else {
		tmp = Math.exp((x * t_1));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = x / (x + y)
	t_1 = math.log(t_0)
	tmp = 0
	if y <= 70.0:
		tmp = t_0 / x
	elif y <= 4.1e+215:
		tmp = t_1
	else:
		tmp = math.exp((x * t_1))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(x / Float64(x + y))
	t_1 = log(t_0)
	tmp = 0.0
	if (y <= 70.0)
		tmp = Float64(t_0 / x);
	elseif (y <= 4.1e+215)
		tmp = t_1;
	else
		tmp = exp(Float64(x * t_1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = x / (x + y);
	t_1 = log(t_0);
	tmp = 0.0;
	if (y <= 70.0)
		tmp = t_0 / x;
	elseif (y <= 4.1e+215)
		tmp = t_1;
	else
		tmp = exp((x * t_1));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(x / N[(x + y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Log[t$95$0], $MachinePrecision]}, If[LessEqual[y, 70.0], N[(t$95$0 / x), $MachinePrecision], If[LessEqual[y, 4.1e+215], t$95$1, N[Exp[N[(x * t$95$1), $MachinePrecision]], $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if y < 70

   1. Initial program 86.4%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 62.4%

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

   if 70 < y < 4.1000000000000004e215

   1. Initial program 34.4%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1 + x \cdot \left(\log x + \left(\log \left(\frac{1}{y}\right) + x \cdot \left(\left(\frac{1}{2} \cdot {\left(\log x + \log \left(\frac{1}{y}\right)\right)}^{2} + x \cdot \left(-1 \cdot \frac{\log x + \log \left(\frac{1}{y}\right)}{y} + \left(\frac{1}{6} \cdot {\left(\log x + \log \left(\frac{1}{y}\right)\right)}^{3} + \frac{1}{2} \cdot \frac{1}{{y}^{2}}\right)\right)\right) - \frac{1}{y}\right)\right)\right)}{x}} \]

   5. Applied rewrites 64.2%

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

   if 4.1000000000000004e215 < y

   1. Initial program 62.7%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 51.7%

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

Alternative 4: 57.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{x + y}\\ \mathbf{if}\;y \leq 70:\\ \;\;\;\;\frac{t\_0}{x}\\ \mathbf{else}:\\ \;\;\;\;\log t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ x (+ x y)))) (if (<= y 70.0) (/ t_0 x) (log t_0))))

C

double code(double x, double y) {
	double t_0 = x / (x + y);
	double tmp;
	if (y <= 70.0) {
		tmp = t_0 / x;
	} else {
		tmp = log(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 = x / (x + y)
    if (y <= 70.0d0) then
        tmp = t_0 / x
    else
        tmp = log(t_0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = x / (x + y);
	double tmp;
	if (y <= 70.0) {
		tmp = t_0 / x;
	} else {
		tmp = Math.log(t_0);
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = x / (x + y)
	tmp = 0
	if y <= 70.0:
		tmp = t_0 / x
	else:
		tmp = math.log(t_0)
	return tmp

Julia

function code(x, y)
	t_0 = Float64(x / Float64(x + y))
	tmp = 0.0
	if (y <= 70.0)
		tmp = Float64(t_0 / x);
	else
		tmp = log(t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = x / (x + y);
	tmp = 0.0;
	if (y <= 70.0)
		tmp = t_0 / x;
	else
		tmp = log(t_0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(x / N[(x + y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, 70.0], N[(t$95$0 / x), $MachinePrecision], N[Log[t$95$0], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < 70

   1. Initial program 86.4%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 62.4%

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

   if 70 < y

   1. Initial program 44.4%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1 + x \cdot \left(\log x + \left(\log \left(\frac{1}{y}\right) + x \cdot \left(\left(\frac{1}{2} \cdot {\left(\log x + \log \left(\frac{1}{y}\right)\right)}^{2} + x \cdot \left(-1 \cdot \frac{\log x + \log \left(\frac{1}{y}\right)}{y} + \left(\frac{1}{6} \cdot {\left(\log x + \log \left(\frac{1}{y}\right)\right)}^{3} + \frac{1}{2} \cdot \frac{1}{{y}^{2}}\right)\right)\right) - \frac{1}{y}\right)\right)\right)}{x}} \]

   5. Applied rewrites 46.1%

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

Alternative 5: 50.9% accurate, 8.9× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{x}{x + y}}{x} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.0%

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

3. Taylor expanded in x around 0

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

5. Applied rewrites 51.1%

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

Alternative 6: 4.5% accurate, 8.9× speedup

Math

\[\begin{array}{l} \\ \frac{x}{\frac{x}{x + y}} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.0%

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

3. Taylor expanded in x around 0

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

5. Applied rewrites 11.7%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 3.1%

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

9. Taylor expanded in x around 0

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

11. Applied rewrites 4.0%

    \[\leadsto \frac{x}{e^{\frac{\frac{x}{x + y}}{x}}} \]

12. Taylor expanded in x around 0

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

14. Applied rewrites 4.5%

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

Alternative 7: 4.5% accurate, 57.8× speedup

Math

\[\begin{array}{l} \\ x + y \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.0%

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

3. Taylor expanded in x around 0

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

5. Applied rewrites 8.3%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 4.5%

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

Developer Target 1: 77.8% 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 2024313 
(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))