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

Percentage Accurate: 90.0% → 97.3%

Time: 9.2s

Alternatives: 6

Speedup: 1.8×

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: 90.0% 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: 97.3% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(\frac{x}{x + y}\right)}^{x}\\ \mathbf{if}\;x \leq -2.2 \cdot 10^{-18}:\\ \;\;\;\;\frac{1}{\frac{x}{t\_0}}\\ \mathbf{elif}\;x \leq 2.2 \cdot 10^{-70}:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \frac{1}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (pow (/ x (+ x y)) x)))
   (if (<= x -2.2e-18)
     (/ 1.0 (/ x t_0))
     (if (<= x 2.2e-70) (/ 1.0 x) (* t_0 (/ 1.0 x))))))

C

double code(double x, double y) {
	double t_0 = pow((x / (x + y)), x);
	double tmp;
	if (x <= -2.2e-18) {
		tmp = 1.0 / (x / t_0);
	} else if (x <= 2.2e-70) {
		tmp = 1.0 / x;
	} else {
		tmp = t_0 * (1.0 / x);
	}
	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) :: tmp
    t_0 = (x / (x + y)) ** x
    if (x <= (-2.2d-18)) then
        tmp = 1.0d0 / (x / t_0)
    else if (x <= 2.2d-70) then
        tmp = 1.0d0 / x
    else
        tmp = t_0 * (1.0d0 / x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = Math.pow((x / (x + y)), x);
	double tmp;
	if (x <= -2.2e-18) {
		tmp = 1.0 / (x / t_0);
	} else if (x <= 2.2e-70) {
		tmp = 1.0 / x;
	} else {
		tmp = t_0 * (1.0 / x);
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.pow((x / (x + y)), x)
	tmp = 0
	if x <= -2.2e-18:
		tmp = 1.0 / (x / t_0)
	elif x <= 2.2e-70:
		tmp = 1.0 / x
	else:
		tmp = t_0 * (1.0 / x)
	return tmp

Julia

function code(x, y)
	t_0 = Float64(x / Float64(x + y)) ^ x
	tmp = 0.0
	if (x <= -2.2e-18)
		tmp = Float64(1.0 / Float64(x / t_0));
	elseif (x <= 2.2e-70)
		tmp = Float64(1.0 / x);
	else
		tmp = Float64(t_0 * Float64(1.0 / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (x / (x + y)) ^ x;
	tmp = 0.0;
	if (x <= -2.2e-18)
		tmp = 1.0 / (x / t_0);
	elseif (x <= 2.2e-70)
		tmp = 1.0 / x;
	else
		tmp = t_0 * (1.0 / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[Power[N[(x / N[(x + y), $MachinePrecision]), $MachinePrecision], x], $MachinePrecision]}, If[LessEqual[x, -2.2e-18], N[(1.0 / N[(x / t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 2.2e-70], N[(1.0 / x), $MachinePrecision], N[(t$95$0 * N[(1.0 / x), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.1999999999999998e-18

   1. Initial program 96.1%

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

      1. exp-prod 95.7%

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

   3. Simplified 95.7%

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

      1. clear-num 95.7%

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

      2. add-exp-log 0.0%

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

      3. add-exp-log 0.0%

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

      4. div-exp 0.0%

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

      5. pow-exp 0.0%

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

      6. add-log-exp 0.0%

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

      7. log-pow 0.0%

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

      8. div-exp 0.0%

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

      9. add-exp-log 96.1%

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

      10. add-exp-log 96.1%

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

      11. associate-/r/ 96.1%

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

      12. *-commutative 96.1%

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

   6. Applied egg-rr 96.1%

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

      1. un-div-inv 96.1%

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

      2. clear-num 96.1%

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

   8. Applied egg-rr 96.1%

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

   if -2.1999999999999998e-18 < x < 2.1999999999999999e-70

   1. Initial program 78.4%

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

      1. exp-prod 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 100.0%

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

   if 2.1999999999999999e-70 < x

   1. Initial program 96.7%

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

      1. exp-prod 96.5%

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

   3. Simplified 96.5%

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

      1. clear-num 96.5%

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

      2. add-exp-log 89.0%

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

      3. add-exp-log 89.0%

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

      4. div-exp 89.0%

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

      5. pow-exp 89.1%

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

      6. add-log-exp 89.1%

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

      7. log-pow 89.1%

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

      8. div-exp 89.2%

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

      9. add-exp-log 96.7%

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

      10. add-exp-log 96.7%

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

      11. associate-/r/ 96.8%

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

      12. *-commutative 96.8%

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

   6. Applied egg-rr 96.8%

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

4. Final simplification 97.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.2 \cdot 10^{-18}:\\ \;\;\;\;\frac{1}{\frac{x}{{\left(\frac{x}{x + y}\right)}^{x}}}\\ \mathbf{elif}\;x \leq 2.2 \cdot 10^{-70}:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{x}{x + y}\right)}^{x} \cdot \frac{1}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 97.3% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

function code(x, y)
	return Float64((exp(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[Power[N[Exp[x], $MachinePrecision], N[Log[N[(x / N[(x + y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] / x), $MachinePrecision]

Derivation

1. Initial program 89.8%

   \[\frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x} \]
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. Final simplification 97.6%

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

Alternative 3: 97.3% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(\frac{x}{x + y}\right)}^{x}\\ \mathbf{if}\;x \leq -2.2 \cdot 10^{-18}:\\ \;\;\;\;\frac{t\_0}{x}\\ \mathbf{elif}\;x \leq 10^{-70}:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \frac{1}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (pow (/ x (+ x y)) x)))
   (if (<= x -2.2e-18)
     (/ t_0 x)
     (if (<= x 1e-70) (/ 1.0 x) (* t_0 (/ 1.0 x))))))

C

double code(double x, double y) {
	double t_0 = pow((x / (x + y)), x);
	double tmp;
	if (x <= -2.2e-18) {
		tmp = t_0 / x;
	} else if (x <= 1e-70) {
		tmp = 1.0 / x;
	} else {
		tmp = t_0 * (1.0 / x);
	}
	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) :: tmp
    t_0 = (x / (x + y)) ** x
    if (x <= (-2.2d-18)) then
        tmp = t_0 / x
    else if (x <= 1d-70) then
        tmp = 1.0d0 / x
    else
        tmp = t_0 * (1.0d0 / x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = Math.pow((x / (x + y)), x);
	double tmp;
	if (x <= -2.2e-18) {
		tmp = t_0 / x;
	} else if (x <= 1e-70) {
		tmp = 1.0 / x;
	} else {
		tmp = t_0 * (1.0 / x);
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.pow((x / (x + y)), x)
	tmp = 0
	if x <= -2.2e-18:
		tmp = t_0 / x
	elif x <= 1e-70:
		tmp = 1.0 / x
	else:
		tmp = t_0 * (1.0 / x)
	return tmp

Julia

function code(x, y)
	t_0 = Float64(x / Float64(x + y)) ^ x
	tmp = 0.0
	if (x <= -2.2e-18)
		tmp = Float64(t_0 / x);
	elseif (x <= 1e-70)
		tmp = Float64(1.0 / x);
	else
		tmp = Float64(t_0 * Float64(1.0 / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (x / (x + y)) ^ x;
	tmp = 0.0;
	if (x <= -2.2e-18)
		tmp = t_0 / x;
	elseif (x <= 1e-70)
		tmp = 1.0 / x;
	else
		tmp = t_0 * (1.0 / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[Power[N[(x / N[(x + y), $MachinePrecision]), $MachinePrecision], x], $MachinePrecision]}, If[LessEqual[x, -2.2e-18], N[(t$95$0 / x), $MachinePrecision], If[LessEqual[x, 1e-70], N[(1.0 / x), $MachinePrecision], N[(t$95$0 * N[(1.0 / x), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.1999999999999998e-18

   1. Initial program 96.1%

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

      1. *-commutative 96.1%

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

      2. exp-to-pow 96.1%

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

   3. Simplified 96.1%

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

   if -2.1999999999999998e-18 < x < 9.99999999999999996e-71

   1. Initial program 78.4%

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

      1. exp-prod 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 100.0%

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

   if 9.99999999999999996e-71 < x

   1. Initial program 96.7%

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

      1. exp-prod 96.5%

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

   3. Simplified 96.5%

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

      1. clear-num 96.5%

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

      2. add-exp-log 89.0%

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

      3. add-exp-log 89.0%

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

      4. div-exp 89.0%

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

      5. pow-exp 89.1%

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

      6. add-log-exp 89.1%

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

      7. log-pow 89.1%

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

      8. div-exp 89.2%

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

      9. add-exp-log 96.7%

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

      10. add-exp-log 96.7%

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

      11. associate-/r/ 96.8%

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

      12. *-commutative 96.8%

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

   6. Applied egg-rr 96.8%

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

4. Final simplification 97.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.2 \cdot 10^{-18}:\\ \;\;\;\;\frac{{\left(\frac{x}{x + y}\right)}^{x}}{x}\\ \mathbf{elif}\;x \leq 10^{-70}:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{x}{x + y}\right)}^{x} \cdot \frac{1}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 89.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.0003 \lor \neg \left(x \leq 0.49\right) \land x \leq 10^{+130}:\\ \;\;\;\;\frac{e^{-y}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -0.0003) (and (not (<= x 0.49)) (<= x 1e+130)))
   (/ (exp (- y)) x)
   (/ 1.0 x)))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -0.0003) || (!(x <= 0.49) && (x <= 1e+130))) {
		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 <= (-0.0003d0)) .or. (.not. (x <= 0.49d0)) .and. (x <= 1d+130)) 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 <= -0.0003) || (!(x <= 0.49) && (x <= 1e+130))) {
		tmp = Math.exp(-y) / x;
	} else {
		tmp = 1.0 / x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -0.0003) or (not (x <= 0.49) and (x <= 1e+130)):
		tmp = math.exp(-y) / x
	else:
		tmp = 1.0 / x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -0.0003) || (!(x <= 0.49) && (x <= 1e+130)))
		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 <= -0.0003) || (~((x <= 0.49)) && (x <= 1e+130)))
		tmp = exp(-y) / x;
	else
		tmp = 1.0 / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -0.0003], And[N[Not[LessEqual[x, 0.49]], $MachinePrecision], LessEqual[x, 1e+130]]], N[(N[Exp[(-y)], $MachinePrecision] / x), $MachinePrecision], N[(1.0 / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2.99999999999999974e-4 or 0.48999999999999999 < x < 1.0000000000000001e130

   1. Initial program 97.2%

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

      1. *-commutative 97.2%

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

      2. exp-to-pow 97.2%

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

   3. Simplified 97.2%

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

   5. Taylor expanded in x around inf 89.4%

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

      1. mul-1-neg 89.4%

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

   7. Simplified 89.4%

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

   if -2.99999999999999974e-4 < x < 0.48999999999999999 or 1.0000000000000001e130 < x

   1. Initial program 84.9%

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

      1. exp-prod 97.8%

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

   3. Simplified 97.8%

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

   5. Taylor expanded in x around 0 95.5%

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

4. Final simplification 93.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.0003 \lor \neg \left(x \leq 0.49\right) \land x \leq 10^{+130}:\\ \;\;\;\;\frac{e^{-y}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 97.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.5 \cdot 10^{-18} \lor \neg \left(x \leq 2.3 \cdot 10^{-70}\right):\\ \;\;\;\;\frac{{\left(\frac{x}{x + y}\right)}^{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -2.5e-18) (not (<= x 2.3e-70)))
   (/ (pow (/ x (+ x y)) x) x)
   (/ 1.0 x)))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -2.5e-18) || !(x <= 2.3e-70)) {
		tmp = pow((x / (x + y)), x) / 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 <= (-2.5d-18)) .or. (.not. (x <= 2.3d-70))) then
        tmp = ((x / (x + y)) ** x) / 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 <= -2.5e-18) || !(x <= 2.3e-70)) {
		tmp = Math.pow((x / (x + y)), x) / x;
	} else {
		tmp = 1.0 / x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -2.5e-18) or not (x <= 2.3e-70):
		tmp = math.pow((x / (x + y)), x) / x
	else:
		tmp = 1.0 / x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -2.5e-18) || !(x <= 2.3e-70))
		tmp = Float64((Float64(x / Float64(x + y)) ^ x) / x);
	else
		tmp = Float64(1.0 / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((x <= -2.5e-18) || ~((x <= 2.3e-70)))
		tmp = ((x / (x + y)) ^ x) / x;
	else
		tmp = 1.0 / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -2.5e-18], N[Not[LessEqual[x, 2.3e-70]], $MachinePrecision]], N[(N[Power[N[(x / N[(x + y), $MachinePrecision]), $MachinePrecision], x], $MachinePrecision] / x), $MachinePrecision], N[(1.0 / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2.50000000000000018e-18 or 2.30000000000000001e-70 < x

   1. Initial program 96.4%

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

      1. *-commutative 96.4%

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

      2. exp-to-pow 96.4%

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

   3. Simplified 96.4%

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

   if -2.50000000000000018e-18 < x < 2.30000000000000001e-70

   1. Initial program 78.4%

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

      1. exp-prod 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 100.0%

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

4. Final simplification 97.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.5 \cdot 10^{-18} \lor \neg \left(x \leq 2.3 \cdot 10^{-70}\right):\\ \;\;\;\;\frac{{\left(\frac{x}{x + y}\right)}^{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 87.0% accurate, 69.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 89.8%

   \[\frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x} \]
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 86.5%

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

6. Final simplification 86.5%

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

Developer target: 83.0% 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 2024031 
(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))