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

Percentage Accurate: 77.8% → 99.6%

Time: 8.3s

Alternatives: 7

Speedup: 41.3×

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: 77.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{+81} \lor \neg \left(x \leq 20\right):\\ \;\;\;\;\frac{e^{-y}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(e^{x}\right)}^{\log \left(\frac{x}{x + y}\right)}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -5e+81) (not (<= x 20.0)))
   (/ (exp (- y)) x)
   (/ (pow (exp x) (log (/ x (+ x y)))) x)))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -5e+81) || !(x <= 20.0)) {
		tmp = exp(-y) / x;
	} else {
		tmp = pow(exp(x), log((x / (x + y)))) / x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((x <= (-5d+81)) .or. (.not. (x <= 20.0d0))) then
        tmp = exp(-y) / x
    else
        tmp = (exp(x) ** log((x / (x + y)))) / x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((x <= -5e+81) || !(x <= 20.0)) {
		tmp = Math.exp(-y) / x;
	} else {
		tmp = Math.pow(Math.exp(x), Math.log((x / (x + y)))) / x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -5e+81) or not (x <= 20.0):
		tmp = math.exp(-y) / x
	else:
		tmp = math.pow(math.exp(x), math.log((x / (x + y)))) / x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -5e+81) || !(x <= 20.0))
		tmp = Float64(exp(Float64(-y)) / x);
	else
		tmp = Float64((exp(x) ^ log(Float64(x / Float64(x + y)))) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((x <= -5e+81) || ~((x <= 20.0)))
		tmp = exp(-y) / x;
	else
		tmp = (exp(x) ^ log((x / (x + y)))) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -5e+81], N[Not[LessEqual[x, 20.0]], $MachinePrecision]], N[(N[Exp[(-y)], $MachinePrecision] / x), $MachinePrecision], N[(N[Power[N[Exp[x], $MachinePrecision], N[Log[N[(x / N[(x + y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -4.9999999999999998e81 or 20 < x

   1. Initial program 74.8%

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

      1. *-commutative 74.8%

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

      2. exp-to-pow 74.8%

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

   3. Simplified 74.8%

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

   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 -4.9999999999999998e81 < x < 20

   1. Initial program 82.5%

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

      1. exp-prod 99.5%

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

   3. Simplified 99.5%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{+81} \lor \neg \left(x \leq 20\right):\\ \;\;\;\;\frac{e^{-y}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(e^{x}\right)}^{\log \left(\frac{x}{x + y}\right)}}{x}\\ \end{array} \]

Alternative 2: 98.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.9 \cdot 10^{+24} \lor \neg \left(x \leq 0.43\right):\\ \;\;\;\;\frac{e^{-y}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -2.9e+24) (not (<= x 0.43))) (/ (exp (- y)) x) (/ 1.0 x)))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -2.9e+24) || !(x <= 0.43)) {
		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 <= (-2.9d+24)) .or. (.not. (x <= 0.43d0))) 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 <= -2.9e+24) || !(x <= 0.43)) {
		tmp = Math.exp(-y) / x;
	} else {
		tmp = 1.0 / x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -2.9e+24) or not (x <= 0.43):
		tmp = math.exp(-y) / x
	else:
		tmp = 1.0 / x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -2.9e+24) || !(x <= 0.43))
		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 <= -2.9e+24) || ~((x <= 0.43)))
		tmp = exp(-y) / x;
	else
		tmp = 1.0 / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -2.9e+24], N[Not[LessEqual[x, 0.43]], $MachinePrecision]], N[(N[Exp[(-y)], $MachinePrecision] / x), $MachinePrecision], N[(1.0 / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2.89999999999999979e24 or 0.429999999999999993 < x

   1. Initial program 76.5%

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

      1. *-commutative 76.5%

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

      2. exp-to-pow 76.5%

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

   3. Simplified 76.5%

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

   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 -2.89999999999999979e24 < x < 0.429999999999999993

   1. Initial program 81.0%

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

      1. exp-prod 99.4%

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

   3. Simplified 99.4%

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

   4. Taylor expanded in x around 0 97.8%

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

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.9 \cdot 10^{+24} \lor \neg \left(x \leq 0.43\right):\\ \;\;\;\;\frac{e^{-y}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x}\\ \end{array} \]

Alternative 3: 83.8% accurate, 13.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\frac{x - x \cdot y}{x}}{x}\\ \mathbf{if}\;x \leq -2.9 \cdot 10^{+24}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 0.47:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{elif}\;x \leq 9.5 \cdot 10^{+224}:\\ \;\;\;\;\frac{1}{x + x \cdot y}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (/ (- x (* x y)) x) x)))
   (if (<= x -2.9e+24)
     t_0
     (if (<= x 0.47)
       (/ 1.0 x)
       (if (<= x 9.5e+224) (/ 1.0 (+ x (* x y))) t_0)))))

C

double code(double x, double y) {
	double t_0 = ((x - (x * y)) / x) / x;
	double tmp;
	if (x <= -2.9e+24) {
		tmp = t_0;
	} else if (x <= 0.47) {
		tmp = 1.0 / x;
	} else if (x <= 9.5e+224) {
		tmp = 1.0 / (x + (x * y));
	} 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 = ((x - (x * y)) / x) / x
    if (x <= (-2.9d+24)) then
        tmp = t_0
    else if (x <= 0.47d0) then
        tmp = 1.0d0 / x
    else if (x <= 9.5d+224) then
        tmp = 1.0d0 / (x + (x * y))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = ((x - (x * y)) / x) / x;
	double tmp;
	if (x <= -2.9e+24) {
		tmp = t_0;
	} else if (x <= 0.47) {
		tmp = 1.0 / x;
	} else if (x <= 9.5e+224) {
		tmp = 1.0 / (x + (x * y));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = ((x - (x * y)) / x) / x
	tmp = 0
	if x <= -2.9e+24:
		tmp = t_0
	elif x <= 0.47:
		tmp = 1.0 / x
	elif x <= 9.5e+224:
		tmp = 1.0 / (x + (x * y))
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(Float64(x - Float64(x * y)) / x) / x)
	tmp = 0.0
	if (x <= -2.9e+24)
		tmp = t_0;
	elseif (x <= 0.47)
		tmp = Float64(1.0 / x);
	elseif (x <= 9.5e+224)
		tmp = Float64(1.0 / Float64(x + Float64(x * y)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = ((x - (x * y)) / x) / x;
	tmp = 0.0;
	if (x <= -2.9e+24)
		tmp = t_0;
	elseif (x <= 0.47)
		tmp = 1.0 / x;
	elseif (x <= 9.5e+224)
		tmp = 1.0 / (x + (x * y));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[(x - N[(x * y), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] / x), $MachinePrecision]}, If[LessEqual[x, -2.9e+24], t$95$0, If[LessEqual[x, 0.47], N[(1.0 / x), $MachinePrecision], If[LessEqual[x, 9.5e+224], N[(1.0 / N[(x + N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.89999999999999979e24 or 9.5000000000000002e224 < x

   1. Initial program 73.3%

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

      1. exp-prod 73.3%

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

   3. Simplified 73.3%

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

   4. Taylor expanded in x around inf 63.5%

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

      1. mul-1-neg 63.5%

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

      2. unsub-neg 63.5%

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

   6. Simplified 63.5%

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

      1. frac-sub 34.1%

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

      2. associate-/r* 82.1%

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

      3. *-un-lft-identity 82.1%

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

      4. *-commutative 82.1%

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

   8. Applied egg-rr 82.1%

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

   if -2.89999999999999979e24 < x < 0.46999999999999997

   1. Initial program 81.0%

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

      1. exp-prod 99.4%

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

   3. Simplified 99.4%

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

   4. Taylor expanded in x around 0 97.8%

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

   if 0.46999999999999997 < x < 9.5000000000000002e224

   1. Initial program 81.4%

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

      1. exp-prod 81.4%

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

   3. Simplified 81.4%

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

      1. clear-num 81.4%

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

      2. inv-pow 81.4%

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

      3. add-exp-log 81.4%

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

      4. log-pow 31.0%

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

      5. add-log-exp 81.4%

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

      6. pow-to-exp 81.4%

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

   5. Applied egg-rr 81.4%

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

      1. unpow-1 81.4%

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

      2. +-commutative 81.4%

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

   7. Simplified 81.4%

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

   8. Taylor expanded in y around 0 68.1%

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

4. Final simplification 85.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.9 \cdot 10^{+24}:\\ \;\;\;\;\frac{\frac{x - x \cdot y}{x}}{x}\\ \mathbf{elif}\;x \leq 0.47:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{elif}\;x \leq 9.5 \cdot 10^{+224}:\\ \;\;\;\;\frac{1}{x + x \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x - x \cdot y}{x}}{x}\\ \end{array} \]

Alternative 4: 80.9% accurate, 18.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4.4 \cdot 10^{+117} \lor \neg \left(x \leq 0.16\right):\\ \;\;\;\;\frac{1}{x + x \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -4.4e+117) (not (<= x 0.16))) (/ 1.0 (+ x (* x y))) (/ 1.0 x)))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -4.4e+117) || !(x <= 0.16)) {
		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 <= (-4.4d+117)) .or. (.not. (x <= 0.16d0))) 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 <= -4.4e+117) || !(x <= 0.16)) {
		tmp = 1.0 / (x + (x * y));
	} else {
		tmp = 1.0 / x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -4.4e+117) or not (x <= 0.16):
		tmp = 1.0 / (x + (x * y))
	else:
		tmp = 1.0 / x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -4.4e+117) || !(x <= 0.16))
		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 <= -4.4e+117) || ~((x <= 0.16)))
		tmp = 1.0 / (x + (x * y));
	else
		tmp = 1.0 / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -4.4e+117], N[Not[LessEqual[x, 0.16]], $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 < -4.40000000000000028e117 or 0.160000000000000003 < x

   1. Initial program 74.0%

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

      1. exp-prod 74.0%

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

   3. Simplified 74.0%

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

      1. clear-num 74.0%

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

      2. inv-pow 74.0%

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

      3. add-exp-log 74.0%

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

      4. log-pow 18.1%

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

      5. add-log-exp 74.0%

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

      6. pow-to-exp 74.0%

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

   5. Applied egg-rr 74.0%

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

      1. unpow-1 74.0%

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

      2. +-commutative 74.0%

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

   7. Simplified 74.0%

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

   8. Taylor expanded in y around 0 68.6%

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

   if -4.40000000000000028e117 < x < 0.160000000000000003

   1. Initial program 82.9%

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

      1. exp-prod 98.8%

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

   3. Simplified 98.8%

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

   4. Taylor expanded in x around 0 94.3%

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

4. Final simplification 81.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.4 \cdot 10^{+117} \lor \neg \left(x \leq 0.16\right):\\ \;\;\;\;\frac{1}{x + x \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x}\\ \end{array} \]

Alternative 5: 24.7% accurate, 41.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 3300:\\ \;\;\;\;\frac{0.5}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= y 3300.0) (/ 0.5 x) (/ 0.0 x)))

C

double code(double x, double y) {
	double tmp;
	if (y <= 3300.0) {
		tmp = 0.5 / x;
	} else {
		tmp = 0.0 / x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 3300.0d0) then
        tmp = 0.5d0 / x
    else
        tmp = 0.0d0 / x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 3300.0) {
		tmp = 0.5 / x;
	} else {
		tmp = 0.0 / x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 3300.0:
		tmp = 0.5 / x
	else:
		tmp = 0.0 / x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 3300.0)
		tmp = Float64(0.5 / x);
	else
		tmp = Float64(0.0 / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 3300.0)
		tmp = 0.5 / x;
	else
		tmp = 0.0 / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 3300.0], N[(0.5 / x), $MachinePrecision], N[(0.0 / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 3300

   1. Initial program 85.7%

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

      1. exp-prod 90.7%

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

   3. Simplified 90.7%

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

   4. Taylor expanded in x around inf 72.6%

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

      1. mul-1-neg 72.6%

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

      2. unsub-neg 72.6%

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

   6. Simplified 72.6%

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

      1. frac-sub 40.1%

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

      2. associate-/r* 81.3%

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

      3. *-un-lft-identity 81.3%

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

      4. *-commutative 81.3%

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

   8. Applied egg-rr 81.3%

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

   10. Applied egg-rr 16.5%

       \[\leadsto \frac{\color{blue}{0.5}}{x} \]

   if 3300 < y

   1. Initial program 52.6%

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

      1. exp-prod 71.5%

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

   3. Simplified 71.5%

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

   4. Taylor expanded in x around inf 2.3%

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

      1. mul-1-neg 2.3%

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

      2. unsub-neg 2.3%

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

   6. Simplified 2.3%

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

      1. frac-sub 6.4%

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

      2. associate-/r* 1.7%

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

      3. *-un-lft-identity 1.7%

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

      4. *-commutative 1.7%

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

   8. Applied egg-rr 1.7%

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

   10. Applied egg-rr 59.8%

       \[\leadsto \frac{\color{blue}{0}}{x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 26.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 3300:\\ \;\;\;\;\frac{0.5}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{x}\\ \end{array} \]

Alternative 6: 78.0% accurate, 41.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 9 \cdot 10^{+21}:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= y 9e+21) (/ 1.0 x) (/ 0.0 x)))

C

double code(double x, double y) {
	double tmp;
	if (y <= 9e+21) {
		tmp = 1.0 / x;
	} else {
		tmp = 0.0 / x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 9d+21) then
        tmp = 1.0d0 / x
    else
        tmp = 0.0d0 / x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 9e+21) {
		tmp = 1.0 / x;
	} else {
		tmp = 0.0 / x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 9e+21:
		tmp = 1.0 / x
	else:
		tmp = 0.0 / x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 9e+21)
		tmp = Float64(1.0 / x);
	else
		tmp = Float64(0.0 / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 9e+21)
		tmp = 1.0 / x;
	else
		tmp = 0.0 / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 9e+21], N[(1.0 / x), $MachinePrecision], N[(0.0 / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 9e21

   1. Initial program 84.9%

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

      1. exp-prod 90.3%

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

   3. Simplified 90.3%

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

   4. Taylor expanded in x around 0 84.9%

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

   if 9e21 < y

   1. Initial program 54.3%

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

      1. exp-prod 72.1%

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

   3. Simplified 72.1%

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

   4. Taylor expanded in x around inf 2.2%

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

      1. mul-1-neg 2.2%

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

      2. unsub-neg 2.2%

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

   6. Simplified 2.2%

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

      1. frac-sub 4.7%

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

      2. associate-/r* 1.6%

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

      3. *-un-lft-identity 1.6%

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

      4. *-commutative 1.6%

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

   8. Applied egg-rr 1.6%

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

   10. Applied egg-rr 60.1%

       \[\leadsto \frac{\color{blue}{0}}{x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 79.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 9 \cdot 10^{+21}:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{x}\\ \end{array} \]

Alternative 7: 14.8% accurate, 69.7× speedup

Math

\[\begin{array}{l} \\ \frac{0}{x} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (/ 0.0 x))

C

double code(double x, double y) {
	return 0.0 / x;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = 0.0d0 / x
end function

Java

public static double code(double x, double y) {
	return 0.0 / x;
}

Python

def code(x, y):
	return 0.0 / x

Julia

function code(x, y)
	return Float64(0.0 / x)
end

MATLAB

function tmp = code(x, y)
	tmp = 0.0 / x;
end

Wolfram

code[x_, y_] := N[(0.0 / x), $MachinePrecision]

Derivation

1. Initial program 78.5%

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

   1. exp-prod 86.5%

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

3. Simplified 86.5%

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

4. Taylor expanded in x around inf 57.2%

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

   1. mul-1-neg 57.2%

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

   2. unsub-neg 57.2%

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

6. Simplified 57.2%

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

   1. frac-sub 32.7%

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

   2. associate-/r* 63.9%

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

   3. *-un-lft-identity 63.9%

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

   4. *-commutative 63.9%

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

8. Applied egg-rr 63.9%

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

10. Applied egg-rr 15.9%

    \[\leadsto \frac{\color{blue}{0}}{x} \]

11. Final simplification 15.9%

    \[\leadsto \frac{0}{x} \]

Developer target: 77.8% 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 2023240 
(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))