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

Percentage Accurate: 77.0% → 99.6%

Time: 8.1s

Alternatives: 6

Speedup: 16.0×

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: 77.0% 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

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{+66} \lor \neg \left(x \leq 20\right):\\ \;\;\;\;\frac{e^{-y}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(e^{x}\right)}^{\log \left(\frac{x}{x + y}\right)}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -1e+66) (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 <= -1e+66) || !(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 <= (-1d+66)) .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 <= -1e+66) || !(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 <= -1e+66) 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 <= -1e+66) || !(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 <= -1e+66) || ~((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, -1e+66], 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 < -9.99999999999999945e65 or 20 < x

   1. Initial program 74.0%

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

   2. Taylor expanded in x around inf 100.0%

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

      1. mul-1-neg 100.0%

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

   4. Simplified 100.0%

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

   if -9.99999999999999945e65 < x < 20

   1. Initial program 93.7%

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

      1. exp-prod 99.6%

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

   3. Simplified 99.6%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{+66} \lor \neg \left(x \leq 20\right):\\ \;\;\;\;\frac{e^{-y}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(e^{x}\right)}^{\log \left(\frac{x}{x + y}\right)}}{x}\\ \end{array} \]

Alternative 2: 99.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -14800 \lor \neg \left(x \leq 0.26\right):\\ \;\;\;\;\frac{e^{-y}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -14800.0) (not (<= x 0.26))) (/ (exp (- y)) x) (/ 1.0 x)))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -14800.0) || !(x <= 0.26)) {
		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 <= (-14800.0d0)) .or. (.not. (x <= 0.26d0))) 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 <= -14800.0) || !(x <= 0.26)) {
		tmp = Math.exp(-y) / x;
	} else {
		tmp = 1.0 / x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -14800.0) or not (x <= 0.26):
		tmp = math.exp(-y) / x
	else:
		tmp = 1.0 / x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -14800.0) || !(x <= 0.26))
		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 <= -14800.0) || ~((x <= 0.26)))
		tmp = exp(-y) / x;
	else
		tmp = 1.0 / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -14800.0], N[Not[LessEqual[x, 0.26]], $MachinePrecision]], N[(N[Exp[(-y)], $MachinePrecision] / x), $MachinePrecision], N[(1.0 / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -14800 or 0.26000000000000001 < x

   1. Initial program 76.1%

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

   2. Taylor expanded in x around inf 100.0%

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

      1. mul-1-neg 100.0%

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

   4. Simplified 100.0%

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

   if -14800 < x < 0.26000000000000001

   1. Initial program 93.0%

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

   2. Taylor expanded in x around 0 97.8%

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

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -14800 \lor \neg \left(x \leq 0.26\right):\\ \;\;\;\;\frac{e^{-y}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x}\\ \end{array} \]

Alternative 3: 77.9% accurate, 16.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.35 \cdot 10^{+154}:\\ \;\;\;\;\frac{\frac{1 - y \cdot y}{x}}{y + 1}\\ \mathbf{elif}\;y \leq 1.5 \cdot 10^{+54}:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.35e+154)
   (/ (/ (- 1.0 (* y y)) x) (+ y 1.0))
   (if (<= y 1.5e+54) (/ 1.0 x) 0.0)))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.35e+154) {
		tmp = ((1.0 - (y * y)) / x) / (y + 1.0);
	} else if (y <= 1.5e+54) {
		tmp = 1.0 / x;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-1.35d+154)) then
        tmp = ((1.0d0 - (y * y)) / x) / (y + 1.0d0)
    else if (y <= 1.5d+54) then
        tmp = 1.0d0 / x
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -1.35e+154) {
		tmp = ((1.0 - (y * y)) / x) / (y + 1.0);
	} else if (y <= 1.5e+54) {
		tmp = 1.0 / x;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.35e+154:
		tmp = ((1.0 - (y * y)) / x) / (y + 1.0)
	elif y <= 1.5e+54:
		tmp = 1.0 / x
	else:
		tmp = 0.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.35e+154)
		tmp = Float64(Float64(Float64(1.0 - Float64(y * y)) / x) / Float64(y + 1.0));
	elseif (y <= 1.5e+54)
		tmp = Float64(1.0 / x);
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -1.35e+154)
		tmp = ((1.0 - (y * y)) / x) / (y + 1.0);
	elseif (y <= 1.5e+54)
		tmp = 1.0 / x;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.35e+154], N[(N[(N[(1.0 - N[(y * y), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] / N[(y + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.5e+54], N[(1.0 / x), $MachinePrecision], 0.0]]

Derivation

1. Split input into 3 regimes

2. if y < -1.35000000000000003e154

   1. Initial program 61.7%

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

      1. exp-prod 76.4%

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

   3. Simplified 76.4%

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

   4. Taylor expanded in x around inf 4.4%

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

      1. mul-1-neg 4.4%

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

      2. unsub-neg 4.4%

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

   6. Simplified 4.4%

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

      1. clear-num 4.4%

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

      2. associate-/r/ 4.4%

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

   8. Applied egg-rr 4.4%

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

      1. flip-- 77.6%

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

      2. associate-*r/ 77.6%

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

      3. metadata-eval 77.6%

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

   10. Applied egg-rr 77.6%

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

   11. Taylor expanded in x around 0 77.6%

       \[\leadsto \frac{\color{blue}{\frac{1 - {y}^{2}}{x}}}{1 + y} \]
   12. Step-by-step derivation

       1. unpow2 77.6%

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

   13. Simplified 77.6%

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

   if -1.35000000000000003e154 < y < 1.4999999999999999e54

   1. Initial program 90.2%

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

   2. Taylor expanded in x around 0 88.1%

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

   if 1.4999999999999999e54 < y

   1. Initial program 66.2%

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

      1. exp-prod 71.6%

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

   3. Simplified 71.6%

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

   4. Taylor expanded in x around inf 2.2%

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

      1. mul-1-neg 2.2%

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

      2. unsub-neg 2.2%

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

   6. Simplified 2.2%

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

      1. clear-num 2.2%

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

      2. associate-/r/ 2.2%

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

   8. Applied egg-rr 2.2%

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

      1. expm1-log1p-u 1.4%

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

      2. expm1-udef 29.1%

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

      3. log1p-udef 29.1%

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

      4. add-exp-log 29.9%

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

      5. associate-*l/ 29.9%

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

      6. *-un-lft-identity 29.9%

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

   10. Applied egg-rr 29.9%

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

   11. Taylor expanded in x around inf 61.0%

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

4. Final simplification 82.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.35 \cdot 10^{+154}:\\ \;\;\;\;\frac{\frac{1 - y \cdot y}{x}}{y + 1}\\ \mathbf{elif}\;y \leq 1.5 \cdot 10^{+54}:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 4: 76.1% accurate, 17.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -8 \cdot 10^{+206}:\\ \;\;\;\;\frac{\frac{-y}{\frac{x}{y}}}{y + 1}\\ \mathbf{elif}\;y \leq 1.5 \cdot 10^{+54}:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -8e+206)
   (/ (/ (- y) (/ x y)) (+ y 1.0))
   (if (<= y 1.5e+54) (/ 1.0 x) 0.0)))

C

double code(double x, double y) {
	double tmp;
	if (y <= -8e+206) {
		tmp = (-y / (x / y)) / (y + 1.0);
	} else if (y <= 1.5e+54) {
		tmp = 1.0 / x;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-8d+206)) then
        tmp = (-y / (x / y)) / (y + 1.0d0)
    else if (y <= 1.5d+54) then
        tmp = 1.0d0 / x
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -8e+206) {
		tmp = (-y / (x / y)) / (y + 1.0);
	} else if (y <= 1.5e+54) {
		tmp = 1.0 / x;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -8e+206:
		tmp = (-y / (x / y)) / (y + 1.0)
	elif y <= 1.5e+54:
		tmp = 1.0 / x
	else:
		tmp = 0.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -8e+206)
		tmp = Float64(Float64(Float64(-y) / Float64(x / y)) / Float64(y + 1.0));
	elseif (y <= 1.5e+54)
		tmp = Float64(1.0 / x);
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -8e+206)
		tmp = (-y / (x / y)) / (y + 1.0);
	elseif (y <= 1.5e+54)
		tmp = 1.0 / x;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -8e+206], N[(N[((-y) / N[(x / y), $MachinePrecision]), $MachinePrecision] / N[(y + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.5e+54], N[(1.0 / x), $MachinePrecision], 0.0]]

Derivation

1. Split input into 3 regimes

2. if y < -8.0000000000000003e206

   1. Initial program 73.0%

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

      1. exp-prod 85.9%

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

   3. Simplified 85.9%

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

   4. Taylor expanded in x around inf 4.8%

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

      1. mul-1-neg 4.8%

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

      2. unsub-neg 4.8%

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

   6. Simplified 4.8%

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

      1. clear-num 4.8%

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

      2. associate-/r/ 4.8%

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

   8. Applied egg-rr 4.8%

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

      1. flip-- 87.1%

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

      2. associate-*r/ 87.1%

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

      3. metadata-eval 87.1%

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

   10. Applied egg-rr 87.1%

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

   11. Taylor expanded in y around inf 87.1%

       \[\leadsto \frac{\color{blue}{-1 \cdot \frac{{y}^{2}}{x}}}{1 + y} \]
   12. Step-by-step derivation

       1. mul-1-neg 87.1%

          \[\leadsto \frac{\color{blue}{-\frac{{y}^{2}}{x}}}{1 + y} \]

       2. unpow2 87.1%

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

       3. associate-/l* 52.4%

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

   13. Simplified 52.4%

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

   if -8.0000000000000003e206 < y < 1.4999999999999999e54

   1. Initial program 87.8%

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

   2. Taylor expanded in x around 0 85.3%

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

   if 1.4999999999999999e54 < y

   1. Initial program 66.2%

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

      1. exp-prod 71.6%

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

   3. Simplified 71.6%

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

   4. Taylor expanded in x around inf 2.2%

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

      1. mul-1-neg 2.2%

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

      2. unsub-neg 2.2%

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

   6. Simplified 2.2%

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

      1. clear-num 2.2%

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

      2. associate-/r/ 2.2%

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

   8. Applied egg-rr 2.2%

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

      1. expm1-log1p-u 1.4%

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

      2. expm1-udef 29.1%

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

      3. log1p-udef 29.1%

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

      4. add-exp-log 29.9%

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

      5. associate-*l/ 29.9%

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

      6. *-un-lft-identity 29.9%

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

   10. Applied egg-rr 29.9%

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

   11. Taylor expanded in x around inf 61.0%

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

4. Final simplification 79.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -8 \cdot 10^{+206}:\\ \;\;\;\;\frac{\frac{-y}{\frac{x}{y}}}{y + 1}\\ \mathbf{elif}\;y \leq 1.5 \cdot 10^{+54}:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 5: 76.9% accurate, 41.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.5 \cdot 10^{+54}:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= y 1.5e+54) (/ 1.0 x) 0.0))

C

double code(double x, double y) {
	double tmp;
	if (y <= 1.5e+54) {
		tmp = 1.0 / x;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 1.5d+54) then
        tmp = 1.0d0 / x
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 1.5e+54) {
		tmp = 1.0 / x;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 1.5e+54:
		tmp = 1.0 / x
	else:
		tmp = 0.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 1.5e+54)
		tmp = Float64(1.0 / x);
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 1.5e+54)
		tmp = 1.0 / x;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 1.5e+54], N[(1.0 / x), $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if y < 1.4999999999999999e54

   1. Initial program 86.8%

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

   2. Taylor expanded in x around 0 80.7%

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

   if 1.4999999999999999e54 < y

   1. Initial program 66.2%

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

      1. exp-prod 71.6%

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

   3. Simplified 71.6%

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

   4. Taylor expanded in x around inf 2.2%

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

      1. mul-1-neg 2.2%

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

      2. unsub-neg 2.2%

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

   6. Simplified 2.2%

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

      1. clear-num 2.2%

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

      2. associate-/r/ 2.2%

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

   8. Applied egg-rr 2.2%

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

      1. expm1-log1p-u 1.4%

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

      2. expm1-udef 29.1%

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

      3. log1p-udef 29.1%

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

      4. add-exp-log 29.9%

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

      5. associate-*l/ 29.9%

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

      6. *-un-lft-identity 29.9%

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

   10. Applied egg-rr 29.9%

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

   11. Taylor expanded in x around inf 61.0%

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

4. Final simplification 77.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.5 \cdot 10^{+54}:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 6: 15.7% accurate, 209.0× speedup

Math

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

FPCore

(FPCore (x y) :precision binary64 0.0)

C

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

Fortran

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

Java

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

Python

def code(x, y):
	return 0.0

Julia

function code(x, y)
	return 0.0
end

MATLAB

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

Wolfram

code[x_, y_] := 0.0

Derivation

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 86.0%

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

3. Simplified 86.0%

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

4. Taylor expanded in x around inf 59.6%

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

   1. mul-1-neg 59.6%

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

   2. unsub-neg 59.6%

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

6. Simplified 59.6%

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

   1. clear-num 59.6%

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

   2. associate-/r/ 59.6%

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

8. Applied egg-rr 59.6%

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

   1. expm1-log1p-u 43.4%

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

   2. expm1-udef 19.3%

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

   3. log1p-udef 19.3%

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

   4. add-exp-log 35.5%

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

   5. associate-*l/ 35.5%

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

   6. *-un-lft-identity 35.5%

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

10. Applied egg-rr 35.5%

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

11. Taylor expanded in x around inf 14.6%

    \[\leadsto \color{blue}{1} - 1 \]

12. Final simplification 14.6%

    \[\leadsto 0 \]

Developer target: 77.3% 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 2023271 
(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))