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

Percentage Accurate: 77.6% → 99.8%

Time: 12.5s

Alternatives: 13

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: 77.6% 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

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4 \cdot 10^{+40} \lor \neg \left(x \leq 1.05\right):\\ \;\;\;\;\frac{1}{x \cdot e^{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(e^{x}\right)}^{\log \left(\frac{x}{x + y}\right)}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -4e+40) (not (<= x 1.05)))
   (/ 1.0 (* x (exp y)))
   (/ (pow (exp x) (log (/ x (+ x y)))) x)))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -4e+40) || !(x <= 1.05)) {
		tmp = 1.0 / (x * exp(y));
	} 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 <= (-4d+40)) .or. (.not. (x <= 1.05d0))) then
        tmp = 1.0d0 / (x * exp(y))
    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 <= -4e+40) || !(x <= 1.05)) {
		tmp = 1.0 / (x * Math.exp(y));
	} else {
		tmp = Math.pow(Math.exp(x), Math.log((x / (x + y)))) / x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -4e+40) or not (x <= 1.05):
		tmp = 1.0 / (x * math.exp(y))
	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 <= -4e+40) || !(x <= 1.05))
		tmp = Float64(1.0 / Float64(x * exp(y)));
	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 <= -4e+40) || ~((x <= 1.05)))
		tmp = 1.0 / (x * exp(y));
	else
		tmp = (exp(x) ^ log((x / (x + y)))) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -4e+40], N[Not[LessEqual[x, 1.05]], $MachinePrecision]], N[(1.0 / N[(x * N[Exp[y], $MachinePrecision]), $MachinePrecision]), $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 < -4.00000000000000012e40 or 1.05000000000000004 < x

   1. Initial program 74.9%

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

      1. *-commutative 74.9%

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

      2. exp-to-pow 74.9%

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

   3. Simplified 74.9%

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

   5. Taylor expanded in x around inf 100.0%

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

      1. mul-1-neg 100.0%

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

   7. Simplified 100.0%

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

      1. clear-num 100.0%

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

      2. inv-pow 100.0%

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

      3. exp-neg 100.0%

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

      4. associate-/r/ 100.0%

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

      5. /-rgt-identity 100.0%

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

   9. Applied egg-rr 100.0%

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

       1. unpow-1 100.0%

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

   11. Simplified 100.0%

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

   if -4.00000000000000012e40 < x < 1.05000000000000004

   1. Initial program 82.2%

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

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4 \cdot 10^{+40} \lor \neg \left(x \leq 1.05\right):\\ \;\;\;\;\frac{1}{x \cdot e^{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(e^{x}\right)}^{\log \left(\frac{x}{x + y}\right)}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 98.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.8 \cdot 10^{+39} \lor \neg \left(x \leq 1.02\right):\\ \;\;\;\;\frac{1}{x \cdot e^{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -2.8e+39) (not (<= x 1.02))) (/ 1.0 (* x (exp y))) (/ 1.0 x)))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -2.8e+39) || !(x <= 1.02)) {
		tmp = 1.0 / (x * exp(y));
	} else {
		tmp = 1.0 / x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((x <= (-2.8d+39)) .or. (.not. (x <= 1.02d0))) then
        tmp = 1.0d0 / (x * exp(y))
    else
        tmp = 1.0d0 / x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((x <= -2.8e+39) || !(x <= 1.02)) {
		tmp = 1.0 / (x * Math.exp(y));
	} else {
		tmp = 1.0 / x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -2.8e+39) or not (x <= 1.02):
		tmp = 1.0 / (x * math.exp(y))
	else:
		tmp = 1.0 / x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -2.8e+39) || !(x <= 1.02))
		tmp = Float64(1.0 / Float64(x * exp(y)));
	else
		tmp = Float64(1.0 / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((x <= -2.8e+39) || ~((x <= 1.02)))
		tmp = 1.0 / (x * exp(y));
	else
		tmp = 1.0 / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -2.8e+39], N[Not[LessEqual[x, 1.02]], $MachinePrecision]], N[(1.0 / N[(x * N[Exp[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2.80000000000000001e39 or 1.02 < x

   1. Initial program 74.9%

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

      1. *-commutative 74.9%

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

      2. exp-to-pow 74.9%

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

   3. Simplified 74.9%

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

   5. Taylor expanded in x around inf 100.0%

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

      1. mul-1-neg 100.0%

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

   7. Simplified 100.0%

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

      1. clear-num 100.0%

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

      2. inv-pow 100.0%

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

      3. exp-neg 100.0%

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

      4. associate-/r/ 100.0%

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

      5. /-rgt-identity 100.0%

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

   9. Applied egg-rr 100.0%

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

       1. unpow-1 100.0%

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

   11. Simplified 100.0%

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

   if -2.80000000000000001e39 < x < 1.02

   1. Initial program 82.2%

      \[\frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x} \]
   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 97.9%

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

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.8 \cdot 10^{+39} \lor \neg \left(x \leq 1.02\right):\\ \;\;\;\;\frac{1}{x \cdot e^{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.8 \cdot 10^{+39} \lor \neg \left(x \leq 0.05\right):\\ \;\;\;\;\frac{e^{-y}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -2.8e+39) (not (<= x 0.05))) (/ (exp (- y)) x) (/ 1.0 x)))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -2.8e+39) || !(x <= 0.05)) {
		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 <= (-2.8d+39)) .or. (.not. (x <= 0.05d0))) 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 <= -2.8e+39) || !(x <= 0.05)) {
		tmp = Math.exp(-y) / x;
	} else {
		tmp = 1.0 / x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -2.8e+39) or not (x <= 0.05):
		tmp = math.exp(-y) / x
	else:
		tmp = 1.0 / x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -2.8e+39) || !(x <= 0.05))
		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 <= -2.8e+39) || ~((x <= 0.05)))
		tmp = exp(-y) / x;
	else
		tmp = 1.0 / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -2.8e+39], N[Not[LessEqual[x, 0.05]], $MachinePrecision]], N[(N[Exp[(-y)], $MachinePrecision] / x), $MachinePrecision], N[(1.0 / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2.80000000000000001e39 or 0.050000000000000003 < x

   1. Initial program 74.9%

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

      1. *-commutative 74.9%

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

      2. exp-to-pow 74.9%

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

   3. Simplified 74.9%

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

   5. Taylor expanded in x around inf 100.0%

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

      1. mul-1-neg 100.0%

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

   7. Simplified 100.0%

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

   if -2.80000000000000001e39 < x < 0.050000000000000003

   1. Initial program 82.2%

      \[\frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x} \]
   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 97.9%

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

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.8 \cdot 10^{+39} \lor \neg \left(x \leq 0.05\right):\\ \;\;\;\;\frac{e^{-y}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 85.5% accurate, 7.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.8 \cdot 10^{+39}:\\ \;\;\;\;\frac{\frac{x - x \cdot y}{x}}{x}\\ \mathbf{elif}\;x \leq 0.042:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x \cdot \left(1 + y \cdot \left(1 + y \cdot \left(0.5 + y \cdot 0.16666666666666666\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -2.8e+39)
   (/ (/ (- x (* x y)) x) x)
   (if (<= x 0.042)
     (/ 1.0 x)
     (/
      1.0
      (* x (+ 1.0 (* y (+ 1.0 (* y (+ 0.5 (* y 0.16666666666666666)))))))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -2.8e+39) {
		tmp = ((x - (x * y)) / x) / x;
	} else if (x <= 0.042) {
		tmp = 1.0 / x;
	} else {
		tmp = 1.0 / (x * (1.0 + (y * (1.0 + (y * (0.5 + (y * 0.16666666666666666)))))));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-2.8d+39)) then
        tmp = ((x - (x * y)) / x) / x
    else if (x <= 0.042d0) then
        tmp = 1.0d0 / x
    else
        tmp = 1.0d0 / (x * (1.0d0 + (y * (1.0d0 + (y * (0.5d0 + (y * 0.16666666666666666d0)))))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -2.8e+39) {
		tmp = ((x - (x * y)) / x) / x;
	} else if (x <= 0.042) {
		tmp = 1.0 / x;
	} else {
		tmp = 1.0 / (x * (1.0 + (y * (1.0 + (y * (0.5 + (y * 0.16666666666666666)))))));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -2.8e+39:
		tmp = ((x - (x * y)) / x) / x
	elif x <= 0.042:
		tmp = 1.0 / x
	else:
		tmp = 1.0 / (x * (1.0 + (y * (1.0 + (y * (0.5 + (y * 0.16666666666666666)))))))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -2.8e+39)
		tmp = Float64(Float64(Float64(x - Float64(x * y)) / x) / x);
	elseif (x <= 0.042)
		tmp = Float64(1.0 / x);
	else
		tmp = Float64(1.0 / Float64(x * Float64(1.0 + Float64(y * Float64(1.0 + Float64(y * Float64(0.5 + Float64(y * 0.16666666666666666))))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -2.8e+39)
		tmp = ((x - (x * y)) / x) / x;
	elseif (x <= 0.042)
		tmp = 1.0 / x;
	else
		tmp = 1.0 / (x * (1.0 + (y * (1.0 + (y * (0.5 + (y * 0.16666666666666666)))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -2.8e+39], N[(N[(N[(x - N[(x * y), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[x, 0.042], N[(1.0 / x), $MachinePrecision], N[(1.0 / N[(x * N[(1.0 + N[(y * N[(1.0 + N[(y * N[(0.5 + N[(y * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.80000000000000001e39

   1. Initial program 69.3%

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

      1. exp-prod 69.3%

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

   3. Simplified 69.3%

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

   5. Taylor expanded in y around 0 59.9%

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

      1. +-commutative 59.9%

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

      2. mul-1-neg 59.9%

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

      3. unsub-neg 59.9%

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

   7. Simplified 59.9%

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

      1. frac-sub 32.6%

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

      2. associate-/r* 73.5%

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

      3. *-un-lft-identity 73.5%

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

      4. *-commutative 73.5%

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

   9. Applied egg-rr 73.5%

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

   if -2.80000000000000001e39 < x < 0.0420000000000000026

   1. Initial program 82.2%

      \[\frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x} \]
   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 97.9%

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

   if 0.0420000000000000026 < x

   1. Initial program 79.9%

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

      1. *-commutative 79.9%

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

      2. exp-to-pow 79.9%

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

   3. Simplified 79.9%

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

   5. Taylor expanded in x around inf 100.0%

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

      1. mul-1-neg 100.0%

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

   7. Simplified 100.0%

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

      1. clear-num 100.0%

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

      2. inv-pow 100.0%

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

      3. exp-neg 100.0%

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

      4. associate-/r/ 100.0%

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

      5. /-rgt-identity 100.0%

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

   9. Applied egg-rr 100.0%

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

       1. unpow-1 100.0%

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

   11. Simplified 100.0%

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

   12. Taylor expanded in y around 0 85.0%

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

       1. *-commutative 85.0%

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

   14. Simplified 85.0%

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

4. Final simplification 87.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.8 \cdot 10^{+39}:\\ \;\;\;\;\frac{\frac{x - x \cdot y}{x}}{x}\\ \mathbf{elif}\;x \leq 0.042:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x \cdot \left(1 + y \cdot \left(1 + y \cdot \left(0.5 + y \cdot 0.16666666666666666\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 82.0% accurate, 8.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.2 \cdot 10^{+178}:\\ \;\;\;\;\frac{1}{x \cdot \left(1 + y\right)}\\ \mathbf{elif}\;x \leq -3 \cdot 10^{+45}:\\ \;\;\;\;\frac{1 + y \cdot \left(y \cdot 0.5\right)}{x}\\ \mathbf{elif}\;x \leq 5.2 \cdot 10^{+53}:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x \cdot \left(\left(y + 2\right) + -1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -3.2e+178)
   (/ 1.0 (* x (+ 1.0 y)))
   (if (<= x -3e+45)
     (/ (+ 1.0 (* y (* y 0.5))) x)
     (if (<= x 5.2e+53) (/ 1.0 x) (/ 1.0 (* x (+ (+ y 2.0) -1.0)))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -3.2e+178) {
		tmp = 1.0 / (x * (1.0 + y));
	} else if (x <= -3e+45) {
		tmp = (1.0 + (y * (y * 0.5))) / x;
	} else if (x <= 5.2e+53) {
		tmp = 1.0 / x;
	} else {
		tmp = 1.0 / (x * ((y + 2.0) + -1.0));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-3.2d+178)) then
        tmp = 1.0d0 / (x * (1.0d0 + y))
    else if (x <= (-3d+45)) then
        tmp = (1.0d0 + (y * (y * 0.5d0))) / x
    else if (x <= 5.2d+53) then
        tmp = 1.0d0 / x
    else
        tmp = 1.0d0 / (x * ((y + 2.0d0) + (-1.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -3.2e+178) {
		tmp = 1.0 / (x * (1.0 + y));
	} else if (x <= -3e+45) {
		tmp = (1.0 + (y * (y * 0.5))) / x;
	} else if (x <= 5.2e+53) {
		tmp = 1.0 / x;
	} else {
		tmp = 1.0 / (x * ((y + 2.0) + -1.0));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -3.2e+178:
		tmp = 1.0 / (x * (1.0 + y))
	elif x <= -3e+45:
		tmp = (1.0 + (y * (y * 0.5))) / x
	elif x <= 5.2e+53:
		tmp = 1.0 / x
	else:
		tmp = 1.0 / (x * ((y + 2.0) + -1.0))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -3.2e+178)
		tmp = Float64(1.0 / Float64(x * Float64(1.0 + y)));
	elseif (x <= -3e+45)
		tmp = Float64(Float64(1.0 + Float64(y * Float64(y * 0.5))) / x);
	elseif (x <= 5.2e+53)
		tmp = Float64(1.0 / x);
	else
		tmp = Float64(1.0 / Float64(x * Float64(Float64(y + 2.0) + -1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -3.2e+178)
		tmp = 1.0 / (x * (1.0 + y));
	elseif (x <= -3e+45)
		tmp = (1.0 + (y * (y * 0.5))) / x;
	elseif (x <= 5.2e+53)
		tmp = 1.0 / x;
	else
		tmp = 1.0 / (x * ((y + 2.0) + -1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -3.2e+178], N[(1.0 / N[(x * N[(1.0 + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -3e+45], N[(N[(1.0 + N[(y * N[(y * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[x, 5.2e+53], N[(1.0 / x), $MachinePrecision], N[(1.0 / N[(x * N[(N[(y + 2.0), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < -3.2e178

   1. Initial program 65.0%

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

      1. exp-prod 65.0%

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

   3. Simplified 65.0%

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

      1. clear-num 65.0%

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

      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 65.0%

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

      10. add-exp-log 65.0%

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

      11. inv-pow 65.0%

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

   6. Applied egg-rr 65.0%

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

      1. unpow-1 65.0%

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

   8. Simplified 65.0%

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

   9. Taylor expanded in y around 0 78.0%

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

       1. *-commutative 78.0%

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

       2. distribute-rgt1-in 78.0%

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

       3. +-commutative 78.0%

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

   11. Simplified 78.0%

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

   if -3.2e178 < x < -3.00000000000000011e45

   1. Initial program 72.7%

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

      1. *-commutative 72.7%

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

      2. exp-to-pow 72.7%

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

   3. Simplified 72.7%

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

   5. Taylor expanded in x around inf 100.0%

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

      1. mul-1-neg 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in y around 0 68.6%

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

   9. Taylor expanded in y around inf 67.0%

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

       1. *-commutative 67.0%

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

   11. Simplified 67.0%

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

   if -3.00000000000000011e45 < x < 5.19999999999999996e53

   1. Initial program 82.2%

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

      1. exp-prod 98.5%

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

   3. Simplified 98.5%

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

   5. Taylor expanded in x around 0 95.2%

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

   if 5.19999999999999996e53 < x

   1. Initial program 79.6%

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

      1. exp-prod 79.6%

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

   3. Simplified 79.6%

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

      1. clear-num 79.6%

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

      2. add-exp-log 74.5%

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

      3. add-exp-log 74.5%

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

      4. div-exp 74.5%

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

      5. pow-exp 74.5%

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

      6. add-log-exp 74.5%

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

      7. log-pow 74.5%

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

      8. div-exp 74.5%

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

      9. add-exp-log 79.6%

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

      10. add-exp-log 79.6%

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

      11. inv-pow 79.6%

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

   6. Applied egg-rr 79.6%

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

      1. unpow-1 79.6%

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

   8. Simplified 79.6%

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

   9. Taylor expanded in y around 0 81.6%

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

       1. *-commutative 81.6%

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

       2. distribute-rgt1-in 81.7%

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

   11. Applied egg-rr 81.7%

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

       1. expm1-log1p-u 81.5%

          \[\leadsto \frac{1}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(y + 1\right)\right)} \cdot x} \]

       2. expm1-undefine 81.5%

          \[\leadsto \frac{1}{\color{blue}{\left(e^{\mathsf{log1p}\left(y + 1\right)} - 1\right)} \cdot x} \]

   13. Applied egg-rr 81.5%

       \[\leadsto \frac{1}{\color{blue}{\left(e^{\mathsf{log1p}\left(y + 1\right)} - 1\right)} \cdot x} \]
   14. Step-by-step derivation

       1. sub-neg 81.5%

          \[\leadsto \frac{1}{\color{blue}{\left(e^{\mathsf{log1p}\left(y + 1\right)} + \left(-1\right)\right)} \cdot x} \]

       2. log1p-undefine 81.5%

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

       3. rem-exp-log 81.7%

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

       4. +-commutative 81.7%

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

       5. associate-+r+ 81.7%

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

       6. metadata-eval 81.7%

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

       7. metadata-eval 81.7%

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

   15. Simplified 81.7%

       \[\leadsto \frac{1}{\color{blue}{\left(\left(2 + y\right) + -1\right)} \cdot x} \]
3. Recombined 4 regimes into one program.

4. Final simplification 85.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.2 \cdot 10^{+178}:\\ \;\;\;\;\frac{1}{x \cdot \left(1 + y\right)}\\ \mathbf{elif}\;x \leq -3 \cdot 10^{+45}:\\ \;\;\;\;\frac{1 + y \cdot \left(y \cdot 0.5\right)}{x}\\ \mathbf{elif}\;x \leq 5.2 \cdot 10^{+53}:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x \cdot \left(\left(y + 2\right) + -1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 84.7% accurate, 9.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.8 \cdot 10^{+39}:\\ \;\;\;\;\frac{\frac{x - x \cdot y}{x}}{x}\\ \mathbf{elif}\;x \leq 0.47:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x \cdot \left(1 + y \cdot \left(1 + y \cdot 0.5\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -2.8e+39)
   (/ (/ (- x (* x y)) x) x)
   (if (<= x 0.47) (/ 1.0 x) (/ 1.0 (* x (+ 1.0 (* y (+ 1.0 (* y 0.5)))))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -2.8e+39) {
		tmp = ((x - (x * y)) / x) / x;
	} else if (x <= 0.47) {
		tmp = 1.0 / x;
	} else {
		tmp = 1.0 / (x * (1.0 + (y * (1.0 + (y * 0.5)))));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-2.8d+39)) then
        tmp = ((x - (x * y)) / x) / x
    else if (x <= 0.47d0) then
        tmp = 1.0d0 / x
    else
        tmp = 1.0d0 / (x * (1.0d0 + (y * (1.0d0 + (y * 0.5d0)))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -2.8e+39) {
		tmp = ((x - (x * y)) / x) / x;
	} else if (x <= 0.47) {
		tmp = 1.0 / x;
	} else {
		tmp = 1.0 / (x * (1.0 + (y * (1.0 + (y * 0.5)))));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -2.8e+39:
		tmp = ((x - (x * y)) / x) / x
	elif x <= 0.47:
		tmp = 1.0 / x
	else:
		tmp = 1.0 / (x * (1.0 + (y * (1.0 + (y * 0.5)))))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -2.8e+39)
		tmp = Float64(Float64(Float64(x - Float64(x * y)) / x) / x);
	elseif (x <= 0.47)
		tmp = Float64(1.0 / x);
	else
		tmp = Float64(1.0 / Float64(x * Float64(1.0 + Float64(y * Float64(1.0 + Float64(y * 0.5))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -2.8e+39)
		tmp = ((x - (x * y)) / x) / x;
	elseif (x <= 0.47)
		tmp = 1.0 / x;
	else
		tmp = 1.0 / (x * (1.0 + (y * (1.0 + (y * 0.5)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -2.8e+39], N[(N[(N[(x - N[(x * y), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[x, 0.47], N[(1.0 / x), $MachinePrecision], N[(1.0 / N[(x * N[(1.0 + N[(y * N[(1.0 + N[(y * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.80000000000000001e39

   1. Initial program 69.3%

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

      1. exp-prod 69.3%

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

   3. Simplified 69.3%

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

   5. Taylor expanded in y around 0 59.9%

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

      1. +-commutative 59.9%

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

      2. mul-1-neg 59.9%

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

      3. unsub-neg 59.9%

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

   7. Simplified 59.9%

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

      1. frac-sub 32.6%

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

      2. associate-/r* 73.5%

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

      3. *-un-lft-identity 73.5%

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

      4. *-commutative 73.5%

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

   9. Applied egg-rr 73.5%

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

   if -2.80000000000000001e39 < x < 0.46999999999999997

   1. Initial program 82.2%

      \[\frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x} \]
   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 97.9%

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

   if 0.46999999999999997 < x

   1. Initial program 79.9%

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

      1. *-commutative 79.9%

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

      2. exp-to-pow 79.9%

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

   3. Simplified 79.9%

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

   5. Taylor expanded in x around inf 100.0%

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

      1. mul-1-neg 100.0%

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

   7. Simplified 100.0%

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

      1. clear-num 100.0%

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

      2. inv-pow 100.0%

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

      3. exp-neg 100.0%

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

      4. associate-/r/ 100.0%

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

      5. /-rgt-identity 100.0%

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

   9. Applied egg-rr 100.0%

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

       1. unpow-1 100.0%

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

   11. Simplified 100.0%

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

   12. Taylor expanded in y around 0 83.6%

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

4. Final simplification 86.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.8 \cdot 10^{+39}:\\ \;\;\;\;\frac{\frac{x - x \cdot y}{x}}{x}\\ \mathbf{elif}\;x \leq 0.47:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x \cdot \left(1 + y \cdot \left(1 + y \cdot 0.5\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 80.6% accurate, 11.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -6.6 \cdot 10^{+40} \lor \neg \left(x \leq 5.2 \cdot 10^{+53}\right):\\ \;\;\;\;\frac{1}{x \cdot \left(\left(y + 2\right) + -1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -6.6e+40) (not (<= x 5.2e+53)))
   (/ 1.0 (* x (+ (+ y 2.0) -1.0)))
   (/ 1.0 x)))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -6.6e+40) || !(x <= 5.2e+53)) {
		tmp = 1.0 / (x * ((y + 2.0) + -1.0));
	} else {
		tmp = 1.0 / x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((x <= (-6.6d+40)) .or. (.not. (x <= 5.2d+53))) then
        tmp = 1.0d0 / (x * ((y + 2.0d0) + (-1.0d0)))
    else
        tmp = 1.0d0 / x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((x <= -6.6e+40) || !(x <= 5.2e+53)) {
		tmp = 1.0 / (x * ((y + 2.0) + -1.0));
	} else {
		tmp = 1.0 / x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -6.6e+40) or not (x <= 5.2e+53):
		tmp = 1.0 / (x * ((y + 2.0) + -1.0))
	else:
		tmp = 1.0 / x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -6.6e+40) || !(x <= 5.2e+53))
		tmp = Float64(1.0 / Float64(x * Float64(Float64(y + 2.0) + -1.0)));
	else
		tmp = Float64(1.0 / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((x <= -6.6e+40) || ~((x <= 5.2e+53)))
		tmp = 1.0 / (x * ((y + 2.0) + -1.0));
	else
		tmp = 1.0 / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -6.6e+40], N[Not[LessEqual[x, 5.2e+53]], $MachinePrecision]], N[(1.0 / N[(x * N[(N[(y + 2.0), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -6.5999999999999997e40 or 5.19999999999999996e53 < x

   1. Initial program 74.6%

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

      1. exp-prod 74.6%

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

   3. Simplified 74.6%

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

      1. clear-num 74.6%

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

      2. add-exp-log 38.3%

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

      3. add-exp-log 38.3%

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

      4. div-exp 38.3%

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

      5. pow-exp 38.3%

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

      6. add-log-exp 38.3%

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

      7. log-pow 38.3%

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

      8. div-exp 38.3%

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

      9. add-exp-log 74.6%

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

      10. add-exp-log 74.6%

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

      11. inv-pow 74.6%

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

   6. Applied egg-rr 74.6%

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

      1. unpow-1 74.6%

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

   8. Simplified 74.6%

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

   9. Taylor expanded in y around 0 73.7%

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

       1. *-commutative 73.7%

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

       2. distribute-rgt1-in 73.7%

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

   11. Applied egg-rr 73.7%

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

       1. expm1-log1p-u 73.4%

          \[\leadsto \frac{1}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(y + 1\right)\right)} \cdot x} \]

       2. expm1-undefine 73.4%

          \[\leadsto \frac{1}{\color{blue}{\left(e^{\mathsf{log1p}\left(y + 1\right)} - 1\right)} \cdot x} \]

   13. Applied egg-rr 73.4%

       \[\leadsto \frac{1}{\color{blue}{\left(e^{\mathsf{log1p}\left(y + 1\right)} - 1\right)} \cdot x} \]
   14. Step-by-step derivation

       1. sub-neg 73.4%

          \[\leadsto \frac{1}{\color{blue}{\left(e^{\mathsf{log1p}\left(y + 1\right)} + \left(-1\right)\right)} \cdot x} \]

       2. log1p-undefine 73.4%

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

       3. rem-exp-log 73.7%

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

       4. +-commutative 73.7%

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

       5. associate-+r+ 73.7%

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

       6. metadata-eval 73.7%

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

       7. metadata-eval 73.7%

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

   15. Simplified 73.7%

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

   if -6.5999999999999997e40 < x < 5.19999999999999996e53

   1. Initial program 82.2%

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

      1. exp-prod 98.8%

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

   3. Simplified 98.8%

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

   5. Taylor expanded in x around 0 96.3%

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

4. Final simplification 83.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -6.6 \cdot 10^{+40} \lor \neg \left(x \leq 5.2 \cdot 10^{+53}\right):\\ \;\;\;\;\frac{1}{x \cdot \left(\left(y + 2\right) + -1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 82.5% accurate, 11.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.8 \cdot 10^{+39}:\\ \;\;\;\;\frac{\frac{x - x \cdot y}{x}}{x}\\ \mathbf{elif}\;x \leq 5.2 \cdot 10^{+53}:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x \cdot \left(\left(y + 2\right) + -1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -2.8e+39)
   (/ (/ (- x (* x y)) x) x)
   (if (<= x 5.2e+53) (/ 1.0 x) (/ 1.0 (* x (+ (+ y 2.0) -1.0))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -2.8e+39) {
		tmp = ((x - (x * y)) / x) / x;
	} else if (x <= 5.2e+53) {
		tmp = 1.0 / x;
	} else {
		tmp = 1.0 / (x * ((y + 2.0) + -1.0));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-2.8d+39)) then
        tmp = ((x - (x * y)) / x) / x
    else if (x <= 5.2d+53) then
        tmp = 1.0d0 / x
    else
        tmp = 1.0d0 / (x * ((y + 2.0d0) + (-1.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -2.8e+39) {
		tmp = ((x - (x * y)) / x) / x;
	} else if (x <= 5.2e+53) {
		tmp = 1.0 / x;
	} else {
		tmp = 1.0 / (x * ((y + 2.0) + -1.0));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -2.8e+39:
		tmp = ((x - (x * y)) / x) / x
	elif x <= 5.2e+53:
		tmp = 1.0 / x
	else:
		tmp = 1.0 / (x * ((y + 2.0) + -1.0))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -2.8e+39)
		tmp = Float64(Float64(Float64(x - Float64(x * y)) / x) / x);
	elseif (x <= 5.2e+53)
		tmp = Float64(1.0 / x);
	else
		tmp = Float64(1.0 / Float64(x * Float64(Float64(y + 2.0) + -1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -2.8e+39)
		tmp = ((x - (x * y)) / x) / x;
	elseif (x <= 5.2e+53)
		tmp = 1.0 / x;
	else
		tmp = 1.0 / (x * ((y + 2.0) + -1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -2.8e+39], N[(N[(N[(x - N[(x * y), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[x, 5.2e+53], N[(1.0 / x), $MachinePrecision], N[(1.0 / N[(x * N[(N[(y + 2.0), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.80000000000000001e39

   1. Initial program 69.3%

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

      1. exp-prod 69.3%

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

   3. Simplified 69.3%

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

   5. Taylor expanded in y around 0 59.9%

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

      1. +-commutative 59.9%

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

      2. mul-1-neg 59.9%

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

      3. unsub-neg 59.9%

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

   7. Simplified 59.9%

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

      1. frac-sub 32.6%

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

      2. associate-/r* 73.5%

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

      3. *-un-lft-identity 73.5%

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

      4. *-commutative 73.5%

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

   9. Applied egg-rr 73.5%

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

   if -2.80000000000000001e39 < x < 5.19999999999999996e53

   1. Initial program 82.2%

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

      1. exp-prod 98.8%

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

   3. Simplified 98.8%

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

   5. Taylor expanded in x around 0 96.3%

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

   if 5.19999999999999996e53 < x

   1. Initial program 79.6%

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

      1. exp-prod 79.6%

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

   3. Simplified 79.6%

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

      1. clear-num 79.6%

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

      2. add-exp-log 74.5%

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

      3. add-exp-log 74.5%

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

      4. div-exp 74.5%

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

      5. pow-exp 74.5%

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

      6. add-log-exp 74.5%

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

      7. log-pow 74.5%

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

      8. div-exp 74.5%

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

      9. add-exp-log 79.6%

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

      10. add-exp-log 79.6%

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

      11. inv-pow 79.6%

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

   6. Applied egg-rr 79.6%

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

      1. unpow-1 79.6%

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

   8. Simplified 79.6%

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

   9. Taylor expanded in y around 0 81.6%

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

       1. *-commutative 81.6%

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

       2. distribute-rgt1-in 81.7%

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

   11. Applied egg-rr 81.7%

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

       1. expm1-log1p-u 81.5%

          \[\leadsto \frac{1}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(y + 1\right)\right)} \cdot x} \]

       2. expm1-undefine 81.5%

          \[\leadsto \frac{1}{\color{blue}{\left(e^{\mathsf{log1p}\left(y + 1\right)} - 1\right)} \cdot x} \]

   13. Applied egg-rr 81.5%

       \[\leadsto \frac{1}{\color{blue}{\left(e^{\mathsf{log1p}\left(y + 1\right)} - 1\right)} \cdot x} \]
   14. Step-by-step derivation

       1. sub-neg 81.5%

          \[\leadsto \frac{1}{\color{blue}{\left(e^{\mathsf{log1p}\left(y + 1\right)} + \left(-1\right)\right)} \cdot x} \]

       2. log1p-undefine 81.5%

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

       3. rem-exp-log 81.7%

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

       4. +-commutative 81.7%

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

       5. associate-+r+ 81.7%

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

       6. metadata-eval 81.7%

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

       7. metadata-eval 81.7%

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

   15. Simplified 81.7%

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

4. Final simplification 85.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.8 \cdot 10^{+39}:\\ \;\;\;\;\frac{\frac{x - x \cdot y}{x}}{x}\\ \mathbf{elif}\;x \leq 5.2 \cdot 10^{+53}:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x \cdot \left(\left(y + 2\right) + -1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 81.8% accurate, 11.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.8 \cdot 10^{+39}:\\ \;\;\;\;\frac{1 + y \cdot \left(y + -1\right)}{x}\\ \mathbf{elif}\;x \leq 5.2 \cdot 10^{+53}:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x \cdot \left(\left(y + 2\right) + -1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -2.8e+39)
   (/ (+ 1.0 (* y (+ y -1.0))) x)
   (if (<= x 5.2e+53) (/ 1.0 x) (/ 1.0 (* x (+ (+ y 2.0) -1.0))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -2.8e+39) {
		tmp = (1.0 + (y * (y + -1.0))) / x;
	} else if (x <= 5.2e+53) {
		tmp = 1.0 / x;
	} else {
		tmp = 1.0 / (x * ((y + 2.0) + -1.0));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-2.8d+39)) then
        tmp = (1.0d0 + (y * (y + (-1.0d0)))) / x
    else if (x <= 5.2d+53) then
        tmp = 1.0d0 / x
    else
        tmp = 1.0d0 / (x * ((y + 2.0d0) + (-1.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -2.8e+39) {
		tmp = (1.0 + (y * (y + -1.0))) / x;
	} else if (x <= 5.2e+53) {
		tmp = 1.0 / x;
	} else {
		tmp = 1.0 / (x * ((y + 2.0) + -1.0));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -2.8e+39:
		tmp = (1.0 + (y * (y + -1.0))) / x
	elif x <= 5.2e+53:
		tmp = 1.0 / x
	else:
		tmp = 1.0 / (x * ((y + 2.0) + -1.0))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -2.8e+39)
		tmp = Float64(Float64(1.0 + Float64(y * Float64(y + -1.0))) / x);
	elseif (x <= 5.2e+53)
		tmp = Float64(1.0 / x);
	else
		tmp = Float64(1.0 / Float64(x * Float64(Float64(y + 2.0) + -1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -2.8e+39)
		tmp = (1.0 + (y * (y + -1.0))) / x;
	elseif (x <= 5.2e+53)
		tmp = 1.0 / x;
	else
		tmp = 1.0 / (x * ((y + 2.0) + -1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -2.8e+39], N[(N[(1.0 + N[(y * N[(y + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[x, 5.2e+53], N[(1.0 / x), $MachinePrecision], N[(1.0 / N[(x * N[(N[(y + 2.0), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.80000000000000001e39

   1. Initial program 69.3%

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

      1. exp-prod 69.3%

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

   3. Simplified 69.3%

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

      1. clear-num 69.3%

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

      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 69.3%

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

      10. add-exp-log 69.3%

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

      11. inv-pow 69.3%

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

   6. Applied egg-rr 69.3%

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

      1. unpow-1 69.3%

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

   8. Simplified 69.3%

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

   9. Taylor expanded in y around 0 65.2%

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

   10. Taylor expanded in y around 0 64.2%

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

   11. Taylor expanded in x around 0 66.6%

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

   if -2.80000000000000001e39 < x < 5.19999999999999996e53

   1. Initial program 82.2%

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

      1. exp-prod 98.8%

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

   3. Simplified 98.8%

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

   5. Taylor expanded in x around 0 96.3%

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

   if 5.19999999999999996e53 < x

   1. Initial program 79.6%

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

      1. exp-prod 79.6%

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

   3. Simplified 79.6%

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

      1. clear-num 79.6%

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

      2. add-exp-log 74.5%

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

      3. add-exp-log 74.5%

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

      4. div-exp 74.5%

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

      5. pow-exp 74.5%

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

      6. add-log-exp 74.5%

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

      7. log-pow 74.5%

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

      8. div-exp 74.5%

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

      9. add-exp-log 79.6%

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

      10. add-exp-log 79.6%

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

      11. inv-pow 79.6%

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

   6. Applied egg-rr 79.6%

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

      1. unpow-1 79.6%

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

   8. Simplified 79.6%

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

   9. Taylor expanded in y around 0 81.6%

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

       1. *-commutative 81.6%

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

       2. distribute-rgt1-in 81.7%

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

   11. Applied egg-rr 81.7%

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

       1. expm1-log1p-u 81.5%

          \[\leadsto \frac{1}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(y + 1\right)\right)} \cdot x} \]

       2. expm1-undefine 81.5%

          \[\leadsto \frac{1}{\color{blue}{\left(e^{\mathsf{log1p}\left(y + 1\right)} - 1\right)} \cdot x} \]

   13. Applied egg-rr 81.5%

       \[\leadsto \frac{1}{\color{blue}{\left(e^{\mathsf{log1p}\left(y + 1\right)} - 1\right)} \cdot x} \]
   14. Step-by-step derivation

       1. sub-neg 81.5%

          \[\leadsto \frac{1}{\color{blue}{\left(e^{\mathsf{log1p}\left(y + 1\right)} + \left(-1\right)\right)} \cdot x} \]

       2. log1p-undefine 81.5%

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

       3. rem-exp-log 81.7%

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

       4. +-commutative 81.7%

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

       5. associate-+r+ 81.7%

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

       6. metadata-eval 81.7%

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

       7. metadata-eval 81.7%

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

   15. Simplified 81.7%

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

4. Final simplification 83.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.8 \cdot 10^{+39}:\\ \;\;\;\;\frac{1 + y \cdot \left(y + -1\right)}{x}\\ \mathbf{elif}\;x \leq 5.2 \cdot 10^{+53}:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x \cdot \left(\left(y + 2\right) + -1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 80.6% accurate, 12.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.1 \cdot 10^{+42} \lor \neg \left(x \leq 5.2 \cdot 10^{+53}\right):\\ \;\;\;\;\frac{1}{x \cdot \left(1 + y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -2.1e+42) (not (<= x 5.2e+53)))
   (/ 1.0 (* x (+ 1.0 y)))
   (/ 1.0 x)))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -2.1e+42) || !(x <= 5.2e+53)) {
		tmp = 1.0 / (x * (1.0 + y));
	} else {
		tmp = 1.0 / x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((x <= (-2.1d+42)) .or. (.not. (x <= 5.2d+53))) then
        tmp = 1.0d0 / (x * (1.0d0 + y))
    else
        tmp = 1.0d0 / x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((x <= -2.1e+42) || !(x <= 5.2e+53)) {
		tmp = 1.0 / (x * (1.0 + y));
	} else {
		tmp = 1.0 / x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -2.1e+42) or not (x <= 5.2e+53):
		tmp = 1.0 / (x * (1.0 + y))
	else:
		tmp = 1.0 / x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -2.1e+42) || !(x <= 5.2e+53))
		tmp = Float64(1.0 / Float64(x * Float64(1.0 + y)));
	else
		tmp = Float64(1.0 / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((x <= -2.1e+42) || ~((x <= 5.2e+53)))
		tmp = 1.0 / (x * (1.0 + y));
	else
		tmp = 1.0 / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -2.1e+42], N[Not[LessEqual[x, 5.2e+53]], $MachinePrecision]], N[(1.0 / N[(x * N[(1.0 + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2.09999999999999995e42 or 5.19999999999999996e53 < x

   1. Initial program 74.6%

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

      1. exp-prod 74.6%

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

   3. Simplified 74.6%

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

      1. clear-num 74.6%

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

      2. add-exp-log 38.3%

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

      3. add-exp-log 38.3%

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

      4. div-exp 38.3%

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

      5. pow-exp 38.3%

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

      6. add-log-exp 38.3%

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

      7. log-pow 38.3%

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

      8. div-exp 38.3%

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

      9. add-exp-log 74.6%

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

      10. add-exp-log 74.6%

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

      11. inv-pow 74.6%

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

   6. Applied egg-rr 74.6%

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

      1. unpow-1 74.6%

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

   8. Simplified 74.6%

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

   9. Taylor expanded in y around 0 73.7%

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

       1. *-commutative 73.7%

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

       2. distribute-rgt1-in 73.7%

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

       3. +-commutative 73.7%

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

   11. Simplified 73.7%

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

   if -2.09999999999999995e42 < x < 5.19999999999999996e53

   1. Initial program 82.2%

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

      1. exp-prod 98.8%

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

   3. Simplified 98.8%

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

   5. Taylor expanded in x around 0 96.3%

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

4. Final simplification 83.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.1 \cdot 10^{+42} \lor \neg \left(x \leq 5.2 \cdot 10^{+53}\right):\\ \;\;\;\;\frac{1}{x \cdot \left(1 + y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 80.6% accurate, 12.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -6.6 \cdot 10^{+43}:\\ \;\;\;\;\frac{1}{x + x \cdot y}\\ \mathbf{elif}\;x \leq 5.2 \cdot 10^{+53}:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x \cdot \left(1 + y\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -6.6e+43)
   (/ 1.0 (+ x (* x y)))
   (if (<= x 5.2e+53) (/ 1.0 x) (/ 1.0 (* x (+ 1.0 y))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -6.6e+43) {
		tmp = 1.0 / (x + (x * y));
	} else if (x <= 5.2e+53) {
		tmp = 1.0 / x;
	} else {
		tmp = 1.0 / (x * (1.0 + y));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-6.6d+43)) then
        tmp = 1.0d0 / (x + (x * y))
    else if (x <= 5.2d+53) then
        tmp = 1.0d0 / x
    else
        tmp = 1.0d0 / (x * (1.0d0 + y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -6.6e+43) {
		tmp = 1.0 / (x + (x * y));
	} else if (x <= 5.2e+53) {
		tmp = 1.0 / x;
	} else {
		tmp = 1.0 / (x * (1.0 + y));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -6.6e+43:
		tmp = 1.0 / (x + (x * y))
	elif x <= 5.2e+53:
		tmp = 1.0 / x
	else:
		tmp = 1.0 / (x * (1.0 + y))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -6.6e+43)
		tmp = Float64(1.0 / Float64(x + Float64(x * y)));
	elseif (x <= 5.2e+53)
		tmp = Float64(1.0 / x);
	else
		tmp = Float64(1.0 / Float64(x * Float64(1.0 + y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -6.6e+43)
		tmp = 1.0 / (x + (x * y));
	elseif (x <= 5.2e+53)
		tmp = 1.0 / x;
	else
		tmp = 1.0 / (x * (1.0 + y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -6.6e+43], N[(1.0 / N[(x + N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 5.2e+53], N[(1.0 / x), $MachinePrecision], N[(1.0 / N[(x * N[(1.0 + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -6.6000000000000003e43

   1. Initial program 69.3%

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

      1. exp-prod 69.3%

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

   3. Simplified 69.3%

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

      1. clear-num 69.3%

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

      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 69.3%

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

      10. add-exp-log 69.3%

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

      11. inv-pow 69.3%

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

   6. Applied egg-rr 69.3%

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

      1. unpow-1 69.3%

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

   8. Simplified 69.3%

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

   9. Taylor expanded in y around 0 65.2%

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

   if -6.6000000000000003e43 < x < 5.19999999999999996e53

   1. Initial program 82.2%

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

      1. exp-prod 98.8%

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

   3. Simplified 98.8%

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

   5. Taylor expanded in x around 0 96.3%

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

   if 5.19999999999999996e53 < x

   1. Initial program 79.6%

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

      1. exp-prod 79.6%

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

   3. Simplified 79.6%

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

      1. clear-num 79.6%

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

      2. add-exp-log 74.5%

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

      3. add-exp-log 74.5%

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

      4. div-exp 74.5%

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

      5. pow-exp 74.5%

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

      6. add-log-exp 74.5%

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

      7. log-pow 74.5%

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

      8. div-exp 74.5%

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

      9. add-exp-log 79.6%

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

      10. add-exp-log 79.6%

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

      11. inv-pow 79.6%

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

   6. Applied egg-rr 79.6%

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

      1. unpow-1 79.6%

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

   8. Simplified 79.6%

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

   9. Taylor expanded in y around 0 81.6%

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

       1. *-commutative 81.6%

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

       2. distribute-rgt1-in 81.7%

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

       3. +-commutative 81.7%

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

   11. Simplified 81.7%

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

4. Final simplification 83.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -6.6 \cdot 10^{+43}:\\ \;\;\;\;\frac{1}{x + x \cdot y}\\ \mathbf{elif}\;x \leq 5.2 \cdot 10^{+53}:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x \cdot \left(1 + y\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 74.8% accurate, 20.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.05 \cdot 10^{+140}:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x \cdot y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 1.05e+140) (/ 1.0 x) (/ 1.0 (* x y))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 1.05e+140) {
		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 <= 1.05d+140) 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 <= 1.05e+140) {
		tmp = 1.0 / x;
	} else {
		tmp = 1.0 / (x * y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 1.05e+140:
		tmp = 1.0 / x
	else:
		tmp = 1.0 / (x * y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 1.05e+140)
		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 <= 1.05e+140)
		tmp = 1.0 / x;
	else
		tmp = 1.0 / (x * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 1.05e+140], N[(1.0 / x), $MachinePrecision], N[(1.0 / N[(x * y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.0500000000000001e140

   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 86.0%

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

   3. Simplified 86.0%

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

   5. Taylor expanded in x around 0 80.2%

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

   if 1.0500000000000001e140 < y

   1. Initial program 64.3%

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

      1. exp-prod 79.3%

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

   3. Simplified 79.3%

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

      1. clear-num 79.3%

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

      2. add-exp-log 76.2%

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

      3. add-exp-log 76.2%

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

      4. div-exp 76.2%

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

      5. pow-exp 62.7%

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

      6. add-log-exp 62.7%

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

      7. log-pow 62.7%

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

      8. div-exp 62.6%

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

      9. add-exp-log 64.4%

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

      10. add-exp-log 64.4%

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

      11. inv-pow 64.4%

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

   6. Applied egg-rr 64.4%

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

      1. unpow-1 64.4%

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

   8. Simplified 64.4%

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

   9. Taylor expanded in y around 0 69.8%

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

   10. Taylor expanded in y around inf 69.8%

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

Alternative 13: 74.2% accurate, 69.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.9%

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

   1. exp-prod 85.2%

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

3. Simplified 85.2%

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

5. Taylor expanded in x around 0 74.3%

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

Developer Target 1: 77.5% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024186 
(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))