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

Percentage Accurate: 77.5% → 99.8%

Time: 8.8s

Alternatives: 5

Speedup: 14.9×

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: 77.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.2 \cdot 10^{+42} \lor \neg \left(x \leq 0.4\right):\\ \;\;\;\;\frac{e^{-y}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(e^{x}\right)}^{\log \left(\frac{x}{x + y}\right)}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -1.2e+42) (not (<= x 0.4)))
   (/ (exp (- y)) x)
   (/ (pow (exp x) (log (/ x (+ x y)))) x)))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -1.2e+42) || !(x <= 0.4)) {
		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 <= (-1.2d+42)) .or. (.not. (x <= 0.4d0))) 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 <= -1.2e+42) || !(x <= 0.4)) {
		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 <= -1.2e+42) or not (x <= 0.4):
		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 <= -1.2e+42) || !(x <= 0.4))
		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 <= -1.2e+42) || ~((x <= 0.4)))
		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, -1.2e+42], N[Not[LessEqual[x, 0.4]], $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 < -1.1999999999999999e42 or 0.40000000000000002 < x

   1. Initial program 73.4%

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

      1. *-commutative 73.4%

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

      2. exp-to-pow 73.4%

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

   3. Simplified 73.4%

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

   5. Taylor expanded in x around inf 100.0%

      \[\leadsto \frac{\color{blue}{e^{-1 \cdot y}}}{x} \]
   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 -1.1999999999999999e42 < x < 0.40000000000000002

   1. Initial program 85.7%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.2 \cdot 10^{+42} \lor \neg \left(x \leq 0.4\right):\\ \;\;\;\;\frac{e^{-y}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(e^{x}\right)}^{\log \left(\frac{x}{x + y}\right)}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.0003 \lor \neg \left(x \leq 1.2\right):\\ \;\;\;\;\frac{e^{-y}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -0.0003) (not (<= x 1.2))) (/ (exp (- y)) x) (/ 1.0 x)))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -0.0003) || !(x <= 1.2)) {
		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 <= (-0.0003d0)) .or. (.not. (x <= 1.2d0))) 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 <= -0.0003) || !(x <= 1.2)) {
		tmp = Math.exp(-y) / x;
	} else {
		tmp = 1.0 / x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -0.0003) or not (x <= 1.2):
		tmp = math.exp(-y) / x
	else:
		tmp = 1.0 / x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -0.0003) || !(x <= 1.2))
		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 <= -0.0003) || ~((x <= 1.2)))
		tmp = exp(-y) / x;
	else
		tmp = 1.0 / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -0.0003], N[Not[LessEqual[x, 1.2]], $MachinePrecision]], N[(N[Exp[(-y)], $MachinePrecision] / x), $MachinePrecision], N[(1.0 / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2.99999999999999974e-4 or 1.19999999999999996 < x

   1. Initial program 75.4%

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

      1. *-commutative 75.4%

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

      2. exp-to-pow 75.4%

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

   3. Simplified 75.4%

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

   5. Taylor expanded in x around inf 99.8%

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

      1. mul-1-neg 99.8%

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

   7. Simplified 99.8%

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

   if -2.99999999999999974e-4 < x < 1.19999999999999996

   1. Initial program 84.3%

      \[\frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x} \]
   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 98.3%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.0003 \lor \neg \left(x \leq 1.2\right):\\ \;\;\;\;\frac{e^{-y}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 82.9% accurate, 11.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.0003:\\ \;\;\;\;\frac{\frac{x - x \cdot y}{x}}{x}\\ \mathbf{elif}\;x \leq 2.8 \cdot 10^{+32}:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x} \cdot \frac{y}{x \cdot y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -0.0003)
   (/ (/ (- x (* x y)) x) x)
   (if (<= x 2.8e+32) (/ 1.0 x) (* (/ x x) (/ y (* x y))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -0.0003) {
		tmp = ((x - (x * y)) / x) / x;
	} else if (x <= 2.8e+32) {
		tmp = 1.0 / x;
	} else {
		tmp = (x / x) * (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 (x <= (-0.0003d0)) then
        tmp = ((x - (x * y)) / x) / x
    else if (x <= 2.8d+32) then
        tmp = 1.0d0 / x
    else
        tmp = (x / x) * (y / (x * y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -0.0003) {
		tmp = ((x - (x * y)) / x) / x;
	} else if (x <= 2.8e+32) {
		tmp = 1.0 / x;
	} else {
		tmp = (x / x) * (y / (x * y));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -0.0003:
		tmp = ((x - (x * y)) / x) / x
	elif x <= 2.8e+32:
		tmp = 1.0 / x
	else:
		tmp = (x / x) * (y / (x * y))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -0.0003)
		tmp = Float64(Float64(Float64(x - Float64(x * y)) / x) / x);
	elseif (x <= 2.8e+32)
		tmp = Float64(1.0 / x);
	else
		tmp = Float64(Float64(x / x) * Float64(y / Float64(x * y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -0.0003)
		tmp = ((x - (x * y)) / x) / x;
	elseif (x <= 2.8e+32)
		tmp = 1.0 / x;
	else
		tmp = (x / x) * (y / (x * y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -0.0003], N[(N[(N[(x - N[(x * y), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[x, 2.8e+32], N[(1.0 / x), $MachinePrecision], N[(N[(x / x), $MachinePrecision] * N[(y / N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.99999999999999974e-4

   1. Initial program 79.2%

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

   5. Taylor expanded in x around inf 61.1%

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

      1. +-commutative 61.1%

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

      2. mul-1-neg 61.1%

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

      3. unsub-neg 61.1%

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

   7. Simplified 61.1%

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

      1. frac-sub 38.8%

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

      2. associate-/r* 76.9%

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

      3. *-un-lft-identity 76.9%

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

      4. *-commutative 76.9%

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

   9. Applied egg-rr 76.9%

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

   if -2.99999999999999974e-4 < x < 2.8e32

   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 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 97.5%

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

   if 2.8e32 < x

   1. Initial program 68.2%

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

      1. exp-prod 68.2%

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

   3. Simplified 68.2%

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

   5. Taylor expanded in x around inf 47.8%

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

      1. +-commutative 47.8%

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

      2. mul-1-neg 47.8%

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

      3. unsub-neg 47.8%

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

   7. Simplified 47.8%

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

      1. clear-num 47.8%

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

      2. frac-sub 14.5%

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

      3. *-un-lft-identity 14.5%

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

      4. *-commutative 14.5%

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

      5. *-un-lft-identity 14.5%

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

   9. Applied egg-rr 14.5%

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

   10. Taylor expanded in y around 0 14.9%

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

       1. div-inv 14.8%

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

       2. times-frac 18.7%

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

       3. *-un-lft-identity 18.7%

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

       4. associate-*l/ 18.7%

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

       5. clear-num 19.7%

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

       6. frac-times 64.4%

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

       7. *-commutative 64.4%

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

       8. *-un-lft-identity 64.4%

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

   12. Applied egg-rr 64.4%

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

4. Final simplification 83.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.0003:\\ \;\;\;\;\frac{\frac{x - x \cdot y}{x}}{x}\\ \mathbf{elif}\;x \leq 2.8 \cdot 10^{+32}:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x} \cdot \frac{y}{x \cdot y}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 80.9% accurate, 14.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2.65 \cdot 10^{-11}:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x} \cdot \frac{y}{x \cdot y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 2.65e-11) (/ 1.0 x) (* (/ x x) (/ y (* x y)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 2.65e-11) {
		tmp = 1.0 / x;
	} else {
		tmp = (x / x) * (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 <= 2.65d-11) then
        tmp = 1.0d0 / x
    else
        tmp = (x / x) * (y / (x * y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 2.65e-11) {
		tmp = 1.0 / x;
	} else {
		tmp = (x / x) * (y / (x * y));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 2.65e-11:
		tmp = 1.0 / x
	else:
		tmp = (x / x) * (y / (x * y))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 2.65e-11)
		tmp = Float64(1.0 / x);
	else
		tmp = Float64(Float64(x / x) * Float64(y / Float64(x * y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 2.65e-11)
		tmp = 1.0 / x;
	else
		tmp = (x / x) * (y / (x * y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 2.65e-11], N[(1.0 / x), $MachinePrecision], N[(N[(x / x), $MachinePrecision] * N[(y / N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2.6499999999999999e-11

   1. Initial program 85.5%

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

      1. exp-prod 90.5%

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

   3. Simplified 90.5%

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

   5. Taylor expanded in x around 0 83.7%

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

   if 2.6499999999999999e-11 < y

   1. Initial program 56.4%

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

      1. exp-prod 67.9%

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

   3. Simplified 67.9%

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

   5. Taylor expanded in x around inf 5.7%

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

      1. +-commutative 5.7%

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

      2. mul-1-neg 5.7%

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

      3. unsub-neg 5.7%

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

   7. Simplified 5.7%

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

      1. clear-num 5.7%

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

      2. frac-sub 20.8%

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

      3. *-un-lft-identity 20.8%

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

      4. *-commutative 20.8%

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

      5. *-un-lft-identity 20.8%

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

   9. Applied egg-rr 20.8%

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

   10. Taylor expanded in y around 0 29.4%

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

       1. div-inv 29.4%

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

       2. times-frac 25.6%

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

       3. *-un-lft-identity 25.6%

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

       4. associate-*l/ 23.8%

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

       5. clear-num 23.8%

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

       6. frac-times 62.0%

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

       7. *-commutative 62.0%

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

       8. *-un-lft-identity 62.0%

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

   12. Applied egg-rr 62.0%

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

4. Final simplification 79.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.65 \cdot 10^{-11}:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x} \cdot \frac{y}{x \cdot y}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 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 79.1%

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

   1. exp-prod 85.6%

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

3. Simplified 85.6%

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

5. Taylor expanded in x around 0 73.9%

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

6. Final simplification 73.9%

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

Developer target: 78.1% 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 2024031 
(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))