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

Percentage Accurate: 77.0% → 98.8%

Time: 12.3s

Alternatives: 9

Speedup: 17.4×

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: 77.0% 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: 98.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{+62} \lor \neg \left(x \leq 7.5 \cdot 10^{-31}\right):\\ \;\;\;\;\frac{e^{-y}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(e^{x}\right)}^{\log \left(\frac{x}{x + y}\right)}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -2e+62) (not (<= x 7.5e-31)))
   (/ (exp (- y)) x)
   (/ (pow (exp x) (log (/ x (+ x y)))) x)))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -2e+62) || !(x <= 7.5e-31)) {
		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 <= (-2d+62)) .or. (.not. (x <= 7.5d-31))) 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 <= -2e+62) || !(x <= 7.5e-31)) {
		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 <= -2e+62) or not (x <= 7.5e-31):
		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 <= -2e+62) || !(x <= 7.5e-31))
		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 <= -2e+62) || ~((x <= 7.5e-31)))
		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, -2e+62], N[Not[LessEqual[x, 7.5e-31]], $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 < -2.00000000000000007e62 or 7.49999999999999975e-31 < x

   1. Initial program 69.3%

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

      1. *-commutative 69.3%

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

      2. exp-to-pow 69.3%

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

   3. Simplified 69.3%

      \[\leadsto \color{blue}{\frac{{\left(\frac{x}{x + y}\right)}^{x}}{x}} \]
   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 -2.00000000000000007e62 < x < 7.49999999999999975e-31

   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 99.9%

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

   3. Simplified 99.9%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{+62} \lor \neg \left(x \leq 7.5 \cdot 10^{-31}\right):\\ \;\;\;\;\frac{e^{-y}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(e^{x}\right)}^{\log \left(\frac{x}{x + y}\right)}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 97.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -6.2 \cdot 10^{+61} \lor \neg \left(x \leq 7.5 \cdot 10^{-31}\right):\\ \;\;\;\;\frac{e^{-y}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -6.2e+61) (not (<= x 7.5e-31))) (/ (exp (- y)) x) (/ 1.0 x)))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -6.2e+61) || !(x <= 7.5e-31)) {
		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 <= (-6.2d+61)) .or. (.not. (x <= 7.5d-31))) 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 <= -6.2e+61) || !(x <= 7.5e-31)) {
		tmp = Math.exp(-y) / x;
	} else {
		tmp = 1.0 / x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -6.2e+61) or not (x <= 7.5e-31):
		tmp = math.exp(-y) / x
	else:
		tmp = 1.0 / x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -6.2e+61) || !(x <= 7.5e-31))
		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 <= -6.2e+61) || ~((x <= 7.5e-31)))
		tmp = exp(-y) / x;
	else
		tmp = 1.0 / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -6.2e+61], N[Not[LessEqual[x, 7.5e-31]], $MachinePrecision]], N[(N[Exp[(-y)], $MachinePrecision] / x), $MachinePrecision], N[(1.0 / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -6.1999999999999998e61 or 7.49999999999999975e-31 < x

   1. Initial program 69.3%

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

      1. *-commutative 69.3%

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

      2. exp-to-pow 69.3%

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

   3. Simplified 69.3%

      \[\leadsto \color{blue}{\frac{{\left(\frac{x}{x + y}\right)}^{x}}{x}} \]
   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 -6.1999999999999998e61 < x < 7.49999999999999975e-31

   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 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in x around 0 99.3%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -6.2 \cdot 10^{+61} \lor \neg \left(x \leq 7.5 \cdot 10^{-31}\right):\\ \;\;\;\;\frac{e^{-y}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 84.1% accurate, 6.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1 + y \cdot \left(\frac{0.5 \cdot \left(x \cdot y\right)}{x} + -1\right)}{x}\\ t_1 := x + x \cdot y\\ \mathbf{if}\;x \leq -6.2 \cdot 10^{+61}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 7.5 \cdot 10^{-31}:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{elif}\;x \leq 3.65 \cdot 10^{+223}:\\ \;\;\;\;\frac{1}{x + y \cdot t\_1}\\ \mathbf{elif}\;x \leq 5 \cdot 10^{+268}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{t\_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (+ 1.0 (* y (+ (/ (* 0.5 (* x y)) x) -1.0))) x))
        (t_1 (+ x (* x y))))
   (if (<= x -6.2e+61)
     t_0
     (if (<= x 7.5e-31)
       (/ 1.0 x)
       (if (<= x 3.65e+223)
         (/ 1.0 (+ x (* y t_1)))
         (if (<= x 5e+268) t_0 (/ 1.0 t_1)))))))

C

double code(double x, double y) {
	double t_0 = (1.0 + (y * (((0.5 * (x * y)) / x) + -1.0))) / x;
	double t_1 = x + (x * y);
	double tmp;
	if (x <= -6.2e+61) {
		tmp = t_0;
	} else if (x <= 7.5e-31) {
		tmp = 1.0 / x;
	} else if (x <= 3.65e+223) {
		tmp = 1.0 / (x + (y * t_1));
	} else if (x <= 5e+268) {
		tmp = t_0;
	} else {
		tmp = 1.0 / 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 = (1.0d0 + (y * (((0.5d0 * (x * y)) / x) + (-1.0d0)))) / x
    t_1 = x + (x * y)
    if (x <= (-6.2d+61)) then
        tmp = t_0
    else if (x <= 7.5d-31) then
        tmp = 1.0d0 / x
    else if (x <= 3.65d+223) then
        tmp = 1.0d0 / (x + (y * t_1))
    else if (x <= 5d+268) then
        tmp = t_0
    else
        tmp = 1.0d0 / t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = (1.0 + (y * (((0.5 * (x * y)) / x) + -1.0))) / x;
	double t_1 = x + (x * y);
	double tmp;
	if (x <= -6.2e+61) {
		tmp = t_0;
	} else if (x <= 7.5e-31) {
		tmp = 1.0 / x;
	} else if (x <= 3.65e+223) {
		tmp = 1.0 / (x + (y * t_1));
	} else if (x <= 5e+268) {
		tmp = t_0;
	} else {
		tmp = 1.0 / t_1;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (1.0 + (y * (((0.5 * (x * y)) / x) + -1.0))) / x
	t_1 = x + (x * y)
	tmp = 0
	if x <= -6.2e+61:
		tmp = t_0
	elif x <= 7.5e-31:
		tmp = 1.0 / x
	elif x <= 3.65e+223:
		tmp = 1.0 / (x + (y * t_1))
	elif x <= 5e+268:
		tmp = t_0
	else:
		tmp = 1.0 / t_1
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(1.0 + Float64(y * Float64(Float64(Float64(0.5 * Float64(x * y)) / x) + -1.0))) / x)
	t_1 = Float64(x + Float64(x * y))
	tmp = 0.0
	if (x <= -6.2e+61)
		tmp = t_0;
	elseif (x <= 7.5e-31)
		tmp = Float64(1.0 / x);
	elseif (x <= 3.65e+223)
		tmp = Float64(1.0 / Float64(x + Float64(y * t_1)));
	elseif (x <= 5e+268)
		tmp = t_0;
	else
		tmp = Float64(1.0 / t_1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (1.0 + (y * (((0.5 * (x * y)) / x) + -1.0))) / x;
	t_1 = x + (x * y);
	tmp = 0.0;
	if (x <= -6.2e+61)
		tmp = t_0;
	elseif (x <= 7.5e-31)
		tmp = 1.0 / x;
	elseif (x <= 3.65e+223)
		tmp = 1.0 / (x + (y * t_1));
	elseif (x <= 5e+268)
		tmp = t_0;
	else
		tmp = 1.0 / t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(1.0 + N[(y * N[(N[(N[(0.5 * N[(x * y), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]}, Block[{t$95$1 = N[(x + N[(x * y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -6.2e+61], t$95$0, If[LessEqual[x, 7.5e-31], N[(1.0 / x), $MachinePrecision], If[LessEqual[x, 3.65e+223], N[(1.0 / N[(x + N[(y * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 5e+268], t$95$0, N[(1.0 / t$95$1), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -6.1999999999999998e61 or 3.64999999999999982e223 < x < 5.0000000000000002e268

   1. Initial program 64.2%

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

      1. exp-prod 64.2%

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

   3. Simplified 64.2%

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

   5. Taylor expanded in y around 0 68.2%

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

   6. Taylor expanded in x around 0 77.5%

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

      1. distribute-lft-out 77.5%

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

   8. Simplified 77.5%

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

   9. Taylor expanded in x around inf 77.5%

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

   if -6.1999999999999998e61 < x < 7.49999999999999975e-31

   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 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in x around 0 99.3%

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

   if 7.49999999999999975e-31 < x < 3.64999999999999982e223

   1. Initial program 80.8%

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

      1. *-commutative 80.8%

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

      2. exp-to-pow 80.8%

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

   3. Simplified 80.8%

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

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

   8. Taylor expanded in y around 0 54.2%

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

      1. neg-mul-1 54.2%

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

      2. +-commutative 54.2%

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

      3. sub-neg 54.2%

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

   10. Simplified 54.2%

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

       1. sub-div 54.2%

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

       2. clear-num 54.2%

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

   12. Applied egg-rr 54.2%

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

   13. Taylor expanded in y around 0 78.9%

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

       1. *-commutative 78.9%

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

       2. neg-mul-1 78.9%

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

   15. Simplified 78.9%

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

   if 5.0000000000000002e268 < x

   1. Initial program 47.5%

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

      1. *-commutative 47.5%

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

      2. exp-to-pow 47.5%

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

   3. Simplified 47.5%

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

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

   8. Taylor expanded in y around 0 44.3%

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

      1. neg-mul-1 44.3%

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

      2. +-commutative 44.3%

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

      3. sub-neg 44.3%

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

   10. Simplified 44.3%

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

       1. sub-div 44.3%

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

       2. clear-num 44.3%

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

   12. Applied egg-rr 44.3%

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

   13. Taylor expanded in y around 0 83.6%

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

       1. *-commutative 83.6%

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

   15. Simplified 83.6%

       \[\leadsto \frac{1}{\color{blue}{x + y \cdot x}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 88.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -6.2 \cdot 10^{+61}:\\ \;\;\;\;\frac{1 + y \cdot \left(\frac{0.5 \cdot \left(x \cdot y\right)}{x} + -1\right)}{x}\\ \mathbf{elif}\;x \leq 7.5 \cdot 10^{-31}:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{elif}\;x \leq 3.65 \cdot 10^{+223}:\\ \;\;\;\;\frac{1}{x + y \cdot \left(x + x \cdot y\right)}\\ \mathbf{elif}\;x \leq 5 \cdot 10^{+268}:\\ \;\;\;\;\frac{1 + y \cdot \left(\frac{0.5 \cdot \left(x \cdot y\right)}{x} + -1\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x + x \cdot y}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 85.1% accurate, 8.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -6.2 \cdot 10^{+61}:\\ \;\;\;\;\frac{1 + y \cdot \left(\frac{0.5 \cdot \left(x \cdot y\right)}{x} + -1\right)}{x}\\ \mathbf{elif}\;x \leq 7.5 \cdot 10^{-31}:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x + y \cdot \left(x + y \cdot \left(x + x \cdot y\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -6.2e+61)
   (/ (+ 1.0 (* y (+ (/ (* 0.5 (* x y)) x) -1.0))) x)
   (if (<= x 7.5e-31)
     (/ 1.0 x)
     (/ 1.0 (+ x (* y (+ x (* y (+ x (* x y))))))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -6.2e+61) {
		tmp = (1.0 + (y * (((0.5 * (x * y)) / x) + -1.0))) / x;
	} else if (x <= 7.5e-31) {
		tmp = 1.0 / x;
	} else {
		tmp = 1.0 / (x + (y * (x + (y * (x + (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 (x <= (-6.2d+61)) then
        tmp = (1.0d0 + (y * (((0.5d0 * (x * y)) / x) + (-1.0d0)))) / x
    else if (x <= 7.5d-31) then
        tmp = 1.0d0 / x
    else
        tmp = 1.0d0 / (x + (y * (x + (y * (x + (x * y))))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -6.2e+61) {
		tmp = (1.0 + (y * (((0.5 * (x * y)) / x) + -1.0))) / x;
	} else if (x <= 7.5e-31) {
		tmp = 1.0 / x;
	} else {
		tmp = 1.0 / (x + (y * (x + (y * (x + (x * y))))));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -6.2e+61:
		tmp = (1.0 + (y * (((0.5 * (x * y)) / x) + -1.0))) / x
	elif x <= 7.5e-31:
		tmp = 1.0 / x
	else:
		tmp = 1.0 / (x + (y * (x + (y * (x + (x * y))))))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -6.2e+61)
		tmp = Float64(Float64(1.0 + Float64(y * Float64(Float64(Float64(0.5 * Float64(x * y)) / x) + -1.0))) / x);
	elseif (x <= 7.5e-31)
		tmp = Float64(1.0 / x);
	else
		tmp = Float64(1.0 / Float64(x + Float64(y * Float64(x + Float64(y * Float64(x + Float64(x * y)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -6.2e+61)
		tmp = (1.0 + (y * (((0.5 * (x * y)) / x) + -1.0))) / x;
	elseif (x <= 7.5e-31)
		tmp = 1.0 / x;
	else
		tmp = 1.0 / (x + (y * (x + (y * (x + (x * y))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -6.2e+61], N[(N[(1.0 + N[(y * N[(N[(N[(0.5 * N[(x * y), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[x, 7.5e-31], N[(1.0 / x), $MachinePrecision], N[(1.0 / N[(x + N[(y * N[(x + N[(y * N[(x + N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -6.1999999999999998e61

   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. Add Preprocessing

   5. Taylor expanded in y around 0 72.6%

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

   6. Taylor expanded in x around 0 77.5%

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

      1. distribute-lft-out 77.5%

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

   8. Simplified 77.5%

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

   9. Taylor expanded in x around inf 77.5%

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

   if -6.1999999999999998e61 < x < 7.49999999999999975e-31

   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 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in x around 0 99.3%

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

   if 7.49999999999999975e-31 < x

   1. Initial program 68.0%

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

      1. *-commutative 68.0%

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

      2. exp-to-pow 68.0%

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

   3. Simplified 68.0%

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

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

   8. Taylor expanded in y around 0 49.2%

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

      1. neg-mul-1 49.2%

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

      2. +-commutative 49.2%

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

      3. sub-neg 49.2%

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

   10. Simplified 49.2%

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

       1. sub-div 49.2%

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

       2. clear-num 49.2%

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

   12. Applied egg-rr 49.2%

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

   13. Taylor expanded in y around 0 75.7%

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

       1. *-commutative 75.7%

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

       2. neg-mul-1 75.7%

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

       3. neg-mul-1 75.7%

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

   15. Simplified 75.7%

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

4. Final simplification 86.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -6.2 \cdot 10^{+61}:\\ \;\;\;\;\frac{1 + y \cdot \left(\frac{0.5 \cdot \left(x \cdot y\right)}{x} + -1\right)}{x}\\ \mathbf{elif}\;x \leq 7.5 \cdot 10^{-31}:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x + y \cdot \left(x + y \cdot \left(x + x \cdot y\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 83.5% accurate, 9.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -6.2 \cdot 10^{+61}:\\ \;\;\;\;\frac{1 - y \cdot \left(1 - y \cdot 0.5\right)}{x}\\ \mathbf{elif}\;x \leq 7.5 \cdot 10^{-31}:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x + y \cdot \left(x + x \cdot y\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -6.2e+61)
   (/ (- 1.0 (* y (- 1.0 (* y 0.5)))) x)
   (if (<= x 7.5e-31) (/ 1.0 x) (/ 1.0 (+ x (* y (+ x (* x y))))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -6.2e+61) {
		tmp = (1.0 - (y * (1.0 - (y * 0.5)))) / x;
	} else if (x <= 7.5e-31) {
		tmp = 1.0 / x;
	} else {
		tmp = 1.0 / (x + (y * (x + (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 (x <= (-6.2d+61)) then
        tmp = (1.0d0 - (y * (1.0d0 - (y * 0.5d0)))) / x
    else if (x <= 7.5d-31) then
        tmp = 1.0d0 / x
    else
        tmp = 1.0d0 / (x + (y * (x + (x * y))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -6.2e+61) {
		tmp = (1.0 - (y * (1.0 - (y * 0.5)))) / x;
	} else if (x <= 7.5e-31) {
		tmp = 1.0 / x;
	} else {
		tmp = 1.0 / (x + (y * (x + (x * y))));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -6.2e+61:
		tmp = (1.0 - (y * (1.0 - (y * 0.5)))) / x
	elif x <= 7.5e-31:
		tmp = 1.0 / x
	else:
		tmp = 1.0 / (x + (y * (x + (x * y))))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -6.2e+61)
		tmp = Float64(Float64(1.0 - Float64(y * Float64(1.0 - Float64(y * 0.5)))) / x);
	elseif (x <= 7.5e-31)
		tmp = Float64(1.0 / x);
	else
		tmp = Float64(1.0 / Float64(x + Float64(y * Float64(x + Float64(x * y)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -6.2e+61)
		tmp = (1.0 - (y * (1.0 - (y * 0.5)))) / x;
	elseif (x <= 7.5e-31)
		tmp = 1.0 / x;
	else
		tmp = 1.0 / (x + (y * (x + (x * y))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -6.2e+61], N[(N[(1.0 - N[(y * N[(1.0 - N[(y * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[x, 7.5e-31], N[(1.0 / x), $MachinePrecision], N[(1.0 / N[(x + N[(y * N[(x + N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -6.1999999999999998e61

   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. Add Preprocessing

   5. Taylor expanded in y around 0 72.6%

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

   6. Taylor expanded in x around inf 72.6%

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

      1. *-commutative 72.6%

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

   8. Simplified 72.6%

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

   if -6.1999999999999998e61 < x < 7.49999999999999975e-31

   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 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in x around 0 99.3%

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

   if 7.49999999999999975e-31 < x

   1. Initial program 68.0%

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

      1. *-commutative 68.0%

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

      2. exp-to-pow 68.0%

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

   3. Simplified 68.0%

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

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

   8. Taylor expanded in y around 0 49.2%

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

      1. neg-mul-1 49.2%

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

      2. +-commutative 49.2%

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

      3. sub-neg 49.2%

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

   10. Simplified 49.2%

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

       1. sub-div 49.2%

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

       2. clear-num 49.2%

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

   12. Applied egg-rr 49.2%

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

   13. Taylor expanded in y around 0 73.3%

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

       1. *-commutative 73.3%

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

       2. neg-mul-1 73.3%

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

   15. Simplified 73.3%

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

4. Final simplification 84.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -6.2 \cdot 10^{+61}:\\ \;\;\;\;\frac{1 - y \cdot \left(1 - y \cdot 0.5\right)}{x}\\ \mathbf{elif}\;x \leq 7.5 \cdot 10^{-31}:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x + y \cdot \left(x + x \cdot y\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 79.7% accurate, 12.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.4 \cdot 10^{+110}:\\ \;\;\;\;\frac{1}{x} + y \cdot \left(y \cdot \frac{0.5}{x}\right)\\ \mathbf{elif}\;x \leq 8.8 \cdot 10^{+21}:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x + x \cdot y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1.4e+110)
   (+ (/ 1.0 x) (* y (* y (/ 0.5 x))))
   (if (<= x 8.8e+21) (/ 1.0 x) (/ 1.0 (+ x (* x y))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.4e+110) {
		tmp = (1.0 / x) + (y * (y * (0.5 / x)));
	} else if (x <= 8.8e+21) {
		tmp = 1.0 / x;
	} else {
		tmp = 1.0 / (x + (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 (x <= (-1.4d+110)) then
        tmp = (1.0d0 / x) + (y * (y * (0.5d0 / x)))
    else if (x <= 8.8d+21) then
        tmp = 1.0d0 / x
    else
        tmp = 1.0d0 / (x + (x * y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -1.4e+110) {
		tmp = (1.0 / x) + (y * (y * (0.5 / x)));
	} else if (x <= 8.8e+21) {
		tmp = 1.0 / x;
	} else {
		tmp = 1.0 / (x + (x * y));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -1.4e+110:
		tmp = (1.0 / x) + (y * (y * (0.5 / x)))
	elif x <= 8.8e+21:
		tmp = 1.0 / x
	else:
		tmp = 1.0 / (x + (x * y))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1.4e+110)
		tmp = Float64(Float64(1.0 / x) + Float64(y * Float64(y * Float64(0.5 / x))));
	elseif (x <= 8.8e+21)
		tmp = Float64(1.0 / x);
	else
		tmp = Float64(1.0 / Float64(x + Float64(x * y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1.4e+110)
		tmp = (1.0 / x) + (y * (y * (0.5 / x)));
	elseif (x <= 8.8e+21)
		tmp = 1.0 / x;
	else
		tmp = 1.0 / (x + (x * y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1.4e+110], N[(N[(1.0 / x), $MachinePrecision] + N[(y * N[(y * N[(0.5 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 8.8e+21], N[(1.0 / x), $MachinePrecision], N[(1.0 / N[(x + N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.39999999999999993e110

   1. Initial program 67.9%

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

      1. *-commutative 67.9%

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

      2. exp-to-pow 67.9%

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

   3. Simplified 67.9%

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

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

   8. Taylor expanded in y around 0 57.8%

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

   9. Taylor expanded in y around inf 56.5%

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

       1. associate-*r/ 56.5%

          \[\leadsto y \cdot \color{blue}{\frac{0.5 \cdot y}{x}} + \frac{1}{x} \]

       2. *-commutative 56.5%

          \[\leadsto y \cdot \frac{\color{blue}{y \cdot 0.5}}{x} + \frac{1}{x} \]

       3. associate-/l* 56.5%

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

   11. Simplified 56.5%

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

   if -1.39999999999999993e110 < x < 8.8e21

   1. Initial program 87.4%

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

      1. exp-prod 98.9%

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

   3. Simplified 98.9%

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

   5. Taylor expanded in x around 0 96.3%

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

   if 8.8e21 < x

   1. Initial program 65.0%

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

      1. *-commutative 65.0%

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

      2. exp-to-pow 65.0%

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

   3. Simplified 65.0%

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

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

   8. Taylor expanded in y around 0 48.4%

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

      1. neg-mul-1 48.4%

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

      2. +-commutative 48.4%

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

      3. sub-neg 48.4%

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

   10. Simplified 48.4%

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

       1. sub-div 48.4%

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

       2. clear-num 48.4%

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

   12. Applied egg-rr 48.4%

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

   13. Taylor expanded in y around 0 68.3%

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

       1. *-commutative 68.3%

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

   15. Simplified 68.3%

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

4. Final simplification 80.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.4 \cdot 10^{+110}:\\ \;\;\;\;\frac{1}{x} + y \cdot \left(y \cdot \frac{0.5}{x}\right)\\ \mathbf{elif}\;x \leq 8.8 \cdot 10^{+21}:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x + x \cdot y}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 82.2% accurate, 12.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -6.2 \cdot 10^{+61}:\\ \;\;\;\;\frac{1 - y \cdot \left(1 - y \cdot 0.5\right)}{x}\\ \mathbf{elif}\;x \leq 1.6 \cdot 10^{+21}:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x + x \cdot y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -6.2e+61)
   (/ (- 1.0 (* y (- 1.0 (* y 0.5)))) x)
   (if (<= x 1.6e+21) (/ 1.0 x) (/ 1.0 (+ x (* x y))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -6.2e+61) {
		tmp = (1.0 - (y * (1.0 - (y * 0.5)))) / x;
	} else if (x <= 1.6e+21) {
		tmp = 1.0 / x;
	} else {
		tmp = 1.0 / (x + (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 (x <= (-6.2d+61)) then
        tmp = (1.0d0 - (y * (1.0d0 - (y * 0.5d0)))) / x
    else if (x <= 1.6d+21) then
        tmp = 1.0d0 / x
    else
        tmp = 1.0d0 / (x + (x * y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -6.2e+61) {
		tmp = (1.0 - (y * (1.0 - (y * 0.5)))) / x;
	} else if (x <= 1.6e+21) {
		tmp = 1.0 / x;
	} else {
		tmp = 1.0 / (x + (x * y));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -6.2e+61:
		tmp = (1.0 - (y * (1.0 - (y * 0.5)))) / x
	elif x <= 1.6e+21:
		tmp = 1.0 / x
	else:
		tmp = 1.0 / (x + (x * y))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -6.2e+61)
		tmp = Float64(Float64(1.0 - Float64(y * Float64(1.0 - Float64(y * 0.5)))) / x);
	elseif (x <= 1.6e+21)
		tmp = Float64(1.0 / x);
	else
		tmp = Float64(1.0 / Float64(x + Float64(x * y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -6.2e+61)
		tmp = (1.0 - (y * (1.0 - (y * 0.5)))) / x;
	elseif (x <= 1.6e+21)
		tmp = 1.0 / x;
	else
		tmp = 1.0 / (x + (x * y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -6.2e+61], N[(N[(1.0 - N[(y * N[(1.0 - N[(y * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[x, 1.6e+21], N[(1.0 / x), $MachinePrecision], N[(1.0 / N[(x + N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -6.1999999999999998e61

   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. Add Preprocessing

   5. Taylor expanded in y around 0 72.6%

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

   6. Taylor expanded in x around inf 72.6%

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

      1. *-commutative 72.6%

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

   8. Simplified 72.6%

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

   if -6.1999999999999998e61 < x < 1.6e21

   1. Initial program 87.5%

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

      1. exp-prod 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in x around 0 97.0%

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

   if 1.6e21 < x

   1. Initial program 65.0%

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

      1. *-commutative 65.0%

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

      2. exp-to-pow 65.0%

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

   3. Simplified 65.0%

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

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

   8. Taylor expanded in y around 0 48.4%

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

      1. neg-mul-1 48.4%

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

      2. +-commutative 48.4%

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

      3. sub-neg 48.4%

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

   10. Simplified 48.4%

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

       1. sub-div 48.4%

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

       2. clear-num 48.4%

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

   12. Applied egg-rr 48.4%

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

   13. Taylor expanded in y around 0 68.3%

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

       1. *-commutative 68.3%

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

   15. Simplified 68.3%

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

4. Final simplification 83.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -6.2 \cdot 10^{+61}:\\ \;\;\;\;\frac{1 - y \cdot \left(1 - y \cdot 0.5\right)}{x}\\ \mathbf{elif}\;x \leq 1.6 \cdot 10^{+21}:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x + x \cdot y}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 78.9% accurate, 17.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.6 \cdot 10^{+21}:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x + x \cdot y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 1.6e+21) (/ 1.0 x) (/ 1.0 (+ x (* x y)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 1.6e+21) {
		tmp = 1.0 / x;
	} else {
		tmp = 1.0 / (x + (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 (x <= 1.6d+21) then
        tmp = 1.0d0 / x
    else
        tmp = 1.0d0 / (x + (x * y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 1.6e+21) {
		tmp = 1.0 / x;
	} else {
		tmp = 1.0 / (x + (x * y));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 1.6e+21:
		tmp = 1.0 / x
	else:
		tmp = 1.0 / (x + (x * y))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 1.6e+21)
		tmp = Float64(1.0 / x);
	else
		tmp = Float64(1.0 / Float64(x + Float64(x * y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 1.6e+21)
		tmp = 1.0 / x;
	else
		tmp = 1.0 / (x + (x * y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 1.6e+21], N[(1.0 / x), $MachinePrecision], N[(1.0 / N[(x + N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.6e21

   1. Initial program 82.1%

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

      1. exp-prod 90.5%

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

   3. Simplified 90.5%

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

   5. Taylor expanded in x around 0 82.6%

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

   if 1.6e21 < x

   1. Initial program 65.0%

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

      1. *-commutative 65.0%

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

      2. exp-to-pow 65.0%

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

   3. Simplified 65.0%

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

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

   8. Taylor expanded in y around 0 48.4%

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

      1. neg-mul-1 48.4%

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

      2. +-commutative 48.4%

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

      3. sub-neg 48.4%

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

   10. Simplified 48.4%

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

       1. sub-div 48.4%

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

       2. clear-num 48.4%

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

   12. Applied egg-rr 48.4%

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

   13. Taylor expanded in y around 0 68.3%

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

       1. *-commutative 68.3%

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

   15. Simplified 68.3%

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

4. Final simplification 78.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.6 \cdot 10^{+21}:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x + x \cdot y}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 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 77.2%

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

   1. exp-prod 83.2%

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

3. Simplified 83.2%

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

5. Taylor expanded in x around 0 73.1%

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

6. Final simplification 73.1%

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

Developer target: 77.9% 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 2024079 
(FPCore (x y)
  :name "Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2, F"
  :precision binary64

  :alt
  (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))