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

Percentage Accurate: 77.6% → 77.6%

Time: 2.6s

Alternatives: 7

Speedup: 1.0×

Specification

\[\mathsf{TRUE}\left(\right)\]

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: 77.6% 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: 77.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \log \left(\frac{x}{x + y}\right)\\ t_1 := \frac{e^{x \cdot t\_0}}{x}\\ \mathbf{if}\;y \leq 2.9 \cdot 10^{+47}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 1.42 \cdot 10^{+131}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (log (/ x (+ x y)))) (t_1 (/ (exp (* x t_0)) x)))
   (if (<= y 2.9e+47) t_1 (if (<= y 1.42e+131) 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 <= 2.9e+47) {
		tmp = t_1;
	} else if (y <= 1.42e+131) {
		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 <= 2.9d+47) then
        tmp = t_1
    else if (y <= 1.42d+131) 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 <= 2.9e+47) {
		tmp = t_1;
	} else if (y <= 1.42e+131) {
		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 <= 2.9e+47:
		tmp = t_1
	elif y <= 1.42e+131:
		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 <= 2.9e+47)
		tmp = t_1;
	elseif (y <= 1.42e+131)
		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 <= 2.9e+47)
		tmp = t_1;
	elseif (y <= 1.42e+131)
		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, 2.9e+47], t$95$1, If[LessEqual[y, 1.42e+131], t$95$0, t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if y < 2.8999999999999998e47 or 1.42e131 < y

   1. Initial program 79.0%

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

   if 2.8999999999999998e47 < y < 1.42e131

   1. Initial program 12.1%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 88.4%

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

Alternative 2: 61.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{x + y}\\ t_1 := x \cdot \log t\_0\\ \mathbf{if}\;y \leq -10500:\\ \;\;\;\;e^{\frac{x}{e^{t\_0}}}\\ \mathbf{elif}\;y \leq 5600:\\ \;\;\;\;\frac{t\_0}{x}\\ \mathbf{elif}\;y \leq 1.15 \cdot 10^{+135}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;e^{t\_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ x (+ x y))) (t_1 (* x (log t_0))))
   (if (<= y -10500.0)
     (exp (/ x (exp t_0)))
     (if (<= y 5600.0) (/ t_0 x) (if (<= y 1.15e+135) t_1 (exp t_1))))))

C

double code(double x, double y) {
	double t_0 = x / (x + y);
	double t_1 = x * log(t_0);
	double tmp;
	if (y <= -10500.0) {
		tmp = exp((x / exp(t_0)));
	} else if (y <= 5600.0) {
		tmp = t_0 / x;
	} else if (y <= 1.15e+135) {
		tmp = t_1;
	} else {
		tmp = exp(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 = x * log(t_0)
    if (y <= (-10500.0d0)) then
        tmp = exp((x / exp(t_0)))
    else if (y <= 5600.0d0) then
        tmp = t_0 / x
    else if (y <= 1.15d+135) then
        tmp = t_1
    else
        tmp = exp(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 = x * Math.log(t_0);
	double tmp;
	if (y <= -10500.0) {
		tmp = Math.exp((x / Math.exp(t_0)));
	} else if (y <= 5600.0) {
		tmp = t_0 / x;
	} else if (y <= 1.15e+135) {
		tmp = t_1;
	} else {
		tmp = Math.exp(t_1);
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = x / (x + y)
	t_1 = x * math.log(t_0)
	tmp = 0
	if y <= -10500.0:
		tmp = math.exp((x / math.exp(t_0)))
	elif y <= 5600.0:
		tmp = t_0 / x
	elif y <= 1.15e+135:
		tmp = t_1
	else:
		tmp = math.exp(t_1)
	return tmp

Julia

function code(x, y)
	t_0 = Float64(x / Float64(x + y))
	t_1 = Float64(x * log(t_0))
	tmp = 0.0
	if (y <= -10500.0)
		tmp = exp(Float64(x / exp(t_0)));
	elseif (y <= 5600.0)
		tmp = Float64(t_0 / x);
	elseif (y <= 1.15e+135)
		tmp = t_1;
	else
		tmp = exp(t_1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = x / (x + y);
	t_1 = x * log(t_0);
	tmp = 0.0;
	if (y <= -10500.0)
		tmp = exp((x / exp(t_0)));
	elseif (y <= 5600.0)
		tmp = t_0 / x;
	elseif (y <= 1.15e+135)
		tmp = t_1;
	else
		tmp = exp(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[(x * N[Log[t$95$0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -10500.0], N[Exp[N[(x / N[Exp[t$95$0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[y, 5600.0], N[(t$95$0 / x), $MachinePrecision], If[LessEqual[y, 1.15e+135], t$95$1, N[Exp[t$95$1], $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if y < -10500

   1. Initial program 44.8%

      \[\frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x} \]
   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 7.0%

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

   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 -10500 < y < 5600

   1. Initial program 98.4%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 84.2%

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

   if 5600 < y < 1.1500000000000001e135

   1. Initial program 31.8%

      \[\frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x} \]
   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 70.6%

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

   if 1.1500000000000001e135 < y

   1. Initial program 54.8%

      \[\frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x} \]
   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 40.2%

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

Alternative 3: 58.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{x + y}\\ t_1 := x \cdot \log t\_0\\ \mathbf{if}\;y \leq 5600:\\ \;\;\;\;\frac{t\_0}{x}\\ \mathbf{elif}\;y \leq 1.15 \cdot 10^{+135}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;e^{t\_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ x (+ x y))) (t_1 (* x (log t_0))))
   (if (<= y 5600.0) (/ t_0 x) (if (<= y 1.15e+135) t_1 (exp t_1)))))

C

double code(double x, double y) {
	double t_0 = x / (x + y);
	double t_1 = x * log(t_0);
	double tmp;
	if (y <= 5600.0) {
		tmp = t_0 / x;
	} else if (y <= 1.15e+135) {
		tmp = t_1;
	} else {
		tmp = exp(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 = x * log(t_0)
    if (y <= 5600.0d0) then
        tmp = t_0 / x
    else if (y <= 1.15d+135) then
        tmp = t_1
    else
        tmp = exp(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 = x * Math.log(t_0);
	double tmp;
	if (y <= 5600.0) {
		tmp = t_0 / x;
	} else if (y <= 1.15e+135) {
		tmp = t_1;
	} else {
		tmp = Math.exp(t_1);
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = x / (x + y)
	t_1 = x * math.log(t_0)
	tmp = 0
	if y <= 5600.0:
		tmp = t_0 / x
	elif y <= 1.15e+135:
		tmp = t_1
	else:
		tmp = math.exp(t_1)
	return tmp

Julia

function code(x, y)
	t_0 = Float64(x / Float64(x + y))
	t_1 = Float64(x * log(t_0))
	tmp = 0.0
	if (y <= 5600.0)
		tmp = Float64(t_0 / x);
	elseif (y <= 1.15e+135)
		tmp = t_1;
	else
		tmp = exp(t_1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = x / (x + y);
	t_1 = x * log(t_0);
	tmp = 0.0;
	if (y <= 5600.0)
		tmp = t_0 / x;
	elseif (y <= 1.15e+135)
		tmp = t_1;
	else
		tmp = exp(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[(x * N[Log[t$95$0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, 5600.0], N[(t$95$0 / x), $MachinePrecision], If[LessEqual[y, 1.15e+135], t$95$1, N[Exp[t$95$1], $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if y < 5600

   1. Initial program 84.7%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 63.4%

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

   if 5600 < y < 1.1500000000000001e135

   1. Initial program 31.8%

      \[\frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x} \]
   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 70.6%

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

   if 1.1500000000000001e135 < y

   1. Initial program 54.8%

      \[\frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x} \]
   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 40.2%

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

Alternative 4: 56.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{x + y}\\ \mathbf{if}\;y \leq 5600:\\ \;\;\;\;\frac{t\_0}{x}\\ \mathbf{else}:\\ \;\;\;\;\log t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ x (+ x y)))) (if (<= y 5600.0) (/ t_0 x) (log t_0))))

C

double code(double x, double y) {
	double t_0 = x / (x + y);
	double tmp;
	if (y <= 5600.0) {
		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 <= 5600.0d0) 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 <= 5600.0) {
		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 <= 5600.0:
		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 <= 5600.0)
		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 <= 5600.0)
		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, 5600.0], N[(t$95$0 / x), $MachinePrecision], N[Log[t$95$0], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < 5600

   1. Initial program 84.7%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 63.4%

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

   if 5600 < y

   1. Initial program 44.0%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 43.2%

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

Alternative 5: 50.0% 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 74.5%

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

3. Taylor expanded in x around 0

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

5. Applied rewrites 48.9%

   \[\leadsto \frac{\color{blue}{\frac{x}{x + y}}}{x} \]
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 74.5%

   \[\frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x} \]
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 13.2%

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

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 3.4%

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

9. Taylor expanded in x around 0

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

11. Applied rewrites 3.8%

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

12. Taylor expanded in x around 0

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

14. Applied rewrites 4.4%

    \[\leadsto \frac{x}{\frac{x}{x + y}} \]
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 74.5%

   \[\frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x} \]
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 10.9%

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

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 4.4%

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

Developer Target 1: 77.9% 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 2024321 
(FPCore (x y)
  :name "Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2, F"
  :precision binary64
  :pre (TRUE)

  :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))