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

Percentage Accurate: 77.4% → 99.2%

Time: 7.5s

Alternatives: 5

Speedup: 18.8×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 77.4% 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: 99.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -500000 \lor \neg \left(x \leq 3.4 \cdot 10^{-25}\right):\\ \;\;\;\;\frac{e^{-y}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(e^{x}\right)}^{\log \left(\frac{x}{x + y}\right)}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -500000.0) (not (<= x 3.4e-25)))
   (/ (exp (- y)) x)
   (/ (pow (exp x) (log (/ x (+ x y)))) x)))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -500000.0) || !(x <= 3.4e-25)) {
		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 <= (-500000.0d0)) .or. (.not. (x <= 3.4d-25))) 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 <= -500000.0) || !(x <= 3.4e-25)) {
		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 <= -500000.0) or not (x <= 3.4e-25):
		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 <= -500000.0) || !(x <= 3.4e-25))
		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 <= -500000.0) || ~((x <= 3.4e-25)))
		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, -500000.0], N[Not[LessEqual[x, 3.4e-25]], $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 < -5e5 or 3.40000000000000002e-25 < x

   1. Initial program 76.9%

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

      1. *-commutative 76.9%

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

      2. exp-to-pow 76.9%

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

   3. Simplified 76.9%

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

   4. Taylor expanded in x around inf 99.9%

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

      1. mul-1-neg 99.9%

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

   6. Simplified 99.9%

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

   if -5e5 < x < 3.40000000000000002e-25

   1. Initial program 74.8%

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

      1. exp-prod 99.9%

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

   3. Simplified 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -500000 \lor \neg \left(x \leq 3.4 \cdot 10^{-25}\right):\\ \;\;\;\;\frac{e^{-y}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(e^{x}\right)}^{\log \left(\frac{x}{x + y}\right)}}{x}\\ \end{array} \]

Alternative 2: 99.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.76 \lor \neg \left(x \leq 3.4 \cdot 10^{-25}\right):\\ \;\;\;\;\frac{e^{-y}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -0.76) (not (<= x 3.4e-25))) (/ (exp (- y)) x) (/ 1.0 x)))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -0.76) || !(x <= 3.4e-25)) {
		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 <= (-0.76d0)) .or. (.not. (x <= 3.4d-25))) 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 <= -0.76) || !(x <= 3.4e-25)) {
		tmp = Math.exp(-y) / x;
	} else {
		tmp = 1.0 / x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -0.76) or not (x <= 3.4e-25):
		tmp = math.exp(-y) / x
	else:
		tmp = 1.0 / x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -0.76) || !(x <= 3.4e-25))
		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 <= -0.76) || ~((x <= 3.4e-25)))
		tmp = exp(-y) / x;
	else
		tmp = 1.0 / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -0.76], N[Not[LessEqual[x, 3.4e-25]], $MachinePrecision]], N[(N[Exp[(-y)], $MachinePrecision] / x), $MachinePrecision], N[(1.0 / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -0.76000000000000001 or 3.40000000000000002e-25 < x

   1. Initial program 76.9%

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

      1. *-commutative 76.9%

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

      2. exp-to-pow 76.9%

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

   3. Simplified 76.9%

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

   4. Taylor expanded in x around inf 99.9%

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

      1. mul-1-neg 99.9%

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

   6. Simplified 99.9%

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

   if -0.76000000000000001 < x < 3.40000000000000002e-25

   1. Initial program 74.8%

      \[\frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x} \]
   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. Taylor expanded in x around 0 99.3%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.76 \lor \neg \left(x \leq 3.4 \cdot 10^{-25}\right):\\ \;\;\;\;\frac{e^{-y}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x}\\ \end{array} \]

Alternative 3: 79.8% accurate, 16.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5.5 \cdot 10^{+19}:\\ \;\;\;\;\frac{-1 - y \cdot y}{x \cdot y - x}\\ \mathbf{elif}\;x \leq 3.4 \cdot 10^{-25}:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x + x \cdot y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -5.5e+19)
   (/ (- -1.0 (* y y)) (- (* x y) x))
   (if (<= x 3.4e-25) (/ 1.0 x) (/ 1.0 (+ x (* x y))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -5.5e+19) {
		tmp = (-1.0 - (y * y)) / ((x * y) - x);
	} else if (x <= 3.4e-25) {
		tmp = 1.0 / x;
	} else {
		tmp = 1.0 / (x + (x * y));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-5.5d+19)) then
        tmp = ((-1.0d0) - (y * y)) / ((x * y) - x)
    else if (x <= 3.4d-25) then
        tmp = 1.0d0 / x
    else
        tmp = 1.0d0 / (x + (x * y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -5.5e+19) {
		tmp = (-1.0 - (y * y)) / ((x * y) - x);
	} else if (x <= 3.4e-25) {
		tmp = 1.0 / x;
	} else {
		tmp = 1.0 / (x + (x * y));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -5.5e+19:
		tmp = (-1.0 - (y * y)) / ((x * y) - x)
	elif x <= 3.4e-25:
		tmp = 1.0 / x
	else:
		tmp = 1.0 / (x + (x * y))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -5.5e+19)
		tmp = Float64(Float64(-1.0 - Float64(y * y)) / Float64(Float64(x * y) - x));
	elseif (x <= 3.4e-25)
		tmp = Float64(1.0 / x);
	else
		tmp = Float64(1.0 / Float64(x + Float64(x * y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -5.5e+19)
		tmp = (-1.0 - (y * y)) / ((x * y) - x);
	elseif (x <= 3.4e-25)
		tmp = 1.0 / x;
	else
		tmp = 1.0 / (x + (x * y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -5.5e+19], N[(N[(-1.0 - N[(y * y), $MachinePrecision]), $MachinePrecision] / N[(N[(x * y), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 3.4e-25], N[(1.0 / x), $MachinePrecision], N[(1.0 / N[(x + N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -5.5e19

   1. Initial program 82.4%

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

      1. *-commutative 82.4%

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

      2. exp-to-pow 82.4%

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

   3. Simplified 82.4%

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

   4. Taylor expanded in x around inf 100.0%

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

      1. mul-1-neg 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in y around 0 64.2%

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

      1. neg-mul-1 64.2%

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

      2. sub-neg 64.2%

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

   9. Simplified 64.2%

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

       1. frac-2neg 64.2%

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

       2. metadata-eval 64.2%

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

       3. frac-2neg 64.2%

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

       4. sub-div 64.2%

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

       5. add-sqr-sqrt 30.7%

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

       6. sqrt-unprod 74.6%

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

       7. sqr-neg 74.6%

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

       8. sqrt-prod 33.5%

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

       9. add-sqr-sqrt 63.4%

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

       10. sub-neg 63.4%

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

       11. add-sqr-sqrt 29.9%

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

       12. sqrt-unprod 63.4%

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

       13. sqr-neg 63.4%

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

       14. sqrt-prod 33.5%

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

       15. add-sqr-sqrt 64.2%

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

       16. *-un-lft-identity 64.2%

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

       17. *-un-lft-identity 64.2%

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

       18. un-div-inv 64.2%

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

       19. flip-+ 74.6%

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

       20. frac-2neg 74.6%

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

       21. metadata-eval 74.6%

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

       22. frac-times 75.7%

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

   11. Applied egg-rr 75.7%

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

       1. *-commutative 75.7%

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

       2. *-commutative 75.7%

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

       3. distribute-lft-in 75.7%

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

       4. metadata-eval 75.7%

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

       5. mul-1-neg 75.7%

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

       6. unsub-neg 75.7%

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

       7. distribute-rgt-in 75.7%

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

       8. neg-mul-1 75.7%

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

       9. unsub-neg 75.7%

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

   13. Simplified 75.7%

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

   if -5.5e19 < x < 3.40000000000000002e-25

   1. Initial program 75.1%

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

      1. exp-prod 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in x around 0 99.1%

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

   if 3.40000000000000002e-25 < x

   1. Initial program 72.7%

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

      1. *-commutative 72.7%

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

      2. exp-to-pow 72.7%

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

   3. Simplified 72.7%

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

   4. Taylor expanded in x around inf 99.9%

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

      1. mul-1-neg 99.9%

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

   6. Simplified 99.9%

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

      1. clear-num 99.9%

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

      2. inv-pow 99.9%

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

      3. div-inv 99.9%

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

      4. add-sqr-sqrt 42.0%

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

      5. sqrt-unprod 77.6%

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

      6. sqr-neg 77.6%

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

      7. *-rgt-identity 77.6%

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

      8. *-rgt-identity 77.6%

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

      9. sqrt-prod 35.4%

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

      10. add-sqr-sqrt 61.9%

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

      11. exp-neg 61.9%

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

      12. add-sqr-sqrt 26.5%

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

      13. sqrt-unprod 84.2%

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

      14. sqr-neg 84.2%

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

      15. *-rgt-identity 84.2%

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

      16. *-rgt-identity 84.2%

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

      17. sqrt-prod 57.7%

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

      18. add-sqr-sqrt 99.9%

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

   8. Applied egg-rr 99.9%

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

   9. Taylor expanded in x around 0 99.9%

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

   10. Taylor expanded in y around 0 72.0%

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

4. Final simplification 83.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.5 \cdot 10^{+19}:\\ \;\;\;\;\frac{-1 - y \cdot y}{x \cdot y - x}\\ \mathbf{elif}\;x \leq 3.4 \cdot 10^{-25}:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x + x \cdot y}\\ \end{array} \]

Alternative 4: 80.4% accurate, 18.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.02 \lor \neg \left(x \leq 3.4 \cdot 10^{-25}\right):\\ \;\;\;\;\frac{1}{x + x \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -1.02) (not (<= x 3.4e-25))) (/ 1.0 (+ x (* x y))) (/ 1.0 x)))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -1.02) || !(x <= 3.4e-25)) {
		tmp = 1.0 / (x + (x * y));
	} else {
		tmp = 1.0 / x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((x <= (-1.02d0)) .or. (.not. (x <= 3.4d-25))) then
        tmp = 1.0d0 / (x + (x * y))
    else
        tmp = 1.0d0 / x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((x <= -1.02) || !(x <= 3.4e-25)) {
		tmp = 1.0 / (x + (x * y));
	} else {
		tmp = 1.0 / x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -1.02) or not (x <= 3.4e-25):
		tmp = 1.0 / (x + (x * y))
	else:
		tmp = 1.0 / x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -1.02) || !(x <= 3.4e-25))
		tmp = Float64(1.0 / Float64(x + Float64(x * y)));
	else
		tmp = Float64(1.0 / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((x <= -1.02) || ~((x <= 3.4e-25)))
		tmp = 1.0 / (x + (x * y));
	else
		tmp = 1.0 / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -1.02], N[Not[LessEqual[x, 3.4e-25]], $MachinePrecision]], N[(1.0 / N[(x + N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.02 or 3.40000000000000002e-25 < x

   1. Initial program 76.9%

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

      1. *-commutative 76.9%

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

      2. exp-to-pow 76.9%

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

   3. Simplified 76.9%

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

   4. Taylor expanded in x around inf 99.9%

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

      1. mul-1-neg 99.9%

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

   6. Simplified 99.9%

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

      1. clear-num 99.9%

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

      2. inv-pow 99.9%

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

      3. div-inv 99.9%

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

      4. add-sqr-sqrt 46.0%

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

      5. sqrt-unprod 81.2%

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

      6. sqr-neg 81.2%

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

      7. *-rgt-identity 81.2%

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

      8. *-rgt-identity 81.2%

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

      9. sqrt-prod 35.1%

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

      10. add-sqr-sqrt 62.8%

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

      11. exp-neg 62.8%

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

      12. add-sqr-sqrt 27.7%

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

      13. sqrt-unprod 81.5%

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

      14. sqr-neg 81.5%

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

      15. *-rgt-identity 81.5%

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

      16. *-rgt-identity 81.5%

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

      17. sqrt-prod 53.8%

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

      18. add-sqr-sqrt 99.9%

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

   8. Applied egg-rr 99.9%

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

   9. Taylor expanded in x around 0 99.9%

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

   10. Taylor expanded in y around 0 72.7%

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

   if -1.02 < x < 3.40000000000000002e-25

   1. Initial program 74.8%

      \[\frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x} \]
   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. Taylor expanded in x around 0 99.3%

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

4. Final simplification 83.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.02 \lor \neg \left(x \leq 3.4 \cdot 10^{-25}\right):\\ \;\;\;\;\frac{1}{x + x \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x}\\ \end{array} \]

Alternative 5: 74.6% 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 76.1%

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

   1. exp-prod 85.6%

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

3. Simplified 85.6%

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

4. Taylor expanded in x around 0 77.6%

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

5. Final simplification 77.6%

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

Developer target: 77.6% 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 2023268 
(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))