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

Percentage Accurate: 77.9% → 99.2%

Time: 12.2s

Alternatives: 8

Speedup: 12.3×

• Could not converge on a ground truth

  1. reg0 = (bf "-2.257388120145214377611084329696508363185e187")
  2. reg1 = (bf "-1.188155307282863516647948613147577492309e-146")

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: 77.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{+75} \lor \neg \left(x \leq 3.35 \cdot 10^{-6}\right):\\ \;\;\;\;\frac{e^{-y}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(e^{x}\right)}^{\log \left(\frac{x}{x + y}\right)}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -1e+75) (not (<= x 3.35e-6)))
   (/ (exp (- y)) x)
   (/ (pow (exp x) (log (/ x (+ x y)))) x)))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -1e+75) || !(x <= 3.35e-6)) {
		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 <= (-1d+75)) .or. (.not. (x <= 3.35d-6))) 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 <= -1e+75) || !(x <= 3.35e-6)) {
		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 <= -1e+75) or not (x <= 3.35e-6):
		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 <= -1e+75) || !(x <= 3.35e-6))
		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 <= -1e+75) || ~((x <= 3.35e-6)))
		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, -1e+75], N[Not[LessEqual[x, 3.35e-6]], $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 < -9.99999999999999927e74 or 3.35e-6 < x

   1. Initial program 75.0%

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

      1. *-commutative 75.0%

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

      2. exp-to-pow 75.0%

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

   3. Simplified 75.0%

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

   5. Taylor expanded in x around inf 100.0%

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

      1. mul-1-neg 100.0%

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

   7. Simplified 100.0%

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

   if -9.99999999999999927e74 < x < 3.35e-6

   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. Add Preprocessing
3. Recombined 2 regimes into one program.

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{+75} \lor \neg \left(x \leq 3.35 \cdot 10^{-6}\right):\\ \;\;\;\;\frac{e^{-y}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(e^{x}\right)}^{\log \left(\frac{x}{x + y}\right)}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 98.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.75 \cdot 10^{+22} \lor \neg \left(x \leq 3.35 \cdot 10^{-6}\right):\\ \;\;\;\;\frac{e^{-y}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -1.75e+22) (not (<= x 3.35e-6))) (/ (exp (- y)) x) (/ 1.0 x)))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -1.75e+22) || !(x <= 3.35e-6)) {
		tmp = exp(-y) / x;
	} else {
		tmp = 1.0 / x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((x <= (-1.75d+22)) .or. (.not. (x <= 3.35d-6))) then
        tmp = exp(-y) / x
    else
        tmp = 1.0d0 / x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((x <= -1.75e+22) || !(x <= 3.35e-6)) {
		tmp = Math.exp(-y) / x;
	} else {
		tmp = 1.0 / x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -1.75e+22) or not (x <= 3.35e-6):
		tmp = math.exp(-y) / x
	else:
		tmp = 1.0 / x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -1.75e+22) || !(x <= 3.35e-6))
		tmp = Float64(exp(Float64(-y)) / x);
	else
		tmp = Float64(1.0 / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((x <= -1.75e+22) || ~((x <= 3.35e-6)))
		tmp = exp(-y) / x;
	else
		tmp = 1.0 / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -1.75e+22], N[Not[LessEqual[x, 3.35e-6]], $MachinePrecision]], N[(N[Exp[(-y)], $MachinePrecision] / x), $MachinePrecision], N[(1.0 / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.75e22 or 3.35e-6 < x

   1. Initial program 76.8%

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

      1. *-commutative 76.8%

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

      2. exp-to-pow 76.8%

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

   3. Simplified 76.8%

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

   5. Taylor expanded in x around inf 100.0%

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

      1. mul-1-neg 100.0%

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

   7. Simplified 100.0%

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

   if -1.75e22 < x < 3.35e-6

   1. Initial program 74.8%

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

      1. exp-prod 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in x around 0 98.9%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.75 \cdot 10^{+22} \lor \neg \left(x \leq 3.35 \cdot 10^{-6}\right):\\ \;\;\;\;\frac{e^{-y}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 86.3% accurate, 6.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.75 \cdot 10^{+22}:\\ \;\;\;\;\frac{1 + y \cdot \left(y \cdot \left(0.5 + y \cdot -0.16666666666666666\right) + -1\right)}{x}\\ \mathbf{elif}\;x \leq 3.35 \cdot 10^{-6}:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x + y \cdot \left(x - y \cdot \left(x \cdot \left(0.5 + \frac{1}{x} \cdot 0.5\right) - x\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1.75e+22)
   (/ (+ 1.0 (* y (+ (* y (+ 0.5 (* y -0.16666666666666666))) -1.0))) x)
   (if (<= x 3.35e-6)
     (/ 1.0 x)
     (/ 1.0 (+ x (* y (- x (* y (- (* x (+ 0.5 (* (/ 1.0 x) 0.5))) x)))))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.75e+22) {
		tmp = (1.0 + (y * ((y * (0.5 + (y * -0.16666666666666666))) + -1.0))) / x;
	} else if (x <= 3.35e-6) {
		tmp = 1.0 / x;
	} else {
		tmp = 1.0 / (x + (y * (x - (y * ((x * (0.5 + ((1.0 / x) * 0.5))) - x)))));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-1.75d+22)) then
        tmp = (1.0d0 + (y * ((y * (0.5d0 + (y * (-0.16666666666666666d0)))) + (-1.0d0)))) / x
    else if (x <= 3.35d-6) then
        tmp = 1.0d0 / x
    else
        tmp = 1.0d0 / (x + (y * (x - (y * ((x * (0.5d0 + ((1.0d0 / x) * 0.5d0))) - x)))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -1.75e+22) {
		tmp = (1.0 + (y * ((y * (0.5 + (y * -0.16666666666666666))) + -1.0))) / x;
	} else if (x <= 3.35e-6) {
		tmp = 1.0 / x;
	} else {
		tmp = 1.0 / (x + (y * (x - (y * ((x * (0.5 + ((1.0 / x) * 0.5))) - x)))));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -1.75e+22:
		tmp = (1.0 + (y * ((y * (0.5 + (y * -0.16666666666666666))) + -1.0))) / x
	elif x <= 3.35e-6:
		tmp = 1.0 / x
	else:
		tmp = 1.0 / (x + (y * (x - (y * ((x * (0.5 + ((1.0 / x) * 0.5))) - x)))))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1.75e+22)
		tmp = Float64(Float64(1.0 + Float64(y * Float64(Float64(y * Float64(0.5 + Float64(y * -0.16666666666666666))) + -1.0))) / x);
	elseif (x <= 3.35e-6)
		tmp = Float64(1.0 / x);
	else
		tmp = Float64(1.0 / Float64(x + Float64(y * Float64(x - Float64(y * Float64(Float64(x * Float64(0.5 + Float64(Float64(1.0 / x) * 0.5))) - x))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1.75e+22)
		tmp = (1.0 + (y * ((y * (0.5 + (y * -0.16666666666666666))) + -1.0))) / x;
	elseif (x <= 3.35e-6)
		tmp = 1.0 / x;
	else
		tmp = 1.0 / (x + (y * (x - (y * ((x * (0.5 + ((1.0 / x) * 0.5))) - x)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1.75e+22], N[(N[(1.0 + N[(y * N[(N[(y * N[(0.5 + N[(y * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[x, 3.35e-6], N[(1.0 / x), $MachinePrecision], N[(1.0 / N[(x + N[(y * N[(x - N[(y * N[(N[(x * N[(0.5 + N[(N[(1.0 / x), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.75e22

   1. Initial program 70.3%

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

      1. *-commutative 70.3%

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

      2. exp-to-pow 70.3%

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

   3. Simplified 70.3%

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

   5. Taylor expanded in x around inf 100.0%

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

      1. mul-1-neg 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in y around 0 72.7%

      \[\leadsto \frac{\color{blue}{1 + y \cdot \left(y \cdot \left(0.5 + -0.16666666666666666 \cdot y\right) - 1\right)}}{x} \]

   if -1.75e22 < x < 3.35e-6

   1. Initial program 74.8%

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

      1. exp-prod 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in x around 0 98.9%

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

   if 3.35e-6 < x

   1. Initial program 82.4%

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

      1. exp-prod 82.4%

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

   3. Simplified 82.4%

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

      1. clear-num 82.4%

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

      2. add-exp-log 77.3%

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

      3. add-exp-log 77.3%

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

      4. div-exp 77.3%

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

      5. pow-exp 77.3%

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

      6. add-log-exp 77.3%

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

      7. log-pow 77.3%

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

      8. div-exp 77.3%

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

      9. add-exp-log 82.4%

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

      10. add-exp-log 82.4%

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

      11. inv-pow 82.4%

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

   6. Applied egg-rr 82.4%

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

      1. unpow-1 82.4%

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

   8. Simplified 82.4%

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

   9. Taylor expanded in y around 0 84.4%

      \[\leadsto \frac{1}{\color{blue}{x + y \cdot \left(-1 \cdot \left(y \cdot \left(-1 \cdot x + x \cdot \left(0.5 + 0.5 \cdot \frac{1}{x}\right)\right)\right) - -1 \cdot x\right)}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 88.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.75 \cdot 10^{+22}:\\ \;\;\;\;\frac{1 + y \cdot \left(y \cdot \left(0.5 + y \cdot -0.16666666666666666\right) + -1\right)}{x}\\ \mathbf{elif}\;x \leq 3.35 \cdot 10^{-6}:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x + y \cdot \left(x - y \cdot \left(x \cdot \left(0.5 + \frac{1}{x} \cdot 0.5\right) - x\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 84.1% accurate, 10.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.75 \cdot 10^{+22}:\\ \;\;\;\;\frac{1 + y \cdot \left(y \cdot \left(0.5 + y \cdot -0.16666666666666666\right) + -1\right)}{x}\\ \mathbf{elif}\;x \leq 3.35 \cdot 10^{-6}:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x + x \cdot y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1.75e+22)
   (/ (+ 1.0 (* y (+ (* y (+ 0.5 (* y -0.16666666666666666))) -1.0))) x)
   (if (<= x 3.35e-6) (/ 1.0 x) (/ 1.0 (+ x (* x y))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.75e+22) {
		tmp = (1.0 + (y * ((y * (0.5 + (y * -0.16666666666666666))) + -1.0))) / x;
	} else if (x <= 3.35e-6) {
		tmp = 1.0 / x;
	} else {
		tmp = 1.0 / (x + (x * y));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-1.75d+22)) then
        tmp = (1.0d0 + (y * ((y * (0.5d0 + (y * (-0.16666666666666666d0)))) + (-1.0d0)))) / x
    else if (x <= 3.35d-6) then
        tmp = 1.0d0 / x
    else
        tmp = 1.0d0 / (x + (x * y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -1.75e+22) {
		tmp = (1.0 + (y * ((y * (0.5 + (y * -0.16666666666666666))) + -1.0))) / x;
	} else if (x <= 3.35e-6) {
		tmp = 1.0 / x;
	} else {
		tmp = 1.0 / (x + (x * y));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -1.75e+22:
		tmp = (1.0 + (y * ((y * (0.5 + (y * -0.16666666666666666))) + -1.0))) / x
	elif x <= 3.35e-6:
		tmp = 1.0 / x
	else:
		tmp = 1.0 / (x + (x * y))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1.75e+22)
		tmp = Float64(Float64(1.0 + Float64(y * Float64(Float64(y * Float64(0.5 + Float64(y * -0.16666666666666666))) + -1.0))) / x);
	elseif (x <= 3.35e-6)
		tmp = Float64(1.0 / x);
	else
		tmp = Float64(1.0 / Float64(x + Float64(x * y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1.75e+22)
		tmp = (1.0 + (y * ((y * (0.5 + (y * -0.16666666666666666))) + -1.0))) / x;
	elseif (x <= 3.35e-6)
		tmp = 1.0 / x;
	else
		tmp = 1.0 / (x + (x * y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1.75e+22], N[(N[(1.0 + N[(y * N[(N[(y * N[(0.5 + N[(y * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[x, 3.35e-6], N[(1.0 / x), $MachinePrecision], N[(1.0 / N[(x + N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.75e22

   1. Initial program 70.3%

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

      1. *-commutative 70.3%

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

      2. exp-to-pow 70.3%

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

   3. Simplified 70.3%

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

   5. Taylor expanded in x around inf 100.0%

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

      1. mul-1-neg 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in y around 0 72.7%

      \[\leadsto \frac{\color{blue}{1 + y \cdot \left(y \cdot \left(0.5 + -0.16666666666666666 \cdot y\right) - 1\right)}}{x} \]

   if -1.75e22 < x < 3.35e-6

   1. Initial program 74.8%

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

      1. exp-prod 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in x around 0 98.9%

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

   if 3.35e-6 < x

   1. Initial program 82.4%

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

      1. exp-prod 82.4%

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

   3. Simplified 82.4%

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

      1. clear-num 82.4%

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

      2. add-exp-log 77.3%

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

      3. add-exp-log 77.3%

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

      4. div-exp 77.3%

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

      5. pow-exp 77.3%

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

      6. add-log-exp 77.3%

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

      7. log-pow 77.3%

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

      8. div-exp 77.3%

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

      9. add-exp-log 82.4%

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

      10. add-exp-log 82.4%

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

      11. inv-pow 82.4%

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

   6. Applied egg-rr 82.4%

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

      1. unpow-1 82.4%

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

   8. Simplified 82.4%

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

   9. Taylor expanded in y around 0 80.7%

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

4. Final simplification 87.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.75 \cdot 10^{+22}:\\ \;\;\;\;\frac{1 + y \cdot \left(y \cdot \left(0.5 + y \cdot -0.16666666666666666\right) + -1\right)}{x}\\ \mathbf{elif}\;x \leq 3.35 \cdot 10^{-6}:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x + x \cdot y}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 84.0% accurate, 11.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.75 \cdot 10^{+22}:\\ \;\;\;\;\frac{1 + y \cdot \left(-1 + y \cdot \left(y \cdot -0.16666666666666666\right)\right)}{x}\\ \mathbf{elif}\;x \leq 3.35 \cdot 10^{-6}:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x + x \cdot y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1.75e+22)
   (/ (+ 1.0 (* y (+ -1.0 (* y (* y -0.16666666666666666))))) x)
   (if (<= x 3.35e-6) (/ 1.0 x) (/ 1.0 (+ x (* x y))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.75e+22) {
		tmp = (1.0 + (y * (-1.0 + (y * (y * -0.16666666666666666))))) / x;
	} else if (x <= 3.35e-6) {
		tmp = 1.0 / x;
	} else {
		tmp = 1.0 / (x + (x * y));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-1.75d+22)) then
        tmp = (1.0d0 + (y * ((-1.0d0) + (y * (y * (-0.16666666666666666d0)))))) / x
    else if (x <= 3.35d-6) then
        tmp = 1.0d0 / x
    else
        tmp = 1.0d0 / (x + (x * y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -1.75e+22) {
		tmp = (1.0 + (y * (-1.0 + (y * (y * -0.16666666666666666))))) / x;
	} else if (x <= 3.35e-6) {
		tmp = 1.0 / x;
	} else {
		tmp = 1.0 / (x + (x * y));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -1.75e+22:
		tmp = (1.0 + (y * (-1.0 + (y * (y * -0.16666666666666666))))) / x
	elif x <= 3.35e-6:
		tmp = 1.0 / x
	else:
		tmp = 1.0 / (x + (x * y))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1.75e+22)
		tmp = Float64(Float64(1.0 + Float64(y * Float64(-1.0 + Float64(y * Float64(y * -0.16666666666666666))))) / x);
	elseif (x <= 3.35e-6)
		tmp = Float64(1.0 / x);
	else
		tmp = Float64(1.0 / Float64(x + Float64(x * y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1.75e+22)
		tmp = (1.0 + (y * (-1.0 + (y * (y * -0.16666666666666666))))) / x;
	elseif (x <= 3.35e-6)
		tmp = 1.0 / x;
	else
		tmp = 1.0 / (x + (x * y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1.75e+22], N[(N[(1.0 + N[(y * N[(-1.0 + N[(y * N[(y * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[x, 3.35e-6], N[(1.0 / x), $MachinePrecision], N[(1.0 / N[(x + N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.75e22

   1. Initial program 70.3%

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

      1. *-commutative 70.3%

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

      2. exp-to-pow 70.3%

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

   3. Simplified 70.3%

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

   5. Taylor expanded in x around inf 100.0%

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

      1. mul-1-neg 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in y around 0 72.7%

      \[\leadsto \frac{\color{blue}{1 + y \cdot \left(y \cdot \left(0.5 + -0.16666666666666666 \cdot y\right) - 1\right)}}{x} \]

   9. Taylor expanded in y around inf 72.1%

      \[\leadsto \frac{1 + y \cdot \left(y \cdot \color{blue}{\left(-0.16666666666666666 \cdot y\right)} - 1\right)}{x} \]
   10. Step-by-step derivation

       1. *-commutative 72.1%

          \[\leadsto \frac{1 + y \cdot \left(y \cdot \color{blue}{\left(y \cdot -0.16666666666666666\right)} - 1\right)}{x} \]

   11. Simplified 72.1%

       \[\leadsto \frac{1 + y \cdot \left(y \cdot \color{blue}{\left(y \cdot -0.16666666666666666\right)} - 1\right)}{x} \]

   if -1.75e22 < x < 3.35e-6

   1. Initial program 74.8%

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

      1. exp-prod 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in x around 0 98.9%

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

   if 3.35e-6 < x

   1. Initial program 82.4%

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

      1. exp-prod 82.4%

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

   3. Simplified 82.4%

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

      1. clear-num 82.4%

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

      2. add-exp-log 77.3%

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

      3. add-exp-log 77.3%

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

      4. div-exp 77.3%

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

      5. pow-exp 77.3%

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

      6. add-log-exp 77.3%

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

      7. log-pow 77.3%

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

      8. div-exp 77.3%

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

      9. add-exp-log 82.4%

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

      10. add-exp-log 82.4%

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

      11. inv-pow 82.4%

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

   6. Applied egg-rr 82.4%

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

      1. unpow-1 82.4%

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

   8. Simplified 82.4%

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

   9. Taylor expanded in y around 0 80.7%

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

4. Final simplification 86.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.75 \cdot 10^{+22}:\\ \;\;\;\;\frac{1 + y \cdot \left(-1 + y \cdot \left(y \cdot -0.16666666666666666\right)\right)}{x}\\ \mathbf{elif}\;x \leq 3.35 \cdot 10^{-6}:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x + x \cdot y}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 81.0% accurate, 12.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4 \cdot 10^{+91} \lor \neg \left(x \leq 3.35 \cdot 10^{-6}\right):\\ \;\;\;\;\frac{1}{x + x \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -4e+91) (not (<= x 3.35e-6))) (/ 1.0 (+ x (* x y))) (/ 1.0 x)))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -4e+91) || !(x <= 3.35e-6)) {
		tmp = 1.0 / (x + (x * y));
	} else {
		tmp = 1.0 / x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((x <= (-4d+91)) .or. (.not. (x <= 3.35d-6))) then
        tmp = 1.0d0 / (x + (x * y))
    else
        tmp = 1.0d0 / x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((x <= -4e+91) || !(x <= 3.35e-6)) {
		tmp = 1.0 / (x + (x * y));
	} else {
		tmp = 1.0 / x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -4e+91) or not (x <= 3.35e-6):
		tmp = 1.0 / (x + (x * y))
	else:
		tmp = 1.0 / x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -4e+91) || !(x <= 3.35e-6))
		tmp = Float64(1.0 / Float64(x + Float64(x * y)));
	else
		tmp = Float64(1.0 / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((x <= -4e+91) || ~((x <= 3.35e-6)))
		tmp = 1.0 / (x + (x * y));
	else
		tmp = 1.0 / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -4e+91], N[Not[LessEqual[x, 3.35e-6]], $MachinePrecision]], N[(1.0 / N[(x + N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -4.00000000000000032e91 or 3.35e-6 < x

   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 76.5%

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

   3. Simplified 76.5%

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

      1. clear-num 76.5%

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

      2. add-exp-log 46.4%

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

      3. add-exp-log 46.4%

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

      4. div-exp 46.4%

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

      5. pow-exp 46.4%

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

      6. add-log-exp 46.4%

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

      7. log-pow 46.4%

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

      8. div-exp 46.4%

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

      9. add-exp-log 76.5%

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

      10. add-exp-log 76.5%

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

      11. inv-pow 76.5%

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

   6. Applied egg-rr 76.5%

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

      1. unpow-1 76.5%

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

   8. Simplified 76.5%

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

   9. Taylor expanded in y around 0 74.0%

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

   if -4.00000000000000032e91 < x < 3.35e-6

   1. Initial program 75.3%

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

      1. exp-prod 97.6%

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

   3. Simplified 97.6%

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

   5. Taylor expanded in x around 0 93.1%

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

4. Final simplification 83.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4 \cdot 10^{+91} \lor \neg \left(x \leq 3.35 \cdot 10^{-6}\right):\\ \;\;\;\;\frac{1}{x + x \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 82.8% accurate, 12.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.75 \cdot 10^{+22}:\\ \;\;\;\;\frac{1 + y \cdot \left(-1 + y \cdot 0.5\right)}{x}\\ \mathbf{elif}\;x \leq 4 \cdot 10^{-7}:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x + x \cdot y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1.75e+22)
   (/ (+ 1.0 (* y (+ -1.0 (* y 0.5)))) x)
   (if (<= x 4e-7) (/ 1.0 x) (/ 1.0 (+ x (* x y))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.75e+22) {
		tmp = (1.0 + (y * (-1.0 + (y * 0.5)))) / x;
	} else if (x <= 4e-7) {
		tmp = 1.0 / x;
	} else {
		tmp = 1.0 / (x + (x * y));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-1.75d+22)) then
        tmp = (1.0d0 + (y * ((-1.0d0) + (y * 0.5d0)))) / x
    else if (x <= 4d-7) then
        tmp = 1.0d0 / x
    else
        tmp = 1.0d0 / (x + (x * y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -1.75e+22) {
		tmp = (1.0 + (y * (-1.0 + (y * 0.5)))) / x;
	} else if (x <= 4e-7) {
		tmp = 1.0 / x;
	} else {
		tmp = 1.0 / (x + (x * y));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -1.75e+22:
		tmp = (1.0 + (y * (-1.0 + (y * 0.5)))) / x
	elif x <= 4e-7:
		tmp = 1.0 / x
	else:
		tmp = 1.0 / (x + (x * y))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1.75e+22)
		tmp = Float64(Float64(1.0 + Float64(y * Float64(-1.0 + Float64(y * 0.5)))) / x);
	elseif (x <= 4e-7)
		tmp = Float64(1.0 / x);
	else
		tmp = Float64(1.0 / Float64(x + Float64(x * y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1.75e+22)
		tmp = (1.0 + (y * (-1.0 + (y * 0.5)))) / x;
	elseif (x <= 4e-7)
		tmp = 1.0 / x;
	else
		tmp = 1.0 / (x + (x * y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1.75e+22], N[(N[(1.0 + N[(y * N[(-1.0 + N[(y * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[x, 4e-7], N[(1.0 / x), $MachinePrecision], N[(1.0 / N[(x + N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.75e22

   1. Initial program 70.3%

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

      1. exp-prod 70.3%

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

   3. Simplified 70.3%

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

   5. Taylor expanded in y around 0 66.5%

      \[\leadsto \frac{\color{blue}{1 + y \cdot \left(y \cdot \left(0.5 + 0.5 \cdot \frac{1}{x}\right) - 1\right)}}{x} \]

   6. Taylor expanded in x around inf 66.5%

      \[\leadsto \frac{1 + y \cdot \left(\color{blue}{0.5 \cdot y} - 1\right)}{x} \]
   7. Step-by-step derivation

      1. *-commutative 66.5%

         \[\leadsto \frac{1 + y \cdot \left(\color{blue}{y \cdot 0.5} - 1\right)}{x} \]

   8. Simplified 66.5%

      \[\leadsto \frac{1 + y \cdot \left(\color{blue}{y \cdot 0.5} - 1\right)}{x} \]

   if -1.75e22 < x < 3.9999999999999998e-7

   1. Initial program 74.8%

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

      1. exp-prod 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in x around 0 98.9%

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

   if 3.9999999999999998e-7 < x

   1. Initial program 82.4%

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

      1. exp-prod 82.4%

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

   3. Simplified 82.4%

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

      1. clear-num 82.4%

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

      2. add-exp-log 77.3%

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

      3. add-exp-log 77.3%

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

      4. div-exp 77.3%

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

      5. pow-exp 77.3%

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

      6. add-log-exp 77.3%

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

      7. log-pow 77.3%

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

      8. div-exp 77.3%

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

      9. add-exp-log 82.4%

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

      10. add-exp-log 82.4%

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

      11. inv-pow 82.4%

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

   6. Applied egg-rr 82.4%

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

      1. unpow-1 82.4%

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

   8. Simplified 82.4%

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

   9. Taylor expanded in y around 0 80.7%

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

4. Final simplification 85.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.75 \cdot 10^{+22}:\\ \;\;\;\;\frac{1 + y \cdot \left(-1 + y \cdot 0.5\right)}{x}\\ \mathbf{elif}\;x \leq 4 \cdot 10^{-7}:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x + x \cdot y}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 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 75.9%

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

   1. exp-prod 87.3%

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

3. Simplified 87.3%

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

5. Taylor expanded in x around 0 77.1%

   \[\leadsto \color{blue}{\frac{1}{x}} \]
6. Add Preprocessing

Developer Target 1: 78.0% 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 2024139 
(FPCore (x y)
  :name "Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2, F"
  :precision binary64

  :alt
  (! :herbie-platform default (if (< y -37311844206647956000000000000000000000000000000000000000000000000000000000000000000000000000000) (/ (exp (/ -1 y)) x) (if (< y 28179592427282880000000000000000000000) (/ (pow (/ x (+ y x)) x) x) (if (< y 23473874151669980000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000) (log (exp (/ (pow (/ x (+ y x)) x) x))) (/ (exp (/ -1 y)) x)))))

  (/ (exp (* x (log (/ x (+ x y))))) x))