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

Percentage Accurate: 78.1% → 98.8%

Time: 11.2s

Alternatives: 6

Speedup: 13.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 78.1% 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: 98.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{+112} \lor \neg \left(x \leq 3.05 \cdot 10^{-5}\right):\\ \;\;\;\;\frac{e^{-y}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(e^{x}\right)}^{\log \left(\frac{x}{x + y}\right)}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -1e+112) (not (<= x 3.05e-5)))
   (/ (exp (- y)) x)
   (/ (pow (exp x) (log (/ x (+ x y)))) x)))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -1e+112) || !(x <= 3.05e-5)) {
		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 <= (-1d+112)) .or. (.not. (x <= 3.05d-5))) 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 <= -1e+112) || !(x <= 3.05e-5)) {
		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 <= -1e+112) or not (x <= 3.05e-5):
		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 <= -1e+112) || !(x <= 3.05e-5))
		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 <= -1e+112) || ~((x <= 3.05e-5)))
		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, -1e+112], N[Not[LessEqual[x, 3.05e-5]], $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 < -9.9999999999999993e111 or 3.04999999999999994e-5 < 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. Add Preprocessing

   5. Taylor expanded in x around inf 100.0%

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

      1. mul-1-neg 100.0%

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

   7. Simplified 100.0%

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

   if -9.9999999999999993e111 < x < 3.04999999999999994e-5

   1. Initial program 84.0%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{+112} \lor \neg \left(x \leq 3.05 \cdot 10^{-5}\right):\\ \;\;\;\;\frac{e^{-y}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(e^{x}\right)}^{\log \left(\frac{x}{x + y}\right)}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 97.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -7.4 \cdot 10^{+49} \lor \neg \left(x \leq 3.05 \cdot 10^{-5}\right):\\ \;\;\;\;\frac{e^{-y}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -7.4e+49) (not (<= x 3.05e-5))) (/ (exp (- y)) x) (/ 1.0 x)))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -7.4e+49) || !(x <= 3.05e-5)) {
		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 <= (-7.4d+49)) .or. (.not. (x <= 3.05d-5))) 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 <= -7.4e+49) || !(x <= 3.05e-5)) {
		tmp = Math.exp(-y) / x;
	} else {
		tmp = 1.0 / x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -7.4e+49) or not (x <= 3.05e-5):
		tmp = math.exp(-y) / x
	else:
		tmp = 1.0 / x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -7.4e+49) || !(x <= 3.05e-5))
		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 <= -7.4e+49) || ~((x <= 3.05e-5)))
		tmp = exp(-y) / x;
	else
		tmp = 1.0 / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -7.4e+49], N[Not[LessEqual[x, 3.05e-5]], $MachinePrecision]], N[(N[Exp[(-y)], $MachinePrecision] / x), $MachinePrecision], N[(1.0 / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -7.40000000000000036e49 or 3.04999999999999994e-5 < x

   1. Initial program 77.7%

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

      1. *-commutative 77.7%

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

      2. exp-to-pow 77.7%

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

   3. Simplified 77.7%

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

   5. Taylor expanded in x around inf 100.0%

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

      1. mul-1-neg 100.0%

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

   7. Simplified 100.0%

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

   if -7.40000000000000036e49 < x < 3.04999999999999994e-5

   1. Initial program 81.8%

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

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

   5. Taylor expanded in x around 0 99.2%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -7.4 \cdot 10^{+49} \lor \neg \left(x \leq 3.05 \cdot 10^{-5}\right):\\ \;\;\;\;\frac{e^{-y}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 81.2% accurate, 7.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -7.4 \cdot 10^{+49} \lor \neg \left(x \leq 1.2 \cdot 10^{+67}\right):\\ \;\;\;\;\frac{1}{x} + \frac{y \cdot \left(y \cdot \left(0.5 + y \cdot -0.16666666666666666\right) + -1\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -7.4e+49) (not (<= x 1.2e+67)))
   (+ (/ 1.0 x) (/ (* y (+ (* y (+ 0.5 (* y -0.16666666666666666))) -1.0)) x))
   (/ 1.0 x)))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -7.4e+49) || !(x <= 1.2e+67)) {
		tmp = (1.0 / x) + ((y * ((y * (0.5 + (y * -0.16666666666666666))) + -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 <= (-7.4d+49)) .or. (.not. (x <= 1.2d+67))) then
        tmp = (1.0d0 / x) + ((y * ((y * (0.5d0 + (y * (-0.16666666666666666d0)))) + (-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 <= -7.4e+49) || !(x <= 1.2e+67)) {
		tmp = (1.0 / x) + ((y * ((y * (0.5 + (y * -0.16666666666666666))) + -1.0)) / x);
	} else {
		tmp = 1.0 / x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -7.4e+49) or not (x <= 1.2e+67):
		tmp = (1.0 / x) + ((y * ((y * (0.5 + (y * -0.16666666666666666))) + -1.0)) / x)
	else:
		tmp = 1.0 / x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -7.4e+49) || !(x <= 1.2e+67))
		tmp = Float64(Float64(1.0 / x) + Float64(Float64(y * Float64(Float64(y * Float64(0.5 + Float64(y * -0.16666666666666666))) + -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 <= -7.4e+49) || ~((x <= 1.2e+67)))
		tmp = (1.0 / x) + ((y * ((y * (0.5 + (y * -0.16666666666666666))) + -1.0)) / x);
	else
		tmp = 1.0 / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -7.4e+49], N[Not[LessEqual[x, 1.2e+67]], $MachinePrecision]], N[(N[(1.0 / x), $MachinePrecision] + N[(N[(y * N[(N[(y * N[(0.5 + N[(y * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], N[(1.0 / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -7.40000000000000036e49 or 1.20000000000000001e67 < x

   1. Initial program 75.4%

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

      1. *-commutative 75.4%

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

      2. exp-to-pow 75.4%

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

   3. Simplified 75.4%

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

   5. Taylor expanded in x around inf 100.0%

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

      1. mul-1-neg 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in y around 0 75.5%

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

   9. Taylor expanded in x around 0 79.5%

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

   if -7.40000000000000036e49 < x < 1.20000000000000001e67

   1. Initial program 83.2%

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

   5. Taylor expanded in x around 0 93.0%

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

4. Final simplification 86.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -7.4 \cdot 10^{+49} \lor \neg \left(x \leq 1.2 \cdot 10^{+67}\right):\\ \;\;\;\;\frac{1}{x} + \frac{y \cdot \left(y \cdot \left(0.5 + y \cdot -0.16666666666666666\right) + -1\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 78.2% accurate, 9.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -7.4 \cdot 10^{+49}:\\ \;\;\;\;y \cdot \left(\frac{y}{x} \cdot \left(0.5 + y \cdot -0.16666666666666666\right)\right) + \frac{1 - y}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -7.4e+49)
   (+ (* y (* (/ y x) (+ 0.5 (* y -0.16666666666666666)))) (/ (- 1.0 y) x))
   (/ 1.0 x)))

C

double code(double x, double y) {
	double tmp;
	if (x <= -7.4e+49) {
		tmp = (y * ((y / x) * (0.5 + (y * -0.16666666666666666)))) + ((1.0 - 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 <= (-7.4d+49)) then
        tmp = (y * ((y / x) * (0.5d0 + (y * (-0.16666666666666666d0))))) + ((1.0d0 - 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 <= -7.4e+49) {
		tmp = (y * ((y / x) * (0.5 + (y * -0.16666666666666666)))) + ((1.0 - y) / x);
	} else {
		tmp = 1.0 / x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -7.4e+49:
		tmp = (y * ((y / x) * (0.5 + (y * -0.16666666666666666)))) + ((1.0 - y) / x)
	else:
		tmp = 1.0 / x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -7.4e+49)
		tmp = Float64(Float64(y * Float64(Float64(y / x) * Float64(0.5 + Float64(y * -0.16666666666666666)))) + Float64(Float64(1.0 - y) / x));
	else
		tmp = Float64(1.0 / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -7.4e+49)
		tmp = (y * ((y / x) * (0.5 + (y * -0.16666666666666666)))) + ((1.0 - y) / x);
	else
		tmp = 1.0 / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -7.4e+49], N[(N[(y * N[(N[(y / x), $MachinePrecision] * N[(0.5 + N[(y * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 - y), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], N[(1.0 / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -7.40000000000000036e49

   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. Add Preprocessing

   5. Taylor expanded in x around inf 100.0%

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

      1. mul-1-neg 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in y around 0 77.2%

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

   10. Simplified 77.2%

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

   if -7.40000000000000036e49 < x

   1. Initial program 81.1%

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

      1. exp-prod 91.8%

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

   3. Simplified 91.8%

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

   5. Taylor expanded in x around 0 86.4%

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

4. Final simplification 84.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -7.4 \cdot 10^{+49}:\\ \;\;\;\;y \cdot \left(\frac{y}{x} \cdot \left(0.5 + y \cdot -0.16666666666666666\right)\right) + \frac{1 - y}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 78.4% accurate, 13.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -7.4 \cdot 10^{+49}:\\ \;\;\;\;\frac{1 + y \cdot \left(y \cdot 0.5 + -1\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -7.4e+49) (/ (+ 1.0 (* y (+ (* y 0.5) -1.0))) x) (/ 1.0 x)))

C

double code(double x, double y) {
	double tmp;
	if (x <= -7.4e+49) {
		tmp = (1.0 + (y * ((y * 0.5) + -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 <= (-7.4d+49)) then
        tmp = (1.0d0 + (y * ((y * 0.5d0) + (-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 <= -7.4e+49) {
		tmp = (1.0 + (y * ((y * 0.5) + -1.0))) / x;
	} else {
		tmp = 1.0 / x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -7.4e+49:
		tmp = (1.0 + (y * ((y * 0.5) + -1.0))) / x
	else:
		tmp = 1.0 / x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -7.4e+49)
		tmp = Float64(Float64(1.0 + Float64(y * Float64(Float64(y * 0.5) + -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 <= -7.4e+49)
		tmp = (1.0 + (y * ((y * 0.5) + -1.0))) / x;
	else
		tmp = 1.0 / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -7.4e+49], N[(N[(1.0 + N[(y * N[(N[(y * 0.5), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], N[(1.0 / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -7.40000000000000036e49

   1. Initial program 74.7%

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

      1. exp-prod 74.7%

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

   3. Simplified 74.7%

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

   5. Taylor expanded in y around 0 76.8%

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

   6. Taylor expanded in x around inf 76.8%

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

      1. *-commutative 76.8%

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

   8. Simplified 76.8%

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

   if -7.40000000000000036e49 < x

   1. Initial program 81.1%

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

      1. exp-prod 91.8%

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

   3. Simplified 91.8%

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

   5. Taylor expanded in x around 0 86.4%

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

4. Final simplification 84.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -7.4 \cdot 10^{+49}:\\ \;\;\;\;\frac{1 + y \cdot \left(y \cdot 0.5 + -1\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 74.9% 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 79.6%

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

   1. exp-prod 87.7%

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

3. Simplified 87.7%

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

5. Taylor expanded in x around 0 78.7%

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

6. Final simplification 78.7%

   \[\leadsto \frac{1}{x} \]
7. Add Preprocessing

Developer target: 78.4% 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 2024131 
(FPCore (x y)
  :name "Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2, F"
  :precision binary64

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

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