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

Percentage Accurate: 77.8% → 99.2%

Time: 10.1s

Alternatives: 7

Speedup: 12.2×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 77.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{+53}:\\ \;\;\;\;\frac{e^{-y}}{x}\\ \mathbf{elif}\;x \leq 5.7 \cdot 10^{-20}:\\ \;\;\;\;\frac{{\left(e^{x}\right)}^{\log \left(\frac{x}{x + y}\right)}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x}}{e^{y}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -2e+53)
   (/ (exp (- y)) x)
   (if (<= x 5.7e-20)
     (/ (pow (exp x) (log (/ x (+ x y)))) x)
     (/ (/ 1.0 x) (exp y)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -2e+53) {
		tmp = exp(-y) / x;
	} else if (x <= 5.7e-20) {
		tmp = pow(exp(x), log((x / (x + y)))) / x;
	} else {
		tmp = (1.0 / x) / exp(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 <= (-2d+53)) then
        tmp = exp(-y) / x
    else if (x <= 5.7d-20) then
        tmp = (exp(x) ** log((x / (x + y)))) / x
    else
        tmp = (1.0d0 / x) / exp(y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -2e+53) {
		tmp = Math.exp(-y) / x;
	} else if (x <= 5.7e-20) {
		tmp = Math.pow(Math.exp(x), Math.log((x / (x + y)))) / x;
	} else {
		tmp = (1.0 / x) / Math.exp(y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -2e+53:
		tmp = math.exp(-y) / x
	elif x <= 5.7e-20:
		tmp = math.pow(math.exp(x), math.log((x / (x + y)))) / x
	else:
		tmp = (1.0 / x) / math.exp(y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -2e+53)
		tmp = Float64(exp(Float64(-y)) / x);
	elseif (x <= 5.7e-20)
		tmp = Float64((exp(x) ^ log(Float64(x / Float64(x + y)))) / x);
	else
		tmp = Float64(Float64(1.0 / x) / exp(y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -2e+53)
		tmp = exp(-y) / x;
	elseif (x <= 5.7e-20)
		tmp = (exp(x) ^ log((x / (x + y)))) / x;
	else
		tmp = (1.0 / x) / exp(y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -2e+53], N[(N[Exp[(-y)], $MachinePrecision] / x), $MachinePrecision], If[LessEqual[x, 5.7e-20], N[(N[Power[N[Exp[x], $MachinePrecision], N[Log[N[(x / N[(x + y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] / x), $MachinePrecision], N[(N[(1.0 / x), $MachinePrecision] / N[Exp[y], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -2e53

   1. Initial program 74.6%

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

      1. *-commutative 74.6%

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

      2. exp-to-pow 74.6%

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

   3. Simplified 74.6%

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

   4. Taylor expanded in x around inf 100.0%

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

      1. mul-1-neg 100.0%

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

   6. Simplified 100.0%

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

   if -2e53 < x < 5.70000000000000024e-20

   1. Initial program 84.3%

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

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

   if 5.70000000000000024e-20 < x

   1. Initial program 77.4%

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

      1. *-commutative 77.4%

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

      2. exp-to-pow 77.4%

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

   3. Simplified 77.4%

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

   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. Step-by-step derivation

      1. clear-num 100.0%

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

      2. inv-pow 100.0%

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

      3. clear-num 99.9%

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

      4. associate-/r/ 100.0%

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

      5. add-exp-log 100.0%

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

      6. log-rec 100.0%

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

      7. add-log-exp 100.0%

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

      8. add-sqr-sqrt 44.8%

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

      9. sqrt-unprod 76.3%

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

      10. sqr-neg 76.3%

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

      11. sqrt-unprod 31.5%

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

      12. add-sqr-sqrt 59.2%

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

      13. add-sqr-sqrt 27.7%

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

      14. sqrt-unprod 82.9%

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

      15. sqr-neg 82.9%

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

      16. sqrt-unprod 55.2%

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

      17. add-sqr-sqrt 100.0%

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

   8. Applied egg-rr 100.0%

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

      1. unpow-1 100.0%

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

      2. associate-/l/ 100.0%

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

   10. Applied egg-rr 100.0%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{+53}:\\ \;\;\;\;\frac{e^{-y}}{x}\\ \mathbf{elif}\;x \leq 5.7 \cdot 10^{-20}:\\ \;\;\;\;\frac{{\left(e^{x}\right)}^{\log \left(\frac{x}{x + y}\right)}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x}}{e^{y}}\\ \end{array} \]

Alternative 2: 99.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.9:\\ \;\;\;\;\frac{e^{-y}}{x}\\ \mathbf{elif}\;x \leq 5.7 \cdot 10^{-20}:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x}}{e^{y}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -0.9)
   (/ (exp (- y)) x)
   (if (<= x 5.7e-20) (/ 1.0 x) (/ (/ 1.0 x) (exp y)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -0.9) {
		tmp = exp(-y) / x;
	} else if (x <= 5.7e-20) {
		tmp = 1.0 / x;
	} else {
		tmp = (1.0 / x) / exp(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 <= (-0.9d0)) then
        tmp = exp(-y) / x
    else if (x <= 5.7d-20) then
        tmp = 1.0d0 / x
    else
        tmp = (1.0d0 / x) / exp(y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -0.9) {
		tmp = Math.exp(-y) / x;
	} else if (x <= 5.7e-20) {
		tmp = 1.0 / x;
	} else {
		tmp = (1.0 / x) / Math.exp(y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -0.9:
		tmp = math.exp(-y) / x
	elif x <= 5.7e-20:
		tmp = 1.0 / x
	else:
		tmp = (1.0 / x) / math.exp(y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -0.9)
		tmp = Float64(exp(Float64(-y)) / x);
	elseif (x <= 5.7e-20)
		tmp = Float64(1.0 / x);
	else
		tmp = Float64(Float64(1.0 / x) / exp(y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -0.9)
		tmp = exp(-y) / x;
	elseif (x <= 5.7e-20)
		tmp = 1.0 / x;
	else
		tmp = (1.0 / x) / exp(y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -0.9], N[(N[Exp[(-y)], $MachinePrecision] / x), $MachinePrecision], If[LessEqual[x, 5.7e-20], N[(1.0 / x), $MachinePrecision], N[(N[(1.0 / x), $MachinePrecision] / N[Exp[y], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.900000000000000022

   1. Initial program 77.3%

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

      1. *-commutative 77.3%

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

      2. exp-to-pow 77.3%

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

   3. Simplified 77.3%

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

   4. Taylor expanded in x around inf 100.0%

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

      1. mul-1-neg 100.0%

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

   6. Simplified 100.0%

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

   if -0.900000000000000022 < x < 5.70000000000000024e-20

   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 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 99.8%

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

   if 5.70000000000000024e-20 < x

   1. Initial program 77.4%

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

      1. *-commutative 77.4%

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

      2. exp-to-pow 77.4%

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

   3. Simplified 77.4%

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

   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. Step-by-step derivation

      1. clear-num 100.0%

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

      2. inv-pow 100.0%

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

      3. clear-num 99.9%

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

      4. associate-/r/ 100.0%

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

      5. add-exp-log 100.0%

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

      6. log-rec 100.0%

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

      7. add-log-exp 100.0%

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

      8. add-sqr-sqrt 44.8%

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

      9. sqrt-unprod 76.3%

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

      10. sqr-neg 76.3%

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

      11. sqrt-unprod 31.5%

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

      12. add-sqr-sqrt 59.2%

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

      13. add-sqr-sqrt 27.7%

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

      14. sqrt-unprod 82.9%

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

      15. sqr-neg 82.9%

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

      16. sqrt-unprod 55.2%

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

      17. add-sqr-sqrt 100.0%

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

   8. Applied egg-rr 100.0%

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

      1. unpow-1 100.0%

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

      2. associate-/l/ 100.0%

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

   10. Applied egg-rr 100.0%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.9:\\ \;\;\;\;\frac{e^{-y}}{x}\\ \mathbf{elif}\;x \leq 5.7 \cdot 10^{-20}:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x}}{e^{y}}\\ \end{array} \]

Alternative 3: 99.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.12 \lor \neg \left(x \leq 5.7 \cdot 10^{-20}\right):\\ \;\;\;\;\frac{e^{-y}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -1.12) (not (<= x 5.7e-20))) (/ (exp (- y)) x) (/ 1.0 x)))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -1.12) || !(x <= 5.7e-20)) {
		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 <= (-1.12d0)) .or. (.not. (x <= 5.7d-20))) 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 <= -1.12) || !(x <= 5.7e-20)) {
		tmp = Math.exp(-y) / x;
	} else {
		tmp = 1.0 / x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -1.12) or not (x <= 5.7e-20):
		tmp = math.exp(-y) / x
	else:
		tmp = 1.0 / x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -1.12) || !(x <= 5.7e-20))
		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 <= -1.12) || ~((x <= 5.7e-20)))
		tmp = exp(-y) / x;
	else
		tmp = 1.0 / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -1.12], N[Not[LessEqual[x, 5.7e-20]], $MachinePrecision]], N[(N[Exp[(-y)], $MachinePrecision] / x), $MachinePrecision], N[(1.0 / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.1200000000000001 or 5.70000000000000024e-20 < x

   1. Initial program 77.3%

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

      1. *-commutative 77.3%

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

      2. exp-to-pow 77.3%

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

   3. Simplified 77.3%

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

   4. Taylor expanded in x around inf 100.0%

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

      1. mul-1-neg 100.0%

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

   6. Simplified 100.0%

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

   if -1.1200000000000001 < x < 5.70000000000000024e-20

   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 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 99.8%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.12 \lor \neg \left(x \leq 5.7 \cdot 10^{-20}\right):\\ \;\;\;\;\frac{e^{-y}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x}\\ \end{array} \]

Alternative 4: 81.2% accurate, 12.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.82:\\ \;\;\;\;\frac{\frac{y + -1}{x} \cdot \left(1 - y\right)}{-1}\\ \mathbf{elif}\;x \leq 9.2 \cdot 10^{+44}:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(1 - y\right) \cdot \frac{1 - y}{x \cdot \left(1 - y\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -0.82)
   (/ (* (/ (+ y -1.0) x) (- 1.0 y)) -1.0)
   (if (<= x 9.2e+44) (/ 1.0 x) (* (- 1.0 y) (/ (- 1.0 y) (* x (- 1.0 y)))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -0.82) {
		tmp = (((y + -1.0) / x) * (1.0 - y)) / -1.0;
	} else if (x <= 9.2e+44) {
		tmp = 1.0 / x;
	} else {
		tmp = (1.0 - y) * ((1.0 - y) / (x * (1.0 - 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 <= (-0.82d0)) then
        tmp = (((y + (-1.0d0)) / x) * (1.0d0 - y)) / (-1.0d0)
    else if (x <= 9.2d+44) then
        tmp = 1.0d0 / x
    else
        tmp = (1.0d0 - y) * ((1.0d0 - y) / (x * (1.0d0 - y)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -0.82) {
		tmp = (((y + -1.0) / x) * (1.0 - y)) / -1.0;
	} else if (x <= 9.2e+44) {
		tmp = 1.0 / x;
	} else {
		tmp = (1.0 - y) * ((1.0 - y) / (x * (1.0 - y)));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -0.82:
		tmp = (((y + -1.0) / x) * (1.0 - y)) / -1.0
	elif x <= 9.2e+44:
		tmp = 1.0 / x
	else:
		tmp = (1.0 - y) * ((1.0 - y) / (x * (1.0 - y)))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -0.82)
		tmp = Float64(Float64(Float64(Float64(y + -1.0) / x) * Float64(1.0 - y)) / -1.0);
	elseif (x <= 9.2e+44)
		tmp = Float64(1.0 / x);
	else
		tmp = Float64(Float64(1.0 - y) * Float64(Float64(1.0 - y) / Float64(x * Float64(1.0 - y))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -0.82)
		tmp = (((y + -1.0) / x) * (1.0 - y)) / -1.0;
	elseif (x <= 9.2e+44)
		tmp = 1.0 / x;
	else
		tmp = (1.0 - y) * ((1.0 - y) / (x * (1.0 - y)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -0.82], N[(N[(N[(N[(y + -1.0), $MachinePrecision] / x), $MachinePrecision] * N[(1.0 - y), $MachinePrecision]), $MachinePrecision] / -1.0), $MachinePrecision], If[LessEqual[x, 9.2e+44], N[(1.0 / x), $MachinePrecision], N[(N[(1.0 - y), $MachinePrecision] * N[(N[(1.0 - y), $MachinePrecision] / N[(x * N[(1.0 - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.819999999999999951

   1. Initial program 77.3%

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

      1. exp-prod 77.3%

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

   3. Simplified 77.3%

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

   4. Taylor expanded in x around inf 57.2%

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

      1. mul-1-neg 57.2%

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

      2. unsub-neg 57.2%

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

   6. Simplified 57.2%

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

      1. frac-2neg 57.2%

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

      2. div-inv 57.2%

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

      3. *-commutative 57.2%

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

      4. neg-sub0 57.2%

         \[\leadsto \frac{1}{-x} \cdot \color{blue}{\left(0 - \left(1 - y\right)\right)} \]

      5. associate--r- 57.2%

         \[\leadsto \frac{1}{-x} \cdot \color{blue}{\left(\left(0 - 1\right) + y\right)} \]

      6. metadata-eval 57.2%

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

      7. +-commutative 57.2%

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

   8. Applied egg-rr 57.2%

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

   10. Applied egg-rr 64.5%

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

   11. Taylor expanded in y around 0 64.7%

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

   if -0.819999999999999951 < x < 9.20000000000000018e44

   1. Initial program 84.1%

      \[\frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x} \]
   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 98.2%

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

   if 9.20000000000000018e44 < x

   1. Initial program 75.2%

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

      1. *-commutative 75.2%

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

      2. exp-to-pow 75.2%

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

   3. Simplified 75.2%

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

   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. Taylor expanded in y around 0 59.6%

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

      1. +-commutative 59.6%

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

      2. mul-1-neg 59.6%

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

      3. sub-neg 59.6%

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

   9. Simplified 59.6%

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

       1. div-inv 59.6%

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

       2. div-inv 59.6%

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

       3. distribute-rgt-out-- 59.6%

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

   11. Applied egg-rr 59.6%

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

       1. associate-*l/ 59.6%

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

       2. *-inverses 59.6%

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

       3. associate-*r/ 59.6%

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

       4. times-frac 59.4%

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

       5. associate-/l* 59.4%

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

       6. frac-2neg 59.4%

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

       7. associate-/l/ 59.4%

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

       8. distribute-neg-in 59.4%

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

       9. metadata-eval 59.4%

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

       10. +-commutative 59.4%

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

       11. sub-neg 59.4%

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

       12. associate-*l/ 73.8%

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

       13. associate-/l* 59.6%

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

   13. Applied egg-rr 59.6%

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

       1. associate-*l/ 73.8%

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

       2. associate-/r/ 73.8%

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

       3. *-inverses 73.8%

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

       4. associate-*l/ 73.8%

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

       5. distribute-rgt-neg-in 73.8%

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

       6. *-commutative 73.8%

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

       7. distribute-neg-frac 73.8%

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

       8. /-rgt-identity 73.8%

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

       9. distribute-rgt-neg-in 73.8%

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

       10. distribute-neg-in 73.8%

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

       11. metadata-eval 73.8%

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

       12. +-commutative 73.8%

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

       13. sub-neg 73.8%

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

   15. Simplified 73.8%

       \[\leadsto \color{blue}{\frac{1 - y}{x \cdot \left(1 - y\right)} \cdot \left(1 - y\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 82.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.82:\\ \;\;\;\;\frac{\frac{y + -1}{x} \cdot \left(1 - y\right)}{-1}\\ \mathbf{elif}\;x \leq 9.2 \cdot 10^{+44}:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(1 - y\right) \cdot \frac{1 - y}{x \cdot \left(1 - y\right)}\\ \end{array} \]

Alternative 5: 75.3% accurate, 16.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -4.5 \cdot 10^{+285}:\\ \;\;\;\;\frac{1}{x} \cdot \frac{\frac{x \cdot y}{x}}{-1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -4.5e+285) (* (/ 1.0 x) (/ (/ (* x y) x) -1.0)) (/ 1.0 x)))

C

double code(double x, double y) {
	double tmp;
	if (y <= -4.5e+285) {
		tmp = (1.0 / x) * (((x * y) / x) / -1.0);
	} 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 (y <= (-4.5d+285)) then
        tmp = (1.0d0 / x) * (((x * y) / x) / (-1.0d0))
    else
        tmp = 1.0d0 / x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -4.5e+285) {
		tmp = (1.0 / x) * (((x * y) / x) / -1.0);
	} else {
		tmp = 1.0 / x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -4.5e+285:
		tmp = (1.0 / x) * (((x * y) / x) / -1.0)
	else:
		tmp = 1.0 / x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -4.5e+285)
		tmp = Float64(Float64(1.0 / x) * Float64(Float64(Float64(x * y) / x) / -1.0));
	else
		tmp = Float64(1.0 / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -4.5e+285)
		tmp = (1.0 / x) * (((x * y) / x) / -1.0);
	else
		tmp = 1.0 / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -4.5e+285], N[(N[(1.0 / x), $MachinePrecision] * N[(N[(N[(x * y), $MachinePrecision] / x), $MachinePrecision] / -1.0), $MachinePrecision]), $MachinePrecision], N[(1.0 / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -4.5e285

   1. Initial program 83.6%

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

      1. *-commutative 83.6%

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

      2. exp-to-pow 83.6%

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

   3. Simplified 83.6%

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

   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 4.1%

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

      1. +-commutative 4.1%

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

      2. mul-1-neg 4.1%

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

      3. sub-neg 4.1%

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

   9. Simplified 4.1%

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

       1. frac-sub 33.3%

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

       2. associate-/r* 100.0%

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

       3. add-sqr-sqrt 33.3%

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

       4. sqrt-unprod 0.0%

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

       5. sqr-neg 0.0%

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

       6. sqrt-unprod 0.0%

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

       7. add-sqr-sqrt 0.0%

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

       8. /-rgt-identity 0.0%

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

       9. neg-mul-1 0.0%

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

       10. associate-/l* 0.0%

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

       11. associate-/r/ 0.0%

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

   11. Applied egg-rr 100.0%

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

   12. Taylor expanded in y around inf 100.0%

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

   if -4.5e285 < y

   1. Initial program 79.9%

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

      1. exp-prod 87.4%

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

   3. Simplified 87.4%

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

   4. Taylor expanded in x around 0 78.5%

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

4. Final simplification 79.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4.5 \cdot 10^{+285}:\\ \;\;\;\;\frac{1}{x} \cdot \frac{\frac{x \cdot y}{x}}{-1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x}\\ \end{array} \]

Alternative 6: 77.3% accurate, 16.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.78:\\ \;\;\;\;\frac{\frac{y + -1}{x} \cdot \left(1 - y\right)}{-1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -0.78) (/ (* (/ (+ y -1.0) x) (- 1.0 y)) -1.0) (/ 1.0 x)))

C

double code(double x, double y) {
	double tmp;
	if (x <= -0.78) {
		tmp = (((y + -1.0) / x) * (1.0 - y)) / -1.0;
	} 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 <= (-0.78d0)) then
        tmp = (((y + (-1.0d0)) / x) * (1.0d0 - y)) / (-1.0d0)
    else
        tmp = 1.0d0 / x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -0.78) {
		tmp = (((y + -1.0) / x) * (1.0 - y)) / -1.0;
	} else {
		tmp = 1.0 / x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -0.78:
		tmp = (((y + -1.0) / x) * (1.0 - y)) / -1.0
	else:
		tmp = 1.0 / x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -0.78)
		tmp = Float64(Float64(Float64(Float64(y + -1.0) / x) * Float64(1.0 - y)) / -1.0);
	else
		tmp = Float64(1.0 / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -0.78)
		tmp = (((y + -1.0) / x) * (1.0 - y)) / -1.0;
	else
		tmp = 1.0 / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -0.78], N[(N[(N[(N[(y + -1.0), $MachinePrecision] / x), $MachinePrecision] * N[(1.0 - y), $MachinePrecision]), $MachinePrecision] / -1.0), $MachinePrecision], N[(1.0 / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -0.78000000000000003

   1. Initial program 77.3%

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

      1. exp-prod 77.3%

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

   3. Simplified 77.3%

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

   4. Taylor expanded in x around inf 57.2%

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

      1. mul-1-neg 57.2%

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

      2. unsub-neg 57.2%

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

   6. Simplified 57.2%

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

      1. frac-2neg 57.2%

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

      2. div-inv 57.2%

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

      3. *-commutative 57.2%

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

      4. neg-sub0 57.2%

         \[\leadsto \frac{1}{-x} \cdot \color{blue}{\left(0 - \left(1 - y\right)\right)} \]

      5. associate--r- 57.2%

         \[\leadsto \frac{1}{-x} \cdot \color{blue}{\left(\left(0 - 1\right) + y\right)} \]

      6. metadata-eval 57.2%

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

      7. +-commutative 57.2%

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

   8. Applied egg-rr 57.2%

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

   10. Applied egg-rr 64.5%

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

   11. Taylor expanded in y around 0 64.7%

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

   if -0.78000000000000003 < x

   1. Initial program 81.1%

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

      1. exp-prod 91.5%

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

   3. Simplified 91.5%

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

   4. Taylor expanded in x around 0 85.1%

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

4. Final simplification 79.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.78:\\ \;\;\;\;\frac{\frac{y + -1}{x} \cdot \left(1 - y\right)}{-1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x}\\ \end{array} \]

Alternative 7: 75.1% accurate, 69.7× speedup

Math

\[\begin{array}{l} \\ \frac{1}{x} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(x, y):
	return 1.0 / x

Julia

function code(x, y)
	return Float64(1.0 / x)
end

MATLAB

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

Wolfram

code[x_, y_] := N[(1.0 / x), $MachinePrecision]

Derivation

1. Initial program 79.9%

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

   1. exp-prod 87.3%

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

3. Simplified 87.3%

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

4. Taylor expanded in x around 0 76.7%

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

5. Final simplification 76.7%

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

Developer target: 77.7% 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 2023301 
(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))