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

Percentage Accurate: 77.6% → 98.9%

Time: 8.4s

Alternatives: 8

Speedup: 18.8×

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: 77.6% 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: 98.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{+43} \lor \neg \left(x \leq 4.2 \cdot 10^{-29}\right):\\ \;\;\;\;\frac{1}{x \cdot e^{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(e^{x}\right)}^{\log \left(\frac{x}{x + y}\right)}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -1e+43) (not (<= x 4.2e-29)))
   (/ 1.0 (* x (exp y)))
   (/ (pow (exp x) (log (/ x (+ x y)))) x)))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -1e+43) || !(x <= 4.2e-29)) {
		tmp = 1.0 / (x * exp(y));
	} else {
		tmp = pow(exp(x), log((x / (x + y)))) / x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((x <= (-1d+43)) .or. (.not. (x <= 4.2d-29))) then
        tmp = 1.0d0 / (x * exp(y))
    else
        tmp = (exp(x) ** log((x / (x + y)))) / x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((x <= -1e+43) || !(x <= 4.2e-29)) {
		tmp = 1.0 / (x * Math.exp(y));
	} else {
		tmp = Math.pow(Math.exp(x), Math.log((x / (x + y)))) / x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -1e+43) or not (x <= 4.2e-29):
		tmp = 1.0 / (x * math.exp(y))
	else:
		tmp = math.pow(math.exp(x), math.log((x / (x + y)))) / x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -1e+43) || !(x <= 4.2e-29))
		tmp = Float64(1.0 / Float64(x * exp(y)));
	else
		tmp = Float64((exp(x) ^ log(Float64(x / Float64(x + y)))) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((x <= -1e+43) || ~((x <= 4.2e-29)))
		tmp = 1.0 / (x * exp(y));
	else
		tmp = (exp(x) ^ log((x / (x + y)))) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -1e+43], N[Not[LessEqual[x, 4.2e-29]], $MachinePrecision]], N[(1.0 / N[(x * N[Exp[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[N[Exp[x], $MachinePrecision], N[Log[N[(x / N[(x + y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.00000000000000001e43 or 4.19999999999999979e-29 < x

   1. Initial program 76.3%

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

      1. *-commutative 76.3%

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

      2. exp-to-pow 76.3%

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

   3. Simplified 76.3%

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

   4. Taylor expanded in x around inf 100.0%

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

      1. mul-1-neg 100.0%

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

   6. Simplified 100.0%

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

      1. clear-num 100.0%

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

      2. inv-pow 100.0%

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

      3. exp-neg 100.0%

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

      4. associate-/r/ 100.0%

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

      5. /-rgt-identity 100.0%

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

   8. Applied egg-rr 100.0%

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

   9. Taylor expanded in x around 0 100.0%

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

   if -1.00000000000000001e43 < x < 4.19999999999999979e-29

   1. Initial program 79.8%

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

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{+43} \lor \neg \left(x \leq 4.2 \cdot 10^{-29}\right):\\ \;\;\;\;\frac{1}{x \cdot e^{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(e^{x}\right)}^{\log \left(\frac{x}{x + y}\right)}}{x}\\ \end{array} \]

Alternative 2: 98.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -6800000 \lor \neg \left(x \leq 4.2 \cdot 10^{-29}\right):\\ \;\;\;\;\frac{1}{x \cdot e^{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -6800000.0) (not (<= x 4.2e-29)))
   (/ 1.0 (* x (exp y)))
   (/ 1.0 x)))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -6800000.0) || !(x <= 4.2e-29)) {
		tmp = 1.0 / (x * exp(y));
	} else {
		tmp = 1.0 / x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((x <= (-6800000.0d0)) .or. (.not. (x <= 4.2d-29))) then
        tmp = 1.0d0 / (x * exp(y))
    else
        tmp = 1.0d0 / x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((x <= -6800000.0) || !(x <= 4.2e-29)) {
		tmp = 1.0 / (x * Math.exp(y));
	} else {
		tmp = 1.0 / x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -6800000.0) or not (x <= 4.2e-29):
		tmp = 1.0 / (x * math.exp(y))
	else:
		tmp = 1.0 / x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -6800000.0) || !(x <= 4.2e-29))
		tmp = Float64(1.0 / Float64(x * exp(y)));
	else
		tmp = Float64(1.0 / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((x <= -6800000.0) || ~((x <= 4.2e-29)))
		tmp = 1.0 / (x * exp(y));
	else
		tmp = 1.0 / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -6800000.0], N[Not[LessEqual[x, 4.2e-29]], $MachinePrecision]], N[(1.0 / N[(x * N[Exp[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -6.8e6 or 4.19999999999999979e-29 < 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. Taylor expanded in x around inf 100.0%

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

      1. mul-1-neg 100.0%

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

   6. Simplified 100.0%

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

      1. clear-num 100.0%

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

      2. inv-pow 100.0%

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

      3. exp-neg 100.0%

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

      4. associate-/r/ 100.0%

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

      5. /-rgt-identity 100.0%

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

   8. Applied egg-rr 100.0%

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

   9. Taylor expanded in x around 0 100.0%

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

   if -6.8e6 < x < 4.19999999999999979e-29

   1. Initial program 79.0%

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

      1. exp-prod 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in x around 0 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -6800000 \lor \neg \left(x \leq 4.2 \cdot 10^{-29}\right):\\ \;\;\;\;\frac{1}{x \cdot e^{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x}\\ \end{array} \]

Alternative 3: 98.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -6800000 \lor \neg \left(x \leq 4.2 \cdot 10^{-29}\right):\\ \;\;\;\;\frac{e^{-y}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -6800000.0) (not (<= x 4.2e-29))) (/ (exp (- y)) x) (/ 1.0 x)))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -6800000.0) || !(x <= 4.2e-29)) {
		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 <= (-6800000.0d0)) .or. (.not. (x <= 4.2d-29))) 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 <= -6800000.0) || !(x <= 4.2e-29)) {
		tmp = Math.exp(-y) / x;
	} else {
		tmp = 1.0 / x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -6800000.0) or not (x <= 4.2e-29):
		tmp = math.exp(-y) / x
	else:
		tmp = 1.0 / x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -6800000.0) || !(x <= 4.2e-29))
		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 <= -6800000.0) || ~((x <= 4.2e-29)))
		tmp = exp(-y) / x;
	else
		tmp = 1.0 / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -6800000.0], N[Not[LessEqual[x, 4.2e-29]], $MachinePrecision]], N[(N[Exp[(-y)], $MachinePrecision] / x), $MachinePrecision], N[(1.0 / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -6.8e6 or 4.19999999999999979e-29 < 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. Taylor expanded in x around inf 100.0%

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

      1. mul-1-neg 100.0%

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

   6. Simplified 100.0%

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

   if -6.8e6 < x < 4.19999999999999979e-29

   1. Initial program 79.0%

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

      1. exp-prod 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in x around 0 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -6800000 \lor \neg \left(x \leq 4.2 \cdot 10^{-29}\right):\\ \;\;\;\;\frac{e^{-y}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x}\\ \end{array} \]

Alternative 4: 83.3% accurate, 13.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -6800000:\\ \;\;\;\;\frac{1}{x} \cdot \left(0.5 \cdot \left(y \cdot y\right) + \left(1 - y\right)\right)\\ \mathbf{elif}\;x \leq 9.5 \cdot 10^{+62}:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x + x \cdot y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -6800000.0)
   (* (/ 1.0 x) (+ (* 0.5 (* y y)) (- 1.0 y)))
   (if (<= x 9.5e+62) (/ 1.0 x) (/ 1.0 (+ x (* x y))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -6800000.0) {
		tmp = (1.0 / x) * ((0.5 * (y * y)) + (1.0 - y));
	} else if (x <= 9.5e+62) {
		tmp = 1.0 / x;
	} else {
		tmp = 1.0 / (x + (x * y));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-6800000.0d0)) then
        tmp = (1.0d0 / x) * ((0.5d0 * (y * y)) + (1.0d0 - y))
    else if (x <= 9.5d+62) then
        tmp = 1.0d0 / x
    else
        tmp = 1.0d0 / (x + (x * y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -6800000.0) {
		tmp = (1.0 / x) * ((0.5 * (y * y)) + (1.0 - y));
	} else if (x <= 9.5e+62) {
		tmp = 1.0 / x;
	} else {
		tmp = 1.0 / (x + (x * y));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -6800000.0:
		tmp = (1.0 / x) * ((0.5 * (y * y)) + (1.0 - y))
	elif x <= 9.5e+62:
		tmp = 1.0 / x
	else:
		tmp = 1.0 / (x + (x * y))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -6800000.0)
		tmp = Float64(Float64(1.0 / x) * Float64(Float64(0.5 * Float64(y * y)) + Float64(1.0 - y)));
	elseif (x <= 9.5e+62)
		tmp = Float64(1.0 / x);
	else
		tmp = Float64(1.0 / Float64(x + Float64(x * y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -6800000.0)
		tmp = (1.0 / x) * ((0.5 * (y * y)) + (1.0 - y));
	elseif (x <= 9.5e+62)
		tmp = 1.0 / x;
	else
		tmp = 1.0 / (x + (x * y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -6800000.0], N[(N[(1.0 / x), $MachinePrecision] * N[(N[(0.5 * N[(y * y), $MachinePrecision]), $MachinePrecision] + N[(1.0 - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 9.5e+62], N[(1.0 / x), $MachinePrecision], N[(1.0 / N[(x + N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -6.8e6

   1. Initial program 85.0%

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

      1. exp-prod 85.0%

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

   3. Simplified 85.0%

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

   4. Taylor expanded in y around 0 83.3%

      \[\leadsto \frac{\color{blue}{{y}^{2} \cdot \left(0.5 + 0.5 \cdot \frac{1}{x}\right) + \left(1 + -1 \cdot y\right)}}{x} \]

   5. Taylor expanded in x around inf 83.3%

      \[\leadsto \color{blue}{\frac{1 + \left(0.5 \cdot {y}^{2} + -1 \cdot y\right)}{x}} \]
   6. Step-by-step derivation

      1. unpow2 83.3%

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

      2. *-commutative 83.3%

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

      3. associate-*r* 83.3%

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

      4. mul-1-neg 83.3%

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

      5. unsub-neg 83.3%

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

   7. Simplified 83.3%

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

      1. frac-2neg 83.3%

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

      2. div-inv 83.3%

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

      3. +-commutative 83.3%

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

      4. associate-+l- 83.3%

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

      5. associate-*r* 83.3%

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

      6. *-commutative 83.3%

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

      7. neg-mul-1 83.3%

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

      8. associate-/r* 83.3%

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

      9. metadata-eval 83.3%

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

   9. Applied egg-rr 83.3%

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

   if -6.8e6 < x < 9.5000000000000003e62

   1. Initial program 80.0%

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

      1. exp-prod 98.5%

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

   3. Simplified 98.5%

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

   4. Taylor expanded in x around 0 97.1%

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

   if 9.5000000000000003e62 < x

   1. Initial program 62.0%

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

      1. *-commutative 62.0%

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

      2. exp-to-pow 62.0%

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

   3. Simplified 62.0%

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

   4. Taylor expanded in x around inf 100.0%

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

      1. mul-1-neg 100.0%

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

   6. Simplified 100.0%

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

      1. clear-num 100.0%

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

      2. inv-pow 100.0%

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

      3. exp-neg 100.0%

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

      4. associate-/r/ 100.0%

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

      5. /-rgt-identity 100.0%

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

   8. Applied egg-rr 100.0%

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

   9. Taylor expanded in x around 0 100.0%

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

   10. Taylor expanded in y around 0 65.3%

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

4. Final simplification 87.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -6800000:\\ \;\;\;\;\frac{1}{x} \cdot \left(0.5 \cdot \left(y \cdot y\right) + \left(1 - y\right)\right)\\ \mathbf{elif}\;x \leq 9.5 \cdot 10^{+62}:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x + x \cdot y}\\ \end{array} \]

Alternative 5: 83.3% accurate, 16.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -6800000:\\ \;\;\;\;\frac{1 + \left(y \cdot \left(y \cdot 0.5\right) - y\right)}{x}\\ \mathbf{elif}\;x \leq 9.5 \cdot 10^{+62}:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x + x \cdot y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -6800000.0)
   (/ (+ 1.0 (- (* y (* y 0.5)) y)) x)
   (if (<= x 9.5e+62) (/ 1.0 x) (/ 1.0 (+ x (* x y))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -6800000.0) {
		tmp = (1.0 + ((y * (y * 0.5)) - y)) / x;
	} else if (x <= 9.5e+62) {
		tmp = 1.0 / x;
	} else {
		tmp = 1.0 / (x + (x * y));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-6800000.0d0)) then
        tmp = (1.0d0 + ((y * (y * 0.5d0)) - y)) / x
    else if (x <= 9.5d+62) then
        tmp = 1.0d0 / x
    else
        tmp = 1.0d0 / (x + (x * y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -6800000.0) {
		tmp = (1.0 + ((y * (y * 0.5)) - y)) / x;
	} else if (x <= 9.5e+62) {
		tmp = 1.0 / x;
	} else {
		tmp = 1.0 / (x + (x * y));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -6800000.0:
		tmp = (1.0 + ((y * (y * 0.5)) - y)) / x
	elif x <= 9.5e+62:
		tmp = 1.0 / x
	else:
		tmp = 1.0 / (x + (x * y))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -6800000.0)
		tmp = Float64(Float64(1.0 + Float64(Float64(y * Float64(y * 0.5)) - y)) / x);
	elseif (x <= 9.5e+62)
		tmp = Float64(1.0 / x);
	else
		tmp = Float64(1.0 / Float64(x + Float64(x * y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -6800000.0)
		tmp = (1.0 + ((y * (y * 0.5)) - y)) / x;
	elseif (x <= 9.5e+62)
		tmp = 1.0 / x;
	else
		tmp = 1.0 / (x + (x * y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -6800000.0], N[(N[(1.0 + N[(N[(y * N[(y * 0.5), $MachinePrecision]), $MachinePrecision] - y), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[x, 9.5e+62], N[(1.0 / x), $MachinePrecision], N[(1.0 / N[(x + N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -6.8e6

   1. Initial program 85.0%

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

      1. exp-prod 85.0%

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

   3. Simplified 85.0%

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

   4. Taylor expanded in y around 0 83.3%

      \[\leadsto \frac{\color{blue}{{y}^{2} \cdot \left(0.5 + 0.5 \cdot \frac{1}{x}\right) + \left(1 + -1 \cdot y\right)}}{x} \]

   5. Taylor expanded in x around inf 83.3%

      \[\leadsto \color{blue}{\frac{1 + \left(0.5 \cdot {y}^{2} + -1 \cdot y\right)}{x}} \]
   6. Step-by-step derivation

      1. unpow2 83.3%

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

      2. *-commutative 83.3%

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

      3. associate-*r* 83.3%

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

      4. mul-1-neg 83.3%

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

      5. unsub-neg 83.3%

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

   7. Simplified 83.3%

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

   if -6.8e6 < x < 9.5000000000000003e62

   1. Initial program 80.0%

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

      1. exp-prod 98.5%

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

   3. Simplified 98.5%

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

   4. Taylor expanded in x around 0 97.1%

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

   if 9.5000000000000003e62 < x

   1. Initial program 62.0%

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

      1. *-commutative 62.0%

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

      2. exp-to-pow 62.0%

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

   3. Simplified 62.0%

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

   4. Taylor expanded in x around inf 100.0%

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

      1. mul-1-neg 100.0%

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

   6. Simplified 100.0%

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

      1. clear-num 100.0%

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

      2. inv-pow 100.0%

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

      3. exp-neg 100.0%

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

      4. associate-/r/ 100.0%

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

      5. /-rgt-identity 100.0%

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

   8. Applied egg-rr 100.0%

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

   9. Taylor expanded in x around 0 100.0%

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

   10. Taylor expanded in y around 0 65.3%

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

4. Final simplification 87.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -6800000:\\ \;\;\;\;\frac{1 + \left(y \cdot \left(y \cdot 0.5\right) - y\right)}{x}\\ \mathbf{elif}\;x \leq 9.5 \cdot 10^{+62}:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x + x \cdot y}\\ \end{array} \]

Alternative 6: 83.2% accurate, 16.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -14600000:\\ \;\;\;\;\frac{y \cdot \left(y \cdot 0.5\right) + \left(1 - y\right)}{x}\\ \mathbf{elif}\;x \leq 2.2 \cdot 10^{+65}:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x + x \cdot y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -14600000.0)
   (/ (+ (* y (* y 0.5)) (- 1.0 y)) x)
   (if (<= x 2.2e+65) (/ 1.0 x) (/ 1.0 (+ x (* x y))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -14600000.0) {
		tmp = ((y * (y * 0.5)) + (1.0 - y)) / x;
	} else if (x <= 2.2e+65) {
		tmp = 1.0 / x;
	} else {
		tmp = 1.0 / (x + (x * y));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-14600000.0d0)) then
        tmp = ((y * (y * 0.5d0)) + (1.0d0 - y)) / x
    else if (x <= 2.2d+65) then
        tmp = 1.0d0 / x
    else
        tmp = 1.0d0 / (x + (x * y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -14600000.0) {
		tmp = ((y * (y * 0.5)) + (1.0 - y)) / x;
	} else if (x <= 2.2e+65) {
		tmp = 1.0 / x;
	} else {
		tmp = 1.0 / (x + (x * y));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -14600000.0:
		tmp = ((y * (y * 0.5)) + (1.0 - y)) / x
	elif x <= 2.2e+65:
		tmp = 1.0 / x
	else:
		tmp = 1.0 / (x + (x * y))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -14600000.0)
		tmp = Float64(Float64(Float64(y * Float64(y * 0.5)) + Float64(1.0 - y)) / x);
	elseif (x <= 2.2e+65)
		tmp = Float64(1.0 / x);
	else
		tmp = Float64(1.0 / Float64(x + Float64(x * y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -14600000.0)
		tmp = ((y * (y * 0.5)) + (1.0 - y)) / x;
	elseif (x <= 2.2e+65)
		tmp = 1.0 / x;
	else
		tmp = 1.0 / (x + (x * y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -14600000.0], N[(N[(N[(y * N[(y * 0.5), $MachinePrecision]), $MachinePrecision] + N[(1.0 - y), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[x, 2.2e+65], N[(1.0 / x), $MachinePrecision], N[(1.0 / N[(x + N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.46e7

   1. Initial program 85.0%

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

      1. exp-prod 85.0%

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

   3. Simplified 85.0%

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

   4. Taylor expanded in y around 0 83.3%

      \[\leadsto \frac{\color{blue}{{y}^{2} \cdot \left(0.5 + 0.5 \cdot \frac{1}{x}\right) + \left(1 + -1 \cdot y\right)}}{x} \]

   5. Taylor expanded in x around inf 83.3%

      \[\leadsto \frac{\color{blue}{0.5 \cdot {y}^{2}} + \left(1 + -1 \cdot y\right)}{x} \]
   6. Step-by-step derivation

      1. unpow2 83.3%

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

      2. *-commutative 83.3%

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

      3. associate-*r* 83.3%

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

   7. Simplified 83.3%

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

   if -1.46e7 < x < 2.1999999999999998e65

   1. Initial program 80.0%

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

      1. exp-prod 98.5%

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

   3. Simplified 98.5%

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

   4. Taylor expanded in x around 0 97.1%

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

   if 2.1999999999999998e65 < x

   1. Initial program 62.0%

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

      1. *-commutative 62.0%

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

      2. exp-to-pow 62.0%

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

   3. Simplified 62.0%

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

   4. Taylor expanded in x around inf 100.0%

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

      1. mul-1-neg 100.0%

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

   6. Simplified 100.0%

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

      1. clear-num 100.0%

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

      2. inv-pow 100.0%

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

      3. exp-neg 100.0%

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

      4. associate-/r/ 100.0%

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

      5. /-rgt-identity 100.0%

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

   8. Applied egg-rr 100.0%

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

   9. Taylor expanded in x around 0 100.0%

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

   10. Taylor expanded in y around 0 65.3%

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

4. Final simplification 87.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -14600000:\\ \;\;\;\;\frac{y \cdot \left(y \cdot 0.5\right) + \left(1 - y\right)}{x}\\ \mathbf{elif}\;x \leq 2.2 \cdot 10^{+65}:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x + x \cdot y}\\ \end{array} \]

Alternative 7: 81.1% accurate, 18.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -8.2 \cdot 10^{+42} \lor \neg \left(x \leq 9.5 \cdot 10^{+62}\right):\\ \;\;\;\;\frac{1}{x + x \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -8.2e+42) (not (<= x 9.5e+62)))
   (/ 1.0 (+ x (* x y)))
   (/ 1.0 x)))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -8.2e+42) || !(x <= 9.5e+62)) {
		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 <= (-8.2d+42)) .or. (.not. (x <= 9.5d+62))) 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 <= -8.2e+42) || !(x <= 9.5e+62)) {
		tmp = 1.0 / (x + (x * y));
	} else {
		tmp = 1.0 / x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -8.2e+42) or not (x <= 9.5e+62):
		tmp = 1.0 / (x + (x * y))
	else:
		tmp = 1.0 / x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -8.2e+42) || !(x <= 9.5e+62))
		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 <= -8.2e+42) || ~((x <= 9.5e+62)))
		tmp = 1.0 / (x + (x * y));
	else
		tmp = 1.0 / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -8.2e+42], N[Not[LessEqual[x, 9.5e+62]], $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 < -8.2000000000000001e42 or 9.5000000000000003e62 < x

   1. Initial program 74.7%

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

      1. *-commutative 74.7%

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

      2. exp-to-pow 74.7%

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

   3. Simplified 74.7%

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

   4. Taylor expanded in x around inf 99.9%

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

      1. mul-1-neg 99.9%

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

   6. Simplified 99.9%

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

      1. clear-num 100.0%

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

      2. inv-pow 100.0%

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

      3. exp-neg 100.0%

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

      4. associate-/r/ 100.0%

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

      5. /-rgt-identity 100.0%

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

   8. Applied egg-rr 100.0%

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

   9. Taylor expanded in x around 0 100.0%

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

   10. Taylor expanded in y around 0 75.5%

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

   if -8.2000000000000001e42 < x < 9.5000000000000003e62

   1. Initial program 80.7%

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

      1. exp-prod 98.6%

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

   3. Simplified 98.6%

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

   4. Taylor expanded in x around 0 96.5%

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

4. Final simplification 87.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -8.2 \cdot 10^{+42} \lor \neg \left(x \leq 9.5 \cdot 10^{+62}\right):\\ \;\;\;\;\frac{1}{x + x \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x}\\ \end{array} \]

Alternative 8: 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 78.1%

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

   1. exp-prod 88.1%

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

3. Simplified 88.1%

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

4. Taylor expanded in x around 0 82.4%

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

5. Final simplification 82.4%

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

Developer target: 78.0% accurate, 0.7× 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 2023224 
(FPCore (x y)
  :name "Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2, F"
  :precision binary64

  :herbie-target
  (if (< y -3.7311844206647956e+94) (/ (exp (/ -1.0 y)) x) (if (< y 2.817959242728288e+37) (/ (pow (/ x (+ y x)) x) x) (if (< y 2.347387415166998e+178) (log (exp (/ (pow (/ x (+ y x)) x) x))) (/ (exp (/ -1.0 y)) x))))

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