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

Percentage Accurate: 78.3% → 99.1%

Time: 12.8s

Alternatives: 7

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: 78.3% 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: 99.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.05 \cdot 10^{+108} \lor \neg \left(x \leq 10\right):\\ \;\;\;\;\frac{1}{x \cdot e^{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(e^{x}\right)}^{\log \left(\frac{x}{x + y}\right)}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -2.05e+108) (not (<= x 10.0)))
   (/ 1.0 (* x (exp y)))
   (/ (pow (exp x) (log (/ x (+ x y)))) x)))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -2.05e+108) || !(x <= 10.0)) {
		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 <= (-2.05d+108)) .or. (.not. (x <= 10.0d0))) 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 <= -2.05e+108) || !(x <= 10.0)) {
		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 <= -2.05e+108) or not (x <= 10.0):
		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 <= -2.05e+108) || !(x <= 10.0))
		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 <= -2.05e+108) || ~((x <= 10.0)))
		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, -2.05e+108], N[Not[LessEqual[x, 10.0]], $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 < -2.05e108 or 10 < x

   1. Initial program 68.8%

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

      1. *-commutative 68.8%

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

      2. exp-to-pow 68.8%

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

   3. Simplified 68.8%

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

      \[\leadsto \frac{\color{blue}{e^{-y}}}{x} \]
   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. Taylor expanded in x around 0 100.0%

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

   if -2.05e108 < x < 10

   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. Add Preprocessing
3. Recombined 2 regimes into one program.

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.05 \cdot 10^{+108} \lor \neg \left(x \leq 10\right):\\ \;\;\;\;\frac{1}{x \cdot e^{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(e^{x}\right)}^{\log \left(\frac{x}{x + y}\right)}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 98.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.95 \lor \neg \left(x \leq 4.4 \cdot 10^{-38}\right):\\ \;\;\;\;\frac{1}{x \cdot e^{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -0.95) (not (<= x 4.4e-38))) (/ 1.0 (* x (exp y))) (/ 1.0 x)))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -0.95) || !(x <= 4.4e-38)) {
		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 <= (-0.95d0)) .or. (.not. (x <= 4.4d-38))) 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 <= -0.95) || !(x <= 4.4e-38)) {
		tmp = 1.0 / (x * Math.exp(y));
	} else {
		tmp = 1.0 / x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -0.95) or not (x <= 4.4e-38):
		tmp = 1.0 / (x * math.exp(y))
	else:
		tmp = 1.0 / x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -0.95) || !(x <= 4.4e-38))
		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 <= -0.95) || ~((x <= 4.4e-38)))
		tmp = 1.0 / (x * exp(y));
	else
		tmp = 1.0 / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -0.95], N[Not[LessEqual[x, 4.4e-38]], $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 < -0.94999999999999996 or 4.40000000000000015e-38 < x

   1. Initial program 73.7%

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

      1. *-commutative 73.7%

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

      2. exp-to-pow 73.7%

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

   3. Simplified 73.7%

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

   5. Taylor expanded in x around inf 99.3%

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

      1. mul-1-neg 99.3%

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

   7. Simplified 99.3%

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

      1. clear-num 99.3%

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

      2. inv-pow 99.3%

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

      3. exp-neg 99.4%

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

      4. associate-/r/ 99.4%

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

      5. /-rgt-identity 99.4%

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

   9. Applied egg-rr 99.4%

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

   10. Taylor expanded in x around 0 99.4%

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

   if -0.94999999999999996 < x < 4.40000000000000015e-38

   1. Initial program 76.2%

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

      1. exp-prod 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in x around 0 99.2%

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

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.95 \lor \neg \left(x \leq 4.4 \cdot 10^{-38}\right):\\ \;\;\;\;\frac{1}{x \cdot e^{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5 \lor \neg \left(x \leq 4.4 \cdot 10^{-38}\right):\\ \;\;\;\;\frac{e^{-y}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -5.0) (not (<= x 4.4e-38))) (/ (exp (- y)) x) (/ 1.0 x)))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -5.0) || !(x <= 4.4e-38)) {
		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 <= (-5.0d0)) .or. (.not. (x <= 4.4d-38))) 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 <= -5.0) || !(x <= 4.4e-38)) {
		tmp = Math.exp(-y) / x;
	} else {
		tmp = 1.0 / x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -5.0) or not (x <= 4.4e-38):
		tmp = math.exp(-y) / x
	else:
		tmp = 1.0 / x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -5.0) || !(x <= 4.4e-38))
		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 <= -5.0) || ~((x <= 4.4e-38)))
		tmp = exp(-y) / x;
	else
		tmp = 1.0 / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -5.0], N[Not[LessEqual[x, 4.4e-38]], $MachinePrecision]], N[(N[Exp[(-y)], $MachinePrecision] / x), $MachinePrecision], N[(1.0 / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -5 or 4.40000000000000015e-38 < x

   1. Initial program 73.7%

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

      1. *-commutative 73.7%

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

      2. exp-to-pow 73.7%

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

   3. Simplified 73.7%

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

   5. Taylor expanded in x around inf 99.3%

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

      1. mul-1-neg 99.3%

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

   7. Simplified 99.3%

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

   if -5 < x < 4.40000000000000015e-38

   1. Initial program 76.2%

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

      1. exp-prod 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in x around 0 99.2%

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

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5 \lor \neg \left(x \leq 4.4 \cdot 10^{-38}\right):\\ \;\;\;\;\frac{e^{-y}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 84.1% accurate, 7.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5.6 \cdot 10^{+76} \lor \neg \left(x \leq 4.4 \cdot 10^{-38}\right):\\ \;\;\;\;\frac{1}{x + y \cdot \left(x + y \cdot \left(0.16666666666666666 \cdot \left(x \cdot y\right) + x \cdot 0.5\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -5.6e+76) (not (<= x 4.4e-38)))
   (/
    1.0
    (+ x (* y (+ x (* y (+ (* 0.16666666666666666 (* x y)) (* x 0.5)))))))
   (/ 1.0 x)))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -5.6e+76) || !(x <= 4.4e-38)) {
		tmp = 1.0 / (x + (y * (x + (y * ((0.16666666666666666 * (x * y)) + (x * 0.5))))));
	} 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 <= (-5.6d+76)) .or. (.not. (x <= 4.4d-38))) then
        tmp = 1.0d0 / (x + (y * (x + (y * ((0.16666666666666666d0 * (x * y)) + (x * 0.5d0))))))
    else
        tmp = 1.0d0 / x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((x <= -5.6e+76) || !(x <= 4.4e-38)) {
		tmp = 1.0 / (x + (y * (x + (y * ((0.16666666666666666 * (x * y)) + (x * 0.5))))));
	} else {
		tmp = 1.0 / x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -5.6e+76) or not (x <= 4.4e-38):
		tmp = 1.0 / (x + (y * (x + (y * ((0.16666666666666666 * (x * y)) + (x * 0.5))))))
	else:
		tmp = 1.0 / x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -5.6e+76) || !(x <= 4.4e-38))
		tmp = Float64(1.0 / Float64(x + Float64(y * Float64(x + Float64(y * Float64(Float64(0.16666666666666666 * Float64(x * y)) + Float64(x * 0.5)))))));
	else
		tmp = Float64(1.0 / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((x <= -5.6e+76) || ~((x <= 4.4e-38)))
		tmp = 1.0 / (x + (y * (x + (y * ((0.16666666666666666 * (x * y)) + (x * 0.5))))));
	else
		tmp = 1.0 / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -5.6e+76], N[Not[LessEqual[x, 4.4e-38]], $MachinePrecision]], N[(1.0 / N[(x + N[(y * N[(x + N[(y * N[(N[(0.16666666666666666 * N[(x * y), $MachinePrecision]), $MachinePrecision] + N[(x * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -5.5999999999999997e76 or 4.40000000000000015e-38 < x

   1. Initial program 71.0%

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

      1. *-commutative 71.0%

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

      2. exp-to-pow 71.0%

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

   3. Simplified 71.0%

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

   5. Taylor expanded in x around inf 99.3%

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

      1. mul-1-neg 99.3%

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

   7. Simplified 99.3%

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

      1. clear-num 99.3%

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

      2. inv-pow 99.3%

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

      3. exp-neg 99.3%

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

      4. associate-/r/ 99.3%

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

      5. /-rgt-identity 99.3%

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

   9. Applied egg-rr 99.3%

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

   10. Taylor expanded in x around 0 99.3%

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

   11. Taylor expanded in y around 0 80.0%

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

   if -5.5999999999999997e76 < x < 4.40000000000000015e-38

   1. Initial program 79.0%

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

      1. exp-prod 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in x around 0 95.2%

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

4. Final simplification 87.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.6 \cdot 10^{+76} \lor \neg \left(x \leq 4.4 \cdot 10^{-38}\right):\\ \;\;\;\;\frac{1}{x + y \cdot \left(x + y \cdot \left(0.16666666666666666 \cdot \left(x \cdot y\right) + x \cdot 0.5\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 83.2% accurate, 9.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5.5 \cdot 10^{+76} \lor \neg \left(x \leq 4.4 \cdot 10^{-38}\right):\\ \;\;\;\;\frac{1}{x + y \cdot \left(x + y \cdot \left(x \cdot 0.5\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -5.5e+76) (not (<= x 4.4e-38)))
   (/ 1.0 (+ x (* y (+ x (* y (* x 0.5))))))
   (/ 1.0 x)))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -5.5e+76) || !(x <= 4.4e-38)) {
		tmp = 1.0 / (x + (y * (x + (y * (x * 0.5)))));
	} 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 <= (-5.5d+76)) .or. (.not. (x <= 4.4d-38))) then
        tmp = 1.0d0 / (x + (y * (x + (y * (x * 0.5d0)))))
    else
        tmp = 1.0d0 / x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((x <= -5.5e+76) || !(x <= 4.4e-38)) {
		tmp = 1.0 / (x + (y * (x + (y * (x * 0.5)))));
	} else {
		tmp = 1.0 / x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -5.5e+76) or not (x <= 4.4e-38):
		tmp = 1.0 / (x + (y * (x + (y * (x * 0.5)))))
	else:
		tmp = 1.0 / x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -5.5e+76) || !(x <= 4.4e-38))
		tmp = Float64(1.0 / Float64(x + Float64(y * Float64(x + Float64(y * Float64(x * 0.5))))));
	else
		tmp = Float64(1.0 / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((x <= -5.5e+76) || ~((x <= 4.4e-38)))
		tmp = 1.0 / (x + (y * (x + (y * (x * 0.5)))));
	else
		tmp = 1.0 / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -5.5e+76], N[Not[LessEqual[x, 4.4e-38]], $MachinePrecision]], N[(1.0 / N[(x + N[(y * N[(x + N[(y * N[(x * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -5.5000000000000001e76 or 4.40000000000000015e-38 < x

   1. Initial program 71.0%

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

      1. *-commutative 71.0%

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

      2. exp-to-pow 71.0%

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

   3. Simplified 71.0%

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

   5. Taylor expanded in x around inf 99.3%

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

      1. mul-1-neg 99.3%

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

   7. Simplified 99.3%

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

      1. clear-num 99.3%

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

      2. inv-pow 99.3%

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

      3. exp-neg 99.3%

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

      4. associate-/r/ 99.3%

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

      5. /-rgt-identity 99.3%

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

   9. Applied egg-rr 99.3%

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

   10. Taylor expanded in x around 0 99.3%

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

   11. Taylor expanded in y around 0 76.5%

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

       1. associate-*r* 76.5%

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

       2. *-commutative 76.5%

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

   13. Simplified 76.5%

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

   if -5.5000000000000001e76 < x < 4.40000000000000015e-38

   1. Initial program 79.0%

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

      1. exp-prod 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in x around 0 95.2%

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

4. Final simplification 85.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.5 \cdot 10^{+76} \lor \neg \left(x \leq 4.4 \cdot 10^{-38}\right):\\ \;\;\;\;\frac{1}{x + y \cdot \left(x + y \cdot \left(x \cdot 0.5\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 80.6% accurate, 12.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5.6 \cdot 10^{+112} \lor \neg \left(x \leq 4.4 \cdot 10^{-38}\right):\\ \;\;\;\;\frac{1}{x + x \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -5.6e+112) (not (<= x 4.4e-38)))
   (/ 1.0 (+ x (* x y)))
   (/ 1.0 x)))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -5.6e+112) || !(x <= 4.4e-38)) {
		tmp = 1.0 / (x + (x * y));
	} else {
		tmp = 1.0 / x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((x <= (-5.6d+112)) .or. (.not. (x <= 4.4d-38))) then
        tmp = 1.0d0 / (x + (x * y))
    else
        tmp = 1.0d0 / x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((x <= -5.6e+112) || !(x <= 4.4e-38)) {
		tmp = 1.0 / (x + (x * y));
	} else {
		tmp = 1.0 / x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -5.6e+112) or not (x <= 4.4e-38):
		tmp = 1.0 / (x + (x * y))
	else:
		tmp = 1.0 / x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -5.6e+112) || !(x <= 4.4e-38))
		tmp = Float64(1.0 / Float64(x + Float64(x * y)));
	else
		tmp = Float64(1.0 / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((x <= -5.6e+112) || ~((x <= 4.4e-38)))
		tmp = 1.0 / (x + (x * y));
	else
		tmp = 1.0 / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -5.6e+112], N[Not[LessEqual[x, 4.4e-38]], $MachinePrecision]], N[(1.0 / N[(x + N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -5.6000000000000003e112 or 4.40000000000000015e-38 < x

   1. Initial program 70.3%

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

      1. *-commutative 70.3%

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

      2. exp-to-pow 70.4%

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

   3. Simplified 70.4%

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

   5. Taylor expanded in x around inf 99.3%

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

      1. mul-1-neg 99.3%

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

   7. Simplified 99.3%

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

      1. clear-num 99.3%

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

      2. inv-pow 99.3%

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

      3. exp-neg 99.3%

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

      4. associate-/r/ 99.3%

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

      5. /-rgt-identity 99.3%

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

   9. Applied egg-rr 99.3%

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

   10. Taylor expanded in x around 0 99.3%

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

   11. Taylor expanded in y around 0 72.9%

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

   if -5.6000000000000003e112 < x < 4.40000000000000015e-38

   1. Initial program 79.5%

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

      1. exp-prod 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in x around 0 93.7%

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

4. Final simplification 82.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.6 \cdot 10^{+112} \lor \neg \left(x \leq 4.4 \cdot 10^{-38}\right):\\ \;\;\;\;\frac{1}{x + x \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 75.1% accurate, 69.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 74.7%

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

   1. exp-prod 84.4%

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

3. Simplified 84.4%

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

5. Taylor expanded in x around 0 75.1%

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

6. Final simplification 75.1%

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

Developer target: 78.2% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024095 
(FPCore (x y)
  :name "Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2, F"
  :precision binary64

  :alt
  (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))