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

Percentage Accurate: 77.3% → 98.8%

Time: 9.8s

Alternatives: 6

Speedup: 20.9×

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: 77.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -7 \cdot 10^{+109} \lor \neg \left(x \leq 7.2 \cdot 10^{-5}\right):\\ \;\;\;\;\frac{e^{-y}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(e^{x}\right)}^{\log \left(\frac{x}{x + y}\right)}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -7e+109) (not (<= x 7.2e-5)))
   (/ (exp (- y)) x)
   (/ (pow (exp x) (log (/ x (+ x y)))) x)))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -7e+109) || !(x <= 7.2e-5)) {
		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 <= (-7d+109)) .or. (.not. (x <= 7.2d-5))) 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 <= -7e+109) || !(x <= 7.2e-5)) {
		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 <= -7e+109) or not (x <= 7.2e-5):
		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 <= -7e+109) || !(x <= 7.2e-5))
		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 <= -7e+109) || ~((x <= 7.2e-5)))
		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, -7e+109], N[Not[LessEqual[x, 7.2e-5]], $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 < -6.99999999999999966e109 or 7.20000000000000018e-5 < x

   1. Initial program 68.7%

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

      1. *-commutative 68.7%

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

      2. exp-to-pow 68.7%

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

   3. Simplified 68.7%

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

   5. Taylor expanded in x around inf 100.0%

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

      1. mul-1-neg 100.0%

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

   7. Simplified 100.0%

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

   if -6.99999999999999966e109 < x < 7.20000000000000018e-5

   1. Initial program 83.9%

      \[\frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x} \]
   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 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -7 \cdot 10^{+109} \lor \neg \left(x \leq 7.2 \cdot 10^{-5}\right):\\ \;\;\;\;\frac{e^{-y}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(e^{x}\right)}^{\log \left(\frac{x}{x + y}\right)}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.8 \lor \neg \left(x \leq 7.2 \cdot 10^{-5}\right):\\ \;\;\;\;\frac{e^{-y}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -0.8) (not (<= x 7.2e-5))) (/ (exp (- y)) x) (/ 1.0 x)))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -0.8) || !(x <= 7.2e-5)) {
		tmp = exp(-y) / x;
	} else {
		tmp = 1.0 / x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((x <= (-0.8d0)) .or. (.not. (x <= 7.2d-5))) then
        tmp = exp(-y) / x
    else
        tmp = 1.0d0 / x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((x <= -0.8) || !(x <= 7.2e-5)) {
		tmp = Math.exp(-y) / x;
	} else {
		tmp = 1.0 / x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -0.8) or not (x <= 7.2e-5):
		tmp = math.exp(-y) / x
	else:
		tmp = 1.0 / x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -0.8) || !(x <= 7.2e-5))
		tmp = Float64(exp(Float64(-y)) / x);
	else
		tmp = Float64(1.0 / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((x <= -0.8) || ~((x <= 7.2e-5)))
		tmp = exp(-y) / x;
	else
		tmp = 1.0 / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -0.8], N[Not[LessEqual[x, 7.2e-5]], $MachinePrecision]], N[(N[Exp[(-y)], $MachinePrecision] / x), $MachinePrecision], N[(1.0 / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -0.80000000000000004 or 7.20000000000000018e-5 < x

   1. Initial program 74.2%

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

      1. *-commutative 74.2%

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

      2. exp-to-pow 74.2%

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

   3. Simplified 74.2%

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

   5. Taylor expanded in x around inf 100.0%

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

      1. mul-1-neg 100.0%

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

   7. Simplified 100.0%

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

   if -0.80000000000000004 < x < 7.20000000000000018e-5

   1. Initial program 79.7%

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

      1. exp-prod 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in x around 0 98.4%

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

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.8 \lor \neg \left(x \leq 7.2 \cdot 10^{-5}\right):\\ \;\;\;\;\frac{e^{-y}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 81.3% accurate, 11.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.32 \lor \neg \left(x \leq 7.2 \cdot 10^{-5}\right):\\ \;\;\;\;\frac{\frac{x - x \cdot y}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -0.32) (not (<= x 7.2e-5)))
   (/ (/ (- x (* x y)) x) x)
   (/ 1.0 x)))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -0.32) || !(x <= 7.2e-5)) {
		tmp = ((x - (x * y)) / 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 ((x <= (-0.32d0)) .or. (.not. (x <= 7.2d-5))) then
        tmp = ((x - (x * y)) / 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 ((x <= -0.32) || !(x <= 7.2e-5)) {
		tmp = ((x - (x * y)) / x) / x;
	} else {
		tmp = 1.0 / x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -0.32) or not (x <= 7.2e-5):
		tmp = ((x - (x * y)) / x) / x
	else:
		tmp = 1.0 / x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -0.32) || !(x <= 7.2e-5))
		tmp = Float64(Float64(Float64(x - Float64(x * y)) / x) / x);
	else
		tmp = Float64(1.0 / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((x <= -0.32) || ~((x <= 7.2e-5)))
		tmp = ((x - (x * y)) / x) / x;
	else
		tmp = 1.0 / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -0.32], N[Not[LessEqual[x, 7.2e-5]], $MachinePrecision]], N[(N[(N[(x - N[(x * y), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] / x), $MachinePrecision], N[(1.0 / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -0.320000000000000007 or 7.20000000000000018e-5 < x

   1. Initial program 74.2%

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

      1. exp-prod 74.2%

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

   3. Simplified 74.2%

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

   5. Taylor expanded in x around inf 57.2%

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

      1. +-commutative 57.2%

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

      2. mul-1-neg 57.2%

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

      3. unsub-neg 57.2%

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

   7. Simplified 57.2%

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

      1. frac-sub 34.6%

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

      2. associate-/r* 69.8%

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

      3. *-un-lft-identity 69.8%

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

      4. *-commutative 69.8%

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

   9. Applied egg-rr 69.8%

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

   if -0.320000000000000007 < x < 7.20000000000000018e-5

   1. Initial program 79.7%

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

      1. exp-prod 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in x around 0 98.4%

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

4. Final simplification 81.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.32 \lor \neg \left(x \leq 7.2 \cdot 10^{-5}\right):\\ \;\;\;\;\frac{\frac{x - x \cdot y}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 77.0% accurate, 13.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3.4 \cdot 10^{+36}:\\ \;\;\;\;\frac{\frac{x \cdot y}{-x}}{x}\\ \mathbf{elif}\;y \leq 85:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -3.4e+36)
   (/ (/ (* x y) (- x)) x)
   (if (<= y 85.0) (/ 1.0 x) (/ x (* x x)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -3.4e+36) {
		tmp = ((x * y) / -x) / x;
	} else if (y <= 85.0) {
		tmp = 1.0 / x;
	} else {
		tmp = x / (x * x);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-3.4d+36)) then
        tmp = ((x * y) / -x) / x
    else if (y <= 85.0d0) then
        tmp = 1.0d0 / x
    else
        tmp = x / (x * x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -3.4e+36) {
		tmp = ((x * y) / -x) / x;
	} else if (y <= 85.0) {
		tmp = 1.0 / x;
	} else {
		tmp = x / (x * x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -3.4e+36:
		tmp = ((x * y) / -x) / x
	elif y <= 85.0:
		tmp = 1.0 / x
	else:
		tmp = x / (x * x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -3.4e+36)
		tmp = Float64(Float64(Float64(x * y) / Float64(-x)) / x);
	elseif (y <= 85.0)
		tmp = Float64(1.0 / x);
	else
		tmp = Float64(x / Float64(x * x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -3.4e+36)
		tmp = ((x * y) / -x) / x;
	elseif (y <= 85.0)
		tmp = 1.0 / x;
	else
		tmp = x / (x * x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -3.4e+36], N[(N[(N[(x * y), $MachinePrecision] / (-x)), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[y, 85.0], N[(1.0 / x), $MachinePrecision], N[(x / N[(x * x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -3.3999999999999998e36

   1. Initial program 38.7%

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

      1. exp-prod 57.4%

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

   3. Simplified 57.4%

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

   5. Taylor expanded in x around inf 4.1%

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

      1. +-commutative 4.1%

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

      2. mul-1-neg 4.1%

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

      3. unsub-neg 4.1%

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

   7. Simplified 4.1%

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

      1. frac-sub 7.3%

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

      2. associate-/r* 48.3%

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

      3. *-un-lft-identity 48.3%

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

      4. *-commutative 48.3%

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

   9. Applied egg-rr 48.3%

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

   10. Taylor expanded in y around inf 48.3%

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

       1. associate-*r* 48.3%

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

       2. mul-1-neg 48.3%

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

   12. Simplified 48.3%

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

   if -3.3999999999999998e36 < y < 85

   1. Initial program 98.2%

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

      1. exp-prod 98.2%

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

   3. Simplified 98.2%

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

   5. Taylor expanded in x around 0 96.8%

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

   if 85 < y

   1. Initial program 45.6%

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

      1. exp-prod 67.7%

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

   3. Simplified 67.7%

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

   5. Taylor expanded in x around inf 2.1%

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

      1. +-commutative 2.1%

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

      2. mul-1-neg 2.1%

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

      3. unsub-neg 2.1%

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

   7. Simplified 2.1%

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

      1. sub-neg 2.1%

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

      2. distribute-neg-frac2 2.1%

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

      3. add-sqr-sqrt 2.1%

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

      4. sqrt-unprod 1.8%

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

      5. sqr-neg 1.8%

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

      6. sqrt-unprod 0.0%

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

      7. add-sqr-sqrt 4.9%

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

      8. frac-add 10.7%

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

      9. *-un-lft-identity 10.7%

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

      10. add-sqr-sqrt 0.0%

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

      11. sqrt-unprod 7.8%

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

      12. sqr-neg 7.8%

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

      13. sqrt-unprod 7.9%

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

      14. add-sqr-sqrt 7.9%

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

      15. *-commutative 7.9%

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

   9. Applied egg-rr 7.9%

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

   10. Taylor expanded in y around 0 57.5%

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

       1. mul-1-neg 57.5%

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

   12. Simplified 57.5%

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

4. Final simplification 79.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.4 \cdot 10^{+36}:\\ \;\;\;\;\frac{\frac{x \cdot y}{-x}}{x}\\ \mathbf{elif}\;y \leq 85:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x \cdot x}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 77.9% accurate, 20.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 85:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= y 85.0) (/ 1.0 x) (/ x (* x x))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 85.0) {
		tmp = 1.0 / x;
	} else {
		tmp = x / (x * x);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 85.0d0) then
        tmp = 1.0d0 / x
    else
        tmp = x / (x * x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 85.0) {
		tmp = 1.0 / x;
	} else {
		tmp = x / (x * x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 85.0:
		tmp = 1.0 / x
	else:
		tmp = x / (x * x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 85.0)
		tmp = Float64(1.0 / x);
	else
		tmp = Float64(x / Float64(x * x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 85.0)
		tmp = 1.0 / x;
	else
		tmp = x / (x * x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 85.0], N[(1.0 / x), $MachinePrecision], N[(x / N[(x * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 85

   1. Initial program 85.1%

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

      1. exp-prod 89.2%

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

   3. Simplified 89.2%

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

   5. Taylor expanded in x around 0 81.6%

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

   if 85 < y

   1. Initial program 45.6%

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

      1. exp-prod 67.7%

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

   3. Simplified 67.7%

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

   5. Taylor expanded in x around inf 2.1%

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

      1. +-commutative 2.1%

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

      2. mul-1-neg 2.1%

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

      3. unsub-neg 2.1%

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

   7. Simplified 2.1%

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

      1. sub-neg 2.1%

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

      2. distribute-neg-frac2 2.1%

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

      3. add-sqr-sqrt 2.1%

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

      4. sqrt-unprod 1.8%

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

      5. sqr-neg 1.8%

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

      6. sqrt-unprod 0.0%

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

      7. add-sqr-sqrt 4.9%

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

      8. frac-add 10.7%

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

      9. *-un-lft-identity 10.7%

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

      10. add-sqr-sqrt 0.0%

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

      11. sqrt-unprod 7.8%

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

      12. sqr-neg 7.8%

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

      13. sqrt-unprod 7.9%

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

      14. add-sqr-sqrt 7.9%

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

      15. *-commutative 7.9%

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

   9. Applied egg-rr 7.9%

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

   10. Taylor expanded in y around 0 57.5%

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

       1. mul-1-neg 57.5%

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

   12. Simplified 57.5%

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

4. Final simplification 76.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 85:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x \cdot x}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 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 76.5%

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

   1. exp-prod 84.5%

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

3. Simplified 84.5%

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

5. Taylor expanded in x around 0 73.8%

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

6. Final simplification 73.8%

   \[\leadsto \frac{1}{x} \]
7. Add Preprocessing

Developer target: 77.8% 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 2024039 
(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))