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

Percentage Accurate: 78.0% → 99.7%

Time: 8.7s

Alternatives: 7

Speedup: 29.6×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 78.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{+26} \lor \neg \left(x \leq 0.098\right):\\ \;\;\;\;\frac{e^{-y}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(e^{x}\right)}^{\log \left(\frac{x}{x + y}\right)}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -2e+26) (not (<= x 0.098)))
   (/ (exp (- y)) x)
   (/ (pow (exp x) (log (/ x (+ x y)))) x)))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -2e+26) || !(x <= 0.098)) {
		tmp = exp(-y) / x;
	} else {
		tmp = pow(exp(x), log((x / (x + y)))) / x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((x <= (-2d+26)) .or. (.not. (x <= 0.098d0))) then
        tmp = exp(-y) / x
    else
        tmp = (exp(x) ** log((x / (x + y)))) / x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((x <= -2e+26) || !(x <= 0.098)) {
		tmp = Math.exp(-y) / x;
	} else {
		tmp = Math.pow(Math.exp(x), Math.log((x / (x + y)))) / x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -2e+26) or not (x <= 0.098):
		tmp = math.exp(-y) / x
	else:
		tmp = math.pow(math.exp(x), math.log((x / (x + y)))) / x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -2e+26) || !(x <= 0.098))
		tmp = Float64(exp(Float64(-y)) / x);
	else
		tmp = Float64((exp(x) ^ log(Float64(x / Float64(x + y)))) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((x <= -2e+26) || ~((x <= 0.098)))
		tmp = exp(-y) / x;
	else
		tmp = (exp(x) ^ log((x / (x + y)))) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -2e+26], N[Not[LessEqual[x, 0.098]], $MachinePrecision]], N[(N[Exp[(-y)], $MachinePrecision] / x), $MachinePrecision], N[(N[Power[N[Exp[x], $MachinePrecision], N[Log[N[(x / N[(x + y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2.0000000000000001e26 or 0.098000000000000004 < x

   1. Initial program 72.6%

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

      1. *-commutative 72.6%

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

      2. exp-to-pow 72.6%

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

   3. Simplified 72.6%

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

   4. Taylor expanded in x around inf 100.0%

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

      1. mul-1-neg 100.0%

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

   6. Simplified 100.0%

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

   if -2.0000000000000001e26 < x < 0.098000000000000004

   1. Initial program 81.0%

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

      1. exp-prod 99.8%

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

   3. Simplified 99.8%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{+26} \lor \neg \left(x \leq 0.098\right):\\ \;\;\;\;\frac{e^{-y}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(e^{x}\right)}^{\log \left(\frac{x}{x + y}\right)}}{x}\\ \end{array} \]

Alternative 2: 99.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.58 \lor \neg \left(x \leq 0.098\right):\\ \;\;\;\;\frac{e^{-y}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -1.58) (not (<= x 0.098))) (/ (exp (- y)) x) (/ 1.0 x)))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -1.58) || !(x <= 0.098)) {
		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 <= (-1.58d0)) .or. (.not. (x <= 0.098d0))) 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 <= -1.58) || !(x <= 0.098)) {
		tmp = Math.exp(-y) / x;
	} else {
		tmp = 1.0 / x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -1.58) or not (x <= 0.098):
		tmp = math.exp(-y) / x
	else:
		tmp = 1.0 / x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -1.58) || !(x <= 0.098))
		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 <= -1.58) || ~((x <= 0.098)))
		tmp = exp(-y) / x;
	else
		tmp = 1.0 / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -1.58], N[Not[LessEqual[x, 0.098]], $MachinePrecision]], N[(N[Exp[(-y)], $MachinePrecision] / x), $MachinePrecision], N[(1.0 / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.5800000000000001 or 0.098000000000000004 < x

   1. Initial program 73.0%

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

      1. *-commutative 73.0%

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

      2. exp-to-pow 73.0%

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

   3. Simplified 73.0%

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

   4. Taylor expanded in x around inf 100.0%

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

      1. mul-1-neg 100.0%

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

   6. Simplified 100.0%

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

   if -1.5800000000000001 < x < 0.098000000000000004

   1. Initial program 80.6%

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

      1. exp-prod 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in x around 0 98.8%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.58 \lor \neg \left(x \leq 0.098\right):\\ \;\;\;\;\frac{e^{-y}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x}\\ \end{array} \]

Alternative 3: 81.8% accurate, 12.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -4 \cdot 10^{+132}:\\ \;\;\;\;\frac{\frac{x \cdot \left(-y\right)}{x}}{x}\\ \mathbf{elif}\;y \leq 160000000000:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{elif}\;y \leq 1.85 \cdot 10^{+145}:\\ \;\;\;\;y \cdot \left(x \cdot \frac{1}{y \cdot \left(x \cdot x\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{x \cdot y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -4e+132)
   (/ (/ (* x (- y)) x) x)
   (if (<= y 160000000000.0)
     (/ 1.0 x)
     (if (<= y 1.85e+145) (* y (* x (/ 1.0 (* y (* x x))))) (/ y (* x y))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -4e+132) {
		tmp = ((x * -y) / x) / x;
	} else if (y <= 160000000000.0) {
		tmp = 1.0 / x;
	} else if (y <= 1.85e+145) {
		tmp = y * (x * (1.0 / (y * (x * x))));
	} else {
		tmp = y / (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 <= (-4d+132)) then
        tmp = ((x * -y) / x) / x
    else if (y <= 160000000000.0d0) then
        tmp = 1.0d0 / x
    else if (y <= 1.85d+145) then
        tmp = y * (x * (1.0d0 / (y * (x * x))))
    else
        tmp = y / (x * y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -4e+132) {
		tmp = ((x * -y) / x) / x;
	} else if (y <= 160000000000.0) {
		tmp = 1.0 / x;
	} else if (y <= 1.85e+145) {
		tmp = y * (x * (1.0 / (y * (x * x))));
	} else {
		tmp = y / (x * y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -4e+132:
		tmp = ((x * -y) / x) / x
	elif y <= 160000000000.0:
		tmp = 1.0 / x
	elif y <= 1.85e+145:
		tmp = y * (x * (1.0 / (y * (x * x))))
	else:
		tmp = y / (x * y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -4e+132)
		tmp = Float64(Float64(Float64(x * Float64(-y)) / x) / x);
	elseif (y <= 160000000000.0)
		tmp = Float64(1.0 / x);
	elseif (y <= 1.85e+145)
		tmp = Float64(y * Float64(x * Float64(1.0 / Float64(y * Float64(x * x)))));
	else
		tmp = Float64(y / Float64(x * y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -4e+132)
		tmp = ((x * -y) / x) / x;
	elseif (y <= 160000000000.0)
		tmp = 1.0 / x;
	elseif (y <= 1.85e+145)
		tmp = y * (x * (1.0 / (y * (x * x))));
	else
		tmp = y / (x * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -4e+132], N[(N[(N[(x * (-y)), $MachinePrecision] / x), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[y, 160000000000.0], N[(1.0 / x), $MachinePrecision], If[LessEqual[y, 1.85e+145], N[(y * N[(x * N[(1.0 / N[(y * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y / N[(x * y), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if y < -3.99999999999999996e132

   1. Initial program 39.9%

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

      1. exp-prod 66.1%

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

   3. Simplified 66.1%

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

   4. Taylor expanded in x around inf 3.7%

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

      1. mul-1-neg 3.7%

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

      2. unsub-neg 3.7%

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

   6. Simplified 3.7%

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

      1. frac-sub 5.3%

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

      2. associate-/r* 63.9%

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

      3. *-un-lft-identity 63.9%

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

      4. *-commutative 63.9%

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

   8. Applied egg-rr 63.9%

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

   9. Taylor expanded in y around inf 63.9%

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

       1. associate-*r* 63.9%

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

       2. neg-mul-1 63.9%

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

   11. Simplified 63.9%

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

   if -3.99999999999999996e132 < y < 1.6e11

   1. Initial program 90.2%

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

      1. exp-prod 90.7%

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

   3. Simplified 90.7%

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

   4. Taylor expanded in x around 0 89.6%

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

   if 1.6e11 < y < 1.84999999999999997e145

   1. Initial program 54.9%

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

      1. exp-prod 58.9%

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

   3. Simplified 58.9%

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

   4. Taylor expanded in x around inf 3.0%

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

      1. mul-1-neg 3.0%

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

      2. unsub-neg 3.0%

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

   6. Simplified 3.0%

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

      1. clear-num 3.0%

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

      2. frac-sub 36.1%

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

      3. *-un-lft-identity 36.1%

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

      4. *-commutative 36.1%

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

      5. *-un-lft-identity 36.1%

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

   8. Applied egg-rr 36.1%

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

      1. associate-*r/ 36.1%

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

      2. associate-/r/ 36.1%

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

   10. Simplified 36.1%

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

   11. Taylor expanded in y around 0 50.7%

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

       1. *-commutative 50.7%

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

       2. associate-/r* 42.5%

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

       3. inv-pow 42.5%

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

       4. metadata-eval 42.5%

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

       5. pow-div 54.8%

          \[\leadsto \frac{\color{blue}{\frac{{x}^{1}}{{x}^{2}}}}{y} \cdot y \]

       6. pow1 54.8%

          \[\leadsto \frac{\frac{\color{blue}{x}}{{x}^{2}}}{y} \cdot y \]

       7. pow2 54.8%

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

       8. associate-/r* 75.5%

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

       9. div-inv 75.5%

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

       10. *-commutative 75.5%

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

   13. Applied egg-rr 75.5%

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

   if 1.84999999999999997e145 < y

   1. Initial program 32.8%

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

      1. exp-prod 79.2%

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

   3. Simplified 79.2%

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

   4. Taylor expanded in x around inf 1.3%

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

      1. mul-1-neg 1.3%

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

      2. unsub-neg 1.3%

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

   6. Simplified 1.3%

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

      1. clear-num 1.3%

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

      2. frac-sub 13.9%

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

      3. *-un-lft-identity 13.9%

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

      4. *-commutative 13.9%

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

      5. *-un-lft-identity 13.9%

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

   8. Applied egg-rr 13.9%

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

      1. associate-*r/ 26.5%

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

      2. associate-/r/ 26.5%

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

   10. Simplified 26.5%

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

   11. Taylor expanded in y around 0 97.7%

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

       1. associate-*l/ 97.8%

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

       2. *-un-lft-identity 97.8%

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

   13. Applied egg-rr 97.8%

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

4. Final simplification 86.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4 \cdot 10^{+132}:\\ \;\;\;\;\frac{\frac{x \cdot \left(-y\right)}{x}}{x}\\ \mathbf{elif}\;y \leq 160000000000:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{elif}\;y \leq 1.85 \cdot 10^{+145}:\\ \;\;\;\;y \cdot \left(x \cdot \frac{1}{y \cdot \left(x \cdot x\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{x \cdot y}\\ \end{array} \]

Alternative 4: 80.6% accurate, 20.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -4.5 \cdot 10^{+132}:\\ \;\;\;\;\frac{\frac{x \cdot \left(-y\right)}{x}}{x}\\ \mathbf{elif}\;y \leq 3.8 \cdot 10^{+44}:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{x \cdot y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -4.5e+132)
   (/ (/ (* x (- y)) x) x)
   (if (<= y 3.8e+44) (/ 1.0 x) (/ y (* x y)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -4.5e+132) {
		tmp = ((x * -y) / x) / x;
	} else if (y <= 3.8e+44) {
		tmp = 1.0 / x;
	} else {
		tmp = y / (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 <= (-4.5d+132)) then
        tmp = ((x * -y) / x) / x
    else if (y <= 3.8d+44) then
        tmp = 1.0d0 / x
    else
        tmp = y / (x * y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -4.5e+132) {
		tmp = ((x * -y) / x) / x;
	} else if (y <= 3.8e+44) {
		tmp = 1.0 / x;
	} else {
		tmp = y / (x * y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -4.5e+132:
		tmp = ((x * -y) / x) / x
	elif y <= 3.8e+44:
		tmp = 1.0 / x
	else:
		tmp = y / (x * y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -4.5e+132)
		tmp = Float64(Float64(Float64(x * Float64(-y)) / x) / x);
	elseif (y <= 3.8e+44)
		tmp = Float64(1.0 / x);
	else
		tmp = Float64(y / Float64(x * y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -4.5e+132)
		tmp = ((x * -y) / x) / x;
	elseif (y <= 3.8e+44)
		tmp = 1.0 / x;
	else
		tmp = y / (x * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -4.5e+132], N[(N[(N[(x * (-y)), $MachinePrecision] / x), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[y, 3.8e+44], N[(1.0 / x), $MachinePrecision], N[(y / N[(x * y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -4.49999999999999972e132

   1. Initial program 39.9%

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

      1. exp-prod 66.1%

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

   3. Simplified 66.1%

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

   4. Taylor expanded in x around inf 3.7%

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

      1. mul-1-neg 3.7%

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

      2. unsub-neg 3.7%

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

   6. Simplified 3.7%

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

      1. frac-sub 5.3%

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

      2. associate-/r* 63.9%

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

      3. *-un-lft-identity 63.9%

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

      4. *-commutative 63.9%

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

   8. Applied egg-rr 63.9%

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

   9. Taylor expanded in y around inf 63.9%

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

       1. associate-*r* 63.9%

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

       2. neg-mul-1 63.9%

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

   11. Simplified 63.9%

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

   if -4.49999999999999972e132 < y < 3.8000000000000002e44

   1. Initial program 89.0%

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

      1. exp-prod 89.5%

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

   3. Simplified 89.5%

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

   4. Taylor expanded in x around 0 88.4%

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

   if 3.8000000000000002e44 < y

   1. Initial program 42.5%

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

      1. exp-prod 71.4%

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

   3. Simplified 71.4%

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

   4. Taylor expanded in x around inf 2.0%

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

      1. mul-1-neg 2.0%

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

      2. unsub-neg 2.0%

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

   6. Simplified 2.0%

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

      1. clear-num 2.0%

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

      2. frac-sub 21.1%

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

      3. *-un-lft-identity 21.1%

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

      4. *-commutative 21.1%

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

      5. *-un-lft-identity 21.1%

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

   8. Applied egg-rr 21.1%

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

      1. associate-*r/ 28.4%

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

      2. associate-/r/ 28.4%

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

   10. Simplified 28.4%

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

   11. Taylor expanded in y around 0 77.4%

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

       1. associate-*l/ 77.5%

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

       2. *-un-lft-identity 77.5%

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

   13. Applied egg-rr 77.5%

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

4. Final simplification 83.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4.5 \cdot 10^{+132}:\\ \;\;\;\;\frac{\frac{x \cdot \left(-y\right)}{x}}{x}\\ \mathbf{elif}\;y \leq 3.8 \cdot 10^{+44}:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{x \cdot y}\\ \end{array} \]

Alternative 5: 77.4% accurate, 22.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 9500000000:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{elif}\;y \leq 8 \cdot 10^{+162}:\\ \;\;\;\;\frac{x}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 9500000000.0) (/ 1.0 x) (if (<= y 8e+162) (/ x (* x x)) (/ 1.0 x))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 9500000000.0) {
		tmp = 1.0 / x;
	} else if (y <= 8e+162) {
		tmp = x / (x * 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 (y <= 9500000000.0d0) then
        tmp = 1.0d0 / x
    else if (y <= 8d+162) then
        tmp = x / (x * x)
    else
        tmp = 1.0d0 / x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 9500000000.0) {
		tmp = 1.0 / x;
	} else if (y <= 8e+162) {
		tmp = x / (x * x);
	} else {
		tmp = 1.0 / x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 9500000000.0:
		tmp = 1.0 / x
	elif y <= 8e+162:
		tmp = x / (x * x)
	else:
		tmp = 1.0 / x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 9500000000.0)
		tmp = Float64(1.0 / x);
	elseif (y <= 8e+162)
		tmp = Float64(x / Float64(x * x));
	else
		tmp = Float64(1.0 / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 9500000000.0)
		tmp = 1.0 / x;
	elseif (y <= 8e+162)
		tmp = x / (x * x);
	else
		tmp = 1.0 / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 9500000000.0], N[(1.0 / x), $MachinePrecision], If[LessEqual[y, 8e+162], N[(x / N[(x * x), $MachinePrecision]), $MachinePrecision], N[(1.0 / x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < 9.5e9 or 7.9999999999999995e162 < y

   1. Initial program 80.1%

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

      1. exp-prod 88.1%

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

   3. Simplified 88.1%

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

   4. Taylor expanded in x around 0 81.1%

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

   if 9.5e9 < y < 7.9999999999999995e162

   1. Initial program 46.5%

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

      1. exp-prod 52.8%

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

   3. Simplified 52.8%

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

   4. Taylor expanded in x around inf 2.9%

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

      1. mul-1-neg 2.9%

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

      2. unsub-neg 2.9%

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

   6. Simplified 2.9%

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

      1. frac-sub 11.1%

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

      2. *-un-lft-identity 11.1%

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

      3. *-commutative 11.1%

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

   8. Applied egg-rr 11.1%

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

   9. Taylor expanded in y around 0 59.0%

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

4. Final simplification 78.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 9500000000:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{elif}\;y \leq 8 \cdot 10^{+162}:\\ \;\;\;\;\frac{x}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x}\\ \end{array} \]

Alternative 6: 80.9% accurate, 29.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.85 \cdot 10^{+42}:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{x \cdot y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= y 1.85e+42) (/ 1.0 x) (/ y (* x y))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 1.85e+42) {
		tmp = 1.0 / x;
	} else {
		tmp = y / (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 <= 1.85d+42) then
        tmp = 1.0d0 / x
    else
        tmp = y / (x * y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 1.85e+42) {
		tmp = 1.0 / x;
	} else {
		tmp = y / (x * y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 1.85e+42:
		tmp = 1.0 / x
	else:
		tmp = y / (x * y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 1.85e+42)
		tmp = Float64(1.0 / x);
	else
		tmp = Float64(y / Float64(x * y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 1.85e+42)
		tmp = 1.0 / x;
	else
		tmp = y / (x * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 1.85e+42], N[(1.0 / x), $MachinePrecision], N[(y / N[(x * y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.84999999999999998e42

   1. Initial program 82.4%

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

      1. exp-prod 86.3%

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

   3. Simplified 86.3%

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

   4. Taylor expanded in x around 0 80.4%

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

   if 1.84999999999999998e42 < y

   1. Initial program 42.5%

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

      1. exp-prod 71.4%

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

   3. Simplified 71.4%

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

   4. Taylor expanded in x around inf 2.0%

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

      1. mul-1-neg 2.0%

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

      2. unsub-neg 2.0%

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

   6. Simplified 2.0%

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

      1. clear-num 2.0%

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

      2. frac-sub 21.1%

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

      3. *-un-lft-identity 21.1%

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

      4. *-commutative 21.1%

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

      5. *-un-lft-identity 21.1%

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

   8. Applied egg-rr 21.1%

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

      1. associate-*r/ 28.4%

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

      2. associate-/r/ 28.4%

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

   10. Simplified 28.4%

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

   11. Taylor expanded in y around 0 77.4%

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

       1. associate-*l/ 77.5%

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

       2. *-un-lft-identity 77.5%

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

   13. Applied egg-rr 77.5%

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

4. Final simplification 80.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.85 \cdot 10^{+42}:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{x \cdot y}\\ \end{array} \]

Alternative 7: 75.0% accurate, 69.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 76.1%

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

   1. exp-prod 84.0%

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

3. Simplified 84.0%

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

4. Taylor expanded in x around 0 75.3%

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

5. Final simplification 75.3%

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

Developer target: 78.5% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2023176 
(FPCore (x y)
  :name "Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2, F"
  :precision binary64

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

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