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

Percentage Accurate: 77.6% → 99.4%

Time: 11.9s

Alternatives: 5

Speedup: 20.9×

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: 77.6% 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: 99.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.68 \lor \neg \left(x \leq 1.02 \cdot 10^{-7}\right):\\ \;\;\;\;\frac{e^{-y}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -0.68) (not (<= x 1.02e-7))) (/ (exp (- y)) x) (/ 1.0 x)))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -0.68) || !(x <= 1.02e-7)) {
		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 <= (-0.68d0)) .or. (.not. (x <= 1.02d-7))) 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 <= -0.68) || !(x <= 1.02e-7)) {
		tmp = Math.exp(-y) / x;
	} else {
		tmp = 1.0 / x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -0.68) or not (x <= 1.02e-7):
		tmp = math.exp(-y) / x
	else:
		tmp = 1.0 / x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -0.68) || !(x <= 1.02e-7))
		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 <= -0.68) || ~((x <= 1.02e-7)))
		tmp = exp(-y) / x;
	else
		tmp = 1.0 / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -0.68], N[Not[LessEqual[x, 1.02e-7]], $MachinePrecision]], N[(N[Exp[(-y)], $MachinePrecision] / x), $MachinePrecision], N[(1.0 / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -0.680000000000000049 or 1.02e-7 < x

   1. Initial program 76.2%

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

      1. *-commutative 76.2%

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

      2. exp-to-pow 76.2%

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

   3. Simplified 76.2%

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

   5. Taylor expanded in x around inf 100.0%

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

      1. mul-1-neg 100.0%

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

   7. Simplified 100.0%

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

   if -0.680000000000000049 < x < 1.02e-7

   1. Initial program 74.6%

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

      1. exp-prod 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 100.0%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.68 \lor \neg \left(x \leq 1.02 \cdot 10^{-7}\right):\\ \;\;\;\;\frac{e^{-y}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 80.5% accurate, 13.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.45 \cdot 10^{+120}:\\ \;\;\;\;\frac{\frac{x \cdot \left(-y\right)}{x}}{x}\\ \mathbf{elif}\;y \leq 1.25 \cdot 10^{+43}:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{x \cdot y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.45e+120)
   (/ (/ (* x (- y)) x) x)
   (if (<= y 1.25e+43) (/ 1.0 x) (/ y (* x y)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.45e+120) {
		tmp = ((x * -y) / x) / x;
	} else if (y <= 1.25e+43) {
		tmp = 1.0 / x;
	} else {
		tmp = y / (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 (y <= (-1.45d+120)) then
        tmp = ((x * -y) / x) / x
    else if (y <= 1.25d+43) then
        tmp = 1.0d0 / x
    else
        tmp = y / (x * y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -1.45e+120) {
		tmp = ((x * -y) / x) / x;
	} else if (y <= 1.25e+43) {
		tmp = 1.0 / x;
	} else {
		tmp = y / (x * y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.45e+120:
		tmp = ((x * -y) / x) / x
	elif y <= 1.25e+43:
		tmp = 1.0 / x
	else:
		tmp = y / (x * y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.45e+120)
		tmp = Float64(Float64(Float64(x * Float64(-y)) / x) / x);
	elseif (y <= 1.25e+43)
		tmp = Float64(1.0 / x);
	else
		tmp = Float64(y / Float64(x * y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -1.45e+120)
		tmp = ((x * -y) / x) / x;
	elseif (y <= 1.25e+43)
		tmp = 1.0 / x;
	else
		tmp = y / (x * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.45e+120], N[(N[(N[(x * (-y)), $MachinePrecision] / x), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[y, 1.25e+43], N[(1.0 / x), $MachinePrecision], N[(y / N[(x * y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.4500000000000001e120

   1. Initial program 42.3%

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

      1. exp-prod 69.1%

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

   3. Simplified 69.1%

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

   5. Taylor expanded in x around inf 4.1%

      \[\leadsto \color{blue}{-1 \cdot \frac{y}{x} + \frac{1}{x}} \]
   6. 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. unsub-neg 4.1%

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

   7. Simplified 4.1%

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

      1. frac-sub 10.8%

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

      2. associate-/r* 59.4%

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

      3. *-un-lft-identity 59.4%

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

      4. *-commutative 59.4%

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

   9. Applied egg-rr 59.4%

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

   10. Taylor expanded in y around inf 59.4%

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

       1. associate-*r* 59.4%

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

       2. mul-1-neg 59.4%

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

   12. Simplified 59.4%

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

   if -1.4500000000000001e120 < y < 1.2500000000000001e43

   1. Initial program 89.0%

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

      1. exp-prod 89.7%

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

   3. Simplified 89.7%

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

   5. Taylor expanded in x around 0 87.7%

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

   if 1.2500000000000001e43 < y

   1. Initial program 48.7%

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

      1. exp-prod 81.4%

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

   3. Simplified 81.4%

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

   5. Taylor expanded in x around inf 1.8%

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

      1. +-commutative 1.8%

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

      2. mul-1-neg 1.8%

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

      3. unsub-neg 1.8%

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

   7. Simplified 1.8%

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

      1. clear-num 1.8%

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

      2. frac-sub 12.3%

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

      3. *-un-lft-identity 12.3%

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

      4. *-commutative 12.3%

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

      5. *-un-lft-identity 12.3%

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

   9. Applied egg-rr 12.3%

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

   10. Taylor expanded in x around 0 1.8%

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

       1. associate-/l* 1.8%

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

       2. sub-neg 1.8%

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

       3. metadata-eval 1.8%

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

   12. Simplified 1.8%

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

   13. Taylor expanded in y around 0 80.6%

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

4. Final simplification 82.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.45 \cdot 10^{+120}:\\ \;\;\;\;\frac{\frac{x \cdot \left(-y\right)}{x}}{x}\\ \mathbf{elif}\;y \leq 1.25 \cdot 10^{+43}:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{x \cdot y}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 82.6% accurate, 13.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.5:\\ \;\;\;\;\frac{\frac{x - x \cdot y}{x}}{x}\\ \mathbf{elif}\;x \leq 2 \cdot 10^{+41}:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{x \cdot y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -0.5)
   (/ (/ (- x (* x y)) x) x)
   (if (<= x 2e+41) (/ 1.0 x) (/ y (* x y)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -0.5) {
		tmp = ((x - (x * y)) / x) / x;
	} else if (x <= 2e+41) {
		tmp = 1.0 / x;
	} else {
		tmp = y / (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 <= (-0.5d0)) then
        tmp = ((x - (x * y)) / x) / x
    else if (x <= 2d+41) then
        tmp = 1.0d0 / x
    else
        tmp = y / (x * y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -0.5) {
		tmp = ((x - (x * y)) / x) / x;
	} else if (x <= 2e+41) {
		tmp = 1.0 / x;
	} else {
		tmp = y / (x * y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -0.5:
		tmp = ((x - (x * y)) / x) / x
	elif x <= 2e+41:
		tmp = 1.0 / x
	else:
		tmp = y / (x * y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -0.5)
		tmp = Float64(Float64(Float64(x - Float64(x * y)) / x) / x);
	elseif (x <= 2e+41)
		tmp = Float64(1.0 / x);
	else
		tmp = Float64(y / Float64(x * y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -0.5)
		tmp = ((x - (x * y)) / x) / x;
	elseif (x <= 2e+41)
		tmp = 1.0 / x;
	else
		tmp = y / (x * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -0.5], N[(N[(N[(x - N[(x * y), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[x, 2e+41], N[(1.0 / x), $MachinePrecision], N[(y / N[(x * y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.5

   1. Initial program 73.2%

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

      1. exp-prod 72.2%

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

   3. Simplified 72.2%

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

   5. Taylor expanded in x around inf 57.0%

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

      1. +-commutative 57.0%

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

      2. mul-1-neg 57.0%

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

      3. unsub-neg 57.0%

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

   7. Simplified 57.0%

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

      1. frac-sub 33.8%

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

      2. associate-/r* 80.8%

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

      3. *-un-lft-identity 80.8%

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

      4. *-commutative 80.8%

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

   9. Applied egg-rr 80.8%

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

   if -0.5 < x < 2.00000000000000001e41

   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 99.3%

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

   3. Simplified 99.3%

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

   5. Taylor expanded in x around 0 98.4%

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

   if 2.00000000000000001e41 < x

   1. Initial program 77.0%

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

      1. exp-prod 77.0%

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

   3. Simplified 77.0%

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

   5. Taylor expanded in x around inf 59.4%

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

      1. +-commutative 59.4%

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

      2. mul-1-neg 59.4%

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

      3. unsub-neg 59.4%

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

   7. Simplified 59.4%

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

      1. clear-num 59.4%

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

      2. frac-sub 17.1%

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

      3. *-un-lft-identity 17.1%

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

      4. *-commutative 17.1%

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

      5. *-un-lft-identity 17.1%

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

   9. Applied egg-rr 17.1%

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

   10. Taylor expanded in x around 0 59.4%

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

       1. associate-/l* 59.2%

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

       2. sub-neg 59.2%

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

       3. metadata-eval 59.2%

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

   12. Simplified 59.2%

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

   13. Taylor expanded in y around 0 72.6%

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

4. Final simplification 86.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.5:\\ \;\;\;\;\frac{\frac{x - x \cdot y}{x}}{x}\\ \mathbf{elif}\;x \leq 2 \cdot 10^{+41}:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{x \cdot y}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 80.2% accurate, 20.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 3.4 \cdot 10^{+54}:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{x \cdot y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= y 3.4e+54) (/ 1.0 x) (/ y (* x y))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 3.4e+54) {
		tmp = 1.0 / x;
	} else {
		tmp = y / (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 (y <= 3.4d+54) then
        tmp = 1.0d0 / x
    else
        tmp = y / (x * y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 3.4e+54) {
		tmp = 1.0 / x;
	} else {
		tmp = y / (x * y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 3.4e+54:
		tmp = 1.0 / x
	else:
		tmp = y / (x * y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 3.4e+54)
		tmp = Float64(1.0 / x);
	else
		tmp = Float64(y / Float64(x * y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 3.4e+54)
		tmp = 1.0 / x;
	else
		tmp = y / (x * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 3.4e+54], N[(1.0 / x), $MachinePrecision], N[(y / N[(x * y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 3.4000000000000001e54

   1. Initial program 80.9%

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

      1. exp-prod 85.9%

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

   3. Simplified 85.9%

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

   5. Taylor expanded in x around 0 79.2%

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

   if 3.4000000000000001e54 < y

   1. Initial program 49.7%

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

      1. exp-prod 83.1%

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

   3. Simplified 83.1%

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

   5. Taylor expanded in x around inf 1.7%

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

      1. +-commutative 1.7%

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

      2. mul-1-neg 1.7%

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

      3. unsub-neg 1.7%

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

   7. Simplified 1.7%

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

      1. clear-num 1.7%

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

      2. frac-sub 12.5%

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

      3. *-un-lft-identity 12.5%

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

      4. *-commutative 12.5%

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

      5. *-un-lft-identity 12.5%

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

   9. Applied egg-rr 12.5%

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

   10. Taylor expanded in x around 0 1.7%

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

       1. associate-/l* 1.7%

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

       2. sub-neg 1.7%

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

       3. metadata-eval 1.7%

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

   12. Simplified 1.7%

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

   13. Taylor expanded in y around 0 82.3%

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

4. Final simplification 79.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 3.4 \cdot 10^{+54}:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{x \cdot y}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 74.5% 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 75.5%

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

   1. exp-prod 85.4%

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

3. Simplified 85.4%

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

5. Taylor expanded in x around 0 75.4%

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

6. Final simplification 75.4%

   \[\leadsto \frac{1}{x} \]
7. Add Preprocessing

Developer target: 77.7% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024024 
(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))