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

Percentage Accurate: 78.2% → 99.9%

Time: 8.8s

Alternatives: 8

Speedup: 18.8×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 78.2% 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: 99.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -200000 \lor \neg \left(x \leq 7\right):\\ \;\;\;\;\frac{e^{-y}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(e^{x}\right)}^{\log \left(\frac{x}{x + y}\right)}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -200000.0) (not (<= x 7.0)))
   (/ (exp (- y)) x)
   (/ (pow (exp x) (log (/ x (+ x y)))) x)))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -200000.0) || !(x <= 7.0)) {
		tmp = exp(-y) / x;
	} else {
		tmp = pow(exp(x), log((x / (x + y)))) / x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((x <= (-200000.0d0)) .or. (.not. (x <= 7.0d0))) then
        tmp = exp(-y) / x
    else
        tmp = (exp(x) ** log((x / (x + y)))) / x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((x <= -200000.0) || !(x <= 7.0)) {
		tmp = Math.exp(-y) / x;
	} else {
		tmp = Math.pow(Math.exp(x), Math.log((x / (x + y)))) / x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -200000.0) or not (x <= 7.0):
		tmp = math.exp(-y) / x
	else:
		tmp = math.pow(math.exp(x), math.log((x / (x + y)))) / x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -200000.0) || !(x <= 7.0))
		tmp = Float64(exp(Float64(-y)) / x);
	else
		tmp = Float64((exp(x) ^ log(Float64(x / Float64(x + y)))) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((x <= -200000.0) || ~((x <= 7.0)))
		tmp = exp(-y) / x;
	else
		tmp = (exp(x) ^ log((x / (x + y)))) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -200000.0], N[Not[LessEqual[x, 7.0]], $MachinePrecision]], N[(N[Exp[(-y)], $MachinePrecision] / x), $MachinePrecision], N[(N[Power[N[Exp[x], $MachinePrecision], N[Log[N[(x / N[(x + y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2e5 or 7 < x

   1. Initial program 75.6%

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

      1. *-commutative 75.6%

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

      2. exp-to-pow 75.6%

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

   3. Simplified 75.6%

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

   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 -2e5 < x < 7

   1. Initial program 76.8%

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

      1. exp-prod 99.8%

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

   3. Simplified 99.8%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -200000 \lor \neg \left(x \leq 7\right):\\ \;\;\;\;\frac{e^{-y}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(e^{x}\right)}^{\log \left(\frac{x}{x + y}\right)}}{x}\\ \end{array} \]

Alternative 2: 99.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.185 \lor \neg \left(x \leq 0.88\right):\\ \;\;\;\;\frac{e^{-y}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -0.185) (not (<= x 0.88))) (/ (exp (- y)) x) (/ 1.0 x)))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -0.185) || !(x <= 0.88)) {
		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 <= (-0.185d0)) .or. (.not. (x <= 0.88d0))) 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 <= -0.185) || !(x <= 0.88)) {
		tmp = Math.exp(-y) / x;
	} else {
		tmp = 1.0 / x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -0.185) or not (x <= 0.88):
		tmp = math.exp(-y) / x
	else:
		tmp = 1.0 / x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -0.185) || !(x <= 0.88))
		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 <= -0.185) || ~((x <= 0.88)))
		tmp = exp(-y) / x;
	else
		tmp = 1.0 / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -0.185], N[Not[LessEqual[x, 0.88]], $MachinePrecision]], N[(N[Exp[(-y)], $MachinePrecision] / x), $MachinePrecision], N[(1.0 / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -0.185 or 0.880000000000000004 < x

   1. Initial program 75.9%

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

      1. *-commutative 75.9%

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

      2. exp-to-pow 75.9%

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

   3. Simplified 75.9%

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

   4. Taylor expanded in x around inf 99.8%

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

      1. mul-1-neg 99.8%

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

   6. Simplified 99.8%

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

   if -0.185 < x < 0.880000000000000004

   1. Initial program 76.5%

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

      1. exp-prod 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in x around 0 98.6%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.185 \lor \neg \left(x \leq 0.88\right):\\ \;\;\;\;\frac{e^{-y}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x}\\ \end{array} \]

Alternative 3: 85.3% accurate, 12.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.195:\\ \;\;\;\;\frac{1 - y}{x} + \frac{y \cdot y}{x}\\ \mathbf{elif}\;x \leq 5.6:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{x \cdot \left(1 + y \cdot y\right)}{y + 1}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -0.195)
   (+ (/ (- 1.0 y) x) (/ (* y y) x))
   (if (<= x 5.6) (/ 1.0 x) (/ 1.0 (/ (* x (+ 1.0 (* y y))) (+ y 1.0))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -0.195) {
		tmp = ((1.0 - y) / x) + ((y * y) / x);
	} else if (x <= 5.6) {
		tmp = 1.0 / x;
	} else {
		tmp = 1.0 / ((x * (1.0 + (y * y))) / (y + 1.0));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-0.195d0)) then
        tmp = ((1.0d0 - y) / x) + ((y * y) / x)
    else if (x <= 5.6d0) then
        tmp = 1.0d0 / x
    else
        tmp = 1.0d0 / ((x * (1.0d0 + (y * y))) / (y + 1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -0.195) {
		tmp = ((1.0 - y) / x) + ((y * y) / x);
	} else if (x <= 5.6) {
		tmp = 1.0 / x;
	} else {
		tmp = 1.0 / ((x * (1.0 + (y * y))) / (y + 1.0));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -0.195:
		tmp = ((1.0 - y) / x) + ((y * y) / x)
	elif x <= 5.6:
		tmp = 1.0 / x
	else:
		tmp = 1.0 / ((x * (1.0 + (y * y))) / (y + 1.0))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -0.195)
		tmp = Float64(Float64(Float64(1.0 - y) / x) + Float64(Float64(y * y) / x));
	elseif (x <= 5.6)
		tmp = Float64(1.0 / x);
	else
		tmp = Float64(1.0 / Float64(Float64(x * Float64(1.0 + Float64(y * y))) / Float64(y + 1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -0.195)
		tmp = ((1.0 - y) / x) + ((y * y) / x);
	elseif (x <= 5.6)
		tmp = 1.0 / x;
	else
		tmp = 1.0 / ((x * (1.0 + (y * y))) / (y + 1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -0.195], N[(N[(N[(1.0 - y), $MachinePrecision] / x), $MachinePrecision] + N[(N[(y * y), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 5.6], N[(1.0 / x), $MachinePrecision], N[(1.0 / N[(N[(x * N[(1.0 + N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.19500000000000001

   1. Initial program 72.7%

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

      1. exp-prod 71.6%

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

   3. Simplified 71.6%

      \[\leadsto \color{blue}{\frac{{\left(e^{x}\right)}^{\log \left(\frac{x}{x + y}\right)}}{x}} \]
   4. Step-by-step derivation

      1. clear-num 71.6%

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

      2. inv-pow 71.6%

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

      3. add-exp-log 71.6%

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

      4. log-pow 13.9%

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

      5. add-log-exp 72.7%

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

      6. pow-to-exp 72.7%

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

   5. Applied egg-rr 72.7%

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

      1. unpow-1 72.7%

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

      2. +-commutative 72.7%

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

   7. Simplified 72.7%

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

   8. Taylor expanded in y around 0 68.7%

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

   9. Taylor expanded in y around 0 77.1%

      \[\leadsto \color{blue}{\frac{{y}^{2}}{x} + \left(\frac{1}{x} + -1 \cdot \frac{y}{x}\right)} \]
   10. Step-by-step derivation

       1. +-commutative 77.1%

          \[\leadsto \color{blue}{\left(\frac{1}{x} + -1 \cdot \frac{y}{x}\right) + \frac{{y}^{2}}{x}} \]

       2. neg-mul-1 77.1%

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

       3. sub-neg 77.1%

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

       4. div-sub 77.1%

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

       5. unpow2 77.1%

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

   11. Simplified 77.1%

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

   if -0.19500000000000001 < x < 5.5999999999999996

   1. Initial program 76.8%

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

      1. exp-prod 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in x around 0 98.1%

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

   if 5.5999999999999996 < x

   1. Initial program 78.8%

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

      1. exp-prod 78.8%

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

   3. Simplified 78.8%

      \[\leadsto \color{blue}{\frac{{\left(e^{x}\right)}^{\log \left(\frac{x}{x + y}\right)}}{x}} \]
   4. Step-by-step derivation

      1. clear-num 78.8%

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

      2. inv-pow 78.8%

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

      3. add-exp-log 78.8%

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

      4. log-pow 23.9%

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

      5. add-log-exp 78.8%

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

      6. pow-to-exp 78.8%

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

   5. Applied egg-rr 78.8%

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

      1. unpow-1 78.8%

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

      2. +-commutative 78.8%

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

   7. Simplified 78.8%

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

   8. Taylor expanded in y around 0 69.4%

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

   10. Applied egg-rr 75.5%

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

4. Final simplification 86.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.195:\\ \;\;\;\;\frac{1 - y}{x} + \frac{y \cdot y}{x}\\ \mathbf{elif}\;x \leq 5.6:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{x \cdot \left(1 + y \cdot y\right)}{y + 1}}\\ \end{array} \]

Alternative 4: 83.5% accurate, 16.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.108:\\ \;\;\;\;\frac{1 - y}{x} + \frac{y \cdot y}{x}\\ \mathbf{elif}\;x \leq 0.24:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x \cdot \left(y + 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -0.108)
   (+ (/ (- 1.0 y) x) (/ (* y y) x))
   (if (<= x 0.24) (/ 1.0 x) (/ 1.0 (* x (+ y 1.0))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -0.108) {
		tmp = ((1.0 - y) / x) + ((y * y) / x);
	} else if (x <= 0.24) {
		tmp = 1.0 / x;
	} else {
		tmp = 1.0 / (x * (y + 1.0));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-0.108d0)) then
        tmp = ((1.0d0 - y) / x) + ((y * y) / x)
    else if (x <= 0.24d0) then
        tmp = 1.0d0 / x
    else
        tmp = 1.0d0 / (x * (y + 1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -0.108) {
		tmp = ((1.0 - y) / x) + ((y * y) / x);
	} else if (x <= 0.24) {
		tmp = 1.0 / x;
	} else {
		tmp = 1.0 / (x * (y + 1.0));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -0.108:
		tmp = ((1.0 - y) / x) + ((y * y) / x)
	elif x <= 0.24:
		tmp = 1.0 / x
	else:
		tmp = 1.0 / (x * (y + 1.0))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -0.108)
		tmp = Float64(Float64(Float64(1.0 - y) / x) + Float64(Float64(y * y) / x));
	elseif (x <= 0.24)
		tmp = Float64(1.0 / x);
	else
		tmp = Float64(1.0 / Float64(x * Float64(y + 1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -0.108)
		tmp = ((1.0 - y) / x) + ((y * y) / x);
	elseif (x <= 0.24)
		tmp = 1.0 / x;
	else
		tmp = 1.0 / (x * (y + 1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -0.108], N[(N[(N[(1.0 - y), $MachinePrecision] / x), $MachinePrecision] + N[(N[(y * y), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.24], N[(1.0 / x), $MachinePrecision], N[(1.0 / N[(x * N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.107999999999999999

   1. Initial program 72.7%

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

      1. exp-prod 71.6%

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

   3. Simplified 71.6%

      \[\leadsto \color{blue}{\frac{{\left(e^{x}\right)}^{\log \left(\frac{x}{x + y}\right)}}{x}} \]
   4. Step-by-step derivation

      1. clear-num 71.6%

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

      2. inv-pow 71.6%

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

      3. add-exp-log 71.6%

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

      4. log-pow 13.9%

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

      5. add-log-exp 72.7%

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

      6. pow-to-exp 72.7%

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

   5. Applied egg-rr 72.7%

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

      1. unpow-1 72.7%

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

      2. +-commutative 72.7%

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

   7. Simplified 72.7%

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

   8. Taylor expanded in y around 0 68.7%

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

   9. Taylor expanded in y around 0 77.1%

      \[\leadsto \color{blue}{\frac{{y}^{2}}{x} + \left(\frac{1}{x} + -1 \cdot \frac{y}{x}\right)} \]
   10. Step-by-step derivation

       1. +-commutative 77.1%

          \[\leadsto \color{blue}{\left(\frac{1}{x} + -1 \cdot \frac{y}{x}\right) + \frac{{y}^{2}}{x}} \]

       2. neg-mul-1 77.1%

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

       3. sub-neg 77.1%

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

       4. div-sub 77.1%

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

       5. unpow2 77.1%

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

   11. Simplified 77.1%

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

   if -0.107999999999999999 < x < 0.23999999999999999

   1. Initial program 76.5%

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

      1. exp-prod 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in x around 0 98.6%

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

   if 0.23999999999999999 < x

   1. Initial program 79.4%

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

      1. exp-prod 79.4%

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

   3. Simplified 79.4%

      \[\leadsto \color{blue}{\frac{{\left(e^{x}\right)}^{\log \left(\frac{x}{x + y}\right)}}{x}} \]
   4. Step-by-step derivation

      1. clear-num 79.4%

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

      2. inv-pow 79.4%

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

      3. add-exp-log 79.4%

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

      4. log-pow 26.1%

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

      5. add-log-exp 79.4%

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

      6. pow-to-exp 79.4%

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

   5. Applied egg-rr 79.4%

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

      1. unpow-1 79.4%

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

      2. +-commutative 79.4%

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

   7. Simplified 79.4%

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

   8. Taylor expanded in y around 0 69.8%

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

      1. distribute-lft1-in 69.8%

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

   10. Applied egg-rr 69.8%

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

4. Final simplification 84.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.108:\\ \;\;\;\;\frac{1 - y}{x} + \frac{y \cdot y}{x}\\ \mathbf{elif}\;x \leq 0.24:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x \cdot \left(y + 1\right)}\\ \end{array} \]

Alternative 5: 81.4% accurate, 18.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -180000 \lor \neg \left(x \leq 0.5\right):\\ \;\;\;\;\frac{1}{x \cdot \left(y + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -180000.0) (not (<= x 0.5))) (/ 1.0 (* x (+ y 1.0))) (/ 1.0 x)))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -180000.0) || !(x <= 0.5)) {
		tmp = 1.0 / (x * (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 <= (-180000.0d0)) .or. (.not. (x <= 0.5d0))) then
        tmp = 1.0d0 / (x * (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 <= -180000.0) || !(x <= 0.5)) {
		tmp = 1.0 / (x * (y + 1.0));
	} else {
		tmp = 1.0 / x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -180000.0) or not (x <= 0.5):
		tmp = 1.0 / (x * (y + 1.0))
	else:
		tmp = 1.0 / x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -180000.0) || !(x <= 0.5))
		tmp = Float64(1.0 / Float64(x * Float64(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 <= -180000.0) || ~((x <= 0.5)))
		tmp = 1.0 / (x * (y + 1.0));
	else
		tmp = 1.0 / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -180000.0], N[Not[LessEqual[x, 0.5]], $MachinePrecision]], N[(1.0 / N[(x * N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.8e5 or 0.5 < x

   1. Initial program 75.9%

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

      1. exp-prod 75.4%

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

   3. Simplified 75.4%

      \[\leadsto \color{blue}{\frac{{\left(e^{x}\right)}^{\log \left(\frac{x}{x + y}\right)}}{x}} \]
   4. Step-by-step derivation

      1. clear-num 75.4%

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

      2. inv-pow 75.4%

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

      3. add-exp-log 75.4%

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

      4. log-pow 19.9%

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

      5. add-log-exp 75.9%

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

      6. pow-to-exp 75.9%

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

   5. Applied egg-rr 75.9%

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

      1. unpow-1 75.9%

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

      2. +-commutative 75.9%

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

   7. Simplified 75.9%

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

   8. Taylor expanded in y around 0 69.2%

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

      1. distribute-lft1-in 69.2%

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

   10. Applied egg-rr 69.2%

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

   if -1.8e5 < x < 0.5

   1. Initial program 76.5%

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

      1. exp-prod 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in x around 0 98.6%

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

4. Final simplification 82.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -180000 \lor \neg \left(x \leq 0.5\right):\\ \;\;\;\;\frac{1}{x \cdot \left(y + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x}\\ \end{array} \]

Alternative 6: 81.4% accurate, 18.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2150:\\ \;\;\;\;\frac{1}{x + x \cdot y}\\ \mathbf{elif}\;x \leq 0.19:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x \cdot \left(y + 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -2150.0)
   (/ 1.0 (+ x (* x y)))
   (if (<= x 0.19) (/ 1.0 x) (/ 1.0 (* x (+ y 1.0))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -2150.0) {
		tmp = 1.0 / (x + (x * y));
	} else if (x <= 0.19) {
		tmp = 1.0 / x;
	} else {
		tmp = 1.0 / (x * (y + 1.0));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-2150.0d0)) then
        tmp = 1.0d0 / (x + (x * y))
    else if (x <= 0.19d0) then
        tmp = 1.0d0 / x
    else
        tmp = 1.0d0 / (x * (y + 1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -2150.0) {
		tmp = 1.0 / (x + (x * y));
	} else if (x <= 0.19) {
		tmp = 1.0 / x;
	} else {
		tmp = 1.0 / (x * (y + 1.0));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -2150.0:
		tmp = 1.0 / (x + (x * y))
	elif x <= 0.19:
		tmp = 1.0 / x
	else:
		tmp = 1.0 / (x * (y + 1.0))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -2150.0)
		tmp = Float64(1.0 / Float64(x + Float64(x * y)));
	elseif (x <= 0.19)
		tmp = Float64(1.0 / x);
	else
		tmp = Float64(1.0 / Float64(x * Float64(y + 1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -2150.0)
		tmp = 1.0 / (x + (x * y));
	elseif (x <= 0.19)
		tmp = 1.0 / x;
	else
		tmp = 1.0 / (x * (y + 1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -2150.0], N[(1.0 / N[(x + N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.19], N[(1.0 / x), $MachinePrecision], N[(1.0 / N[(x * N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -2150

   1. Initial program 72.7%

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

      1. exp-prod 71.6%

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

   3. Simplified 71.6%

      \[\leadsto \color{blue}{\frac{{\left(e^{x}\right)}^{\log \left(\frac{x}{x + y}\right)}}{x}} \]
   4. Step-by-step derivation

      1. clear-num 71.6%

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

      2. inv-pow 71.6%

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

      3. add-exp-log 71.6%

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

      4. log-pow 13.9%

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

      5. add-log-exp 72.7%

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

      6. pow-to-exp 72.7%

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

   5. Applied egg-rr 72.7%

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

      1. unpow-1 72.7%

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

      2. +-commutative 72.7%

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

   7. Simplified 72.7%

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

   8. Taylor expanded in y around 0 68.7%

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

   if -2150 < x < 0.19

   1. Initial program 76.5%

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

      1. exp-prod 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in x around 0 98.6%

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

   if 0.19 < x

   1. Initial program 79.4%

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

      1. exp-prod 79.4%

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

   3. Simplified 79.4%

      \[\leadsto \color{blue}{\frac{{\left(e^{x}\right)}^{\log \left(\frac{x}{x + y}\right)}}{x}} \]
   4. Step-by-step derivation

      1. clear-num 79.4%

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

      2. inv-pow 79.4%

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

      3. add-exp-log 79.4%

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

      4. log-pow 26.1%

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

      5. add-log-exp 79.4%

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

      6. pow-to-exp 79.4%

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

   5. Applied egg-rr 79.4%

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

      1. unpow-1 79.4%

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

      2. +-commutative 79.4%

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

   7. Simplified 79.4%

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

   8. Taylor expanded in y around 0 69.8%

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

      1. distribute-lft1-in 69.8%

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

   10. Applied egg-rr 69.8%

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

4. Final simplification 82.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2150:\\ \;\;\;\;\frac{1}{x + x \cdot y}\\ \mathbf{elif}\;x \leq 0.19:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x \cdot \left(y + 1\right)}\\ \end{array} \]

Alternative 7: 83.5% accurate, 18.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.082:\\ \;\;\;\;\frac{\frac{x - x \cdot y}{x}}{x}\\ \mathbf{elif}\;x \leq 0.3:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x \cdot \left(y + 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -0.082)
   (/ (/ (- x (* x y)) x) x)
   (if (<= x 0.3) (/ 1.0 x) (/ 1.0 (* x (+ y 1.0))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -0.082) {
		tmp = ((x - (x * y)) / x) / x;
	} else if (x <= 0.3) {
		tmp = 1.0 / x;
	} else {
		tmp = 1.0 / (x * (y + 1.0));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-0.082d0)) then
        tmp = ((x - (x * y)) / x) / x
    else if (x <= 0.3d0) then
        tmp = 1.0d0 / x
    else
        tmp = 1.0d0 / (x * (y + 1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -0.082) {
		tmp = ((x - (x * y)) / x) / x;
	} else if (x <= 0.3) {
		tmp = 1.0 / x;
	} else {
		tmp = 1.0 / (x * (y + 1.0));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -0.082:
		tmp = ((x - (x * y)) / x) / x
	elif x <= 0.3:
		tmp = 1.0 / x
	else:
		tmp = 1.0 / (x * (y + 1.0))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -0.082)
		tmp = Float64(Float64(Float64(x - Float64(x * y)) / x) / x);
	elseif (x <= 0.3)
		tmp = Float64(1.0 / x);
	else
		tmp = Float64(1.0 / Float64(x * Float64(y + 1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -0.082)
		tmp = ((x - (x * y)) / x) / x;
	elseif (x <= 0.3)
		tmp = 1.0 / x;
	else
		tmp = 1.0 / (x * (y + 1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -0.082], N[(N[(N[(x - N[(x * y), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[x, 0.3], N[(1.0 / x), $MachinePrecision], N[(1.0 / N[(x * N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.0820000000000000034

   1. Initial program 72.7%

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

      1. exp-prod 71.6%

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

   3. Simplified 71.6%

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

   4. Taylor expanded in x around inf 59.8%

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

      1. mul-1-neg 59.8%

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

      2. unsub-neg 59.8%

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

   6. Simplified 59.8%

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

      1. frac-sub 30.6%

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

      2. associate-/r* 75.7%

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

      3. *-un-lft-identity 75.7%

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

      4. *-commutative 75.7%

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

   8. Applied egg-rr 75.7%

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

   if -0.0820000000000000034 < x < 0.299999999999999989

   1. Initial program 76.5%

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

      1. exp-prod 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in x around 0 98.6%

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

   if 0.299999999999999989 < x

   1. Initial program 79.4%

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

      1. exp-prod 79.4%

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

   3. Simplified 79.4%

      \[\leadsto \color{blue}{\frac{{\left(e^{x}\right)}^{\log \left(\frac{x}{x + y}\right)}}{x}} \]
   4. Step-by-step derivation

      1. clear-num 79.4%

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

      2. inv-pow 79.4%

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

      3. add-exp-log 79.4%

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

      4. log-pow 26.1%

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

      5. add-log-exp 79.4%

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

      6. pow-to-exp 79.4%

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

   5. Applied egg-rr 79.4%

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

      1. unpow-1 79.4%

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

      2. +-commutative 79.4%

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

   7. Simplified 79.4%

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

   8. Taylor expanded in y around 0 69.8%

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

      1. distribute-lft1-in 69.8%

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

   10. Applied egg-rr 69.8%

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

4. Final simplification 84.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.082:\\ \;\;\;\;\frac{\frac{x - x \cdot y}{x}}{x}\\ \mathbf{elif}\;x \leq 0.3:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x \cdot \left(y + 1\right)}\\ \end{array} \]

Alternative 8: 74.8% accurate, 69.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 76.2%

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

   1. exp-prod 86.4%

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

3. Simplified 86.4%

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

4. Taylor expanded in x around 0 76.4%

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

5. Final simplification 76.4%

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

Developer target: 77.8% 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 2023199 
(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))