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

Percentage Accurate: 77.9% → 99.1%

Time: 12.2s

Alternatives: 7

Speedup: 12.3×

• Could not converge on a ground truth

  1. reg0 = (bf "1.398979927068672119381632279149807974144e122")
  2. reg1 = (bf "5.336883156464011360260023197228840846601e-76")

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 77.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.1 \lor \neg \left(x \leq 4.8 \cdot 10^{-27}\right):\\ \;\;\;\;\frac{e^{-y}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -1.1) (not (<= x 4.8e-27))) (/ (exp (- y)) x) (/ 1.0 x)))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -1.1) || !(x <= 4.8e-27)) {
		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 <= (-1.1d0)) .or. (.not. (x <= 4.8d-27))) 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 <= -1.1) || !(x <= 4.8e-27)) {
		tmp = Math.exp(-y) / x;
	} else {
		tmp = 1.0 / x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -1.1) or not (x <= 4.8e-27):
		tmp = math.exp(-y) / x
	else:
		tmp = 1.0 / x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -1.1) || !(x <= 4.8e-27))
		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 <= -1.1) || ~((x <= 4.8e-27)))
		tmp = exp(-y) / x;
	else
		tmp = 1.0 / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -1.1], N[Not[LessEqual[x, 4.8e-27]], $MachinePrecision]], N[(N[Exp[(-y)], $MachinePrecision] / x), $MachinePrecision], N[(1.0 / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.1000000000000001 or 4.80000000000000004e-27 < x

   1. Initial program 77.2%

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

      1. *-commutative 77.2%

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

      2. exp-to-pow 77.2%

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

   3. Simplified 77.2%

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

   5. Taylor expanded in x around inf 100.0%

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

      1. mul-1-neg 100.0%

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

   7. Simplified 100.0%

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

   if -1.1000000000000001 < x < 4.80000000000000004e-27

   1. Initial program 78.9%

      \[\frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x} \]
   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 99.9%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.1 \lor \neg \left(x \leq 4.8 \cdot 10^{-27}\right):\\ \;\;\;\;\frac{e^{-y}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 87.8% accurate, 8.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y \cdot \left(\left(1 + \frac{1}{y}\right) + -1\right)}{x}\\ \mathbf{if}\;x \leq -7.5 \cdot 10^{+186}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq -0.38:\\ \;\;\;\;\frac{1 + y \cdot \left(y \cdot \left(0.5 + y \cdot -0.16666666666666666\right) + -1\right)}{x}\\ \mathbf{elif}\;x \leq 4.8 \cdot 10^{-27}:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* y (+ (+ 1.0 (/ 1.0 y)) -1.0)) x)))
   (if (<= x -7.5e+186)
     t_0
     (if (<= x -0.38)
       (/ (+ 1.0 (* y (+ (* y (+ 0.5 (* y -0.16666666666666666))) -1.0))) x)
       (if (<= x 4.8e-27) (/ 1.0 x) t_0)))))

C

double code(double x, double y) {
	double t_0 = (y * ((1.0 + (1.0 / y)) + -1.0)) / x;
	double tmp;
	if (x <= -7.5e+186) {
		tmp = t_0;
	} else if (x <= -0.38) {
		tmp = (1.0 + (y * ((y * (0.5 + (y * -0.16666666666666666))) + -1.0))) / x;
	} else if (x <= 4.8e-27) {
		tmp = 1.0 / x;
	} 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) :: tmp
    t_0 = (y * ((1.0d0 + (1.0d0 / y)) + (-1.0d0))) / x
    if (x <= (-7.5d+186)) then
        tmp = t_0
    else if (x <= (-0.38d0)) then
        tmp = (1.0d0 + (y * ((y * (0.5d0 + (y * (-0.16666666666666666d0)))) + (-1.0d0)))) / x
    else if (x <= 4.8d-27) then
        tmp = 1.0d0 / x
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = (y * ((1.0 + (1.0 / y)) + -1.0)) / x;
	double tmp;
	if (x <= -7.5e+186) {
		tmp = t_0;
	} else if (x <= -0.38) {
		tmp = (1.0 + (y * ((y * (0.5 + (y * -0.16666666666666666))) + -1.0))) / x;
	} else if (x <= 4.8e-27) {
		tmp = 1.0 / x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (y * ((1.0 + (1.0 / y)) + -1.0)) / x
	tmp = 0
	if x <= -7.5e+186:
		tmp = t_0
	elif x <= -0.38:
		tmp = (1.0 + (y * ((y * (0.5 + (y * -0.16666666666666666))) + -1.0))) / x
	elif x <= 4.8e-27:
		tmp = 1.0 / x
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(y * Float64(Float64(1.0 + Float64(1.0 / y)) + -1.0)) / x)
	tmp = 0.0
	if (x <= -7.5e+186)
		tmp = t_0;
	elseif (x <= -0.38)
		tmp = Float64(Float64(1.0 + Float64(y * Float64(Float64(y * Float64(0.5 + Float64(y * -0.16666666666666666))) + -1.0))) / x);
	elseif (x <= 4.8e-27)
		tmp = Float64(1.0 / x);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (y * ((1.0 + (1.0 / y)) + -1.0)) / x;
	tmp = 0.0;
	if (x <= -7.5e+186)
		tmp = t_0;
	elseif (x <= -0.38)
		tmp = (1.0 + (y * ((y * (0.5 + (y * -0.16666666666666666))) + -1.0))) / x;
	elseif (x <= 4.8e-27)
		tmp = 1.0 / x;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(y * N[(N[(1.0 + N[(1.0 / y), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]}, If[LessEqual[x, -7.5e+186], t$95$0, If[LessEqual[x, -0.38], N[(N[(1.0 + N[(y * N[(N[(y * N[(0.5 + N[(y * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[x, 4.8e-27], N[(1.0 / x), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if x < -7.4999999999999998e186 or 4.80000000000000004e-27 < x

   1. Initial program 73.0%

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

      1. exp-prod 73.0%

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

   3. Simplified 73.0%

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

   5. Taylor expanded in x around inf 62.9%

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

      1. mul-1-neg 62.9%

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

      2. unsub-neg 62.9%

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

   7. Simplified 62.9%

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

   8. Taylor expanded in y around inf 62.8%

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

   9. Taylor expanded in y around 0 63.3%

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

       1. expm1-log1p-u 30.8%

          \[\leadsto \frac{y \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{y}\right)\right)}}{x} \]

       2. expm1-undefine 56.8%

          \[\leadsto \frac{y \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{1}{y}\right)} - 1\right)}}{x} \]

       3. log1p-undefine 56.8%

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

       4. *-rgt-identity 56.8%

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

       5. add-exp-log 89.3%

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

       6. *-rgt-identity 89.3%

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

   11. Applied egg-rr 89.3%

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

   if -7.4999999999999998e186 < x < -0.38

   1. Initial program 86.3%

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

      1. *-commutative 86.3%

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

      2. exp-to-pow 86.3%

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

   3. Simplified 86.3%

      \[\leadsto \color{blue}{\frac{{\left(\frac{x}{x + y}\right)}^{x}}{x}} \]
   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 84.2%

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

   if -0.38 < x < 4.80000000000000004e-27

   1. Initial program 78.9%

      \[\frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x} \]
   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 99.9%

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

4. Final simplification 92.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -7.5 \cdot 10^{+186}:\\ \;\;\;\;\frac{y \cdot \left(\left(1 + \frac{1}{y}\right) + -1\right)}{x}\\ \mathbf{elif}\;x \leq -0.38:\\ \;\;\;\;\frac{1 + y \cdot \left(y \cdot \left(0.5 + y \cdot -0.16666666666666666\right) + -1\right)}{x}\\ \mathbf{elif}\;x \leq 4.8 \cdot 10^{-27}:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot \left(\left(1 + \frac{1}{y}\right) + -1\right)}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 87.7% accurate, 8.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y \cdot \left(\left(1 + \frac{1}{y}\right) + -1\right)}{x}\\ \mathbf{if}\;x \leq -1.32 \cdot 10^{+183}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq -0.52:\\ \;\;\;\;\frac{1 + y \cdot \left(y \cdot \left(y \cdot -0.16666666666666666\right) + -1\right)}{x}\\ \mathbf{elif}\;x \leq 4.8 \cdot 10^{-27}:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* y (+ (+ 1.0 (/ 1.0 y)) -1.0)) x)))
   (if (<= x -1.32e+183)
     t_0
     (if (<= x -0.52)
       (/ (+ 1.0 (* y (+ (* y (* y -0.16666666666666666)) -1.0))) x)
       (if (<= x 4.8e-27) (/ 1.0 x) t_0)))))

C

double code(double x, double y) {
	double t_0 = (y * ((1.0 + (1.0 / y)) + -1.0)) / x;
	double tmp;
	if (x <= -1.32e+183) {
		tmp = t_0;
	} else if (x <= -0.52) {
		tmp = (1.0 + (y * ((y * (y * -0.16666666666666666)) + -1.0))) / x;
	} else if (x <= 4.8e-27) {
		tmp = 1.0 / x;
	} 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) :: tmp
    t_0 = (y * ((1.0d0 + (1.0d0 / y)) + (-1.0d0))) / x
    if (x <= (-1.32d+183)) then
        tmp = t_0
    else if (x <= (-0.52d0)) then
        tmp = (1.0d0 + (y * ((y * (y * (-0.16666666666666666d0))) + (-1.0d0)))) / x
    else if (x <= 4.8d-27) then
        tmp = 1.0d0 / x
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = (y * ((1.0 + (1.0 / y)) + -1.0)) / x;
	double tmp;
	if (x <= -1.32e+183) {
		tmp = t_0;
	} else if (x <= -0.52) {
		tmp = (1.0 + (y * ((y * (y * -0.16666666666666666)) + -1.0))) / x;
	} else if (x <= 4.8e-27) {
		tmp = 1.0 / x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (y * ((1.0 + (1.0 / y)) + -1.0)) / x
	tmp = 0
	if x <= -1.32e+183:
		tmp = t_0
	elif x <= -0.52:
		tmp = (1.0 + (y * ((y * (y * -0.16666666666666666)) + -1.0))) / x
	elif x <= 4.8e-27:
		tmp = 1.0 / x
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(y * Float64(Float64(1.0 + Float64(1.0 / y)) + -1.0)) / x)
	tmp = 0.0
	if (x <= -1.32e+183)
		tmp = t_0;
	elseif (x <= -0.52)
		tmp = Float64(Float64(1.0 + Float64(y * Float64(Float64(y * Float64(y * -0.16666666666666666)) + -1.0))) / x);
	elseif (x <= 4.8e-27)
		tmp = Float64(1.0 / x);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (y * ((1.0 + (1.0 / y)) + -1.0)) / x;
	tmp = 0.0;
	if (x <= -1.32e+183)
		tmp = t_0;
	elseif (x <= -0.52)
		tmp = (1.0 + (y * ((y * (y * -0.16666666666666666)) + -1.0))) / x;
	elseif (x <= 4.8e-27)
		tmp = 1.0 / x;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(y * N[(N[(1.0 + N[(1.0 / y), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]}, If[LessEqual[x, -1.32e+183], t$95$0, If[LessEqual[x, -0.52], 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, 4.8e-27], N[(1.0 / x), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.32e183 or 4.80000000000000004e-27 < x

   1. Initial program 73.0%

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

      1. exp-prod 73.0%

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

   3. Simplified 73.0%

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

   5. Taylor expanded in x around inf 62.9%

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

      1. mul-1-neg 62.9%

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

      2. unsub-neg 62.9%

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

   7. Simplified 62.9%

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

   8. Taylor expanded in y around inf 62.8%

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

   9. Taylor expanded in y around 0 63.3%

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

       1. expm1-log1p-u 30.8%

          \[\leadsto \frac{y \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{y}\right)\right)}}{x} \]

       2. expm1-undefine 56.8%

          \[\leadsto \frac{y \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{1}{y}\right)} - 1\right)}}{x} \]

       3. log1p-undefine 56.8%

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

       4. *-rgt-identity 56.8%

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

       5. add-exp-log 89.3%

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

       6. *-rgt-identity 89.3%

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

   11. Applied egg-rr 89.3%

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

   if -1.32e183 < x < -0.52000000000000002

   1. Initial program 86.3%

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

      1. *-commutative 86.3%

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

      2. exp-to-pow 86.3%

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

   3. Simplified 86.3%

      \[\leadsto \color{blue}{\frac{{\left(\frac{x}{x + y}\right)}^{x}}{x}} \]
   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 84.2%

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

   9. Taylor expanded in y around inf 84.2%

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

       1. *-commutative 84.2%

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

   11. Simplified 84.2%

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

   if -0.52000000000000002 < x < 4.80000000000000004e-27

   1. Initial program 78.9%

      \[\frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x} \]
   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 99.9%

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

4. Final simplification 92.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.32 \cdot 10^{+183}:\\ \;\;\;\;\frac{y \cdot \left(\left(1 + \frac{1}{y}\right) + -1\right)}{x}\\ \mathbf{elif}\;x \leq -0.52:\\ \;\;\;\;\frac{1 + y \cdot \left(y \cdot \left(y \cdot -0.16666666666666666\right) + -1\right)}{x}\\ \mathbf{elif}\;x \leq 4.8 \cdot 10^{-27}:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot \left(\left(1 + \frac{1}{y}\right) + -1\right)}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 86.9% accurate, 8.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y \cdot \left(\left(1 + \frac{1}{y}\right) + -1\right)}{x}\\ \mathbf{if}\;x \leq -4.3 \cdot 10^{+154}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq -0.76:\\ \;\;\;\;\frac{1 + y \cdot \left(y \cdot 0.5 + -1\right)}{x}\\ \mathbf{elif}\;x \leq 4.8 \cdot 10^{-27}:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* y (+ (+ 1.0 (/ 1.0 y)) -1.0)) x)))
   (if (<= x -4.3e+154)
     t_0
     (if (<= x -0.76)
       (/ (+ 1.0 (* y (+ (* y 0.5) -1.0))) x)
       (if (<= x 4.8e-27) (/ 1.0 x) t_0)))))

C

double code(double x, double y) {
	double t_0 = (y * ((1.0 + (1.0 / y)) + -1.0)) / x;
	double tmp;
	if (x <= -4.3e+154) {
		tmp = t_0;
	} else if (x <= -0.76) {
		tmp = (1.0 + (y * ((y * 0.5) + -1.0))) / x;
	} else if (x <= 4.8e-27) {
		tmp = 1.0 / x;
	} 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) :: tmp
    t_0 = (y * ((1.0d0 + (1.0d0 / y)) + (-1.0d0))) / x
    if (x <= (-4.3d+154)) then
        tmp = t_0
    else if (x <= (-0.76d0)) then
        tmp = (1.0d0 + (y * ((y * 0.5d0) + (-1.0d0)))) / x
    else if (x <= 4.8d-27) then
        tmp = 1.0d0 / x
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = (y * ((1.0 + (1.0 / y)) + -1.0)) / x;
	double tmp;
	if (x <= -4.3e+154) {
		tmp = t_0;
	} else if (x <= -0.76) {
		tmp = (1.0 + (y * ((y * 0.5) + -1.0))) / x;
	} else if (x <= 4.8e-27) {
		tmp = 1.0 / x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (y * ((1.0 + (1.0 / y)) + -1.0)) / x
	tmp = 0
	if x <= -4.3e+154:
		tmp = t_0
	elif x <= -0.76:
		tmp = (1.0 + (y * ((y * 0.5) + -1.0))) / x
	elif x <= 4.8e-27:
		tmp = 1.0 / x
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(y * Float64(Float64(1.0 + Float64(1.0 / y)) + -1.0)) / x)
	tmp = 0.0
	if (x <= -4.3e+154)
		tmp = t_0;
	elseif (x <= -0.76)
		tmp = Float64(Float64(1.0 + Float64(y * Float64(Float64(y * 0.5) + -1.0))) / x);
	elseif (x <= 4.8e-27)
		tmp = Float64(1.0 / x);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (y * ((1.0 + (1.0 / y)) + -1.0)) / x;
	tmp = 0.0;
	if (x <= -4.3e+154)
		tmp = t_0;
	elseif (x <= -0.76)
		tmp = (1.0 + (y * ((y * 0.5) + -1.0))) / x;
	elseif (x <= 4.8e-27)
		tmp = 1.0 / x;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(y * N[(N[(1.0 + N[(1.0 / y), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]}, If[LessEqual[x, -4.3e+154], t$95$0, If[LessEqual[x, -0.76], N[(N[(1.0 + N[(y * N[(N[(y * 0.5), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[x, 4.8e-27], N[(1.0 / x), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if x < -4.2999999999999998e154 or 4.80000000000000004e-27 < x

   1. Initial program 72.2%

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

      1. exp-prod 72.2%

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

   3. Simplified 72.2%

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

   5. Taylor expanded in x around inf 60.1%

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

      1. mul-1-neg 60.1%

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

      2. unsub-neg 60.1%

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

   7. Simplified 60.1%

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

   8. Taylor expanded in y around inf 60.1%

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

   9. Taylor expanded in y around 0 60.5%

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

       1. expm1-log1p-u 29.1%

          \[\leadsto \frac{y \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{y}\right)\right)}}{x} \]

       2. expm1-undefine 56.0%

          \[\leadsto \frac{y \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{1}{y}\right)} - 1\right)}}{x} \]

       3. log1p-undefine 56.0%

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

       4. *-rgt-identity 56.0%

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

       5. add-exp-log 87.4%

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

       6. *-rgt-identity 87.4%

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

   11. Applied egg-rr 87.4%

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

   if -4.2999999999999998e154 < x < -0.76000000000000001

   1. Initial program 90.8%

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

      1. exp-prod 90.8%

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

   3. Simplified 90.8%

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

   5. Taylor expanded in y around 0 79.4%

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

   6. Taylor expanded in x around inf 79.4%

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

      1. *-commutative 79.4%

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

   8. Simplified 79.4%

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

   if -0.76000000000000001 < x < 4.80000000000000004e-27

   1. Initial program 78.9%

      \[\frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x} \]
   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 99.9%

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

4. Final simplification 90.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.3 \cdot 10^{+154}:\\ \;\;\;\;\frac{y \cdot \left(\left(1 + \frac{1}{y}\right) + -1\right)}{x}\\ \mathbf{elif}\;x \leq -0.76:\\ \;\;\;\;\frac{1 + y \cdot \left(y \cdot 0.5 + -1\right)}{x}\\ \mathbf{elif}\;x \leq 4.8 \cdot 10^{-27}:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot \left(\left(1 + \frac{1}{y}\right) + -1\right)}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 85.4% accurate, 9.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5.1 \cdot 10^{+85} \lor \neg \left(x \leq 4.8 \cdot 10^{-27}\right):\\ \;\;\;\;\frac{y \cdot \left(\left(1 + \frac{1}{y}\right) + -1\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -5.1e+85) (not (<= x 4.8e-27)))
   (/ (* y (+ (+ 1.0 (/ 1.0 y)) -1.0)) x)
   (/ 1.0 x)))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -5.1e+85) || !(x <= 4.8e-27)) {
		tmp = (y * ((1.0 + (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 <= (-5.1d+85)) .or. (.not. (x <= 4.8d-27))) then
        tmp = (y * ((1.0d0 + (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 <= -5.1e+85) || !(x <= 4.8e-27)) {
		tmp = (y * ((1.0 + (1.0 / y)) + -1.0)) / x;
	} else {
		tmp = 1.0 / x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -5.1e+85) or not (x <= 4.8e-27):
		tmp = (y * ((1.0 + (1.0 / y)) + -1.0)) / x
	else:
		tmp = 1.0 / x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -5.1e+85) || !(x <= 4.8e-27))
		tmp = Float64(Float64(y * Float64(Float64(1.0 + Float64(1.0 / 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 <= -5.1e+85) || ~((x <= 4.8e-27)))
		tmp = (y * ((1.0 + (1.0 / y)) + -1.0)) / x;
	else
		tmp = 1.0 / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -5.1e+85], N[Not[LessEqual[x, 4.8e-27]], $MachinePrecision]], N[(N[(y * N[(N[(1.0 + N[(1.0 / y), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], N[(1.0 / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -5.0999999999999998e85 or 4.80000000000000004e-27 < x

   1. Initial program 73.8%

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

      1. exp-prod 73.8%

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

   3. Simplified 73.8%

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

   5. Taylor expanded in x around inf 58.8%

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

      1. mul-1-neg 58.8%

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

      2. unsub-neg 58.8%

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

   7. Simplified 58.8%

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

   8. Taylor expanded in y around inf 58.7%

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

   9. Taylor expanded in y around 0 59.0%

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

       1. expm1-log1p-u 28.7%

          \[\leadsto \frac{y \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{y}\right)\right)}}{x} \]

       2. expm1-undefine 52.6%

          \[\leadsto \frac{y \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{1}{y}\right)} - 1\right)}}{x} \]

       3. log1p-undefine 52.6%

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

       4. *-rgt-identity 52.6%

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

       5. add-exp-log 83.0%

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

       6. *-rgt-identity 83.0%

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

   11. Applied egg-rr 83.0%

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

   if -5.0999999999999998e85 < x < 4.80000000000000004e-27

   1. Initial program 82.4%

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

      1. exp-prod 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 96.7%

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

4. Final simplification 89.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.1 \cdot 10^{+85} \lor \neg \left(x \leq 4.8 \cdot 10^{-27}\right):\\ \;\;\;\;\frac{y \cdot \left(\left(1 + \frac{1}{y}\right) + -1\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 80.7% accurate, 12.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{+178} \lor \neg \left(x \leq 4.8 \cdot 10^{-27}\right):\\ \;\;\;\;\frac{1}{x + x \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -2e+178) (not (<= x 4.8e-27))) (/ 1.0 (+ x (* x y))) (/ 1.0 x)))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -2e+178) || !(x <= 4.8e-27)) {
		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 <= (-2d+178)) .or. (.not. (x <= 4.8d-27))) 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 <= -2e+178) || !(x <= 4.8e-27)) {
		tmp = 1.0 / (x + (x * y));
	} else {
		tmp = 1.0 / x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -2e+178) or not (x <= 4.8e-27):
		tmp = 1.0 / (x + (x * y))
	else:
		tmp = 1.0 / x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -2e+178) || !(x <= 4.8e-27))
		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 <= -2e+178) || ~((x <= 4.8e-27)))
		tmp = 1.0 / (x + (x * y));
	else
		tmp = 1.0 / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -2e+178], N[Not[LessEqual[x, 4.8e-27]], $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 < -2.0000000000000001e178 or 4.80000000000000004e-27 < x

   1. Initial program 72.9%

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

      1. exp-prod 72.9%

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

   3. Simplified 72.9%

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

   5. Taylor expanded in x around inf 62.2%

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

      1. mul-1-neg 62.2%

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

      2. unsub-neg 62.2%

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

   7. Simplified 62.2%

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

   8. Taylor expanded in y around inf 62.1%

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

      1. clear-num 62.1%

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

      2. inv-pow 62.1%

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

      3. sub-neg 62.1%

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

      4. metadata-eval 62.1%

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

   10. Applied egg-rr 62.1%

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

       1. unpow-1 62.1%

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

       2. distribute-rgt-in 62.1%

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

       3. lft-mult-inverse 62.2%

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

       4. neg-mul-1 62.2%

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

       5. sub-neg 62.2%

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

   12. Simplified 62.2%

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

   13. Taylor expanded in y around 0 82.3%

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

   if -2.0000000000000001e178 < x < 4.80000000000000004e-27

   1. Initial program 81.6%

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

      1. exp-prod 96.0%

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

   3. Simplified 96.0%

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

   5. Taylor expanded in x around 0 87.2%

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

4. Final simplification 85.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{+178} \lor \neg \left(x \leq 4.8 \cdot 10^{-27}\right):\\ \;\;\;\;\frac{1}{x + x \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 74.6% 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 77.8%

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

   1. exp-prod 86.0%

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

3. Simplified 86.0%

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

5. Taylor expanded in x around 0 76.5%

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

Developer Target 1: 78.2% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024144 
(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))