2nthrt (problem 3.4.6)

Percentage Accurate: 53.8% → 86.2%

Time: 1.6min

Alternatives: 17

Speedup: 1.8×

• could not determine a ground truth

  1. x = -2
  2. n = 4.94066e-324

Specification

Math

\[\begin{array}{l} \\ {\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (- (pow (+ x 1.0) (/ 1.0 n)) (pow x (/ 1.0 n))))

C

double code(double x, double n) {
	return pow((x + 1.0), (1.0 / n)) - pow(x, (1.0 / n));
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    code = ((x + 1.0d0) ** (1.0d0 / n)) - (x ** (1.0d0 / n))
end function

Java

public static double code(double x, double n) {
	return Math.pow((x + 1.0), (1.0 / n)) - Math.pow(x, (1.0 / n));
}

Python

def code(x, n):
	return math.pow((x + 1.0), (1.0 / n)) - math.pow(x, (1.0 / n))

Julia

function code(x, n)
	return Float64((Float64(x + 1.0) ^ Float64(1.0 / n)) - (x ^ Float64(1.0 / n)))
end

MATLAB

function tmp = code(x, n)
	tmp = ((x + 1.0) ^ (1.0 / n)) - (x ^ (1.0 / n));
end

Wolfram

code[x_, n_] := N[(N[Power[N[(x + 1.0), $MachinePrecision], N[(1.0 / n), $MachinePrecision]], $MachinePrecision] - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Initial Program: 53.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ {\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (- (pow (+ x 1.0) (/ 1.0 n)) (pow x (/ 1.0 n))))

C

double code(double x, double n) {
	return pow((x + 1.0), (1.0 / n)) - pow(x, (1.0 / n));
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    code = ((x + 1.0d0) ** (1.0d0 / n)) - (x ** (1.0d0 / n))
end function

Java

public static double code(double x, double n) {
	return Math.pow((x + 1.0), (1.0 / n)) - Math.pow(x, (1.0 / n));
}

Python

def code(x, n):
	return math.pow((x + 1.0), (1.0 / n)) - math.pow(x, (1.0 / n))

Julia

function code(x, n)
	return Float64((Float64(x + 1.0) ^ Float64(1.0 / n)) - (x ^ Float64(1.0 / n)))
end

MATLAB

function tmp = code(x, n)
	tmp = ((x + 1.0) ^ (1.0 / n)) - (x ^ (1.0 / n));
end

Wolfram

code[x_, n_] := N[(N[Power[N[(x + 1.0), $MachinePrecision], N[(1.0 / n), $MachinePrecision]], $MachinePrecision] - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Alternative 1: 86.2% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ t_1 := e^{\frac{\log x}{n}}\\ \mathbf{if}\;\frac{1}{n} \leq -4 \cdot 10^{-12}:\\ \;\;\;\;\frac{t\_0}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-36}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 100000:\\ \;\;\;\;\frac{t\_1}{{x}^{2}} \cdot \left(\frac{0.5}{{n}^{2}} + \frac{-0.5}{n}\right) + \left(\frac{t\_1}{n \cdot x} + \frac{t\_1}{{x}^{3}} \cdot \left(\frac{0.16666666666666666}{{n}^{3}} + \left(\frac{0.3333333333333333}{n} + \frac{-0.5}{{n}^{2}}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{x}{n}} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))) (t_1 (exp (/ (log x) n))))
   (if (<= (/ 1.0 n) -4e-12)
     (/ t_0 (* n x))
     (if (<= (/ 1.0 n) 5e-36)
       (/ (log (/ (+ 1.0 x) x)) n)
       (if (<= (/ 1.0 n) 100000.0)
         (+
          (* (/ t_1 (pow x 2.0)) (+ (/ 0.5 (pow n 2.0)) (/ -0.5 n)))
          (+
           (/ t_1 (* n x))
           (*
            (/ t_1 (pow x 3.0))
            (+
             (/ 0.16666666666666666 (pow n 3.0))
             (+ (/ 0.3333333333333333 n) (/ -0.5 (pow n 2.0)))))))
         (- (exp (/ x n)) t_0))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double t_1 = exp((log(x) / n));
	double tmp;
	if ((1.0 / n) <= -4e-12) {
		tmp = t_0 / (n * x);
	} else if ((1.0 / n) <= 5e-36) {
		tmp = log(((1.0 + x) / x)) / n;
	} else if ((1.0 / n) <= 100000.0) {
		tmp = ((t_1 / pow(x, 2.0)) * ((0.5 / pow(n, 2.0)) + (-0.5 / n))) + ((t_1 / (n * x)) + ((t_1 / pow(x, 3.0)) * ((0.16666666666666666 / pow(n, 3.0)) + ((0.3333333333333333 / n) + (-0.5 / pow(n, 2.0))))));
	} else {
		tmp = exp((x / n)) - t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = x ** (1.0d0 / n)
    t_1 = exp((log(x) / n))
    if ((1.0d0 / n) <= (-4d-12)) then
        tmp = t_0 / (n * x)
    else if ((1.0d0 / n) <= 5d-36) then
        tmp = log(((1.0d0 + x) / x)) / n
    else if ((1.0d0 / n) <= 100000.0d0) then
        tmp = ((t_1 / (x ** 2.0d0)) * ((0.5d0 / (n ** 2.0d0)) + ((-0.5d0) / n))) + ((t_1 / (n * x)) + ((t_1 / (x ** 3.0d0)) * ((0.16666666666666666d0 / (n ** 3.0d0)) + ((0.3333333333333333d0 / n) + ((-0.5d0) / (n ** 2.0d0))))))
    else
        tmp = exp((x / n)) - t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, (1.0 / n));
	double t_1 = Math.exp((Math.log(x) / n));
	double tmp;
	if ((1.0 / n) <= -4e-12) {
		tmp = t_0 / (n * x);
	} else if ((1.0 / n) <= 5e-36) {
		tmp = Math.log(((1.0 + x) / x)) / n;
	} else if ((1.0 / n) <= 100000.0) {
		tmp = ((t_1 / Math.pow(x, 2.0)) * ((0.5 / Math.pow(n, 2.0)) + (-0.5 / n))) + ((t_1 / (n * x)) + ((t_1 / Math.pow(x, 3.0)) * ((0.16666666666666666 / Math.pow(n, 3.0)) + ((0.3333333333333333 / n) + (-0.5 / Math.pow(n, 2.0))))));
	} else {
		tmp = Math.exp((x / n)) - t_0;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	t_1 = math.exp((math.log(x) / n))
	tmp = 0
	if (1.0 / n) <= -4e-12:
		tmp = t_0 / (n * x)
	elif (1.0 / n) <= 5e-36:
		tmp = math.log(((1.0 + x) / x)) / n
	elif (1.0 / n) <= 100000.0:
		tmp = ((t_1 / math.pow(x, 2.0)) * ((0.5 / math.pow(n, 2.0)) + (-0.5 / n))) + ((t_1 / (n * x)) + ((t_1 / math.pow(x, 3.0)) * ((0.16666666666666666 / math.pow(n, 3.0)) + ((0.3333333333333333 / n) + (-0.5 / math.pow(n, 2.0))))))
	else:
		tmp = math.exp((x / n)) - t_0
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	t_1 = exp(Float64(log(x) / n))
	tmp = 0.0
	if (Float64(1.0 / n) <= -4e-12)
		tmp = Float64(t_0 / Float64(n * x));
	elseif (Float64(1.0 / n) <= 5e-36)
		tmp = Float64(log(Float64(Float64(1.0 + x) / x)) / n);
	elseif (Float64(1.0 / n) <= 100000.0)
		tmp = Float64(Float64(Float64(t_1 / (x ^ 2.0)) * Float64(Float64(0.5 / (n ^ 2.0)) + Float64(-0.5 / n))) + Float64(Float64(t_1 / Float64(n * x)) + Float64(Float64(t_1 / (x ^ 3.0)) * Float64(Float64(0.16666666666666666 / (n ^ 3.0)) + Float64(Float64(0.3333333333333333 / n) + Float64(-0.5 / (n ^ 2.0)))))));
	else
		tmp = Float64(exp(Float64(x / n)) - t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = x ^ (1.0 / n);
	t_1 = exp((log(x) / n));
	tmp = 0.0;
	if ((1.0 / n) <= -4e-12)
		tmp = t_0 / (n * x);
	elseif ((1.0 / n) <= 5e-36)
		tmp = log(((1.0 + x) / x)) / n;
	elseif ((1.0 / n) <= 100000.0)
		tmp = ((t_1 / (x ^ 2.0)) * ((0.5 / (n ^ 2.0)) + (-0.5 / n))) + ((t_1 / (n * x)) + ((t_1 / (x ^ 3.0)) * ((0.16666666666666666 / (n ^ 3.0)) + ((0.3333333333333333 / n) + (-0.5 / (n ^ 2.0))))));
	else
		tmp = exp((x / n)) - t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Exp[N[(N[Log[x], $MachinePrecision] / n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -4e-12], N[(t$95$0 / N[(n * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e-36], N[(N[Log[N[(N[(1.0 + x), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 100000.0], N[(N[(N[(t$95$1 / N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(0.5 / N[Power[n, 2.0], $MachinePrecision]), $MachinePrecision] + N[(-0.5 / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(t$95$1 / N[(n * x), $MachinePrecision]), $MachinePrecision] + N[(N[(t$95$1 / N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision] * N[(N[(0.16666666666666666 / N[Power[n, 3.0], $MachinePrecision]), $MachinePrecision] + N[(N[(0.3333333333333333 / n), $MachinePrecision] + N[(-0.5 / N[Power[n, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Exp[N[(x / n), $MachinePrecision]], $MachinePrecision] - t$95$0), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 1 n) < -3.99999999999999992e-12

   1. Initial program 96.1%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 100.0%

      \[\leadsto \color{blue}{\frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}}{n \cdot x}} \]
   4. Step-by-step derivation

      1. mul-1-neg 100.0%

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

      2. log-rec 100.0%

         \[\leadsto \frac{e^{-\frac{\color{blue}{-\log x}}{n}}}{n \cdot x} \]

      3. mul-1-neg 100.0%

         \[\leadsto \frac{e^{-\frac{\color{blue}{-1 \cdot \log x}}{n}}}{n \cdot x} \]

      4. distribute-neg-frac 100.0%

         \[\leadsto \frac{e^{\color{blue}{\frac{--1 \cdot \log x}{n}}}}{n \cdot x} \]

      5. mul-1-neg 100.0%

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

      6. remove-double-neg 100.0%

         \[\leadsto \frac{e^{\frac{\color{blue}{\log x}}{n}}}{n \cdot x} \]

      7. *-commutative 100.0%

         \[\leadsto \frac{e^{\frac{\log x}{n}}}{\color{blue}{x \cdot n}} \]

   5. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{e^{\frac{\log x}{n}}}{x \cdot n}} \]

   6. Taylor expanded in x around 0 100.0%

      \[\leadsto \color{blue}{\frac{e^{\frac{\log x}{n}}}{n \cdot x}} \]
   7. Step-by-step derivation

      1. *-rgt-identity 100.0%

         \[\leadsto \frac{e^{\color{blue}{\frac{\log x}{n} \cdot 1}}}{n \cdot x} \]

      2. associate-*l/ 100.0%

         \[\leadsto \frac{e^{\color{blue}{\frac{\log x \cdot 1}{n}}}}{n \cdot x} \]

      3. associate-*r/ 100.0%

         \[\leadsto \frac{e^{\color{blue}{\log x \cdot \frac{1}{n}}}}{n \cdot x} \]

      4. exp-to-pow 100.0%

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

      5. *-commutative 100.0%

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

   8. Simplified 100.0%

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

   if -3.99999999999999992e-12 < (/.f64 1 n) < 5.00000000000000004e-36

   1. Initial program 39.4%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in n around inf 81.5%

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
   4. Step-by-step derivation

      1. log1p-def 81.5%

         \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(x\right)} - \log x}{n} \]

   5. Simplified 81.5%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) - \log x}{n}} \]
   6. Step-by-step derivation

      1. log1p-udef 81.5%

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

      2. diff-log 81.8%

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

      3. +-commutative 81.8%

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

   7. Applied egg-rr 81.8%

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

   if 5.00000000000000004e-36 < (/.f64 1 n) < 1e5

   1. Initial program 17.5%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 71.5%

      \[\leadsto \color{blue}{\frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}}{n \cdot x} + \left(\frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}} \cdot \left(0.5 \cdot \frac{1}{{n}^{2}} - 0.5 \cdot \frac{1}{n}\right)}{{x}^{2}} + \frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}} \cdot \left(\left(0.16666666666666666 \cdot \frac{1}{{n}^{3}} + 0.3333333333333333 \cdot \frac{1}{n}\right) - 0.5 \cdot \frac{1}{{n}^{2}}\right)}{{x}^{3}}\right)} \]

   5. Simplified 71.5%

      \[\leadsto \color{blue}{\frac{e^{\frac{\log x}{n}}}{{x}^{2}} \cdot \left(\frac{0.5}{{n}^{2}} + \frac{-0.5}{n}\right) + \left(\frac{e^{\frac{\log x}{n}}}{x \cdot n} + \frac{e^{\frac{\log x}{n}}}{{x}^{3}} \cdot \left(\frac{0.16666666666666666}{{n}^{3}} + \left(\frac{0.3333333333333333}{n} + \frac{-0.5}{{n}^{2}}\right)\right)\right)} \]

   if 1e5 < (/.f64 1 n)

   1. Initial program 42.2%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in n around 0 42.2%

      \[\leadsto \color{blue}{e^{\frac{\log \left(1 + x\right)}{n}}} - {x}^{\left(\frac{1}{n}\right)} \]
   4. Step-by-step derivation

      1. log1p-def 100.0%

         \[\leadsto e^{\frac{\color{blue}{\mathsf{log1p}\left(x\right)}}{n}} - {x}^{\left(\frac{1}{n}\right)} \]

   5. Simplified 100.0%

      \[\leadsto \color{blue}{e^{\frac{\mathsf{log1p}\left(x\right)}{n}}} - {x}^{\left(\frac{1}{n}\right)} \]

   6. Taylor expanded in x around 0 100.0%

      \[\leadsto e^{\color{blue}{\frac{x}{n}}} - {x}^{\left(\frac{1}{n}\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 88.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -4 \cdot 10^{-12}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-36}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 100000:\\ \;\;\;\;\frac{e^{\frac{\log x}{n}}}{{x}^{2}} \cdot \left(\frac{0.5}{{n}^{2}} + \frac{-0.5}{n}\right) + \left(\frac{e^{\frac{\log x}{n}}}{n \cdot x} + \frac{e^{\frac{\log x}{n}}}{{x}^{3}} \cdot \left(\frac{0.16666666666666666}{{n}^{3}} + \left(\frac{0.3333333333333333}{n} + \frac{-0.5}{{n}^{2}}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{x}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 86.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\frac{\log x}{n}}\\ t_1 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -4 \cdot 10^{-12}:\\ \;\;\;\;\frac{t\_1}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-36}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 3 \cdot 10^{-15}:\\ \;\;\;\;\frac{t\_0}{n \cdot x} + \frac{t\_0}{\frac{{x}^{2}}{\frac{0.5}{{n}^{2}} + \frac{-0.5}{n}}}\\ \mathbf{else}:\\ \;\;\;\;\log \left(e^{e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - t\_1}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (exp (/ (log x) n))) (t_1 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -4e-12)
     (/ t_1 (* n x))
     (if (<= (/ 1.0 n) 5e-36)
       (/ (log (/ (+ 1.0 x) x)) n)
       (if (<= (/ 1.0 n) 3e-15)
         (+
          (/ t_0 (* n x))
          (/ t_0 (/ (pow x 2.0) (+ (/ 0.5 (pow n 2.0)) (/ -0.5 n)))))
         (log (exp (- (exp (/ (log1p x) n)) t_1))))))))

C

double code(double x, double n) {
	double t_0 = exp((log(x) / n));
	double t_1 = pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -4e-12) {
		tmp = t_1 / (n * x);
	} else if ((1.0 / n) <= 5e-36) {
		tmp = log(((1.0 + x) / x)) / n;
	} else if ((1.0 / n) <= 3e-15) {
		tmp = (t_0 / (n * x)) + (t_0 / (pow(x, 2.0) / ((0.5 / pow(n, 2.0)) + (-0.5 / n))));
	} else {
		tmp = log(exp((exp((log1p(x) / n)) - t_1)));
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double t_0 = Math.exp((Math.log(x) / n));
	double t_1 = Math.pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -4e-12) {
		tmp = t_1 / (n * x);
	} else if ((1.0 / n) <= 5e-36) {
		tmp = Math.log(((1.0 + x) / x)) / n;
	} else if ((1.0 / n) <= 3e-15) {
		tmp = (t_0 / (n * x)) + (t_0 / (Math.pow(x, 2.0) / ((0.5 / Math.pow(n, 2.0)) + (-0.5 / n))));
	} else {
		tmp = Math.log(Math.exp((Math.exp((Math.log1p(x) / n)) - t_1)));
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.exp((math.log(x) / n))
	t_1 = math.pow(x, (1.0 / n))
	tmp = 0
	if (1.0 / n) <= -4e-12:
		tmp = t_1 / (n * x)
	elif (1.0 / n) <= 5e-36:
		tmp = math.log(((1.0 + x) / x)) / n
	elif (1.0 / n) <= 3e-15:
		tmp = (t_0 / (n * x)) + (t_0 / (math.pow(x, 2.0) / ((0.5 / math.pow(n, 2.0)) + (-0.5 / n))))
	else:
		tmp = math.log(math.exp((math.exp((math.log1p(x) / n)) - t_1)))
	return tmp

Julia

function code(x, n)
	t_0 = exp(Float64(log(x) / n))
	t_1 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -4e-12)
		tmp = Float64(t_1 / Float64(n * x));
	elseif (Float64(1.0 / n) <= 5e-36)
		tmp = Float64(log(Float64(Float64(1.0 + x) / x)) / n);
	elseif (Float64(1.0 / n) <= 3e-15)
		tmp = Float64(Float64(t_0 / Float64(n * x)) + Float64(t_0 / Float64((x ^ 2.0) / Float64(Float64(0.5 / (n ^ 2.0)) + Float64(-0.5 / n)))));
	else
		tmp = log(exp(Float64(exp(Float64(log1p(x) / n)) - t_1)));
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Exp[N[(N[Log[x], $MachinePrecision] / n), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -4e-12], N[(t$95$1 / N[(n * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e-36], N[(N[Log[N[(N[(1.0 + x), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 3e-15], N[(N[(t$95$0 / N[(n * x), $MachinePrecision]), $MachinePrecision] + N[(t$95$0 / N[(N[Power[x, 2.0], $MachinePrecision] / N[(N[(0.5 / N[Power[n, 2.0], $MachinePrecision]), $MachinePrecision] + N[(-0.5 / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Log[N[Exp[N[(N[Exp[N[(N[Log[1 + x], $MachinePrecision] / n), $MachinePrecision]], $MachinePrecision] - t$95$1), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 1 n) < -3.99999999999999992e-12

   1. Initial program 96.1%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 100.0%

      \[\leadsto \color{blue}{\frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}}{n \cdot x}} \]
   4. Step-by-step derivation

      1. mul-1-neg 100.0%

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

      2. log-rec 100.0%

         \[\leadsto \frac{e^{-\frac{\color{blue}{-\log x}}{n}}}{n \cdot x} \]

      3. mul-1-neg 100.0%

         \[\leadsto \frac{e^{-\frac{\color{blue}{-1 \cdot \log x}}{n}}}{n \cdot x} \]

      4. distribute-neg-frac 100.0%

         \[\leadsto \frac{e^{\color{blue}{\frac{--1 \cdot \log x}{n}}}}{n \cdot x} \]

      5. mul-1-neg 100.0%

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

      6. remove-double-neg 100.0%

         \[\leadsto \frac{e^{\frac{\color{blue}{\log x}}{n}}}{n \cdot x} \]

      7. *-commutative 100.0%

         \[\leadsto \frac{e^{\frac{\log x}{n}}}{\color{blue}{x \cdot n}} \]

   5. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{e^{\frac{\log x}{n}}}{x \cdot n}} \]

   6. Taylor expanded in x around 0 100.0%

      \[\leadsto \color{blue}{\frac{e^{\frac{\log x}{n}}}{n \cdot x}} \]
   7. Step-by-step derivation

      1. *-rgt-identity 100.0%

         \[\leadsto \frac{e^{\color{blue}{\frac{\log x}{n} \cdot 1}}}{n \cdot x} \]

      2. associate-*l/ 100.0%

         \[\leadsto \frac{e^{\color{blue}{\frac{\log x \cdot 1}{n}}}}{n \cdot x} \]

      3. associate-*r/ 100.0%

         \[\leadsto \frac{e^{\color{blue}{\log x \cdot \frac{1}{n}}}}{n \cdot x} \]

      4. exp-to-pow 100.0%

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

      5. *-commutative 100.0%

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

   8. Simplified 100.0%

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

   if -3.99999999999999992e-12 < (/.f64 1 n) < 5.00000000000000004e-36

   1. Initial program 39.4%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in n around inf 81.5%

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
   4. Step-by-step derivation

      1. log1p-def 81.5%

         \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(x\right)} - \log x}{n} \]

   5. Simplified 81.5%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) - \log x}{n}} \]
   6. Step-by-step derivation

      1. log1p-udef 81.5%

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

      2. diff-log 81.8%

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

      3. +-commutative 81.8%

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

   7. Applied egg-rr 81.8%

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

   if 5.00000000000000004e-36 < (/.f64 1 n) < 3e-15

   1. Initial program 5.1%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 88.4%

      \[\leadsto \color{blue}{\frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}}{n \cdot x} + \frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}} \cdot \left(0.5 \cdot \frac{1}{{n}^{2}} - 0.5 \cdot \frac{1}{n}\right)}{{x}^{2}}} \]
   4. Step-by-step derivation

      1. mul-1-neg 88.4%

         \[\leadsto \frac{e^{\color{blue}{-\frac{\log \left(\frac{1}{x}\right)}{n}}}}{n \cdot x} + \frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}} \cdot \left(0.5 \cdot \frac{1}{{n}^{2}} - 0.5 \cdot \frac{1}{n}\right)}{{x}^{2}} \]

      2. log-rec 88.4%

         \[\leadsto \frac{e^{-\frac{\color{blue}{-\log x}}{n}}}{n \cdot x} + \frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}} \cdot \left(0.5 \cdot \frac{1}{{n}^{2}} - 0.5 \cdot \frac{1}{n}\right)}{{x}^{2}} \]

      3. mul-1-neg 88.4%

         \[\leadsto \frac{e^{-\frac{\color{blue}{-1 \cdot \log x}}{n}}}{n \cdot x} + \frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}} \cdot \left(0.5 \cdot \frac{1}{{n}^{2}} - 0.5 \cdot \frac{1}{n}\right)}{{x}^{2}} \]

      4. distribute-neg-frac 88.4%

         \[\leadsto \frac{e^{\color{blue}{\frac{--1 \cdot \log x}{n}}}}{n \cdot x} + \frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}} \cdot \left(0.5 \cdot \frac{1}{{n}^{2}} - 0.5 \cdot \frac{1}{n}\right)}{{x}^{2}} \]

      5. mul-1-neg 88.4%

         \[\leadsto \frac{e^{\frac{-\color{blue}{\left(-\log x\right)}}{n}}}{n \cdot x} + \frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}} \cdot \left(0.5 \cdot \frac{1}{{n}^{2}} - 0.5 \cdot \frac{1}{n}\right)}{{x}^{2}} \]

      6. remove-double-neg 88.4%

         \[\leadsto \frac{e^{\frac{\color{blue}{\log x}}{n}}}{n \cdot x} + \frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}} \cdot \left(0.5 \cdot \frac{1}{{n}^{2}} - 0.5 \cdot \frac{1}{n}\right)}{{x}^{2}} \]

      7. *-commutative 88.4%

         \[\leadsto \frac{e^{\frac{\log x}{n}}}{\color{blue}{x \cdot n}} + \frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}} \cdot \left(0.5 \cdot \frac{1}{{n}^{2}} - 0.5 \cdot \frac{1}{n}\right)}{{x}^{2}} \]

      8. associate-/l* 88.4%

         \[\leadsto \frac{e^{\frac{\log x}{n}}}{x \cdot n} + \color{blue}{\frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}}{\frac{{x}^{2}}{0.5 \cdot \frac{1}{{n}^{2}} - 0.5 \cdot \frac{1}{n}}}} \]

   5. Simplified 88.4%

      \[\leadsto \color{blue}{\frac{e^{\frac{\log x}{n}}}{x \cdot n} + \frac{e^{\frac{\log x}{n}}}{\frac{{x}^{2}}{\frac{0.5}{{n}^{2}} + \frac{-0.5}{n}}}} \]

   if 3e-15 < (/.f64 1 n)

   1. Initial program 41.9%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. add-log-exp 42.0%

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

      2. add-exp-log 42.0%

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

      3. log-pow 42.0%

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

      4. +-commutative 42.0%

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

      5. log1p-udef 92.4%

         \[\leadsto \log \left(e^{e^{\frac{1}{n} \cdot \color{blue}{\mathsf{log1p}\left(x\right)}} - {x}^{\left(\frac{1}{n}\right)}}\right) \]

      6. *-commutative 92.4%

         \[\leadsto \log \left(e^{e^{\color{blue}{\mathsf{log1p}\left(x\right) \cdot \frac{1}{n}}} - {x}^{\left(\frac{1}{n}\right)}}\right) \]

      7. un-div-inv 92.4%

         \[\leadsto \log \left(e^{e^{\color{blue}{\frac{\mathsf{log1p}\left(x\right)}{n}}} - {x}^{\left(\frac{1}{n}\right)}}\right) \]

   4. Applied egg-rr 92.4%

      \[\leadsto \color{blue}{\log \left(e^{e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{1}{n}\right)}}\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 88.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -4 \cdot 10^{-12}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-36}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 3 \cdot 10^{-15}:\\ \;\;\;\;\frac{e^{\frac{\log x}{n}}}{n \cdot x} + \frac{e^{\frac{\log x}{n}}}{\frac{{x}^{2}}{\frac{0.5}{{n}^{2}} + \frac{-0.5}{n}}}\\ \mathbf{else}:\\ \;\;\;\;\log \left(e^{e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{1}{n}\right)}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 81.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -4 \cdot 10^{-12}:\\ \;\;\;\;\frac{t\_0}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-77}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 100000:\\ \;\;\;\;\frac{1}{x} \cdot \frac{t\_0}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{+93}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - t\_0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{{\left(n \cdot x\right)}^{-2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -4e-12)
     (/ t_0 (* n x))
     (if (<= (/ 1.0 n) 2e-77)
       (/ (log (/ (+ 1.0 x) x)) n)
       (if (<= (/ 1.0 n) 100000.0)
         (* (/ 1.0 x) (/ t_0 n))
         (if (<= (/ 1.0 n) 2e+93)
           (- (+ 1.0 (/ x n)) t_0)
           (sqrt (pow (* n x) -2.0))))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -4e-12) {
		tmp = t_0 / (n * x);
	} else if ((1.0 / n) <= 2e-77) {
		tmp = log(((1.0 + x) / x)) / n;
	} else if ((1.0 / n) <= 100000.0) {
		tmp = (1.0 / x) * (t_0 / n);
	} else if ((1.0 / n) <= 2e+93) {
		tmp = (1.0 + (x / n)) - t_0;
	} else {
		tmp = sqrt(pow((n * x), -2.0));
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x ** (1.0d0 / n)
    if ((1.0d0 / n) <= (-4d-12)) then
        tmp = t_0 / (n * x)
    else if ((1.0d0 / n) <= 2d-77) then
        tmp = log(((1.0d0 + x) / x)) / n
    else if ((1.0d0 / n) <= 100000.0d0) then
        tmp = (1.0d0 / x) * (t_0 / n)
    else if ((1.0d0 / n) <= 2d+93) then
        tmp = (1.0d0 + (x / n)) - t_0
    else
        tmp = sqrt(((n * x) ** (-2.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -4e-12) {
		tmp = t_0 / (n * x);
	} else if ((1.0 / n) <= 2e-77) {
		tmp = Math.log(((1.0 + x) / x)) / n;
	} else if ((1.0 / n) <= 100000.0) {
		tmp = (1.0 / x) * (t_0 / n);
	} else if ((1.0 / n) <= 2e+93) {
		tmp = (1.0 + (x / n)) - t_0;
	} else {
		tmp = Math.sqrt(Math.pow((n * x), -2.0));
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	tmp = 0
	if (1.0 / n) <= -4e-12:
		tmp = t_0 / (n * x)
	elif (1.0 / n) <= 2e-77:
		tmp = math.log(((1.0 + x) / x)) / n
	elif (1.0 / n) <= 100000.0:
		tmp = (1.0 / x) * (t_0 / n)
	elif (1.0 / n) <= 2e+93:
		tmp = (1.0 + (x / n)) - t_0
	else:
		tmp = math.sqrt(math.pow((n * x), -2.0))
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -4e-12)
		tmp = Float64(t_0 / Float64(n * x));
	elseif (Float64(1.0 / n) <= 2e-77)
		tmp = Float64(log(Float64(Float64(1.0 + x) / x)) / n);
	elseif (Float64(1.0 / n) <= 100000.0)
		tmp = Float64(Float64(1.0 / x) * Float64(t_0 / n));
	elseif (Float64(1.0 / n) <= 2e+93)
		tmp = Float64(Float64(1.0 + Float64(x / n)) - t_0);
	else
		tmp = sqrt((Float64(n * x) ^ -2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = x ^ (1.0 / n);
	tmp = 0.0;
	if ((1.0 / n) <= -4e-12)
		tmp = t_0 / (n * x);
	elseif ((1.0 / n) <= 2e-77)
		tmp = log(((1.0 + x) / x)) / n;
	elseif ((1.0 / n) <= 100000.0)
		tmp = (1.0 / x) * (t_0 / n);
	elseif ((1.0 / n) <= 2e+93)
		tmp = (1.0 + (x / n)) - t_0;
	else
		tmp = sqrt(((n * x) ^ -2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -4e-12], N[(t$95$0 / N[(n * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-77], N[(N[Log[N[(N[(1.0 + x), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 100000.0], N[(N[(1.0 / x), $MachinePrecision] * N[(t$95$0 / n), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e+93], N[(N[(1.0 + N[(x / n), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], N[Sqrt[N[Power[N[(n * x), $MachinePrecision], -2.0], $MachinePrecision]], $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if (/.f64 1 n) < -3.99999999999999992e-12

   1. Initial program 96.1%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 100.0%

      \[\leadsto \color{blue}{\frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}}{n \cdot x}} \]
   4. Step-by-step derivation

      1. mul-1-neg 100.0%

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

      2. log-rec 100.0%

         \[\leadsto \frac{e^{-\frac{\color{blue}{-\log x}}{n}}}{n \cdot x} \]

      3. mul-1-neg 100.0%

         \[\leadsto \frac{e^{-\frac{\color{blue}{-1 \cdot \log x}}{n}}}{n \cdot x} \]

      4. distribute-neg-frac 100.0%

         \[\leadsto \frac{e^{\color{blue}{\frac{--1 \cdot \log x}{n}}}}{n \cdot x} \]

      5. mul-1-neg 100.0%

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

      6. remove-double-neg 100.0%

         \[\leadsto \frac{e^{\frac{\color{blue}{\log x}}{n}}}{n \cdot x} \]

      7. *-commutative 100.0%

         \[\leadsto \frac{e^{\frac{\log x}{n}}}{\color{blue}{x \cdot n}} \]

   5. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{e^{\frac{\log x}{n}}}{x \cdot n}} \]

   6. Taylor expanded in x around 0 100.0%

      \[\leadsto \color{blue}{\frac{e^{\frac{\log x}{n}}}{n \cdot x}} \]
   7. Step-by-step derivation

      1. *-rgt-identity 100.0%

         \[\leadsto \frac{e^{\color{blue}{\frac{\log x}{n} \cdot 1}}}{n \cdot x} \]

      2. associate-*l/ 100.0%

         \[\leadsto \frac{e^{\color{blue}{\frac{\log x \cdot 1}{n}}}}{n \cdot x} \]

      3. associate-*r/ 100.0%

         \[\leadsto \frac{e^{\color{blue}{\log x \cdot \frac{1}{n}}}}{n \cdot x} \]

      4. exp-to-pow 100.0%

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

      5. *-commutative 100.0%

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

   8. Simplified 100.0%

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

   if -3.99999999999999992e-12 < (/.f64 1 n) < 1.9999999999999999e-77

   1. Initial program 41.1%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in n around inf 82.8%

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
   4. Step-by-step derivation

      1. log1p-def 82.8%

         \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(x\right)} - \log x}{n} \]

   5. Simplified 82.8%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) - \log x}{n}} \]
   6. Step-by-step derivation

      1. log1p-udef 82.8%

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

      2. diff-log 83.1%

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

      3. +-commutative 83.1%

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

   7. Applied egg-rr 83.1%

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

   if 1.9999999999999999e-77 < (/.f64 1 n) < 1e5

   1. Initial program 13.5%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 65.7%

      \[\leadsto \color{blue}{\frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}}{n \cdot x}} \]
   4. Step-by-step derivation

      1. mul-1-neg 65.7%

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

      2. log-rec 65.7%

         \[\leadsto \frac{e^{-\frac{\color{blue}{-\log x}}{n}}}{n \cdot x} \]

      3. mul-1-neg 65.7%

         \[\leadsto \frac{e^{-\frac{\color{blue}{-1 \cdot \log x}}{n}}}{n \cdot x} \]

      4. distribute-neg-frac 65.7%

         \[\leadsto \frac{e^{\color{blue}{\frac{--1 \cdot \log x}{n}}}}{n \cdot x} \]

      5. mul-1-neg 65.7%

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

      6. remove-double-neg 65.7%

         \[\leadsto \frac{e^{\frac{\color{blue}{\log x}}{n}}}{n \cdot x} \]

      7. *-commutative 65.7%

         \[\leadsto \frac{e^{\frac{\log x}{n}}}{\color{blue}{x \cdot n}} \]

   5. Simplified 65.7%

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

      1. div-inv 65.7%

         \[\leadsto \frac{e^{\color{blue}{\log x \cdot \frac{1}{n}}}}{x \cdot n} \]

      2. pow-to-exp 65.7%

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

      3. *-un-lft-identity 65.7%

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

      4. times-frac 65.7%

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

   7. Applied egg-rr 65.7%

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

   if 1e5 < (/.f64 1 n) < 2.00000000000000009e93

   1. Initial program 91.2%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 91.2%

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

   if 2.00000000000000009e93 < (/.f64 1 n)

   1. Initial program 18.8%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 0.8%

      \[\leadsto \color{blue}{\frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}}{n \cdot x}} \]
   4. Step-by-step derivation

      1. mul-1-neg 0.8%

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

      2. log-rec 0.8%

         \[\leadsto \frac{e^{-\frac{\color{blue}{-\log x}}{n}}}{n \cdot x} \]

      3. mul-1-neg 0.8%

         \[\leadsto \frac{e^{-\frac{\color{blue}{-1 \cdot \log x}}{n}}}{n \cdot x} \]

      4. distribute-neg-frac 0.8%

         \[\leadsto \frac{e^{\color{blue}{\frac{--1 \cdot \log x}{n}}}}{n \cdot x} \]

      5. mul-1-neg 0.8%

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

      6. remove-double-neg 0.8%

         \[\leadsto \frac{e^{\frac{\color{blue}{\log x}}{n}}}{n \cdot x} \]

      7. *-commutative 0.8%

         \[\leadsto \frac{e^{\frac{\log x}{n}}}{\color{blue}{x \cdot n}} \]

   5. Simplified 0.8%

      \[\leadsto \color{blue}{\frac{e^{\frac{\log x}{n}}}{x \cdot n}} \]

   6. Taylor expanded in n around inf 46.5%

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

      1. *-commutative 46.5%

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

   8. Simplified 46.5%

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

      1. add-sqr-sqrt 46.5%

         \[\leadsto \color{blue}{\sqrt{\frac{1}{x \cdot n}} \cdot \sqrt{\frac{1}{x \cdot n}}} \]

      2. sqrt-unprod 83.2%

         \[\leadsto \color{blue}{\sqrt{\frac{1}{x \cdot n} \cdot \frac{1}{x \cdot n}}} \]

      3. inv-pow 83.2%

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

      4. inv-pow 83.2%

         \[\leadsto \sqrt{{\left(x \cdot n\right)}^{-1} \cdot \color{blue}{{\left(x \cdot n\right)}^{-1}}} \]

      5. pow-prod-up 83.2%

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

      6. metadata-eval 83.2%

         \[\leadsto \sqrt{{\left(x \cdot n\right)}^{\color{blue}{-2}}} \]

   10. Applied egg-rr 83.2%

       \[\leadsto \color{blue}{\sqrt{{\left(x \cdot n\right)}^{-2}}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 86.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -4 \cdot 10^{-12}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-77}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 100000:\\ \;\;\;\;\frac{1}{x} \cdot \frac{{x}^{\left(\frac{1}{n}\right)}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{+93}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{{\left(n \cdot x\right)}^{-2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 86.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -4 \cdot 10^{-12}:\\ \;\;\;\;\frac{t\_0}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-77}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 100000:\\ \;\;\;\;\frac{1}{x} \cdot \frac{t\_0}{n}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{x}{n}} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -4e-12)
     (/ t_0 (* n x))
     (if (<= (/ 1.0 n) 2e-77)
       (/ (log (/ (+ 1.0 x) x)) n)
       (if (<= (/ 1.0 n) 100000.0)
         (* (/ 1.0 x) (/ t_0 n))
         (- (exp (/ x n)) t_0))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -4e-12) {
		tmp = t_0 / (n * x);
	} else if ((1.0 / n) <= 2e-77) {
		tmp = log(((1.0 + x) / x)) / n;
	} else if ((1.0 / n) <= 100000.0) {
		tmp = (1.0 / x) * (t_0 / n);
	} else {
		tmp = exp((x / n)) - t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x ** (1.0d0 / n)
    if ((1.0d0 / n) <= (-4d-12)) then
        tmp = t_0 / (n * x)
    else if ((1.0d0 / n) <= 2d-77) then
        tmp = log(((1.0d0 + x) / x)) / n
    else if ((1.0d0 / n) <= 100000.0d0) then
        tmp = (1.0d0 / x) * (t_0 / n)
    else
        tmp = exp((x / n)) - t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -4e-12) {
		tmp = t_0 / (n * x);
	} else if ((1.0 / n) <= 2e-77) {
		tmp = Math.log(((1.0 + x) / x)) / n;
	} else if ((1.0 / n) <= 100000.0) {
		tmp = (1.0 / x) * (t_0 / n);
	} else {
		tmp = Math.exp((x / n)) - t_0;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	tmp = 0
	if (1.0 / n) <= -4e-12:
		tmp = t_0 / (n * x)
	elif (1.0 / n) <= 2e-77:
		tmp = math.log(((1.0 + x) / x)) / n
	elif (1.0 / n) <= 100000.0:
		tmp = (1.0 / x) * (t_0 / n)
	else:
		tmp = math.exp((x / n)) - t_0
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -4e-12)
		tmp = Float64(t_0 / Float64(n * x));
	elseif (Float64(1.0 / n) <= 2e-77)
		tmp = Float64(log(Float64(Float64(1.0 + x) / x)) / n);
	elseif (Float64(1.0 / n) <= 100000.0)
		tmp = Float64(Float64(1.0 / x) * Float64(t_0 / n));
	else
		tmp = Float64(exp(Float64(x / n)) - t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = x ^ (1.0 / n);
	tmp = 0.0;
	if ((1.0 / n) <= -4e-12)
		tmp = t_0 / (n * x);
	elseif ((1.0 / n) <= 2e-77)
		tmp = log(((1.0 + x) / x)) / n;
	elseif ((1.0 / n) <= 100000.0)
		tmp = (1.0 / x) * (t_0 / n);
	else
		tmp = exp((x / n)) - t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -4e-12], N[(t$95$0 / N[(n * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-77], N[(N[Log[N[(N[(1.0 + x), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 100000.0], N[(N[(1.0 / x), $MachinePrecision] * N[(t$95$0 / n), $MachinePrecision]), $MachinePrecision], N[(N[Exp[N[(x / n), $MachinePrecision]], $MachinePrecision] - t$95$0), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 1 n) < -3.99999999999999992e-12

   1. Initial program 96.1%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 100.0%

      \[\leadsto \color{blue}{\frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}}{n \cdot x}} \]
   4. Step-by-step derivation

      1. mul-1-neg 100.0%

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

      2. log-rec 100.0%

         \[\leadsto \frac{e^{-\frac{\color{blue}{-\log x}}{n}}}{n \cdot x} \]

      3. mul-1-neg 100.0%

         \[\leadsto \frac{e^{-\frac{\color{blue}{-1 \cdot \log x}}{n}}}{n \cdot x} \]

      4. distribute-neg-frac 100.0%

         \[\leadsto \frac{e^{\color{blue}{\frac{--1 \cdot \log x}{n}}}}{n \cdot x} \]

      5. mul-1-neg 100.0%

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

      6. remove-double-neg 100.0%

         \[\leadsto \frac{e^{\frac{\color{blue}{\log x}}{n}}}{n \cdot x} \]

      7. *-commutative 100.0%

         \[\leadsto \frac{e^{\frac{\log x}{n}}}{\color{blue}{x \cdot n}} \]

   5. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{e^{\frac{\log x}{n}}}{x \cdot n}} \]

   6. Taylor expanded in x around 0 100.0%

      \[\leadsto \color{blue}{\frac{e^{\frac{\log x}{n}}}{n \cdot x}} \]
   7. Step-by-step derivation

      1. *-rgt-identity 100.0%

         \[\leadsto \frac{e^{\color{blue}{\frac{\log x}{n} \cdot 1}}}{n \cdot x} \]

      2. associate-*l/ 100.0%

         \[\leadsto \frac{e^{\color{blue}{\frac{\log x \cdot 1}{n}}}}{n \cdot x} \]

      3. associate-*r/ 100.0%

         \[\leadsto \frac{e^{\color{blue}{\log x \cdot \frac{1}{n}}}}{n \cdot x} \]

      4. exp-to-pow 100.0%

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

      5. *-commutative 100.0%

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

   8. Simplified 100.0%

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

   if -3.99999999999999992e-12 < (/.f64 1 n) < 1.9999999999999999e-77

   1. Initial program 41.1%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in n around inf 82.8%

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
   4. Step-by-step derivation

      1. log1p-def 82.8%

         \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(x\right)} - \log x}{n} \]

   5. Simplified 82.8%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) - \log x}{n}} \]
   6. Step-by-step derivation

      1. log1p-udef 82.8%

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

      2. diff-log 83.1%

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

      3. +-commutative 83.1%

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

   7. Applied egg-rr 83.1%

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

   if 1.9999999999999999e-77 < (/.f64 1 n) < 1e5

   1. Initial program 13.5%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 65.7%

      \[\leadsto \color{blue}{\frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}}{n \cdot x}} \]
   4. Step-by-step derivation

      1. mul-1-neg 65.7%

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

      2. log-rec 65.7%

         \[\leadsto \frac{e^{-\frac{\color{blue}{-\log x}}{n}}}{n \cdot x} \]

      3. mul-1-neg 65.7%

         \[\leadsto \frac{e^{-\frac{\color{blue}{-1 \cdot \log x}}{n}}}{n \cdot x} \]

      4. distribute-neg-frac 65.7%

         \[\leadsto \frac{e^{\color{blue}{\frac{--1 \cdot \log x}{n}}}}{n \cdot x} \]

      5. mul-1-neg 65.7%

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

      6. remove-double-neg 65.7%

         \[\leadsto \frac{e^{\frac{\color{blue}{\log x}}{n}}}{n \cdot x} \]

      7. *-commutative 65.7%

         \[\leadsto \frac{e^{\frac{\log x}{n}}}{\color{blue}{x \cdot n}} \]

   5. Simplified 65.7%

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

      1. div-inv 65.7%

         \[\leadsto \frac{e^{\color{blue}{\log x \cdot \frac{1}{n}}}}{x \cdot n} \]

      2. pow-to-exp 65.7%

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

      3. *-un-lft-identity 65.7%

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

      4. times-frac 65.7%

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

   7. Applied egg-rr 65.7%

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

   if 1e5 < (/.f64 1 n)

   1. Initial program 42.2%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in n around 0 42.2%

      \[\leadsto \color{blue}{e^{\frac{\log \left(1 + x\right)}{n}}} - {x}^{\left(\frac{1}{n}\right)} \]
   4. Step-by-step derivation

      1. log1p-def 100.0%

         \[\leadsto e^{\frac{\color{blue}{\mathsf{log1p}\left(x\right)}}{n}} - {x}^{\left(\frac{1}{n}\right)} \]

   5. Simplified 100.0%

      \[\leadsto \color{blue}{e^{\frac{\mathsf{log1p}\left(x\right)}{n}}} - {x}^{\left(\frac{1}{n}\right)} \]

   6. Taylor expanded in x around 0 100.0%

      \[\leadsto e^{\color{blue}{\frac{x}{n}}} - {x}^{\left(\frac{1}{n}\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 88.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -4 \cdot 10^{-12}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-77}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 100000:\\ \;\;\;\;\frac{1}{x} \cdot \frac{{x}^{\left(\frac{1}{n}\right)}}{n}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{x}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 66.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{n \cdot \left(\left(1 + x\right) + -1\right)}\\ t_1 := 1 - {x}^{\left(\frac{1}{n}\right)}\\ t_2 := \frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{+42}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\frac{1}{n} \leq -1 \cdot 10^{+14}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;\frac{1}{n} \leq -4 \cdot 10^{-12}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-77}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;\frac{1}{n} \leq 3 \cdot 10^{-15}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{+93}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (/ 1.0 (* n (+ (+ 1.0 x) -1.0))))
        (t_1 (- 1.0 (pow x (/ 1.0 n))))
        (t_2 (/ (log (/ (+ 1.0 x) x)) n)))
   (if (<= (/ 1.0 n) -5e+42)
     t_1
     (if (<= (/ 1.0 n) -1e+14)
       t_2
       (if (<= (/ 1.0 n) -4e-12)
         t_0
         (if (<= (/ 1.0 n) 2e-77)
           t_2
           (if (<= (/ 1.0 n) 3e-15)
             (/ (/ 1.0 x) n)
             (if (<= (/ 1.0 n) 2e+93) t_1 t_0))))))))

C

double code(double x, double n) {
	double t_0 = 1.0 / (n * ((1.0 + x) + -1.0));
	double t_1 = 1.0 - pow(x, (1.0 / n));
	double t_2 = log(((1.0 + x) / x)) / n;
	double tmp;
	if ((1.0 / n) <= -5e+42) {
		tmp = t_1;
	} else if ((1.0 / n) <= -1e+14) {
		tmp = t_2;
	} else if ((1.0 / n) <= -4e-12) {
		tmp = t_0;
	} else if ((1.0 / n) <= 2e-77) {
		tmp = t_2;
	} else if ((1.0 / n) <= 3e-15) {
		tmp = (1.0 / x) / n;
	} else if ((1.0 / n) <= 2e+93) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = 1.0d0 / (n * ((1.0d0 + x) + (-1.0d0)))
    t_1 = 1.0d0 - (x ** (1.0d0 / n))
    t_2 = log(((1.0d0 + x) / x)) / n
    if ((1.0d0 / n) <= (-5d+42)) then
        tmp = t_1
    else if ((1.0d0 / n) <= (-1d+14)) then
        tmp = t_2
    else if ((1.0d0 / n) <= (-4d-12)) then
        tmp = t_0
    else if ((1.0d0 / n) <= 2d-77) then
        tmp = t_2
    else if ((1.0d0 / n) <= 3d-15) then
        tmp = (1.0d0 / x) / n
    else if ((1.0d0 / n) <= 2d+93) then
        tmp = t_1
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = 1.0 / (n * ((1.0 + x) + -1.0));
	double t_1 = 1.0 - Math.pow(x, (1.0 / n));
	double t_2 = Math.log(((1.0 + x) / x)) / n;
	double tmp;
	if ((1.0 / n) <= -5e+42) {
		tmp = t_1;
	} else if ((1.0 / n) <= -1e+14) {
		tmp = t_2;
	} else if ((1.0 / n) <= -4e-12) {
		tmp = t_0;
	} else if ((1.0 / n) <= 2e-77) {
		tmp = t_2;
	} else if ((1.0 / n) <= 3e-15) {
		tmp = (1.0 / x) / n;
	} else if ((1.0 / n) <= 2e+93) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = 1.0 / (n * ((1.0 + x) + -1.0))
	t_1 = 1.0 - math.pow(x, (1.0 / n))
	t_2 = math.log(((1.0 + x) / x)) / n
	tmp = 0
	if (1.0 / n) <= -5e+42:
		tmp = t_1
	elif (1.0 / n) <= -1e+14:
		tmp = t_2
	elif (1.0 / n) <= -4e-12:
		tmp = t_0
	elif (1.0 / n) <= 2e-77:
		tmp = t_2
	elif (1.0 / n) <= 3e-15:
		tmp = (1.0 / x) / n
	elif (1.0 / n) <= 2e+93:
		tmp = t_1
	else:
		tmp = t_0
	return tmp

Julia

function code(x, n)
	t_0 = Float64(1.0 / Float64(n * Float64(Float64(1.0 + x) + -1.0)))
	t_1 = Float64(1.0 - (x ^ Float64(1.0 / n)))
	t_2 = Float64(log(Float64(Float64(1.0 + x) / x)) / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -5e+42)
		tmp = t_1;
	elseif (Float64(1.0 / n) <= -1e+14)
		tmp = t_2;
	elseif (Float64(1.0 / n) <= -4e-12)
		tmp = t_0;
	elseif (Float64(1.0 / n) <= 2e-77)
		tmp = t_2;
	elseif (Float64(1.0 / n) <= 3e-15)
		tmp = Float64(Float64(1.0 / x) / n);
	elseif (Float64(1.0 / n) <= 2e+93)
		tmp = t_1;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = 1.0 / (n * ((1.0 + x) + -1.0));
	t_1 = 1.0 - (x ^ (1.0 / n));
	t_2 = log(((1.0 + x) / x)) / n;
	tmp = 0.0;
	if ((1.0 / n) <= -5e+42)
		tmp = t_1;
	elseif ((1.0 / n) <= -1e+14)
		tmp = t_2;
	elseif ((1.0 / n) <= -4e-12)
		tmp = t_0;
	elseif ((1.0 / n) <= 2e-77)
		tmp = t_2;
	elseif ((1.0 / n) <= 3e-15)
		tmp = (1.0 / x) / n;
	elseif ((1.0 / n) <= 2e+93)
		tmp = t_1;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(1.0 / N[(n * N[(N[(1.0 + x), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Log[N[(N[(1.0 + x), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -5e+42], t$95$1, If[LessEqual[N[(1.0 / n), $MachinePrecision], -1e+14], t$95$2, If[LessEqual[N[(1.0 / n), $MachinePrecision], -4e-12], t$95$0, If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-77], t$95$2, If[LessEqual[N[(1.0 / n), $MachinePrecision], 3e-15], N[(N[(1.0 / x), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e+93], t$95$1, t$95$0]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 1 n) < -5.00000000000000007e42 or 3e-15 < (/.f64 1 n) < 2.00000000000000009e93

   1. Initial program 94.7%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 63.7%

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

   if -5.00000000000000007e42 < (/.f64 1 n) < -1e14 or -3.99999999999999992e-12 < (/.f64 1 n) < 1.9999999999999999e-77

   1. Initial program 44.1%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in n around inf 83.0%

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
   4. Step-by-step derivation

      1. log1p-def 83.0%

         \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(x\right)} - \log x}{n} \]

   5. Simplified 83.0%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) - \log x}{n}} \]
   6. Step-by-step derivation

      1. log1p-udef 83.0%

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

      2. diff-log 83.3%

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

      3. +-commutative 83.3%

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

   7. Applied egg-rr 83.3%

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

   if -1e14 < (/.f64 1 n) < -3.99999999999999992e-12 or 2.00000000000000009e93 < (/.f64 1 n)

   1. Initial program 25.8%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 21.3%

      \[\leadsto \color{blue}{\frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}}{n \cdot x}} \]
   4. Step-by-step derivation

      1. mul-1-neg 21.3%

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

      2. log-rec 21.3%

         \[\leadsto \frac{e^{-\frac{\color{blue}{-\log x}}{n}}}{n \cdot x} \]

      3. mul-1-neg 21.3%

         \[\leadsto \frac{e^{-\frac{\color{blue}{-1 \cdot \log x}}{n}}}{n \cdot x} \]

      4. distribute-neg-frac 21.3%

         \[\leadsto \frac{e^{\color{blue}{\frac{--1 \cdot \log x}{n}}}}{n \cdot x} \]

      5. mul-1-neg 21.3%

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

      6. remove-double-neg 21.3%

         \[\leadsto \frac{e^{\frac{\color{blue}{\log x}}{n}}}{n \cdot x} \]

      7. *-commutative 21.3%

         \[\leadsto \frac{e^{\frac{\log x}{n}}}{\color{blue}{x \cdot n}} \]

   5. Simplified 21.3%

      \[\leadsto \color{blue}{\frac{e^{\frac{\log x}{n}}}{x \cdot n}} \]

   6. Taylor expanded in n around inf 42.4%

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

      1. *-commutative 42.4%

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

   8. Simplified 42.4%

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

      1. expm1-log1p-u 42.4%

         \[\leadsto \frac{1}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(x\right)\right)} \cdot n} \]

      2. expm1-udef 77.7%

         \[\leadsto \frac{1}{\color{blue}{\left(e^{\mathsf{log1p}\left(x\right)} - 1\right)} \cdot n} \]

      3. log1p-udef 77.7%

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

      4. rem-exp-log 77.7%

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

      5. +-commutative 77.7%

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

   10. Applied egg-rr 77.7%

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

   if 1.9999999999999999e-77 < (/.f64 1 n) < 3e-15

   1. Initial program 4.7%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in n around inf 32.6%

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
   4. Step-by-step derivation

      1. log1p-def 32.6%

         \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(x\right)} - \log x}{n} \]

   5. Simplified 32.6%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) - \log x}{n}} \]

   6. Taylor expanded in x around inf 70.6%

      \[\leadsto \frac{\color{blue}{\frac{1}{x}}}{n} \]
3. Recombined 4 regimes into one program.

4. Final simplification 76.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{+42}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;\frac{1}{n} \leq -1 \cdot 10^{+14}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq -4 \cdot 10^{-12}:\\ \;\;\;\;\frac{1}{n \cdot \left(\left(1 + x\right) + -1\right)}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-77}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 3 \cdot 10^{-15}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{+93}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{n \cdot \left(\left(1 + x\right) + -1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 81.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -4 \cdot 10^{-12}:\\ \;\;\;\;\frac{t\_0}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-77}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 100000:\\ \;\;\;\;\frac{1}{x} \cdot \frac{t\_0}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{+93}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{n \cdot \left(\left(1 + x\right) + -1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -4e-12)
     (/ t_0 (* n x))
     (if (<= (/ 1.0 n) 2e-77)
       (/ (log (/ (+ 1.0 x) x)) n)
       (if (<= (/ 1.0 n) 100000.0)
         (* (/ 1.0 x) (/ t_0 n))
         (if (<= (/ 1.0 n) 2e+93)
           (- (+ 1.0 (/ x n)) t_0)
           (/ 1.0 (* n (+ (+ 1.0 x) -1.0)))))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -4e-12) {
		tmp = t_0 / (n * x);
	} else if ((1.0 / n) <= 2e-77) {
		tmp = log(((1.0 + x) / x)) / n;
	} else if ((1.0 / n) <= 100000.0) {
		tmp = (1.0 / x) * (t_0 / n);
	} else if ((1.0 / n) <= 2e+93) {
		tmp = (1.0 + (x / n)) - t_0;
	} else {
		tmp = 1.0 / (n * ((1.0 + x) + -1.0));
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x ** (1.0d0 / n)
    if ((1.0d0 / n) <= (-4d-12)) then
        tmp = t_0 / (n * x)
    else if ((1.0d0 / n) <= 2d-77) then
        tmp = log(((1.0d0 + x) / x)) / n
    else if ((1.0d0 / n) <= 100000.0d0) then
        tmp = (1.0d0 / x) * (t_0 / n)
    else if ((1.0d0 / n) <= 2d+93) then
        tmp = (1.0d0 + (x / n)) - t_0
    else
        tmp = 1.0d0 / (n * ((1.0d0 + x) + (-1.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -4e-12) {
		tmp = t_0 / (n * x);
	} else if ((1.0 / n) <= 2e-77) {
		tmp = Math.log(((1.0 + x) / x)) / n;
	} else if ((1.0 / n) <= 100000.0) {
		tmp = (1.0 / x) * (t_0 / n);
	} else if ((1.0 / n) <= 2e+93) {
		tmp = (1.0 + (x / n)) - t_0;
	} else {
		tmp = 1.0 / (n * ((1.0 + x) + -1.0));
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	tmp = 0
	if (1.0 / n) <= -4e-12:
		tmp = t_0 / (n * x)
	elif (1.0 / n) <= 2e-77:
		tmp = math.log(((1.0 + x) / x)) / n
	elif (1.0 / n) <= 100000.0:
		tmp = (1.0 / x) * (t_0 / n)
	elif (1.0 / n) <= 2e+93:
		tmp = (1.0 + (x / n)) - t_0
	else:
		tmp = 1.0 / (n * ((1.0 + x) + -1.0))
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -4e-12)
		tmp = Float64(t_0 / Float64(n * x));
	elseif (Float64(1.0 / n) <= 2e-77)
		tmp = Float64(log(Float64(Float64(1.0 + x) / x)) / n);
	elseif (Float64(1.0 / n) <= 100000.0)
		tmp = Float64(Float64(1.0 / x) * Float64(t_0 / n));
	elseif (Float64(1.0 / n) <= 2e+93)
		tmp = Float64(Float64(1.0 + Float64(x / n)) - t_0);
	else
		tmp = Float64(1.0 / Float64(n * Float64(Float64(1.0 + x) + -1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = x ^ (1.0 / n);
	tmp = 0.0;
	if ((1.0 / n) <= -4e-12)
		tmp = t_0 / (n * x);
	elseif ((1.0 / n) <= 2e-77)
		tmp = log(((1.0 + x) / x)) / n;
	elseif ((1.0 / n) <= 100000.0)
		tmp = (1.0 / x) * (t_0 / n);
	elseif ((1.0 / n) <= 2e+93)
		tmp = (1.0 + (x / n)) - t_0;
	else
		tmp = 1.0 / (n * ((1.0 + x) + -1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -4e-12], N[(t$95$0 / N[(n * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-77], N[(N[Log[N[(N[(1.0 + x), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 100000.0], N[(N[(1.0 / x), $MachinePrecision] * N[(t$95$0 / n), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e+93], N[(N[(1.0 + N[(x / n), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], N[(1.0 / N[(n * N[(N[(1.0 + x), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if (/.f64 1 n) < -3.99999999999999992e-12

   1. Initial program 96.1%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 100.0%

      \[\leadsto \color{blue}{\frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}}{n \cdot x}} \]
   4. Step-by-step derivation

      1. mul-1-neg 100.0%

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

      2. log-rec 100.0%

         \[\leadsto \frac{e^{-\frac{\color{blue}{-\log x}}{n}}}{n \cdot x} \]

      3. mul-1-neg 100.0%

         \[\leadsto \frac{e^{-\frac{\color{blue}{-1 \cdot \log x}}{n}}}{n \cdot x} \]

      4. distribute-neg-frac 100.0%

         \[\leadsto \frac{e^{\color{blue}{\frac{--1 \cdot \log x}{n}}}}{n \cdot x} \]

      5. mul-1-neg 100.0%

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

      6. remove-double-neg 100.0%

         \[\leadsto \frac{e^{\frac{\color{blue}{\log x}}{n}}}{n \cdot x} \]

      7. *-commutative 100.0%

         \[\leadsto \frac{e^{\frac{\log x}{n}}}{\color{blue}{x \cdot n}} \]

   5. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{e^{\frac{\log x}{n}}}{x \cdot n}} \]

   6. Taylor expanded in x around 0 100.0%

      \[\leadsto \color{blue}{\frac{e^{\frac{\log x}{n}}}{n \cdot x}} \]
   7. Step-by-step derivation

      1. *-rgt-identity 100.0%

         \[\leadsto \frac{e^{\color{blue}{\frac{\log x}{n} \cdot 1}}}{n \cdot x} \]

      2. associate-*l/ 100.0%

         \[\leadsto \frac{e^{\color{blue}{\frac{\log x \cdot 1}{n}}}}{n \cdot x} \]

      3. associate-*r/ 100.0%

         \[\leadsto \frac{e^{\color{blue}{\log x \cdot \frac{1}{n}}}}{n \cdot x} \]

      4. exp-to-pow 100.0%

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

      5. *-commutative 100.0%

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

   8. Simplified 100.0%

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

   if -3.99999999999999992e-12 < (/.f64 1 n) < 1.9999999999999999e-77

   1. Initial program 41.1%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in n around inf 82.8%

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
   4. Step-by-step derivation

      1. log1p-def 82.8%

         \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(x\right)} - \log x}{n} \]

   5. Simplified 82.8%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) - \log x}{n}} \]
   6. Step-by-step derivation

      1. log1p-udef 82.8%

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

      2. diff-log 83.1%

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

      3. +-commutative 83.1%

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

   7. Applied egg-rr 83.1%

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

   if 1.9999999999999999e-77 < (/.f64 1 n) < 1e5

   1. Initial program 13.5%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 65.7%

      \[\leadsto \color{blue}{\frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}}{n \cdot x}} \]
   4. Step-by-step derivation

      1. mul-1-neg 65.7%

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

      2. log-rec 65.7%

         \[\leadsto \frac{e^{-\frac{\color{blue}{-\log x}}{n}}}{n \cdot x} \]

      3. mul-1-neg 65.7%

         \[\leadsto \frac{e^{-\frac{\color{blue}{-1 \cdot \log x}}{n}}}{n \cdot x} \]

      4. distribute-neg-frac 65.7%

         \[\leadsto \frac{e^{\color{blue}{\frac{--1 \cdot \log x}{n}}}}{n \cdot x} \]

      5. mul-1-neg 65.7%

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

      6. remove-double-neg 65.7%

         \[\leadsto \frac{e^{\frac{\color{blue}{\log x}}{n}}}{n \cdot x} \]

      7. *-commutative 65.7%

         \[\leadsto \frac{e^{\frac{\log x}{n}}}{\color{blue}{x \cdot n}} \]

   5. Simplified 65.7%

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

      1. div-inv 65.7%

         \[\leadsto \frac{e^{\color{blue}{\log x \cdot \frac{1}{n}}}}{x \cdot n} \]

      2. pow-to-exp 65.7%

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

      3. *-un-lft-identity 65.7%

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

      4. times-frac 65.7%

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

   7. Applied egg-rr 65.7%

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

   if 1e5 < (/.f64 1 n) < 2.00000000000000009e93

   1. Initial program 91.2%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 91.2%

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

   if 2.00000000000000009e93 < (/.f64 1 n)

   1. Initial program 18.8%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 0.8%

      \[\leadsto \color{blue}{\frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}}{n \cdot x}} \]
   4. Step-by-step derivation

      1. mul-1-neg 0.8%

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

      2. log-rec 0.8%

         \[\leadsto \frac{e^{-\frac{\color{blue}{-\log x}}{n}}}{n \cdot x} \]

      3. mul-1-neg 0.8%

         \[\leadsto \frac{e^{-\frac{\color{blue}{-1 \cdot \log x}}{n}}}{n \cdot x} \]

      4. distribute-neg-frac 0.8%

         \[\leadsto \frac{e^{\color{blue}{\frac{--1 \cdot \log x}{n}}}}{n \cdot x} \]

      5. mul-1-neg 0.8%

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

      6. remove-double-neg 0.8%

         \[\leadsto \frac{e^{\frac{\color{blue}{\log x}}{n}}}{n \cdot x} \]

      7. *-commutative 0.8%

         \[\leadsto \frac{e^{\frac{\log x}{n}}}{\color{blue}{x \cdot n}} \]

   5. Simplified 0.8%

      \[\leadsto \color{blue}{\frac{e^{\frac{\log x}{n}}}{x \cdot n}} \]

   6. Taylor expanded in n around inf 46.5%

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

      1. *-commutative 46.5%

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

   8. Simplified 46.5%

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

      1. expm1-log1p-u 46.5%

         \[\leadsto \frac{1}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(x\right)\right)} \cdot n} \]

      2. expm1-udef 79.2%

         \[\leadsto \frac{1}{\color{blue}{\left(e^{\mathsf{log1p}\left(x\right)} - 1\right)} \cdot n} \]

      3. log1p-udef 79.2%

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

      4. rem-exp-log 79.2%

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

      5. +-commutative 79.2%

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

   10. Applied egg-rr 79.2%

       \[\leadsto \frac{1}{\color{blue}{\left(\left(x + 1\right) - 1\right)} \cdot n} \]
3. Recombined 5 regimes into one program.

4. Final simplification 86.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -4 \cdot 10^{-12}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-77}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 100000:\\ \;\;\;\;\frac{1}{x} \cdot \frac{{x}^{\left(\frac{1}{n}\right)}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{+93}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{n \cdot \left(\left(1 + x\right) + -1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 57.6% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - {x}^{\left(\frac{1}{n}\right)}\\ t_1 := \frac{-\log x}{n}\\ \mathbf{if}\;x \leq 2.65 \cdot 10^{-275}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 2.8 \cdot 10^{-223}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 8 \cdot 10^{-218}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 4.8 \cdot 10^{-111}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 2.8 \cdot 10^{-87}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 7.4 \cdot 10^{-60}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 1.08 \cdot 10^{+216}:\\ \;\;\;\;\frac{\frac{1}{\left(1 + x\right) + -1}}{n}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (- 1.0 (pow x (/ 1.0 n)))) (t_1 (/ (- (log x)) n)))
   (if (<= x 2.65e-275)
     t_0
     (if (<= x 2.8e-223)
       t_1
       (if (<= x 8e-218)
         t_0
         (if (<= x 4.8e-111)
           t_1
           (if (<= x 2.8e-87)
             t_0
             (if (<= x 7.4e-60)
               t_1
               (if (<= x 1.08e+216)
                 (/ (/ 1.0 (+ (+ 1.0 x) -1.0)) n)
                 0.0)))))))))

C

double code(double x, double n) {
	double t_0 = 1.0 - pow(x, (1.0 / n));
	double t_1 = -log(x) / n;
	double tmp;
	if (x <= 2.65e-275) {
		tmp = t_0;
	} else if (x <= 2.8e-223) {
		tmp = t_1;
	} else if (x <= 8e-218) {
		tmp = t_0;
	} else if (x <= 4.8e-111) {
		tmp = t_1;
	} else if (x <= 2.8e-87) {
		tmp = t_0;
	} else if (x <= 7.4e-60) {
		tmp = t_1;
	} else if (x <= 1.08e+216) {
		tmp = (1.0 / ((1.0 + x) + -1.0)) / n;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = 1.0d0 - (x ** (1.0d0 / n))
    t_1 = -log(x) / n
    if (x <= 2.65d-275) then
        tmp = t_0
    else if (x <= 2.8d-223) then
        tmp = t_1
    else if (x <= 8d-218) then
        tmp = t_0
    else if (x <= 4.8d-111) then
        tmp = t_1
    else if (x <= 2.8d-87) then
        tmp = t_0
    else if (x <= 7.4d-60) then
        tmp = t_1
    else if (x <= 1.08d+216) then
        tmp = (1.0d0 / ((1.0d0 + x) + (-1.0d0))) / n
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = 1.0 - Math.pow(x, (1.0 / n));
	double t_1 = -Math.log(x) / n;
	double tmp;
	if (x <= 2.65e-275) {
		tmp = t_0;
	} else if (x <= 2.8e-223) {
		tmp = t_1;
	} else if (x <= 8e-218) {
		tmp = t_0;
	} else if (x <= 4.8e-111) {
		tmp = t_1;
	} else if (x <= 2.8e-87) {
		tmp = t_0;
	} else if (x <= 7.4e-60) {
		tmp = t_1;
	} else if (x <= 1.08e+216) {
		tmp = (1.0 / ((1.0 + x) + -1.0)) / n;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = 1.0 - math.pow(x, (1.0 / n))
	t_1 = -math.log(x) / n
	tmp = 0
	if x <= 2.65e-275:
		tmp = t_0
	elif x <= 2.8e-223:
		tmp = t_1
	elif x <= 8e-218:
		tmp = t_0
	elif x <= 4.8e-111:
		tmp = t_1
	elif x <= 2.8e-87:
		tmp = t_0
	elif x <= 7.4e-60:
		tmp = t_1
	elif x <= 1.08e+216:
		tmp = (1.0 / ((1.0 + x) + -1.0)) / n
	else:
		tmp = 0.0
	return tmp

Julia

function code(x, n)
	t_0 = Float64(1.0 - (x ^ Float64(1.0 / n)))
	t_1 = Float64(Float64(-log(x)) / n)
	tmp = 0.0
	if (x <= 2.65e-275)
		tmp = t_0;
	elseif (x <= 2.8e-223)
		tmp = t_1;
	elseif (x <= 8e-218)
		tmp = t_0;
	elseif (x <= 4.8e-111)
		tmp = t_1;
	elseif (x <= 2.8e-87)
		tmp = t_0;
	elseif (x <= 7.4e-60)
		tmp = t_1;
	elseif (x <= 1.08e+216)
		tmp = Float64(Float64(1.0 / Float64(Float64(1.0 + x) + -1.0)) / n);
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = 1.0 - (x ^ (1.0 / n));
	t_1 = -log(x) / n;
	tmp = 0.0;
	if (x <= 2.65e-275)
		tmp = t_0;
	elseif (x <= 2.8e-223)
		tmp = t_1;
	elseif (x <= 8e-218)
		tmp = t_0;
	elseif (x <= 4.8e-111)
		tmp = t_1;
	elseif (x <= 2.8e-87)
		tmp = t_0;
	elseif (x <= 7.4e-60)
		tmp = t_1;
	elseif (x <= 1.08e+216)
		tmp = (1.0 / ((1.0 + x) + -1.0)) / n;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(1.0 - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[((-N[Log[x], $MachinePrecision]) / n), $MachinePrecision]}, If[LessEqual[x, 2.65e-275], t$95$0, If[LessEqual[x, 2.8e-223], t$95$1, If[LessEqual[x, 8e-218], t$95$0, If[LessEqual[x, 4.8e-111], t$95$1, If[LessEqual[x, 2.8e-87], t$95$0, If[LessEqual[x, 7.4e-60], t$95$1, If[LessEqual[x, 1.08e+216], N[(N[(1.0 / N[(N[(1.0 + x), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], 0.0]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if x < 2.64999999999999993e-275 or 2.80000000000000015e-223 < x < 8.0000000000000003e-218 or 4.8000000000000001e-111 < x < 2.8000000000000001e-87

   1. Initial program 85.4%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 85.4%

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

   if 2.64999999999999993e-275 < x < 2.80000000000000015e-223 or 8.0000000000000003e-218 < x < 4.8000000000000001e-111 or 2.8000000000000001e-87 < x < 7.4000000000000005e-60

   1. Initial program 34.4%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 34.4%

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

   4. Taylor expanded in n around inf 60.9%

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

      1. neg-mul-1 60.9%

         \[\leadsto \color{blue}{-\frac{\log x}{n}} \]

      2. distribute-neg-frac 60.9%

         \[\leadsto \color{blue}{\frac{-\log x}{n}} \]

   6. Simplified 60.9%

      \[\leadsto \color{blue}{\frac{-\log x}{n}} \]

   if 7.4000000000000005e-60 < x < 1.08e216

   1. Initial program 50.0%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in n around inf 48.6%

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
   4. Step-by-step derivation

      1. log1p-def 48.6%

         \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(x\right)} - \log x}{n} \]

   5. Simplified 48.6%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) - \log x}{n}} \]

   6. Taylor expanded in x around inf 59.2%

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

      1. expm1-log1p-u 56.6%

         \[\leadsto \frac{1}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(x\right)\right)} \cdot n} \]

      2. expm1-udef 67.0%

         \[\leadsto \frac{1}{\color{blue}{\left(e^{\mathsf{log1p}\left(x\right)} - 1\right)} \cdot n} \]

      3. log1p-udef 67.0%

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

      4. rem-exp-log 69.0%

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

      5. +-commutative 69.0%

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

   8. Applied egg-rr 69.5%

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

   if 1.08e216 < x

   1. Initial program 93.0%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. add-log-exp 93.0%

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

      2. add-exp-log 93.0%

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

      3. log-pow 93.0%

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

      4. +-commutative 93.0%

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

      5. log1p-udef 93.0%

         \[\leadsto \log \left(e^{e^{\frac{1}{n} \cdot \color{blue}{\mathsf{log1p}\left(x\right)}} - {x}^{\left(\frac{1}{n}\right)}}\right) \]

      6. *-commutative 93.0%

         \[\leadsto \log \left(e^{e^{\color{blue}{\mathsf{log1p}\left(x\right) \cdot \frac{1}{n}}} - {x}^{\left(\frac{1}{n}\right)}}\right) \]

      7. un-div-inv 93.0%

         \[\leadsto \log \left(e^{e^{\color{blue}{\frac{\mathsf{log1p}\left(x\right)}{n}}} - {x}^{\left(\frac{1}{n}\right)}}\right) \]

   4. Applied egg-rr 93.0%

      \[\leadsto \color{blue}{\log \left(e^{e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{1}{n}\right)}}\right)} \]

   5. Taylor expanded in x around inf 93.0%

      \[\leadsto \log \color{blue}{1} \]
3. Recombined 4 regimes into one program.

4. Final simplification 71.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.65 \cdot 10^{-275}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;x \leq 2.8 \cdot 10^{-223}:\\ \;\;\;\;\frac{-\log x}{n}\\ \mathbf{elif}\;x \leq 8 \cdot 10^{-218}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;x \leq 4.8 \cdot 10^{-111}:\\ \;\;\;\;\frac{-\log x}{n}\\ \mathbf{elif}\;x \leq 2.8 \cdot 10^{-87}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;x \leq 7.4 \cdot 10^{-60}:\\ \;\;\;\;\frac{-\log x}{n}\\ \mathbf{elif}\;x \leq 1.08 \cdot 10^{+216}:\\ \;\;\;\;\frac{\frac{1}{\left(1 + x\right) + -1}}{n}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 81.5% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ t_1 := \frac{\frac{t\_0}{n}}{x}\\ \mathbf{if}\;\frac{1}{n} \leq -4 \cdot 10^{-12}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-77}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 100000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{+93}:\\ \;\;\;\;1 - t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{n \cdot \left(\left(1 + x\right) + -1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))) (t_1 (/ (/ t_0 n) x)))
   (if (<= (/ 1.0 n) -4e-12)
     t_1
     (if (<= (/ 1.0 n) 2e-77)
       (/ (log (/ (+ 1.0 x) x)) n)
       (if (<= (/ 1.0 n) 100000.0)
         t_1
         (if (<= (/ 1.0 n) 2e+93)
           (- 1.0 t_0)
           (/ 1.0 (* n (+ (+ 1.0 x) -1.0)))))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double t_1 = (t_0 / n) / x;
	double tmp;
	if ((1.0 / n) <= -4e-12) {
		tmp = t_1;
	} else if ((1.0 / n) <= 2e-77) {
		tmp = log(((1.0 + x) / x)) / n;
	} else if ((1.0 / n) <= 100000.0) {
		tmp = t_1;
	} else if ((1.0 / n) <= 2e+93) {
		tmp = 1.0 - t_0;
	} else {
		tmp = 1.0 / (n * ((1.0 + x) + -1.0));
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = x ** (1.0d0 / n)
    t_1 = (t_0 / n) / x
    if ((1.0d0 / n) <= (-4d-12)) then
        tmp = t_1
    else if ((1.0d0 / n) <= 2d-77) then
        tmp = log(((1.0d0 + x) / x)) / n
    else if ((1.0d0 / n) <= 100000.0d0) then
        tmp = t_1
    else if ((1.0d0 / n) <= 2d+93) then
        tmp = 1.0d0 - t_0
    else
        tmp = 1.0d0 / (n * ((1.0d0 + x) + (-1.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, (1.0 / n));
	double t_1 = (t_0 / n) / x;
	double tmp;
	if ((1.0 / n) <= -4e-12) {
		tmp = t_1;
	} else if ((1.0 / n) <= 2e-77) {
		tmp = Math.log(((1.0 + x) / x)) / n;
	} else if ((1.0 / n) <= 100000.0) {
		tmp = t_1;
	} else if ((1.0 / n) <= 2e+93) {
		tmp = 1.0 - t_0;
	} else {
		tmp = 1.0 / (n * ((1.0 + x) + -1.0));
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	t_1 = (t_0 / n) / x
	tmp = 0
	if (1.0 / n) <= -4e-12:
		tmp = t_1
	elif (1.0 / n) <= 2e-77:
		tmp = math.log(((1.0 + x) / x)) / n
	elif (1.0 / n) <= 100000.0:
		tmp = t_1
	elif (1.0 / n) <= 2e+93:
		tmp = 1.0 - t_0
	else:
		tmp = 1.0 / (n * ((1.0 + x) + -1.0))
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	t_1 = Float64(Float64(t_0 / n) / x)
	tmp = 0.0
	if (Float64(1.0 / n) <= -4e-12)
		tmp = t_1;
	elseif (Float64(1.0 / n) <= 2e-77)
		tmp = Float64(log(Float64(Float64(1.0 + x) / x)) / n);
	elseif (Float64(1.0 / n) <= 100000.0)
		tmp = t_1;
	elseif (Float64(1.0 / n) <= 2e+93)
		tmp = Float64(1.0 - t_0);
	else
		tmp = Float64(1.0 / Float64(n * Float64(Float64(1.0 + x) + -1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = x ^ (1.0 / n);
	t_1 = (t_0 / n) / x;
	tmp = 0.0;
	if ((1.0 / n) <= -4e-12)
		tmp = t_1;
	elseif ((1.0 / n) <= 2e-77)
		tmp = log(((1.0 + x) / x)) / n;
	elseif ((1.0 / n) <= 100000.0)
		tmp = t_1;
	elseif ((1.0 / n) <= 2e+93)
		tmp = 1.0 - t_0;
	else
		tmp = 1.0 / (n * ((1.0 + x) + -1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(t$95$0 / n), $MachinePrecision] / x), $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -4e-12], t$95$1, If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-77], N[(N[Log[N[(N[(1.0 + x), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 100000.0], t$95$1, If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e+93], N[(1.0 - t$95$0), $MachinePrecision], N[(1.0 / N[(n * N[(N[(1.0 + x), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 1 n) < -3.99999999999999992e-12 or 1.9999999999999999e-77 < (/.f64 1 n) < 1e5

   1. Initial program 78.1%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 92.5%

      \[\leadsto \color{blue}{\frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}}{n \cdot x}} \]
   4. Step-by-step derivation

      1. mul-1-neg 92.5%

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

      2. log-rec 92.5%

         \[\leadsto \frac{e^{-\frac{\color{blue}{-\log x}}{n}}}{n \cdot x} \]

      3. mul-1-neg 92.5%

         \[\leadsto \frac{e^{-\frac{\color{blue}{-1 \cdot \log x}}{n}}}{n \cdot x} \]

      4. distribute-neg-frac 92.5%

         \[\leadsto \frac{e^{\color{blue}{\frac{--1 \cdot \log x}{n}}}}{n \cdot x} \]

      5. mul-1-neg 92.5%

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

      6. remove-double-neg 92.5%

         \[\leadsto \frac{e^{\frac{\color{blue}{\log x}}{n}}}{n \cdot x} \]

      7. *-commutative 92.5%

         \[\leadsto \frac{e^{\frac{\log x}{n}}}{\color{blue}{x \cdot n}} \]

   5. Simplified 92.5%

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

      1. div-inv 92.5%

         \[\leadsto \frac{e^{\color{blue}{\log x \cdot \frac{1}{n}}}}{x \cdot n} \]

      2. pow-to-exp 92.5%

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

      3. *-un-lft-identity 92.5%

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

      4. times-frac 92.6%

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

   7. Applied egg-rr 92.6%

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

      1. associate-*l/ 92.5%

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

      2. *-un-lft-identity 92.5%

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

   9. Applied egg-rr 92.5%

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

   if -3.99999999999999992e-12 < (/.f64 1 n) < 1.9999999999999999e-77

   1. Initial program 41.1%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in n around inf 82.8%

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
   4. Step-by-step derivation

      1. log1p-def 82.8%

         \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(x\right)} - \log x}{n} \]

   5. Simplified 82.8%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) - \log x}{n}} \]
   6. Step-by-step derivation

      1. log1p-udef 82.8%

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

      2. diff-log 83.1%

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

      3. +-commutative 83.1%

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

   7. Applied egg-rr 83.1%

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

   if 1e5 < (/.f64 1 n) < 2.00000000000000009e93

   1. Initial program 91.2%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 91.2%

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

   if 2.00000000000000009e93 < (/.f64 1 n)

   1. Initial program 18.8%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 0.8%

      \[\leadsto \color{blue}{\frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}}{n \cdot x}} \]
   4. Step-by-step derivation

      1. mul-1-neg 0.8%

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

      2. log-rec 0.8%

         \[\leadsto \frac{e^{-\frac{\color{blue}{-\log x}}{n}}}{n \cdot x} \]

      3. mul-1-neg 0.8%

         \[\leadsto \frac{e^{-\frac{\color{blue}{-1 \cdot \log x}}{n}}}{n \cdot x} \]

      4. distribute-neg-frac 0.8%

         \[\leadsto \frac{e^{\color{blue}{\frac{--1 \cdot \log x}{n}}}}{n \cdot x} \]

      5. mul-1-neg 0.8%

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

      6. remove-double-neg 0.8%

         \[\leadsto \frac{e^{\frac{\color{blue}{\log x}}{n}}}{n \cdot x} \]

      7. *-commutative 0.8%

         \[\leadsto \frac{e^{\frac{\log x}{n}}}{\color{blue}{x \cdot n}} \]

   5. Simplified 0.8%

      \[\leadsto \color{blue}{\frac{e^{\frac{\log x}{n}}}{x \cdot n}} \]

   6. Taylor expanded in n around inf 46.5%

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

      1. *-commutative 46.5%

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

   8. Simplified 46.5%

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

      1. expm1-log1p-u 46.5%

         \[\leadsto \frac{1}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(x\right)\right)} \cdot n} \]

      2. expm1-udef 79.2%

         \[\leadsto \frac{1}{\color{blue}{\left(e^{\mathsf{log1p}\left(x\right)} - 1\right)} \cdot n} \]

      3. log1p-udef 79.2%

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

      4. rem-exp-log 79.2%

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

      5. +-commutative 79.2%

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

   10. Applied egg-rr 79.2%

       \[\leadsto \frac{1}{\color{blue}{\left(\left(x + 1\right) - 1\right)} \cdot n} \]
3. Recombined 4 regimes into one program.

4. Final simplification 86.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -4 \cdot 10^{-12}:\\ \;\;\;\;\frac{\frac{{x}^{\left(\frac{1}{n}\right)}}{n}}{x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-77}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 100000:\\ \;\;\;\;\frac{\frac{{x}^{\left(\frac{1}{n}\right)}}{n}}{x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{+93}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{n \cdot \left(\left(1 + x\right) + -1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 81.5% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -4 \cdot 10^{-12}:\\ \;\;\;\;\frac{t\_0}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-77}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 100000:\\ \;\;\;\;\frac{\frac{t\_0}{n}}{x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{+93}:\\ \;\;\;\;1 - t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{n \cdot \left(\left(1 + x\right) + -1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -4e-12)
     (/ t_0 (* n x))
     (if (<= (/ 1.0 n) 2e-77)
       (/ (log (/ (+ 1.0 x) x)) n)
       (if (<= (/ 1.0 n) 100000.0)
         (/ (/ t_0 n) x)
         (if (<= (/ 1.0 n) 2e+93)
           (- 1.0 t_0)
           (/ 1.0 (* n (+ (+ 1.0 x) -1.0)))))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -4e-12) {
		tmp = t_0 / (n * x);
	} else if ((1.0 / n) <= 2e-77) {
		tmp = log(((1.0 + x) / x)) / n;
	} else if ((1.0 / n) <= 100000.0) {
		tmp = (t_0 / n) / x;
	} else if ((1.0 / n) <= 2e+93) {
		tmp = 1.0 - t_0;
	} else {
		tmp = 1.0 / (n * ((1.0 + x) + -1.0));
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x ** (1.0d0 / n)
    if ((1.0d0 / n) <= (-4d-12)) then
        tmp = t_0 / (n * x)
    else if ((1.0d0 / n) <= 2d-77) then
        tmp = log(((1.0d0 + x) / x)) / n
    else if ((1.0d0 / n) <= 100000.0d0) then
        tmp = (t_0 / n) / x
    else if ((1.0d0 / n) <= 2d+93) then
        tmp = 1.0d0 - t_0
    else
        tmp = 1.0d0 / (n * ((1.0d0 + x) + (-1.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -4e-12) {
		tmp = t_0 / (n * x);
	} else if ((1.0 / n) <= 2e-77) {
		tmp = Math.log(((1.0 + x) / x)) / n;
	} else if ((1.0 / n) <= 100000.0) {
		tmp = (t_0 / n) / x;
	} else if ((1.0 / n) <= 2e+93) {
		tmp = 1.0 - t_0;
	} else {
		tmp = 1.0 / (n * ((1.0 + x) + -1.0));
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	tmp = 0
	if (1.0 / n) <= -4e-12:
		tmp = t_0 / (n * x)
	elif (1.0 / n) <= 2e-77:
		tmp = math.log(((1.0 + x) / x)) / n
	elif (1.0 / n) <= 100000.0:
		tmp = (t_0 / n) / x
	elif (1.0 / n) <= 2e+93:
		tmp = 1.0 - t_0
	else:
		tmp = 1.0 / (n * ((1.0 + x) + -1.0))
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -4e-12)
		tmp = Float64(t_0 / Float64(n * x));
	elseif (Float64(1.0 / n) <= 2e-77)
		tmp = Float64(log(Float64(Float64(1.0 + x) / x)) / n);
	elseif (Float64(1.0 / n) <= 100000.0)
		tmp = Float64(Float64(t_0 / n) / x);
	elseif (Float64(1.0 / n) <= 2e+93)
		tmp = Float64(1.0 - t_0);
	else
		tmp = Float64(1.0 / Float64(n * Float64(Float64(1.0 + x) + -1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = x ^ (1.0 / n);
	tmp = 0.0;
	if ((1.0 / n) <= -4e-12)
		tmp = t_0 / (n * x);
	elseif ((1.0 / n) <= 2e-77)
		tmp = log(((1.0 + x) / x)) / n;
	elseif ((1.0 / n) <= 100000.0)
		tmp = (t_0 / n) / x;
	elseif ((1.0 / n) <= 2e+93)
		tmp = 1.0 - t_0;
	else
		tmp = 1.0 / (n * ((1.0 + x) + -1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -4e-12], N[(t$95$0 / N[(n * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-77], N[(N[Log[N[(N[(1.0 + x), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 100000.0], N[(N[(t$95$0 / n), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e+93], N[(1.0 - t$95$0), $MachinePrecision], N[(1.0 / N[(n * N[(N[(1.0 + x), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if (/.f64 1 n) < -3.99999999999999992e-12

   1. Initial program 96.1%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 100.0%

      \[\leadsto \color{blue}{\frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}}{n \cdot x}} \]
   4. Step-by-step derivation

      1. mul-1-neg 100.0%

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

      2. log-rec 100.0%

         \[\leadsto \frac{e^{-\frac{\color{blue}{-\log x}}{n}}}{n \cdot x} \]

      3. mul-1-neg 100.0%

         \[\leadsto \frac{e^{-\frac{\color{blue}{-1 \cdot \log x}}{n}}}{n \cdot x} \]

      4. distribute-neg-frac 100.0%

         \[\leadsto \frac{e^{\color{blue}{\frac{--1 \cdot \log x}{n}}}}{n \cdot x} \]

      5. mul-1-neg 100.0%

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

      6. remove-double-neg 100.0%

         \[\leadsto \frac{e^{\frac{\color{blue}{\log x}}{n}}}{n \cdot x} \]

      7. *-commutative 100.0%

         \[\leadsto \frac{e^{\frac{\log x}{n}}}{\color{blue}{x \cdot n}} \]

   5. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{e^{\frac{\log x}{n}}}{x \cdot n}} \]

   6. Taylor expanded in x around 0 100.0%

      \[\leadsto \color{blue}{\frac{e^{\frac{\log x}{n}}}{n \cdot x}} \]
   7. Step-by-step derivation

      1. *-rgt-identity 100.0%

         \[\leadsto \frac{e^{\color{blue}{\frac{\log x}{n} \cdot 1}}}{n \cdot x} \]

      2. associate-*l/ 100.0%

         \[\leadsto \frac{e^{\color{blue}{\frac{\log x \cdot 1}{n}}}}{n \cdot x} \]

      3. associate-*r/ 100.0%

         \[\leadsto \frac{e^{\color{blue}{\log x \cdot \frac{1}{n}}}}{n \cdot x} \]

      4. exp-to-pow 100.0%

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

      5. *-commutative 100.0%

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

   8. Simplified 100.0%

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

   if -3.99999999999999992e-12 < (/.f64 1 n) < 1.9999999999999999e-77

   1. Initial program 41.1%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in n around inf 82.8%

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
   4. Step-by-step derivation

      1. log1p-def 82.8%

         \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(x\right)} - \log x}{n} \]

   5. Simplified 82.8%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) - \log x}{n}} \]
   6. Step-by-step derivation

      1. log1p-udef 82.8%

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

      2. diff-log 83.1%

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

      3. +-commutative 83.1%

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

   7. Applied egg-rr 83.1%

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

   if 1.9999999999999999e-77 < (/.f64 1 n) < 1e5

   1. Initial program 13.5%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 65.7%

      \[\leadsto \color{blue}{\frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}}{n \cdot x}} \]
   4. Step-by-step derivation

      1. mul-1-neg 65.7%

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

      2. log-rec 65.7%

         \[\leadsto \frac{e^{-\frac{\color{blue}{-\log x}}{n}}}{n \cdot x} \]

      3. mul-1-neg 65.7%

         \[\leadsto \frac{e^{-\frac{\color{blue}{-1 \cdot \log x}}{n}}}{n \cdot x} \]

      4. distribute-neg-frac 65.7%

         \[\leadsto \frac{e^{\color{blue}{\frac{--1 \cdot \log x}{n}}}}{n \cdot x} \]

      5. mul-1-neg 65.7%

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

      6. remove-double-neg 65.7%

         \[\leadsto \frac{e^{\frac{\color{blue}{\log x}}{n}}}{n \cdot x} \]

      7. *-commutative 65.7%

         \[\leadsto \frac{e^{\frac{\log x}{n}}}{\color{blue}{x \cdot n}} \]

   5. Simplified 65.7%

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

      1. div-inv 65.7%

         \[\leadsto \frac{e^{\color{blue}{\log x \cdot \frac{1}{n}}}}{x \cdot n} \]

      2. pow-to-exp 65.7%

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

      3. *-un-lft-identity 65.7%

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

      4. times-frac 65.7%

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

   7. Applied egg-rr 65.7%

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

      1. associate-*l/ 65.7%

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

      2. *-un-lft-identity 65.7%

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

   9. Applied egg-rr 65.7%

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

   if 1e5 < (/.f64 1 n) < 2.00000000000000009e93

   1. Initial program 91.2%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 91.2%

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

   if 2.00000000000000009e93 < (/.f64 1 n)

   1. Initial program 18.8%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 0.8%

      \[\leadsto \color{blue}{\frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}}{n \cdot x}} \]
   4. Step-by-step derivation

      1. mul-1-neg 0.8%

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

      2. log-rec 0.8%

         \[\leadsto \frac{e^{-\frac{\color{blue}{-\log x}}{n}}}{n \cdot x} \]

      3. mul-1-neg 0.8%

         \[\leadsto \frac{e^{-\frac{\color{blue}{-1 \cdot \log x}}{n}}}{n \cdot x} \]

      4. distribute-neg-frac 0.8%

         \[\leadsto \frac{e^{\color{blue}{\frac{--1 \cdot \log x}{n}}}}{n \cdot x} \]

      5. mul-1-neg 0.8%

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

      6. remove-double-neg 0.8%

         \[\leadsto \frac{e^{\frac{\color{blue}{\log x}}{n}}}{n \cdot x} \]

      7. *-commutative 0.8%

         \[\leadsto \frac{e^{\frac{\log x}{n}}}{\color{blue}{x \cdot n}} \]

   5. Simplified 0.8%

      \[\leadsto \color{blue}{\frac{e^{\frac{\log x}{n}}}{x \cdot n}} \]

   6. Taylor expanded in n around inf 46.5%

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

      1. *-commutative 46.5%

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

   8. Simplified 46.5%

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

      1. expm1-log1p-u 46.5%

         \[\leadsto \frac{1}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(x\right)\right)} \cdot n} \]

      2. expm1-udef 79.2%

         \[\leadsto \frac{1}{\color{blue}{\left(e^{\mathsf{log1p}\left(x\right)} - 1\right)} \cdot n} \]

      3. log1p-udef 79.2%

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

      4. rem-exp-log 79.2%

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

      5. +-commutative 79.2%

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

   10. Applied egg-rr 79.2%

       \[\leadsto \frac{1}{\color{blue}{\left(\left(x + 1\right) - 1\right)} \cdot n} \]
3. Recombined 5 regimes into one program.

4. Final simplification 86.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -4 \cdot 10^{-12}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-77}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 100000:\\ \;\;\;\;\frac{\frac{{x}^{\left(\frac{1}{n}\right)}}{n}}{x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{+93}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{n \cdot \left(\left(1 + x\right) + -1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 81.5% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -4 \cdot 10^{-12}:\\ \;\;\;\;\frac{t\_0}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-77}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 100000:\\ \;\;\;\;\frac{1}{x} \cdot \frac{t\_0}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{+93}:\\ \;\;\;\;1 - t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{n \cdot \left(\left(1 + x\right) + -1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -4e-12)
     (/ t_0 (* n x))
     (if (<= (/ 1.0 n) 2e-77)
       (/ (log (/ (+ 1.0 x) x)) n)
       (if (<= (/ 1.0 n) 100000.0)
         (* (/ 1.0 x) (/ t_0 n))
         (if (<= (/ 1.0 n) 2e+93)
           (- 1.0 t_0)
           (/ 1.0 (* n (+ (+ 1.0 x) -1.0)))))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -4e-12) {
		tmp = t_0 / (n * x);
	} else if ((1.0 / n) <= 2e-77) {
		tmp = log(((1.0 + x) / x)) / n;
	} else if ((1.0 / n) <= 100000.0) {
		tmp = (1.0 / x) * (t_0 / n);
	} else if ((1.0 / n) <= 2e+93) {
		tmp = 1.0 - t_0;
	} else {
		tmp = 1.0 / (n * ((1.0 + x) + -1.0));
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x ** (1.0d0 / n)
    if ((1.0d0 / n) <= (-4d-12)) then
        tmp = t_0 / (n * x)
    else if ((1.0d0 / n) <= 2d-77) then
        tmp = log(((1.0d0 + x) / x)) / n
    else if ((1.0d0 / n) <= 100000.0d0) then
        tmp = (1.0d0 / x) * (t_0 / n)
    else if ((1.0d0 / n) <= 2d+93) then
        tmp = 1.0d0 - t_0
    else
        tmp = 1.0d0 / (n * ((1.0d0 + x) + (-1.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -4e-12) {
		tmp = t_0 / (n * x);
	} else if ((1.0 / n) <= 2e-77) {
		tmp = Math.log(((1.0 + x) / x)) / n;
	} else if ((1.0 / n) <= 100000.0) {
		tmp = (1.0 / x) * (t_0 / n);
	} else if ((1.0 / n) <= 2e+93) {
		tmp = 1.0 - t_0;
	} else {
		tmp = 1.0 / (n * ((1.0 + x) + -1.0));
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	tmp = 0
	if (1.0 / n) <= -4e-12:
		tmp = t_0 / (n * x)
	elif (1.0 / n) <= 2e-77:
		tmp = math.log(((1.0 + x) / x)) / n
	elif (1.0 / n) <= 100000.0:
		tmp = (1.0 / x) * (t_0 / n)
	elif (1.0 / n) <= 2e+93:
		tmp = 1.0 - t_0
	else:
		tmp = 1.0 / (n * ((1.0 + x) + -1.0))
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -4e-12)
		tmp = Float64(t_0 / Float64(n * x));
	elseif (Float64(1.0 / n) <= 2e-77)
		tmp = Float64(log(Float64(Float64(1.0 + x) / x)) / n);
	elseif (Float64(1.0 / n) <= 100000.0)
		tmp = Float64(Float64(1.0 / x) * Float64(t_0 / n));
	elseif (Float64(1.0 / n) <= 2e+93)
		tmp = Float64(1.0 - t_0);
	else
		tmp = Float64(1.0 / Float64(n * Float64(Float64(1.0 + x) + -1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = x ^ (1.0 / n);
	tmp = 0.0;
	if ((1.0 / n) <= -4e-12)
		tmp = t_0 / (n * x);
	elseif ((1.0 / n) <= 2e-77)
		tmp = log(((1.0 + x) / x)) / n;
	elseif ((1.0 / n) <= 100000.0)
		tmp = (1.0 / x) * (t_0 / n);
	elseif ((1.0 / n) <= 2e+93)
		tmp = 1.0 - t_0;
	else
		tmp = 1.0 / (n * ((1.0 + x) + -1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -4e-12], N[(t$95$0 / N[(n * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-77], N[(N[Log[N[(N[(1.0 + x), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 100000.0], N[(N[(1.0 / x), $MachinePrecision] * N[(t$95$0 / n), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e+93], N[(1.0 - t$95$0), $MachinePrecision], N[(1.0 / N[(n * N[(N[(1.0 + x), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if (/.f64 1 n) < -3.99999999999999992e-12

   1. Initial program 96.1%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 100.0%

      \[\leadsto \color{blue}{\frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}}{n \cdot x}} \]
   4. Step-by-step derivation

      1. mul-1-neg 100.0%

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

      2. log-rec 100.0%

         \[\leadsto \frac{e^{-\frac{\color{blue}{-\log x}}{n}}}{n \cdot x} \]

      3. mul-1-neg 100.0%

         \[\leadsto \frac{e^{-\frac{\color{blue}{-1 \cdot \log x}}{n}}}{n \cdot x} \]

      4. distribute-neg-frac 100.0%

         \[\leadsto \frac{e^{\color{blue}{\frac{--1 \cdot \log x}{n}}}}{n \cdot x} \]

      5. mul-1-neg 100.0%

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

      6. remove-double-neg 100.0%

         \[\leadsto \frac{e^{\frac{\color{blue}{\log x}}{n}}}{n \cdot x} \]

      7. *-commutative 100.0%

         \[\leadsto \frac{e^{\frac{\log x}{n}}}{\color{blue}{x \cdot n}} \]

   5. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{e^{\frac{\log x}{n}}}{x \cdot n}} \]

   6. Taylor expanded in x around 0 100.0%

      \[\leadsto \color{blue}{\frac{e^{\frac{\log x}{n}}}{n \cdot x}} \]
   7. Step-by-step derivation

      1. *-rgt-identity 100.0%

         \[\leadsto \frac{e^{\color{blue}{\frac{\log x}{n} \cdot 1}}}{n \cdot x} \]

      2. associate-*l/ 100.0%

         \[\leadsto \frac{e^{\color{blue}{\frac{\log x \cdot 1}{n}}}}{n \cdot x} \]

      3. associate-*r/ 100.0%

         \[\leadsto \frac{e^{\color{blue}{\log x \cdot \frac{1}{n}}}}{n \cdot x} \]

      4. exp-to-pow 100.0%

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

      5. *-commutative 100.0%

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

   8. Simplified 100.0%

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

   if -3.99999999999999992e-12 < (/.f64 1 n) < 1.9999999999999999e-77

   1. Initial program 41.1%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in n around inf 82.8%

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
   4. Step-by-step derivation

      1. log1p-def 82.8%

         \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(x\right)} - \log x}{n} \]

   5. Simplified 82.8%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) - \log x}{n}} \]
   6. Step-by-step derivation

      1. log1p-udef 82.8%

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

      2. diff-log 83.1%

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

      3. +-commutative 83.1%

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

   7. Applied egg-rr 83.1%

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

   if 1.9999999999999999e-77 < (/.f64 1 n) < 1e5

   1. Initial program 13.5%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 65.7%

      \[\leadsto \color{blue}{\frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}}{n \cdot x}} \]
   4. Step-by-step derivation

      1. mul-1-neg 65.7%

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

      2. log-rec 65.7%

         \[\leadsto \frac{e^{-\frac{\color{blue}{-\log x}}{n}}}{n \cdot x} \]

      3. mul-1-neg 65.7%

         \[\leadsto \frac{e^{-\frac{\color{blue}{-1 \cdot \log x}}{n}}}{n \cdot x} \]

      4. distribute-neg-frac 65.7%

         \[\leadsto \frac{e^{\color{blue}{\frac{--1 \cdot \log x}{n}}}}{n \cdot x} \]

      5. mul-1-neg 65.7%

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

      6. remove-double-neg 65.7%

         \[\leadsto \frac{e^{\frac{\color{blue}{\log x}}{n}}}{n \cdot x} \]

      7. *-commutative 65.7%

         \[\leadsto \frac{e^{\frac{\log x}{n}}}{\color{blue}{x \cdot n}} \]

   5. Simplified 65.7%

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

      1. div-inv 65.7%

         \[\leadsto \frac{e^{\color{blue}{\log x \cdot \frac{1}{n}}}}{x \cdot n} \]

      2. pow-to-exp 65.7%

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

      3. *-un-lft-identity 65.7%

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

      4. times-frac 65.7%

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

   7. Applied egg-rr 65.7%

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

   if 1e5 < (/.f64 1 n) < 2.00000000000000009e93

   1. Initial program 91.2%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 91.2%

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

   if 2.00000000000000009e93 < (/.f64 1 n)

   1. Initial program 18.8%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 0.8%

      \[\leadsto \color{blue}{\frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}}{n \cdot x}} \]
   4. Step-by-step derivation

      1. mul-1-neg 0.8%

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

      2. log-rec 0.8%

         \[\leadsto \frac{e^{-\frac{\color{blue}{-\log x}}{n}}}{n \cdot x} \]

      3. mul-1-neg 0.8%

         \[\leadsto \frac{e^{-\frac{\color{blue}{-1 \cdot \log x}}{n}}}{n \cdot x} \]

      4. distribute-neg-frac 0.8%

         \[\leadsto \frac{e^{\color{blue}{\frac{--1 \cdot \log x}{n}}}}{n \cdot x} \]

      5. mul-1-neg 0.8%

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

      6. remove-double-neg 0.8%

         \[\leadsto \frac{e^{\frac{\color{blue}{\log x}}{n}}}{n \cdot x} \]

      7. *-commutative 0.8%

         \[\leadsto \frac{e^{\frac{\log x}{n}}}{\color{blue}{x \cdot n}} \]

   5. Simplified 0.8%

      \[\leadsto \color{blue}{\frac{e^{\frac{\log x}{n}}}{x \cdot n}} \]

   6. Taylor expanded in n around inf 46.5%

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

      1. *-commutative 46.5%

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

   8. Simplified 46.5%

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

      1. expm1-log1p-u 46.5%

         \[\leadsto \frac{1}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(x\right)\right)} \cdot n} \]

      2. expm1-udef 79.2%

         \[\leadsto \frac{1}{\color{blue}{\left(e^{\mathsf{log1p}\left(x\right)} - 1\right)} \cdot n} \]

      3. log1p-udef 79.2%

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

      4. rem-exp-log 79.2%

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

      5. +-commutative 79.2%

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

   10. Applied egg-rr 79.2%

       \[\leadsto \frac{1}{\color{blue}{\left(\left(x + 1\right) - 1\right)} \cdot n} \]
3. Recombined 5 regimes into one program.

4. Final simplification 86.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -4 \cdot 10^{-12}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-77}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 100000:\\ \;\;\;\;\frac{1}{x} \cdot \frac{{x}^{\left(\frac{1}{n}\right)}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{+93}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{n \cdot \left(\left(1 + x\right) + -1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 57.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(1 + x\right) + -1\\ t_1 := \frac{-\log x}{n}\\ \mathbf{if}\;x \leq 6.3 \cdot 10^{-109}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 10^{-82}:\\ \;\;\;\;\frac{1}{n \cdot t\_0}\\ \mathbf{elif}\;x \leq 8.8 \cdot 10^{-60}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 2.2 \cdot 10^{+216}:\\ \;\;\;\;\frac{\frac{1}{t\_0}}{n}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (+ (+ 1.0 x) -1.0)) (t_1 (/ (- (log x)) n)))
   (if (<= x 6.3e-109)
     t_1
     (if (<= x 1e-82)
       (/ 1.0 (* n t_0))
       (if (<= x 8.8e-60) t_1 (if (<= x 2.2e+216) (/ (/ 1.0 t_0) n) 0.0))))))

C

double code(double x, double n) {
	double t_0 = (1.0 + x) + -1.0;
	double t_1 = -log(x) / n;
	double tmp;
	if (x <= 6.3e-109) {
		tmp = t_1;
	} else if (x <= 1e-82) {
		tmp = 1.0 / (n * t_0);
	} else if (x <= 8.8e-60) {
		tmp = t_1;
	} else if (x <= 2.2e+216) {
		tmp = (1.0 / t_0) / n;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (1.0d0 + x) + (-1.0d0)
    t_1 = -log(x) / n
    if (x <= 6.3d-109) then
        tmp = t_1
    else if (x <= 1d-82) then
        tmp = 1.0d0 / (n * t_0)
    else if (x <= 8.8d-60) then
        tmp = t_1
    else if (x <= 2.2d+216) then
        tmp = (1.0d0 / t_0) / n
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = (1.0 + x) + -1.0;
	double t_1 = -Math.log(x) / n;
	double tmp;
	if (x <= 6.3e-109) {
		tmp = t_1;
	} else if (x <= 1e-82) {
		tmp = 1.0 / (n * t_0);
	} else if (x <= 8.8e-60) {
		tmp = t_1;
	} else if (x <= 2.2e+216) {
		tmp = (1.0 / t_0) / n;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = (1.0 + x) + -1.0
	t_1 = -math.log(x) / n
	tmp = 0
	if x <= 6.3e-109:
		tmp = t_1
	elif x <= 1e-82:
		tmp = 1.0 / (n * t_0)
	elif x <= 8.8e-60:
		tmp = t_1
	elif x <= 2.2e+216:
		tmp = (1.0 / t_0) / n
	else:
		tmp = 0.0
	return tmp

Julia

function code(x, n)
	t_0 = Float64(Float64(1.0 + x) + -1.0)
	t_1 = Float64(Float64(-log(x)) / n)
	tmp = 0.0
	if (x <= 6.3e-109)
		tmp = t_1;
	elseif (x <= 1e-82)
		tmp = Float64(1.0 / Float64(n * t_0));
	elseif (x <= 8.8e-60)
		tmp = t_1;
	elseif (x <= 2.2e+216)
		tmp = Float64(Float64(1.0 / t_0) / n);
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = (1.0 + x) + -1.0;
	t_1 = -log(x) / n;
	tmp = 0.0;
	if (x <= 6.3e-109)
		tmp = t_1;
	elseif (x <= 1e-82)
		tmp = 1.0 / (n * t_0);
	elseif (x <= 8.8e-60)
		tmp = t_1;
	elseif (x <= 2.2e+216)
		tmp = (1.0 / t_0) / n;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(N[(1.0 + x), $MachinePrecision] + -1.0), $MachinePrecision]}, Block[{t$95$1 = N[((-N[Log[x], $MachinePrecision]) / n), $MachinePrecision]}, If[LessEqual[x, 6.3e-109], t$95$1, If[LessEqual[x, 1e-82], N[(1.0 / N[(n * t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 8.8e-60], t$95$1, If[LessEqual[x, 2.2e+216], N[(N[(1.0 / t$95$0), $MachinePrecision] / n), $MachinePrecision], 0.0]]]]]]

Derivation

1. Split input into 4 regimes

2. if x < 6.3000000000000001e-109 or 1e-82 < x < 8.7999999999999995e-60

   1. Initial program 42.8%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 42.8%

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

   4. Taylor expanded in n around inf 55.4%

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

      1. neg-mul-1 55.4%

         \[\leadsto \color{blue}{-\frac{\log x}{n}} \]

      2. distribute-neg-frac 55.4%

         \[\leadsto \color{blue}{\frac{-\log x}{n}} \]

   6. Simplified 55.4%

      \[\leadsto \color{blue}{\frac{-\log x}{n}} \]

   if 6.3000000000000001e-109 < x < 1e-82

   1. Initial program 83.9%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 67.2%

      \[\leadsto \color{blue}{\frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}}{n \cdot x}} \]
   4. Step-by-step derivation

      1. mul-1-neg 67.2%

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

      2. log-rec 67.2%

         \[\leadsto \frac{e^{-\frac{\color{blue}{-\log x}}{n}}}{n \cdot x} \]

      3. mul-1-neg 67.2%

         \[\leadsto \frac{e^{-\frac{\color{blue}{-1 \cdot \log x}}{n}}}{n \cdot x} \]

      4. distribute-neg-frac 67.2%

         \[\leadsto \frac{e^{\color{blue}{\frac{--1 \cdot \log x}{n}}}}{n \cdot x} \]

      5. mul-1-neg 67.2%

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

      6. remove-double-neg 67.2%

         \[\leadsto \frac{e^{\frac{\color{blue}{\log x}}{n}}}{n \cdot x} \]

      7. *-commutative 67.2%

         \[\leadsto \frac{e^{\frac{\log x}{n}}}{\color{blue}{x \cdot n}} \]

   5. Simplified 67.2%

      \[\leadsto \color{blue}{\frac{e^{\frac{\log x}{n}}}{x \cdot n}} \]

   6. Taylor expanded in n around inf 37.6%

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

      1. *-commutative 37.6%

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

   8. Simplified 37.6%

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

      1. expm1-log1p-u 37.6%

         \[\leadsto \frac{1}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(x\right)\right)} \cdot n} \]

      2. expm1-udef 83.9%

         \[\leadsto \frac{1}{\color{blue}{\left(e^{\mathsf{log1p}\left(x\right)} - 1\right)} \cdot n} \]

      3. log1p-udef 83.9%

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

      4. rem-exp-log 83.9%

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

      5. +-commutative 83.9%

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

   10. Applied egg-rr 83.9%

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

   if 8.7999999999999995e-60 < x < 2.2e216

   1. Initial program 50.0%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in n around inf 48.6%

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
   4. Step-by-step derivation

      1. log1p-def 48.6%

         \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(x\right)} - \log x}{n} \]

   5. Simplified 48.6%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) - \log x}{n}} \]

   6. Taylor expanded in x around inf 59.2%

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

      1. expm1-log1p-u 56.6%

         \[\leadsto \frac{1}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(x\right)\right)} \cdot n} \]

      2. expm1-udef 67.0%

         \[\leadsto \frac{1}{\color{blue}{\left(e^{\mathsf{log1p}\left(x\right)} - 1\right)} \cdot n} \]

      3. log1p-udef 67.0%

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

      4. rem-exp-log 69.0%

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

      5. +-commutative 69.0%

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

   8. Applied egg-rr 69.5%

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

   if 2.2e216 < x

   1. Initial program 93.0%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. add-log-exp 93.0%

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

      2. add-exp-log 93.0%

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

      3. log-pow 93.0%

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

      4. +-commutative 93.0%

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

      5. log1p-udef 93.0%

         \[\leadsto \log \left(e^{e^{\frac{1}{n} \cdot \color{blue}{\mathsf{log1p}\left(x\right)}} - {x}^{\left(\frac{1}{n}\right)}}\right) \]

      6. *-commutative 93.0%

         \[\leadsto \log \left(e^{e^{\color{blue}{\mathsf{log1p}\left(x\right) \cdot \frac{1}{n}}} - {x}^{\left(\frac{1}{n}\right)}}\right) \]

      7. un-div-inv 93.0%

         \[\leadsto \log \left(e^{e^{\color{blue}{\frac{\mathsf{log1p}\left(x\right)}{n}}} - {x}^{\left(\frac{1}{n}\right)}}\right) \]

   4. Applied egg-rr 93.0%

      \[\leadsto \color{blue}{\log \left(e^{e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{1}{n}\right)}}\right)} \]

   5. Taylor expanded in x around inf 93.0%

      \[\leadsto \log \color{blue}{1} \]
3. Recombined 4 regimes into one program.

4. Final simplification 67.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 6.3 \cdot 10^{-109}:\\ \;\;\;\;\frac{-\log x}{n}\\ \mathbf{elif}\;x \leq 10^{-82}:\\ \;\;\;\;\frac{1}{n \cdot \left(\left(1 + x\right) + -1\right)}\\ \mathbf{elif}\;x \leq 8.8 \cdot 10^{-60}:\\ \;\;\;\;\frac{-\log x}{n}\\ \mathbf{elif}\;x \leq 2.2 \cdot 10^{+216}:\\ \;\;\;\;\frac{\frac{1}{\left(1 + x\right) + -1}}{n}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 51.8% accurate, 15.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 8.2 \cdot 10^{+215}:\\ \;\;\;\;\frac{\frac{1}{\left(1 + x\right) + -1}}{n}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 8.2e+215) (/ (/ 1.0 (+ (+ 1.0 x) -1.0)) n) 0.0))

C

double code(double x, double n) {
	double tmp;
	if (x <= 8.2e+215) {
		tmp = (1.0 / ((1.0 + x) + -1.0)) / n;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if (x <= 8.2d+215) then
        tmp = (1.0d0 / ((1.0d0 + x) + (-1.0d0))) / n
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (x <= 8.2e+215) {
		tmp = (1.0 / ((1.0 + x) + -1.0)) / n;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 8.2e+215:
		tmp = (1.0 / ((1.0 + x) + -1.0)) / n
	else:
		tmp = 0.0
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 8.2e+215)
		tmp = Float64(Float64(1.0 / Float64(Float64(1.0 + x) + -1.0)) / n);
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (x <= 8.2e+215)
		tmp = (1.0 / ((1.0 + x) + -1.0)) / n;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 8.2e+215], N[(N[(1.0 / N[(N[(1.0 + x), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if x < 8.2000000000000007e215

   1. Initial program 47.4%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in n around inf 50.7%

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
   4. Step-by-step derivation

      1. log1p-def 50.7%

         \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(x\right)} - \log x}{n} \]

   5. Simplified 50.7%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) - \log x}{n}} \]

   6. Taylor expanded in x around inf 41.8%

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

      1. expm1-log1p-u 40.6%

         \[\leadsto \frac{1}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(x\right)\right)} \cdot n} \]

      2. expm1-udef 52.3%

         \[\leadsto \frac{1}{\color{blue}{\left(e^{\mathsf{log1p}\left(x\right)} - 1\right)} \cdot n} \]

      3. log1p-udef 52.3%

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

      4. rem-exp-log 53.3%

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

      5. +-commutative 53.3%

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

   8. Applied egg-rr 53.6%

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

   if 8.2000000000000007e215 < x

   1. Initial program 93.0%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. add-log-exp 93.0%

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

      2. add-exp-log 93.0%

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

      3. log-pow 93.0%

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

      4. +-commutative 93.0%

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

      5. log1p-udef 93.0%

         \[\leadsto \log \left(e^{e^{\frac{1}{n} \cdot \color{blue}{\mathsf{log1p}\left(x\right)}} - {x}^{\left(\frac{1}{n}\right)}}\right) \]

      6. *-commutative 93.0%

         \[\leadsto \log \left(e^{e^{\color{blue}{\mathsf{log1p}\left(x\right) \cdot \frac{1}{n}}} - {x}^{\left(\frac{1}{n}\right)}}\right) \]

      7. un-div-inv 93.0%

         \[\leadsto \log \left(e^{e^{\color{blue}{\frac{\mathsf{log1p}\left(x\right)}{n}}} - {x}^{\left(\frac{1}{n}\right)}}\right) \]

   4. Applied egg-rr 93.0%

      \[\leadsto \color{blue}{\log \left(e^{e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{1}{n}\right)}}\right)} \]

   5. Taylor expanded in x around inf 93.0%

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

4. Final simplification 59.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 8.2 \cdot 10^{+215}:\\ \;\;\;\;\frac{\frac{1}{\left(1 + x\right) + -1}}{n}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 48.0% accurate, 23.4× speedup

Math

\[\begin{array}{l} \\ \frac{1}{n \cdot \left(\left(1 + x\right) + -1\right)} \end{array} \]

FPCore

(FPCore (x n) :precision binary64 (/ 1.0 (* n (+ (+ 1.0 x) -1.0))))

C

double code(double x, double n) {
	return 1.0 / (n * ((1.0 + x) + -1.0));
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    code = 1.0d0 / (n * ((1.0d0 + x) + (-1.0d0)))
end function

Java

public static double code(double x, double n) {
	return 1.0 / (n * ((1.0 + x) + -1.0));
}

Python

def code(x, n):
	return 1.0 / (n * ((1.0 + x) + -1.0))

Julia

function code(x, n)
	return Float64(1.0 / Float64(n * Float64(Float64(1.0 + x) + -1.0)))
end

MATLAB

function tmp = code(x, n)
	tmp = 1.0 / (n * ((1.0 + x) + -1.0));
end

Wolfram

code[x_, n_] := N[(1.0 / N[(n * N[(N[(1.0 + x), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 54.5%

   \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
2. Add Preprocessing

3. Taylor expanded in x around inf 62.6%

   \[\leadsto \color{blue}{\frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}}{n \cdot x}} \]
4. Step-by-step derivation

   1. mul-1-neg 62.6%

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

   2. log-rec 62.6%

      \[\leadsto \frac{e^{-\frac{\color{blue}{-\log x}}{n}}}{n \cdot x} \]

   3. mul-1-neg 62.6%

      \[\leadsto \frac{e^{-\frac{\color{blue}{-1 \cdot \log x}}{n}}}{n \cdot x} \]

   4. distribute-neg-frac 62.6%

      \[\leadsto \frac{e^{\color{blue}{\frac{--1 \cdot \log x}{n}}}}{n \cdot x} \]

   5. mul-1-neg 62.6%

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

   6. remove-double-neg 62.6%

      \[\leadsto \frac{e^{\frac{\color{blue}{\log x}}{n}}}{n \cdot x} \]

   7. *-commutative 62.6%

      \[\leadsto \frac{e^{\frac{\log x}{n}}}{\color{blue}{x \cdot n}} \]

5. Simplified 62.6%

   \[\leadsto \color{blue}{\frac{e^{\frac{\log x}{n}}}{x \cdot n}} \]

6. Taylor expanded in n around inf 46.8%

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

   1. *-commutative 46.8%

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

8. Simplified 46.8%

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

   1. expm1-log1p-u 46.0%

      \[\leadsto \frac{1}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(x\right)\right)} \cdot n} \]

   2. expm1-udef 55.8%

      \[\leadsto \frac{1}{\color{blue}{\left(e^{\mathsf{log1p}\left(x\right)} - 1\right)} \cdot n} \]

   3. log1p-udef 55.8%

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

   4. rem-exp-log 56.7%

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

   5. +-commutative 56.7%

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

10. Applied egg-rr 56.7%

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

11. Final simplification 56.7%

    \[\leadsto \frac{1}{n \cdot \left(\left(1 + x\right) + -1\right)} \]
12. Add Preprocessing

Alternative 14: 48.6% accurate, 23.4× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{1}{\left(1 + x\right) + -1}}{n} \end{array} \]

FPCore

(FPCore (x n) :precision binary64 (/ (/ 1.0 (+ (+ 1.0 x) -1.0)) n))

C

double code(double x, double n) {
	return (1.0 / ((1.0 + x) + -1.0)) / n;
}

Fortran

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

Java

public static double code(double x, double n) {
	return (1.0 / ((1.0 + x) + -1.0)) / n;
}

Python

def code(x, n):
	return (1.0 / ((1.0 + x) + -1.0)) / n

Julia

function code(x, n)
	return Float64(Float64(1.0 / Float64(Float64(1.0 + x) + -1.0)) / n)
end

MATLAB

function tmp = code(x, n)
	tmp = (1.0 / ((1.0 + x) + -1.0)) / n;
end

Wolfram

code[x_, n_] := N[(N[(1.0 / N[(N[(1.0 + x), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision]

Derivation

1. Initial program 54.5%

   \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
2. Add Preprocessing

3. Taylor expanded in n around inf 57.3%

   \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
4. Step-by-step derivation

   1. log1p-def 57.3%

      \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(x\right)} - \log x}{n} \]

5. Simplified 57.3%

   \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) - \log x}{n}} \]

6. Taylor expanded in x around inf 47.0%

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

   1. expm1-log1p-u 46.0%

      \[\leadsto \frac{1}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(x\right)\right)} \cdot n} \]

   2. expm1-udef 55.8%

      \[\leadsto \frac{1}{\color{blue}{\left(e^{\mathsf{log1p}\left(x\right)} - 1\right)} \cdot n} \]

   3. log1p-udef 55.8%

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

   4. rem-exp-log 56.7%

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

   5. +-commutative 56.7%

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

8. Applied egg-rr 56.9%

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

9. Final simplification 56.9%

   \[\leadsto \frac{\frac{1}{\left(1 + x\right) + -1}}{n} \]
10. Add Preprocessing

Alternative 15: 40.4% accurate, 42.2× speedup

Math

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

FPCore

(FPCore (x n) :precision binary64 (/ 1.0 (* n x)))

C

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

Fortran

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

Java

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

Python

def code(x, n):
	return 1.0 / (n * x)

Julia

function code(x, n)
	return Float64(1.0 / Float64(n * x))
end

MATLAB

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

Wolfram

code[x_, n_] := N[(1.0 / N[(n * x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 54.5%

   \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
2. Add Preprocessing

3. Taylor expanded in x around inf 62.6%

   \[\leadsto \color{blue}{\frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}}{n \cdot x}} \]
4. Step-by-step derivation

   1. mul-1-neg 62.6%

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

   2. log-rec 62.6%

      \[\leadsto \frac{e^{-\frac{\color{blue}{-\log x}}{n}}}{n \cdot x} \]

   3. mul-1-neg 62.6%

      \[\leadsto \frac{e^{-\frac{\color{blue}{-1 \cdot \log x}}{n}}}{n \cdot x} \]

   4. distribute-neg-frac 62.6%

      \[\leadsto \frac{e^{\color{blue}{\frac{--1 \cdot \log x}{n}}}}{n \cdot x} \]

   5. mul-1-neg 62.6%

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

   6. remove-double-neg 62.6%

      \[\leadsto \frac{e^{\frac{\color{blue}{\log x}}{n}}}{n \cdot x} \]

   7. *-commutative 62.6%

      \[\leadsto \frac{e^{\frac{\log x}{n}}}{\color{blue}{x \cdot n}} \]

5. Simplified 62.6%

   \[\leadsto \color{blue}{\frac{e^{\frac{\log x}{n}}}{x \cdot n}} \]

6. Taylor expanded in n around inf 46.8%

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

   1. *-commutative 46.8%

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

8. Simplified 46.8%

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

9. Final simplification 46.8%

   \[\leadsto \frac{1}{n \cdot x} \]
10. Add Preprocessing

Alternative 16: 41.0% accurate, 42.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 54.5%

   \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
2. Add Preprocessing

3. Taylor expanded in x around inf 62.6%

   \[\leadsto \color{blue}{\frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}}{n \cdot x}} \]
4. Step-by-step derivation

   1. mul-1-neg 62.6%

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

   2. log-rec 62.6%

      \[\leadsto \frac{e^{-\frac{\color{blue}{-\log x}}{n}}}{n \cdot x} \]

   3. mul-1-neg 62.6%

      \[\leadsto \frac{e^{-\frac{\color{blue}{-1 \cdot \log x}}{n}}}{n \cdot x} \]

   4. distribute-neg-frac 62.6%

      \[\leadsto \frac{e^{\color{blue}{\frac{--1 \cdot \log x}{n}}}}{n \cdot x} \]

   5. mul-1-neg 62.6%

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

   6. remove-double-neg 62.6%

      \[\leadsto \frac{e^{\frac{\color{blue}{\log x}}{n}}}{n \cdot x} \]

   7. *-commutative 62.6%

      \[\leadsto \frac{e^{\frac{\log x}{n}}}{\color{blue}{x \cdot n}} \]

5. Simplified 62.6%

   \[\leadsto \color{blue}{\frac{e^{\frac{\log x}{n}}}{x \cdot n}} \]

6. Taylor expanded in n around inf 46.8%

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

   1. *-commutative 46.8%

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

8. Simplified 46.8%

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

   1. add-sqr-sqrt 32.0%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{x \cdot n}} \cdot \sqrt{\frac{1}{x \cdot n}}} \]

   2. pow2 32.0%

      \[\leadsto \color{blue}{{\left(\sqrt{\frac{1}{x \cdot n}}\right)}^{2}} \]

   3. inv-pow 32.0%

      \[\leadsto {\left(\sqrt{\color{blue}{{\left(x \cdot n\right)}^{-1}}}\right)}^{2} \]

   4. sqrt-pow1 32.0%

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

   5. metadata-eval 32.0%

      \[\leadsto {\left({\left(x \cdot n\right)}^{\color{blue}{-0.5}}\right)}^{2} \]

10. Applied egg-rr 32.0%

    \[\leadsto \color{blue}{{\left({\left(x \cdot n\right)}^{-0.5}\right)}^{2}} \]
11. Step-by-step derivation

    1. pow-pow 46.8%

       \[\leadsto \color{blue}{{\left(x \cdot n\right)}^{\left(-0.5 \cdot 2\right)}} \]

    2. metadata-eval 46.8%

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

    3. inv-pow 46.8%

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

    4. *-commutative 46.8%

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

    5. associate-/r* 47.0%

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

12. Applied egg-rr 47.0%

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

13. Final simplification 47.0%

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

Alternative 17: 4.5% accurate, 70.3× speedup

Math

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

FPCore

(FPCore (x n) :precision binary64 (/ x n))

C

double code(double x, double n) {
	return x / n;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    code = x / n
end function

Java

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

Python

def code(x, n):
	return x / n

Julia

function code(x, n)
	return Float64(x / n)
end

MATLAB

function tmp = code(x, n)
	tmp = x / n;
end

Wolfram

code[x_, n_] := N[(x / n), $MachinePrecision]

Derivation

1. Initial program 54.5%

   \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
2. Add Preprocessing

3. Taylor expanded in n around inf 57.3%

   \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
4. Step-by-step derivation

   1. log1p-def 57.3%

      \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(x\right)} - \log x}{n} \]

5. Simplified 57.3%

   \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) - \log x}{n}} \]

6. Taylor expanded in x around 0 27.3%

   \[\leadsto \frac{\color{blue}{x + -1 \cdot \log x}}{n} \]
7. Step-by-step derivation

   1. neg-mul-1 27.3%

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

   2. unsub-neg 27.3%

      \[\leadsto \frac{\color{blue}{x - \log x}}{n} \]

8. Simplified 27.3%

   \[\leadsto \frac{\color{blue}{x - \log x}}{n} \]

9. Taylor expanded in x around inf 4.4%

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

10. Final simplification 4.4%

    \[\leadsto \frac{x}{n} \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024101 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  :precision binary64
  (- (pow (+ x 1.0) (/ 1.0 n)) (pow x (/ 1.0 n))))