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

Percentage Accurate: 76.9% → 99.5%

Time: 11.4s

Alternatives: 12

Speedup: 20.9×

• Could not converge on a ground truth

  1. x = -1.7851232068946496e+188
  2. y = -3.777494873085752e-107

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: 76.9% 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: 99.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{+56}:\\ \;\;\;\;\frac{1}{x \cdot e^{y}}\\ \mathbf{elif}\;x \leq 0.0032:\\ \;\;\;\;\frac{{\left(e^{x}\right)}^{\log \left(\frac{x}{x + y}\right)}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{-y}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1e+56)
   (/ 1.0 (* x (exp y)))
   (if (<= x 0.0032)
     (/ (pow (exp x) (log (/ x (+ x y)))) x)
     (/ (exp (- y)) x))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1e+56) {
		tmp = 1.0 / (x * exp(y));
	} else if (x <= 0.0032) {
		tmp = pow(exp(x), log((x / (x + y)))) / x;
	} else {
		tmp = exp(-y) / x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-1d+56)) then
        tmp = 1.0d0 / (x * exp(y))
    else if (x <= 0.0032d0) then
        tmp = (exp(x) ** log((x / (x + y)))) / x
    else
        tmp = exp(-y) / x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -1e+56) {
		tmp = 1.0 / (x * Math.exp(y));
	} else if (x <= 0.0032) {
		tmp = Math.pow(Math.exp(x), Math.log((x / (x + y)))) / x;
	} else {
		tmp = Math.exp(-y) / x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -1e+56:
		tmp = 1.0 / (x * math.exp(y))
	elif x <= 0.0032:
		tmp = math.pow(math.exp(x), math.log((x / (x + y)))) / x
	else:
		tmp = math.exp(-y) / x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1e+56)
		tmp = Float64(1.0 / Float64(x * exp(y)));
	elseif (x <= 0.0032)
		tmp = Float64((exp(x) ^ log(Float64(x / Float64(x + y)))) / x);
	else
		tmp = Float64(exp(Float64(-y)) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1e+56)
		tmp = 1.0 / (x * exp(y));
	elseif (x <= 0.0032)
		tmp = (exp(x) ^ log((x / (x + y)))) / x;
	else
		tmp = exp(-y) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1e+56], N[(1.0 / N[(x * N[Exp[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.0032], N[(N[Power[N[Exp[x], $MachinePrecision], N[Log[N[(x / N[(x + y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] / x), $MachinePrecision], N[(N[Exp[(-y)], $MachinePrecision] / x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.00000000000000009e56

   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. Add Preprocessing

   5. Taylor expanded in x around inf 99.9%

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

      1. mul-1-neg 99.9%

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

   7. Simplified 99.9%

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

      1. clear-num 99.9%

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

      2. inv-pow 99.9%

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

      3. exp-neg 99.9%

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

      4. associate-/r/ 100.0%

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

      5. /-rgt-identity 100.0%

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

   9. Applied egg-rr 100.0%

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

   10. Taylor expanded in x around 0 100.0%

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

   if -1.00000000000000009e56 < x < 0.00320000000000000015

   1. Initial program 90.5%

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

   if 0.00320000000000000015 < x

   1. Initial program 69.5%

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

      1. *-commutative 69.5%

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

      2. exp-to-pow 69.5%

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

   3. Simplified 69.5%

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

   5. Taylor expanded in x around inf 100.0%

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

      1. mul-1-neg 100.0%

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

   7. Simplified 100.0%

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

Alternative 2: 99.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4400 \lor \neg \left(x \leq 0.0032\right):\\ \;\;\;\;\frac{e^{-y}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -4400.0) (not (<= x 0.0032))) (/ (exp (- y)) x) (/ 1.0 x)))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -4400.0) || !(x <= 0.0032)) {
		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 <= (-4400.0d0)) .or. (.not. (x <= 0.0032d0))) 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 <= -4400.0) || !(x <= 0.0032)) {
		tmp = Math.exp(-y) / x;
	} else {
		tmp = 1.0 / x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -4400.0) or not (x <= 0.0032):
		tmp = math.exp(-y) / x
	else:
		tmp = 1.0 / x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -4400.0) || !(x <= 0.0032))
		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 <= -4400.0) || ~((x <= 0.0032)))
		tmp = exp(-y) / x;
	else
		tmp = 1.0 / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -4400.0], N[Not[LessEqual[x, 0.0032]], $MachinePrecision]], N[(N[Exp[(-y)], $MachinePrecision] / x), $MachinePrecision], N[(1.0 / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -4400 or 0.00320000000000000015 < x

   1. Initial program 71.1%

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

      1. *-commutative 71.1%

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

      2. exp-to-pow 71.1%

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

   3. Simplified 71.1%

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

   5. Taylor expanded in x around inf 100.0%

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

      1. mul-1-neg 100.0%

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

   7. Simplified 100.0%

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

   if -4400 < x < 0.00320000000000000015

   1. Initial program 88.8%

      \[\frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x} \]
   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 98.8%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4400 \lor \neg \left(x \leq 0.0032\right):\\ \;\;\;\;\frac{e^{-y}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4400:\\ \;\;\;\;\frac{1}{x \cdot e^{y}}\\ \mathbf{elif}\;x \leq 0.0032:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{-y}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -4400.0)
   (/ 1.0 (* x (exp y)))
   (if (<= x 0.0032) (/ 1.0 x) (/ (exp (- y)) x))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -4400.0) {
		tmp = 1.0 / (x * exp(y));
	} else if (x <= 0.0032) {
		tmp = 1.0 / x;
	} else {
		tmp = exp(-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 <= (-4400.0d0)) then
        tmp = 1.0d0 / (x * exp(y))
    else if (x <= 0.0032d0) then
        tmp = 1.0d0 / x
    else
        tmp = exp(-y) / x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -4400.0) {
		tmp = 1.0 / (x * Math.exp(y));
	} else if (x <= 0.0032) {
		tmp = 1.0 / x;
	} else {
		tmp = Math.exp(-y) / x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -4400.0:
		tmp = 1.0 / (x * math.exp(y))
	elif x <= 0.0032:
		tmp = 1.0 / x
	else:
		tmp = math.exp(-y) / x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -4400.0)
		tmp = Float64(1.0 / Float64(x * exp(y)));
	elseif (x <= 0.0032)
		tmp = Float64(1.0 / x);
	else
		tmp = Float64(exp(Float64(-y)) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -4400.0)
		tmp = 1.0 / (x * exp(y));
	elseif (x <= 0.0032)
		tmp = 1.0 / x;
	else
		tmp = exp(-y) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -4400.0], N[(1.0 / N[(x * N[Exp[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.0032], N[(1.0 / x), $MachinePrecision], N[(N[Exp[(-y)], $MachinePrecision] / x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -4400

   1. Initial program 72.7%

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

      1. *-commutative 72.7%

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

      2. exp-to-pow 72.7%

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

   3. Simplified 72.7%

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

   5. Taylor expanded in x around inf 100.0%

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

      1. mul-1-neg 100.0%

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

   7. Simplified 100.0%

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

      1. clear-num 100.0%

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

      2. inv-pow 100.0%

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

      3. exp-neg 100.0%

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

      4. associate-/r/ 100.0%

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

      5. /-rgt-identity 100.0%

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

   9. Applied egg-rr 100.0%

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

   10. Taylor expanded in x around 0 100.0%

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

   if -4400 < x < 0.00320000000000000015

   1. Initial program 88.8%

      \[\frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x} \]
   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 98.8%

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

   if 0.00320000000000000015 < x

   1. Initial program 69.5%

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

      1. *-commutative 69.5%

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

      2. exp-to-pow 69.5%

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

   3. Simplified 69.5%

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

   5. Taylor expanded in x around inf 100.0%

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

      1. mul-1-neg 100.0%

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

   7. Simplified 100.0%

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

Alternative 4: 87.3% accurate, 6.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{x \cdot \left(1 + y \cdot \left(1 + y \cdot \left(0.5 + y \cdot 0.16666666666666666\right)\right)\right)}\\ \mathbf{if}\;x \leq -5.2 \cdot 10^{+241}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq -4400:\\ \;\;\;\;\frac{1 + y \cdot \left(y \cdot \left(0.5 + y \cdot -0.16666666666666666\right) + -1\right)}{x}\\ \mathbf{elif}\;x \leq 0.00305:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0
         (/
          1.0
          (*
           x
           (+ 1.0 (* y (+ 1.0 (* y (+ 0.5 (* y 0.16666666666666666))))))))))
   (if (<= x -5.2e+241)
     t_0
     (if (<= x -4400.0)
       (/ (+ 1.0 (* y (+ (* y (+ 0.5 (* y -0.16666666666666666))) -1.0))) x)
       (if (<= x 0.00305) (/ 1.0 x) t_0)))))

C

double code(double x, double y) {
	double t_0 = 1.0 / (x * (1.0 + (y * (1.0 + (y * (0.5 + (y * 0.16666666666666666)))))));
	double tmp;
	if (x <= -5.2e+241) {
		tmp = t_0;
	} else if (x <= -4400.0) {
		tmp = (1.0 + (y * ((y * (0.5 + (y * -0.16666666666666666))) + -1.0))) / x;
	} else if (x <= 0.00305) {
		tmp = 1.0 / x;
	} 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 = 1.0d0 / (x * (1.0d0 + (y * (1.0d0 + (y * (0.5d0 + (y * 0.16666666666666666d0)))))))
    if (x <= (-5.2d+241)) then
        tmp = t_0
    else if (x <= (-4400.0d0)) then
        tmp = (1.0d0 + (y * ((y * (0.5d0 + (y * (-0.16666666666666666d0)))) + (-1.0d0)))) / x
    else if (x <= 0.00305d0) then
        tmp = 1.0d0 / x
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = 1.0 / (x * (1.0 + (y * (1.0 + (y * (0.5 + (y * 0.16666666666666666)))))));
	double tmp;
	if (x <= -5.2e+241) {
		tmp = t_0;
	} else if (x <= -4400.0) {
		tmp = (1.0 + (y * ((y * (0.5 + (y * -0.16666666666666666))) + -1.0))) / x;
	} else if (x <= 0.00305) {
		tmp = 1.0 / x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 1.0 / (x * (1.0 + (y * (1.0 + (y * (0.5 + (y * 0.16666666666666666)))))))
	tmp = 0
	if x <= -5.2e+241:
		tmp = t_0
	elif x <= -4400.0:
		tmp = (1.0 + (y * ((y * (0.5 + (y * -0.16666666666666666))) + -1.0))) / x
	elif x <= 0.00305:
		tmp = 1.0 / x
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(1.0 / Float64(x * Float64(1.0 + Float64(y * Float64(1.0 + Float64(y * Float64(0.5 + Float64(y * 0.16666666666666666))))))))
	tmp = 0.0
	if (x <= -5.2e+241)
		tmp = t_0;
	elseif (x <= -4400.0)
		tmp = Float64(Float64(1.0 + Float64(y * Float64(Float64(y * Float64(0.5 + Float64(y * -0.16666666666666666))) + -1.0))) / x);
	elseif (x <= 0.00305)
		tmp = Float64(1.0 / x);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = 1.0 / (x * (1.0 + (y * (1.0 + (y * (0.5 + (y * 0.16666666666666666)))))));
	tmp = 0.0;
	if (x <= -5.2e+241)
		tmp = t_0;
	elseif (x <= -4400.0)
		tmp = (1.0 + (y * ((y * (0.5 + (y * -0.16666666666666666))) + -1.0))) / x;
	elseif (x <= 0.00305)
		tmp = 1.0 / x;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(1.0 / N[(x * N[(1.0 + N[(y * N[(1.0 + N[(y * N[(0.5 + N[(y * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -5.2e+241], t$95$0, If[LessEqual[x, -4400.0], N[(N[(1.0 + N[(y * N[(N[(y * N[(0.5 + N[(y * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[x, 0.00305], N[(1.0 / x), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if x < -5.20000000000000015e241 or 0.00305000000000000019 < x

   1. Initial program 65.3%

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

      1. *-commutative 65.3%

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

      2. exp-to-pow 65.3%

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

   3. Simplified 65.3%

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

   5. Taylor expanded in x around inf 100.0%

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

      1. mul-1-neg 100.0%

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

   7. Simplified 100.0%

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

      1. clear-num 100.0%

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

      2. inv-pow 100.0%

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

      3. exp-neg 100.0%

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

      4. associate-/r/ 100.0%

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

      5. /-rgt-identity 100.0%

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

   9. Applied egg-rr 100.0%

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

   10. Taylor expanded in x around 0 100.0%

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

   11. Taylor expanded in y around 0 78.2%

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

       1. *-commutative 78.2%

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

   13. Simplified 78.2%

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

   if -5.20000000000000015e241 < x < -4400

   1. Initial program 79.9%

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

      1. *-commutative 79.9%

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

      2. exp-to-pow 79.9%

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

   3. Simplified 79.9%

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

   5. Taylor expanded in x around inf 99.9%

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

      1. mul-1-neg 99.9%

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

   7. Simplified 99.9%

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

   8. Taylor expanded in y around 0 79.0%

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

   if -4400 < x < 0.00305000000000000019

   1. Initial program 88.8%

      \[\frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x} \]
   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 98.8%

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

4. Final simplification 86.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.2 \cdot 10^{+241}:\\ \;\;\;\;\frac{1}{x \cdot \left(1 + y \cdot \left(1 + y \cdot \left(0.5 + y \cdot 0.16666666666666666\right)\right)\right)}\\ \mathbf{elif}\;x \leq -4400:\\ \;\;\;\;\frac{1 + y \cdot \left(y \cdot \left(0.5 + y \cdot -0.16666666666666666\right) + -1\right)}{x}\\ \mathbf{elif}\;x \leq 0.00305:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x \cdot \left(1 + y \cdot \left(1 + y \cdot \left(0.5 + y \cdot 0.16666666666666666\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 86.6% accurate, 7.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-1}{x \cdot \left(-1 - y \cdot \left(1 + y \cdot 0.5\right)\right)}\\ \mathbf{if}\;x \leq -3.6 \cdot 10^{+243}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq -4400:\\ \;\;\;\;\frac{1 + y \cdot \left(y \cdot \left(0.5 + y \cdot -0.16666666666666666\right) + -1\right)}{x}\\ \mathbf{elif}\;x \leq 0.0029:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ -1.0 (* x (- -1.0 (* y (+ 1.0 (* y 0.5))))))))
   (if (<= x -3.6e+243)
     t_0
     (if (<= x -4400.0)
       (/ (+ 1.0 (* y (+ (* y (+ 0.5 (* y -0.16666666666666666))) -1.0))) x)
       (if (<= x 0.0029) (/ 1.0 x) t_0)))))

C

double code(double x, double y) {
	double t_0 = -1.0 / (x * (-1.0 - (y * (1.0 + (y * 0.5)))));
	double tmp;
	if (x <= -3.6e+243) {
		tmp = t_0;
	} else if (x <= -4400.0) {
		tmp = (1.0 + (y * ((y * (0.5 + (y * -0.16666666666666666))) + -1.0))) / x;
	} else if (x <= 0.0029) {
		tmp = 1.0 / x;
	} 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 = (-1.0d0) / (x * ((-1.0d0) - (y * (1.0d0 + (y * 0.5d0)))))
    if (x <= (-3.6d+243)) then
        tmp = t_0
    else if (x <= (-4400.0d0)) then
        tmp = (1.0d0 + (y * ((y * (0.5d0 + (y * (-0.16666666666666666d0)))) + (-1.0d0)))) / x
    else if (x <= 0.0029d0) then
        tmp = 1.0d0 / x
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = -1.0 / (x * (-1.0 - (y * (1.0 + (y * 0.5)))));
	double tmp;
	if (x <= -3.6e+243) {
		tmp = t_0;
	} else if (x <= -4400.0) {
		tmp = (1.0 + (y * ((y * (0.5 + (y * -0.16666666666666666))) + -1.0))) / x;
	} else if (x <= 0.0029) {
		tmp = 1.0 / x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = -1.0 / (x * (-1.0 - (y * (1.0 + (y * 0.5)))))
	tmp = 0
	if x <= -3.6e+243:
		tmp = t_0
	elif x <= -4400.0:
		tmp = (1.0 + (y * ((y * (0.5 + (y * -0.16666666666666666))) + -1.0))) / x
	elif x <= 0.0029:
		tmp = 1.0 / x
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(-1.0 / Float64(x * Float64(-1.0 - Float64(y * Float64(1.0 + Float64(y * 0.5))))))
	tmp = 0.0
	if (x <= -3.6e+243)
		tmp = t_0;
	elseif (x <= -4400.0)
		tmp = Float64(Float64(1.0 + Float64(y * Float64(Float64(y * Float64(0.5 + Float64(y * -0.16666666666666666))) + -1.0))) / x);
	elseif (x <= 0.0029)
		tmp = Float64(1.0 / x);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = -1.0 / (x * (-1.0 - (y * (1.0 + (y * 0.5)))));
	tmp = 0.0;
	if (x <= -3.6e+243)
		tmp = t_0;
	elseif (x <= -4400.0)
		tmp = (1.0 + (y * ((y * (0.5 + (y * -0.16666666666666666))) + -1.0))) / x;
	elseif (x <= 0.0029)
		tmp = 1.0 / x;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(-1.0 / N[(x * N[(-1.0 - N[(y * N[(1.0 + N[(y * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -3.6e+243], t$95$0, If[LessEqual[x, -4400.0], N[(N[(1.0 + N[(y * N[(N[(y * N[(0.5 + N[(y * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[x, 0.0029], N[(1.0 / x), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if x < -3.5999999999999997e243 or 0.0029 < x

   1. Initial program 65.3%

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

      1. *-commutative 65.3%

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

      2. exp-to-pow 65.3%

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

   3. Simplified 65.3%

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

   5. Taylor expanded in x around inf 100.0%

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

      1. mul-1-neg 100.0%

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

   7. Simplified 100.0%

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

      1. clear-num 100.0%

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

      2. inv-pow 100.0%

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

      3. exp-neg 100.0%

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

      4. associate-/r/ 100.0%

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

      5. /-rgt-identity 100.0%

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

   9. Applied egg-rr 100.0%

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

   10. Taylor expanded in x around 0 100.0%

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

   11. Taylor expanded in y around 0 76.0%

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

       1. *-commutative 76.0%

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

   13. Simplified 76.0%

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

   if -3.5999999999999997e243 < x < -4400

   1. Initial program 79.9%

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

      1. *-commutative 79.9%

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

      2. exp-to-pow 79.9%

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

   3. Simplified 79.9%

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

   5. Taylor expanded in x around inf 99.9%

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

      1. mul-1-neg 99.9%

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

   7. Simplified 99.9%

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

   8. Taylor expanded in y around 0 79.0%

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

   if -4400 < x < 0.0029

   1. Initial program 88.8%

      \[\frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x} \]
   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 98.8%

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

4. Final simplification 85.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.6 \cdot 10^{+243}:\\ \;\;\;\;\frac{-1}{x \cdot \left(-1 - y \cdot \left(1 + y \cdot 0.5\right)\right)}\\ \mathbf{elif}\;x \leq -4400:\\ \;\;\;\;\frac{1 + y \cdot \left(y \cdot \left(0.5 + y \cdot -0.16666666666666666\right) + -1\right)}{x}\\ \mathbf{elif}\;x \leq 0.0029:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{x \cdot \left(-1 - y \cdot \left(1 + y \cdot 0.5\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 86.5% accurate, 7.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-1}{x \cdot \left(-1 - y \cdot \left(1 + y \cdot 0.5\right)\right)}\\ \mathbf{if}\;x \leq -4 \cdot 10^{+242}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq -4400:\\ \;\;\;\;\frac{1 + y \cdot \left(y \cdot \left(y \cdot -0.16666666666666666\right) + -1\right)}{x}\\ \mathbf{elif}\;x \leq 0.0032:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ -1.0 (* x (- -1.0 (* y (+ 1.0 (* y 0.5))))))))
   (if (<= x -4e+242)
     t_0
     (if (<= x -4400.0)
       (/ (+ 1.0 (* y (+ (* y (* y -0.16666666666666666)) -1.0))) x)
       (if (<= x 0.0032) (/ 1.0 x) t_0)))))

C

double code(double x, double y) {
	double t_0 = -1.0 / (x * (-1.0 - (y * (1.0 + (y * 0.5)))));
	double tmp;
	if (x <= -4e+242) {
		tmp = t_0;
	} else if (x <= -4400.0) {
		tmp = (1.0 + (y * ((y * (y * -0.16666666666666666)) + -1.0))) / x;
	} else if (x <= 0.0032) {
		tmp = 1.0 / x;
	} 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 = (-1.0d0) / (x * ((-1.0d0) - (y * (1.0d0 + (y * 0.5d0)))))
    if (x <= (-4d+242)) then
        tmp = t_0
    else if (x <= (-4400.0d0)) then
        tmp = (1.0d0 + (y * ((y * (y * (-0.16666666666666666d0))) + (-1.0d0)))) / x
    else if (x <= 0.0032d0) then
        tmp = 1.0d0 / x
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = -1.0 / (x * (-1.0 - (y * (1.0 + (y * 0.5)))));
	double tmp;
	if (x <= -4e+242) {
		tmp = t_0;
	} else if (x <= -4400.0) {
		tmp = (1.0 + (y * ((y * (y * -0.16666666666666666)) + -1.0))) / x;
	} else if (x <= 0.0032) {
		tmp = 1.0 / x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = -1.0 / (x * (-1.0 - (y * (1.0 + (y * 0.5)))))
	tmp = 0
	if x <= -4e+242:
		tmp = t_0
	elif x <= -4400.0:
		tmp = (1.0 + (y * ((y * (y * -0.16666666666666666)) + -1.0))) / x
	elif x <= 0.0032:
		tmp = 1.0 / x
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(-1.0 / Float64(x * Float64(-1.0 - Float64(y * Float64(1.0 + Float64(y * 0.5))))))
	tmp = 0.0
	if (x <= -4e+242)
		tmp = t_0;
	elseif (x <= -4400.0)
		tmp = Float64(Float64(1.0 + Float64(y * Float64(Float64(y * Float64(y * -0.16666666666666666)) + -1.0))) / x);
	elseif (x <= 0.0032)
		tmp = Float64(1.0 / x);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = -1.0 / (x * (-1.0 - (y * (1.0 + (y * 0.5)))));
	tmp = 0.0;
	if (x <= -4e+242)
		tmp = t_0;
	elseif (x <= -4400.0)
		tmp = (1.0 + (y * ((y * (y * -0.16666666666666666)) + -1.0))) / x;
	elseif (x <= 0.0032)
		tmp = 1.0 / x;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(-1.0 / N[(x * N[(-1.0 - N[(y * N[(1.0 + N[(y * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -4e+242], t$95$0, If[LessEqual[x, -4400.0], N[(N[(1.0 + N[(y * N[(N[(y * N[(y * -0.16666666666666666), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[x, 0.0032], N[(1.0 / x), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if x < -4.0000000000000002e242 or 0.00320000000000000015 < x

   1. Initial program 65.3%

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

      1. *-commutative 65.3%

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

      2. exp-to-pow 65.3%

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

   3. Simplified 65.3%

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

   5. Taylor expanded in x around inf 100.0%

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

      1. mul-1-neg 100.0%

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

   7. Simplified 100.0%

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

      1. clear-num 100.0%

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

      2. inv-pow 100.0%

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

      3. exp-neg 100.0%

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

      4. associate-/r/ 100.0%

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

      5. /-rgt-identity 100.0%

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

   9. Applied egg-rr 100.0%

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

   10. Taylor expanded in x around 0 100.0%

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

   11. Taylor expanded in y around 0 76.0%

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

       1. *-commutative 76.0%

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

   13. Simplified 76.0%

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

   if -4.0000000000000002e242 < x < -4400

   1. Initial program 79.9%

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

      1. *-commutative 79.9%

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

      2. exp-to-pow 79.9%

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

   3. Simplified 79.9%

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

   5. Taylor expanded in x around inf 99.9%

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

      1. mul-1-neg 99.9%

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

   7. Simplified 99.9%

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

   8. Taylor expanded in y around 0 79.0%

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

   9. Taylor expanded in y around inf 77.9%

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

       1. *-commutative 77.9%

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

   11. Simplified 77.9%

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

   if -4400 < x < 0.00320000000000000015

   1. Initial program 88.8%

      \[\frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x} \]
   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 98.8%

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

4. Final simplification 85.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4 \cdot 10^{+242}:\\ \;\;\;\;\frac{-1}{x \cdot \left(-1 - y \cdot \left(1 + y \cdot 0.5\right)\right)}\\ \mathbf{elif}\;x \leq -4400:\\ \;\;\;\;\frac{1 + y \cdot \left(y \cdot \left(y \cdot -0.16666666666666666\right) + -1\right)}{x}\\ \mathbf{elif}\;x \leq 0.0032:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{x \cdot \left(-1 - y \cdot \left(1 + y \cdot 0.5\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 85.5% accurate, 7.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-1}{x \cdot \left(-1 - y \cdot \left(1 + y \cdot 0.5\right)\right)}\\ \mathbf{if}\;x \leq -1.26 \cdot 10^{+169}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq -4400:\\ \;\;\;\;\frac{1 + y \cdot \left(y \cdot 0.5 + -1\right)}{x}\\ \mathbf{elif}\;x \leq 0.0032:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ -1.0 (* x (- -1.0 (* y (+ 1.0 (* y 0.5))))))))
   (if (<= x -1.26e+169)
     t_0
     (if (<= x -4400.0)
       (/ (+ 1.0 (* y (+ (* y 0.5) -1.0))) x)
       (if (<= x 0.0032) (/ 1.0 x) t_0)))))

C

double code(double x, double y) {
	double t_0 = -1.0 / (x * (-1.0 - (y * (1.0 + (y * 0.5)))));
	double tmp;
	if (x <= -1.26e+169) {
		tmp = t_0;
	} else if (x <= -4400.0) {
		tmp = (1.0 + (y * ((y * 0.5) + -1.0))) / x;
	} else if (x <= 0.0032) {
		tmp = 1.0 / x;
	} 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 = (-1.0d0) / (x * ((-1.0d0) - (y * (1.0d0 + (y * 0.5d0)))))
    if (x <= (-1.26d+169)) then
        tmp = t_0
    else if (x <= (-4400.0d0)) then
        tmp = (1.0d0 + (y * ((y * 0.5d0) + (-1.0d0)))) / x
    else if (x <= 0.0032d0) then
        tmp = 1.0d0 / x
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = -1.0 / (x * (-1.0 - (y * (1.0 + (y * 0.5)))));
	double tmp;
	if (x <= -1.26e+169) {
		tmp = t_0;
	} else if (x <= -4400.0) {
		tmp = (1.0 + (y * ((y * 0.5) + -1.0))) / x;
	} else if (x <= 0.0032) {
		tmp = 1.0 / x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = -1.0 / (x * (-1.0 - (y * (1.0 + (y * 0.5)))))
	tmp = 0
	if x <= -1.26e+169:
		tmp = t_0
	elif x <= -4400.0:
		tmp = (1.0 + (y * ((y * 0.5) + -1.0))) / x
	elif x <= 0.0032:
		tmp = 1.0 / x
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(-1.0 / Float64(x * Float64(-1.0 - Float64(y * Float64(1.0 + Float64(y * 0.5))))))
	tmp = 0.0
	if (x <= -1.26e+169)
		tmp = t_0;
	elseif (x <= -4400.0)
		tmp = Float64(Float64(1.0 + Float64(y * Float64(Float64(y * 0.5) + -1.0))) / x);
	elseif (x <= 0.0032)
		tmp = Float64(1.0 / x);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = -1.0 / (x * (-1.0 - (y * (1.0 + (y * 0.5)))));
	tmp = 0.0;
	if (x <= -1.26e+169)
		tmp = t_0;
	elseif (x <= -4400.0)
		tmp = (1.0 + (y * ((y * 0.5) + -1.0))) / x;
	elseif (x <= 0.0032)
		tmp = 1.0 / x;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(-1.0 / N[(x * N[(-1.0 - N[(y * N[(1.0 + N[(y * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.26e+169], t$95$0, If[LessEqual[x, -4400.0], N[(N[(1.0 + N[(y * N[(N[(y * 0.5), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[x, 0.0032], N[(1.0 / x), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.2599999999999999e169 or 0.00320000000000000015 < x

   1. Initial program 63.9%

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

      1. *-commutative 63.9%

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

      2. exp-to-pow 63.9%

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

   3. Simplified 63.9%

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

   5. Taylor expanded in x around inf 100.0%

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

      1. mul-1-neg 100.0%

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

   7. Simplified 100.0%

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

      1. clear-num 100.0%

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

      2. inv-pow 100.0%

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

      3. exp-neg 100.0%

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

      4. associate-/r/ 100.0%

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

      5. /-rgt-identity 100.0%

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

   9. Applied egg-rr 100.0%

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

   10. Taylor expanded in x around 0 100.0%

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

   11. Taylor expanded in y around 0 75.1%

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

       1. *-commutative 75.1%

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

   13. Simplified 75.1%

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

   if -1.2599999999999999e169 < x < -4400

   1. Initial program 89.0%

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

      1. exp-prod 89.0%

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

   3. Simplified 89.0%

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

   5. Taylor expanded in y around 0 75.0%

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

   6. Taylor expanded in x around inf 75.0%

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

      1. *-commutative 75.0%

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

   8. Simplified 75.0%

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

   if -4400 < x < 0.00320000000000000015

   1. Initial program 88.8%

      \[\frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x} \]
   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 98.8%

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

4. Final simplification 84.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.26 \cdot 10^{+169}:\\ \;\;\;\;\frac{-1}{x \cdot \left(-1 - y \cdot \left(1 + y \cdot 0.5\right)\right)}\\ \mathbf{elif}\;x \leq -4400:\\ \;\;\;\;\frac{1 + y \cdot \left(y \cdot 0.5 + -1\right)}{x}\\ \mathbf{elif}\;x \leq 0.0032:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{x \cdot \left(-1 - y \cdot \left(1 + y \cdot 0.5\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 83.1% accurate, 9.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{x \cdot \left(1 + y\right)}\\ \mathbf{if}\;x \leq -3.1 \cdot 10^{+245}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq -4400:\\ \;\;\;\;\frac{\frac{x - x \cdot y}{x}}{x}\\ \mathbf{elif}\;x \leq 0.0032:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ 1.0 (* x (+ 1.0 y)))))
   (if (<= x -3.1e+245)
     t_0
     (if (<= x -4400.0)
       (/ (/ (- x (* x y)) x) x)
       (if (<= x 0.0032) (/ 1.0 x) t_0)))))

C

double code(double x, double y) {
	double t_0 = 1.0 / (x * (1.0 + y));
	double tmp;
	if (x <= -3.1e+245) {
		tmp = t_0;
	} else if (x <= -4400.0) {
		tmp = ((x - (x * y)) / x) / x;
	} else if (x <= 0.0032) {
		tmp = 1.0 / x;
	} 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 = 1.0d0 / (x * (1.0d0 + y))
    if (x <= (-3.1d+245)) then
        tmp = t_0
    else if (x <= (-4400.0d0)) then
        tmp = ((x - (x * y)) / x) / x
    else if (x <= 0.0032d0) then
        tmp = 1.0d0 / x
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = 1.0 / (x * (1.0 + y));
	double tmp;
	if (x <= -3.1e+245) {
		tmp = t_0;
	} else if (x <= -4400.0) {
		tmp = ((x - (x * y)) / x) / x;
	} else if (x <= 0.0032) {
		tmp = 1.0 / x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 1.0 / (x * (1.0 + y))
	tmp = 0
	if x <= -3.1e+245:
		tmp = t_0
	elif x <= -4400.0:
		tmp = ((x - (x * y)) / x) / x
	elif x <= 0.0032:
		tmp = 1.0 / x
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(1.0 / Float64(x * Float64(1.0 + y)))
	tmp = 0.0
	if (x <= -3.1e+245)
		tmp = t_0;
	elseif (x <= -4400.0)
		tmp = Float64(Float64(Float64(x - Float64(x * y)) / x) / x);
	elseif (x <= 0.0032)
		tmp = Float64(1.0 / x);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = 1.0 / (x * (1.0 + y));
	tmp = 0.0;
	if (x <= -3.1e+245)
		tmp = t_0;
	elseif (x <= -4400.0)
		tmp = ((x - (x * y)) / x) / x;
	elseif (x <= 0.0032)
		tmp = 1.0 / x;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(1.0 / N[(x * N[(1.0 + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -3.1e+245], t$95$0, If[LessEqual[x, -4400.0], N[(N[(N[(x - N[(x * y), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[x, 0.0032], N[(1.0 / x), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if x < -3.0999999999999999e245 or 0.00320000000000000015 < x

   1. Initial program 65.3%

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

      1. *-commutative 65.3%

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

      2. exp-to-pow 65.3%

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

   3. Simplified 65.3%

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

   5. Taylor expanded in x around inf 100.0%

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

      1. mul-1-neg 100.0%

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

   7. Simplified 100.0%

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

      1. clear-num 100.0%

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

      2. inv-pow 100.0%

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

      3. exp-neg 100.0%

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

      4. associate-/r/ 100.0%

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

      5. /-rgt-identity 100.0%

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

   9. Applied egg-rr 100.0%

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

   10. Taylor expanded in x around 0 100.0%

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

   11. Taylor expanded in y around 0 71.7%

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

       1. *-rgt-identity 71.7%

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

       2. distribute-lft-in 71.7%

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

   13. Simplified 71.7%

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

   if -3.0999999999999999e245 < x < -4400

   1. Initial program 79.9%

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

      1. exp-prod 79.9%

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

   3. Simplified 79.9%

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

   5. Taylor expanded in y around 0 54.1%

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

      1. +-commutative 54.1%

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

      2. mul-1-neg 54.1%

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

      3. unsub-neg 54.1%

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

   7. Simplified 54.1%

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

      1. frac-sub 51.4%

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

      2. associate-/r* 73.0%

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

      3. *-un-lft-identity 73.0%

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

      4. *-commutative 73.0%

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

   9. Applied egg-rr 73.0%

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

   if -4400 < x < 0.00320000000000000015

   1. Initial program 88.8%

      \[\frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x} \]
   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 98.8%

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

4. Final simplification 82.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.1 \cdot 10^{+245}:\\ \;\;\;\;\frac{1}{x \cdot \left(1 + y\right)}\\ \mathbf{elif}\;x \leq -4400:\\ \;\;\;\;\frac{\frac{x - x \cdot y}{x}}{x}\\ \mathbf{elif}\;x \leq 0.0032:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x \cdot \left(1 + y\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 81.0% accurate, 12.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -7.2 \cdot 10^{+103} \lor \neg \left(x \leq 0.0032\right):\\ \;\;\;\;\frac{1}{x \cdot \left(1 + y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -7.2e+103) (not (<= x 0.0032)))
   (/ 1.0 (* x (+ 1.0 y)))
   (/ 1.0 x)))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -7.2e+103) || !(x <= 0.0032)) {
		tmp = 1.0 / (x * (1.0 + 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 <= (-7.2d+103)) .or. (.not. (x <= 0.0032d0))) then
        tmp = 1.0d0 / (x * (1.0d0 + 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 <= -7.2e+103) || !(x <= 0.0032)) {
		tmp = 1.0 / (x * (1.0 + y));
	} else {
		tmp = 1.0 / x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -7.2e+103) or not (x <= 0.0032):
		tmp = 1.0 / (x * (1.0 + y))
	else:
		tmp = 1.0 / x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -7.2e+103) || !(x <= 0.0032))
		tmp = Float64(1.0 / Float64(x * Float64(1.0 + y)));
	else
		tmp = Float64(1.0 / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((x <= -7.2e+103) || ~((x <= 0.0032)))
		tmp = 1.0 / (x * (1.0 + y));
	else
		tmp = 1.0 / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -7.2e+103], N[Not[LessEqual[x, 0.0032]], $MachinePrecision]], N[(1.0 / N[(x * N[(1.0 + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -7.20000000000000033e103 or 0.00320000000000000015 < x

   1. Initial program 65.7%

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

      1. *-commutative 65.7%

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

      2. exp-to-pow 65.7%

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

   3. Simplified 65.7%

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

   5. Taylor expanded in x around inf 100.0%

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

      1. mul-1-neg 100.0%

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

   7. Simplified 100.0%

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

      1. clear-num 99.9%

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

      2. inv-pow 99.9%

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

      3. exp-neg 99.9%

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

      4. associate-/r/ 100.0%

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

      5. /-rgt-identity 100.0%

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

   9. Applied egg-rr 100.0%

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

   10. Taylor expanded in x around 0 100.0%

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

   11. Taylor expanded in y around 0 66.2%

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

       1. *-rgt-identity 66.2%

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

       2. distribute-lft-in 66.2%

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

   13. Simplified 66.2%

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

   if -7.20000000000000033e103 < x < 0.00320000000000000015

   1. Initial program 89.8%

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

      1. exp-prod 98.4%

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

   3. Simplified 98.4%

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

   5. Taylor expanded in x around 0 89.6%

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

4. Final simplification 78.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -7.2 \cdot 10^{+103} \lor \neg \left(x \leq 0.0032\right):\\ \;\;\;\;\frac{1}{x \cdot \left(1 + y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 77.3% accurate, 13.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3.8 \cdot 10^{+72}:\\ \;\;\;\;\frac{\frac{x \cdot y}{-x}}{x}\\ \mathbf{elif}\;y \leq 2.6 \cdot 10^{+21}:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -3.8e+72)
   (/ (/ (* x y) (- x)) x)
   (if (<= y 2.6e+21) (/ 1.0 x) (/ x (* x x)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -3.8e+72) {
		tmp = ((x * y) / -x) / x;
	} else if (y <= 2.6e+21) {
		tmp = 1.0 / x;
	} else {
		tmp = x / (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 (y <= (-3.8d+72)) then
        tmp = ((x * y) / -x) / x
    else if (y <= 2.6d+21) then
        tmp = 1.0d0 / x
    else
        tmp = x / (x * x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -3.8e+72) {
		tmp = ((x * y) / -x) / x;
	} else if (y <= 2.6e+21) {
		tmp = 1.0 / x;
	} else {
		tmp = x / (x * x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -3.8e+72:
		tmp = ((x * y) / -x) / x
	elif y <= 2.6e+21:
		tmp = 1.0 / x
	else:
		tmp = x / (x * x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -3.8e+72)
		tmp = Float64(Float64(Float64(x * y) / Float64(-x)) / x);
	elseif (y <= 2.6e+21)
		tmp = Float64(1.0 / x);
	else
		tmp = Float64(x / Float64(x * x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -3.8e+72)
		tmp = ((x * y) / -x) / x;
	elseif (y <= 2.6e+21)
		tmp = 1.0 / x;
	else
		tmp = x / (x * x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -3.8e+72], N[(N[(N[(x * y), $MachinePrecision] / (-x)), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[y, 2.6e+21], N[(1.0 / x), $MachinePrecision], N[(x / N[(x * x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -3.80000000000000006e72

   1. Initial program 51.4%

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

      1. exp-prod 65.4%

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

   3. Simplified 65.4%

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

   5. Taylor expanded in y around 0 3.9%

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

      1. +-commutative 3.9%

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

      2. mul-1-neg 3.9%

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

      3. unsub-neg 3.9%

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

   7. Simplified 3.9%

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

      1. frac-sub 14.7%

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

      2. associate-/r* 49.7%

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

      3. *-un-lft-identity 49.7%

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

      4. *-commutative 49.7%

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

   9. Applied egg-rr 49.7%

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

   10. Taylor expanded in y around inf 49.7%

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

       1. neg-mul-1 49.7%

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

       2. distribute-lft-neg-in 49.7%

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

   12. Simplified 49.7%

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

   if -3.80000000000000006e72 < y < 2.6e21

   1. Initial program 93.5%

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

      1. exp-prod 93.5%

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

   3. Simplified 93.5%

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

   5. Taylor expanded in x around 0 92.3%

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

   if 2.6e21 < y

   1. Initial program 49.0%

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

      1. exp-prod 60.7%

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

   3. Simplified 60.7%

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

   5. Taylor expanded in y around 0 2.5%

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

      1. +-commutative 2.5%

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

      2. mul-1-neg 2.5%

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

      3. unsub-neg 2.5%

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

   7. Simplified 2.5%

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

      1. frac-2neg 2.5%

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

      2. frac-sub 12.8%

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

      3. *-un-lft-identity 12.8%

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

      4. add-sqr-sqrt 0.0%

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

      5. sqrt-unprod 14.7%

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

      6. sqr-neg 14.7%

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

      7. sqrt-unprod 14.8%

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

      8. add-sqr-sqrt 14.8%

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

      9. *-commutative 14.8%

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

   9. Applied egg-rr 14.8%

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

   10. Taylor expanded in x around 0 14.8%

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

       1. distribute-lft-in 14.8%

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

       2. *-rgt-identity 14.8%

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

       3. distribute-lft-in 14.8%

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

       4. neg-mul-1 14.8%

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

       5. sub-neg 14.8%

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

       6. *-commutative 14.8%

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

       7. distribute-rgt-out-- 14.8%

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

   12. Simplified 14.8%

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

   13. Taylor expanded in y around 0 67.2%

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

       1. mul-1-neg 67.2%

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

   15. Simplified 67.2%

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

4. Final simplification 80.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.8 \cdot 10^{+72}:\\ \;\;\;\;\frac{\frac{x \cdot y}{-x}}{x}\\ \mathbf{elif}\;y \leq 2.6 \cdot 10^{+21}:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x \cdot x}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 77.6% accurate, 20.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.26 \cdot 10^{+20}:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= y 1.26e+20) (/ 1.0 x) (/ x (* x x))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 1.26e+20) {
		tmp = 1.0 / x;
	} else {
		tmp = x / (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 (y <= 1.26d+20) then
        tmp = 1.0d0 / x
    else
        tmp = x / (x * x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 1.26e+20) {
		tmp = 1.0 / x;
	} else {
		tmp = x / (x * x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 1.26e+20:
		tmp = 1.0 / x
	else:
		tmp = x / (x * x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 1.26e+20)
		tmp = Float64(1.0 / x);
	else
		tmp = Float64(x / Float64(x * x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 1.26e+20)
		tmp = 1.0 / x;
	else
		tmp = x / (x * x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 1.26e+20], N[(1.0 / x), $MachinePrecision], N[(x / N[(x * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.26e20

   1. Initial program 85.3%

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

      1. exp-prod 88.0%

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

   3. Simplified 88.0%

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

   5. Taylor expanded in x around 0 79.7%

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

   if 1.26e20 < y

   1. Initial program 49.0%

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

      1. exp-prod 60.7%

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

   3. Simplified 60.7%

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

   5. Taylor expanded in y around 0 2.5%

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

      1. +-commutative 2.5%

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

      2. mul-1-neg 2.5%

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

      3. unsub-neg 2.5%

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

   7. Simplified 2.5%

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

      1. frac-2neg 2.5%

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

      2. frac-sub 12.8%

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

      3. *-un-lft-identity 12.8%

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

      4. add-sqr-sqrt 0.0%

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

      5. sqrt-unprod 14.7%

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

      6. sqr-neg 14.7%

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

      7. sqrt-unprod 14.8%

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

      8. add-sqr-sqrt 14.8%

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

      9. *-commutative 14.8%

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

   9. Applied egg-rr 14.8%

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

   10. Taylor expanded in x around 0 14.8%

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

       1. distribute-lft-in 14.8%

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

       2. *-rgt-identity 14.8%

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

       3. distribute-lft-in 14.8%

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

       4. neg-mul-1 14.8%

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

       5. sub-neg 14.8%

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

       6. *-commutative 14.8%

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

       7. distribute-rgt-out-- 14.8%

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

   12. Simplified 14.8%

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

   13. Taylor expanded in y around 0 67.2%

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

       1. mul-1-neg 67.2%

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

   15. Simplified 67.2%

       \[\leadsto \frac{\color{blue}{-x}}{x \cdot \left(-x\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 77.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.26 \cdot 10^{+20}:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x \cdot x}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 75.0% 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 78.2%

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

   1. exp-prod 82.7%

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

3. Simplified 82.7%

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

5. Taylor expanded in x around 0 72.1%

   \[\leadsto \color{blue}{\frac{1}{x}} \]
6. Add Preprocessing

Developer Target 1: 77.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{e^{\frac{-1}{y}}}{x}\\ t_1 := \frac{{\left(\frac{x}{y + x}\right)}^{x}}{x}\\ \mathbf{if}\;y < -3.7311844206647956 \cdot 10^{+94}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y < 2.817959242728288 \cdot 10^{+37}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y < 2.347387415166998 \cdot 10^{+178}:\\ \;\;\;\;\log \left(e^{t\_1}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (exp (/ -1.0 y)) x)) (t_1 (/ (pow (/ x (+ y x)) x) x)))
   (if (< y -3.7311844206647956e+94)
     t_0
     (if (< y 2.817959242728288e+37)
       t_1
       (if (< y 2.347387415166998e+178) (log (exp t_1)) t_0)))))

C

double code(double x, double y) {
	double t_0 = exp((-1.0 / y)) / x;
	double t_1 = pow((x / (y + x)), x) / x;
	double tmp;
	if (y < -3.7311844206647956e+94) {
		tmp = t_0;
	} else if (y < 2.817959242728288e+37) {
		tmp = t_1;
	} else if (y < 2.347387415166998e+178) {
		tmp = log(exp(t_1));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = exp(((-1.0d0) / y)) / x
    t_1 = ((x / (y + x)) ** x) / x
    if (y < (-3.7311844206647956d+94)) then
        tmp = t_0
    else if (y < 2.817959242728288d+37) then
        tmp = t_1
    else if (y < 2.347387415166998d+178) then
        tmp = log(exp(t_1))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = Math.exp((-1.0 / y)) / x;
	double t_1 = Math.pow((x / (y + x)), x) / x;
	double tmp;
	if (y < -3.7311844206647956e+94) {
		tmp = t_0;
	} else if (y < 2.817959242728288e+37) {
		tmp = t_1;
	} else if (y < 2.347387415166998e+178) {
		tmp = Math.log(Math.exp(t_1));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.exp((-1.0 / y)) / x
	t_1 = math.pow((x / (y + x)), x) / x
	tmp = 0
	if y < -3.7311844206647956e+94:
		tmp = t_0
	elif y < 2.817959242728288e+37:
		tmp = t_1
	elif y < 2.347387415166998e+178:
		tmp = math.log(math.exp(t_1))
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(exp(Float64(-1.0 / y)) / x)
	t_1 = Float64((Float64(x / Float64(y + x)) ^ x) / x)
	tmp = 0.0
	if (y < -3.7311844206647956e+94)
		tmp = t_0;
	elseif (y < 2.817959242728288e+37)
		tmp = t_1;
	elseif (y < 2.347387415166998e+178)
		tmp = log(exp(t_1));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = exp((-1.0 / y)) / x;
	t_1 = ((x / (y + x)) ^ x) / x;
	tmp = 0.0;
	if (y < -3.7311844206647956e+94)
		tmp = t_0;
	elseif (y < 2.817959242728288e+37)
		tmp = t_1;
	elseif (y < 2.347387415166998e+178)
		tmp = log(exp(t_1));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Exp[N[(-1.0 / y), $MachinePrecision]], $MachinePrecision] / x), $MachinePrecision]}, Block[{t$95$1 = N[(N[Power[N[(x / N[(y + x), $MachinePrecision]), $MachinePrecision], x], $MachinePrecision] / x), $MachinePrecision]}, If[Less[y, -3.7311844206647956e+94], t$95$0, If[Less[y, 2.817959242728288e+37], t$95$1, If[Less[y, 2.347387415166998e+178], N[Log[N[Exp[t$95$1], $MachinePrecision]], $MachinePrecision], t$95$0]]]]]

Reproduce

herbie shell --seed 2024180 
(FPCore (x y)
  :name "Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2, F"
  :precision binary64

  :alt
  (! :herbie-platform default (if (< y -37311844206647956000000000000000000000000000000000000000000000000000000000000000000000000000000) (/ (exp (/ -1 y)) x) (if (< y 28179592427282880000000000000000000000) (/ (pow (/ x (+ y x)) x) x) (if (< y 23473874151669980000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000) (log (exp (/ (pow (/ x (+ y x)) x) x))) (/ (exp (/ -1 y)) x)))))

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