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

Percentage Accurate: 77.8% → 99.4%

Time: 7.8s

Alternatives: 6

Speedup: 18.8×

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: 77.8% 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: 99.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3 \cdot 10^{+89} \lor \neg \left(x \leq 0.24\right):\\ \;\;\;\;\frac{e^{-y}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(e^{x}\right)}^{\log \left(\frac{x}{x + y}\right)}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -3e+89) (not (<= x 0.24)))
   (/ (exp (- y)) x)
   (/ (pow (exp x) (log (/ x (+ x y)))) x)))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -3e+89) || !(x <= 0.24)) {
		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 <= (-3d+89)) .or. (.not. (x <= 0.24d0))) 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 <= -3e+89) || !(x <= 0.24)) {
		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 <= -3e+89) or not (x <= 0.24):
		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 <= -3e+89) || !(x <= 0.24))
		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 <= -3e+89) || ~((x <= 0.24)))
		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, -3e+89], N[Not[LessEqual[x, 0.24]], $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 < -3.00000000000000013e89 or 0.23999999999999999 < x

   1. Initial program 64.1%

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

      1. *-commutative 64.1%

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

      2. exp-to-pow 64.1%

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

   3. Simplified 64.1%

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

   4. Taylor expanded in x around inf 100.0%

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

      1. mul-1-neg 100.0%

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

   6. Simplified 100.0%

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

   if -3.00000000000000013e89 < x < 0.23999999999999999

   1. Initial program 86.7%

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

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3 \cdot 10^{+89} \lor \neg \left(x \leq 0.24\right):\\ \;\;\;\;\frac{e^{-y}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(e^{x}\right)}^{\log \left(\frac{x}{x + y}\right)}}{x}\\ \end{array} \]

Alternative 2: 99.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -500000000 \lor \neg \left(x \leq 0.02\right):\\ \;\;\;\;\frac{e^{-y}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -500000000.0) (not (<= x 0.02))) (/ (exp (- y)) x) (/ 1.0 x)))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -500000000.0) || !(x <= 0.02)) {
		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 <= (-500000000.0d0)) .or. (.not. (x <= 0.02d0))) 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 <= -500000000.0) || !(x <= 0.02)) {
		tmp = Math.exp(-y) / x;
	} else {
		tmp = 1.0 / x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -500000000.0) or not (x <= 0.02):
		tmp = math.exp(-y) / x
	else:
		tmp = 1.0 / x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -500000000.0) || !(x <= 0.02))
		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 <= -500000000.0) || ~((x <= 0.02)))
		tmp = exp(-y) / x;
	else
		tmp = 1.0 / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -500000000.0], N[Not[LessEqual[x, 0.02]], $MachinePrecision]], N[(N[Exp[(-y)], $MachinePrecision] / x), $MachinePrecision], N[(1.0 / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -5e8 or 0.0200000000000000004 < x

   1. Initial program 68.6%

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

      1. *-commutative 68.6%

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

      2. exp-to-pow 68.6%

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

   3. Simplified 68.6%

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

   4. Taylor expanded in x around inf 100.0%

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

      1. mul-1-neg 100.0%

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

   6. Simplified 100.0%

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

   if -5e8 < x < 0.0200000000000000004

   1. Initial program 84.6%

      \[\frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x} \]
   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. Taylor expanded in x around 0 98.0%

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

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -500000000 \lor \neg \left(x \leq 0.02\right):\\ \;\;\;\;\frac{e^{-y}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x}\\ \end{array} \]

Alternative 3: 83.0% accurate, 12.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\frac{x - x \cdot y}{x}}{x}\\ \mathbf{if}\;x \leq -500000000:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 0.19:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{elif}\;x \leq 5 \cdot 10^{+240} \lor \neg \left(x \leq 6.4 \cdot 10^{+299}\right):\\ \;\;\;\;\frac{1}{x + x \cdot y}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (/ (- x (* x y)) x) x)))
   (if (<= x -500000000.0)
     t_0
     (if (<= x 0.19)
       (/ 1.0 x)
       (if (or (<= x 5e+240) (not (<= x 6.4e+299)))
         (/ 1.0 (+ x (* x y)))
         t_0)))))

C

double code(double x, double y) {
	double t_0 = ((x - (x * y)) / x) / x;
	double tmp;
	if (x <= -500000000.0) {
		tmp = t_0;
	} else if (x <= 0.19) {
		tmp = 1.0 / x;
	} else if ((x <= 5e+240) || !(x <= 6.4e+299)) {
		tmp = 1.0 / (x + (x * y));
	} 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 = ((x - (x * y)) / x) / x
    if (x <= (-500000000.0d0)) then
        tmp = t_0
    else if (x <= 0.19d0) then
        tmp = 1.0d0 / x
    else if ((x <= 5d+240) .or. (.not. (x <= 6.4d+299))) then
        tmp = 1.0d0 / (x + (x * y))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = ((x - (x * y)) / x) / x;
	double tmp;
	if (x <= -500000000.0) {
		tmp = t_0;
	} else if (x <= 0.19) {
		tmp = 1.0 / x;
	} else if ((x <= 5e+240) || !(x <= 6.4e+299)) {
		tmp = 1.0 / (x + (x * y));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = ((x - (x * y)) / x) / x
	tmp = 0
	if x <= -500000000.0:
		tmp = t_0
	elif x <= 0.19:
		tmp = 1.0 / x
	elif (x <= 5e+240) or not (x <= 6.4e+299):
		tmp = 1.0 / (x + (x * y))
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(Float64(x - Float64(x * y)) / x) / x)
	tmp = 0.0
	if (x <= -500000000.0)
		tmp = t_0;
	elseif (x <= 0.19)
		tmp = Float64(1.0 / x);
	elseif ((x <= 5e+240) || !(x <= 6.4e+299))
		tmp = Float64(1.0 / Float64(x + Float64(x * y)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = ((x - (x * y)) / x) / x;
	tmp = 0.0;
	if (x <= -500000000.0)
		tmp = t_0;
	elseif (x <= 0.19)
		tmp = 1.0 / x;
	elseif ((x <= 5e+240) || ~((x <= 6.4e+299)))
		tmp = 1.0 / (x + (x * y));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[(x - N[(x * y), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] / x), $MachinePrecision]}, If[LessEqual[x, -500000000.0], t$95$0, If[LessEqual[x, 0.19], N[(1.0 / x), $MachinePrecision], If[Or[LessEqual[x, 5e+240], N[Not[LessEqual[x, 6.4e+299]], $MachinePrecision]], N[(1.0 / N[(x + N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if x < -5e8 or 5.0000000000000003e240 < x < 6.3999999999999997e299

   1. Initial program 67.4%

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

      1. *-commutative 67.4%

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

      2. exp-to-pow 67.4%

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

   3. Simplified 67.4%

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

   4. Taylor expanded in x around inf 100.0%

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

      1. mul-1-neg 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in y around 0 53.7%

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

      1. mul-1-neg 53.7%

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

      2. sub-neg 53.7%

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

   9. Simplified 53.7%

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

       1. frac-sub 32.0%

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

       2. associate-/r* 72.6%

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

       3. *-un-lft-identity 72.6%

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

   11. Applied egg-rr 72.6%

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

   if -5e8 < x < 0.19

   1. Initial program 84.6%

      \[\frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x} \]
   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. Taylor expanded in x around 0 98.0%

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

   if 0.19 < x < 5.0000000000000003e240 or 6.3999999999999997e299 < x

   1. Initial program 70.3%

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

      1. *-commutative 70.3%

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

      2. exp-to-pow 70.3%

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

   3. Simplified 70.3%

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

   4. Taylor expanded in x around inf 99.9%

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

      1. mul-1-neg 99.9%

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

   6. Simplified 99.9%

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

   7. Taylor expanded in y around 0 52.3%

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

      1. mul-1-neg 52.3%

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

      2. sub-neg 52.3%

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

   9. Simplified 52.3%

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

       1. sub-div 52.2%

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

       2. clear-num 52.2%

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

   11. Applied egg-rr 52.2%

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

   12. Taylor expanded in y around 0 64.3%

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

4. Final simplification 81.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -500000000:\\ \;\;\;\;\frac{\frac{x - x \cdot y}{x}}{x}\\ \mathbf{elif}\;x \leq 0.19:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{elif}\;x \leq 5 \cdot 10^{+240} \lor \neg \left(x \leq 6.4 \cdot 10^{+299}\right):\\ \;\;\;\;\frac{1}{x + x \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x - x \cdot y}{x}}{x}\\ \end{array} \]

Alternative 4: 82.9% accurate, 12.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -500000000:\\ \;\;\;\;\frac{\left(1 + y \cdot \left(y \cdot 0.5\right)\right) - y}{x}\\ \mathbf{elif}\;x \leq 0.185:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{elif}\;x \leq 7 \cdot 10^{+240} \lor \neg \left(x \leq 2.35 \cdot 10^{+300}\right):\\ \;\;\;\;\frac{1}{x + x \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x - x \cdot y}{x}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -500000000.0)
   (/ (- (+ 1.0 (* y (* y 0.5))) y) x)
   (if (<= x 0.185)
     (/ 1.0 x)
     (if (or (<= x 7e+240) (not (<= x 2.35e+300)))
       (/ 1.0 (+ x (* x y)))
       (/ (/ (- x (* x y)) x) x)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -500000000.0) {
		tmp = ((1.0 + (y * (y * 0.5))) - y) / x;
	} else if (x <= 0.185) {
		tmp = 1.0 / x;
	} else if ((x <= 7e+240) || !(x <= 2.35e+300)) {
		tmp = 1.0 / (x + (x * y));
	} else {
		tmp = ((x - (x * y)) / x) / 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 <= (-500000000.0d0)) then
        tmp = ((1.0d0 + (y * (y * 0.5d0))) - y) / x
    else if (x <= 0.185d0) then
        tmp = 1.0d0 / x
    else if ((x <= 7d+240) .or. (.not. (x <= 2.35d+300))) then
        tmp = 1.0d0 / (x + (x * y))
    else
        tmp = ((x - (x * y)) / x) / x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -500000000.0) {
		tmp = ((1.0 + (y * (y * 0.5))) - y) / x;
	} else if (x <= 0.185) {
		tmp = 1.0 / x;
	} else if ((x <= 7e+240) || !(x <= 2.35e+300)) {
		tmp = 1.0 / (x + (x * y));
	} else {
		tmp = ((x - (x * y)) / x) / x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -500000000.0:
		tmp = ((1.0 + (y * (y * 0.5))) - y) / x
	elif x <= 0.185:
		tmp = 1.0 / x
	elif (x <= 7e+240) or not (x <= 2.35e+300):
		tmp = 1.0 / (x + (x * y))
	else:
		tmp = ((x - (x * y)) / x) / x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -500000000.0)
		tmp = Float64(Float64(Float64(1.0 + Float64(y * Float64(y * 0.5))) - y) / x);
	elseif (x <= 0.185)
		tmp = Float64(1.0 / x);
	elseif ((x <= 7e+240) || !(x <= 2.35e+300))
		tmp = Float64(1.0 / Float64(x + Float64(x * y)));
	else
		tmp = Float64(Float64(Float64(x - Float64(x * y)) / x) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -500000000.0)
		tmp = ((1.0 + (y * (y * 0.5))) - y) / x;
	elseif (x <= 0.185)
		tmp = 1.0 / x;
	elseif ((x <= 7e+240) || ~((x <= 2.35e+300)))
		tmp = 1.0 / (x + (x * y));
	else
		tmp = ((x - (x * y)) / x) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -500000000.0], N[(N[(N[(1.0 + N[(y * N[(y * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - y), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[x, 0.185], N[(1.0 / x), $MachinePrecision], If[Or[LessEqual[x, 7e+240], N[Not[LessEqual[x, 2.35e+300]], $MachinePrecision]], N[(1.0 / N[(x + N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x - N[(x * y), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] / x), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < -5e8

   1. Initial program 71.1%

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

      1. exp-prod 71.1%

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

   3. Simplified 71.1%

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

   4. Taylor expanded in y around 0 77.4%

      \[\leadsto \frac{\color{blue}{{y}^{2} \cdot \left(0.5 + 0.5 \cdot \frac{1}{x}\right) + \left(1 + \left(-1 \cdot \left({y}^{3} \cdot \left(0.5 \cdot \frac{1}{x} + \left(0.16666666666666666 + 0.3333333333333333 \cdot \frac{1}{{x}^{2}}\right)\right)\right) + -1 \cdot y\right)\right)}}{x} \]
   5. Step-by-step derivation

      1. fma-def 77.4%

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({y}^{2}, 0.5 + 0.5 \cdot \frac{1}{x}, 1 + \left(-1 \cdot \left({y}^{3} \cdot \left(0.5 \cdot \frac{1}{x} + \left(0.16666666666666666 + 0.3333333333333333 \cdot \frac{1}{{x}^{2}}\right)\right)\right) + -1 \cdot y\right)\right)}}{x} \]

      2. unpow2 77.4%

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{y \cdot y}, 0.5 + 0.5 \cdot \frac{1}{x}, 1 + \left(-1 \cdot \left({y}^{3} \cdot \left(0.5 \cdot \frac{1}{x} + \left(0.16666666666666666 + 0.3333333333333333 \cdot \frac{1}{{x}^{2}}\right)\right)\right) + -1 \cdot y\right)\right)}{x} \]

      3. associate-*r/ 77.4%

         \[\leadsto \frac{\mathsf{fma}\left(y \cdot y, 0.5 + \color{blue}{\frac{0.5 \cdot 1}{x}}, 1 + \left(-1 \cdot \left({y}^{3} \cdot \left(0.5 \cdot \frac{1}{x} + \left(0.16666666666666666 + 0.3333333333333333 \cdot \frac{1}{{x}^{2}}\right)\right)\right) + -1 \cdot y\right)\right)}{x} \]

      4. metadata-eval 77.4%

         \[\leadsto \frac{\mathsf{fma}\left(y \cdot y, 0.5 + \frac{\color{blue}{0.5}}{x}, 1 + \left(-1 \cdot \left({y}^{3} \cdot \left(0.5 \cdot \frac{1}{x} + \left(0.16666666666666666 + 0.3333333333333333 \cdot \frac{1}{{x}^{2}}\right)\right)\right) + -1 \cdot y\right)\right)}{x} \]

      5. distribute-lft-out 77.4%

         \[\leadsto \frac{\mathsf{fma}\left(y \cdot y, 0.5 + \frac{0.5}{x}, 1 + \color{blue}{-1 \cdot \left({y}^{3} \cdot \left(0.5 \cdot \frac{1}{x} + \left(0.16666666666666666 + 0.3333333333333333 \cdot \frac{1}{{x}^{2}}\right)\right) + y\right)}\right)}{x} \]

      6. associate-*r/ 77.4%

         \[\leadsto \frac{\mathsf{fma}\left(y \cdot y, 0.5 + \frac{0.5}{x}, 1 + -1 \cdot \left({y}^{3} \cdot \left(\color{blue}{\frac{0.5 \cdot 1}{x}} + \left(0.16666666666666666 + 0.3333333333333333 \cdot \frac{1}{{x}^{2}}\right)\right) + y\right)\right)}{x} \]

      7. metadata-eval 77.4%

         \[\leadsto \frac{\mathsf{fma}\left(y \cdot y, 0.5 + \frac{0.5}{x}, 1 + -1 \cdot \left({y}^{3} \cdot \left(\frac{\color{blue}{0.5}}{x} + \left(0.16666666666666666 + 0.3333333333333333 \cdot \frac{1}{{x}^{2}}\right)\right) + y\right)\right)}{x} \]

      8. associate-*r/ 77.4%

         \[\leadsto \frac{\mathsf{fma}\left(y \cdot y, 0.5 + \frac{0.5}{x}, 1 + -1 \cdot \left({y}^{3} \cdot \left(\frac{0.5}{x} + \left(0.16666666666666666 + \color{blue}{\frac{0.3333333333333333 \cdot 1}{{x}^{2}}}\right)\right) + y\right)\right)}{x} \]

      9. metadata-eval 77.4%

         \[\leadsto \frac{\mathsf{fma}\left(y \cdot y, 0.5 + \frac{0.5}{x}, 1 + -1 \cdot \left({y}^{3} \cdot \left(\frac{0.5}{x} + \left(0.16666666666666666 + \frac{\color{blue}{0.3333333333333333}}{{x}^{2}}\right)\right) + y\right)\right)}{x} \]

      10. unpow2 77.4%

          \[\leadsto \frac{\mathsf{fma}\left(y \cdot y, 0.5 + \frac{0.5}{x}, 1 + -1 \cdot \left({y}^{3} \cdot \left(\frac{0.5}{x} + \left(0.16666666666666666 + \frac{0.3333333333333333}{\color{blue}{x \cdot x}}\right)\right) + y\right)\right)}{x} \]

   6. Simplified 77.4%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(y \cdot y, 0.5 + \frac{0.5}{x}, 1 + -1 \cdot \left({y}^{3} \cdot \left(\frac{0.5}{x} + \left(0.16666666666666666 + \frac{0.3333333333333333}{x \cdot x}\right)\right) + y\right)\right)}}{x} \]

   7. Taylor expanded in x around inf 77.4%

      \[\leadsto \frac{\color{blue}{1 + \left(0.5 \cdot {y}^{2} + -1 \cdot \left(y + 0.16666666666666666 \cdot {y}^{3}\right)\right)}}{x} \]
   8. Step-by-step derivation

      1. associate-+r+ 77.4%

         \[\leadsto \frac{\color{blue}{\left(1 + 0.5 \cdot {y}^{2}\right) + -1 \cdot \left(y + 0.16666666666666666 \cdot {y}^{3}\right)}}{x} \]

      2. mul-1-neg 77.4%

         \[\leadsto \frac{\left(1 + 0.5 \cdot {y}^{2}\right) + \color{blue}{\left(-\left(y + 0.16666666666666666 \cdot {y}^{3}\right)\right)}}{x} \]

      3. unsub-neg 77.4%

         \[\leadsto \frac{\color{blue}{\left(1 + 0.5 \cdot {y}^{2}\right) - \left(y + 0.16666666666666666 \cdot {y}^{3}\right)}}{x} \]

      4. unpow2 77.4%

         \[\leadsto \frac{\left(1 + 0.5 \cdot \color{blue}{\left(y \cdot y\right)}\right) - \left(y + 0.16666666666666666 \cdot {y}^{3}\right)}{x} \]

      5. associate-*r* 77.4%

         \[\leadsto \frac{\left(1 + \color{blue}{\left(0.5 \cdot y\right) \cdot y}\right) - \left(y + 0.16666666666666666 \cdot {y}^{3}\right)}{x} \]

      6. +-commutative 77.4%

         \[\leadsto \frac{\left(1 + \left(0.5 \cdot y\right) \cdot y\right) - \color{blue}{\left(0.16666666666666666 \cdot {y}^{3} + y\right)}}{x} \]

      7. *-commutative 77.4%

         \[\leadsto \frac{\left(1 + \left(0.5 \cdot y\right) \cdot y\right) - \left(\color{blue}{{y}^{3} \cdot 0.16666666666666666} + y\right)}{x} \]

      8. fma-def 77.4%

         \[\leadsto \frac{\left(1 + \left(0.5 \cdot y\right) \cdot y\right) - \color{blue}{\mathsf{fma}\left({y}^{3}, 0.16666666666666666, y\right)}}{x} \]

   9. Simplified 77.4%

      \[\leadsto \frac{\color{blue}{\left(1 + \left(0.5 \cdot y\right) \cdot y\right) - \mathsf{fma}\left({y}^{3}, 0.16666666666666666, y\right)}}{x} \]

   10. Taylor expanded in y around 0 73.8%

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

   if -5e8 < x < 0.185

   1. Initial program 84.6%

      \[\frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x} \]
   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. Taylor expanded in x around 0 98.0%

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

   if 0.185 < x < 7.00000000000000065e240 or 2.35e300 < x

   1. Initial program 70.3%

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

      1. *-commutative 70.3%

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

      2. exp-to-pow 70.3%

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

   3. Simplified 70.3%

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

   4. Taylor expanded in x around inf 99.9%

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

      1. mul-1-neg 99.9%

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

   6. Simplified 99.9%

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

   7. Taylor expanded in y around 0 52.3%

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

      1. mul-1-neg 52.3%

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

      2. sub-neg 52.3%

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

   9. Simplified 52.3%

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

       1. sub-div 52.2%

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

       2. clear-num 52.2%

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

   11. Applied egg-rr 52.2%

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

   12. Taylor expanded in y around 0 64.3%

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

   if 7.00000000000000065e240 < x < 2.35e300

   1. Initial program 46.6%

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

      1. *-commutative 46.6%

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

      2. exp-to-pow 46.6%

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

   3. Simplified 46.6%

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

   4. Taylor expanded in x around inf 100.0%

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

      1. mul-1-neg 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in y around 0 51.7%

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

      1. mul-1-neg 51.7%

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

      2. sub-neg 51.7%

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

   9. Simplified 51.7%

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

       1. frac-sub 4.6%

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

       2. associate-/r* 89.2%

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

       3. *-un-lft-identity 89.2%

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

   11. Applied egg-rr 89.2%

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

4. Final simplification 82.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -500000000:\\ \;\;\;\;\frac{\left(1 + y \cdot \left(y \cdot 0.5\right)\right) - y}{x}\\ \mathbf{elif}\;x \leq 0.185:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{elif}\;x \leq 7 \cdot 10^{+240} \lor \neg \left(x \leq 2.35 \cdot 10^{+300}\right):\\ \;\;\;\;\frac{1}{x + x \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x - x \cdot y}{x}}{x}\\ \end{array} \]

Alternative 5: 81.5% accurate, 18.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.35 \cdot 10^{+89} \lor \neg \left(x \leq 0.04\right):\\ \;\;\;\;\frac{1}{x + x \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -2.35e+89) (not (<= x 0.04))) (/ 1.0 (+ x (* x y))) (/ 1.0 x)))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -2.35e+89) || !(x <= 0.04)) {
		tmp = 1.0 / (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 ((x <= (-2.35d+89)) .or. (.not. (x <= 0.04d0))) then
        tmp = 1.0d0 / (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 ((x <= -2.35e+89) || !(x <= 0.04)) {
		tmp = 1.0 / (x + (x * y));
	} else {
		tmp = 1.0 / x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -2.35e+89) or not (x <= 0.04):
		tmp = 1.0 / (x + (x * y))
	else:
		tmp = 1.0 / x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -2.35e+89) || !(x <= 0.04))
		tmp = Float64(1.0 / Float64(x + Float64(x * y)));
	else
		tmp = Float64(1.0 / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((x <= -2.35e+89) || ~((x <= 0.04)))
		tmp = 1.0 / (x + (x * y));
	else
		tmp = 1.0 / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -2.35e+89], N[Not[LessEqual[x, 0.04]], $MachinePrecision]], N[(1.0 / N[(x + N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2.35000000000000011e89 or 0.0400000000000000008 < x

   1. Initial program 64.1%

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

      1. *-commutative 64.1%

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

      2. exp-to-pow 64.1%

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

   3. Simplified 64.1%

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

   4. Taylor expanded in x around inf 100.0%

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

      1. mul-1-neg 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in y around 0 52.4%

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

      1. mul-1-neg 52.4%

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

      2. sub-neg 52.4%

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

   9. Simplified 52.4%

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

       1. sub-div 52.4%

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

       2. clear-num 52.4%

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

   11. Applied egg-rr 52.4%

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

   12. Taylor expanded in y around 0 62.6%

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

   if -2.35000000000000011e89 < x < 0.0400000000000000008

   1. Initial program 86.7%

      \[\frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x} \]
   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. Taylor expanded in x around 0 92.2%

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

4. Final simplification 77.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.35 \cdot 10^{+89} \lor \neg \left(x \leq 0.04\right):\\ \;\;\;\;\frac{1}{x + x \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x}\\ \end{array} \]

Alternative 6: 75.4% 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 75.5%

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

   1. exp-prod 81.9%

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

3. Simplified 81.9%

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

4. Taylor expanded in x around 0 71.9%

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

5. Final simplification 71.9%

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

Developer target: 78.3% 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 2023195 
(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))