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

Percentage Accurate: 78.5% → 99.7%

Time: 9.0s

Alternatives: 5

Speedup: 12.3×

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: 99.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{+25} \lor \neg \left(x \leq 0.1\right):\\ \;\;\;\;\frac{e^{-y}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(e^{x}\right)}^{\log \left(\frac{x}{x + y}\right)}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -1e+25) (not (<= x 0.1)))
   (/ (exp (- y)) x)
   (/ (pow (exp x) (log (/ x (+ x y)))) x)))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -1e+25) || !(x <= 0.1)) {
		tmp = exp(-y) / x;
	} else {
		tmp = pow(exp(x), log((x / (x + y)))) / 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 <= (-1d+25)) .or. (.not. (x <= 0.1d0))) then
        tmp = exp(-y) / x
    else
        tmp = (exp(x) ** log((x / (x + y)))) / x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((x <= -1e+25) || !(x <= 0.1)) {
		tmp = Math.exp(-y) / x;
	} else {
		tmp = Math.pow(Math.exp(x), Math.log((x / (x + y)))) / x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -1e+25) or not (x <= 0.1):
		tmp = math.exp(-y) / x
	else:
		tmp = math.pow(math.exp(x), math.log((x / (x + y)))) / x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -1e+25) || !(x <= 0.1))
		tmp = Float64(exp(Float64(-y)) / x);
	else
		tmp = Float64((exp(x) ^ log(Float64(x / Float64(x + y)))) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((x <= -1e+25) || ~((x <= 0.1)))
		tmp = exp(-y) / x;
	else
		tmp = (exp(x) ^ log((x / (x + y)))) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -1e+25], N[Not[LessEqual[x, 0.1]], $MachinePrecision]], N[(N[Exp[(-y)], $MachinePrecision] / x), $MachinePrecision], N[(N[Power[N[Exp[x], $MachinePrecision], N[Log[N[(x / N[(x + y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.00000000000000009e25 or 0.10000000000000001 < x

   1. Initial program 72.1%

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

      1. *-commutative 72.1%

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

      2. exp-to-pow 72.1%

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

   3. Simplified 72.1%

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

   5. Taylor expanded in x around inf 100.0%

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

      1. mul-1-neg 100.0%

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

   7. Simplified 100.0%

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

   if -1.00000000000000009e25 < x < 0.10000000000000001

   1. Initial program 89.9%

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

      1. exp-prod 99.6%

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

   3. Simplified 99.6%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{+25} \lor \neg \left(x \leq 0.1\right):\\ \;\;\;\;\frac{e^{-y}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(e^{x}\right)}^{\log \left(\frac{x}{x + y}\right)}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.72 \lor \neg \left(x \leq 1.65\right):\\ \;\;\;\;\frac{e^{-y}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -0.72) (not (<= x 1.65))) (/ (exp (- y)) x) (/ 1.0 x)))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -0.72) || !(x <= 1.65)) {
		tmp = exp(-y) / 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 <= (-0.72d0)) .or. (.not. (x <= 1.65d0))) then
        tmp = exp(-y) / 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 <= -0.72) || !(x <= 1.65)) {
		tmp = Math.exp(-y) / x;
	} else {
		tmp = 1.0 / x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -0.72) or not (x <= 1.65):
		tmp = math.exp(-y) / x
	else:
		tmp = 1.0 / x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -0.72) || !(x <= 1.65))
		tmp = Float64(exp(Float64(-y)) / x);
	else
		tmp = Float64(1.0 / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((x <= -0.72) || ~((x <= 1.65)))
		tmp = exp(-y) / x;
	else
		tmp = 1.0 / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -0.72], N[Not[LessEqual[x, 1.65]], $MachinePrecision]], N[(N[Exp[(-y)], $MachinePrecision] / x), $MachinePrecision], N[(1.0 / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -0.71999999999999997 or 1.6499999999999999 < x

   1. Initial program 73.7%

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

      1. *-commutative 73.7%

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

      2. exp-to-pow 73.8%

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

   3. Simplified 73.8%

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

   5. Taylor expanded in x around inf 99.4%

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

      1. mul-1-neg 99.4%

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

   7. Simplified 99.4%

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

   if -0.71999999999999997 < x < 1.6499999999999999

   1. Initial program 89.1%

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

      1. exp-prod 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in x around 0 98.6%

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

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.72 \lor \neg \left(x \leq 1.65\right):\\ \;\;\;\;\frac{e^{-y}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 80.5% accurate, 12.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5.5 \cdot 10^{+26} \lor \neg \left(x \leq 1.6 \cdot 10^{+73}\right):\\ \;\;\;\;\frac{\frac{1}{x}}{y + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -5.5e+26) (not (<= x 1.6e+73)))
   (/ (/ 1.0 x) (+ y 1.0))
   (/ 1.0 x)))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -5.5e+26) || !(x <= 1.6e+73)) {
		tmp = (1.0 / x) / (y + 1.0);
	} 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 <= (-5.5d+26)) .or. (.not. (x <= 1.6d+73))) then
        tmp = (1.0d0 / x) / (y + 1.0d0)
    else
        tmp = 1.0d0 / x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((x <= -5.5e+26) || !(x <= 1.6e+73)) {
		tmp = (1.0 / x) / (y + 1.0);
	} else {
		tmp = 1.0 / x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -5.5e+26) or not (x <= 1.6e+73):
		tmp = (1.0 / x) / (y + 1.0)
	else:
		tmp = 1.0 / x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -5.5e+26) || !(x <= 1.6e+73))
		tmp = Float64(Float64(1.0 / x) / Float64(y + 1.0));
	else
		tmp = Float64(1.0 / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((x <= -5.5e+26) || ~((x <= 1.6e+73)))
		tmp = (1.0 / x) / (y + 1.0);
	else
		tmp = 1.0 / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -5.5e+26], N[Not[LessEqual[x, 1.6e+73]], $MachinePrecision]], N[(N[(1.0 / x), $MachinePrecision] / N[(y + 1.0), $MachinePrecision]), $MachinePrecision], N[(1.0 / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -5.4999999999999997e26 or 1.59999999999999991e73 < x

   1. Initial program 69.1%

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

      1. exp-prod 69.1%

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

   3. Simplified 69.1%

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

      1. clear-num 69.1%

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

      2. inv-pow 69.1%

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

      3. add-exp-log 32.5%

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

      4. add-exp-log 32.5%

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

      5. div-exp 32.5%

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

      6. pow-exp 32.5%

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

      7. add-log-exp 32.5%

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

      8. log-pow 32.5%

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

      9. div-exp 32.5%

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

      10. add-exp-log 69.1%

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

      11. add-exp-log 69.1%

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

   6. Applied egg-rr 69.1%

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

      1. div-inv 69.1%

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

      2. unpow-prod-down 69.1%

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

      3. inv-pow 69.1%

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

      4. pow-flip 69.1%

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

   8. Applied egg-rr 69.1%

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

      1. associate-*l/ 69.1%

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

      2. *-lft-identity 69.1%

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

      3. unpow-1 69.1%

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

      4. associate-/r* 69.1%

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

      5. associate-/l/ 69.1%

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

   10. Simplified 69.1%

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

   11. Taylor expanded in y around 0 66.6%

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

   if -5.4999999999999997e26 < x < 1.59999999999999991e73

   1. Initial program 91.0%

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

      1. exp-prod 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in x around 0 95.8%

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

4. Final simplification 81.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.5 \cdot 10^{+26} \lor \neg \left(x \leq 1.6 \cdot 10^{+73}\right):\\ \;\;\;\;\frac{\frac{1}{x}}{y + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 75.0% accurate, 20.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 6.5 \cdot 10^{+108}:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x \cdot y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 6.5e+108) (/ 1.0 x) (/ 1.0 (* x y))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 6.5e+108) {
		tmp = 1.0 / x;
	} else {
		tmp = 1.0 / (x * y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 6.5d+108) then
        tmp = 1.0d0 / x
    else
        tmp = 1.0d0 / (x * y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 6.5e+108) {
		tmp = 1.0 / x;
	} else {
		tmp = 1.0 / (x * y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 6.5e+108:
		tmp = 1.0 / x
	else:
		tmp = 1.0 / (x * y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 6.5e+108)
		tmp = Float64(1.0 / x);
	else
		tmp = Float64(1.0 / Float64(x * y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 6.5e+108)
		tmp = 1.0 / x;
	else
		tmp = 1.0 / (x * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 6.5e+108], N[(1.0 / x), $MachinePrecision], N[(1.0 / N[(x * y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 6.4999999999999996e108

   1. Initial program 83.6%

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

      1. exp-prod 86.0%

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

   3. Simplified 86.0%

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

   5. Taylor expanded in x around 0 79.1%

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

   if 6.4999999999999996e108 < y

   1. Initial program 55.8%

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

      1. exp-prod 73.8%

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

   3. Simplified 73.8%

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

      1. clear-num 73.8%

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

      2. inv-pow 73.8%

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

      3. add-exp-log 67.1%

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

      4. add-exp-log 67.1%

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

      5. div-exp 67.1%

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

      6. pow-exp 50.7%

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

      7. add-log-exp 50.7%

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

      8. log-pow 50.7%

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

      9. div-exp 50.7%

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

      10. add-exp-log 55.8%

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

      11. add-exp-log 55.8%

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

   6. Applied egg-rr 55.8%

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

      1. div-inv 55.7%

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

      2. unpow-prod-down 55.7%

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

      3. inv-pow 55.7%

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

      4. pow-flip 55.7%

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

   8. Applied egg-rr 55.7%

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

      1. associate-*l/ 55.7%

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

      2. *-lft-identity 55.7%

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

      3. unpow-1 55.7%

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

      4. associate-/r* 55.7%

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

      5. associate-/l/ 55.7%

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

   10. Simplified 55.7%

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

   11. Taylor expanded in y around 0 58.3%

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

   12. Taylor expanded in y around inf 58.3%

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

4. Final simplification 76.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 6.5 \cdot 10^{+108}:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x \cdot y}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 74.6% accurate, 69.7× 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 80.1%

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

   1. exp-prod 84.5%

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

3. Simplified 84.5%

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

5. Taylor expanded in x around 0 73.5%

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

6. Final simplification 73.5%

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

Developer target: 77.5% accurate, 0.6× 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 2024018 
(FPCore (x y)
  :name "Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2, F"
  :precision binary64

  :herbie-target
  (if (< y -3.7311844206647956e+94) (/ (exp (/ -1.0 y)) x) (if (< y 2.817959242728288e+37) (/ (pow (/ x (+ y x)) x) x) (if (< y 2.347387415166998e+178) (log (exp (/ (pow (/ x (+ y x)) x) x))) (/ (exp (/ -1.0 y)) x))))

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