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

Percentage Accurate: 78.0% → 98.1%

Time: 13.0s

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: 98.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.3 \cdot 10^{+114} \lor \neg \left(x \leq 2.3 \cdot 10^{-32}\right):\\ \;\;\;\;\frac{e^{-y}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(e^{x}\right)}^{\log \left(\frac{x}{x + y}\right)}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -1.3e+114) (not (<= x 2.3e-32)))
   (/ (exp (- y)) x)
   (/ (pow (exp x) (log (/ x (+ x y)))) x)))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -1.3e+114) || !(x <= 2.3e-32)) {
		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 <= (-1.3d+114)) .or. (.not. (x <= 2.3d-32))) 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 <= -1.3e+114) || !(x <= 2.3e-32)) {
		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 <= -1.3e+114) or not (x <= 2.3e-32):
		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 <= -1.3e+114) || !(x <= 2.3e-32))
		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 <= -1.3e+114) || ~((x <= 2.3e-32)))
		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, -1.3e+114], N[Not[LessEqual[x, 2.3e-32]], $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 < -1.3e114 or 2.3000000000000001e-32 < x

   1. Initial program 67.4%

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

      1. *-commutative 67.4%

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

      2. exp-to-pow 67.4%

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

   3. Simplified 67.4%

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

   5. Taylor expanded in x around inf 99.8%

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

      1. mul-1-neg 99.8%

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

   7. Simplified 99.8%

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

   if -1.3e114 < x < 2.3000000000000001e-32

   1. Initial program 85.4%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.3 \cdot 10^{+114} \lor \neg \left(x \leq 2.3 \cdot 10^{-32}\right):\\ \;\;\;\;\frac{e^{-y}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(e^{x}\right)}^{\log \left(\frac{x}{x + y}\right)}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 98.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.4 \lor \neg \left(x \leq 2.3 \cdot 10^{-32}\right):\\ \;\;\;\;\frac{e^{-y}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -0.4) (not (<= x 2.3e-32))) (/ (exp (- y)) x) (/ 1.0 x)))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -0.4) || !(x <= 2.3e-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 <= (-0.4d0)) .or. (.not. (x <= 2.3d-32))) 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 <= -0.4) || !(x <= 2.3e-32)) {
		tmp = Math.exp(-y) / x;
	} else {
		tmp = 1.0 / x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -0.4) or not (x <= 2.3e-32):
		tmp = math.exp(-y) / x
	else:
		tmp = 1.0 / x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -0.4) || !(x <= 2.3e-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 <= -0.4) || ~((x <= 2.3e-32)))
		tmp = exp(-y) / x;
	else
		tmp = 1.0 / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -0.4], N[Not[LessEqual[x, 2.3e-32]], $MachinePrecision]], N[(N[Exp[(-y)], $MachinePrecision] / x), $MachinePrecision], N[(1.0 / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -0.40000000000000002 or 2.3000000000000001e-32 < x

   1. Initial program 72.7%

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

      1. *-commutative 72.7%

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

      2. exp-to-pow 72.7%

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

   3. Simplified 72.7%

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

   5. Taylor expanded in x around inf 99.9%

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

      1. mul-1-neg 99.9%

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

   7. Simplified 99.9%

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

   if -0.40000000000000002 < x < 2.3000000000000001e-32

   1. Initial program 82.1%

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

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.4 \lor \neg \left(x \leq 2.3 \cdot 10^{-32}\right):\\ \;\;\;\;\frac{e^{-y}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 86.5% accurate, 7.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.78:\\ \;\;\;\;\frac{1 + y \cdot \left(\frac{0.5 \cdot \left(x \cdot y\right)}{x} + -1\right)}{x}\\ \mathbf{elif}\;x \leq 2.3 \cdot 10^{-32}:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{1 + y \cdot \left(1 + y \cdot \left(0.5 + y \cdot 0.16666666666666666\right)\right)}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -0.78)
   (/ (+ 1.0 (* y (+ (/ (* 0.5 (* x y)) x) -1.0))) x)
   (if (<= x 2.3e-32)
     (/ 1.0 x)
     (/
      (/ 1.0 (+ 1.0 (* y (+ 1.0 (* y (+ 0.5 (* y 0.16666666666666666)))))))
      x))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -0.78) {
		tmp = (1.0 + (y * (((0.5 * (x * y)) / x) + -1.0))) / x;
	} else if (x <= 2.3e-32) {
		tmp = 1.0 / x;
	} else {
		tmp = (1.0 / (1.0 + (y * (1.0 + (y * (0.5 + (y * 0.16666666666666666))))))) / 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 <= (-0.78d0)) then
        tmp = (1.0d0 + (y * (((0.5d0 * (x * y)) / x) + (-1.0d0)))) / x
    else if (x <= 2.3d-32) then
        tmp = 1.0d0 / x
    else
        tmp = (1.0d0 / (1.0d0 + (y * (1.0d0 + (y * (0.5d0 + (y * 0.16666666666666666d0))))))) / x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -0.78) {
		tmp = (1.0 + (y * (((0.5 * (x * y)) / x) + -1.0))) / x;
	} else if (x <= 2.3e-32) {
		tmp = 1.0 / x;
	} else {
		tmp = (1.0 / (1.0 + (y * (1.0 + (y * (0.5 + (y * 0.16666666666666666))))))) / x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -0.78:
		tmp = (1.0 + (y * (((0.5 * (x * y)) / x) + -1.0))) / x
	elif x <= 2.3e-32:
		tmp = 1.0 / x
	else:
		tmp = (1.0 / (1.0 + (y * (1.0 + (y * (0.5 + (y * 0.16666666666666666))))))) / x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -0.78)
		tmp = Float64(Float64(1.0 + Float64(y * Float64(Float64(Float64(0.5 * Float64(x * y)) / x) + -1.0))) / x);
	elseif (x <= 2.3e-32)
		tmp = Float64(1.0 / x);
	else
		tmp = Float64(Float64(1.0 / Float64(1.0 + Float64(y * Float64(1.0 + Float64(y * Float64(0.5 + Float64(y * 0.16666666666666666))))))) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -0.78)
		tmp = (1.0 + (y * (((0.5 * (x * y)) / x) + -1.0))) / x;
	elseif (x <= 2.3e-32)
		tmp = 1.0 / x;
	else
		tmp = (1.0 / (1.0 + (y * (1.0 + (y * (0.5 + (y * 0.16666666666666666))))))) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -0.78], N[(N[(1.0 + N[(y * N[(N[(N[(0.5 * N[(x * y), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[x, 2.3e-32], N[(1.0 / x), $MachinePrecision], N[(N[(1.0 / N[(1.0 + N[(y * N[(1.0 + N[(y * N[(0.5 + N[(y * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.78000000000000003

   1. Initial program 72.3%

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

      1. exp-prod 72.3%

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

   3. Simplified 72.3%

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

   5. Taylor expanded in y around 0 73.6%

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

   6. Taylor expanded in x around 0 80.5%

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

      1. distribute-lft-out 80.5%

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

   8. Simplified 80.5%

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

   9. Taylor expanded in x around inf 80.5%

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

   if -0.78000000000000003 < x < 2.3000000000000001e-32

   1. Initial program 82.1%

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

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

   if 2.3000000000000001e-32 < x

   1. Initial program 73.1%

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

      1. *-commutative 73.1%

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

      2. exp-to-pow 73.1%

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

   3. Simplified 73.1%

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

   5. Taylor expanded in x around inf 99.8%

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

      1. mul-1-neg 99.8%

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

   7. Simplified 99.8%

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

      1. exp-neg 99.8%

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

   9. Applied egg-rr 99.8%

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

   10. Taylor expanded in y around 0 83.9%

       \[\leadsto \frac{\frac{1}{\color{blue}{1 + y \cdot \left(1 + y \cdot \left(0.5 + 0.16666666666666666 \cdot y\right)\right)}}}{x} \]
   11. Step-by-step derivation

       1. *-commutative 83.9%

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

   12. Simplified 83.9%

       \[\leadsto \frac{\frac{1}{\color{blue}{1 + y \cdot \left(1 + y \cdot \left(0.5 + y \cdot 0.16666666666666666\right)\right)}}}{x} \]
3. Recombined 3 regimes into one program.

4. Final simplification 89.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.78:\\ \;\;\;\;\frac{1 + y \cdot \left(\frac{0.5 \cdot \left(x \cdot y\right)}{x} + -1\right)}{x}\\ \mathbf{elif}\;x \leq 2.3 \cdot 10^{-32}:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{1 + y \cdot \left(1 + y \cdot \left(0.5 + y \cdot 0.16666666666666666\right)\right)}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 86.0% accurate, 9.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.77:\\ \;\;\;\;\frac{1 + y \cdot \left(\frac{0.5 \cdot \left(x \cdot y\right)}{x} + -1\right)}{x}\\ \mathbf{elif}\;x \leq 2.3 \cdot 10^{-32}:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{1 + y \cdot \left(1 + y \cdot 0.5\right)}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -0.77)
   (/ (+ 1.0 (* y (+ (/ (* 0.5 (* x y)) x) -1.0))) x)
   (if (<= x 2.3e-32)
     (/ 1.0 x)
     (/ (/ 1.0 (+ 1.0 (* y (+ 1.0 (* y 0.5))))) x))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -0.77) {
		tmp = (1.0 + (y * (((0.5 * (x * y)) / x) + -1.0))) / x;
	} else if (x <= 2.3e-32) {
		tmp = 1.0 / x;
	} else {
		tmp = (1.0 / (1.0 + (y * (1.0 + (y * 0.5))))) / 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 <= (-0.77d0)) then
        tmp = (1.0d0 + (y * (((0.5d0 * (x * y)) / x) + (-1.0d0)))) / x
    else if (x <= 2.3d-32) then
        tmp = 1.0d0 / x
    else
        tmp = (1.0d0 / (1.0d0 + (y * (1.0d0 + (y * 0.5d0))))) / x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -0.77) {
		tmp = (1.0 + (y * (((0.5 * (x * y)) / x) + -1.0))) / x;
	} else if (x <= 2.3e-32) {
		tmp = 1.0 / x;
	} else {
		tmp = (1.0 / (1.0 + (y * (1.0 + (y * 0.5))))) / x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -0.77:
		tmp = (1.0 + (y * (((0.5 * (x * y)) / x) + -1.0))) / x
	elif x <= 2.3e-32:
		tmp = 1.0 / x
	else:
		tmp = (1.0 / (1.0 + (y * (1.0 + (y * 0.5))))) / x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -0.77)
		tmp = Float64(Float64(1.0 + Float64(y * Float64(Float64(Float64(0.5 * Float64(x * y)) / x) + -1.0))) / x);
	elseif (x <= 2.3e-32)
		tmp = Float64(1.0 / x);
	else
		tmp = Float64(Float64(1.0 / Float64(1.0 + Float64(y * Float64(1.0 + Float64(y * 0.5))))) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -0.77)
		tmp = (1.0 + (y * (((0.5 * (x * y)) / x) + -1.0))) / x;
	elseif (x <= 2.3e-32)
		tmp = 1.0 / x;
	else
		tmp = (1.0 / (1.0 + (y * (1.0 + (y * 0.5))))) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -0.77], N[(N[(1.0 + N[(y * N[(N[(N[(0.5 * N[(x * y), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[x, 2.3e-32], N[(1.0 / x), $MachinePrecision], N[(N[(1.0 / N[(1.0 + N[(y * N[(1.0 + N[(y * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.77000000000000002

   1. Initial program 72.3%

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

      1. exp-prod 72.3%

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

   3. Simplified 72.3%

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

   5. Taylor expanded in y around 0 73.6%

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

   6. Taylor expanded in x around 0 80.5%

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

      1. distribute-lft-out 80.5%

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

   8. Simplified 80.5%

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

   9. Taylor expanded in x around inf 80.5%

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

   if -0.77000000000000002 < x < 2.3000000000000001e-32

   1. Initial program 82.1%

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

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

   if 2.3000000000000001e-32 < x

   1. Initial program 73.1%

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

      1. *-commutative 73.1%

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

      2. exp-to-pow 73.1%

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

   3. Simplified 73.1%

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

   5. Taylor expanded in x around inf 99.8%

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

      1. mul-1-neg 99.8%

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

   7. Simplified 99.8%

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

      1. exp-neg 99.8%

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

   9. Applied egg-rr 99.8%

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

   10. Taylor expanded in y around 0 83.6%

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

       1. *-commutative 83.6%

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

   12. Simplified 83.6%

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

4. Final simplification 89.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.77:\\ \;\;\;\;\frac{1 + y \cdot \left(\frac{0.5 \cdot \left(x \cdot y\right)}{x} + -1\right)}{x}\\ \mathbf{elif}\;x \leq 2.3 \cdot 10^{-32}:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{1 + y \cdot \left(1 + y \cdot 0.5\right)}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 86.2% accurate, 9.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.88:\\ \;\;\;\;\frac{1 + y \cdot \left(y \cdot \left(y \cdot -0.16666666666666666\right) + -1\right)}{x}\\ \mathbf{elif}\;x \leq 2.3 \cdot 10^{-32}:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{1 + y \cdot \left(1 + y \cdot 0.5\right)}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -0.88)
   (/ (+ 1.0 (* y (+ (* y (* y -0.16666666666666666)) -1.0))) x)
   (if (<= x 2.3e-32)
     (/ 1.0 x)
     (/ (/ 1.0 (+ 1.0 (* y (+ 1.0 (* y 0.5))))) x))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -0.88) {
		tmp = (1.0 + (y * ((y * (y * -0.16666666666666666)) + -1.0))) / x;
	} else if (x <= 2.3e-32) {
		tmp = 1.0 / x;
	} else {
		tmp = (1.0 / (1.0 + (y * (1.0 + (y * 0.5))))) / 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 <= (-0.88d0)) then
        tmp = (1.0d0 + (y * ((y * (y * (-0.16666666666666666d0))) + (-1.0d0)))) / x
    else if (x <= 2.3d-32) then
        tmp = 1.0d0 / x
    else
        tmp = (1.0d0 / (1.0d0 + (y * (1.0d0 + (y * 0.5d0))))) / x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -0.88) {
		tmp = (1.0 + (y * ((y * (y * -0.16666666666666666)) + -1.0))) / x;
	} else if (x <= 2.3e-32) {
		tmp = 1.0 / x;
	} else {
		tmp = (1.0 / (1.0 + (y * (1.0 + (y * 0.5))))) / x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -0.88:
		tmp = (1.0 + (y * ((y * (y * -0.16666666666666666)) + -1.0))) / x
	elif x <= 2.3e-32:
		tmp = 1.0 / x
	else:
		tmp = (1.0 / (1.0 + (y * (1.0 + (y * 0.5))))) / x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -0.88)
		tmp = Float64(Float64(1.0 + Float64(y * Float64(Float64(y * Float64(y * -0.16666666666666666)) + -1.0))) / x);
	elseif (x <= 2.3e-32)
		tmp = Float64(1.0 / x);
	else
		tmp = Float64(Float64(1.0 / Float64(1.0 + Float64(y * Float64(1.0 + Float64(y * 0.5))))) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -0.88)
		tmp = (1.0 + (y * ((y * (y * -0.16666666666666666)) + -1.0))) / x;
	elseif (x <= 2.3e-32)
		tmp = 1.0 / x;
	else
		tmp = (1.0 / (1.0 + (y * (1.0 + (y * 0.5))))) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -0.88], N[(N[(1.0 + N[(y * N[(N[(y * N[(y * -0.16666666666666666), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[x, 2.3e-32], N[(1.0 / x), $MachinePrecision], N[(N[(1.0 / N[(1.0 + N[(y * N[(1.0 + N[(y * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.880000000000000004

   1. Initial program 72.3%

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

      1. *-commutative 72.3%

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

      2. exp-to-pow 72.3%

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

   3. Simplified 72.3%

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

   5. Taylor expanded in x around inf 100.0%

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

      1. mul-1-neg 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in y around 0 77.7%

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

   9. Taylor expanded in y around inf 77.7%

      \[\leadsto \frac{1 + y \cdot \left(y \cdot \color{blue}{\left(-0.16666666666666666 \cdot y\right)} - 1\right)}{x} \]
   10. Step-by-step derivation

       1. *-commutative 77.7%

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

   11. Simplified 77.7%

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

   if -0.880000000000000004 < x < 2.3000000000000001e-32

   1. Initial program 82.1%

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

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

   if 2.3000000000000001e-32 < x

   1. Initial program 73.1%

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

      1. *-commutative 73.1%

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

      2. exp-to-pow 73.1%

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

   3. Simplified 73.1%

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

   5. Taylor expanded in x around inf 99.8%

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

      1. mul-1-neg 99.8%

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

   7. Simplified 99.8%

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

      1. exp-neg 99.8%

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

   9. Applied egg-rr 99.8%

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

   10. Taylor expanded in y around 0 83.6%

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

       1. *-commutative 83.6%

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

   12. Simplified 83.6%

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

4. Final simplification 88.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.88:\\ \;\;\;\;\frac{1 + y \cdot \left(y \cdot \left(y \cdot -0.16666666666666666\right) + -1\right)}{x}\\ \mathbf{elif}\;x \leq 2.3 \cdot 10^{-32}:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{1 + y \cdot \left(1 + y \cdot 0.5\right)}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 84.1% accurate, 11.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.9:\\ \;\;\;\;\frac{1 + y \cdot \left(y \cdot \left(y \cdot -0.16666666666666666\right) + -1\right)}{x}\\ \mathbf{elif}\;x \leq 2.3 \cdot 10^{-32}:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{y + 1}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -0.9)
   (/ (+ 1.0 (* y (+ (* y (* y -0.16666666666666666)) -1.0))) x)
   (if (<= x 2.3e-32) (/ 1.0 x) (/ (/ 1.0 (+ y 1.0)) x))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -0.9) {
		tmp = (1.0 + (y * ((y * (y * -0.16666666666666666)) + -1.0))) / x;
	} else if (x <= 2.3e-32) {
		tmp = 1.0 / x;
	} else {
		tmp = (1.0 / (y + 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 <= (-0.9d0)) then
        tmp = (1.0d0 + (y * ((y * (y * (-0.16666666666666666d0))) + (-1.0d0)))) / x
    else if (x <= 2.3d-32) then
        tmp = 1.0d0 / x
    else
        tmp = (1.0d0 / (y + 1.0d0)) / x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -0.9) {
		tmp = (1.0 + (y * ((y * (y * -0.16666666666666666)) + -1.0))) / x;
	} else if (x <= 2.3e-32) {
		tmp = 1.0 / x;
	} else {
		tmp = (1.0 / (y + 1.0)) / x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -0.9:
		tmp = (1.0 + (y * ((y * (y * -0.16666666666666666)) + -1.0))) / x
	elif x <= 2.3e-32:
		tmp = 1.0 / x
	else:
		tmp = (1.0 / (y + 1.0)) / x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -0.9)
		tmp = Float64(Float64(1.0 + Float64(y * Float64(Float64(y * Float64(y * -0.16666666666666666)) + -1.0))) / x);
	elseif (x <= 2.3e-32)
		tmp = Float64(1.0 / x);
	else
		tmp = Float64(Float64(1.0 / Float64(y + 1.0)) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -0.9)
		tmp = (1.0 + (y * ((y * (y * -0.16666666666666666)) + -1.0))) / x;
	elseif (x <= 2.3e-32)
		tmp = 1.0 / x;
	else
		tmp = (1.0 / (y + 1.0)) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -0.9], N[(N[(1.0 + N[(y * N[(N[(y * N[(y * -0.16666666666666666), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[x, 2.3e-32], N[(1.0 / x), $MachinePrecision], N[(N[(1.0 / N[(y + 1.0), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.900000000000000022

   1. Initial program 72.3%

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

      1. *-commutative 72.3%

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

      2. exp-to-pow 72.3%

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

   3. Simplified 72.3%

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

   5. Taylor expanded in x around inf 100.0%

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

      1. mul-1-neg 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in y around 0 77.7%

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

   9. Taylor expanded in y around inf 77.7%

      \[\leadsto \frac{1 + y \cdot \left(y \cdot \color{blue}{\left(-0.16666666666666666 \cdot y\right)} - 1\right)}{x} \]
   10. Step-by-step derivation

       1. *-commutative 77.7%

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

   11. Simplified 77.7%

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

   if -0.900000000000000022 < x < 2.3000000000000001e-32

   1. Initial program 82.1%

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

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

   if 2.3000000000000001e-32 < x

   1. Initial program 73.1%

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

      1. *-commutative 73.1%

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

      2. exp-to-pow 73.1%

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

   3. Simplified 73.1%

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

   5. Taylor expanded in x around inf 99.8%

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

      1. mul-1-neg 99.8%

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

   7. Simplified 99.8%

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

      1. exp-neg 99.8%

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

   9. Applied egg-rr 99.8%

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

   10. Taylor expanded in y around 0 77.4%

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

4. Final simplification 86.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.9:\\ \;\;\;\;\frac{1 + y \cdot \left(y \cdot \left(y \cdot -0.16666666666666666\right) + -1\right)}{x}\\ \mathbf{elif}\;x \leq 2.3 \cdot 10^{-32}:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{y + 1}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 80.6% accurate, 12.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{+172} \lor \neg \left(x \leq 2.3 \cdot 10^{-32}\right):\\ \;\;\;\;\frac{\frac{1}{y + 1}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -2e+172) (not (<= x 2.3e-32)))
   (/ (/ 1.0 (+ y 1.0)) x)
   (/ 1.0 x)))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -2e+172) || !(x <= 2.3e-32)) {
		tmp = (1.0 / (y + 1.0)) / 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 <= (-2d+172)) .or. (.not. (x <= 2.3d-32))) then
        tmp = (1.0d0 / (y + 1.0d0)) / 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 <= -2e+172) || !(x <= 2.3e-32)) {
		tmp = (1.0 / (y + 1.0)) / x;
	} else {
		tmp = 1.0 / x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -2e+172) or not (x <= 2.3e-32):
		tmp = (1.0 / (y + 1.0)) / x
	else:
		tmp = 1.0 / x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -2e+172) || !(x <= 2.3e-32))
		tmp = Float64(Float64(1.0 / Float64(y + 1.0)) / x);
	else
		tmp = Float64(1.0 / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((x <= -2e+172) || ~((x <= 2.3e-32)))
		tmp = (1.0 / (y + 1.0)) / x;
	else
		tmp = 1.0 / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -2e+172], N[Not[LessEqual[x, 2.3e-32]], $MachinePrecision]], N[(N[(1.0 / N[(y + 1.0), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], N[(1.0 / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2.0000000000000002e172 or 2.3000000000000001e-32 < x

   1. Initial program 67.7%

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

      1. *-commutative 67.7%

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

      2. exp-to-pow 67.7%

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

   3. Simplified 67.7%

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

   5. Taylor expanded in x around inf 99.8%

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

      1. mul-1-neg 99.8%

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

   7. Simplified 99.8%

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

      1. exp-neg 99.8%

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

   9. Applied egg-rr 99.8%

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

   10. Taylor expanded in y around 0 71.5%

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

   if -2.0000000000000002e172 < x < 2.3000000000000001e-32

   1. Initial program 83.8%

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

      1. exp-prod 97.2%

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

   3. Simplified 97.2%

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

   5. Taylor expanded in x around 0 89.9%

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

4. Final simplification 81.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{+172} \lor \neg \left(x \leq 2.3 \cdot 10^{-32}\right):\\ \;\;\;\;\frac{\frac{1}{y + 1}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 82.9% accurate, 12.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.54:\\ \;\;\;\;\frac{1 + y \cdot \left(y \cdot 0.5 + -1\right)}{x}\\ \mathbf{elif}\;x \leq 2.3 \cdot 10^{-32}:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{y + 1}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -0.54)
   (/ (+ 1.0 (* y (+ (* y 0.5) -1.0))) x)
   (if (<= x 2.3e-32) (/ 1.0 x) (/ (/ 1.0 (+ y 1.0)) x))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -0.54) {
		tmp = (1.0 + (y * ((y * 0.5) + -1.0))) / x;
	} else if (x <= 2.3e-32) {
		tmp = 1.0 / x;
	} else {
		tmp = (1.0 / (y + 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 <= (-0.54d0)) then
        tmp = (1.0d0 + (y * ((y * 0.5d0) + (-1.0d0)))) / x
    else if (x <= 2.3d-32) then
        tmp = 1.0d0 / x
    else
        tmp = (1.0d0 / (y + 1.0d0)) / x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -0.54) {
		tmp = (1.0 + (y * ((y * 0.5) + -1.0))) / x;
	} else if (x <= 2.3e-32) {
		tmp = 1.0 / x;
	} else {
		tmp = (1.0 / (y + 1.0)) / x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -0.54:
		tmp = (1.0 + (y * ((y * 0.5) + -1.0))) / x
	elif x <= 2.3e-32:
		tmp = 1.0 / x
	else:
		tmp = (1.0 / (y + 1.0)) / x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -0.54)
		tmp = Float64(Float64(1.0 + Float64(y * Float64(Float64(y * 0.5) + -1.0))) / x);
	elseif (x <= 2.3e-32)
		tmp = Float64(1.0 / x);
	else
		tmp = Float64(Float64(1.0 / Float64(y + 1.0)) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -0.54)
		tmp = (1.0 + (y * ((y * 0.5) + -1.0))) / x;
	elseif (x <= 2.3e-32)
		tmp = 1.0 / x;
	else
		tmp = (1.0 / (y + 1.0)) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -0.54], N[(N[(1.0 + N[(y * N[(N[(y * 0.5), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[x, 2.3e-32], N[(1.0 / x), $MachinePrecision], N[(N[(1.0 / N[(y + 1.0), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.54000000000000004

   1. Initial program 72.3%

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

      1. exp-prod 72.3%

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

   3. Simplified 72.3%

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

   5. Taylor expanded in y around 0 73.6%

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

   6. Taylor expanded in x around inf 73.6%

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

      1. *-commutative 73.6%

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

   8. Simplified 73.6%

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

   if -0.54000000000000004 < x < 2.3000000000000001e-32

   1. Initial program 82.1%

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

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

   if 2.3000000000000001e-32 < x

   1. Initial program 73.1%

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

      1. *-commutative 73.1%

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

      2. exp-to-pow 73.1%

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

   3. Simplified 73.1%

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

   5. Taylor expanded in x around inf 99.8%

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

      1. mul-1-neg 99.8%

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

   7. Simplified 99.8%

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

      1. exp-neg 99.8%

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

   9. Applied egg-rr 99.8%

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

   10. Taylor expanded in y around 0 77.4%

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

4. Final simplification 85.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.54:\\ \;\;\;\;\frac{1 + y \cdot \left(y \cdot 0.5 + -1\right)}{x}\\ \mathbf{elif}\;x \leq 2.3 \cdot 10^{-32}:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{y + 1}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 75.2% accurate, 69.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 76.7%

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

   1. exp-prod 84.0%

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

3. Simplified 84.0%

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

5. Taylor expanded in x around 0 74.7%

   \[\leadsto \color{blue}{\frac{1}{x}} \]
6. Add Preprocessing

Developer Target 1: 77.7% 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 2024157 
(FPCore (x y)
  :name "Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2, F"
  :precision binary64

  :alt
  (! :herbie-platform default (if (< y -37311844206647956000000000000000000000000000000000000000000000000000000000000000000000000000000) (/ (exp (/ -1 y)) x) (if (< y 28179592427282880000000000000000000000) (/ (pow (/ x (+ y x)) x) x) (if (< y 23473874151669980000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000) (log (exp (/ (pow (/ x (+ y x)) x) x))) (/ (exp (/ -1 y)) x)))))

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