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

Percentage Accurate: 78.4% → 98.6%

Time: 17.4s

Alternatives: 10

Speedup: 18.8×

Specification

Math

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

FPCore

(FPCore (x y) :precision binary64 (/ (exp (* x (log (/ x (+ x y))))) x))

C

double code(double x, double y) {
	return exp((x * log((x / (x + y))))) / x;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = exp((x * log((x / (x + y))))) / x
end function

Java

public static double code(double x, double y) {
	return Math.exp((x * Math.log((x / (x + y))))) / x;
}

Python

def code(x, y):
	return math.exp((x * math.log((x / (x + y))))) / x

Julia

function code(x, y)
	return Float64(exp(Float64(x * log(Float64(x / Float64(x + y))))) / x)
end

MATLAB

function tmp = code(x, y)
	tmp = exp((x * log((x / (x + y))))) / x;
end

Wolfram

code[x_, y_] := N[(N[Exp[N[(x * N[Log[N[(x / N[(x + y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / x), $MachinePrecision]

Initial Program: 78.4% 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: 98.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-y}\\ \mathbf{if}\;x \leq -5.8 \cdot 10^{+128}:\\ \;\;\;\;\frac{t_0}{x}\\ \mathbf{elif}\;x \leq 3.8 \cdot 10^{-5}:\\ \;\;\;\;\frac{{\left(e^{x}\right)}^{\log \left(\frac{x}{x + y}\right)}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{x}{t_0}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (exp (- y))))
   (if (<= x -5.8e+128)
     (/ t_0 x)
     (if (<= x 3.8e-5)
       (/ (pow (exp x) (log (/ x (+ x y)))) x)
       (/ 1.0 (/ x t_0))))))

C

double code(double x, double y) {
	double t_0 = exp(-y);
	double tmp;
	if (x <= -5.8e+128) {
		tmp = t_0 / x;
	} else if (x <= 3.8e-5) {
		tmp = pow(exp(x), log((x / (x + y)))) / x;
	} else {
		tmp = 1.0 / (x / 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 = exp(-y)
    if (x <= (-5.8d+128)) then
        tmp = t_0 / x
    else if (x <= 3.8d-5) then
        tmp = (exp(x) ** log((x / (x + y)))) / x
    else
        tmp = 1.0d0 / (x / t_0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = Math.exp(-y);
	double tmp;
	if (x <= -5.8e+128) {
		tmp = t_0 / x;
	} else if (x <= 3.8e-5) {
		tmp = Math.pow(Math.exp(x), Math.log((x / (x + y)))) / x;
	} else {
		tmp = 1.0 / (x / t_0);
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.exp(-y)
	tmp = 0
	if x <= -5.8e+128:
		tmp = t_0 / x
	elif x <= 3.8e-5:
		tmp = math.pow(math.exp(x), math.log((x / (x + y)))) / x
	else:
		tmp = 1.0 / (x / t_0)
	return tmp

Julia

function code(x, y)
	t_0 = exp(Float64(-y))
	tmp = 0.0
	if (x <= -5.8e+128)
		tmp = Float64(t_0 / x);
	elseif (x <= 3.8e-5)
		tmp = Float64((exp(x) ^ log(Float64(x / Float64(x + y)))) / x);
	else
		tmp = Float64(1.0 / Float64(x / t_0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = exp(-y);
	tmp = 0.0;
	if (x <= -5.8e+128)
		tmp = t_0 / x;
	elseif (x <= 3.8e-5)
		tmp = (exp(x) ^ log((x / (x + y)))) / x;
	else
		tmp = 1.0 / (x / t_0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[Exp[(-y)], $MachinePrecision]}, If[LessEqual[x, -5.8e+128], N[(t$95$0 / x), $MachinePrecision], If[LessEqual[x, 3.8e-5], N[(N[Power[N[Exp[x], $MachinePrecision], N[Log[N[(x / N[(x + y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] / x), $MachinePrecision], N[(1.0 / N[(x / t$95$0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -5.8000000000000001e128

   1. Initial program 50.7%

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

      1. *-commutative 50.7%

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

      2. exp-to-pow 50.7%

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

   3. Simplified 50.7%

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

   4. Taylor expanded in x around inf 100.0%

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

      1. mul-1-neg 100.0%

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

   6. Simplified 100.0%

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

   if -5.8000000000000001e128 < x < 3.8000000000000002e-5

   1. Initial program 87.2%

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

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

   if 3.8000000000000002e-5 < x

   1. Initial program 79.6%

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

      1. exp-prod 79.6%

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

   3. Simplified 79.6%

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

      1. clear-num 79.6%

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

      2. inv-pow 79.6%

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

      3. add-exp-log 79.6%

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

      4. log-pow 22.4%

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

      5. add-log-exp 79.6%

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

      6. pow-to-exp 79.6%

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

   5. Applied egg-rr 79.6%

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

      1. unpow-1 79.6%

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

   7. Simplified 79.6%

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

   8. Taylor expanded in x around inf 100.0%

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

      1. mul-1-neg 100.0%

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

   10. Simplified 100.0%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.8 \cdot 10^{+128}:\\ \;\;\;\;\frac{e^{-y}}{x}\\ \mathbf{elif}\;x \leq 3.8 \cdot 10^{-5}:\\ \;\;\;\;\frac{{\left(e^{x}\right)}^{\log \left(\frac{x}{x + y}\right)}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{x}{e^{-y}}}\\ \end{array} \]

Alternative 2: 98.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-y}\\ \mathbf{if}\;x \leq -9 \cdot 10^{+27}:\\ \;\;\;\;\frac{t_0}{x}\\ \mathbf{elif}\;x \leq 3.8 \cdot 10^{-5}:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{x}{t_0}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (exp (- y))))
   (if (<= x -9e+27)
     (/ t_0 x)
     (if (<= x 3.8e-5) (/ 1.0 x) (/ 1.0 (/ x t_0))))))

C

double code(double x, double y) {
	double t_0 = exp(-y);
	double tmp;
	if (x <= -9e+27) {
		tmp = t_0 / x;
	} else if (x <= 3.8e-5) {
		tmp = 1.0 / x;
	} else {
		tmp = 1.0 / (x / 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 = exp(-y)
    if (x <= (-9d+27)) then
        tmp = t_0 / x
    else if (x <= 3.8d-5) then
        tmp = 1.0d0 / x
    else
        tmp = 1.0d0 / (x / t_0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = Math.exp(-y);
	double tmp;
	if (x <= -9e+27) {
		tmp = t_0 / x;
	} else if (x <= 3.8e-5) {
		tmp = 1.0 / x;
	} else {
		tmp = 1.0 / (x / t_0);
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.exp(-y)
	tmp = 0
	if x <= -9e+27:
		tmp = t_0 / x
	elif x <= 3.8e-5:
		tmp = 1.0 / x
	else:
		tmp = 1.0 / (x / t_0)
	return tmp

Julia

function code(x, y)
	t_0 = exp(Float64(-y))
	tmp = 0.0
	if (x <= -9e+27)
		tmp = Float64(t_0 / x);
	elseif (x <= 3.8e-5)
		tmp = Float64(1.0 / x);
	else
		tmp = Float64(1.0 / Float64(x / t_0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = exp(-y);
	tmp = 0.0;
	if (x <= -9e+27)
		tmp = t_0 / x;
	elseif (x <= 3.8e-5)
		tmp = 1.0 / x;
	else
		tmp = 1.0 / (x / t_0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[Exp[(-y)], $MachinePrecision]}, If[LessEqual[x, -9e+27], N[(t$95$0 / x), $MachinePrecision], If[LessEqual[x, 3.8e-5], N[(1.0 / x), $MachinePrecision], N[(1.0 / N[(x / t$95$0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -8.9999999999999998e27

   1. Initial program 67.6%

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

      1. *-commutative 67.6%

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

      2. exp-to-pow 67.6%

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

   3. Simplified 67.6%

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

   4. Taylor expanded in x around inf 100.0%

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

      1. mul-1-neg 100.0%

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

   6. Simplified 100.0%

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

   if -8.9999999999999998e27 < x < 3.8000000000000002e-5

   1. Initial program 84.4%

      \[\frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x} \]
   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. Taylor expanded in x around 0 99.8%

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

   if 3.8000000000000002e-5 < x

   1. Initial program 79.6%

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

      1. exp-prod 79.6%

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

   3. Simplified 79.6%

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

      1. clear-num 79.6%

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

      2. inv-pow 79.6%

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

      3. add-exp-log 79.6%

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

      4. log-pow 22.4%

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

      5. add-log-exp 79.6%

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

      6. pow-to-exp 79.6%

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

   5. Applied egg-rr 79.6%

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

      1. unpow-1 79.6%

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

   7. Simplified 79.6%

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

   8. Taylor expanded in x around inf 100.0%

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

      1. mul-1-neg 100.0%

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

   10. Simplified 100.0%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -9 \cdot 10^{+27}:\\ \;\;\;\;\frac{e^{-y}}{x}\\ \mathbf{elif}\;x \leq 3.8 \cdot 10^{-5}:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{x}{e^{-y}}}\\ \end{array} \]

Alternative 3: 98.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -9 \cdot 10^{+27} \lor \neg \left(x \leq 3.8 \cdot 10^{-5}\right):\\ \;\;\;\;\frac{e^{-y}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -9e+27) (not (<= x 3.8e-5))) (/ (exp (- y)) x) (/ 1.0 x)))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -9e+27) || !(x <= 3.8e-5)) {
		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 <= (-9d+27)) .or. (.not. (x <= 3.8d-5))) 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 <= -9e+27) || !(x <= 3.8e-5)) {
		tmp = Math.exp(-y) / x;
	} else {
		tmp = 1.0 / x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -9e+27) or not (x <= 3.8e-5):
		tmp = math.exp(-y) / x
	else:
		tmp = 1.0 / x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -9e+27) || !(x <= 3.8e-5))
		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 <= -9e+27) || ~((x <= 3.8e-5)))
		tmp = exp(-y) / x;
	else
		tmp = 1.0 / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -9e+27], N[Not[LessEqual[x, 3.8e-5]], $MachinePrecision]], N[(N[Exp[(-y)], $MachinePrecision] / x), $MachinePrecision], N[(1.0 / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -8.9999999999999998e27 or 3.8000000000000002e-5 < x

   1. Initial program 74.3%

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

      1. *-commutative 74.3%

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

      2. exp-to-pow 74.3%

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

   3. Simplified 74.3%

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

   4. Taylor expanded in x around inf 100.0%

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

      1. mul-1-neg 100.0%

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

   6. Simplified 100.0%

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

   if -8.9999999999999998e27 < x < 3.8000000000000002e-5

   1. Initial program 84.4%

      \[\frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x} \]
   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. Taylor expanded in x around 0 99.8%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -9 \cdot 10^{+27} \lor \neg \left(x \leq 3.8 \cdot 10^{-5}\right):\\ \;\;\;\;\frac{e^{-y}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x}\\ \end{array} \]

Alternative 4: 85.3% accurate, 9.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -9 \cdot 10^{+27}:\\ \;\;\;\;\frac{y \cdot y}{x} - \frac{y + -1}{x}\\ \mathbf{elif}\;x \leq 3.8 \cdot 10^{-5}:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x + \left(x \cdot y + x \cdot \left(\left(y \cdot y\right) \cdot \left(0.5 + y \cdot 0.16666666666666666\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -9e+27)
   (- (/ (* y y) x) (/ (+ y -1.0) x))
   (if (<= x 3.8e-5)
     (/ 1.0 x)
     (/
      1.0
      (+ x (+ (* x y) (* x (* (* y y) (+ 0.5 (* y 0.16666666666666666))))))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -9e+27) {
		tmp = ((y * y) / x) - ((y + -1.0) / x);
	} else if (x <= 3.8e-5) {
		tmp = 1.0 / x;
	} else {
		tmp = 1.0 / (x + ((x * y) + (x * ((y * y) * (0.5 + (y * 0.16666666666666666))))));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-9d+27)) then
        tmp = ((y * y) / x) - ((y + (-1.0d0)) / x)
    else if (x <= 3.8d-5) then
        tmp = 1.0d0 / x
    else
        tmp = 1.0d0 / (x + ((x * y) + (x * ((y * y) * (0.5d0 + (y * 0.16666666666666666d0))))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -9e+27) {
		tmp = ((y * y) / x) - ((y + -1.0) / x);
	} else if (x <= 3.8e-5) {
		tmp = 1.0 / x;
	} else {
		tmp = 1.0 / (x + ((x * y) + (x * ((y * y) * (0.5 + (y * 0.16666666666666666))))));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -9e+27:
		tmp = ((y * y) / x) - ((y + -1.0) / x)
	elif x <= 3.8e-5:
		tmp = 1.0 / x
	else:
		tmp = 1.0 / (x + ((x * y) + (x * ((y * y) * (0.5 + (y * 0.16666666666666666))))))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -9e+27)
		tmp = Float64(Float64(Float64(y * y) / x) - Float64(Float64(y + -1.0) / x));
	elseif (x <= 3.8e-5)
		tmp = Float64(1.0 / x);
	else
		tmp = Float64(1.0 / Float64(x + Float64(Float64(x * y) + Float64(x * Float64(Float64(y * y) * Float64(0.5 + Float64(y * 0.16666666666666666)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -9e+27)
		tmp = ((y * y) / x) - ((y + -1.0) / x);
	elseif (x <= 3.8e-5)
		tmp = 1.0 / x;
	else
		tmp = 1.0 / (x + ((x * y) + (x * ((y * y) * (0.5 + (y * 0.16666666666666666))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -9e+27], N[(N[(N[(y * y), $MachinePrecision] / x), $MachinePrecision] - N[(N[(y + -1.0), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 3.8e-5], N[(1.0 / x), $MachinePrecision], N[(1.0 / N[(x + N[(N[(x * y), $MachinePrecision] + N[(x * N[(N[(y * y), $MachinePrecision] * N[(0.5 + N[(y * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -8.9999999999999998e27

   1. Initial program 67.6%

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

      1. exp-prod 67.6%

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

   3. Simplified 67.6%

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

      1. clear-num 67.6%

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

      2. inv-pow 67.6%

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

      3. add-exp-log 67.6%

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

      4. log-pow 19.4%

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

      5. add-log-exp 67.6%

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

      6. pow-to-exp 67.6%

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

   5. Applied egg-rr 67.6%

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

      1. unpow-1 67.6%

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

   7. Simplified 67.6%

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

   8. Taylor expanded in x around inf 100.0%

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

      1. mul-1-neg 100.0%

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

   10. Simplified 100.0%

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

   11. Taylor expanded in y around 0 57.1%

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

   12. Taylor expanded in y around 0 68.7%

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

       1. associate-+r+ 68.7%

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

       2. associate-*r/ 68.7%

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

       3. associate-*l/ 68.7%

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

       4. metadata-eval 68.7%

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

       5. associate-*l/ 68.7%

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

       6. distribute-lft-in 68.7%

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

       7. associate-*l/ 68.7%

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

       8. mul-1-neg 68.7%

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

       9. distribute-frac-neg 68.7%

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

       10. unpow2 68.7%

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

   14. Simplified 68.7%

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

   if -8.9999999999999998e27 < x < 3.8000000000000002e-5

   1. Initial program 84.4%

      \[\frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x} \]
   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. Taylor expanded in x around 0 99.8%

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

   if 3.8000000000000002e-5 < x

   1. Initial program 79.6%

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

      1. exp-prod 79.6%

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

   3. Simplified 79.6%

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

      1. clear-num 79.6%

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

      2. inv-pow 79.6%

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

      3. add-exp-log 79.6%

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

      4. log-pow 22.4%

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

      5. add-log-exp 79.6%

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

      6. pow-to-exp 79.6%

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

   5. Applied egg-rr 79.6%

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

      1. unpow-1 79.6%

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

   7. Simplified 79.6%

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

   8. Taylor expanded in x around inf 100.0%

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

      1. mul-1-neg 100.0%

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

   10. Simplified 100.0%

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

   11. Taylor expanded in y around 0 86.1%

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

       1. associate-+r+ 86.1%

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

       2. +-commutative 86.1%

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

       3. distribute-lft-out 86.1%

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

       4. neg-mul-1 86.1%

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

       5. unsub-neg 86.1%

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

       6. unpow2 86.1%

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

       7. unpow3 86.1%

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

       8. associate-*l* 86.1%

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

       9. distribute-lft-out 86.2%

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

   13. Simplified 86.2%

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

   14. Taylor expanded in x around -inf 86.2%

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

       1. associate-*r* 86.2%

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

       2. neg-mul-1 86.2%

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

       3. unpow2 86.2%

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

       4. *-commutative 86.2%

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

       5. *-commutative 86.2%

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

   16. Simplified 86.2%

       \[\leadsto \frac{1}{x + \left(x \cdot y - \color{blue}{\left(-x\right) \cdot \left(\left(0.5 + y \cdot 0.16666666666666666\right) \cdot \left(y \cdot y\right)\right)}\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 87.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -9 \cdot 10^{+27}:\\ \;\;\;\;\frac{y \cdot y}{x} - \frac{y + -1}{x}\\ \mathbf{elif}\;x \leq 3.8 \cdot 10^{-5}:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x + \left(x \cdot y + x \cdot \left(\left(y \cdot y\right) \cdot \left(0.5 + y \cdot 0.16666666666666666\right)\right)\right)}\\ \end{array} \]

Alternative 5: 84.8% accurate, 10.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -9 \cdot 10^{+27}:\\ \;\;\;\;\frac{y \cdot y}{x} - \frac{y + -1}{x}\\ \mathbf{elif}\;x \leq 4 \cdot 10^{-6}:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x + \left(x \cdot y - -0.5 \cdot \left(x \cdot \left(y \cdot y\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -9e+27)
   (- (/ (* y y) x) (/ (+ y -1.0) x))
   (if (<= x 4e-6)
     (/ 1.0 x)
     (/ 1.0 (+ x (- (* x y) (* -0.5 (* x (* y y)))))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -9e+27) {
		tmp = ((y * y) / x) - ((y + -1.0) / x);
	} else if (x <= 4e-6) {
		tmp = 1.0 / x;
	} else {
		tmp = 1.0 / (x + ((x * y) - (-0.5 * (x * (y * 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 <= (-9d+27)) then
        tmp = ((y * y) / x) - ((y + (-1.0d0)) / x)
    else if (x <= 4d-6) then
        tmp = 1.0d0 / x
    else
        tmp = 1.0d0 / (x + ((x * y) - ((-0.5d0) * (x * (y * y)))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -9e+27) {
		tmp = ((y * y) / x) - ((y + -1.0) / x);
	} else if (x <= 4e-6) {
		tmp = 1.0 / x;
	} else {
		tmp = 1.0 / (x + ((x * y) - (-0.5 * (x * (y * y)))));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -9e+27:
		tmp = ((y * y) / x) - ((y + -1.0) / x)
	elif x <= 4e-6:
		tmp = 1.0 / x
	else:
		tmp = 1.0 / (x + ((x * y) - (-0.5 * (x * (y * y)))))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -9e+27)
		tmp = Float64(Float64(Float64(y * y) / x) - Float64(Float64(y + -1.0) / x));
	elseif (x <= 4e-6)
		tmp = Float64(1.0 / x);
	else
		tmp = Float64(1.0 / Float64(x + Float64(Float64(x * y) - Float64(-0.5 * Float64(x * Float64(y * y))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -9e+27)
		tmp = ((y * y) / x) - ((y + -1.0) / x);
	elseif (x <= 4e-6)
		tmp = 1.0 / x;
	else
		tmp = 1.0 / (x + ((x * y) - (-0.5 * (x * (y * y)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -9e+27], N[(N[(N[(y * y), $MachinePrecision] / x), $MachinePrecision] - N[(N[(y + -1.0), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 4e-6], N[(1.0 / x), $MachinePrecision], N[(1.0 / N[(x + N[(N[(x * y), $MachinePrecision] - N[(-0.5 * N[(x * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -8.9999999999999998e27

   1. Initial program 67.6%

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

      1. exp-prod 67.6%

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

   3. Simplified 67.6%

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

      1. clear-num 67.6%

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

      2. inv-pow 67.6%

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

      3. add-exp-log 67.6%

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

      4. log-pow 19.4%

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

      5. add-log-exp 67.6%

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

      6. pow-to-exp 67.6%

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

   5. Applied egg-rr 67.6%

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

      1. unpow-1 67.6%

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

   7. Simplified 67.6%

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

   8. Taylor expanded in x around inf 100.0%

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

      1. mul-1-neg 100.0%

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

   10. Simplified 100.0%

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

   11. Taylor expanded in y around 0 57.1%

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

   12. Taylor expanded in y around 0 68.7%

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

       1. associate-+r+ 68.7%

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

       2. associate-*r/ 68.7%

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

       3. associate-*l/ 68.7%

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

       4. metadata-eval 68.7%

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

       5. associate-*l/ 68.7%

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

       6. distribute-lft-in 68.7%

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

       7. associate-*l/ 68.7%

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

       8. mul-1-neg 68.7%

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

       9. distribute-frac-neg 68.7%

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

       10. unpow2 68.7%

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

   14. Simplified 68.7%

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

   if -8.9999999999999998e27 < x < 3.99999999999999982e-6

   1. Initial program 84.4%

      \[\frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x} \]
   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. Taylor expanded in x around 0 99.8%

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

   if 3.99999999999999982e-6 < x

   1. Initial program 79.6%

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

      1. exp-prod 79.6%

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

   3. Simplified 79.6%

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

      1. clear-num 79.6%

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

      2. inv-pow 79.6%

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

      3. add-exp-log 79.6%

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

      4. log-pow 22.4%

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

      5. add-log-exp 79.6%

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

      6. pow-to-exp 79.6%

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

   5. Applied egg-rr 79.6%

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

      1. unpow-1 79.6%

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

   7. Simplified 79.6%

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

   8. Taylor expanded in x around inf 100.0%

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

      1. mul-1-neg 100.0%

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

   10. Simplified 100.0%

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

   11. Taylor expanded in y around 0 86.1%

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

       1. associate-+r+ 86.1%

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

       2. +-commutative 86.1%

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

       3. distribute-lft-out 86.1%

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

       4. neg-mul-1 86.1%

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

       5. unsub-neg 86.1%

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

       6. unpow2 86.1%

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

       7. unpow3 86.1%

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

       8. associate-*l* 86.1%

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

       9. distribute-lft-out 86.2%

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

   13. Simplified 86.2%

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

   14. Taylor expanded in y around 0 82.9%

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

       1. unpow2 82.9%

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

   16. Simplified 82.9%

       \[\leadsto \frac{1}{x + \left(x \cdot y - \color{blue}{-0.5 \cdot \left(x \cdot \left(y \cdot y\right)\right)}\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 86.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -9 \cdot 10^{+27}:\\ \;\;\;\;\frac{y \cdot y}{x} - \frac{y + -1}{x}\\ \mathbf{elif}\;x \leq 4 \cdot 10^{-6}:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x + \left(x \cdot y - -0.5 \cdot \left(x \cdot \left(y \cdot y\right)\right)\right)}\\ \end{array} \]

Alternative 6: 82.8% accurate, 16.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -9 \cdot 10^{+27}:\\ \;\;\;\;\frac{y \cdot y}{x} - \frac{y + -1}{x}\\ \mathbf{elif}\;x \leq 3.6 \cdot 10^{-5}:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x + x \cdot y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -9e+27)
   (- (/ (* y y) x) (/ (+ y -1.0) x))
   (if (<= x 3.6e-5) (/ 1.0 x) (/ 1.0 (+ x (* x y))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -9e+27) {
		tmp = ((y * y) / x) - ((y + -1.0) / x);
	} else if (x <= 3.6e-5) {
		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 <= (-9d+27)) then
        tmp = ((y * y) / x) - ((y + (-1.0d0)) / x)
    else if (x <= 3.6d-5) 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 <= -9e+27) {
		tmp = ((y * y) / x) - ((y + -1.0) / x);
	} else if (x <= 3.6e-5) {
		tmp = 1.0 / x;
	} else {
		tmp = 1.0 / (x + (x * y));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -9e+27:
		tmp = ((y * y) / x) - ((y + -1.0) / x)
	elif x <= 3.6e-5:
		tmp = 1.0 / x
	else:
		tmp = 1.0 / (x + (x * y))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -9e+27)
		tmp = Float64(Float64(Float64(y * y) / x) - Float64(Float64(y + -1.0) / x));
	elseif (x <= 3.6e-5)
		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 <= -9e+27)
		tmp = ((y * y) / x) - ((y + -1.0) / x);
	elseif (x <= 3.6e-5)
		tmp = 1.0 / x;
	else
		tmp = 1.0 / (x + (x * y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -9e+27], N[(N[(N[(y * y), $MachinePrecision] / x), $MachinePrecision] - N[(N[(y + -1.0), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 3.6e-5], 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 < -8.9999999999999998e27

   1. Initial program 67.6%

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

      1. exp-prod 67.6%

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

   3. Simplified 67.6%

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

      1. clear-num 67.6%

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

      2. inv-pow 67.6%

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

      3. add-exp-log 67.6%

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

      4. log-pow 19.4%

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

      5. add-log-exp 67.6%

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

      6. pow-to-exp 67.6%

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

   5. Applied egg-rr 67.6%

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

      1. unpow-1 67.6%

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

   7. Simplified 67.6%

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

   8. Taylor expanded in x around inf 100.0%

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

      1. mul-1-neg 100.0%

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

   10. Simplified 100.0%

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

   11. Taylor expanded in y around 0 57.1%

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

   12. Taylor expanded in y around 0 68.7%

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

       1. associate-+r+ 68.7%

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

       2. associate-*r/ 68.7%

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

       3. associate-*l/ 68.7%

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

       4. metadata-eval 68.7%

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

       5. associate-*l/ 68.7%

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

       6. distribute-lft-in 68.7%

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

       7. associate-*l/ 68.7%

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

       8. mul-1-neg 68.7%

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

       9. distribute-frac-neg 68.7%

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

       10. unpow2 68.7%

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

   14. Simplified 68.7%

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

   if -8.9999999999999998e27 < x < 3.60000000000000009e-5

   1. Initial program 84.4%

      \[\frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x} \]
   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. Taylor expanded in x around 0 99.8%

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

   if 3.60000000000000009e-5 < x

   1. Initial program 79.6%

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

      1. exp-prod 79.6%

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

   3. Simplified 79.6%

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

      1. clear-num 79.6%

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

      2. inv-pow 79.6%

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

      3. add-exp-log 79.6%

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

      4. log-pow 22.4%

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

      5. add-log-exp 79.6%

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

      6. pow-to-exp 79.6%

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

   5. Applied egg-rr 79.6%

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

      1. unpow-1 79.6%

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

   7. Simplified 79.6%

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

   8. Taylor expanded in x around inf 100.0%

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

      1. mul-1-neg 100.0%

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

   10. Simplified 100.0%

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

   11. Taylor expanded in y around 0 78.5%

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

4. Final simplification 84.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -9 \cdot 10^{+27}:\\ \;\;\;\;\frac{y \cdot y}{x} - \frac{y + -1}{x}\\ \mathbf{elif}\;x \leq 3.6 \cdot 10^{-5}:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x + x \cdot y}\\ \end{array} \]

Alternative 7: 81.1% accurate, 18.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.16 \cdot 10^{+140} \lor \neg \left(x \leq 3.8 \cdot 10^{-5}\right):\\ \;\;\;\;\frac{1}{x + x \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -1.16e+140) (not (<= x 3.8e-5)))
   (/ 1.0 (+ x (* x y)))
   (/ 1.0 x)))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -1.16e+140) || !(x <= 3.8e-5)) {
		tmp = 1.0 / (x + (x * y));
	} else {
		tmp = 1.0 / x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((x <= (-1.16d+140)) .or. (.not. (x <= 3.8d-5))) then
        tmp = 1.0d0 / (x + (x * y))
    else
        tmp = 1.0d0 / x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((x <= -1.16e+140) || !(x <= 3.8e-5)) {
		tmp = 1.0 / (x + (x * y));
	} else {
		tmp = 1.0 / x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -1.16e+140) or not (x <= 3.8e-5):
		tmp = 1.0 / (x + (x * y))
	else:
		tmp = 1.0 / x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -1.16e+140) || !(x <= 3.8e-5))
		tmp = Float64(1.0 / Float64(x + Float64(x * y)));
	else
		tmp = Float64(1.0 / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((x <= -1.16e+140) || ~((x <= 3.8e-5)))
		tmp = 1.0 / (x + (x * y));
	else
		tmp = 1.0 / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -1.16e+140], N[Not[LessEqual[x, 3.8e-5]], $MachinePrecision]], N[(1.0 / N[(x + N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.16e140 or 3.8000000000000002e-5 < x

   1. Initial program 69.8%

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

      1. exp-prod 69.8%

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

   3. Simplified 69.8%

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

      1. clear-num 69.8%

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

      2. inv-pow 69.8%

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

      3. add-exp-log 69.8%

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

      4. log-pow 17.5%

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

      5. add-log-exp 69.8%

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

      6. pow-to-exp 69.8%

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

   5. Applied egg-rr 69.8%

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

      1. unpow-1 69.8%

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

   7. Simplified 69.8%

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

   8. Taylor expanded in x around inf 100.0%

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

      1. mul-1-neg 100.0%

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

   10. Simplified 100.0%

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

   11. Taylor expanded in y around 0 71.3%

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

   if -1.16e140 < x < 3.8000000000000002e-5

   1. Initial program 86.8%

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

      1. exp-prod 99.2%

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

   3. Simplified 99.2%

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

   4. Taylor expanded in x around 0 91.6%

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

4. Final simplification 81.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.16 \cdot 10^{+140} \lor \neg \left(x \leq 3.8 \cdot 10^{-5}\right):\\ \;\;\;\;\frac{1}{x + x \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x}\\ \end{array} \]

Alternative 8: 82.6% accurate, 18.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -9 \cdot 10^{+27}:\\ \;\;\;\;\frac{-1 - y \cdot y}{-x}\\ \mathbf{elif}\;x \leq 3.8 \cdot 10^{-5}:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x + x \cdot y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -9e+27)
   (/ (- -1.0 (* y y)) (- x))
   (if (<= x 3.8e-5) (/ 1.0 x) (/ 1.0 (+ x (* x y))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -9e+27) {
		tmp = (-1.0 - (y * y)) / -x;
	} else if (x <= 3.8e-5) {
		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 <= (-9d+27)) then
        tmp = ((-1.0d0) - (y * y)) / -x
    else if (x <= 3.8d-5) 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 <= -9e+27) {
		tmp = (-1.0 - (y * y)) / -x;
	} else if (x <= 3.8e-5) {
		tmp = 1.0 / x;
	} else {
		tmp = 1.0 / (x + (x * y));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -9e+27:
		tmp = (-1.0 - (y * y)) / -x
	elif x <= 3.8e-5:
		tmp = 1.0 / x
	else:
		tmp = 1.0 / (x + (x * y))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -9e+27)
		tmp = Float64(Float64(-1.0 - Float64(y * y)) / Float64(-x));
	elseif (x <= 3.8e-5)
		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 <= -9e+27)
		tmp = (-1.0 - (y * y)) / -x;
	elseif (x <= 3.8e-5)
		tmp = 1.0 / x;
	else
		tmp = 1.0 / (x + (x * y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -9e+27], N[(N[(-1.0 - N[(y * y), $MachinePrecision]), $MachinePrecision] / (-x)), $MachinePrecision], If[LessEqual[x, 3.8e-5], 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 < -8.9999999999999998e27

   1. Initial program 67.6%

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

      1. exp-prod 67.6%

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

   3. Simplified 67.6%

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

   4. Taylor expanded in x around inf 48.7%

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

      1. mul-1-neg 48.7%

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

      2. unsub-neg 48.7%

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

   6. Simplified 48.7%

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

      1. frac-2neg 48.7%

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

      2. div-inv 48.7%

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

      3. sub-neg 48.7%

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

      4. distribute-neg-in 48.7%

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

      5. metadata-eval 48.7%

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

      6. add-sqr-sqrt 19.3%

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

      7. sqrt-unprod 68.7%

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

      8. sqr-neg 68.7%

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

      9. sqrt-unprod 29.3%

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

      10. add-sqr-sqrt 47.7%

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

      11. add-sqr-sqrt 18.4%

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

      12. sqrt-unprod 47.5%

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

      13. sqr-neg 47.5%

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

      14. sqrt-unprod 29.3%

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

      15. add-sqr-sqrt 48.7%

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

   8. Applied egg-rr 48.7%

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

      1. flip-+ 68.7%

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

      2. frac-2neg 68.7%

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

      3. metadata-eval 68.7%

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

      4. frac-times 55.0%

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

      5. metadata-eval 55.0%

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

      6. cancel-sign-sub-inv 55.0%

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

      7. add-sqr-sqrt 23.0%

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

      8. sqrt-unprod 55.0%

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

      9. sqr-neg 55.0%

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

      10. sqrt-unprod 32.0%

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

      11. add-sqr-sqrt 50.3%

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

      12. sub-neg 50.3%

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

      13. add-sqr-sqrt 18.3%

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

      14. sqrt-unprod 50.3%

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

      15. sqr-neg 50.3%

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

      16. sqrt-unprod 32.0%

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

      17. add-sqr-sqrt 54.9%

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

      18. remove-double-neg 54.9%

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

   10. Applied egg-rr 54.9%

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

       1. *-commutative 54.9%

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

       2. *-commutative 54.9%

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

       3. mul-1-neg 54.9%

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

   12. Simplified 54.9%

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

   13. Taylor expanded in y around 0 68.6%

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

       1. neg-mul-1 68.6%

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

   15. Simplified 68.6%

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

   if -8.9999999999999998e27 < x < 3.8000000000000002e-5

   1. Initial program 84.4%

      \[\frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x} \]
   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. Taylor expanded in x around 0 99.8%

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

   if 3.8000000000000002e-5 < x

   1. Initial program 79.6%

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

      1. exp-prod 79.6%

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

   3. Simplified 79.6%

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

      1. clear-num 79.6%

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

      2. inv-pow 79.6%

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

      3. add-exp-log 79.6%

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

      4. log-pow 22.4%

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

      5. add-log-exp 79.6%

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

      6. pow-to-exp 79.6%

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

   5. Applied egg-rr 79.6%

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

      1. unpow-1 79.6%

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

   7. Simplified 79.6%

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

   8. Taylor expanded in x around inf 100.0%

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

      1. mul-1-neg 100.0%

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

   10. Simplified 100.0%

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

   11. Taylor expanded in y around 0 78.5%

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

4. Final simplification 84.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -9 \cdot 10^{+27}:\\ \;\;\;\;\frac{-1 - y \cdot y}{-x}\\ \mathbf{elif}\;x \leq 3.8 \cdot 10^{-5}:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x + x \cdot y}\\ \end{array} \]

Alternative 9: 75.1% accurate, 29.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 5 \cdot 10^{+232}:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x \cdot y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= y 5e+232) (/ 1.0 x) (/ 1.0 (* x y))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 5e+232) {
		tmp = 1.0 / x;
	} else {
		tmp = 1.0 / (x * y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 5d+232) then
        tmp = 1.0d0 / x
    else
        tmp = 1.0d0 / (x * y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 5e+232) {
		tmp = 1.0 / x;
	} else {
		tmp = 1.0 / (x * y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 5e+232:
		tmp = 1.0 / x
	else:
		tmp = 1.0 / (x * y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 5e+232)
		tmp = Float64(1.0 / x);
	else
		tmp = Float64(1.0 / Float64(x * y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 5e+232)
		tmp = 1.0 / x;
	else
		tmp = 1.0 / (x * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 5e+232], N[(1.0 / x), $MachinePrecision], N[(1.0 / N[(x * y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 4.99999999999999987e232

   1. Initial program 77.5%

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

      1. exp-prod 84.2%

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

   3. Simplified 84.2%

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

   4. Taylor expanded in x around 0 77.3%

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

   if 4.99999999999999987e232 < y

   1. Initial program 93.9%

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

      1. exp-prod 93.3%

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

   3. Simplified 93.3%

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

      1. clear-num 93.3%

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

      2. inv-pow 93.3%

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

      3. add-exp-log 93.3%

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

      4. log-pow 92.6%

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

      5. add-log-exp 94.0%

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

      6. pow-to-exp 94.0%

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

   5. Applied egg-rr 94.0%

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

      1. unpow-1 94.0%

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

   7. Simplified 94.0%

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

   8. Taylor expanded in x around inf 93.5%

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

      1. mul-1-neg 93.5%

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

   10. Simplified 93.5%

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

   11. Taylor expanded in y around 0 81.6%

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

   12. Taylor expanded in y around inf 81.6%

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

4. Final simplification 77.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 5 \cdot 10^{+232}:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x \cdot y}\\ \end{array} \]

Alternative 10: 75.1% 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 78.4%

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

   1. exp-prod 84.7%

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

3. Simplified 84.7%

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

4. Taylor expanded in x around 0 73.4%

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

5. Final simplification 73.4%

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

Developer target: 77.8% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2023275 
(FPCore (x y)
  :name "Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2, F"
  :precision binary64

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

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