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

Percentage Accurate: 78.3% → 99.5%

Time: 10.5s

Alternatives: 5

Speedup: 23.1×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 78.3% 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: 99.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.5 \cdot 10^{+71} \lor \neg \left(x \leq 0.13\right):\\ \;\;\;\;\frac{e^{-y}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(e^{x}\right)}^{\log \left(\frac{x}{x + y}\right)}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -3.5e+71) (not (<= x 0.13)))
   (/ (exp (- y)) x)
   (/ (pow (exp x) (log (/ x (+ x y)))) x)))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -3.5e+71) || !(x <= 0.13)) {
		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 <= (-3.5d+71)) .or. (.not. (x <= 0.13d0))) 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 <= -3.5e+71) || !(x <= 0.13)) {
		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 <= -3.5e+71) or not (x <= 0.13):
		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 <= -3.5e+71) || !(x <= 0.13))
		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 <= -3.5e+71) || ~((x <= 0.13)))
		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, -3.5e+71], N[Not[LessEqual[x, 0.13]], $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 < -3.4999999999999999e71 or 0.13 < x

   1. Initial program 74.4%

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

      1. *-commutative 74.4%

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

      2. exp-to-pow 74.4%

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

   3. Simplified 74.4%

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

   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%

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

   if -3.4999999999999999e71 < x < 0.13

   1. Initial program 84.8%

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

      1. exp-prod 99.5%

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

   3. Simplified 99.5%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.5 \cdot 10^{+71} \lor \neg \left(x \leq 0.13\right):\\ \;\;\;\;\frac{e^{-y}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(e^{x}\right)}^{\log \left(\frac{x}{x + y}\right)}}{x}\\ \end{array} \]

Alternative 2: 99.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1000000000000 \lor \neg \left(x \leq 0.122\right):\\ \;\;\;\;\frac{e^{-y}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -1000000000000.0) (not (<= x 0.122)))
   (/ (exp (- y)) x)
   (/ 1.0 x)))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -1000000000000.0) || !(x <= 0.122)) {
		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 <= (-1000000000000.0d0)) .or. (.not. (x <= 0.122d0))) 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 <= -1000000000000.0) || !(x <= 0.122)) {
		tmp = Math.exp(-y) / x;
	} else {
		tmp = 1.0 / x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -1000000000000.0) or not (x <= 0.122):
		tmp = math.exp(-y) / x
	else:
		tmp = 1.0 / x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -1000000000000.0) || !(x <= 0.122))
		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 <= -1000000000000.0) || ~((x <= 0.122)))
		tmp = exp(-y) / x;
	else
		tmp = 1.0 / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -1000000000000.0], N[Not[LessEqual[x, 0.122]], $MachinePrecision]], N[(N[Exp[(-y)], $MachinePrecision] / x), $MachinePrecision], N[(1.0 / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1e12 or 0.122 < x

   1. Initial program 76.1%

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

      1. *-commutative 76.1%

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

      2. exp-to-pow 76.1%

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

   3. Simplified 76.1%

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

   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 -1e12 < x < 0.122

   1. Initial program 83.6%

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

      1. exp-prod 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in x around 0 96.6%

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

4. Final simplification 98.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1000000000000 \lor \neg \left(x \leq 0.122\right):\\ \;\;\;\;\frac{e^{-y}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x}\\ \end{array} \]

Alternative 3: 82.8% accurate, 13.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -900000000000:\\ \;\;\;\;\frac{\frac{-1}{x} \cdot \left(1 - y \cdot y\right)}{-1 - y}\\ \mathbf{elif}\;x \leq 0.13:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-1}{x}}{-1 - y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -900000000000.0)
   (/ (* (/ -1.0 x) (- 1.0 (* y y))) (- -1.0 y))
   (if (<= x 0.13) (/ 1.0 x) (/ (/ -1.0 x) (- -1.0 y)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -900000000000.0) {
		tmp = ((-1.0 / x) * (1.0 - (y * y))) / (-1.0 - y);
	} else if (x <= 0.13) {
		tmp = 1.0 / x;
	} else {
		tmp = (-1.0 / 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 <= (-900000000000.0d0)) then
        tmp = (((-1.0d0) / x) * (1.0d0 - (y * y))) / ((-1.0d0) - y)
    else if (x <= 0.13d0) then
        tmp = 1.0d0 / x
    else
        tmp = ((-1.0d0) / x) / ((-1.0d0) - y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -900000000000.0) {
		tmp = ((-1.0 / x) * (1.0 - (y * y))) / (-1.0 - y);
	} else if (x <= 0.13) {
		tmp = 1.0 / x;
	} else {
		tmp = (-1.0 / x) / (-1.0 - y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -900000000000.0:
		tmp = ((-1.0 / x) * (1.0 - (y * y))) / (-1.0 - y)
	elif x <= 0.13:
		tmp = 1.0 / x
	else:
		tmp = (-1.0 / x) / (-1.0 - y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -900000000000.0)
		tmp = Float64(Float64(Float64(-1.0 / x) * Float64(1.0 - Float64(y * y))) / Float64(-1.0 - y));
	elseif (x <= 0.13)
		tmp = Float64(1.0 / x);
	else
		tmp = Float64(Float64(-1.0 / x) / Float64(-1.0 - y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -900000000000.0)
		tmp = ((-1.0 / x) * (1.0 - (y * y))) / (-1.0 - y);
	elseif (x <= 0.13)
		tmp = 1.0 / x;
	else
		tmp = (-1.0 / x) / (-1.0 - y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -900000000000.0], N[(N[(N[(-1.0 / x), $MachinePrecision] * N[(1.0 - N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(-1.0 - y), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.13], N[(1.0 / x), $MachinePrecision], N[(N[(-1.0 / x), $MachinePrecision] / N[(-1.0 - y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -9e11

   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 81.1%

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

   3. Simplified 81.1%

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

   4. Taylor expanded in x around inf 66.9%

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

      1. mul-1-neg 66.9%

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

      2. unsub-neg 66.9%

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

   6. Simplified 66.9%

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

      1. frac-2neg 66.9%

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

      2. div-inv 66.9%

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

      3. sub-neg 66.9%

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

      4. distribute-neg-in 66.9%

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

      5. metadata-eval 66.9%

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

      6. add-sqr-sqrt 36.6%

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

      7. sqrt-unprod 75.4%

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

      8. sqr-neg 75.4%

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

      9. sqrt-unprod 30.4%

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

      10. add-sqr-sqrt 65.8%

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

      11. add-sqr-sqrt 35.5%

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

      12. sqrt-unprod 65.7%

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

      13. sqr-neg 65.7%

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

      14. sqrt-unprod 30.4%

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

      15. add-sqr-sqrt 66.9%

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

   8. Applied egg-rr 66.9%

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

      1. *-commutative 66.9%

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

      2. flip-+ 75.4%

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

      3. associate-*r/ 75.4%

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

      4. neg-mul-1 75.4%

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

      5. associate-/r* 75.4%

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

      6. metadata-eval 75.4%

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

      7. metadata-eval 75.4%

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

   10. Applied egg-rr 75.4%

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

   if -9e11 < x < 0.13

   1. Initial program 83.6%

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

      1. exp-prod 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in x around 0 96.6%

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

   if 0.13 < x

   1. Initial program 71.4%

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

      1. exp-prod 71.4%

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

   3. Simplified 71.4%

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

   4. Taylor expanded in x around inf 56.7%

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

      1. mul-1-neg 56.7%

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

      2. unsub-neg 56.7%

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

   6. Simplified 56.7%

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

      1. frac-2neg 56.7%

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

      2. div-inv 56.7%

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

      3. sub-neg 56.7%

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

      4. distribute-neg-in 56.7%

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

      5. metadata-eval 56.7%

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

      6. add-sqr-sqrt 25.1%

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

      7. sqrt-unprod 56.6%

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

      8. sqr-neg 56.6%

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

      9. sqrt-unprod 31.6%

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

      10. add-sqr-sqrt 55.9%

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

      11. add-sqr-sqrt 24.3%

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

      12. sqrt-unprod 55.8%

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

      13. sqr-neg 55.8%

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

      14. sqrt-unprod 31.6%

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

      15. add-sqr-sqrt 56.7%

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

   8. Applied egg-rr 56.7%

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

      1. *-commutative 56.7%

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

      2. flip-+ 56.6%

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

      3. associate-*r/ 56.6%

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

      4. neg-mul-1 56.6%

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

      5. associate-/r* 56.6%

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

      6. metadata-eval 56.6%

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

      7. metadata-eval 56.6%

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

   10. Applied egg-rr 56.6%

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

   11. Taylor expanded in y around 0 70.0%

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

4. Final simplification 83.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -900000000000:\\ \;\;\;\;\frac{\frac{-1}{x} \cdot \left(1 - y \cdot y\right)}{-1 - y}\\ \mathbf{elif}\;x \leq 0.13:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-1}{x}}{-1 - y}\\ \end{array} \]

Alternative 4: 79.0% accurate, 23.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.13:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-1}{x}}{-1 - y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 0.13) (/ 1.0 x) (/ (/ -1.0 x) (- -1.0 y))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 0.13) {
		tmp = 1.0 / x;
	} else {
		tmp = (-1.0 / 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 <= 0.13d0) then
        tmp = 1.0d0 / x
    else
        tmp = ((-1.0d0) / x) / ((-1.0d0) - y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 0.13) {
		tmp = 1.0 / x;
	} else {
		tmp = (-1.0 / x) / (-1.0 - y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 0.13:
		tmp = 1.0 / x
	else:
		tmp = (-1.0 / x) / (-1.0 - y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 0.13)
		tmp = Float64(1.0 / x);
	else
		tmp = Float64(Float64(-1.0 / x) / Float64(-1.0 - y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 0.13)
		tmp = 1.0 / x;
	else
		tmp = (-1.0 / x) / (-1.0 - y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 0.13], N[(1.0 / x), $MachinePrecision], N[(N[(-1.0 / x), $MachinePrecision] / N[(-1.0 - y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.13

   1. Initial program 82.7%

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

      1. exp-prod 92.8%

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

   3. Simplified 92.8%

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

   4. Taylor expanded in x around 0 85.7%

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

   if 0.13 < x

   1. Initial program 71.4%

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

      1. exp-prod 71.4%

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

   3. Simplified 71.4%

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

   4. Taylor expanded in x around inf 56.7%

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

      1. mul-1-neg 56.7%

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

      2. unsub-neg 56.7%

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

   6. Simplified 56.7%

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

      1. frac-2neg 56.7%

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

      2. div-inv 56.7%

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

      3. sub-neg 56.7%

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

      4. distribute-neg-in 56.7%

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

      5. metadata-eval 56.7%

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

      6. add-sqr-sqrt 25.1%

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

      7. sqrt-unprod 56.6%

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

      8. sqr-neg 56.6%

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

      9. sqrt-unprod 31.6%

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

      10. add-sqr-sqrt 55.9%

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

      11. add-sqr-sqrt 24.3%

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

      12. sqrt-unprod 55.8%

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

      13. sqr-neg 55.8%

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

      14. sqrt-unprod 31.6%

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

      15. add-sqr-sqrt 56.7%

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

   8. Applied egg-rr 56.7%

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

      1. *-commutative 56.7%

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

      2. flip-+ 56.6%

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

      3. associate-*r/ 56.6%

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

      4. neg-mul-1 56.6%

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

      5. associate-/r* 56.6%

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

      6. metadata-eval 56.6%

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

      7. metadata-eval 56.6%

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

   10. Applied egg-rr 56.6%

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

   11. Taylor expanded in y around 0 70.0%

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

4. Final simplification 81.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.13:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-1}{x}}{-1 - y}\\ \end{array} \]

Alternative 5: 75.0% 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 79.5%

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

   1. exp-prod 86.8%

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

3. Simplified 86.8%

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

4. Taylor expanded in x around 0 77.5%

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

5. Final simplification 77.5%

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

Developer target: 78.3% 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 2023274 
(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))