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

Percentage Accurate: 78.0% → 99.6%

Time: 12.6s

Alternatives: 9

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: 78.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{+65} \lor \neg \left(x \leq 10\right):\\ \;\;\;\;\frac{e^{-y}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(e^{x}\right)}^{\log \left(\frac{x}{x + y}\right)}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -5e+65) (not (<= x 10.0)))
   (/ (exp (- y)) x)
   (/ (pow (exp x) (log (/ x (+ x y)))) x)))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -5e+65) || !(x <= 10.0)) {
		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 <= (-5d+65)) .or. (.not. (x <= 10.0d0))) 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 <= -5e+65) || !(x <= 10.0)) {
		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 <= -5e+65) or not (x <= 10.0):
		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 <= -5e+65) || !(x <= 10.0))
		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 <= -5e+65) || ~((x <= 10.0)))
		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, -5e+65], N[Not[LessEqual[x, 10.0]], $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 < -4.99999999999999973e65 or 10 < x

   1. Initial program 70.1%

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

   3. Taylor expanded in x around inf 100.0%

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

      1. mul-1-neg 100.0%

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

   5. Simplified 100.0%

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

   if -4.99999999999999973e65 < x < 10

   1. Initial program 87.8%

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

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{+65} \lor \neg \left(x \leq 10\right):\\ \;\;\;\;\frac{e^{-y}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(e^{x}\right)}^{\log \left(\frac{x}{x + y}\right)}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.86 \lor \neg \left(x \leq 0.32\right):\\ \;\;\;\;\frac{e^{-y}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -0.86) (not (<= x 0.32))) (/ (exp (- y)) x) (/ 1.0 x)))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -0.86) || !(x <= 0.32)) {
		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 <= (-0.86d0)) .or. (.not. (x <= 0.32d0))) 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 <= -0.86) || !(x <= 0.32)) {
		tmp = Math.exp(-y) / x;
	} else {
		tmp = 1.0 / x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -0.86) or not (x <= 0.32):
		tmp = math.exp(-y) / x
	else:
		tmp = 1.0 / x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -0.86) || !(x <= 0.32))
		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 <= -0.86) || ~((x <= 0.32)))
		tmp = exp(-y) / x;
	else
		tmp = 1.0 / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -0.86], N[Not[LessEqual[x, 0.32]], $MachinePrecision]], N[(N[Exp[(-y)], $MachinePrecision] / x), $MachinePrecision], N[(1.0 / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -0.859999999999999987 or 0.320000000000000007 < x

   1. Initial program 72.6%

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

   3. Taylor expanded in x around inf 100.0%

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

      1. mul-1-neg 100.0%

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

   5. Simplified 100.0%

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

   if -0.859999999999999987 < x < 0.320000000000000007

   1. Initial program 86.5%

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

   3. Taylor expanded in x around 0 98.2%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.86 \lor \neg \left(x \leq 0.32\right):\\ \;\;\;\;\frac{e^{-y}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 87.2% accurate, 6.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.75:\\ \;\;\;\;\frac{1}{x} + \frac{y \cdot \left(y \cdot \left(0.5 + y \cdot -0.16666666666666666\right) + -1\right)}{x}\\ \mathbf{elif}\;x \leq 0.032:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x + y \cdot \left(x + y \cdot \left(x - x \cdot \left(0.5 + \frac{1}{x} \cdot 0.5\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -0.75)
   (+ (/ 1.0 x) (/ (* y (+ (* y (+ 0.5 (* y -0.16666666666666666))) -1.0)) x))
   (if (<= x 0.032)
     (/ 1.0 x)
     (/ 1.0 (+ x (* y (+ x (* y (- x (* x (+ 0.5 (* (/ 1.0 x) 0.5))))))))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -0.75) {
		tmp = (1.0 / x) + ((y * ((y * (0.5 + (y * -0.16666666666666666))) + -1.0)) / x);
	} else if (x <= 0.032) {
		tmp = 1.0 / x;
	} else {
		tmp = 1.0 / (x + (y * (x + (y * (x - (x * (0.5 + ((1.0 / x) * 0.5))))))));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-0.75d0)) then
        tmp = (1.0d0 / x) + ((y * ((y * (0.5d0 + (y * (-0.16666666666666666d0)))) + (-1.0d0))) / x)
    else if (x <= 0.032d0) then
        tmp = 1.0d0 / x
    else
        tmp = 1.0d0 / (x + (y * (x + (y * (x - (x * (0.5d0 + ((1.0d0 / x) * 0.5d0))))))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -0.75) {
		tmp = (1.0 / x) + ((y * ((y * (0.5 + (y * -0.16666666666666666))) + -1.0)) / x);
	} else if (x <= 0.032) {
		tmp = 1.0 / x;
	} else {
		tmp = 1.0 / (x + (y * (x + (y * (x - (x * (0.5 + ((1.0 / x) * 0.5))))))));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -0.75:
		tmp = (1.0 / x) + ((y * ((y * (0.5 + (y * -0.16666666666666666))) + -1.0)) / x)
	elif x <= 0.032:
		tmp = 1.0 / x
	else:
		tmp = 1.0 / (x + (y * (x + (y * (x - (x * (0.5 + ((1.0 / x) * 0.5))))))))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -0.75)
		tmp = Float64(Float64(1.0 / x) + Float64(Float64(y * Float64(Float64(y * Float64(0.5 + Float64(y * -0.16666666666666666))) + -1.0)) / x));
	elseif (x <= 0.032)
		tmp = Float64(1.0 / x);
	else
		tmp = Float64(1.0 / Float64(x + Float64(y * Float64(x + Float64(y * Float64(x - Float64(x * Float64(0.5 + Float64(Float64(1.0 / x) * 0.5)))))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -0.75)
		tmp = (1.0 / x) + ((y * ((y * (0.5 + (y * -0.16666666666666666))) + -1.0)) / x);
	elseif (x <= 0.032)
		tmp = 1.0 / x;
	else
		tmp = 1.0 / (x + (y * (x + (y * (x - (x * (0.5 + ((1.0 / x) * 0.5))))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -0.75], N[(N[(1.0 / x), $MachinePrecision] + N[(N[(y * N[(N[(y * N[(0.5 + N[(y * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.032], N[(1.0 / x), $MachinePrecision], N[(1.0 / N[(x + N[(y * N[(x + N[(y * N[(x - N[(x * N[(0.5 + N[(N[(1.0 / x), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.75

   1. Initial program 71.7%

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

   3. Taylor expanded in x around inf 100.0%

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

      1. mul-1-neg 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in y around 0 71.0%

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

   7. Taylor expanded in x around 0 76.2%

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

   if -0.75 < x < 0.032000000000000001

   1. Initial program 86.5%

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

   3. Taylor expanded in x around 0 98.2%

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

   if 0.032000000000000001 < x

   1. Initial program 73.5%

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

      1. pow-exp 73.5%

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

      2. clear-num 73.5%

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

      3. inv-pow 73.5%

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

      4. pow-exp 73.5%

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

      5. *-commutative 73.5%

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

      6. pow-to-exp 73.5%

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

   4. Applied egg-rr 73.5%

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

      1. unpow-1 73.5%

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

   6. Simplified 73.5%

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

   7. Taylor expanded in y around 0 79.1%

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

4. Final simplification 86.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.75:\\ \;\;\;\;\frac{1}{x} + \frac{y \cdot \left(y \cdot \left(0.5 + y \cdot -0.16666666666666666\right) + -1\right)}{x}\\ \mathbf{elif}\;x \leq 0.032:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x + y \cdot \left(x + y \cdot \left(x - x \cdot \left(0.5 + \frac{1}{x} \cdot 0.5\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 85.2% accurate, 9.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.47:\\ \;\;\;\;\frac{1}{x} + \frac{y \cdot \left(y \cdot \left(0.5 + y \cdot -0.16666666666666666\right) + -1\right)}{x}\\ \mathbf{elif}\;x \leq 0.46:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x + x \cdot y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -0.47)
   (+ (/ 1.0 x) (/ (* y (+ (* y (+ 0.5 (* y -0.16666666666666666))) -1.0)) x))
   (if (<= x 0.46) (/ 1.0 x) (/ 1.0 (+ x (* x y))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -0.47) {
		tmp = (1.0 / x) + ((y * ((y * (0.5 + (y * -0.16666666666666666))) + -1.0)) / x);
	} else if (x <= 0.46) {
		tmp = 1.0 / x;
	} else {
		tmp = 1.0 / (x + (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 <= (-0.47d0)) then
        tmp = (1.0d0 / x) + ((y * ((y * (0.5d0 + (y * (-0.16666666666666666d0)))) + (-1.0d0))) / x)
    else if (x <= 0.46d0) then
        tmp = 1.0d0 / x
    else
        tmp = 1.0d0 / (x + (x * y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -0.47) {
		tmp = (1.0 / x) + ((y * ((y * (0.5 + (y * -0.16666666666666666))) + -1.0)) / x);
	} else if (x <= 0.46) {
		tmp = 1.0 / x;
	} else {
		tmp = 1.0 / (x + (x * y));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -0.47:
		tmp = (1.0 / x) + ((y * ((y * (0.5 + (y * -0.16666666666666666))) + -1.0)) / x)
	elif x <= 0.46:
		tmp = 1.0 / x
	else:
		tmp = 1.0 / (x + (x * y))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -0.47)
		tmp = Float64(Float64(1.0 / x) + Float64(Float64(y * Float64(Float64(y * Float64(0.5 + Float64(y * -0.16666666666666666))) + -1.0)) / x));
	elseif (x <= 0.46)
		tmp = Float64(1.0 / x);
	else
		tmp = Float64(1.0 / Float64(x + Float64(x * y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -0.47)
		tmp = (1.0 / x) + ((y * ((y * (0.5 + (y * -0.16666666666666666))) + -1.0)) / x);
	elseif (x <= 0.46)
		tmp = 1.0 / x;
	else
		tmp = 1.0 / (x + (x * y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -0.47], N[(N[(1.0 / x), $MachinePrecision] + N[(N[(y * N[(N[(y * N[(0.5 + N[(y * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.46], N[(1.0 / x), $MachinePrecision], N[(1.0 / N[(x + N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.46999999999999997

   1. Initial program 71.7%

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

   3. Taylor expanded in x around inf 100.0%

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

      1. mul-1-neg 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in y around 0 71.0%

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

   7. Taylor expanded in x around 0 76.2%

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

   if -0.46999999999999997 < x < 0.46000000000000002

   1. Initial program 86.5%

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

   3. Taylor expanded in x around 0 98.2%

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

   if 0.46000000000000002 < x

   1. Initial program 73.5%

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

      1. pow-exp 73.5%

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

      2. clear-num 73.5%

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

      3. inv-pow 73.5%

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

      4. pow-exp 73.5%

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

      5. *-commutative 73.5%

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

      6. pow-to-exp 73.5%

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

   4. Applied egg-rr 73.5%

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

      1. unpow-1 73.5%

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

   6. Simplified 73.5%

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

   7. Taylor expanded in y around 0 73.0%

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

4. Final simplification 85.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.47:\\ \;\;\;\;\frac{1}{x} + \frac{y \cdot \left(y \cdot \left(0.5 + y \cdot -0.16666666666666666\right) + -1\right)}{x}\\ \mathbf{elif}\;x \leq 0.46:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x + x \cdot y}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 84.8% accurate, 10.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.215:\\ \;\;\;\;\frac{1 + y \cdot \left(-1 + \frac{0.5 \cdot \left(x \cdot y\right)}{x}\right)}{x}\\ \mathbf{elif}\;x \leq 0.08:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x + x \cdot y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -0.215)
   (/ (+ 1.0 (* y (+ -1.0 (/ (* 0.5 (* x y)) x)))) x)
   (if (<= x 0.08) (/ 1.0 x) (/ 1.0 (+ x (* x y))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -0.215) {
		tmp = (1.0 + (y * (-1.0 + ((0.5 * (x * y)) / x)))) / x;
	} else if (x <= 0.08) {
		tmp = 1.0 / x;
	} else {
		tmp = 1.0 / (x + (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 <= (-0.215d0)) then
        tmp = (1.0d0 + (y * ((-1.0d0) + ((0.5d0 * (x * y)) / x)))) / x
    else if (x <= 0.08d0) then
        tmp = 1.0d0 / x
    else
        tmp = 1.0d0 / (x + (x * y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -0.215) {
		tmp = (1.0 + (y * (-1.0 + ((0.5 * (x * y)) / x)))) / x;
	} else if (x <= 0.08) {
		tmp = 1.0 / x;
	} else {
		tmp = 1.0 / (x + (x * y));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -0.215:
		tmp = (1.0 + (y * (-1.0 + ((0.5 * (x * y)) / x)))) / x
	elif x <= 0.08:
		tmp = 1.0 / x
	else:
		tmp = 1.0 / (x + (x * y))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -0.215)
		tmp = Float64(Float64(1.0 + Float64(y * Float64(-1.0 + Float64(Float64(0.5 * Float64(x * y)) / x)))) / x);
	elseif (x <= 0.08)
		tmp = Float64(1.0 / x);
	else
		tmp = Float64(1.0 / Float64(x + Float64(x * y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -0.215)
		tmp = (1.0 + (y * (-1.0 + ((0.5 * (x * y)) / x)))) / x;
	elseif (x <= 0.08)
		tmp = 1.0 / x;
	else
		tmp = 1.0 / (x + (x * y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -0.215], N[(N[(1.0 + N[(y * N[(-1.0 + N[(N[(0.5 * N[(x * y), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[x, 0.08], N[(1.0 / x), $MachinePrecision], N[(1.0 / N[(x + N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.214999999999999997

   1. Initial program 71.7%

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

      1. exp-prod 71.7%

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

   3. Simplified 71.7%

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

   5. Taylor expanded in y around 0 69.6%

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

   6. Taylor expanded in x around 0 76.2%

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

      1. distribute-lft-out 76.2%

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

   8. Simplified 76.2%

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

   9. Taylor expanded in x around inf 76.2%

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

   if -0.214999999999999997 < x < 0.0800000000000000017

   1. Initial program 86.5%

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

   3. Taylor expanded in x around 0 98.2%

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

   if 0.0800000000000000017 < x

   1. Initial program 73.5%

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

      1. pow-exp 73.5%

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

      2. clear-num 73.5%

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

      3. inv-pow 73.5%

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

      4. pow-exp 73.5%

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

      5. *-commutative 73.5%

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

      6. pow-to-exp 73.5%

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

   4. Applied egg-rr 73.5%

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

      1. unpow-1 73.5%

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

   6. Simplified 73.5%

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

   7. Taylor expanded in y around 0 73.0%

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

4. Final simplification 85.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.215:\\ \;\;\;\;\frac{1 + y \cdot \left(-1 + \frac{0.5 \cdot \left(x \cdot y\right)}{x}\right)}{x}\\ \mathbf{elif}\;x \leq 0.08:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x + x \cdot y}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 81.5% accurate, 12.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.8 \cdot 10^{+79} \lor \neg \left(x \leq 0.108\right):\\ \;\;\;\;\frac{1}{x + x \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -3.8e+79) (not (<= x 0.108))) (/ 1.0 (+ x (* x y))) (/ 1.0 x)))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -3.8e+79) || !(x <= 0.108)) {
		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 <= (-3.8d+79)) .or. (.not. (x <= 0.108d0))) 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 <= -3.8e+79) || !(x <= 0.108)) {
		tmp = 1.0 / (x + (x * y));
	} else {
		tmp = 1.0 / x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -3.8e+79) or not (x <= 0.108):
		tmp = 1.0 / (x + (x * y))
	else:
		tmp = 1.0 / x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -3.8e+79) || !(x <= 0.108))
		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 <= -3.8e+79) || ~((x <= 0.108)))
		tmp = 1.0 / (x + (x * y));
	else
		tmp = 1.0 / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -3.8e+79], N[Not[LessEqual[x, 0.108]], $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 < -3.8000000000000002e79 or 0.107999999999999999 < x

   1. Initial program 69.9%

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

      1. pow-exp 69.9%

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

      2. clear-num 69.9%

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

      3. inv-pow 69.9%

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

      4. pow-exp 69.9%

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

      5. *-commutative 69.9%

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

      6. pow-to-exp 69.9%

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

   4. Applied egg-rr 69.9%

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

      1. unpow-1 69.9%

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

   6. Simplified 69.9%

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

   7. Taylor expanded in y around 0 69.1%

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

   if -3.8000000000000002e79 < x < 0.107999999999999999

   1. Initial program 87.4%

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

   3. Taylor expanded in x around 0 93.2%

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

4. Final simplification 81.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.8 \cdot 10^{+79} \lor \neg \left(x \leq 0.108\right):\\ \;\;\;\;\frac{1}{x + x \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 81.5% accurate, 12.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.95:\\ \;\;\;\;\frac{1}{x} + y \cdot \left(0.5 \cdot \frac{y}{x}\right)\\ \mathbf{elif}\;x \leq 0.3:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x + x \cdot y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -0.95)
   (+ (/ 1.0 x) (* y (* 0.5 (/ y x))))
   (if (<= x 0.3) (/ 1.0 x) (/ 1.0 (+ x (* x y))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -0.95) {
		tmp = (1.0 / x) + (y * (0.5 * (y / x)));
	} else if (x <= 0.3) {
		tmp = 1.0 / x;
	} else {
		tmp = 1.0 / (x + (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 <= (-0.95d0)) then
        tmp = (1.0d0 / x) + (y * (0.5d0 * (y / x)))
    else if (x <= 0.3d0) then
        tmp = 1.0d0 / x
    else
        tmp = 1.0d0 / (x + (x * y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -0.95) {
		tmp = (1.0 / x) + (y * (0.5 * (y / x)));
	} else if (x <= 0.3) {
		tmp = 1.0 / x;
	} else {
		tmp = 1.0 / (x + (x * y));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -0.95:
		tmp = (1.0 / x) + (y * (0.5 * (y / x)))
	elif x <= 0.3:
		tmp = 1.0 / x
	else:
		tmp = 1.0 / (x + (x * y))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -0.95)
		tmp = Float64(Float64(1.0 / x) + Float64(y * Float64(0.5 * Float64(y / x))));
	elseif (x <= 0.3)
		tmp = Float64(1.0 / x);
	else
		tmp = Float64(1.0 / Float64(x + Float64(x * y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -0.95)
		tmp = (1.0 / x) + (y * (0.5 * (y / x)));
	elseif (x <= 0.3)
		tmp = 1.0 / x;
	else
		tmp = 1.0 / (x + (x * y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -0.95], N[(N[(1.0 / x), $MachinePrecision] + N[(y * N[(0.5 * N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.3], N[(1.0 / x), $MachinePrecision], N[(1.0 / N[(x + N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.94999999999999996

   1. Initial program 71.7%

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

   3. Taylor expanded in x around inf 100.0%

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

      1. mul-1-neg 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in y around 0 65.9%

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

   7. Taylor expanded in y around inf 65.2%

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

   if -0.94999999999999996 < x < 0.299999999999999989

   1. Initial program 86.5%

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

   3. Taylor expanded in x around 0 98.2%

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

   if 0.299999999999999989 < x

   1. Initial program 73.5%

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

      1. pow-exp 73.5%

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

      2. clear-num 73.5%

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

      3. inv-pow 73.5%

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

      4. pow-exp 73.5%

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

      5. *-commutative 73.5%

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

      6. pow-to-exp 73.5%

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

   4. Applied egg-rr 73.5%

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

      1. unpow-1 73.5%

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

   6. Simplified 73.5%

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

   7. Taylor expanded in y around 0 73.0%

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

4. Final simplification 81.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.95:\\ \;\;\;\;\frac{1}{x} + y \cdot \left(0.5 \cdot \frac{y}{x}\right)\\ \mathbf{elif}\;x \leq 0.3:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x + x \cdot y}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 83.9% accurate, 12.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.44:\\ \;\;\;\;\frac{1 + y \cdot \left(-1 + y \cdot 0.5\right)}{x}\\ \mathbf{elif}\;x \leq 0.4:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x + x \cdot y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -0.44)
   (/ (+ 1.0 (* y (+ -1.0 (* y 0.5)))) x)
   (if (<= x 0.4) (/ 1.0 x) (/ 1.0 (+ x (* x y))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -0.44) {
		tmp = (1.0 + (y * (-1.0 + (y * 0.5)))) / x;
	} else if (x <= 0.4) {
		tmp = 1.0 / x;
	} else {
		tmp = 1.0 / (x + (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 <= (-0.44d0)) then
        tmp = (1.0d0 + (y * ((-1.0d0) + (y * 0.5d0)))) / x
    else if (x <= 0.4d0) then
        tmp = 1.0d0 / x
    else
        tmp = 1.0d0 / (x + (x * y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -0.44) {
		tmp = (1.0 + (y * (-1.0 + (y * 0.5)))) / x;
	} else if (x <= 0.4) {
		tmp = 1.0 / x;
	} else {
		tmp = 1.0 / (x + (x * y));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -0.44:
		tmp = (1.0 + (y * (-1.0 + (y * 0.5)))) / x
	elif x <= 0.4:
		tmp = 1.0 / x
	else:
		tmp = 1.0 / (x + (x * y))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -0.44)
		tmp = Float64(Float64(1.0 + Float64(y * Float64(-1.0 + Float64(y * 0.5)))) / x);
	elseif (x <= 0.4)
		tmp = Float64(1.0 / x);
	else
		tmp = Float64(1.0 / Float64(x + Float64(x * y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -0.44)
		tmp = (1.0 + (y * (-1.0 + (y * 0.5)))) / x;
	elseif (x <= 0.4)
		tmp = 1.0 / x;
	else
		tmp = 1.0 / (x + (x * y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -0.44], N[(N[(1.0 + N[(y * N[(-1.0 + N[(y * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[x, 0.4], N[(1.0 / x), $MachinePrecision], N[(1.0 / N[(x + N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.440000000000000002

   1. Initial program 71.7%

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

      1. exp-prod 71.7%

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

   3. Simplified 71.7%

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

   5. Taylor expanded in y around 0 69.6%

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

   6. Taylor expanded in x around inf 69.6%

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

      1. *-commutative 69.6%

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

   8. Simplified 69.6%

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

   if -0.440000000000000002 < x < 0.40000000000000002

   1. Initial program 86.5%

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

   3. Taylor expanded in x around 0 98.2%

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

   if 0.40000000000000002 < x

   1. Initial program 73.5%

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

      1. pow-exp 73.5%

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

      2. clear-num 73.5%

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

      3. inv-pow 73.5%

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

      4. pow-exp 73.5%

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

      5. *-commutative 73.5%

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

      6. pow-to-exp 73.5%

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

   4. Applied egg-rr 73.5%

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

      1. unpow-1 73.5%

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

   6. Simplified 73.5%

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

   7. Taylor expanded in y around 0 73.0%

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

4. Final simplification 83.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.44:\\ \;\;\;\;\frac{1 + y \cdot \left(-1 + y \cdot 0.5\right)}{x}\\ \mathbf{elif}\;x \leq 0.4:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x + x \cdot y}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 75.6% 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 78.7%

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

3. Taylor expanded in x around 0 75.2%

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

4. Final simplification 75.2%

   \[\leadsto \frac{1}{x} \]
5. 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 2024130 
(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))