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

Percentage Accurate: 77.1% → 97.5%

Time: 15.5s

Alternatives: 10

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: 77.1% 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: 97.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4.7 \cdot 10^{+163} \lor \neg \left(x \leq 4.3 \cdot 10^{-20}\right):\\ \;\;\;\;\frac{e^{-y}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(e^{x}\right)}^{\log \left(\frac{x}{x + y}\right)}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -4.7e+163) (not (<= x 4.3e-20)))
   (/ (exp (- y)) x)
   (/ (pow (exp x) (log (/ x (+ x y)))) x)))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -4.7e+163) || !(x <= 4.3e-20)) {
		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 <= (-4.7d+163)) .or. (.not. (x <= 4.3d-20))) 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 <= -4.7e+163) || !(x <= 4.3e-20)) {
		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 <= -4.7e+163) or not (x <= 4.3e-20):
		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 <= -4.7e+163) || !(x <= 4.3e-20))
		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 <= -4.7e+163) || ~((x <= 4.3e-20)))
		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, -4.7e+163], N[Not[LessEqual[x, 4.3e-20]], $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 < -4.70000000000000019e163 or 4.30000000000000011e-20 < x

   1. Initial program 71.5%

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

      1. *-commutative 71.5%

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

      2. exp-to-pow 71.5%

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

   3. Simplified 71.5%

      \[\leadsto \color{blue}{\frac{{\left(\frac{x}{x + y}\right)}^{x}}{x}} \]
   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 -4.70000000000000019e163 < x < 4.30000000000000011e-20

   1. Initial program 88.1%

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

      1. exp-prod 97.7%

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

   3. Simplified 97.7%

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

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.7 \cdot 10^{+163} \lor \neg \left(x \leq 4.3 \cdot 10^{-20}\right):\\ \;\;\;\;\frac{e^{-y}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(e^{x}\right)}^{\log \left(\frac{x}{x + y}\right)}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 97.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{e^{-y}}{x}\\ \mathbf{if}\;x \leq -5 \cdot 10^{+59}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq -8 \cdot 10^{-68}:\\ \;\;\;\;\frac{{\left(\frac{x}{x + y}\right)}^{x}}{x}\\ \mathbf{elif}\;x \leq 2.1 \cdot 10^{-100}:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (exp (- y)) x)))
   (if (<= x -5e+59)
     t_0
     (if (<= x -8e-68)
       (/ (pow (/ x (+ x y)) x) x)
       (if (<= x 2.1e-100) (/ 1.0 x) t_0)))))

C

double code(double x, double y) {
	double t_0 = exp(-y) / x;
	double tmp;
	if (x <= -5e+59) {
		tmp = t_0;
	} else if (x <= -8e-68) {
		tmp = pow((x / (x + y)), x) / x;
	} else if (x <= 2.1e-100) {
		tmp = 1.0 / x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = exp(-y) / x
    if (x <= (-5d+59)) then
        tmp = t_0
    else if (x <= (-8d-68)) then
        tmp = ((x / (x + y)) ** x) / x
    else if (x <= 2.1d-100) then
        tmp = 1.0d0 / x
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = Math.exp(-y) / x;
	double tmp;
	if (x <= -5e+59) {
		tmp = t_0;
	} else if (x <= -8e-68) {
		tmp = Math.pow((x / (x + y)), x) / x;
	} else if (x <= 2.1e-100) {
		tmp = 1.0 / x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.exp(-y) / x
	tmp = 0
	if x <= -5e+59:
		tmp = t_0
	elif x <= -8e-68:
		tmp = math.pow((x / (x + y)), x) / x
	elif x <= 2.1e-100:
		tmp = 1.0 / x
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(exp(Float64(-y)) / x)
	tmp = 0.0
	if (x <= -5e+59)
		tmp = t_0;
	elseif (x <= -8e-68)
		tmp = Float64((Float64(x / Float64(x + y)) ^ x) / x);
	elseif (x <= 2.1e-100)
		tmp = Float64(1.0 / x);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = exp(-y) / x;
	tmp = 0.0;
	if (x <= -5e+59)
		tmp = t_0;
	elseif (x <= -8e-68)
		tmp = ((x / (x + y)) ^ x) / x;
	elseif (x <= 2.1e-100)
		tmp = 1.0 / x;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Exp[(-y)], $MachinePrecision] / x), $MachinePrecision]}, If[LessEqual[x, -5e+59], t$95$0, If[LessEqual[x, -8e-68], N[(N[Power[N[(x / N[(x + y), $MachinePrecision]), $MachinePrecision], x], $MachinePrecision] / x), $MachinePrecision], If[LessEqual[x, 2.1e-100], N[(1.0 / x), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if x < -4.9999999999999997e59 or 2.10000000000000009e-100 < x

   1. Initial program 74.9%

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

      1. *-commutative 74.9%

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

      2. exp-to-pow 74.9%

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

   3. Simplified 74.9%

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

   5. Taylor expanded in x around inf 96.8%

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

      1. mul-1-neg 96.8%

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

   7. Simplified 96.8%

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

   if -4.9999999999999997e59 < x < -8.00000000000000053e-68

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. exp-to-pow 100.0%

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

   3. Simplified 100.0%

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

   if -8.00000000000000053e-68 < x < 2.10000000000000009e-100

   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 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 100.0%

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

Alternative 3: 97.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5 \lor \neg \left(x \leq 6 \cdot 10^{-96}\right):\\ \;\;\;\;\frac{e^{-y}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -5.0) (not (<= x 6e-96))) (/ (exp (- y)) x) (/ 1.0 x)))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -5.0) || !(x <= 6e-96)) {
		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 <= (-5.0d0)) .or. (.not. (x <= 6d-96))) 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 <= -5.0) || !(x <= 6e-96)) {
		tmp = Math.exp(-y) / x;
	} else {
		tmp = 1.0 / x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -5.0) or not (x <= 6e-96):
		tmp = math.exp(-y) / x
	else:
		tmp = 1.0 / x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -5.0) || !(x <= 6e-96))
		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 <= -5.0) || ~((x <= 6e-96)))
		tmp = exp(-y) / x;
	else
		tmp = 1.0 / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -5.0], N[Not[LessEqual[x, 6e-96]], $MachinePrecision]], N[(N[Exp[(-y)], $MachinePrecision] / x), $MachinePrecision], N[(1.0 / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -5 or 6e-96 < x

   1. Initial program 76.6%

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

      1. *-commutative 76.6%

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

      2. exp-to-pow 76.6%

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

   3. Simplified 76.6%

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

   5. Taylor expanded in x around inf 98.2%

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

      1. mul-1-neg 98.2%

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

   7. Simplified 98.2%

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

   if -5 < x < 6e-96

   1. Initial program 87.5%

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

      1. exp-prod 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 99.3%

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

4. Final simplification 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5 \lor \neg \left(x \leq 6 \cdot 10^{-96}\right):\\ \;\;\;\;\frac{e^{-y}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 79.3% accurate, 10.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1300000000000:\\ \;\;\;\;\frac{1 + y \cdot \left(\frac{y \cdot \left(x \cdot 0.5\right)}{x} + -1\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1300000000000.0)
   (/ (+ 1.0 (* y (+ (/ (* y (* x 0.5)) x) -1.0))) x)
   (/ 1.0 x)))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1300000000000.0) {
		tmp = (1.0 + (y * (((y * (x * 0.5)) / x) + -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 <= (-1300000000000.0d0)) then
        tmp = (1.0d0 + (y * (((y * (x * 0.5d0)) / x) + (-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 <= -1300000000000.0) {
		tmp = (1.0 + (y * (((y * (x * 0.5)) / x) + -1.0))) / x;
	} else {
		tmp = 1.0 / x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -1300000000000.0:
		tmp = (1.0 + (y * (((y * (x * 0.5)) / x) + -1.0))) / x
	else:
		tmp = 1.0 / x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1300000000000.0)
		tmp = Float64(Float64(1.0 + Float64(y * Float64(Float64(Float64(y * Float64(x * 0.5)) / x) + -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 <= -1300000000000.0)
		tmp = (1.0 + (y * (((y * (x * 0.5)) / x) + -1.0))) / x;
	else
		tmp = 1.0 / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1300000000000.0], N[(N[(1.0 + N[(y * N[(N[(N[(y * N[(x * 0.5), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], N[(1.0 / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.3e12

   1. Initial program 70.4%

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

      1. exp-prod 70.4%

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

   3. Simplified 70.4%

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

   5. Taylor expanded in y around 0 67.7%

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

   6. Taylor expanded in x around 0 72.9%

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

      1. associate-*r* 72.9%

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

      2. distribute-rgt-out 72.9%

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

      3. *-commutative 72.9%

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

   8. Simplified 72.9%

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

   9. Taylor expanded in x around inf 72.9%

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

       1. associate-*r* 72.9%

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

       2. *-commutative 72.9%

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

       3. *-commutative 72.9%

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

   11. Simplified 72.9%

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

   if -1.3e12 < x

   1. Initial program 84.7%

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

      1. exp-prod 92.1%

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

   3. Simplified 92.1%

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

   5. Taylor expanded in x around 0 84.4%

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

4. Final simplification 81.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1300000000000:\\ \;\;\;\;\frac{1 + y \cdot \left(\frac{y \cdot \left(x \cdot 0.5\right)}{x} + -1\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 77.8% accurate, 11.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -6.2 \cdot 10^{+43}:\\ \;\;\;\;\frac{1}{x} + \frac{y \cdot \left(y \cdot 0.5 + -1\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -6.2e+43) (+ (/ 1.0 x) (/ (* y (+ (* y 0.5) -1.0)) x)) (/ 1.0 x)))

C

double code(double x, double y) {
	double tmp;
	if (x <= -6.2e+43) {
		tmp = (1.0 / x) + ((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 <= (-6.2d+43)) then
        tmp = (1.0d0 / x) + ((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 <= -6.2e+43) {
		tmp = (1.0 / x) + ((y * ((y * 0.5) + -1.0)) / x);
	} else {
		tmp = 1.0 / x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -6.2e+43:
		tmp = (1.0 / x) + ((y * ((y * 0.5) + -1.0)) / x)
	else:
		tmp = 1.0 / x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -6.2e+43)
		tmp = Float64(Float64(1.0 / x) + Float64(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 <= -6.2e+43)
		tmp = (1.0 / x) + ((y * ((y * 0.5) + -1.0)) / x);
	else
		tmp = 1.0 / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -6.2e+43], N[(N[(1.0 / x), $MachinePrecision] + N[(N[(y * N[(N[(y * 0.5), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], N[(1.0 / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -6.2000000000000003e43

   1. Initial program 68.2%

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

      1. *-commutative 68.2%

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

      2. exp-to-pow 68.2%

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

   3. Simplified 68.2%

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

   5. Taylor expanded in x around inf 100.0%

      \[\leadsto \frac{\color{blue}{e^{-1 \cdot y}}}{x} \]
   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 58.8%

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

   9. Taylor expanded in x around 0 66.8%

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

   if -6.2000000000000003e43 < x

   1. Initial program 85.1%

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

      1. exp-prod 92.3%

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

   3. Simplified 92.3%

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

   5. Taylor expanded in x around 0 83.8%

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

4. Final simplification 79.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -6.2 \cdot 10^{+43}:\\ \;\;\;\;\frac{1}{x} + \frac{y \cdot \left(y \cdot 0.5 + -1\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 77.8% accurate, 13.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -6.2 \cdot 10^{+43}:\\ \;\;\;\;\frac{1 + y \cdot \left(y \cdot 0.5 + -1\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -6.2e+43) (/ (+ 1.0 (* y (+ (* y 0.5) -1.0))) x) (/ 1.0 x)))

C

double code(double x, double y) {
	double tmp;
	if (x <= -6.2e+43) {
		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 <= (-6.2d+43)) 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 <= -6.2e+43) {
		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 <= -6.2e+43:
		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 <= -6.2e+43)
		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 <= -6.2e+43)
		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, -6.2e+43], 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 < -6.2000000000000003e43

   1. Initial program 68.2%

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

      1. exp-prod 68.2%

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

   3. Simplified 68.2%

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

   5. Taylor expanded in y around 0 66.8%

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

   6. Taylor expanded in x around inf 66.8%

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

      1. *-commutative 66.8%

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

   8. Simplified 66.8%

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

   if -6.2000000000000003e43 < x

   1. Initial program 85.1%

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

      1. exp-prod 92.3%

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

   3. Simplified 92.3%

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

   5. Taylor expanded in x around 0 83.8%

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

4. Final simplification 79.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -6.2 \cdot 10^{+43}:\\ \;\;\;\;\frac{1 + y \cdot \left(y \cdot 0.5 + -1\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 76.8% accurate, 20.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 108:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= y 108.0) (/ 1.0 x) (/ x (* x x))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 108.0) {
		tmp = 1.0 / x;
	} else {
		tmp = x / (x * 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 <= 108.0d0) then
        tmp = 1.0d0 / x
    else
        tmp = x / (x * x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 108.0) {
		tmp = 1.0 / x;
	} else {
		tmp = x / (x * x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 108.0:
		tmp = 1.0 / x
	else:
		tmp = x / (x * x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 108.0)
		tmp = Float64(1.0 / x);
	else
		tmp = Float64(x / Float64(x * x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 108.0)
		tmp = 1.0 / x;
	else
		tmp = x / (x * x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 108.0], N[(1.0 / x), $MachinePrecision], N[(x / N[(x * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 108

   1. Initial program 88.1%

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

      1. exp-prod 90.4%

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

   3. Simplified 90.4%

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

   5. Taylor expanded in x around 0 83.4%

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

   if 108 < y

   1. Initial program 50.8%

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

      1. *-commutative 50.8%

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

      2. exp-to-pow 50.8%

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

   3. Simplified 50.8%

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

   5. Taylor expanded in x around inf 58.6%

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

      1. mul-1-neg 58.6%

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

   7. Simplified 58.6%

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

   8. Taylor expanded in y around 0 2.3%

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

      1. neg-mul-1 2.3%

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

      2. +-commutative 2.3%

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

      3. sub-neg 2.3%

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

   10. Simplified 2.3%

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

       1. frac-2neg 2.3%

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

       2. frac-sub 10.5%

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

       3. metadata-eval 10.5%

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

   12. Applied egg-rr 10.5%

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

   13. Taylor expanded in y around 0 54.7%

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

       1. mul-1-neg 54.7%

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

   15. Simplified 54.7%

       \[\leadsto \frac{\color{blue}{-x}}{\left(-x\right) \cdot x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 77.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 108:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x \cdot x}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 73.9% 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 80.5%

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

   1. exp-prod 85.8%

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

3. Simplified 85.8%

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

5. Taylor expanded in x around 0 75.7%

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

Alternative 9: 4.2% accurate, 209.0× speedup

Math

\[\begin{array}{l} \\ y \end{array} \]

FPCore

(FPCore (x y) :precision binary64 y)

C

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

Fortran

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

Java

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

Python

def code(x, y):
	return y

Julia

function code(x, y)
	return y
end

MATLAB

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

Wolfram

code[x_, y_] := y

Derivation

1. Initial program 80.5%

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

   1. *-commutative 80.5%

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

   2. exp-to-pow 80.5%

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

3. Simplified 80.5%

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

5. Taylor expanded in x around inf 85.9%

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

   1. mul-1-neg 85.9%

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

7. Simplified 85.9%

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

8. Taylor expanded in y around 0 61.7%

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

   1. neg-mul-1 61.7%

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

   2. +-commutative 61.7%

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

   3. sub-neg 61.7%

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

10. Simplified 61.7%

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

    1. frac-2neg 61.7%

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

    2. frac-sub 39.2%

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

    3. metadata-eval 39.2%

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

12. Applied egg-rr 39.2%

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

13. Taylor expanded in y around inf 6.5%

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

    1. *-commutative 6.5%

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

15. Simplified 6.5%

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

17. Applied egg-rr 2.2%

    \[\leadsto \color{blue}{\sqrt{y} \cdot \sqrt{y}} \]
18. Step-by-step derivation

    1. rem-square-sqrt 4.2%

       \[\leadsto \color{blue}{y} \]

19. Simplified 4.2%

    \[\leadsto \color{blue}{y} \]
20. Add Preprocessing

Alternative 10: 3.3% accurate, 209.0× speedup

Math

\[\begin{array}{l} \\ -1 \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(x, y):
	return -1.0

Julia

function code(x, y)
	return -1.0
end

MATLAB

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

Wolfram

code[x_, y_] := -1.0

Derivation

1. Initial program 80.5%

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

   1. *-commutative 80.5%

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

   2. exp-to-pow 80.5%

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

3. Simplified 80.5%

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

5. Taylor expanded in x around inf 85.9%

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

   1. mul-1-neg 85.9%

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

7. Simplified 85.9%

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

8. Taylor expanded in y around 0 61.7%

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

   1. neg-mul-1 61.7%

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

   2. +-commutative 61.7%

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

   3. sub-neg 61.7%

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

10. Simplified 61.7%

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

    1. frac-2neg 61.7%

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

    2. frac-sub 39.2%

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

    3. metadata-eval 39.2%

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

12. Applied egg-rr 39.2%

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

13. Taylor expanded in y around 0 49.6%

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

    1. mul-1-neg 49.6%

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

15. Simplified 49.6%

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

17. Applied egg-rr 3.5%

    \[\leadsto \color{blue}{-1} \]
18. Add Preprocessing

Developer target: 76.2% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024107 
(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))