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

Percentage Accurate: 77.9% → 99.8%

Time: 12.3s

Alternatives: 12

Speedup: 12.3×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 77.9% 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: 99.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -6 \cdot 10^{+44} \lor \neg \left(x \leq 5\right):\\ \;\;\;\;\frac{e^{-y}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(e^{x}\right)}^{\log \left(\frac{x}{x + y}\right)}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -6e+44) (not (<= x 5.0)))
   (/ (exp (- y)) x)
   (/ (pow (exp x) (log (/ x (+ x y)))) x)))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -6e+44) || !(x <= 5.0)) {
		tmp = exp(-y) / x;
	} else {
		tmp = pow(exp(x), log((x / (x + y)))) / x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((x <= (-6d+44)) .or. (.not. (x <= 5.0d0))) then
        tmp = exp(-y) / x
    else
        tmp = (exp(x) ** log((x / (x + y)))) / x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((x <= -6e+44) || !(x <= 5.0)) {
		tmp = Math.exp(-y) / x;
	} else {
		tmp = Math.pow(Math.exp(x), Math.log((x / (x + y)))) / x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -6e+44) or not (x <= 5.0):
		tmp = math.exp(-y) / x
	else:
		tmp = math.pow(math.exp(x), math.log((x / (x + y)))) / x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -6e+44) || !(x <= 5.0))
		tmp = Float64(exp(Float64(-y)) / x);
	else
		tmp = Float64((exp(x) ^ log(Float64(x / Float64(x + y)))) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((x <= -6e+44) || ~((x <= 5.0)))
		tmp = exp(-y) / x;
	else
		tmp = (exp(x) ^ log((x / (x + y)))) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -6e+44], N[Not[LessEqual[x, 5.0]], $MachinePrecision]], N[(N[Exp[(-y)], $MachinePrecision] / x), $MachinePrecision], N[(N[Power[N[Exp[x], $MachinePrecision], N[Log[N[(x / N[(x + y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -5.99999999999999974e44 or 5 < x

   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%

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

   if -5.99999999999999974e44 < x < 5

   1. Initial program 84.5%

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

      1. exp-prod 99.7%

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

   3. Simplified 99.7%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -6 \cdot 10^{+44} \lor \neg \left(x \leq 5\right):\\ \;\;\;\;\frac{e^{-y}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(e^{x}\right)}^{\log \left(\frac{x}{x + y}\right)}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -650000 \lor \neg \left(x \leq 0.65\right):\\ \;\;\;\;\frac{e^{-y}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -650000.0) (not (<= x 0.65))) (/ (exp (- y)) x) (/ 1.0 x)))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -650000.0) || !(x <= 0.65)) {
		tmp = exp(-y) / x;
	} else {
		tmp = 1.0 / x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((x <= (-650000.0d0)) .or. (.not. (x <= 0.65d0))) then
        tmp = exp(-y) / x
    else
        tmp = 1.0d0 / x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((x <= -650000.0) || !(x <= 0.65)) {
		tmp = Math.exp(-y) / x;
	} else {
		tmp = 1.0 / x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -650000.0) or not (x <= 0.65):
		tmp = math.exp(-y) / x
	else:
		tmp = 1.0 / x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -650000.0) || !(x <= 0.65))
		tmp = Float64(exp(Float64(-y)) / x);
	else
		tmp = Float64(1.0 / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((x <= -650000.0) || ~((x <= 0.65)))
		tmp = exp(-y) / x;
	else
		tmp = 1.0 / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -650000.0], N[Not[LessEqual[x, 0.65]], $MachinePrecision]], N[(N[Exp[(-y)], $MachinePrecision] / x), $MachinePrecision], N[(1.0 / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -6.5e5 or 0.650000000000000022 < x

   1. Initial program 74.4%

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

      1. *-commutative 74.4%

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

      2. exp-to-pow 74.4%

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

   3. Simplified 74.4%

      \[\leadsto \color{blue}{\frac{{\left(\frac{x}{x + y}\right)}^{x}}{x}} \]
   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 -6.5e5 < x < 0.650000000000000022

   1. Initial program 83.3%

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

      1. exp-prod 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in x around 0 99.0%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -650000 \lor \neg \left(x \leq 0.65\right):\\ \;\;\;\;\frac{e^{-y}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 86.4% accurate, 6.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.6 \cdot 10^{+145}:\\ \;\;\;\;\frac{1}{x + y \cdot \left(x + y \cdot \left(x - x \cdot \left(0.5 - 0.5 \cdot \frac{-1}{x}\right)\right)\right)}\\ \mathbf{elif}\;x \leq -650000:\\ \;\;\;\;\frac{1}{x} + y \cdot \left(y \cdot \left(-0.16666666666666666 \cdot \frac{y}{x} - \frac{-0.5}{x}\right) + \frac{-1}{x}\right)\\ \mathbf{elif}\;x \leq 3.8 \cdot 10^{+41}:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x + y \cdot \left(x + y \cdot \left(x + x \cdot y\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1.6e+145)
   (/ 1.0 (+ x (* y (+ x (* y (- x (* x (- 0.5 (* 0.5 (/ -1.0 x))))))))))
   (if (<= x -650000.0)
     (+
      (/ 1.0 x)
      (*
       y
       (+ (* y (- (* -0.16666666666666666 (/ y x)) (/ -0.5 x))) (/ -1.0 x))))
     (if (<= x 3.8e+41)
       (/ 1.0 x)
       (/ 1.0 (+ x (* y (+ x (* y (+ x (* x y)))))))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.6e+145) {
		tmp = 1.0 / (x + (y * (x + (y * (x - (x * (0.5 - (0.5 * (-1.0 / x)))))))));
	} else if (x <= -650000.0) {
		tmp = (1.0 / x) + (y * ((y * ((-0.16666666666666666 * (y / x)) - (-0.5 / x))) + (-1.0 / x)));
	} else if (x <= 3.8e+41) {
		tmp = 1.0 / x;
	} else {
		tmp = 1.0 / (x + (y * (x + (y * (x + (x * y))))));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-1.6d+145)) then
        tmp = 1.0d0 / (x + (y * (x + (y * (x - (x * (0.5d0 - (0.5d0 * ((-1.0d0) / x)))))))))
    else if (x <= (-650000.0d0)) then
        tmp = (1.0d0 / x) + (y * ((y * (((-0.16666666666666666d0) * (y / x)) - ((-0.5d0) / x))) + ((-1.0d0) / x)))
    else if (x <= 3.8d+41) then
        tmp = 1.0d0 / x
    else
        tmp = 1.0d0 / (x + (y * (x + (y * (x + (x * y))))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -1.6e+145) {
		tmp = 1.0 / (x + (y * (x + (y * (x - (x * (0.5 - (0.5 * (-1.0 / x)))))))));
	} else if (x <= -650000.0) {
		tmp = (1.0 / x) + (y * ((y * ((-0.16666666666666666 * (y / x)) - (-0.5 / x))) + (-1.0 / x)));
	} else if (x <= 3.8e+41) {
		tmp = 1.0 / x;
	} else {
		tmp = 1.0 / (x + (y * (x + (y * (x + (x * y))))));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -1.6e+145:
		tmp = 1.0 / (x + (y * (x + (y * (x - (x * (0.5 - (0.5 * (-1.0 / x)))))))))
	elif x <= -650000.0:
		tmp = (1.0 / x) + (y * ((y * ((-0.16666666666666666 * (y / x)) - (-0.5 / x))) + (-1.0 / x)))
	elif x <= 3.8e+41:
		tmp = 1.0 / x
	else:
		tmp = 1.0 / (x + (y * (x + (y * (x + (x * y))))))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1.6e+145)
		tmp = Float64(1.0 / Float64(x + Float64(y * Float64(x + Float64(y * Float64(x - Float64(x * Float64(0.5 - Float64(0.5 * Float64(-1.0 / x))))))))));
	elseif (x <= -650000.0)
		tmp = Float64(Float64(1.0 / x) + Float64(y * Float64(Float64(y * Float64(Float64(-0.16666666666666666 * Float64(y / x)) - Float64(-0.5 / x))) + Float64(-1.0 / x))));
	elseif (x <= 3.8e+41)
		tmp = Float64(1.0 / x);
	else
		tmp = Float64(1.0 / Float64(x + Float64(y * Float64(x + Float64(y * Float64(x + Float64(x * y)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1.6e+145)
		tmp = 1.0 / (x + (y * (x + (y * (x - (x * (0.5 - (0.5 * (-1.0 / x)))))))));
	elseif (x <= -650000.0)
		tmp = (1.0 / x) + (y * ((y * ((-0.16666666666666666 * (y / x)) - (-0.5 / x))) + (-1.0 / x)));
	elseif (x <= 3.8e+41)
		tmp = 1.0 / x;
	else
		tmp = 1.0 / (x + (y * (x + (y * (x + (x * y))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1.6e+145], N[(1.0 / N[(x + N[(y * N[(x + N[(y * N[(x - N[(x * N[(0.5 - N[(0.5 * N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -650000.0], N[(N[(1.0 / x), $MachinePrecision] + N[(y * N[(N[(y * N[(N[(-0.16666666666666666 * N[(y / x), $MachinePrecision]), $MachinePrecision] - N[(-0.5 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 3.8e+41], N[(1.0 / x), $MachinePrecision], N[(1.0 / N[(x + N[(y * N[(x + N[(y * N[(x + N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < -1.60000000000000004e145

   1. Initial program 62.1%

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

      1. exp-prod 62.1%

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

   3. Simplified 62.1%

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

      1. clear-num 62.1%

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

      2. add-exp-log 0.0%

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

      3. add-exp-log 0.0%

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

      4. div-exp 0.0%

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

      5. pow-exp 0.0%

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

      6. add-log-exp 0.0%

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

      7. log-pow 0.0%

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

      8. div-exp 0.0%

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

      9. add-exp-log 62.1%

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

      10. add-exp-log 62.1%

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

      11. inv-pow 62.1%

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

   6. Applied egg-rr 62.1%

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

      1. unpow-1 62.1%

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

   8. Simplified 62.1%

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

   9. Taylor expanded in y around 0 83.2%

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

   if -1.60000000000000004e145 < x < -6.5e5

   1. Initial program 85.5%

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

      1. *-commutative 85.5%

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

      2. exp-to-pow 85.5%

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

   3. Simplified 85.5%

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

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

   8. Taylor expanded in y around 0 81.2%

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

      1. un-div-inv 81.2%

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

      2. frac-2neg 81.2%

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

      3. metadata-eval 81.2%

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

   10. Applied egg-rr 81.2%

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

   if -6.5e5 < x < 3.8000000000000001e41

   1. Initial program 83.2%

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

      1. exp-prod 98.9%

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

   3. Simplified 98.9%

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

   5. Taylor expanded in x around 0 98.2%

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

   if 3.8000000000000001e41 < x

   1. Initial program 76.4%

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

      1. exp-prod 76.4%

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

   3. Simplified 76.4%

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

   5. Taylor expanded in x around inf 53.9%

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

      1. mul-1-neg 53.9%

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

      2. unsub-neg 53.9%

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

   7. Simplified 53.9%

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

   8. Taylor expanded in y around inf 53.8%

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

      1. clear-num 53.8%

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

      2. inv-pow 53.8%

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

      3. sub-neg 53.8%

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

      4. metadata-eval 53.8%

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

   10. Applied egg-rr 53.8%

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

       1. unpow-1 53.8%

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

       2. distribute-lft-in 53.8%

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

       3. *-commutative 53.8%

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

       4. rgt-mult-inverse 53.9%

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

       5. mul-1-neg 53.9%

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

       6. unsub-neg 53.9%

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

   12. Simplified 53.9%

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

   13. Taylor expanded in y around 0 84.0%

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

       1. neg-mul-1 84.0%

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

       2. neg-mul-1 84.0%

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

   15. Simplified 84.0%

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

4. Final simplification 90.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.6 \cdot 10^{+145}:\\ \;\;\;\;\frac{1}{x + y \cdot \left(x + y \cdot \left(x - x \cdot \left(0.5 - 0.5 \cdot \frac{-1}{x}\right)\right)\right)}\\ \mathbf{elif}\;x \leq -650000:\\ \;\;\;\;\frac{1}{x} + y \cdot \left(y \cdot \left(-0.16666666666666666 \cdot \frac{y}{x} - \frac{-0.5}{x}\right) + \frac{-1}{x}\right)\\ \mathbf{elif}\;x \leq 3.8 \cdot 10^{+41}:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x + y \cdot \left(x + y \cdot \left(x + x \cdot y\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 86.4% accurate, 6.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x + y \cdot \left(x + x \cdot y\right)\\ \mathbf{if}\;x \leq -3.2 \cdot 10^{+146}:\\ \;\;\;\;\frac{1}{t\_0}\\ \mathbf{elif}\;x \leq -650000:\\ \;\;\;\;\frac{1}{x} + y \cdot \left(y \cdot \left(-0.16666666666666666 \cdot \frac{y}{x} - \frac{-0.5}{x}\right) + \frac{-1}{x}\right)\\ \mathbf{elif}\;x \leq 2.9 \cdot 10^{+41}:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x + y \cdot t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ x (* y (+ x (* x y))))))
   (if (<= x -3.2e+146)
     (/ 1.0 t_0)
     (if (<= x -650000.0)
       (+
        (/ 1.0 x)
        (*
         y
         (+ (* y (- (* -0.16666666666666666 (/ y x)) (/ -0.5 x))) (/ -1.0 x))))
       (if (<= x 2.9e+41) (/ 1.0 x) (/ 1.0 (+ x (* y t_0))))))))

C

double code(double x, double y) {
	double t_0 = x + (y * (x + (x * y)));
	double tmp;
	if (x <= -3.2e+146) {
		tmp = 1.0 / t_0;
	} else if (x <= -650000.0) {
		tmp = (1.0 / x) + (y * ((y * ((-0.16666666666666666 * (y / x)) - (-0.5 / x))) + (-1.0 / x)));
	} else if (x <= 2.9e+41) {
		tmp = 1.0 / x;
	} else {
		tmp = 1.0 / (x + (y * t_0));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x + (y * (x + (x * y)))
    if (x <= (-3.2d+146)) then
        tmp = 1.0d0 / t_0
    else if (x <= (-650000.0d0)) then
        tmp = (1.0d0 / x) + (y * ((y * (((-0.16666666666666666d0) * (y / x)) - ((-0.5d0) / x))) + ((-1.0d0) / x)))
    else if (x <= 2.9d+41) then
        tmp = 1.0d0 / x
    else
        tmp = 1.0d0 / (x + (y * t_0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = x + (y * (x + (x * y)));
	double tmp;
	if (x <= -3.2e+146) {
		tmp = 1.0 / t_0;
	} else if (x <= -650000.0) {
		tmp = (1.0 / x) + (y * ((y * ((-0.16666666666666666 * (y / x)) - (-0.5 / x))) + (-1.0 / x)));
	} else if (x <= 2.9e+41) {
		tmp = 1.0 / x;
	} else {
		tmp = 1.0 / (x + (y * t_0));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = x + (y * (x + (x * y)))
	tmp = 0
	if x <= -3.2e+146:
		tmp = 1.0 / t_0
	elif x <= -650000.0:
		tmp = (1.0 / x) + (y * ((y * ((-0.16666666666666666 * (y / x)) - (-0.5 / x))) + (-1.0 / x)))
	elif x <= 2.9e+41:
		tmp = 1.0 / x
	else:
		tmp = 1.0 / (x + (y * t_0))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(x + Float64(y * Float64(x + Float64(x * y))))
	tmp = 0.0
	if (x <= -3.2e+146)
		tmp = Float64(1.0 / t_0);
	elseif (x <= -650000.0)
		tmp = Float64(Float64(1.0 / x) + Float64(y * Float64(Float64(y * Float64(Float64(-0.16666666666666666 * Float64(y / x)) - Float64(-0.5 / x))) + Float64(-1.0 / x))));
	elseif (x <= 2.9e+41)
		tmp = Float64(1.0 / x);
	else
		tmp = Float64(1.0 / Float64(x + Float64(y * t_0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = x + (y * (x + (x * y)));
	tmp = 0.0;
	if (x <= -3.2e+146)
		tmp = 1.0 / t_0;
	elseif (x <= -650000.0)
		tmp = (1.0 / x) + (y * ((y * ((-0.16666666666666666 * (y / x)) - (-0.5 / x))) + (-1.0 / x)));
	elseif (x <= 2.9e+41)
		tmp = 1.0 / x;
	else
		tmp = 1.0 / (x + (y * t_0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(x + N[(y * N[(x + N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -3.2e+146], N[(1.0 / t$95$0), $MachinePrecision], If[LessEqual[x, -650000.0], N[(N[(1.0 / x), $MachinePrecision] + N[(y * N[(N[(y * N[(N[(-0.16666666666666666 * N[(y / x), $MachinePrecision]), $MachinePrecision] - N[(-0.5 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 2.9e+41], N[(1.0 / x), $MachinePrecision], N[(1.0 / N[(x + N[(y * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -3.2e146

   1. Initial program 62.1%

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

      1. exp-prod 62.1%

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

   3. Simplified 62.1%

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

   5. Taylor expanded in x around inf 60.0%

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

      1. mul-1-neg 60.0%

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

      2. unsub-neg 60.0%

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

   7. Simplified 60.0%

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

   8. Taylor expanded in y around inf 59.9%

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

      1. clear-num 59.8%

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

      2. inv-pow 59.8%

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

      3. sub-neg 59.8%

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

      4. metadata-eval 59.8%

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

   10. Applied egg-rr 59.8%

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

       1. unpow-1 59.8%

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

       2. distribute-lft-in 59.8%

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

       3. *-commutative 59.8%

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

       4. rgt-mult-inverse 60.0%

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

       5. mul-1-neg 60.0%

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

       6. unsub-neg 60.0%

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

   12. Simplified 60.0%

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

   13. Taylor expanded in y around 0 83.2%

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

       1. neg-mul-1 83.2%

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

   15. Simplified 83.2%

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

   if -3.2e146 < x < -6.5e5

   1. Initial program 85.5%

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

      1. *-commutative 85.5%

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

      2. exp-to-pow 85.5%

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

   3. Simplified 85.5%

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

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

   8. Taylor expanded in y around 0 81.2%

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

      1. un-div-inv 81.2%

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

      2. frac-2neg 81.2%

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

      3. metadata-eval 81.2%

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

   10. Applied egg-rr 81.2%

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

   if -6.5e5 < x < 2.89999999999999988e41

   1. Initial program 83.2%

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

      1. exp-prod 98.9%

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

   3. Simplified 98.9%

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

   5. Taylor expanded in x around 0 98.2%

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

   if 2.89999999999999988e41 < x

   1. Initial program 76.4%

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

      1. exp-prod 76.4%

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

   3. Simplified 76.4%

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

   5. Taylor expanded in x around inf 53.9%

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

      1. mul-1-neg 53.9%

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

      2. unsub-neg 53.9%

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

   7. Simplified 53.9%

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

   8. Taylor expanded in y around inf 53.8%

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

      1. clear-num 53.8%

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

      2. inv-pow 53.8%

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

      3. sub-neg 53.8%

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

      4. metadata-eval 53.8%

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

   10. Applied egg-rr 53.8%

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

       1. unpow-1 53.8%

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

       2. distribute-lft-in 53.8%

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

       3. *-commutative 53.8%

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

       4. rgt-mult-inverse 53.9%

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

       5. mul-1-neg 53.9%

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

       6. unsub-neg 53.9%

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

   12. Simplified 53.9%

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

   13. Taylor expanded in y around 0 84.0%

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

       1. neg-mul-1 84.0%

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

       2. neg-mul-1 84.0%

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

   15. Simplified 84.0%

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

4. Final simplification 90.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.2 \cdot 10^{+146}:\\ \;\;\;\;\frac{1}{x + y \cdot \left(x + x \cdot y\right)}\\ \mathbf{elif}\;x \leq -650000:\\ \;\;\;\;\frac{1}{x} + y \cdot \left(y \cdot \left(-0.16666666666666666 \cdot \frac{y}{x} - \frac{-0.5}{x}\right) + \frac{-1}{x}\right)\\ \mathbf{elif}\;x \leq 2.9 \cdot 10^{+41}:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x + y \cdot \left(x + y \cdot \left(x + x \cdot y\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 86.7% accurate, 7.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x + y \cdot \left(x + x \cdot y\right)\\ \mathbf{if}\;x \leq -8 \cdot 10^{+146}:\\ \;\;\;\;\frac{1}{t\_0}\\ \mathbf{elif}\;x \leq -650000:\\ \;\;\;\;\frac{1}{x} + \frac{y \cdot \left(-1 + y \cdot \left(0.5 + y \cdot -0.16666666666666666\right)\right)}{x}\\ \mathbf{elif}\;x \leq 2.9 \cdot 10^{+41}:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x + y \cdot t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ x (* y (+ x (* x y))))))
   (if (<= x -8e+146)
     (/ 1.0 t_0)
     (if (<= x -650000.0)
       (+
        (/ 1.0 x)
        (/ (* y (+ -1.0 (* y (+ 0.5 (* y -0.16666666666666666))))) x))
       (if (<= x 2.9e+41) (/ 1.0 x) (/ 1.0 (+ x (* y t_0))))))))

C

double code(double x, double y) {
	double t_0 = x + (y * (x + (x * y)));
	double tmp;
	if (x <= -8e+146) {
		tmp = 1.0 / t_0;
	} else if (x <= -650000.0) {
		tmp = (1.0 / x) + ((y * (-1.0 + (y * (0.5 + (y * -0.16666666666666666))))) / x);
	} else if (x <= 2.9e+41) {
		tmp = 1.0 / x;
	} else {
		tmp = 1.0 / (x + (y * t_0));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x + (y * (x + (x * y)))
    if (x <= (-8d+146)) then
        tmp = 1.0d0 / t_0
    else if (x <= (-650000.0d0)) then
        tmp = (1.0d0 / x) + ((y * ((-1.0d0) + (y * (0.5d0 + (y * (-0.16666666666666666d0)))))) / x)
    else if (x <= 2.9d+41) then
        tmp = 1.0d0 / x
    else
        tmp = 1.0d0 / (x + (y * t_0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = x + (y * (x + (x * y)));
	double tmp;
	if (x <= -8e+146) {
		tmp = 1.0 / t_0;
	} else if (x <= -650000.0) {
		tmp = (1.0 / x) + ((y * (-1.0 + (y * (0.5 + (y * -0.16666666666666666))))) / x);
	} else if (x <= 2.9e+41) {
		tmp = 1.0 / x;
	} else {
		tmp = 1.0 / (x + (y * t_0));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = x + (y * (x + (x * y)))
	tmp = 0
	if x <= -8e+146:
		tmp = 1.0 / t_0
	elif x <= -650000.0:
		tmp = (1.0 / x) + ((y * (-1.0 + (y * (0.5 + (y * -0.16666666666666666))))) / x)
	elif x <= 2.9e+41:
		tmp = 1.0 / x
	else:
		tmp = 1.0 / (x + (y * t_0))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(x + Float64(y * Float64(x + Float64(x * y))))
	tmp = 0.0
	if (x <= -8e+146)
		tmp = Float64(1.0 / t_0);
	elseif (x <= -650000.0)
		tmp = Float64(Float64(1.0 / x) + Float64(Float64(y * Float64(-1.0 + Float64(y * Float64(0.5 + Float64(y * -0.16666666666666666))))) / x));
	elseif (x <= 2.9e+41)
		tmp = Float64(1.0 / x);
	else
		tmp = Float64(1.0 / Float64(x + Float64(y * t_0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = x + (y * (x + (x * y)));
	tmp = 0.0;
	if (x <= -8e+146)
		tmp = 1.0 / t_0;
	elseif (x <= -650000.0)
		tmp = (1.0 / x) + ((y * (-1.0 + (y * (0.5 + (y * -0.16666666666666666))))) / x);
	elseif (x <= 2.9e+41)
		tmp = 1.0 / x;
	else
		tmp = 1.0 / (x + (y * t_0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(x + N[(y * N[(x + N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -8e+146], N[(1.0 / t$95$0), $MachinePrecision], If[LessEqual[x, -650000.0], N[(N[(1.0 / x), $MachinePrecision] + N[(N[(y * N[(-1.0 + N[(y * N[(0.5 + N[(y * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 2.9e+41], N[(1.0 / x), $MachinePrecision], N[(1.0 / N[(x + N[(y * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -7.99999999999999947e146

   1. Initial program 62.1%

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

      1. exp-prod 62.1%

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

   3. Simplified 62.1%

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

   5. Taylor expanded in x around inf 60.0%

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

      1. mul-1-neg 60.0%

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

      2. unsub-neg 60.0%

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

   7. Simplified 60.0%

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

   8. Taylor expanded in y around inf 59.9%

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

      1. clear-num 59.8%

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

      2. inv-pow 59.8%

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

      3. sub-neg 59.8%

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

      4. metadata-eval 59.8%

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

   10. Applied egg-rr 59.8%

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

       1. unpow-1 59.8%

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

       2. distribute-lft-in 59.8%

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

       3. *-commutative 59.8%

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

       4. rgt-mult-inverse 60.0%

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

       5. mul-1-neg 60.0%

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

       6. unsub-neg 60.0%

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

   12. Simplified 60.0%

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

   13. Taylor expanded in y around 0 83.2%

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

       1. neg-mul-1 83.2%

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

   15. Simplified 83.2%

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

   if -7.99999999999999947e146 < x < -6.5e5

   1. Initial program 85.5%

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

      1. *-commutative 85.5%

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

      2. exp-to-pow 85.5%

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

   3. Simplified 85.5%

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

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

   8. Taylor expanded in y around 0 81.2%

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

   9. Taylor expanded in x around 0 81.2%

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

   if -6.5e5 < x < 2.89999999999999988e41

   1. Initial program 83.2%

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

      1. exp-prod 98.9%

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

   3. Simplified 98.9%

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

   5. Taylor expanded in x around 0 98.2%

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

   if 2.89999999999999988e41 < x

   1. Initial program 76.4%

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

      1. exp-prod 76.4%

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

   3. Simplified 76.4%

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

   5. Taylor expanded in x around inf 53.9%

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

      1. mul-1-neg 53.9%

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

      2. unsub-neg 53.9%

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

   7. Simplified 53.9%

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

   8. Taylor expanded in y around inf 53.8%

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

      1. clear-num 53.8%

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

      2. inv-pow 53.8%

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

      3. sub-neg 53.8%

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

      4. metadata-eval 53.8%

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

   10. Applied egg-rr 53.8%

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

       1. unpow-1 53.8%

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

       2. distribute-lft-in 53.8%

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

       3. *-commutative 53.8%

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

       4. rgt-mult-inverse 53.9%

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

       5. mul-1-neg 53.9%

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

       6. unsub-neg 53.9%

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

   12. Simplified 53.9%

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

   13. Taylor expanded in y around 0 84.0%

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

       1. neg-mul-1 84.0%

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

       2. neg-mul-1 84.0%

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

   15. Simplified 84.0%

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

4. Final simplification 90.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -8 \cdot 10^{+146}:\\ \;\;\;\;\frac{1}{x + y \cdot \left(x + x \cdot y\right)}\\ \mathbf{elif}\;x \leq -650000:\\ \;\;\;\;\frac{1}{x} + \frac{y \cdot \left(-1 + y \cdot \left(0.5 + y \cdot -0.16666666666666666\right)\right)}{x}\\ \mathbf{elif}\;x \leq 2.9 \cdot 10^{+41}:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x + y \cdot \left(x + y \cdot \left(x + x \cdot y\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 86.1% accurate, 7.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{x + y \cdot \left(x + x \cdot y\right)}\\ \mathbf{if}\;x \leq -7.4 \cdot 10^{+146}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq -650000:\\ \;\;\;\;\frac{1}{x} + \frac{y \cdot \left(-1 + y \cdot \left(0.5 + y \cdot -0.16666666666666666\right)\right)}{x}\\ \mathbf{elif}\;x \leq 2.9 \cdot 10^{+41}:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ 1.0 (+ x (* y (+ x (* x y)))))))
   (if (<= x -7.4e+146)
     t_0
     (if (<= x -650000.0)
       (+
        (/ 1.0 x)
        (/ (* y (+ -1.0 (* y (+ 0.5 (* y -0.16666666666666666))))) x))
       (if (<= x 2.9e+41) (/ 1.0 x) t_0)))))

C

double code(double x, double y) {
	double t_0 = 1.0 / (x + (y * (x + (x * y))));
	double tmp;
	if (x <= -7.4e+146) {
		tmp = t_0;
	} else if (x <= -650000.0) {
		tmp = (1.0 / x) + ((y * (-1.0 + (y * (0.5 + (y * -0.16666666666666666))))) / x);
	} else if (x <= 2.9e+41) {
		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 + (y * (x + (x * y))))
    if (x <= (-7.4d+146)) then
        tmp = t_0
    else if (x <= (-650000.0d0)) then
        tmp = (1.0d0 / x) + ((y * ((-1.0d0) + (y * (0.5d0 + (y * (-0.16666666666666666d0)))))) / x)
    else if (x <= 2.9d+41) 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 + (y * (x + (x * y))));
	double tmp;
	if (x <= -7.4e+146) {
		tmp = t_0;
	} else if (x <= -650000.0) {
		tmp = (1.0 / x) + ((y * (-1.0 + (y * (0.5 + (y * -0.16666666666666666))))) / x);
	} else if (x <= 2.9e+41) {
		tmp = 1.0 / x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 1.0 / (x + (y * (x + (x * y))))
	tmp = 0
	if x <= -7.4e+146:
		tmp = t_0
	elif x <= -650000.0:
		tmp = (1.0 / x) + ((y * (-1.0 + (y * (0.5 + (y * -0.16666666666666666))))) / x)
	elif x <= 2.9e+41:
		tmp = 1.0 / x
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(1.0 / Float64(x + Float64(y * Float64(x + Float64(x * y)))))
	tmp = 0.0
	if (x <= -7.4e+146)
		tmp = t_0;
	elseif (x <= -650000.0)
		tmp = Float64(Float64(1.0 / x) + Float64(Float64(y * Float64(-1.0 + Float64(y * Float64(0.5 + Float64(y * -0.16666666666666666))))) / x));
	elseif (x <= 2.9e+41)
		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 + (y * (x + (x * y))));
	tmp = 0.0;
	if (x <= -7.4e+146)
		tmp = t_0;
	elseif (x <= -650000.0)
		tmp = (1.0 / x) + ((y * (-1.0 + (y * (0.5 + (y * -0.16666666666666666))))) / x);
	elseif (x <= 2.9e+41)
		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[(y * N[(x + N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -7.4e+146], t$95$0, If[LessEqual[x, -650000.0], N[(N[(1.0 / x), $MachinePrecision] + N[(N[(y * N[(-1.0 + N[(y * N[(0.5 + N[(y * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 2.9e+41], N[(1.0 / x), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if x < -7.40000000000000009e146 or 2.89999999999999988e41 < x

   1. Initial program 70.8%

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

      1. exp-prod 70.8%

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

   3. Simplified 70.8%

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

   5. Taylor expanded in x around inf 56.2%

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

      1. mul-1-neg 56.2%

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

      2. unsub-neg 56.2%

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

   7. Simplified 56.2%

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

   8. Taylor expanded in y around inf 56.2%

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

      1. clear-num 56.1%

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

      2. inv-pow 56.1%

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

      3. sub-neg 56.1%

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

      4. metadata-eval 56.1%

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

   10. Applied egg-rr 56.1%

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

       1. unpow-1 56.1%

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

       2. distribute-lft-in 56.1%

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

       3. *-commutative 56.1%

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

       4. rgt-mult-inverse 56.2%

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

       5. mul-1-neg 56.2%

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

       6. unsub-neg 56.2%

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

   12. Simplified 56.2%

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

   13. Taylor expanded in y around 0 82.9%

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

       1. neg-mul-1 82.9%

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

   15. Simplified 82.9%

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

   if -7.40000000000000009e146 < x < -6.5e5

   1. Initial program 85.5%

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

      1. *-commutative 85.5%

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

      2. exp-to-pow 85.5%

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

   3. Simplified 85.5%

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

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

   8. Taylor expanded in y around 0 81.2%

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

   9. Taylor expanded in x around 0 81.2%

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

   if -6.5e5 < x < 2.89999999999999988e41

   1. Initial program 83.2%

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

      1. exp-prod 98.9%

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

   3. Simplified 98.9%

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

   5. Taylor expanded in x around 0 98.2%

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

4. Final simplification 89.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -7.4 \cdot 10^{+146}:\\ \;\;\;\;\frac{1}{x + y \cdot \left(x + x \cdot y\right)}\\ \mathbf{elif}\;x \leq -650000:\\ \;\;\;\;\frac{1}{x} + \frac{y \cdot \left(-1 + y \cdot \left(0.5 + y \cdot -0.16666666666666666\right)\right)}{x}\\ \mathbf{elif}\;x \leq 2.9 \cdot 10^{+41}:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x + y \cdot \left(x + x \cdot y\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 86.1% accurate, 8.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{x + y \cdot \left(x + x \cdot y\right)}\\ \mathbf{if}\;x \leq -6.1 \cdot 10^{+146}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq -650000:\\ \;\;\;\;\frac{1 + y \cdot \left(-1 + y \cdot \left(0.5 + y \cdot -0.16666666666666666\right)\right)}{x}\\ \mathbf{elif}\;x \leq 2.9 \cdot 10^{+41}:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ 1.0 (+ x (* y (+ x (* x y)))))))
   (if (<= x -6.1e+146)
     t_0
     (if (<= x -650000.0)
       (/ (+ 1.0 (* y (+ -1.0 (* y (+ 0.5 (* y -0.16666666666666666)))))) x)
       (if (<= x 2.9e+41) (/ 1.0 x) t_0)))))

C

double code(double x, double y) {
	double t_0 = 1.0 / (x + (y * (x + (x * y))));
	double tmp;
	if (x <= -6.1e+146) {
		tmp = t_0;
	} else if (x <= -650000.0) {
		tmp = (1.0 + (y * (-1.0 + (y * (0.5 + (y * -0.16666666666666666)))))) / x;
	} else if (x <= 2.9e+41) {
		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 + (y * (x + (x * y))))
    if (x <= (-6.1d+146)) then
        tmp = t_0
    else if (x <= (-650000.0d0)) then
        tmp = (1.0d0 + (y * ((-1.0d0) + (y * (0.5d0 + (y * (-0.16666666666666666d0))))))) / x
    else if (x <= 2.9d+41) 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 + (y * (x + (x * y))));
	double tmp;
	if (x <= -6.1e+146) {
		tmp = t_0;
	} else if (x <= -650000.0) {
		tmp = (1.0 + (y * (-1.0 + (y * (0.5 + (y * -0.16666666666666666)))))) / x;
	} else if (x <= 2.9e+41) {
		tmp = 1.0 / x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 1.0 / (x + (y * (x + (x * y))))
	tmp = 0
	if x <= -6.1e+146:
		tmp = t_0
	elif x <= -650000.0:
		tmp = (1.0 + (y * (-1.0 + (y * (0.5 + (y * -0.16666666666666666)))))) / x
	elif x <= 2.9e+41:
		tmp = 1.0 / x
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(1.0 / Float64(x + Float64(y * Float64(x + Float64(x * y)))))
	tmp = 0.0
	if (x <= -6.1e+146)
		tmp = t_0;
	elseif (x <= -650000.0)
		tmp = Float64(Float64(1.0 + Float64(y * Float64(-1.0 + Float64(y * Float64(0.5 + Float64(y * -0.16666666666666666)))))) / x);
	elseif (x <= 2.9e+41)
		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 + (y * (x + (x * y))));
	tmp = 0.0;
	if (x <= -6.1e+146)
		tmp = t_0;
	elseif (x <= -650000.0)
		tmp = (1.0 + (y * (-1.0 + (y * (0.5 + (y * -0.16666666666666666)))))) / x;
	elseif (x <= 2.9e+41)
		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[(y * N[(x + N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -6.1e+146], t$95$0, If[LessEqual[x, -650000.0], N[(N[(1.0 + N[(y * N[(-1.0 + N[(y * N[(0.5 + N[(y * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[x, 2.9e+41], N[(1.0 / x), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if x < -6.09999999999999979e146 or 2.89999999999999988e41 < x

   1. Initial program 70.8%

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

      1. exp-prod 70.8%

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

   3. Simplified 70.8%

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

   5. Taylor expanded in x around inf 56.2%

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

      1. mul-1-neg 56.2%

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

      2. unsub-neg 56.2%

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

   7. Simplified 56.2%

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

   8. Taylor expanded in y around inf 56.2%

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

      1. clear-num 56.1%

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

      2. inv-pow 56.1%

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

      3. sub-neg 56.1%

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

      4. metadata-eval 56.1%

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

   10. Applied egg-rr 56.1%

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

       1. unpow-1 56.1%

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

       2. distribute-lft-in 56.1%

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

       3. *-commutative 56.1%

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

       4. rgt-mult-inverse 56.2%

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

       5. mul-1-neg 56.2%

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

       6. unsub-neg 56.2%

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

   12. Simplified 56.2%

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

   13. Taylor expanded in y around 0 82.9%

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

       1. neg-mul-1 82.9%

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

   15. Simplified 82.9%

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

   if -6.09999999999999979e146 < x < -6.5e5

   1. Initial program 85.5%

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

      1. *-commutative 85.5%

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

      2. exp-to-pow 85.5%

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

   3. Simplified 85.5%

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

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

   8. Taylor expanded in y around 0 81.2%

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

   if -6.5e5 < x < 2.89999999999999988e41

   1. Initial program 83.2%

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

      1. exp-prod 98.9%

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

   3. Simplified 98.9%

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

   5. Taylor expanded in x around 0 98.2%

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

4. Final simplification 89.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -6.1 \cdot 10^{+146}:\\ \;\;\;\;\frac{1}{x + y \cdot \left(x + x \cdot y\right)}\\ \mathbf{elif}\;x \leq -650000:\\ \;\;\;\;\frac{1 + y \cdot \left(-1 + y \cdot \left(0.5 + y \cdot -0.16666666666666666\right)\right)}{x}\\ \mathbf{elif}\;x \leq 2.9 \cdot 10^{+41}:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x + y \cdot \left(x + x \cdot y\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 86.1% accurate, 8.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{x + y \cdot \left(x + x \cdot y\right)}\\ \mathbf{if}\;x \leq -4.8 \cdot 10^{+146}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq -650000:\\ \;\;\;\;\frac{1 + y \cdot \left(-1 + y \cdot \left(y \cdot -0.16666666666666666\right)\right)}{x}\\ \mathbf{elif}\;x \leq 2.9 \cdot 10^{+41}:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ 1.0 (+ x (* y (+ x (* x y)))))))
   (if (<= x -4.8e+146)
     t_0
     (if (<= x -650000.0)
       (/ (+ 1.0 (* y (+ -1.0 (* y (* y -0.16666666666666666))))) x)
       (if (<= x 2.9e+41) (/ 1.0 x) t_0)))))

C

double code(double x, double y) {
	double t_0 = 1.0 / (x + (y * (x + (x * y))));
	double tmp;
	if (x <= -4.8e+146) {
		tmp = t_0;
	} else if (x <= -650000.0) {
		tmp = (1.0 + (y * (-1.0 + (y * (y * -0.16666666666666666))))) / x;
	} else if (x <= 2.9e+41) {
		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 + (y * (x + (x * y))))
    if (x <= (-4.8d+146)) then
        tmp = t_0
    else if (x <= (-650000.0d0)) then
        tmp = (1.0d0 + (y * ((-1.0d0) + (y * (y * (-0.16666666666666666d0)))))) / x
    else if (x <= 2.9d+41) 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 + (y * (x + (x * y))));
	double tmp;
	if (x <= -4.8e+146) {
		tmp = t_0;
	} else if (x <= -650000.0) {
		tmp = (1.0 + (y * (-1.0 + (y * (y * -0.16666666666666666))))) / x;
	} else if (x <= 2.9e+41) {
		tmp = 1.0 / x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 1.0 / (x + (y * (x + (x * y))))
	tmp = 0
	if x <= -4.8e+146:
		tmp = t_0
	elif x <= -650000.0:
		tmp = (1.0 + (y * (-1.0 + (y * (y * -0.16666666666666666))))) / x
	elif x <= 2.9e+41:
		tmp = 1.0 / x
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(1.0 / Float64(x + Float64(y * Float64(x + Float64(x * y)))))
	tmp = 0.0
	if (x <= -4.8e+146)
		tmp = t_0;
	elseif (x <= -650000.0)
		tmp = Float64(Float64(1.0 + Float64(y * Float64(-1.0 + Float64(y * Float64(y * -0.16666666666666666))))) / x);
	elseif (x <= 2.9e+41)
		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 + (y * (x + (x * y))));
	tmp = 0.0;
	if (x <= -4.8e+146)
		tmp = t_0;
	elseif (x <= -650000.0)
		tmp = (1.0 + (y * (-1.0 + (y * (y * -0.16666666666666666))))) / x;
	elseif (x <= 2.9e+41)
		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[(y * N[(x + N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -4.8e+146], t$95$0, If[LessEqual[x, -650000.0], N[(N[(1.0 + N[(y * N[(-1.0 + N[(y * N[(y * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[x, 2.9e+41], N[(1.0 / x), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if x < -4.8000000000000004e146 or 2.89999999999999988e41 < x

   1. Initial program 70.8%

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

      1. exp-prod 70.8%

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

   3. Simplified 70.8%

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

   5. Taylor expanded in x around inf 56.2%

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

      1. mul-1-neg 56.2%

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

      2. unsub-neg 56.2%

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

   7. Simplified 56.2%

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

   8. Taylor expanded in y around inf 56.2%

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

      1. clear-num 56.1%

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

      2. inv-pow 56.1%

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

      3. sub-neg 56.1%

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

      4. metadata-eval 56.1%

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

   10. Applied egg-rr 56.1%

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

       1. unpow-1 56.1%

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

       2. distribute-lft-in 56.1%

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

       3. *-commutative 56.1%

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

       4. rgt-mult-inverse 56.2%

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

       5. mul-1-neg 56.2%

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

       6. unsub-neg 56.2%

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

   12. Simplified 56.2%

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

   13. Taylor expanded in y around 0 82.9%

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

       1. neg-mul-1 82.9%

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

   15. Simplified 82.9%

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

   if -4.8000000000000004e146 < x < -6.5e5

   1. Initial program 85.5%

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

      1. *-commutative 85.5%

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

      2. exp-to-pow 85.5%

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

   3. Simplified 85.5%

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

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

   8. Taylor expanded in y around 0 81.2%

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

   9. Taylor expanded in y around inf 80.1%

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

       1. *-commutative 80.1%

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

   11. Simplified 80.1%

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

   if -6.5e5 < x < 2.89999999999999988e41

   1. Initial program 83.2%

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

      1. exp-prod 98.9%

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

   3. Simplified 98.9%

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

   5. Taylor expanded in x around 0 98.2%

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

4. Final simplification 89.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.8 \cdot 10^{+146}:\\ \;\;\;\;\frac{1}{x + y \cdot \left(x + x \cdot y\right)}\\ \mathbf{elif}\;x \leq -650000:\\ \;\;\;\;\frac{1 + y \cdot \left(-1 + y \cdot \left(y \cdot -0.16666666666666666\right)\right)}{x}\\ \mathbf{elif}\;x \leq 2.9 \cdot 10^{+41}:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x + y \cdot \left(x + x \cdot y\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 83.8% accurate, 9.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.2 \cdot 10^{+93} \lor \neg \left(x \leq 2.9 \cdot 10^{+41}\right):\\ \;\;\;\;\frac{1}{x + y \cdot \left(x + x \cdot y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -1.2e+93) (not (<= x 2.9e+41)))
   (/ 1.0 (+ x (* y (+ x (* x y)))))
   (/ 1.0 x)))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -1.2e+93) || !(x <= 2.9e+41)) {
		tmp = 1.0 / (x + (y * (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 <= (-1.2d+93)) .or. (.not. (x <= 2.9d+41))) then
        tmp = 1.0d0 / (x + (y * (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 <= -1.2e+93) || !(x <= 2.9e+41)) {
		tmp = 1.0 / (x + (y * (x + (x * y))));
	} else {
		tmp = 1.0 / x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -1.2e+93) or not (x <= 2.9e+41):
		tmp = 1.0 / (x + (y * (x + (x * y))))
	else:
		tmp = 1.0 / x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -1.2e+93) || !(x <= 2.9e+41))
		tmp = Float64(1.0 / Float64(x + Float64(y * 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 <= -1.2e+93) || ~((x <= 2.9e+41)))
		tmp = 1.0 / (x + (y * (x + (x * y))));
	else
		tmp = 1.0 / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -1.2e+93], N[Not[LessEqual[x, 2.9e+41]], $MachinePrecision]], N[(1.0 / N[(x + N[(y * N[(x + N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.20000000000000005e93 or 2.89999999999999988e41 < x

   1. Initial program 71.1%

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

      1. exp-prod 71.1%

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

   3. Simplified 71.1%

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

   5. Taylor expanded in x around inf 57.1%

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

      1. mul-1-neg 57.1%

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

      2. unsub-neg 57.1%

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

   7. Simplified 57.1%

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

   8. Taylor expanded in y around inf 57.0%

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

      1. clear-num 57.0%

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

      2. inv-pow 57.0%

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

      3. sub-neg 57.0%

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

      4. metadata-eval 57.0%

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

   10. Applied egg-rr 57.0%

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

       1. unpow-1 57.0%

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

       2. distribute-lft-in 57.0%

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

       3. *-commutative 57.0%

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

       4. rgt-mult-inverse 57.1%

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

       5. mul-1-neg 57.1%

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

       6. unsub-neg 57.1%

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

   12. Simplified 57.1%

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

   13. Taylor expanded in y around 0 81.3%

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

       1. neg-mul-1 81.3%

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

   15. Simplified 81.3%

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

   if -1.20000000000000005e93 < x < 2.89999999999999988e41

   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 97.8%

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

   3. Simplified 97.8%

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

   5. Taylor expanded in x around 0 93.7%

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

4. Final simplification 88.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.2 \cdot 10^{+93} \lor \neg \left(x \leq 2.9 \cdot 10^{+41}\right):\\ \;\;\;\;\frac{1}{x + y \cdot \left(x + x \cdot y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 81.8% accurate, 12.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.3 \cdot 10^{+93} \lor \neg \left(x \leq 2.9 \cdot 10^{+41}\right):\\ \;\;\;\;\frac{1}{x + x \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -3.3e+93) (not (<= x 2.9e+41)))
   (/ 1.0 (+ x (* x y)))
   (/ 1.0 x)))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -3.3e+93) || !(x <= 2.9e+41)) {
		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 <= (-3.3d+93)) .or. (.not. (x <= 2.9d+41))) 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 <= -3.3e+93) || !(x <= 2.9e+41)) {
		tmp = 1.0 / (x + (x * y));
	} else {
		tmp = 1.0 / x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -3.3e+93) or not (x <= 2.9e+41):
		tmp = 1.0 / (x + (x * y))
	else:
		tmp = 1.0 / x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -3.3e+93) || !(x <= 2.9e+41))
		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 <= -3.3e+93) || ~((x <= 2.9e+41)))
		tmp = 1.0 / (x + (x * y));
	else
		tmp = 1.0 / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -3.3e+93], N[Not[LessEqual[x, 2.9e+41]], $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 < -3.30000000000000009e93 or 2.89999999999999988e41 < x

   1. Initial program 71.1%

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

      1. exp-prod 71.1%

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

   3. Simplified 71.1%

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

      1. clear-num 71.1%

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

      2. add-exp-log 39.7%

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

      3. add-exp-log 39.7%

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

      4. div-exp 39.7%

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

      5. pow-exp 39.7%

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

      6. add-log-exp 39.7%

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

      7. log-pow 39.7%

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

      8. div-exp 39.7%

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

      9. add-exp-log 71.1%

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

      10. add-exp-log 71.1%

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

      11. inv-pow 71.1%

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

   6. Applied egg-rr 71.1%

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

      1. unpow-1 71.1%

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

   8. Simplified 71.1%

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

   9. Taylor expanded in y around 0 76.5%

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

   if -3.30000000000000009e93 < x < 2.89999999999999988e41

   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 97.8%

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

   3. Simplified 97.8%

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

   5. Taylor expanded in x around 0 93.7%

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

4. Final simplification 86.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.3 \cdot 10^{+93} \lor \neg \left(x \leq 2.9 \cdot 10^{+41}\right):\\ \;\;\;\;\frac{1}{x + x \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 75.9% accurate, 20.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.55 \cdot 10^{+196}:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x \cdot y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 1.55e+196) (/ 1.0 x) (/ 1.0 (* x y))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 1.55e+196) {
		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 <= 1.55d+196) 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 <= 1.55e+196) {
		tmp = 1.0 / x;
	} else {
		tmp = 1.0 / (x * y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 1.55e+196:
		tmp = 1.0 / x
	else:
		tmp = 1.0 / (x * y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 1.55e+196)
		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 <= 1.55e+196)
		tmp = 1.0 / x;
	else
		tmp = 1.0 / (x * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 1.55e+196], N[(1.0 / x), $MachinePrecision], N[(1.0 / N[(x * y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.55000000000000005e196

   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 85.9%

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

   3. Simplified 85.9%

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

   5. Taylor expanded in x around 0 81.4%

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

   if 1.55000000000000005e196 < y

   1. Initial program 63.0%

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

      1. exp-prod 84.3%

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

   3. Simplified 84.3%

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

      1. clear-num 84.3%

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

      2. add-exp-log 80.5%

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

      3. add-exp-log 80.5%

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

      4. div-exp 80.5%

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

      5. pow-exp 61.3%

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

      6. add-log-exp 61.3%

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

      7. log-pow 61.3%

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

      8. div-exp 61.3%

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

      9. add-exp-log 63.0%

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

      10. add-exp-log 63.0%

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

      11. inv-pow 63.0%

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

   6. Applied egg-rr 63.0%

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

      1. unpow-1 63.0%

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

   8. Simplified 63.0%

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

   9. Taylor expanded in y around 0 66.8%

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

   10. Taylor expanded in y around inf 66.8%

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

Alternative 12: 75.6% accurate, 69.7× speedup

Math

\[\begin{array}{l} \\ \frac{1}{x} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (/ 1.0 x))

C

double code(double x, double y) {
	return 1.0 / x;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = 1.0d0 / x
end function

Java

public static double code(double x, double y) {
	return 1.0 / x;
}

Python

def code(x, y):
	return 1.0 / x

Julia

function code(x, y)
	return Float64(1.0 / x)
end

MATLAB

function tmp = code(x, y)
	tmp = 1.0 / x;
end

Wolfram

code[x_, y_] := N[(1.0 / x), $MachinePrecision]

Derivation

1. Initial program 78.4%

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

   1. exp-prod 85.8%

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

3. Simplified 85.8%

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

5. Taylor expanded in x around 0 77.2%

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

Developer Target 1: 78.2% 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 2024129 
(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))