2nthrt (problem 3.4.6)

Percentage Accurate: 54.8% → 85.8%

Time: 28.5s

Alternatives: 18

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: 54.8% 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: 85.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -21000000000 \lor \neg \left(n \leq 10^{+24}\right):\\ \;\;\;\;\frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (or (<= n -21000000000.0) (not (<= n 1e+24)))
   (/ (log (/ (+ x 1.0) x)) n)
   (- (exp (/ (log1p x) n)) (pow x (/ 1.0 n)))))

C

double code(double x, double n) {
	double tmp;
	if ((n <= -21000000000.0) || !(n <= 1e+24)) {
		tmp = log(((x + 1.0) / x)) / n;
	} else {
		tmp = exp((log1p(x) / n)) - pow(x, (1.0 / n));
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double tmp;
	if ((n <= -21000000000.0) || !(n <= 1e+24)) {
		tmp = Math.log(((x + 1.0) / x)) / n;
	} else {
		tmp = Math.exp((Math.log1p(x) / n)) - Math.pow(x, (1.0 / n));
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if (n <= -21000000000.0) or not (n <= 1e+24):
		tmp = math.log(((x + 1.0) / x)) / n
	else:
		tmp = math.exp((math.log1p(x) / n)) - math.pow(x, (1.0 / n))
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if ((n <= -21000000000.0) || !(n <= 1e+24))
		tmp = Float64(log(Float64(Float64(x + 1.0) / x)) / n);
	else
		tmp = Float64(exp(Float64(log1p(x) / n)) - (x ^ Float64(1.0 / n)));
	end
	return tmp
end

Wolfram

code[x_, n_] := If[Or[LessEqual[n, -21000000000.0], N[Not[LessEqual[n, 1e+24]], $MachinePrecision]], N[(N[Log[N[(N[(x + 1.0), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], N[(N[Exp[N[(N[Log[1 + x], $MachinePrecision] / n), $MachinePrecision]], $MachinePrecision] - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < -2.1e10 or 9.9999999999999998e23 < n

   1. Initial program 24.3%

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

   3. Taylor expanded in n around inf 78.7%

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

      1. diff-log 78.8%

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

   5. Applied egg-rr 78.8%

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

      1. +-commutative 78.8%

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

   7. Simplified 78.8%

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

   if -2.1e10 < n < 9.9999999999999998e23

   1. Initial program 84.4%

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

   3. Taylor expanded in n around 0 84.4%

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

      1. log1p-define 98.0%

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

      2. *-rgt-identity 98.0%

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

      3. associate-*l/ 98.0%

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

      4. associate-/l* 98.0%

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

      5. exp-to-pow 98.0%

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

   5. Simplified 98.0%

      \[\leadsto \color{blue}{e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{1}{n}\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 87.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -21000000000 \lor \neg \left(n \leq 10^{+24}\right):\\ \;\;\;\;\frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 82.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{-5}:\\ \;\;\;\;{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - t\_0\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-5}:\\ \;\;\;\;\frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + x \cdot \left(\frac{1}{n} + \frac{x \cdot -0.5 + 0.5 \cdot \frac{x}{n}}{n}\right)\right) - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -5e-5)
     (- (pow (+ x 1.0) (/ 1.0 n)) t_0)
     (if (<= (/ 1.0 n) 5e-5)
       (/ (log (/ (+ x 1.0) x)) n)
       (-
        (+ 1.0 (* x (+ (/ 1.0 n) (/ (+ (* x -0.5) (* 0.5 (/ x n))) n))))
        t_0)))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -5e-5) {
		tmp = pow((x + 1.0), (1.0 / n)) - t_0;
	} else if ((1.0 / n) <= 5e-5) {
		tmp = log(((x + 1.0) / x)) / n;
	} else {
		tmp = (1.0 + (x * ((1.0 / n) + (((x * -0.5) + (0.5 * (x / n))) / 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) <= (-5d-5)) then
        tmp = ((x + 1.0d0) ** (1.0d0 / n)) - t_0
    else if ((1.0d0 / n) <= 5d-5) then
        tmp = log(((x + 1.0d0) / x)) / n
    else
        tmp = (1.0d0 + (x * ((1.0d0 / n) + (((x * (-0.5d0)) + (0.5d0 * (x / n))) / 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) <= -5e-5) {
		tmp = Math.pow((x + 1.0), (1.0 / n)) - t_0;
	} else if ((1.0 / n) <= 5e-5) {
		tmp = Math.log(((x + 1.0) / x)) / n;
	} else {
		tmp = (1.0 + (x * ((1.0 / n) + (((x * -0.5) + (0.5 * (x / n))) / n)))) - t_0;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	tmp = 0
	if (1.0 / n) <= -5e-5:
		tmp = math.pow((x + 1.0), (1.0 / n)) - t_0
	elif (1.0 / n) <= 5e-5:
		tmp = math.log(((x + 1.0) / x)) / n
	else:
		tmp = (1.0 + (x * ((1.0 / n) + (((x * -0.5) + (0.5 * (x / n))) / 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) <= -5e-5)
		tmp = Float64((Float64(x + 1.0) ^ Float64(1.0 / n)) - t_0);
	elseif (Float64(1.0 / n) <= 5e-5)
		tmp = Float64(log(Float64(Float64(x + 1.0) / x)) / n);
	else
		tmp = Float64(Float64(1.0 + Float64(x * Float64(Float64(1.0 / n) + Float64(Float64(Float64(x * -0.5) + Float64(0.5 * Float64(x / n))) / 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) <= -5e-5)
		tmp = ((x + 1.0) ^ (1.0 / n)) - t_0;
	elseif ((1.0 / n) <= 5e-5)
		tmp = log(((x + 1.0) / x)) / n;
	else
		tmp = (1.0 + (x * ((1.0 / n) + (((x * -0.5) + (0.5 * (x / n))) / 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], -5e-5], N[(N[Power[N[(x + 1.0), $MachinePrecision], N[(1.0 / n), $MachinePrecision]], $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e-5], N[(N[Log[N[(N[(x + 1.0), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], N[(N[(1.0 + N[(x * N[(N[(1.0 / n), $MachinePrecision] + N[(N[(N[(x * -0.5), $MachinePrecision] + N[(0.5 * N[(x / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 1 n) < -5.00000000000000024e-5

   1. Initial program 99.7%

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

   if -5.00000000000000024e-5 < (/.f64 1 n) < 5.00000000000000024e-5

   1. Initial program 25.0%

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

   3. Taylor expanded in n around inf 77.9%

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

      1. diff-log 78.0%

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

   5. Applied egg-rr 78.0%

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

      1. +-commutative 78.0%

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

   7. Simplified 78.0%

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

   if 5.00000000000000024e-5 < (/.f64 1 n)

   1. Initial program 63.4%

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

   3. Taylor expanded in x around 0 70.8%

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

   4. Taylor expanded in n around inf 77.7%

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

4. Final simplification 83.7%

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

Alternative 3: 59.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := n \cdot -0.3333333333333333 + n \cdot 0.25\\ t_1 := 1 - {x}^{\left(\frac{1}{n}\right)}\\ t_2 := \frac{\log x}{-n}\\ \mathbf{if}\;x \leq 6.8 \cdot 10^{-298}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 2.6 \cdot 10^{-283}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;x \leq 3.6 \cdot 10^{-257}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 2.8 \cdot 10^{-163}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;x \leq 1.4 \cdot 10^{-152}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 1.2 \cdot 10^{-135}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;x \leq 9.2 \cdot 10^{-94}:\\ \;\;\;\;\frac{\frac{1}{n} - \frac{\frac{1}{n} \cdot 0.5 - 0.3333333333333333 \cdot \frac{1}{n \cdot x}}{x}}{x}\\ \mathbf{elif}\;x \leq 5 \cdot 10^{-79}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;x \leq 2.1 \cdot 10^{-43}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 0.5:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{elif}\;x \leq 3.7 \cdot 10^{+73}:\\ \;\;\;\;\frac{1}{x \cdot \left(n - \frac{\frac{\left(-0.5 \cdot \frac{t\_0}{x} + \left(\frac{n}{x} \cdot -0.25 + \frac{n}{x} \cdot 0.16666666666666666\right)\right) - t\_0}{x} + n \cdot -0.5}{x}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (+ (* n -0.3333333333333333) (* n 0.25)))
        (t_1 (- 1.0 (pow x (/ 1.0 n))))
        (t_2 (/ (log x) (- n))))
   (if (<= x 6.8e-298)
     t_1
     (if (<= x 2.6e-283)
       t_2
       (if (<= x 3.6e-257)
         t_1
         (if (<= x 2.8e-163)
           t_2
           (if (<= x 1.4e-152)
             t_1
             (if (<= x 1.2e-135)
               t_2
               (if (<= x 9.2e-94)
                 (/
                  (-
                   (/ 1.0 n)
                   (/
                    (-
                     (* (/ 1.0 n) 0.5)
                     (* 0.3333333333333333 (/ 1.0 (* n x))))
                    x))
                  x)
                 (if (<= x 5e-79)
                   t_2
                   (if (<= x 2.1e-43)
                     t_1
                     (if (<= x 0.5)
                       (/ (- x (log x)) n)
                       (if (<= x 3.7e+73)
                         (/
                          1.0
                          (*
                           x
                           (-
                            n
                            (/
                             (+
                              (/
                               (-
                                (+
                                 (* -0.5 (/ t_0 x))
                                 (+
                                  (* (/ n x) -0.25)
                                  (* (/ n x) 0.16666666666666666)))
                                t_0)
                               x)
                              (* n -0.5))
                             x))))
                         0.0)))))))))))))

C

double code(double x, double n) {
	double t_0 = (n * -0.3333333333333333) + (n * 0.25);
	double t_1 = 1.0 - pow(x, (1.0 / n));
	double t_2 = log(x) / -n;
	double tmp;
	if (x <= 6.8e-298) {
		tmp = t_1;
	} else if (x <= 2.6e-283) {
		tmp = t_2;
	} else if (x <= 3.6e-257) {
		tmp = t_1;
	} else if (x <= 2.8e-163) {
		tmp = t_2;
	} else if (x <= 1.4e-152) {
		tmp = t_1;
	} else if (x <= 1.2e-135) {
		tmp = t_2;
	} else if (x <= 9.2e-94) {
		tmp = ((1.0 / n) - ((((1.0 / n) * 0.5) - (0.3333333333333333 * (1.0 / (n * x)))) / x)) / x;
	} else if (x <= 5e-79) {
		tmp = t_2;
	} else if (x <= 2.1e-43) {
		tmp = t_1;
	} else if (x <= 0.5) {
		tmp = (x - log(x)) / n;
	} else if (x <= 3.7e+73) {
		tmp = 1.0 / (x * (n - ((((((-0.5 * (t_0 / x)) + (((n / x) * -0.25) + ((n / x) * 0.16666666666666666))) - t_0) / x) + (n * -0.5)) / x)));
	} 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) :: t_2
    real(8) :: tmp
    t_0 = (n * (-0.3333333333333333d0)) + (n * 0.25d0)
    t_1 = 1.0d0 - (x ** (1.0d0 / n))
    t_2 = log(x) / -n
    if (x <= 6.8d-298) then
        tmp = t_1
    else if (x <= 2.6d-283) then
        tmp = t_2
    else if (x <= 3.6d-257) then
        tmp = t_1
    else if (x <= 2.8d-163) then
        tmp = t_2
    else if (x <= 1.4d-152) then
        tmp = t_1
    else if (x <= 1.2d-135) then
        tmp = t_2
    else if (x <= 9.2d-94) then
        tmp = ((1.0d0 / n) - ((((1.0d0 / n) * 0.5d0) - (0.3333333333333333d0 * (1.0d0 / (n * x)))) / x)) / x
    else if (x <= 5d-79) then
        tmp = t_2
    else if (x <= 2.1d-43) then
        tmp = t_1
    else if (x <= 0.5d0) then
        tmp = (x - log(x)) / n
    else if (x <= 3.7d+73) then
        tmp = 1.0d0 / (x * (n - (((((((-0.5d0) * (t_0 / x)) + (((n / x) * (-0.25d0)) + ((n / x) * 0.16666666666666666d0))) - t_0) / x) + (n * (-0.5d0))) / x)))
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = (n * -0.3333333333333333) + (n * 0.25);
	double t_1 = 1.0 - Math.pow(x, (1.0 / n));
	double t_2 = Math.log(x) / -n;
	double tmp;
	if (x <= 6.8e-298) {
		tmp = t_1;
	} else if (x <= 2.6e-283) {
		tmp = t_2;
	} else if (x <= 3.6e-257) {
		tmp = t_1;
	} else if (x <= 2.8e-163) {
		tmp = t_2;
	} else if (x <= 1.4e-152) {
		tmp = t_1;
	} else if (x <= 1.2e-135) {
		tmp = t_2;
	} else if (x <= 9.2e-94) {
		tmp = ((1.0 / n) - ((((1.0 / n) * 0.5) - (0.3333333333333333 * (1.0 / (n * x)))) / x)) / x;
	} else if (x <= 5e-79) {
		tmp = t_2;
	} else if (x <= 2.1e-43) {
		tmp = t_1;
	} else if (x <= 0.5) {
		tmp = (x - Math.log(x)) / n;
	} else if (x <= 3.7e+73) {
		tmp = 1.0 / (x * (n - ((((((-0.5 * (t_0 / x)) + (((n / x) * -0.25) + ((n / x) * 0.16666666666666666))) - t_0) / x) + (n * -0.5)) / x)));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = (n * -0.3333333333333333) + (n * 0.25)
	t_1 = 1.0 - math.pow(x, (1.0 / n))
	t_2 = math.log(x) / -n
	tmp = 0
	if x <= 6.8e-298:
		tmp = t_1
	elif x <= 2.6e-283:
		tmp = t_2
	elif x <= 3.6e-257:
		tmp = t_1
	elif x <= 2.8e-163:
		tmp = t_2
	elif x <= 1.4e-152:
		tmp = t_1
	elif x <= 1.2e-135:
		tmp = t_2
	elif x <= 9.2e-94:
		tmp = ((1.0 / n) - ((((1.0 / n) * 0.5) - (0.3333333333333333 * (1.0 / (n * x)))) / x)) / x
	elif x <= 5e-79:
		tmp = t_2
	elif x <= 2.1e-43:
		tmp = t_1
	elif x <= 0.5:
		tmp = (x - math.log(x)) / n
	elif x <= 3.7e+73:
		tmp = 1.0 / (x * (n - ((((((-0.5 * (t_0 / x)) + (((n / x) * -0.25) + ((n / x) * 0.16666666666666666))) - t_0) / x) + (n * -0.5)) / x)))
	else:
		tmp = 0.0
	return tmp

Julia

function code(x, n)
	t_0 = Float64(Float64(n * -0.3333333333333333) + Float64(n * 0.25))
	t_1 = Float64(1.0 - (x ^ Float64(1.0 / n)))
	t_2 = Float64(log(x) / Float64(-n))
	tmp = 0.0
	if (x <= 6.8e-298)
		tmp = t_1;
	elseif (x <= 2.6e-283)
		tmp = t_2;
	elseif (x <= 3.6e-257)
		tmp = t_1;
	elseif (x <= 2.8e-163)
		tmp = t_2;
	elseif (x <= 1.4e-152)
		tmp = t_1;
	elseif (x <= 1.2e-135)
		tmp = t_2;
	elseif (x <= 9.2e-94)
		tmp = Float64(Float64(Float64(1.0 / n) - Float64(Float64(Float64(Float64(1.0 / n) * 0.5) - Float64(0.3333333333333333 * Float64(1.0 / Float64(n * x)))) / x)) / x);
	elseif (x <= 5e-79)
		tmp = t_2;
	elseif (x <= 2.1e-43)
		tmp = t_1;
	elseif (x <= 0.5)
		tmp = Float64(Float64(x - log(x)) / n);
	elseif (x <= 3.7e+73)
		tmp = Float64(1.0 / Float64(x * Float64(n - Float64(Float64(Float64(Float64(Float64(Float64(-0.5 * Float64(t_0 / x)) + Float64(Float64(Float64(n / x) * -0.25) + Float64(Float64(n / x) * 0.16666666666666666))) - t_0) / x) + Float64(n * -0.5)) / x))));
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = (n * -0.3333333333333333) + (n * 0.25);
	t_1 = 1.0 - (x ^ (1.0 / n));
	t_2 = log(x) / -n;
	tmp = 0.0;
	if (x <= 6.8e-298)
		tmp = t_1;
	elseif (x <= 2.6e-283)
		tmp = t_2;
	elseif (x <= 3.6e-257)
		tmp = t_1;
	elseif (x <= 2.8e-163)
		tmp = t_2;
	elseif (x <= 1.4e-152)
		tmp = t_1;
	elseif (x <= 1.2e-135)
		tmp = t_2;
	elseif (x <= 9.2e-94)
		tmp = ((1.0 / n) - ((((1.0 / n) * 0.5) - (0.3333333333333333 * (1.0 / (n * x)))) / x)) / x;
	elseif (x <= 5e-79)
		tmp = t_2;
	elseif (x <= 2.1e-43)
		tmp = t_1;
	elseif (x <= 0.5)
		tmp = (x - log(x)) / n;
	elseif (x <= 3.7e+73)
		tmp = 1.0 / (x * (n - ((((((-0.5 * (t_0 / x)) + (((n / x) * -0.25) + ((n / x) * 0.16666666666666666))) - t_0) / x) + (n * -0.5)) / x)));
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(N[(n * -0.3333333333333333), $MachinePrecision] + N[(n * 0.25), $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[x], $MachinePrecision] / (-n)), $MachinePrecision]}, If[LessEqual[x, 6.8e-298], t$95$1, If[LessEqual[x, 2.6e-283], t$95$2, If[LessEqual[x, 3.6e-257], t$95$1, If[LessEqual[x, 2.8e-163], t$95$2, If[LessEqual[x, 1.4e-152], t$95$1, If[LessEqual[x, 1.2e-135], t$95$2, If[LessEqual[x, 9.2e-94], N[(N[(N[(1.0 / n), $MachinePrecision] - N[(N[(N[(N[(1.0 / n), $MachinePrecision] * 0.5), $MachinePrecision] - N[(0.3333333333333333 * N[(1.0 / N[(n * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[x, 5e-79], t$95$2, If[LessEqual[x, 2.1e-43], t$95$1, If[LessEqual[x, 0.5], N[(N[(x - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[x, 3.7e+73], N[(1.0 / N[(x * N[(n - N[(N[(N[(N[(N[(N[(-0.5 * N[(t$95$0 / x), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(n / x), $MachinePrecision] * -0.25), $MachinePrecision] + N[(N[(n / x), $MachinePrecision] * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision] / x), $MachinePrecision] + N[(n * -0.5), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0]]]]]]]]]]]]]]

Derivation

1. Split input into 6 regimes

2. if x < 6.8e-298 or 2.6000000000000001e-283 < x < 3.60000000000000007e-257 or 2.8e-163 < x < 1.39999999999999992e-152 or 4.99999999999999999e-79 < x < 2.1000000000000001e-43

   1. Initial program 79.9%

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

   3. Taylor expanded in x around 0 80.0%

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

      1. *-rgt-identity 80.0%

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

      2. associate-*l/ 80.0%

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

      3. associate-/l* 80.0%

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

      4. exp-to-pow 79.9%

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

   5. Simplified 79.9%

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

   if 6.8e-298 < x < 2.6000000000000001e-283 or 3.60000000000000007e-257 < x < 2.8e-163 or 1.39999999999999992e-152 < x < 1.1999999999999999e-135 or 9.1999999999999997e-94 < x < 4.99999999999999999e-79

   1. Initial program 24.6%

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

   3. Taylor expanded in n around inf 76.5%

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

   4. Taylor expanded in x around 0 76.5%

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

      1. neg-mul-1 76.5%

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

   6. Simplified 76.5%

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

   if 1.1999999999999999e-135 < x < 9.1999999999999997e-94

   1. Initial program 35.9%

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

   3. Taylor expanded in n around inf 23.1%

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

   4. Taylor expanded in x around -inf 73.1%

      \[\leadsto \color{blue}{-1 \cdot \frac{-1 \cdot \frac{0.3333333333333333 \cdot \frac{1}{n \cdot x} - 0.5 \cdot \frac{1}{n}}{x} - \frac{1}{n}}{x}} \]

   if 2.1000000000000001e-43 < x < 0.5

   1. Initial program 33.7%

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

   3. Taylor expanded in n around inf 58.8%

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

   4. Taylor expanded in x around 0 53.7%

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

   if 0.5 < x < 3.69999999999999973e73

   1. Initial program 29.1%

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

   3. Taylor expanded in n around inf 31.0%

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

      1. clear-num 31.0%

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

      2. inv-pow 31.0%

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

      3. log1p-define 31.0%

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

   5. Applied egg-rr 31.0%

      \[\leadsto \color{blue}{{\left(\frac{n}{\mathsf{log1p}\left(x\right) - \log x}\right)}^{-1}} \]
   6. Step-by-step derivation

      1. unpow-1 31.0%

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

   7. Simplified 31.0%

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

   8. Taylor expanded in x around -inf 77.7%

      \[\leadsto \frac{1}{\color{blue}{-1 \cdot \left(x \cdot \left(-1 \cdot n + -1 \cdot \frac{-1 \cdot \frac{\left(-0.5 \cdot \frac{-0.3333333333333333 \cdot n + 0.25 \cdot n}{x} + \left(-0.25 \cdot \frac{n}{x} + 0.16666666666666666 \cdot \frac{n}{x}\right)\right) - \left(-0.3333333333333333 \cdot n + 0.25 \cdot n\right)}{x} - -0.5 \cdot n}{x}\right)\right)}} \]

   if 3.69999999999999973e73 < x

   1. Initial program 80.4%

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

   3. Taylor expanded in x around 0 46.0%

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

   4. Taylor expanded in n around inf 80.4%

      \[\leadsto 1 - \color{blue}{1} \]
3. Recombined 6 regimes into one program.

4. Final simplification 76.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 6.8 \cdot 10^{-298}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;x \leq 2.6 \cdot 10^{-283}:\\ \;\;\;\;\frac{\log x}{-n}\\ \mathbf{elif}\;x \leq 3.6 \cdot 10^{-257}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;x \leq 2.8 \cdot 10^{-163}:\\ \;\;\;\;\frac{\log x}{-n}\\ \mathbf{elif}\;x \leq 1.4 \cdot 10^{-152}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;x \leq 1.2 \cdot 10^{-135}:\\ \;\;\;\;\frac{\log x}{-n}\\ \mathbf{elif}\;x \leq 9.2 \cdot 10^{-94}:\\ \;\;\;\;\frac{\frac{1}{n} - \frac{\frac{1}{n} \cdot 0.5 - 0.3333333333333333 \cdot \frac{1}{n \cdot x}}{x}}{x}\\ \mathbf{elif}\;x \leq 5 \cdot 10^{-79}:\\ \;\;\;\;\frac{\log x}{-n}\\ \mathbf{elif}\;x \leq 2.1 \cdot 10^{-43}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;x \leq 0.5:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{elif}\;x \leq 3.7 \cdot 10^{+73}:\\ \;\;\;\;\frac{1}{x \cdot \left(n - \frac{\frac{\left(-0.5 \cdot \frac{n \cdot -0.3333333333333333 + n \cdot 0.25}{x} + \left(\frac{n}{x} \cdot -0.25 + \frac{n}{x} \cdot 0.16666666666666666\right)\right) - \left(n \cdot -0.3333333333333333 + n \cdot 0.25\right)}{x} + n \cdot -0.5}{x}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 59.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := n \cdot -0.3333333333333333 + n \cdot 0.25\\ t_1 := 1 - {x}^{\left(\frac{1}{n}\right)}\\ t_2 := \frac{-1}{\frac{n}{\log x}}\\ t_3 := \frac{\log x}{-n}\\ \mathbf{if}\;x \leq 5.8 \cdot 10^{-298}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 4.8 \cdot 10^{-282}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;x \leq 1.9 \cdot 10^{-257}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 2.75 \cdot 10^{-163}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;x \leq 1.5 \cdot 10^{-152}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 1.2 \cdot 10^{-135}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;x \leq 3.1 \cdot 10^{-94}:\\ \;\;\;\;\frac{\frac{1}{n} - \frac{\frac{1}{n} \cdot 0.5 - 0.3333333333333333 \cdot \frac{1}{n \cdot x}}{x}}{x}\\ \mathbf{elif}\;x \leq 6 \cdot 10^{-79}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;x \leq 7.5 \cdot 10^{-43}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 0.5:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{elif}\;x \leq 1.12 \cdot 10^{+74}:\\ \;\;\;\;\frac{1}{x \cdot \left(n - \frac{\frac{\left(-0.5 \cdot \frac{t\_0}{x} + \left(\frac{n}{x} \cdot -0.25 + \frac{n}{x} \cdot 0.16666666666666666\right)\right) - t\_0}{x} + n \cdot -0.5}{x}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (+ (* n -0.3333333333333333) (* n 0.25)))
        (t_1 (- 1.0 (pow x (/ 1.0 n))))
        (t_2 (/ -1.0 (/ n (log x))))
        (t_3 (/ (log x) (- n))))
   (if (<= x 5.8e-298)
     t_1
     (if (<= x 4.8e-282)
       t_2
       (if (<= x 1.9e-257)
         t_1
         (if (<= x 2.75e-163)
           t_2
           (if (<= x 1.5e-152)
             t_1
             (if (<= x 1.2e-135)
               t_3
               (if (<= x 3.1e-94)
                 (/
                  (-
                   (/ 1.0 n)
                   (/
                    (-
                     (* (/ 1.0 n) 0.5)
                     (* 0.3333333333333333 (/ 1.0 (* n x))))
                    x))
                  x)
                 (if (<= x 6e-79)
                   t_3
                   (if (<= x 7.5e-43)
                     t_1
                     (if (<= x 0.5)
                       (/ (- x (log x)) n)
                       (if (<= x 1.12e+74)
                         (/
                          1.0
                          (*
                           x
                           (-
                            n
                            (/
                             (+
                              (/
                               (-
                                (+
                                 (* -0.5 (/ t_0 x))
                                 (+
                                  (* (/ n x) -0.25)
                                  (* (/ n x) 0.16666666666666666)))
                                t_0)
                               x)
                              (* n -0.5))
                             x))))
                         0.0)))))))))))))

C

double code(double x, double n) {
	double t_0 = (n * -0.3333333333333333) + (n * 0.25);
	double t_1 = 1.0 - pow(x, (1.0 / n));
	double t_2 = -1.0 / (n / log(x));
	double t_3 = log(x) / -n;
	double tmp;
	if (x <= 5.8e-298) {
		tmp = t_1;
	} else if (x <= 4.8e-282) {
		tmp = t_2;
	} else if (x <= 1.9e-257) {
		tmp = t_1;
	} else if (x <= 2.75e-163) {
		tmp = t_2;
	} else if (x <= 1.5e-152) {
		tmp = t_1;
	} else if (x <= 1.2e-135) {
		tmp = t_3;
	} else if (x <= 3.1e-94) {
		tmp = ((1.0 / n) - ((((1.0 / n) * 0.5) - (0.3333333333333333 * (1.0 / (n * x)))) / x)) / x;
	} else if (x <= 6e-79) {
		tmp = t_3;
	} else if (x <= 7.5e-43) {
		tmp = t_1;
	} else if (x <= 0.5) {
		tmp = (x - log(x)) / n;
	} else if (x <= 1.12e+74) {
		tmp = 1.0 / (x * (n - ((((((-0.5 * (t_0 / x)) + (((n / x) * -0.25) + ((n / x) * 0.16666666666666666))) - t_0) / x) + (n * -0.5)) / x)));
	} 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) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_0 = (n * (-0.3333333333333333d0)) + (n * 0.25d0)
    t_1 = 1.0d0 - (x ** (1.0d0 / n))
    t_2 = (-1.0d0) / (n / log(x))
    t_3 = log(x) / -n
    if (x <= 5.8d-298) then
        tmp = t_1
    else if (x <= 4.8d-282) then
        tmp = t_2
    else if (x <= 1.9d-257) then
        tmp = t_1
    else if (x <= 2.75d-163) then
        tmp = t_2
    else if (x <= 1.5d-152) then
        tmp = t_1
    else if (x <= 1.2d-135) then
        tmp = t_3
    else if (x <= 3.1d-94) then
        tmp = ((1.0d0 / n) - ((((1.0d0 / n) * 0.5d0) - (0.3333333333333333d0 * (1.0d0 / (n * x)))) / x)) / x
    else if (x <= 6d-79) then
        tmp = t_3
    else if (x <= 7.5d-43) then
        tmp = t_1
    else if (x <= 0.5d0) then
        tmp = (x - log(x)) / n
    else if (x <= 1.12d+74) then
        tmp = 1.0d0 / (x * (n - (((((((-0.5d0) * (t_0 / x)) + (((n / x) * (-0.25d0)) + ((n / x) * 0.16666666666666666d0))) - t_0) / x) + (n * (-0.5d0))) / x)))
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = (n * -0.3333333333333333) + (n * 0.25);
	double t_1 = 1.0 - Math.pow(x, (1.0 / n));
	double t_2 = -1.0 / (n / Math.log(x));
	double t_3 = Math.log(x) / -n;
	double tmp;
	if (x <= 5.8e-298) {
		tmp = t_1;
	} else if (x <= 4.8e-282) {
		tmp = t_2;
	} else if (x <= 1.9e-257) {
		tmp = t_1;
	} else if (x <= 2.75e-163) {
		tmp = t_2;
	} else if (x <= 1.5e-152) {
		tmp = t_1;
	} else if (x <= 1.2e-135) {
		tmp = t_3;
	} else if (x <= 3.1e-94) {
		tmp = ((1.0 / n) - ((((1.0 / n) * 0.5) - (0.3333333333333333 * (1.0 / (n * x)))) / x)) / x;
	} else if (x <= 6e-79) {
		tmp = t_3;
	} else if (x <= 7.5e-43) {
		tmp = t_1;
	} else if (x <= 0.5) {
		tmp = (x - Math.log(x)) / n;
	} else if (x <= 1.12e+74) {
		tmp = 1.0 / (x * (n - ((((((-0.5 * (t_0 / x)) + (((n / x) * -0.25) + ((n / x) * 0.16666666666666666))) - t_0) / x) + (n * -0.5)) / x)));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = (n * -0.3333333333333333) + (n * 0.25)
	t_1 = 1.0 - math.pow(x, (1.0 / n))
	t_2 = -1.0 / (n / math.log(x))
	t_3 = math.log(x) / -n
	tmp = 0
	if x <= 5.8e-298:
		tmp = t_1
	elif x <= 4.8e-282:
		tmp = t_2
	elif x <= 1.9e-257:
		tmp = t_1
	elif x <= 2.75e-163:
		tmp = t_2
	elif x <= 1.5e-152:
		tmp = t_1
	elif x <= 1.2e-135:
		tmp = t_3
	elif x <= 3.1e-94:
		tmp = ((1.0 / n) - ((((1.0 / n) * 0.5) - (0.3333333333333333 * (1.0 / (n * x)))) / x)) / x
	elif x <= 6e-79:
		tmp = t_3
	elif x <= 7.5e-43:
		tmp = t_1
	elif x <= 0.5:
		tmp = (x - math.log(x)) / n
	elif x <= 1.12e+74:
		tmp = 1.0 / (x * (n - ((((((-0.5 * (t_0 / x)) + (((n / x) * -0.25) + ((n / x) * 0.16666666666666666))) - t_0) / x) + (n * -0.5)) / x)))
	else:
		tmp = 0.0
	return tmp

Julia

function code(x, n)
	t_0 = Float64(Float64(n * -0.3333333333333333) + Float64(n * 0.25))
	t_1 = Float64(1.0 - (x ^ Float64(1.0 / n)))
	t_2 = Float64(-1.0 / Float64(n / log(x)))
	t_3 = Float64(log(x) / Float64(-n))
	tmp = 0.0
	if (x <= 5.8e-298)
		tmp = t_1;
	elseif (x <= 4.8e-282)
		tmp = t_2;
	elseif (x <= 1.9e-257)
		tmp = t_1;
	elseif (x <= 2.75e-163)
		tmp = t_2;
	elseif (x <= 1.5e-152)
		tmp = t_1;
	elseif (x <= 1.2e-135)
		tmp = t_3;
	elseif (x <= 3.1e-94)
		tmp = Float64(Float64(Float64(1.0 / n) - Float64(Float64(Float64(Float64(1.0 / n) * 0.5) - Float64(0.3333333333333333 * Float64(1.0 / Float64(n * x)))) / x)) / x);
	elseif (x <= 6e-79)
		tmp = t_3;
	elseif (x <= 7.5e-43)
		tmp = t_1;
	elseif (x <= 0.5)
		tmp = Float64(Float64(x - log(x)) / n);
	elseif (x <= 1.12e+74)
		tmp = Float64(1.0 / Float64(x * Float64(n - Float64(Float64(Float64(Float64(Float64(Float64(-0.5 * Float64(t_0 / x)) + Float64(Float64(Float64(n / x) * -0.25) + Float64(Float64(n / x) * 0.16666666666666666))) - t_0) / x) + Float64(n * -0.5)) / x))));
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = (n * -0.3333333333333333) + (n * 0.25);
	t_1 = 1.0 - (x ^ (1.0 / n));
	t_2 = -1.0 / (n / log(x));
	t_3 = log(x) / -n;
	tmp = 0.0;
	if (x <= 5.8e-298)
		tmp = t_1;
	elseif (x <= 4.8e-282)
		tmp = t_2;
	elseif (x <= 1.9e-257)
		tmp = t_1;
	elseif (x <= 2.75e-163)
		tmp = t_2;
	elseif (x <= 1.5e-152)
		tmp = t_1;
	elseif (x <= 1.2e-135)
		tmp = t_3;
	elseif (x <= 3.1e-94)
		tmp = ((1.0 / n) - ((((1.0 / n) * 0.5) - (0.3333333333333333 * (1.0 / (n * x)))) / x)) / x;
	elseif (x <= 6e-79)
		tmp = t_3;
	elseif (x <= 7.5e-43)
		tmp = t_1;
	elseif (x <= 0.5)
		tmp = (x - log(x)) / n;
	elseif (x <= 1.12e+74)
		tmp = 1.0 / (x * (n - ((((((-0.5 * (t_0 / x)) + (((n / x) * -0.25) + ((n / x) * 0.16666666666666666))) - t_0) / x) + (n * -0.5)) / x)));
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(N[(n * -0.3333333333333333), $MachinePrecision] + N[(n * 0.25), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(-1.0 / N[(n / N[Log[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[Log[x], $MachinePrecision] / (-n)), $MachinePrecision]}, If[LessEqual[x, 5.8e-298], t$95$1, If[LessEqual[x, 4.8e-282], t$95$2, If[LessEqual[x, 1.9e-257], t$95$1, If[LessEqual[x, 2.75e-163], t$95$2, If[LessEqual[x, 1.5e-152], t$95$1, If[LessEqual[x, 1.2e-135], t$95$3, If[LessEqual[x, 3.1e-94], N[(N[(N[(1.0 / n), $MachinePrecision] - N[(N[(N[(N[(1.0 / n), $MachinePrecision] * 0.5), $MachinePrecision] - N[(0.3333333333333333 * N[(1.0 / N[(n * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[x, 6e-79], t$95$3, If[LessEqual[x, 7.5e-43], t$95$1, If[LessEqual[x, 0.5], N[(N[(x - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[x, 1.12e+74], N[(1.0 / N[(x * N[(n - N[(N[(N[(N[(N[(N[(-0.5 * N[(t$95$0 / x), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(n / x), $MachinePrecision] * -0.25), $MachinePrecision] + N[(N[(n / x), $MachinePrecision] * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision] / x), $MachinePrecision] + N[(n * -0.5), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0]]]]]]]]]]]]]]]

Derivation

1. Split input into 7 regimes

2. if x < 5.8000000000000003e-298 or 4.79999999999999994e-282 < x < 1.9000000000000002e-257 or 2.7499999999999999e-163 < x < 1.5e-152 or 5.99999999999999999e-79 < x < 7.50000000000000068e-43

   1. Initial program 79.9%

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

   3. Taylor expanded in x around 0 80.0%

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

      1. *-rgt-identity 80.0%

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

      2. associate-*l/ 80.0%

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

      3. associate-/l* 80.0%

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

      4. exp-to-pow 79.9%

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

   5. Simplified 79.9%

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

   if 5.8000000000000003e-298 < x < 4.79999999999999994e-282 or 1.9000000000000002e-257 < x < 2.7499999999999999e-163

   1. Initial program 29.6%

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

   3. Taylor expanded in n around inf 71.8%

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

      1. clear-num 71.9%

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

      2. inv-pow 71.9%

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

      3. log1p-define 71.9%

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

   5. Applied egg-rr 71.9%

      \[\leadsto \color{blue}{{\left(\frac{n}{\mathsf{log1p}\left(x\right) - \log x}\right)}^{-1}} \]
   6. Step-by-step derivation

      1. unpow-1 71.9%

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

   7. Simplified 71.9%

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

   8. Taylor expanded in x around 0 71.9%

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

      1. associate-*r/ 71.9%

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

      2. neg-mul-1 71.9%

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

   10. Simplified 71.9%

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

   if 1.5e-152 < x < 1.1999999999999999e-135 or 3.0999999999999998e-94 < x < 5.99999999999999999e-79

   1. Initial program 10.1%

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

   3. Taylor expanded in n around inf 90.1%

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

   4. Taylor expanded in x around 0 90.1%

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

      1. neg-mul-1 90.1%

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

   6. Simplified 90.1%

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

   if 1.1999999999999999e-135 < x < 3.0999999999999998e-94

   1. Initial program 35.9%

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

   3. Taylor expanded in n around inf 23.1%

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

   4. Taylor expanded in x around -inf 73.1%

      \[\leadsto \color{blue}{-1 \cdot \frac{-1 \cdot \frac{0.3333333333333333 \cdot \frac{1}{n \cdot x} - 0.5 \cdot \frac{1}{n}}{x} - \frac{1}{n}}{x}} \]

   if 7.50000000000000068e-43 < x < 0.5

   1. Initial program 33.7%

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

   3. Taylor expanded in n around inf 58.8%

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

   4. Taylor expanded in x around 0 53.7%

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

   if 0.5 < x < 1.12000000000000003e74

   1. Initial program 29.1%

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

   3. Taylor expanded in n around inf 31.0%

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

      1. clear-num 31.0%

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

      2. inv-pow 31.0%

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

      3. log1p-define 31.0%

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

   5. Applied egg-rr 31.0%

      \[\leadsto \color{blue}{{\left(\frac{n}{\mathsf{log1p}\left(x\right) - \log x}\right)}^{-1}} \]
   6. Step-by-step derivation

      1. unpow-1 31.0%

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

   7. Simplified 31.0%

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

   8. Taylor expanded in x around -inf 77.7%

      \[\leadsto \frac{1}{\color{blue}{-1 \cdot \left(x \cdot \left(-1 \cdot n + -1 \cdot \frac{-1 \cdot \frac{\left(-0.5 \cdot \frac{-0.3333333333333333 \cdot n + 0.25 \cdot n}{x} + \left(-0.25 \cdot \frac{n}{x} + 0.16666666666666666 \cdot \frac{n}{x}\right)\right) - \left(-0.3333333333333333 \cdot n + 0.25 \cdot n\right)}{x} - -0.5 \cdot n}{x}\right)\right)}} \]

   if 1.12000000000000003e74 < x

   1. Initial program 80.4%

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

   3. Taylor expanded in x around 0 46.0%

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

   4. Taylor expanded in n around inf 80.4%

      \[\leadsto 1 - \color{blue}{1} \]
3. Recombined 7 regimes into one program.

4. Final simplification 76.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 5.8 \cdot 10^{-298}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;x \leq 4.8 \cdot 10^{-282}:\\ \;\;\;\;\frac{-1}{\frac{n}{\log x}}\\ \mathbf{elif}\;x \leq 1.9 \cdot 10^{-257}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;x \leq 2.75 \cdot 10^{-163}:\\ \;\;\;\;\frac{-1}{\frac{n}{\log x}}\\ \mathbf{elif}\;x \leq 1.5 \cdot 10^{-152}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;x \leq 1.2 \cdot 10^{-135}:\\ \;\;\;\;\frac{\log x}{-n}\\ \mathbf{elif}\;x \leq 3.1 \cdot 10^{-94}:\\ \;\;\;\;\frac{\frac{1}{n} - \frac{\frac{1}{n} \cdot 0.5 - 0.3333333333333333 \cdot \frac{1}{n \cdot x}}{x}}{x}\\ \mathbf{elif}\;x \leq 6 \cdot 10^{-79}:\\ \;\;\;\;\frac{\log x}{-n}\\ \mathbf{elif}\;x \leq 7.5 \cdot 10^{-43}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;x \leq 0.5:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{elif}\;x \leq 1.12 \cdot 10^{+74}:\\ \;\;\;\;\frac{1}{x \cdot \left(n - \frac{\frac{\left(-0.5 \cdot \frac{n \cdot -0.3333333333333333 + n \cdot 0.25}{x} + \left(\frac{n}{x} \cdot -0.25 + \frac{n}{x} \cdot 0.16666666666666666\right)\right) - \left(n \cdot -0.3333333333333333 + n \cdot 0.25\right)}{x} + n \cdot -0.5}{x}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 82.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{-27}:\\ \;\;\;\;\frac{\frac{t\_0}{n}}{x}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-5}:\\ \;\;\;\;\frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + x \cdot \left(\frac{1}{n} + \frac{x \cdot -0.5 + 0.5 \cdot \frac{x}{n}}{n}\right)\right) - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -1e-27)
     (/ (/ t_0 n) x)
     (if (<= (/ 1.0 n) 5e-5)
       (/ (log (/ (+ x 1.0) x)) n)
       (-
        (+ 1.0 (* x (+ (/ 1.0 n) (/ (+ (* x -0.5) (* 0.5 (/ x n))) n))))
        t_0)))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -1e-27) {
		tmp = (t_0 / n) / x;
	} else if ((1.0 / n) <= 5e-5) {
		tmp = log(((x + 1.0) / x)) / n;
	} else {
		tmp = (1.0 + (x * ((1.0 / n) + (((x * -0.5) + (0.5 * (x / n))) / 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) <= (-1d-27)) then
        tmp = (t_0 / n) / x
    else if ((1.0d0 / n) <= 5d-5) then
        tmp = log(((x + 1.0d0) / x)) / n
    else
        tmp = (1.0d0 + (x * ((1.0d0 / n) + (((x * (-0.5d0)) + (0.5d0 * (x / n))) / 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) <= -1e-27) {
		tmp = (t_0 / n) / x;
	} else if ((1.0 / n) <= 5e-5) {
		tmp = Math.log(((x + 1.0) / x)) / n;
	} else {
		tmp = (1.0 + (x * ((1.0 / n) + (((x * -0.5) + (0.5 * (x / n))) / n)))) - t_0;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	tmp = 0
	if (1.0 / n) <= -1e-27:
		tmp = (t_0 / n) / x
	elif (1.0 / n) <= 5e-5:
		tmp = math.log(((x + 1.0) / x)) / n
	else:
		tmp = (1.0 + (x * ((1.0 / n) + (((x * -0.5) + (0.5 * (x / n))) / 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) <= -1e-27)
		tmp = Float64(Float64(t_0 / n) / x);
	elseif (Float64(1.0 / n) <= 5e-5)
		tmp = Float64(log(Float64(Float64(x + 1.0) / x)) / n);
	else
		tmp = Float64(Float64(1.0 + Float64(x * Float64(Float64(1.0 / n) + Float64(Float64(Float64(x * -0.5) + Float64(0.5 * Float64(x / n))) / 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) <= -1e-27)
		tmp = (t_0 / n) / x;
	elseif ((1.0 / n) <= 5e-5)
		tmp = log(((x + 1.0) / x)) / n;
	else
		tmp = (1.0 + (x * ((1.0 / n) + (((x * -0.5) + (0.5 * (x / n))) / 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], -1e-27], N[(N[(t$95$0 / n), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e-5], N[(N[Log[N[(N[(x + 1.0), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], N[(N[(1.0 + N[(x * N[(N[(1.0 / n), $MachinePrecision] + N[(N[(N[(x * -0.5), $MachinePrecision] + N[(0.5 * N[(x / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 1 n) < -1e-27

   1. Initial program 98.3%

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

   3. Taylor expanded in x around inf 94.5%

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

      1. associate-/r* 94.5%

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

      2. mul-1-neg 94.5%

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

      3. log-rec 94.5%

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

      4. mul-1-neg 94.5%

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

      5. distribute-neg-frac 94.5%

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

      6. mul-1-neg 94.5%

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

      7. remove-double-neg 94.5%

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

      8. *-rgt-identity 94.5%

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

      9. associate-/l* 94.5%

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

      10. exp-to-pow 94.5%

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

   5. Simplified 94.5%

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

   if -1e-27 < (/.f64 1 n) < 5.00000000000000024e-5

   1. Initial program 25.1%

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

   3. Taylor expanded in n around inf 78.4%

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

      1. diff-log 78.5%

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

   5. Applied egg-rr 78.5%

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

      1. +-commutative 78.5%

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

   7. Simplified 78.5%

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

   if 5.00000000000000024e-5 < (/.f64 1 n)

   1. Initial program 63.4%

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

   3. Taylor expanded in x around 0 70.8%

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

   4. Taylor expanded in n around inf 77.7%

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

4. Final simplification 82.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{-27}:\\ \;\;\;\;\frac{\frac{{x}^{\left(\frac{1}{n}\right)}}{n}}{x}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-5}:\\ \;\;\;\;\frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + x \cdot \left(\frac{1}{n} + \frac{x \cdot -0.5 + 0.5 \cdot \frac{x}{n}}{n}\right)\right) - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 82.0% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{-27}:\\ \;\;\;\;\frac{\frac{t\_0}{n}}{x}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-5}:\\ \;\;\;\;\frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{+218}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{-1 + \left(1 - \frac{n}{\log x}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -1e-27)
     (/ (/ t_0 n) x)
     (if (<= (/ 1.0 n) 5e-5)
       (/ (log (/ (+ x 1.0) x)) n)
       (if (<= (/ 1.0 n) 5e+218)
         (- (+ 1.0 (/ x n)) t_0)
         (/ 1.0 (+ -1.0 (- 1.0 (/ n (log x))))))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -1e-27) {
		tmp = (t_0 / n) / x;
	} else if ((1.0 / n) <= 5e-5) {
		tmp = log(((x + 1.0) / x)) / n;
	} else if ((1.0 / n) <= 5e+218) {
		tmp = (1.0 + (x / n)) - t_0;
	} else {
		tmp = 1.0 / (-1.0 + (1.0 - (n / log(x))));
	}
	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) <= (-1d-27)) then
        tmp = (t_0 / n) / x
    else if ((1.0d0 / n) <= 5d-5) then
        tmp = log(((x + 1.0d0) / x)) / n
    else if ((1.0d0 / n) <= 5d+218) then
        tmp = (1.0d0 + (x / n)) - t_0
    else
        tmp = 1.0d0 / ((-1.0d0) + (1.0d0 - (n / log(x))))
    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) <= -1e-27) {
		tmp = (t_0 / n) / x;
	} else if ((1.0 / n) <= 5e-5) {
		tmp = Math.log(((x + 1.0) / x)) / n;
	} else if ((1.0 / n) <= 5e+218) {
		tmp = (1.0 + (x / n)) - t_0;
	} else {
		tmp = 1.0 / (-1.0 + (1.0 - (n / Math.log(x))));
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	tmp = 0
	if (1.0 / n) <= -1e-27:
		tmp = (t_0 / n) / x
	elif (1.0 / n) <= 5e-5:
		tmp = math.log(((x + 1.0) / x)) / n
	elif (1.0 / n) <= 5e+218:
		tmp = (1.0 + (x / n)) - t_0
	else:
		tmp = 1.0 / (-1.0 + (1.0 - (n / math.log(x))))
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -1e-27)
		tmp = Float64(Float64(t_0 / n) / x);
	elseif (Float64(1.0 / n) <= 5e-5)
		tmp = Float64(log(Float64(Float64(x + 1.0) / x)) / n);
	elseif (Float64(1.0 / n) <= 5e+218)
		tmp = Float64(Float64(1.0 + Float64(x / n)) - t_0);
	else
		tmp = Float64(1.0 / Float64(-1.0 + Float64(1.0 - Float64(n / log(x)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = x ^ (1.0 / n);
	tmp = 0.0;
	if ((1.0 / n) <= -1e-27)
		tmp = (t_0 / n) / x;
	elseif ((1.0 / n) <= 5e-5)
		tmp = log(((x + 1.0) / x)) / n;
	elseif ((1.0 / n) <= 5e+218)
		tmp = (1.0 + (x / n)) - t_0;
	else
		tmp = 1.0 / (-1.0 + (1.0 - (n / log(x))));
	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], -1e-27], N[(N[(t$95$0 / n), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e-5], N[(N[Log[N[(N[(x + 1.0), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e+218], N[(N[(1.0 + N[(x / n), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], N[(1.0 / N[(-1.0 + N[(1.0 - N[(n / N[Log[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 1 n) < -1e-27

   1. Initial program 98.3%

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

   3. Taylor expanded in x around inf 94.5%

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

      1. associate-/r* 94.5%

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

      2. mul-1-neg 94.5%

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

      3. log-rec 94.5%

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

      4. mul-1-neg 94.5%

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

      5. distribute-neg-frac 94.5%

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

      6. mul-1-neg 94.5%

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

      7. remove-double-neg 94.5%

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

      8. *-rgt-identity 94.5%

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

      9. associate-/l* 94.5%

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

      10. exp-to-pow 94.5%

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

   5. Simplified 94.5%

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

   if -1e-27 < (/.f64 1 n) < 5.00000000000000024e-5

   1. Initial program 25.1%

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

   3. Taylor expanded in n around inf 78.4%

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

      1. diff-log 78.5%

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

   5. Applied egg-rr 78.5%

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

      1. +-commutative 78.5%

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

   7. Simplified 78.5%

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

   if 5.00000000000000024e-5 < (/.f64 1 n) < 4.99999999999999983e218

   1. Initial program 79.8%

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

   3. Taylor expanded in x around 0 74.2%

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

   if 4.99999999999999983e218 < (/.f64 1 n)

   1. Initial program 25.5%

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

   3. Taylor expanded in n around inf 8.0%

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

      1. clear-num 8.0%

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

      2. inv-pow 8.0%

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

      3. log1p-define 8.0%

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

   5. Applied egg-rr 8.0%

      \[\leadsto \color{blue}{{\left(\frac{n}{\mathsf{log1p}\left(x\right) - \log x}\right)}^{-1}} \]
   6. Step-by-step derivation

      1. unpow-1 8.0%

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

   7. Simplified 8.0%

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

   8. Taylor expanded in x around 0 8.0%

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

      1. associate-*r/ 8.0%

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

      2. neg-mul-1 8.0%

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

   10. Simplified 8.0%

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

       1. expm1-log1p-u 8.0%

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

       2. expm1-undefine 92.5%

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

   12. Applied egg-rr 92.5%

       \[\leadsto \frac{1}{\color{blue}{e^{\mathsf{log1p}\left(\frac{-n}{\log x}\right)} - 1}} \]
   13. Step-by-step derivation

       1. sub-neg 92.5%

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

       2. metadata-eval 92.5%

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

       3. +-commutative 92.5%

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

       4. log1p-undefine 92.5%

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

       5. rem-exp-log 92.5%

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

       6. distribute-frac-neg 92.5%

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

       7. unsub-neg 92.5%

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

   14. Simplified 92.5%

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

4. Final simplification 83.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{-27}:\\ \;\;\;\;\frac{\frac{{x}^{\left(\frac{1}{n}\right)}}{n}}{x}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-5}:\\ \;\;\;\;\frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{+218}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{-1 + \left(1 - \frac{n}{\log x}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 61.3% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x - \log x}{n}\\ \mathbf{if}\;n \leq -2.5:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n \leq 2.1 \cdot 10^{-219}:\\ \;\;\;\;\frac{\frac{0.25}{n}}{{x}^{4}}\\ \mathbf{elif}\;n \leq 10^{+24}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;n \leq 3 \cdot 10^{+139} \lor \neg \left(n \leq 2.85 \cdot 10^{+197}\right):\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x \cdot \left(n + 0.5 \cdot \frac{n}{x}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (/ (- x (log x)) n)))
   (if (<= n -2.5)
     t_0
     (if (<= n 2.1e-219)
       (/ (/ 0.25 n) (pow x 4.0))
       (if (<= n 1e+24)
         (- 1.0 (pow x (/ 1.0 n)))
         (if (or (<= n 3e+139) (not (<= n 2.85e+197)))
           t_0
           (/ 1.0 (* x (+ n (* 0.5 (/ n x)))))))))))

C

double code(double x, double n) {
	double t_0 = (x - log(x)) / n;
	double tmp;
	if (n <= -2.5) {
		tmp = t_0;
	} else if (n <= 2.1e-219) {
		tmp = (0.25 / n) / pow(x, 4.0);
	} else if (n <= 1e+24) {
		tmp = 1.0 - pow(x, (1.0 / n));
	} else if ((n <= 3e+139) || !(n <= 2.85e+197)) {
		tmp = t_0;
	} else {
		tmp = 1.0 / (x * (n + (0.5 * (n / x))));
	}
	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 - log(x)) / n
    if (n <= (-2.5d0)) then
        tmp = t_0
    else if (n <= 2.1d-219) then
        tmp = (0.25d0 / n) / (x ** 4.0d0)
    else if (n <= 1d+24) then
        tmp = 1.0d0 - (x ** (1.0d0 / n))
    else if ((n <= 3d+139) .or. (.not. (n <= 2.85d+197))) then
        tmp = t_0
    else
        tmp = 1.0d0 / (x * (n + (0.5d0 * (n / x))))
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = (x - Math.log(x)) / n;
	double tmp;
	if (n <= -2.5) {
		tmp = t_0;
	} else if (n <= 2.1e-219) {
		tmp = (0.25 / n) / Math.pow(x, 4.0);
	} else if (n <= 1e+24) {
		tmp = 1.0 - Math.pow(x, (1.0 / n));
	} else if ((n <= 3e+139) || !(n <= 2.85e+197)) {
		tmp = t_0;
	} else {
		tmp = 1.0 / (x * (n + (0.5 * (n / x))));
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = (x - math.log(x)) / n
	tmp = 0
	if n <= -2.5:
		tmp = t_0
	elif n <= 2.1e-219:
		tmp = (0.25 / n) / math.pow(x, 4.0)
	elif n <= 1e+24:
		tmp = 1.0 - math.pow(x, (1.0 / n))
	elif (n <= 3e+139) or not (n <= 2.85e+197):
		tmp = t_0
	else:
		tmp = 1.0 / (x * (n + (0.5 * (n / x))))
	return tmp

Julia

function code(x, n)
	t_0 = Float64(Float64(x - log(x)) / n)
	tmp = 0.0
	if (n <= -2.5)
		tmp = t_0;
	elseif (n <= 2.1e-219)
		tmp = Float64(Float64(0.25 / n) / (x ^ 4.0));
	elseif (n <= 1e+24)
		tmp = Float64(1.0 - (x ^ Float64(1.0 / n)));
	elseif ((n <= 3e+139) || !(n <= 2.85e+197))
		tmp = t_0;
	else
		tmp = Float64(1.0 / Float64(x * Float64(n + Float64(0.5 * Float64(n / x)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = (x - log(x)) / n;
	tmp = 0.0;
	if (n <= -2.5)
		tmp = t_0;
	elseif (n <= 2.1e-219)
		tmp = (0.25 / n) / (x ^ 4.0);
	elseif (n <= 1e+24)
		tmp = 1.0 - (x ^ (1.0 / n));
	elseif ((n <= 3e+139) || ~((n <= 2.85e+197)))
		tmp = t_0;
	else
		tmp = 1.0 / (x * (n + (0.5 * (n / x))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(N[(x - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision]}, If[LessEqual[n, -2.5], t$95$0, If[LessEqual[n, 2.1e-219], N[(N[(0.25 / n), $MachinePrecision] / N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 1e+24], N[(1.0 - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[n, 3e+139], N[Not[LessEqual[n, 2.85e+197]], $MachinePrecision]], t$95$0, N[(1.0 / N[(x * N[(n + N[(0.5 * N[(n / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if n < -2.5 or 9.9999999999999998e23 < n < 3e139 or 2.85000000000000011e197 < n

   1. Initial program 22.8%

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

   3. Taylor expanded in n around inf 75.9%

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

   4. Taylor expanded in x around 0 59.7%

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

   if -2.5 < n < 2.1e-219

   1. Initial program 87.4%

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

   3. Taylor expanded in n around inf 41.1%

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

   4. Taylor expanded in x around -inf 1.2%

      \[\leadsto \frac{\color{blue}{-1 \cdot \frac{-1 \cdot \frac{-1 \cdot \frac{0.25 \cdot \frac{1}{x} - 0.3333333333333333}{x} - 0.5}{x} - 1}{x}}}{n} \]
   5. Step-by-step derivation

      1. add-sqr-sqrt 0.0%

         \[\leadsto \frac{-1 \cdot \color{blue}{\left(\sqrt{\frac{-1 \cdot \frac{-1 \cdot \frac{0.25 \cdot \frac{1}{x} - 0.3333333333333333}{x} - 0.5}{x} - 1}{x}} \cdot \sqrt{\frac{-1 \cdot \frac{-1 \cdot \frac{0.25 \cdot \frac{1}{x} - 0.3333333333333333}{x} - 0.5}{x} - 1}{x}}\right)}}{n} \]

      2. sqrt-unprod 17.5%

         \[\leadsto \frac{-1 \cdot \color{blue}{\sqrt{\frac{-1 \cdot \frac{-1 \cdot \frac{0.25 \cdot \frac{1}{x} - 0.3333333333333333}{x} - 0.5}{x} - 1}{x} \cdot \frac{-1 \cdot \frac{-1 \cdot \frac{0.25 \cdot \frac{1}{x} - 0.3333333333333333}{x} - 0.5}{x} - 1}{x}}}}{n} \]

      3. sqr-neg 17.5%

         \[\leadsto \frac{-1 \cdot \sqrt{\color{blue}{\left(-\frac{-1 \cdot \frac{-1 \cdot \frac{0.25 \cdot \frac{1}{x} - 0.3333333333333333}{x} - 0.5}{x} - 1}{x}\right) \cdot \left(-\frac{-1 \cdot \frac{-1 \cdot \frac{0.25 \cdot \frac{1}{x} - 0.3333333333333333}{x} - 0.5}{x} - 1}{x}\right)}}}{n} \]

      4. mul-1-neg 17.5%

         \[\leadsto \frac{-1 \cdot \sqrt{\color{blue}{\left(-1 \cdot \frac{-1 \cdot \frac{-1 \cdot \frac{0.25 \cdot \frac{1}{x} - 0.3333333333333333}{x} - 0.5}{x} - 1}{x}\right)} \cdot \left(-\frac{-1 \cdot \frac{-1 \cdot \frac{0.25 \cdot \frac{1}{x} - 0.3333333333333333}{x} - 0.5}{x} - 1}{x}\right)}}{n} \]

      5. mul-1-neg 17.5%

         \[\leadsto \frac{-1 \cdot \sqrt{\left(-1 \cdot \frac{-1 \cdot \frac{-1 \cdot \frac{0.25 \cdot \frac{1}{x} - 0.3333333333333333}{x} - 0.5}{x} - 1}{x}\right) \cdot \color{blue}{\left(-1 \cdot \frac{-1 \cdot \frac{-1 \cdot \frac{0.25 \cdot \frac{1}{x} - 0.3333333333333333}{x} - 0.5}{x} - 1}{x}\right)}}}{n} \]

      6. sqrt-unprod 1.2%

         \[\leadsto \frac{-1 \cdot \color{blue}{\left(\sqrt{-1 \cdot \frac{-1 \cdot \frac{-1 \cdot \frac{0.25 \cdot \frac{1}{x} - 0.3333333333333333}{x} - 0.5}{x} - 1}{x}} \cdot \sqrt{-1 \cdot \frac{-1 \cdot \frac{-1 \cdot \frac{0.25 \cdot \frac{1}{x} - 0.3333333333333333}{x} - 0.5}{x} - 1}{x}}\right)}}{n} \]

      7. add-sqr-sqrt 53.9%

         \[\leadsto \frac{-1 \cdot \color{blue}{\left(-1 \cdot \frac{-1 \cdot \frac{-1 \cdot \frac{0.25 \cdot \frac{1}{x} - 0.3333333333333333}{x} - 0.5}{x} - 1}{x}\right)}}{n} \]

   6. Applied egg-rr 53.9%

      \[\leadsto \frac{-1 \cdot \color{blue}{\left(0 - \frac{\frac{\frac{\frac{0.25}{x} + -0.3333333333333333}{x} + -0.5}{x} + -1}{x}\right)}}{n} \]
   7. Step-by-step derivation

      1. neg-sub0 53.9%

         \[\leadsto \frac{-1 \cdot \color{blue}{\left(-\frac{\frac{\frac{\frac{0.25}{x} + -0.3333333333333333}{x} + -0.5}{x} + -1}{x}\right)}}{n} \]

      2. distribute-neg-frac 53.9%

         \[\leadsto \frac{-1 \cdot \color{blue}{\frac{-\left(\frac{\frac{\frac{0.25}{x} + -0.3333333333333333}{x} + -0.5}{x} + -1\right)}{x}}}{n} \]

   8. Simplified 53.9%

      \[\leadsto \frac{-1 \cdot \color{blue}{\frac{1 + \frac{0.5 + \frac{0.3333333333333333 + \frac{-0.25}{x}}{x}}{x}}{x}}}{n} \]

   9. Taylor expanded in x around 0 81.5%

      \[\leadsto \color{blue}{\frac{0.25}{n \cdot {x}^{4}}} \]
   10. Step-by-step derivation

       1. associate-/r* 81.5%

          \[\leadsto \color{blue}{\frac{\frac{0.25}{n}}{{x}^{4}}} \]

   11. Simplified 81.5%

       \[\leadsto \color{blue}{\frac{\frac{0.25}{n}}{{x}^{4}}} \]

   if 2.1e-219 < n < 9.9999999999999998e23

   1. Initial program 76.2%

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

   3. Taylor expanded in x around 0 70.3%

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

      1. *-rgt-identity 70.3%

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

      2. associate-*l/ 70.3%

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

      3. associate-/l* 70.3%

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

      4. exp-to-pow 70.3%

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

   5. Simplified 70.3%

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

   if 3e139 < n < 2.85000000000000011e197

   1. Initial program 55.3%

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

   3. Taylor expanded in n around inf 86.5%

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

      1. clear-num 86.6%

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

      2. inv-pow 86.6%

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

      3. log1p-define 86.6%

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

   5. Applied egg-rr 86.6%

      \[\leadsto \color{blue}{{\left(\frac{n}{\mathsf{log1p}\left(x\right) - \log x}\right)}^{-1}} \]
   6. Step-by-step derivation

      1. unpow-1 86.6%

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

   7. Simplified 86.6%

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

   8. Taylor expanded in x around inf 70.9%

      \[\leadsto \frac{1}{\color{blue}{x \cdot \left(n + 0.5 \cdot \frac{n}{x}\right)}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 68.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -2.5:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{elif}\;n \leq 2.1 \cdot 10^{-219}:\\ \;\;\;\;\frac{\frac{0.25}{n}}{{x}^{4}}\\ \mathbf{elif}\;n \leq 10^{+24}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;n \leq 3 \cdot 10^{+139} \lor \neg \left(n \leq 2.85 \cdot 10^{+197}\right):\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x \cdot \left(n + 0.5 \cdot \frac{n}{x}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 81.9% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{-27}:\\ \;\;\;\;\frac{\frac{t\_0}{n}}{x}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-5}:\\ \;\;\;\;\frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{+218}:\\ \;\;\;\;1 - t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{-1 + \left(1 - \frac{n}{\log x}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -1e-27)
     (/ (/ t_0 n) x)
     (if (<= (/ 1.0 n) 5e-5)
       (/ (log (/ (+ x 1.0) x)) n)
       (if (<= (/ 1.0 n) 5e+218)
         (- 1.0 t_0)
         (/ 1.0 (+ -1.0 (- 1.0 (/ n (log x))))))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -1e-27) {
		tmp = (t_0 / n) / x;
	} else if ((1.0 / n) <= 5e-5) {
		tmp = log(((x + 1.0) / x)) / n;
	} else if ((1.0 / n) <= 5e+218) {
		tmp = 1.0 - t_0;
	} else {
		tmp = 1.0 / (-1.0 + (1.0 - (n / log(x))));
	}
	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) <= (-1d-27)) then
        tmp = (t_0 / n) / x
    else if ((1.0d0 / n) <= 5d-5) then
        tmp = log(((x + 1.0d0) / x)) / n
    else if ((1.0d0 / n) <= 5d+218) then
        tmp = 1.0d0 - t_0
    else
        tmp = 1.0d0 / ((-1.0d0) + (1.0d0 - (n / log(x))))
    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) <= -1e-27) {
		tmp = (t_0 / n) / x;
	} else if ((1.0 / n) <= 5e-5) {
		tmp = Math.log(((x + 1.0) / x)) / n;
	} else if ((1.0 / n) <= 5e+218) {
		tmp = 1.0 - t_0;
	} else {
		tmp = 1.0 / (-1.0 + (1.0 - (n / Math.log(x))));
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	tmp = 0
	if (1.0 / n) <= -1e-27:
		tmp = (t_0 / n) / x
	elif (1.0 / n) <= 5e-5:
		tmp = math.log(((x + 1.0) / x)) / n
	elif (1.0 / n) <= 5e+218:
		tmp = 1.0 - t_0
	else:
		tmp = 1.0 / (-1.0 + (1.0 - (n / math.log(x))))
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -1e-27)
		tmp = Float64(Float64(t_0 / n) / x);
	elseif (Float64(1.0 / n) <= 5e-5)
		tmp = Float64(log(Float64(Float64(x + 1.0) / x)) / n);
	elseif (Float64(1.0 / n) <= 5e+218)
		tmp = Float64(1.0 - t_0);
	else
		tmp = Float64(1.0 / Float64(-1.0 + Float64(1.0 - Float64(n / log(x)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = x ^ (1.0 / n);
	tmp = 0.0;
	if ((1.0 / n) <= -1e-27)
		tmp = (t_0 / n) / x;
	elseif ((1.0 / n) <= 5e-5)
		tmp = log(((x + 1.0) / x)) / n;
	elseif ((1.0 / n) <= 5e+218)
		tmp = 1.0 - t_0;
	else
		tmp = 1.0 / (-1.0 + (1.0 - (n / log(x))));
	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], -1e-27], N[(N[(t$95$0 / n), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e-5], N[(N[Log[N[(N[(x + 1.0), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e+218], N[(1.0 - t$95$0), $MachinePrecision], N[(1.0 / N[(-1.0 + N[(1.0 - N[(n / N[Log[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 1 n) < -1e-27

   1. Initial program 98.3%

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

   3. Taylor expanded in x around inf 94.5%

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

      1. associate-/r* 94.5%

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

      2. mul-1-neg 94.5%

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

      3. log-rec 94.5%

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

      4. mul-1-neg 94.5%

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

      5. distribute-neg-frac 94.5%

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

      6. mul-1-neg 94.5%

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

      7. remove-double-neg 94.5%

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

      8. *-rgt-identity 94.5%

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

      9. associate-/l* 94.5%

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

      10. exp-to-pow 94.5%

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

   5. Simplified 94.5%

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

   if -1e-27 < (/.f64 1 n) < 5.00000000000000024e-5

   1. Initial program 25.1%

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

   3. Taylor expanded in n around inf 78.4%

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

      1. diff-log 78.5%

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

   5. Applied egg-rr 78.5%

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

      1. +-commutative 78.5%

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

   7. Simplified 78.5%

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

   if 5.00000000000000024e-5 < (/.f64 1 n) < 4.99999999999999983e218

   1. Initial program 79.8%

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

   3. Taylor expanded in x around 0 73.3%

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

      1. *-rgt-identity 73.3%

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

      2. associate-*l/ 73.3%

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

      3. associate-/l* 73.3%

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

      4. exp-to-pow 73.3%

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

   5. Simplified 73.3%

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

   if 4.99999999999999983e218 < (/.f64 1 n)

   1. Initial program 25.5%

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

   3. Taylor expanded in n around inf 8.0%

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

      1. clear-num 8.0%

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

      2. inv-pow 8.0%

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

      3. log1p-define 8.0%

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

   5. Applied egg-rr 8.0%

      \[\leadsto \color{blue}{{\left(\frac{n}{\mathsf{log1p}\left(x\right) - \log x}\right)}^{-1}} \]
   6. Step-by-step derivation

      1. unpow-1 8.0%

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

   7. Simplified 8.0%

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

   8. Taylor expanded in x around 0 8.0%

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

      1. associate-*r/ 8.0%

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

      2. neg-mul-1 8.0%

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

   10. Simplified 8.0%

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

       1. expm1-log1p-u 8.0%

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

       2. expm1-undefine 92.5%

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

   12. Applied egg-rr 92.5%

       \[\leadsto \frac{1}{\color{blue}{e^{\mathsf{log1p}\left(\frac{-n}{\log x}\right)} - 1}} \]
   13. Step-by-step derivation

       1. sub-neg 92.5%

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

       2. metadata-eval 92.5%

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

       3. +-commutative 92.5%

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

       4. log1p-undefine 92.5%

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

       5. rem-exp-log 92.5%

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

       6. distribute-frac-neg 92.5%

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

       7. unsub-neg 92.5%

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

   14. Simplified 92.5%

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

4. Final simplification 82.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{-27}:\\ \;\;\;\;\frac{\frac{{x}^{\left(\frac{1}{n}\right)}}{n}}{x}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-5}:\\ \;\;\;\;\frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{+218}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{-1 + \left(1 - \frac{n}{\log x}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 75.9% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\frac{0.25}{n}}{{x}^{4}}\\ \mathbf{if}\;\frac{1}{n} \leq -2000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-5}:\\ \;\;\;\;\frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{+218}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (/ (/ 0.25 n) (pow x 4.0))))
   (if (<= (/ 1.0 n) -2000000.0)
     t_0
     (if (<= (/ 1.0 n) 5e-5)
       (/ (log (/ (+ x 1.0) x)) n)
       (if (<= (/ 1.0 n) 5e+218) (- 1.0 (pow x (/ 1.0 n))) t_0)))))

C

double code(double x, double n) {
	double t_0 = (0.25 / n) / pow(x, 4.0);
	double tmp;
	if ((1.0 / n) <= -2000000.0) {
		tmp = t_0;
	} else if ((1.0 / n) <= 5e-5) {
		tmp = log(((x + 1.0) / x)) / n;
	} else if ((1.0 / n) <= 5e+218) {
		tmp = 1.0 - pow(x, (1.0 / n));
	} 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) :: tmp
    t_0 = (0.25d0 / n) / (x ** 4.0d0)
    if ((1.0d0 / n) <= (-2000000.0d0)) then
        tmp = t_0
    else if ((1.0d0 / n) <= 5d-5) then
        tmp = log(((x + 1.0d0) / x)) / n
    else if ((1.0d0 / n) <= 5d+218) then
        tmp = 1.0d0 - (x ** (1.0d0 / n))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = (0.25 / n) / Math.pow(x, 4.0);
	double tmp;
	if ((1.0 / n) <= -2000000.0) {
		tmp = t_0;
	} else if ((1.0 / n) <= 5e-5) {
		tmp = Math.log(((x + 1.0) / x)) / n;
	} else if ((1.0 / n) <= 5e+218) {
		tmp = 1.0 - Math.pow(x, (1.0 / n));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = (0.25 / n) / math.pow(x, 4.0)
	tmp = 0
	if (1.0 / n) <= -2000000.0:
		tmp = t_0
	elif (1.0 / n) <= 5e-5:
		tmp = math.log(((x + 1.0) / x)) / n
	elif (1.0 / n) <= 5e+218:
		tmp = 1.0 - math.pow(x, (1.0 / n))
	else:
		tmp = t_0
	return tmp

Julia

function code(x, n)
	t_0 = Float64(Float64(0.25 / n) / (x ^ 4.0))
	tmp = 0.0
	if (Float64(1.0 / n) <= -2000000.0)
		tmp = t_0;
	elseif (Float64(1.0 / n) <= 5e-5)
		tmp = Float64(log(Float64(Float64(x + 1.0) / x)) / n);
	elseif (Float64(1.0 / n) <= 5e+218)
		tmp = Float64(1.0 - (x ^ Float64(1.0 / n)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = (0.25 / n) / (x ^ 4.0);
	tmp = 0.0;
	if ((1.0 / n) <= -2000000.0)
		tmp = t_0;
	elseif ((1.0 / n) <= 5e-5)
		tmp = log(((x + 1.0) / x)) / n;
	elseif ((1.0 / n) <= 5e+218)
		tmp = 1.0 - (x ^ (1.0 / n));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(N[(0.25 / n), $MachinePrecision] / N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -2000000.0], t$95$0, If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e-5], N[(N[Log[N[(N[(x + 1.0), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e+218], N[(1.0 - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 1 n) < -2e6 or 4.99999999999999983e218 < (/.f64 1 n)

   1. Initial program 87.4%

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

   3. Taylor expanded in n around inf 41.1%

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

   4. Taylor expanded in x around -inf 1.2%

      \[\leadsto \frac{\color{blue}{-1 \cdot \frac{-1 \cdot \frac{-1 \cdot \frac{0.25 \cdot \frac{1}{x} - 0.3333333333333333}{x} - 0.5}{x} - 1}{x}}}{n} \]
   5. Step-by-step derivation

      1. add-sqr-sqrt 0.0%

         \[\leadsto \frac{-1 \cdot \color{blue}{\left(\sqrt{\frac{-1 \cdot \frac{-1 \cdot \frac{0.25 \cdot \frac{1}{x} - 0.3333333333333333}{x} - 0.5}{x} - 1}{x}} \cdot \sqrt{\frac{-1 \cdot \frac{-1 \cdot \frac{0.25 \cdot \frac{1}{x} - 0.3333333333333333}{x} - 0.5}{x} - 1}{x}}\right)}}{n} \]

      2. sqrt-unprod 17.5%

         \[\leadsto \frac{-1 \cdot \color{blue}{\sqrt{\frac{-1 \cdot \frac{-1 \cdot \frac{0.25 \cdot \frac{1}{x} - 0.3333333333333333}{x} - 0.5}{x} - 1}{x} \cdot \frac{-1 \cdot \frac{-1 \cdot \frac{0.25 \cdot \frac{1}{x} - 0.3333333333333333}{x} - 0.5}{x} - 1}{x}}}}{n} \]

      3. sqr-neg 17.5%

         \[\leadsto \frac{-1 \cdot \sqrt{\color{blue}{\left(-\frac{-1 \cdot \frac{-1 \cdot \frac{0.25 \cdot \frac{1}{x} - 0.3333333333333333}{x} - 0.5}{x} - 1}{x}\right) \cdot \left(-\frac{-1 \cdot \frac{-1 \cdot \frac{0.25 \cdot \frac{1}{x} - 0.3333333333333333}{x} - 0.5}{x} - 1}{x}\right)}}}{n} \]

      4. mul-1-neg 17.5%

         \[\leadsto \frac{-1 \cdot \sqrt{\color{blue}{\left(-1 \cdot \frac{-1 \cdot \frac{-1 \cdot \frac{0.25 \cdot \frac{1}{x} - 0.3333333333333333}{x} - 0.5}{x} - 1}{x}\right)} \cdot \left(-\frac{-1 \cdot \frac{-1 \cdot \frac{0.25 \cdot \frac{1}{x} - 0.3333333333333333}{x} - 0.5}{x} - 1}{x}\right)}}{n} \]

      5. mul-1-neg 17.5%

         \[\leadsto \frac{-1 \cdot \sqrt{\left(-1 \cdot \frac{-1 \cdot \frac{-1 \cdot \frac{0.25 \cdot \frac{1}{x} - 0.3333333333333333}{x} - 0.5}{x} - 1}{x}\right) \cdot \color{blue}{\left(-1 \cdot \frac{-1 \cdot \frac{-1 \cdot \frac{0.25 \cdot \frac{1}{x} - 0.3333333333333333}{x} - 0.5}{x} - 1}{x}\right)}}}{n} \]

      6. sqrt-unprod 1.2%

         \[\leadsto \frac{-1 \cdot \color{blue}{\left(\sqrt{-1 \cdot \frac{-1 \cdot \frac{-1 \cdot \frac{0.25 \cdot \frac{1}{x} - 0.3333333333333333}{x} - 0.5}{x} - 1}{x}} \cdot \sqrt{-1 \cdot \frac{-1 \cdot \frac{-1 \cdot \frac{0.25 \cdot \frac{1}{x} - 0.3333333333333333}{x} - 0.5}{x} - 1}{x}}\right)}}{n} \]

      7. add-sqr-sqrt 53.9%

         \[\leadsto \frac{-1 \cdot \color{blue}{\left(-1 \cdot \frac{-1 \cdot \frac{-1 \cdot \frac{0.25 \cdot \frac{1}{x} - 0.3333333333333333}{x} - 0.5}{x} - 1}{x}\right)}}{n} \]

   6. Applied egg-rr 53.9%

      \[\leadsto \frac{-1 \cdot \color{blue}{\left(0 - \frac{\frac{\frac{\frac{0.25}{x} + -0.3333333333333333}{x} + -0.5}{x} + -1}{x}\right)}}{n} \]
   7. Step-by-step derivation

      1. neg-sub0 53.9%

         \[\leadsto \frac{-1 \cdot \color{blue}{\left(-\frac{\frac{\frac{\frac{0.25}{x} + -0.3333333333333333}{x} + -0.5}{x} + -1}{x}\right)}}{n} \]

      2. distribute-neg-frac 53.9%

         \[\leadsto \frac{-1 \cdot \color{blue}{\frac{-\left(\frac{\frac{\frac{0.25}{x} + -0.3333333333333333}{x} + -0.5}{x} + -1\right)}{x}}}{n} \]

   8. Simplified 53.9%

      \[\leadsto \frac{-1 \cdot \color{blue}{\frac{1 + \frac{0.5 + \frac{0.3333333333333333 + \frac{-0.25}{x}}{x}}{x}}{x}}}{n} \]

   9. Taylor expanded in x around 0 81.5%

      \[\leadsto \color{blue}{\frac{0.25}{n \cdot {x}^{4}}} \]
   10. Step-by-step derivation

       1. associate-/r* 81.5%

          \[\leadsto \color{blue}{\frac{\frac{0.25}{n}}{{x}^{4}}} \]

   11. Simplified 81.5%

       \[\leadsto \color{blue}{\frac{\frac{0.25}{n}}{{x}^{4}}} \]

   if -2e6 < (/.f64 1 n) < 5.00000000000000024e-5

   1. Initial program 26.8%

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

   3. Taylor expanded in n around inf 76.3%

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

      1. diff-log 76.4%

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

   5. Applied egg-rr 76.4%

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

      1. +-commutative 76.4%

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

   7. Simplified 76.4%

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

   if 5.00000000000000024e-5 < (/.f64 1 n) < 4.99999999999999983e218

   1. Initial program 79.8%

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

   3. Taylor expanded in x around 0 73.3%

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

      1. *-rgt-identity 73.3%

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

      2. associate-*l/ 73.3%

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

      3. associate-/l* 73.3%

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

      4. exp-to-pow 73.3%

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

   5. Simplified 73.3%

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

4. Final simplification 77.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -2000000:\\ \;\;\;\;\frac{\frac{0.25}{n}}{{x}^{4}}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-5}:\\ \;\;\;\;\frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{+218}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.25}{n}}{{x}^{4}}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 81.7% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{-27}:\\ \;\;\;\;\frac{\frac{t\_0}{n}}{x}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-5}:\\ \;\;\;\;\frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{+218}:\\ \;\;\;\;1 - t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.25}{n}}{{x}^{4}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -1e-27)
     (/ (/ t_0 n) x)
     (if (<= (/ 1.0 n) 5e-5)
       (/ (log (/ (+ x 1.0) x)) n)
       (if (<= (/ 1.0 n) 5e+218) (- 1.0 t_0) (/ (/ 0.25 n) (pow x 4.0)))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -1e-27) {
		tmp = (t_0 / n) / x;
	} else if ((1.0 / n) <= 5e-5) {
		tmp = log(((x + 1.0) / x)) / n;
	} else if ((1.0 / n) <= 5e+218) {
		tmp = 1.0 - t_0;
	} else {
		tmp = (0.25 / n) / pow(x, 4.0);
	}
	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) <= (-1d-27)) then
        tmp = (t_0 / n) / x
    else if ((1.0d0 / n) <= 5d-5) then
        tmp = log(((x + 1.0d0) / x)) / n
    else if ((1.0d0 / n) <= 5d+218) then
        tmp = 1.0d0 - t_0
    else
        tmp = (0.25d0 / n) / (x ** 4.0d0)
    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) <= -1e-27) {
		tmp = (t_0 / n) / x;
	} else if ((1.0 / n) <= 5e-5) {
		tmp = Math.log(((x + 1.0) / x)) / n;
	} else if ((1.0 / n) <= 5e+218) {
		tmp = 1.0 - t_0;
	} else {
		tmp = (0.25 / n) / Math.pow(x, 4.0);
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	tmp = 0
	if (1.0 / n) <= -1e-27:
		tmp = (t_0 / n) / x
	elif (1.0 / n) <= 5e-5:
		tmp = math.log(((x + 1.0) / x)) / n
	elif (1.0 / n) <= 5e+218:
		tmp = 1.0 - t_0
	else:
		tmp = (0.25 / n) / math.pow(x, 4.0)
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -1e-27)
		tmp = Float64(Float64(t_0 / n) / x);
	elseif (Float64(1.0 / n) <= 5e-5)
		tmp = Float64(log(Float64(Float64(x + 1.0) / x)) / n);
	elseif (Float64(1.0 / n) <= 5e+218)
		tmp = Float64(1.0 - t_0);
	else
		tmp = Float64(Float64(0.25 / n) / (x ^ 4.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = x ^ (1.0 / n);
	tmp = 0.0;
	if ((1.0 / n) <= -1e-27)
		tmp = (t_0 / n) / x;
	elseif ((1.0 / n) <= 5e-5)
		tmp = log(((x + 1.0) / x)) / n;
	elseif ((1.0 / n) <= 5e+218)
		tmp = 1.0 - t_0;
	else
		tmp = (0.25 / n) / (x ^ 4.0);
	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], -1e-27], N[(N[(t$95$0 / n), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e-5], N[(N[Log[N[(N[(x + 1.0), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e+218], N[(1.0 - t$95$0), $MachinePrecision], N[(N[(0.25 / n), $MachinePrecision] / N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 1 n) < -1e-27

   1. Initial program 98.3%

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

   3. Taylor expanded in x around inf 94.5%

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

      1. associate-/r* 94.5%

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

      2. mul-1-neg 94.5%

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

      3. log-rec 94.5%

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

      4. mul-1-neg 94.5%

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

      5. distribute-neg-frac 94.5%

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

      6. mul-1-neg 94.5%

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

      7. remove-double-neg 94.5%

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

      8. *-rgt-identity 94.5%

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

      9. associate-/l* 94.5%

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

      10. exp-to-pow 94.5%

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

   5. Simplified 94.5%

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

   if -1e-27 < (/.f64 1 n) < 5.00000000000000024e-5

   1. Initial program 25.1%

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

   3. Taylor expanded in n around inf 78.4%

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

      1. diff-log 78.5%

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

   5. Applied egg-rr 78.5%

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

      1. +-commutative 78.5%

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

   7. Simplified 78.5%

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

   if 5.00000000000000024e-5 < (/.f64 1 n) < 4.99999999999999983e218

   1. Initial program 79.8%

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

   3. Taylor expanded in x around 0 73.3%

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

      1. *-rgt-identity 73.3%

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

      2. associate-*l/ 73.3%

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

      3. associate-/l* 73.3%

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

      4. exp-to-pow 73.3%

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

   5. Simplified 73.3%

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

   if 4.99999999999999983e218 < (/.f64 1 n)

   1. Initial program 25.5%

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

   3. Taylor expanded in n around inf 8.0%

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

   4. Taylor expanded in x around -inf 0.1%

      \[\leadsto \frac{\color{blue}{-1 \cdot \frac{-1 \cdot \frac{-1 \cdot \frac{0.25 \cdot \frac{1}{x} - 0.3333333333333333}{x} - 0.5}{x} - 1}{x}}}{n} \]
   5. Step-by-step derivation

      1. add-sqr-sqrt 0.1%

         \[\leadsto \frac{-1 \cdot \color{blue}{\left(\sqrt{\frac{-1 \cdot \frac{-1 \cdot \frac{0.25 \cdot \frac{1}{x} - 0.3333333333333333}{x} - 0.5}{x} - 1}{x}} \cdot \sqrt{\frac{-1 \cdot \frac{-1 \cdot \frac{0.25 \cdot \frac{1}{x} - 0.3333333333333333}{x} - 0.5}{x} - 1}{x}}\right)}}{n} \]

      2. sqrt-unprod 0.1%

         \[\leadsto \frac{-1 \cdot \color{blue}{\sqrt{\frac{-1 \cdot \frac{-1 \cdot \frac{0.25 \cdot \frac{1}{x} - 0.3333333333333333}{x} - 0.5}{x} - 1}{x} \cdot \frac{-1 \cdot \frac{-1 \cdot \frac{0.25 \cdot \frac{1}{x} - 0.3333333333333333}{x} - 0.5}{x} - 1}{x}}}}{n} \]

      3. sqr-neg 0.1%

         \[\leadsto \frac{-1 \cdot \sqrt{\color{blue}{\left(-\frac{-1 \cdot \frac{-1 \cdot \frac{0.25 \cdot \frac{1}{x} - 0.3333333333333333}{x} - 0.5}{x} - 1}{x}\right) \cdot \left(-\frac{-1 \cdot \frac{-1 \cdot \frac{0.25 \cdot \frac{1}{x} - 0.3333333333333333}{x} - 0.5}{x} - 1}{x}\right)}}}{n} \]

      4. mul-1-neg 0.1%

         \[\leadsto \frac{-1 \cdot \sqrt{\color{blue}{\left(-1 \cdot \frac{-1 \cdot \frac{-1 \cdot \frac{0.25 \cdot \frac{1}{x} - 0.3333333333333333}{x} - 0.5}{x} - 1}{x}\right)} \cdot \left(-\frac{-1 \cdot \frac{-1 \cdot \frac{0.25 \cdot \frac{1}{x} - 0.3333333333333333}{x} - 0.5}{x} - 1}{x}\right)}}{n} \]

      5. mul-1-neg 0.1%

         \[\leadsto \frac{-1 \cdot \sqrt{\left(-1 \cdot \frac{-1 \cdot \frac{-1 \cdot \frac{0.25 \cdot \frac{1}{x} - 0.3333333333333333}{x} - 0.5}{x} - 1}{x}\right) \cdot \color{blue}{\left(-1 \cdot \frac{-1 \cdot \frac{-1 \cdot \frac{0.25 \cdot \frac{1}{x} - 0.3333333333333333}{x} - 0.5}{x} - 1}{x}\right)}}}{n} \]

      6. sqrt-unprod 0.0%

         \[\leadsto \frac{-1 \cdot \color{blue}{\left(\sqrt{-1 \cdot \frac{-1 \cdot \frac{-1 \cdot \frac{0.25 \cdot \frac{1}{x} - 0.3333333333333333}{x} - 0.5}{x} - 1}{x}} \cdot \sqrt{-1 \cdot \frac{-1 \cdot \frac{-1 \cdot \frac{0.25 \cdot \frac{1}{x} - 0.3333333333333333}{x} - 0.5}{x} - 1}{x}}\right)}}{n} \]

      7. add-sqr-sqrt 79.0%

         \[\leadsto \frac{-1 \cdot \color{blue}{\left(-1 \cdot \frac{-1 \cdot \frac{-1 \cdot \frac{0.25 \cdot \frac{1}{x} - 0.3333333333333333}{x} - 0.5}{x} - 1}{x}\right)}}{n} \]

   6. Applied egg-rr 79.0%

      \[\leadsto \frac{-1 \cdot \color{blue}{\left(0 - \frac{\frac{\frac{\frac{0.25}{x} + -0.3333333333333333}{x} + -0.5}{x} + -1}{x}\right)}}{n} \]
   7. Step-by-step derivation

      1. neg-sub0 79.0%

         \[\leadsto \frac{-1 \cdot \color{blue}{\left(-\frac{\frac{\frac{\frac{0.25}{x} + -0.3333333333333333}{x} + -0.5}{x} + -1}{x}\right)}}{n} \]

      2. distribute-neg-frac 79.0%

         \[\leadsto \frac{-1 \cdot \color{blue}{\frac{-\left(\frac{\frac{\frac{0.25}{x} + -0.3333333333333333}{x} + -0.5}{x} + -1\right)}{x}}}{n} \]

   8. Simplified 79.0%

      \[\leadsto \frac{-1 \cdot \color{blue}{\frac{1 + \frac{0.5 + \frac{0.3333333333333333 + \frac{-0.25}{x}}{x}}{x}}{x}}}{n} \]

   9. Taylor expanded in x around 0 79.0%

      \[\leadsto \color{blue}{\frac{0.25}{n \cdot {x}^{4}}} \]
   10. Step-by-step derivation

       1. associate-/r* 79.0%

          \[\leadsto \color{blue}{\frac{\frac{0.25}{n}}{{x}^{4}}} \]

   11. Simplified 79.0%

       \[\leadsto \color{blue}{\frac{\frac{0.25}{n}}{{x}^{4}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 82.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{-27}:\\ \;\;\;\;\frac{\frac{{x}^{\left(\frac{1}{n}\right)}}{n}}{x}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-5}:\\ \;\;\;\;\frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{+218}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.25}{n}}{{x}^{4}}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 59.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := n \cdot -0.3333333333333333 + n \cdot 0.25\\ \mathbf{if}\;x \leq 9.5 \cdot 10^{-136}:\\ \;\;\;\;\frac{\log x}{-n}\\ \mathbf{elif}\;x \leq 4.5 \cdot 10^{-94}:\\ \;\;\;\;\frac{\frac{1}{n} - \frac{\frac{1}{n} \cdot 0.5 - 0.3333333333333333 \cdot \frac{1}{n \cdot x}}{x}}{x}\\ \mathbf{elif}\;x \leq 0.5:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{elif}\;x \leq 8.8 \cdot 10^{+73}:\\ \;\;\;\;\frac{1}{x \cdot \left(n - \frac{\frac{\left(-0.5 \cdot \frac{t\_0}{x} + \left(\frac{n}{x} \cdot -0.25 + \frac{n}{x} \cdot 0.16666666666666666\right)\right) - t\_0}{x} + n \cdot -0.5}{x}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (+ (* n -0.3333333333333333) (* n 0.25))))
   (if (<= x 9.5e-136)
     (/ (log x) (- n))
     (if (<= x 4.5e-94)
       (/
        (-
         (/ 1.0 n)
         (/ (- (* (/ 1.0 n) 0.5) (* 0.3333333333333333 (/ 1.0 (* n x)))) x))
        x)
       (if (<= x 0.5)
         (/ (- x (log x)) n)
         (if (<= x 8.8e+73)
           (/
            1.0
            (*
             x
             (-
              n
              (/
               (+
                (/
                 (-
                  (+
                   (* -0.5 (/ t_0 x))
                   (+ (* (/ n x) -0.25) (* (/ n x) 0.16666666666666666)))
                  t_0)
                 x)
                (* n -0.5))
               x))))
           0.0))))))

C

double code(double x, double n) {
	double t_0 = (n * -0.3333333333333333) + (n * 0.25);
	double tmp;
	if (x <= 9.5e-136) {
		tmp = log(x) / -n;
	} else if (x <= 4.5e-94) {
		tmp = ((1.0 / n) - ((((1.0 / n) * 0.5) - (0.3333333333333333 * (1.0 / (n * x)))) / x)) / x;
	} else if (x <= 0.5) {
		tmp = (x - log(x)) / n;
	} else if (x <= 8.8e+73) {
		tmp = 1.0 / (x * (n - ((((((-0.5 * (t_0 / x)) + (((n / x) * -0.25) + ((n / x) * 0.16666666666666666))) - t_0) / x) + (n * -0.5)) / x)));
	} 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) :: tmp
    t_0 = (n * (-0.3333333333333333d0)) + (n * 0.25d0)
    if (x <= 9.5d-136) then
        tmp = log(x) / -n
    else if (x <= 4.5d-94) then
        tmp = ((1.0d0 / n) - ((((1.0d0 / n) * 0.5d0) - (0.3333333333333333d0 * (1.0d0 / (n * x)))) / x)) / x
    else if (x <= 0.5d0) then
        tmp = (x - log(x)) / n
    else if (x <= 8.8d+73) then
        tmp = 1.0d0 / (x * (n - (((((((-0.5d0) * (t_0 / x)) + (((n / x) * (-0.25d0)) + ((n / x) * 0.16666666666666666d0))) - t_0) / x) + (n * (-0.5d0))) / x)))
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = (n * -0.3333333333333333) + (n * 0.25);
	double tmp;
	if (x <= 9.5e-136) {
		tmp = Math.log(x) / -n;
	} else if (x <= 4.5e-94) {
		tmp = ((1.0 / n) - ((((1.0 / n) * 0.5) - (0.3333333333333333 * (1.0 / (n * x)))) / x)) / x;
	} else if (x <= 0.5) {
		tmp = (x - Math.log(x)) / n;
	} else if (x <= 8.8e+73) {
		tmp = 1.0 / (x * (n - ((((((-0.5 * (t_0 / x)) + (((n / x) * -0.25) + ((n / x) * 0.16666666666666666))) - t_0) / x) + (n * -0.5)) / x)));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = (n * -0.3333333333333333) + (n * 0.25)
	tmp = 0
	if x <= 9.5e-136:
		tmp = math.log(x) / -n
	elif x <= 4.5e-94:
		tmp = ((1.0 / n) - ((((1.0 / n) * 0.5) - (0.3333333333333333 * (1.0 / (n * x)))) / x)) / x
	elif x <= 0.5:
		tmp = (x - math.log(x)) / n
	elif x <= 8.8e+73:
		tmp = 1.0 / (x * (n - ((((((-0.5 * (t_0 / x)) + (((n / x) * -0.25) + ((n / x) * 0.16666666666666666))) - t_0) / x) + (n * -0.5)) / x)))
	else:
		tmp = 0.0
	return tmp

Julia

function code(x, n)
	t_0 = Float64(Float64(n * -0.3333333333333333) + Float64(n * 0.25))
	tmp = 0.0
	if (x <= 9.5e-136)
		tmp = Float64(log(x) / Float64(-n));
	elseif (x <= 4.5e-94)
		tmp = Float64(Float64(Float64(1.0 / n) - Float64(Float64(Float64(Float64(1.0 / n) * 0.5) - Float64(0.3333333333333333 * Float64(1.0 / Float64(n * x)))) / x)) / x);
	elseif (x <= 0.5)
		tmp = Float64(Float64(x - log(x)) / n);
	elseif (x <= 8.8e+73)
		tmp = Float64(1.0 / Float64(x * Float64(n - Float64(Float64(Float64(Float64(Float64(Float64(-0.5 * Float64(t_0 / x)) + Float64(Float64(Float64(n / x) * -0.25) + Float64(Float64(n / x) * 0.16666666666666666))) - t_0) / x) + Float64(n * -0.5)) / x))));
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = (n * -0.3333333333333333) + (n * 0.25);
	tmp = 0.0;
	if (x <= 9.5e-136)
		tmp = log(x) / -n;
	elseif (x <= 4.5e-94)
		tmp = ((1.0 / n) - ((((1.0 / n) * 0.5) - (0.3333333333333333 * (1.0 / (n * x)))) / x)) / x;
	elseif (x <= 0.5)
		tmp = (x - log(x)) / n;
	elseif (x <= 8.8e+73)
		tmp = 1.0 / (x * (n - ((((((-0.5 * (t_0 / x)) + (((n / x) * -0.25) + ((n / x) * 0.16666666666666666))) - t_0) / x) + (n * -0.5)) / x)));
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(N[(n * -0.3333333333333333), $MachinePrecision] + N[(n * 0.25), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 9.5e-136], N[(N[Log[x], $MachinePrecision] / (-n)), $MachinePrecision], If[LessEqual[x, 4.5e-94], N[(N[(N[(1.0 / n), $MachinePrecision] - N[(N[(N[(N[(1.0 / n), $MachinePrecision] * 0.5), $MachinePrecision] - N[(0.3333333333333333 * N[(1.0 / N[(n * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[x, 0.5], N[(N[(x - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[x, 8.8e+73], N[(1.0 / N[(x * N[(n - N[(N[(N[(N[(N[(N[(-0.5 * N[(t$95$0 / x), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(n / x), $MachinePrecision] * -0.25), $MachinePrecision] + N[(N[(n / x), $MachinePrecision] * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision] / x), $MachinePrecision] + N[(n * -0.5), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0]]]]]

Derivation

1. Split input into 5 regimes

2. if x < 9.5000000000000007e-136

   1. Initial program 45.1%

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

   3. Taylor expanded in n around inf 58.3%

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

   4. Taylor expanded in x around 0 58.3%

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

      1. neg-mul-1 58.3%

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

   6. Simplified 58.3%

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

   if 9.5000000000000007e-136 < x < 4.5000000000000002e-94

   1. Initial program 35.9%

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

   3. Taylor expanded in n around inf 23.1%

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

   4. Taylor expanded in x around -inf 73.1%

      \[\leadsto \color{blue}{-1 \cdot \frac{-1 \cdot \frac{0.3333333333333333 \cdot \frac{1}{n \cdot x} - 0.5 \cdot \frac{1}{n}}{x} - \frac{1}{n}}{x}} \]

   if 4.5000000000000002e-94 < x < 0.5

   1. Initial program 43.1%

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

   3. Taylor expanded in n around inf 54.1%

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

   4. Taylor expanded in x around 0 51.6%

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

   if 0.5 < x < 8.8e73

   1. Initial program 29.1%

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

   3. Taylor expanded in n around inf 31.0%

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

      1. clear-num 31.0%

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

      2. inv-pow 31.0%

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

      3. log1p-define 31.0%

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

   5. Applied egg-rr 31.0%

      \[\leadsto \color{blue}{{\left(\frac{n}{\mathsf{log1p}\left(x\right) - \log x}\right)}^{-1}} \]
   6. Step-by-step derivation

      1. unpow-1 31.0%

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

   7. Simplified 31.0%

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

   8. Taylor expanded in x around -inf 77.7%

      \[\leadsto \frac{1}{\color{blue}{-1 \cdot \left(x \cdot \left(-1 \cdot n + -1 \cdot \frac{-1 \cdot \frac{\left(-0.5 \cdot \frac{-0.3333333333333333 \cdot n + 0.25 \cdot n}{x} + \left(-0.25 \cdot \frac{n}{x} + 0.16666666666666666 \cdot \frac{n}{x}\right)\right) - \left(-0.3333333333333333 \cdot n + 0.25 \cdot n\right)}{x} - -0.5 \cdot n}{x}\right)\right)}} \]

   if 8.8e73 < x

   1. Initial program 80.4%

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

   3. Taylor expanded in x around 0 46.0%

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

   4. Taylor expanded in n around inf 80.4%

      \[\leadsto 1 - \color{blue}{1} \]
3. Recombined 5 regimes into one program.

4. Final simplification 65.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 9.5 \cdot 10^{-136}:\\ \;\;\;\;\frac{\log x}{-n}\\ \mathbf{elif}\;x \leq 4.5 \cdot 10^{-94}:\\ \;\;\;\;\frac{\frac{1}{n} - \frac{\frac{1}{n} \cdot 0.5 - 0.3333333333333333 \cdot \frac{1}{n \cdot x}}{x}}{x}\\ \mathbf{elif}\;x \leq 0.5:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{elif}\;x \leq 8.8 \cdot 10^{+73}:\\ \;\;\;\;\frac{1}{x \cdot \left(n - \frac{\frac{\left(-0.5 \cdot \frac{n \cdot -0.3333333333333333 + n \cdot 0.25}{x} + \left(\frac{n}{x} \cdot -0.25 + \frac{n}{x} \cdot 0.16666666666666666\right)\right) - \left(n \cdot -0.3333333333333333 + n \cdot 0.25\right)}{x} + n \cdot -0.5}{x}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 59.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\log x}{-n}\\ t_1 := n \cdot -0.3333333333333333 + n \cdot 0.25\\ \mathbf{if}\;x \leq 1.2 \cdot 10^{-135}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 3.3 \cdot 10^{-94}:\\ \;\;\;\;\frac{\frac{1}{n} - \frac{\frac{1}{n} \cdot 0.5 - 0.3333333333333333 \cdot \frac{1}{n \cdot x}}{x}}{x}\\ \mathbf{elif}\;x \leq 0.35:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 1.16 \cdot 10^{+73}:\\ \;\;\;\;\frac{1}{x \cdot \left(n - \frac{\frac{\left(-0.5 \cdot \frac{t\_1}{x} + \left(\frac{n}{x} \cdot -0.25 + \frac{n}{x} \cdot 0.16666666666666666\right)\right) - t\_1}{x} + n \cdot -0.5}{x}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (/ (log x) (- n))) (t_1 (+ (* n -0.3333333333333333) (* n 0.25))))
   (if (<= x 1.2e-135)
     t_0
     (if (<= x 3.3e-94)
       (/
        (-
         (/ 1.0 n)
         (/ (- (* (/ 1.0 n) 0.5) (* 0.3333333333333333 (/ 1.0 (* n x)))) x))
        x)
       (if (<= x 0.35)
         t_0
         (if (<= x 1.16e+73)
           (/
            1.0
            (*
             x
             (-
              n
              (/
               (+
                (/
                 (-
                  (+
                   (* -0.5 (/ t_1 x))
                   (+ (* (/ n x) -0.25) (* (/ n x) 0.16666666666666666)))
                  t_1)
                 x)
                (* n -0.5))
               x))))
           0.0))))))

C

double code(double x, double n) {
	double t_0 = log(x) / -n;
	double t_1 = (n * -0.3333333333333333) + (n * 0.25);
	double tmp;
	if (x <= 1.2e-135) {
		tmp = t_0;
	} else if (x <= 3.3e-94) {
		tmp = ((1.0 / n) - ((((1.0 / n) * 0.5) - (0.3333333333333333 * (1.0 / (n * x)))) / x)) / x;
	} else if (x <= 0.35) {
		tmp = t_0;
	} else if (x <= 1.16e+73) {
		tmp = 1.0 / (x * (n - ((((((-0.5 * (t_1 / x)) + (((n / x) * -0.25) + ((n / x) * 0.16666666666666666))) - t_1) / x) + (n * -0.5)) / x)));
	} 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 = log(x) / -n
    t_1 = (n * (-0.3333333333333333d0)) + (n * 0.25d0)
    if (x <= 1.2d-135) then
        tmp = t_0
    else if (x <= 3.3d-94) then
        tmp = ((1.0d0 / n) - ((((1.0d0 / n) * 0.5d0) - (0.3333333333333333d0 * (1.0d0 / (n * x)))) / x)) / x
    else if (x <= 0.35d0) then
        tmp = t_0
    else if (x <= 1.16d+73) then
        tmp = 1.0d0 / (x * (n - (((((((-0.5d0) * (t_1 / x)) + (((n / x) * (-0.25d0)) + ((n / x) * 0.16666666666666666d0))) - t_1) / x) + (n * (-0.5d0))) / x)))
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = Math.log(x) / -n;
	double t_1 = (n * -0.3333333333333333) + (n * 0.25);
	double tmp;
	if (x <= 1.2e-135) {
		tmp = t_0;
	} else if (x <= 3.3e-94) {
		tmp = ((1.0 / n) - ((((1.0 / n) * 0.5) - (0.3333333333333333 * (1.0 / (n * x)))) / x)) / x;
	} else if (x <= 0.35) {
		tmp = t_0;
	} else if (x <= 1.16e+73) {
		tmp = 1.0 / (x * (n - ((((((-0.5 * (t_1 / x)) + (((n / x) * -0.25) + ((n / x) * 0.16666666666666666))) - t_1) / x) + (n * -0.5)) / x)));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.log(x) / -n
	t_1 = (n * -0.3333333333333333) + (n * 0.25)
	tmp = 0
	if x <= 1.2e-135:
		tmp = t_0
	elif x <= 3.3e-94:
		tmp = ((1.0 / n) - ((((1.0 / n) * 0.5) - (0.3333333333333333 * (1.0 / (n * x)))) / x)) / x
	elif x <= 0.35:
		tmp = t_0
	elif x <= 1.16e+73:
		tmp = 1.0 / (x * (n - ((((((-0.5 * (t_1 / x)) + (((n / x) * -0.25) + ((n / x) * 0.16666666666666666))) - t_1) / x) + (n * -0.5)) / x)))
	else:
		tmp = 0.0
	return tmp

Julia

function code(x, n)
	t_0 = Float64(log(x) / Float64(-n))
	t_1 = Float64(Float64(n * -0.3333333333333333) + Float64(n * 0.25))
	tmp = 0.0
	if (x <= 1.2e-135)
		tmp = t_0;
	elseif (x <= 3.3e-94)
		tmp = Float64(Float64(Float64(1.0 / n) - Float64(Float64(Float64(Float64(1.0 / n) * 0.5) - Float64(0.3333333333333333 * Float64(1.0 / Float64(n * x)))) / x)) / x);
	elseif (x <= 0.35)
		tmp = t_0;
	elseif (x <= 1.16e+73)
		tmp = Float64(1.0 / Float64(x * Float64(n - Float64(Float64(Float64(Float64(Float64(Float64(-0.5 * Float64(t_1 / x)) + Float64(Float64(Float64(n / x) * -0.25) + Float64(Float64(n / x) * 0.16666666666666666))) - t_1) / x) + Float64(n * -0.5)) / x))));
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = log(x) / -n;
	t_1 = (n * -0.3333333333333333) + (n * 0.25);
	tmp = 0.0;
	if (x <= 1.2e-135)
		tmp = t_0;
	elseif (x <= 3.3e-94)
		tmp = ((1.0 / n) - ((((1.0 / n) * 0.5) - (0.3333333333333333 * (1.0 / (n * x)))) / x)) / x;
	elseif (x <= 0.35)
		tmp = t_0;
	elseif (x <= 1.16e+73)
		tmp = 1.0 / (x * (n - ((((((-0.5 * (t_1 / x)) + (((n / x) * -0.25) + ((n / x) * 0.16666666666666666))) - t_1) / x) + (n * -0.5)) / x)));
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(N[Log[x], $MachinePrecision] / (-n)), $MachinePrecision]}, Block[{t$95$1 = N[(N[(n * -0.3333333333333333), $MachinePrecision] + N[(n * 0.25), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 1.2e-135], t$95$0, If[LessEqual[x, 3.3e-94], N[(N[(N[(1.0 / n), $MachinePrecision] - N[(N[(N[(N[(1.0 / n), $MachinePrecision] * 0.5), $MachinePrecision] - N[(0.3333333333333333 * N[(1.0 / N[(n * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[x, 0.35], t$95$0, If[LessEqual[x, 1.16e+73], N[(1.0 / N[(x * N[(n - N[(N[(N[(N[(N[(N[(-0.5 * N[(t$95$1 / x), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(n / x), $MachinePrecision] * -0.25), $MachinePrecision] + N[(N[(n / x), $MachinePrecision] * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision] / x), $MachinePrecision] + N[(n * -0.5), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0]]]]]]

Derivation

1. Split input into 4 regimes

2. if x < 1.1999999999999999e-135 or 3.3000000000000001e-94 < x < 0.34999999999999998

   1. Initial program 44.5%

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

   3. Taylor expanded in n around inf 56.9%

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

   4. Taylor expanded in x around 0 55.4%

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

      1. neg-mul-1 55.4%

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

   6. Simplified 55.4%

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

   if 1.1999999999999999e-135 < x < 3.3000000000000001e-94

   1. Initial program 35.9%

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

   3. Taylor expanded in n around inf 23.1%

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

   4. Taylor expanded in x around -inf 73.1%

      \[\leadsto \color{blue}{-1 \cdot \frac{-1 \cdot \frac{0.3333333333333333 \cdot \frac{1}{n \cdot x} - 0.5 \cdot \frac{1}{n}}{x} - \frac{1}{n}}{x}} \]

   if 0.34999999999999998 < x < 1.16000000000000007e73

   1. Initial program 29.1%

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

   3. Taylor expanded in n around inf 31.0%

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

      1. clear-num 31.0%

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

      2. inv-pow 31.0%

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

      3. log1p-define 31.0%

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

   5. Applied egg-rr 31.0%

      \[\leadsto \color{blue}{{\left(\frac{n}{\mathsf{log1p}\left(x\right) - \log x}\right)}^{-1}} \]
   6. Step-by-step derivation

      1. unpow-1 31.0%

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

   7. Simplified 31.0%

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

   8. Taylor expanded in x around -inf 77.7%

      \[\leadsto \frac{1}{\color{blue}{-1 \cdot \left(x \cdot \left(-1 \cdot n + -1 \cdot \frac{-1 \cdot \frac{\left(-0.5 \cdot \frac{-0.3333333333333333 \cdot n + 0.25 \cdot n}{x} + \left(-0.25 \cdot \frac{n}{x} + 0.16666666666666666 \cdot \frac{n}{x}\right)\right) - \left(-0.3333333333333333 \cdot n + 0.25 \cdot n\right)}{x} - -0.5 \cdot n}{x}\right)\right)}} \]

   if 1.16000000000000007e73 < x

   1. Initial program 80.4%

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

   3. Taylor expanded in x around 0 46.0%

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

   4. Taylor expanded in n around inf 80.4%

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

4. Final simplification 65.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.2 \cdot 10^{-135}:\\ \;\;\;\;\frac{\log x}{-n}\\ \mathbf{elif}\;x \leq 3.3 \cdot 10^{-94}:\\ \;\;\;\;\frac{\frac{1}{n} - \frac{\frac{1}{n} \cdot 0.5 - 0.3333333333333333 \cdot \frac{1}{n \cdot x}}{x}}{x}\\ \mathbf{elif}\;x \leq 0.35:\\ \;\;\;\;\frac{\log x}{-n}\\ \mathbf{elif}\;x \leq 1.16 \cdot 10^{+73}:\\ \;\;\;\;\frac{1}{x \cdot \left(n - \frac{\frac{\left(-0.5 \cdot \frac{n \cdot -0.3333333333333333 + n \cdot 0.25}{x} + \left(\frac{n}{x} \cdot -0.25 + \frac{n}{x} \cdot 0.16666666666666666\right)\right) - \left(n \cdot -0.3333333333333333 + n \cdot 0.25\right)}{x} + n \cdot -0.5}{x}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 52.0% accurate, 9.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.12 \cdot 10^{+74}:\\ \;\;\;\;\frac{1 + \frac{-0.5 + \frac{-0.3333333333333333 + \frac{0.25}{x}}{x}}{x}}{n \cdot x}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 1.12e+74)
   (/ (+ 1.0 (/ (+ -0.5 (/ (+ -0.3333333333333333 (/ 0.25 x)) x)) x)) (* n x))
   0.0))

C

double code(double x, double n) {
	double tmp;
	if (x <= 1.12e+74) {
		tmp = (1.0 + ((-0.5 + ((-0.3333333333333333 + (0.25 / x)) / x)) / x)) / (n * x);
	} 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 <= 1.12d+74) then
        tmp = (1.0d0 + (((-0.5d0) + (((-0.3333333333333333d0) + (0.25d0 / x)) / x)) / x)) / (n * x)
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (x <= 1.12e+74) {
		tmp = (1.0 + ((-0.5 + ((-0.3333333333333333 + (0.25 / x)) / x)) / x)) / (n * x);
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 1.12e+74:
		tmp = (1.0 + ((-0.5 + ((-0.3333333333333333 + (0.25 / x)) / x)) / x)) / (n * x)
	else:
		tmp = 0.0
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 1.12e+74)
		tmp = Float64(Float64(1.0 + Float64(Float64(-0.5 + Float64(Float64(-0.3333333333333333 + Float64(0.25 / x)) / x)) / x)) / Float64(n * x));
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (x <= 1.12e+74)
		tmp = (1.0 + ((-0.5 + ((-0.3333333333333333 + (0.25 / x)) / x)) / x)) / (n * x);
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 1.12e+74], N[(N[(1.0 + N[(N[(-0.5 + N[(N[(-0.3333333333333333 + N[(0.25 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / N[(n * x), $MachinePrecision]), $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if x < 1.12000000000000003e74

   1. Initial program 41.3%

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

   3. Taylor expanded in n around inf 49.8%

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

   4. Taylor expanded in x around -inf 12.5%

      \[\leadsto \frac{\color{blue}{-1 \cdot \frac{-1 \cdot \frac{-1 \cdot \frac{0.25 \cdot \frac{1}{x} - 0.3333333333333333}{x} - 0.5}{x} - 1}{x}}}{n} \]

   6. Applied egg-rr 36.9%

      \[\leadsto \color{blue}{1 \cdot \frac{1 + \frac{\frac{\frac{0.25}{x} + -0.3333333333333333}{x} + -0.5}{x}}{x \cdot n}} \]
   7. Step-by-step derivation

      1. *-lft-identity 36.9%

         \[\leadsto \color{blue}{\frac{1 + \frac{\frac{\frac{0.25}{x} + -0.3333333333333333}{x} + -0.5}{x}}{x \cdot n}} \]

      2. +-commutative 36.9%

         \[\leadsto \frac{1 + \frac{\color{blue}{-0.5 + \frac{\frac{0.25}{x} + -0.3333333333333333}{x}}}{x}}{x \cdot n} \]

   8. Simplified 36.9%

      \[\leadsto \color{blue}{\frac{1 + \frac{-0.5 + \frac{\frac{0.25}{x} + -0.3333333333333333}{x}}{x}}{x \cdot n}} \]

   if 1.12000000000000003e74 < x

   1. Initial program 80.4%

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

   3. Taylor expanded in x around 0 46.0%

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

   4. Taylor expanded in n around inf 80.4%

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

4. Final simplification 47.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.12 \cdot 10^{+74}:\\ \;\;\;\;\frac{1 + \frac{-0.5 + \frac{-0.3333333333333333 + \frac{0.25}{x}}{x}}{x}}{n \cdot x}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 51.5% accurate, 11.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 8.8 \cdot 10^{+73}:\\ \;\;\;\;\frac{\frac{-1 + \frac{0.5 + \frac{-0.3333333333333333}{x}}{x}}{-x}}{n}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 8.8e+73)
   (/ (/ (+ -1.0 (/ (+ 0.5 (/ -0.3333333333333333 x)) x)) (- x)) n)
   0.0))

C

double code(double x, double n) {
	double tmp;
	if (x <= 8.8e+73) {
		tmp = ((-1.0 + ((0.5 + (-0.3333333333333333 / x)) / x)) / -x) / 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 <= 8.8d+73) then
        tmp = (((-1.0d0) + ((0.5d0 + ((-0.3333333333333333d0) / x)) / x)) / -x) / n
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (x <= 8.8e+73) {
		tmp = ((-1.0 + ((0.5 + (-0.3333333333333333 / x)) / x)) / -x) / n;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 8.8e+73:
		tmp = ((-1.0 + ((0.5 + (-0.3333333333333333 / x)) / x)) / -x) / n
	else:
		tmp = 0.0
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 8.8e+73)
		tmp = Float64(Float64(Float64(-1.0 + Float64(Float64(0.5 + Float64(-0.3333333333333333 / x)) / x)) / Float64(-x)) / n);
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (x <= 8.8e+73)
		tmp = ((-1.0 + ((0.5 + (-0.3333333333333333 / x)) / x)) / -x) / n;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 8.8e+73], N[(N[(N[(-1.0 + N[(N[(0.5 + N[(-0.3333333333333333 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / (-x)), $MachinePrecision] / n), $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if x < 8.8e73

   1. Initial program 41.3%

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

   3. Taylor expanded in n around inf 49.8%

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

      1. diff-log 49.9%

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

   5. Applied egg-rr 49.9%

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

      1. +-commutative 49.9%

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

   7. Simplified 49.9%

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

   8. Taylor expanded in x around -inf 35.5%

      \[\leadsto \frac{\color{blue}{-1 \cdot \frac{-1 \cdot \frac{0.3333333333333333 \cdot \frac{1}{x} - 0.5}{x} - 1}{x}}}{n} \]
   9. Step-by-step derivation

      1. mul-1-neg 35.5%

         \[\leadsto \frac{\color{blue}{-\frac{-1 \cdot \frac{0.3333333333333333 \cdot \frac{1}{x} - 0.5}{x} - 1}{x}}}{n} \]

      2. distribute-neg-frac2 35.5%

         \[\leadsto \frac{\color{blue}{\frac{-1 \cdot \frac{0.3333333333333333 \cdot \frac{1}{x} - 0.5}{x} - 1}{-x}}}{n} \]

      3. sub-neg 35.5%

         \[\leadsto \frac{\frac{\color{blue}{-1 \cdot \frac{0.3333333333333333 \cdot \frac{1}{x} - 0.5}{x} + \left(-1\right)}}{-x}}{n} \]

      4. associate-*r/ 35.5%

         \[\leadsto \frac{\frac{\color{blue}{\frac{-1 \cdot \left(0.3333333333333333 \cdot \frac{1}{x} - 0.5\right)}{x}} + \left(-1\right)}{-x}}{n} \]

      5. sub-neg 35.5%

         \[\leadsto \frac{\frac{\frac{-1 \cdot \color{blue}{\left(0.3333333333333333 \cdot \frac{1}{x} + \left(-0.5\right)\right)}}{x} + \left(-1\right)}{-x}}{n} \]

      6. metadata-eval 35.5%

         \[\leadsto \frac{\frac{\frac{-1 \cdot \left(0.3333333333333333 \cdot \frac{1}{x} + \color{blue}{-0.5}\right)}{x} + \left(-1\right)}{-x}}{n} \]

      7. distribute-lft-in 35.5%

         \[\leadsto \frac{\frac{\frac{\color{blue}{-1 \cdot \left(0.3333333333333333 \cdot \frac{1}{x}\right) + -1 \cdot -0.5}}{x} + \left(-1\right)}{-x}}{n} \]

      8. associate-*r/ 35.5%

         \[\leadsto \frac{\frac{\frac{-1 \cdot \color{blue}{\frac{0.3333333333333333 \cdot 1}{x}} + -1 \cdot -0.5}{x} + \left(-1\right)}{-x}}{n} \]

      9. metadata-eval 35.5%

         \[\leadsto \frac{\frac{\frac{-1 \cdot \frac{\color{blue}{0.3333333333333333}}{x} + -1 \cdot -0.5}{x} + \left(-1\right)}{-x}}{n} \]

      10. associate-*r/ 35.5%

          \[\leadsto \frac{\frac{\frac{\color{blue}{\frac{-1 \cdot 0.3333333333333333}{x}} + -1 \cdot -0.5}{x} + \left(-1\right)}{-x}}{n} \]

      11. metadata-eval 35.5%

          \[\leadsto \frac{\frac{\frac{\frac{\color{blue}{-0.3333333333333333}}{x} + -1 \cdot -0.5}{x} + \left(-1\right)}{-x}}{n} \]

      12. metadata-eval 35.5%

          \[\leadsto \frac{\frac{\frac{\frac{-0.3333333333333333}{x} + \color{blue}{0.5}}{x} + \left(-1\right)}{-x}}{n} \]

      13. metadata-eval 35.5%

          \[\leadsto \frac{\frac{\frac{\frac{-0.3333333333333333}{x} + 0.5}{x} + \color{blue}{-1}}{-x}}{n} \]

   10. Simplified 35.5%

       \[\leadsto \frac{\color{blue}{\frac{\frac{\frac{-0.3333333333333333}{x} + 0.5}{x} + -1}{-x}}}{n} \]

   if 8.8e73 < x

   1. Initial program 80.4%

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

   3. Taylor expanded in x around 0 46.0%

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

   4. Taylor expanded in n around inf 80.4%

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

4. Final simplification 46.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 8.8 \cdot 10^{+73}:\\ \;\;\;\;\frac{\frac{-1 + \frac{0.5 + \frac{-0.3333333333333333}{x}}{x}}{-x}}{n}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 49.5% accurate, 13.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2.8 \cdot 10^{+73}:\\ \;\;\;\;\frac{\frac{1}{n} + \frac{0.5}{n \cdot x}}{x}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 2.8e+73) (/ (+ (/ 1.0 n) (/ 0.5 (* n x))) x) 0.0))

C

double code(double x, double n) {
	double tmp;
	if (x <= 2.8e+73) {
		tmp = ((1.0 / n) + (0.5 / (n * x))) / x;
	} 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 <= 2.8d+73) then
        tmp = ((1.0d0 / n) + (0.5d0 / (n * x))) / x
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (x <= 2.8e+73) {
		tmp = ((1.0 / n) + (0.5 / (n * x))) / x;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 2.8e+73:
		tmp = ((1.0 / n) + (0.5 / (n * x))) / x
	else:
		tmp = 0.0
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 2.8e+73)
		tmp = Float64(Float64(Float64(1.0 / n) + Float64(0.5 / Float64(n * x))) / x);
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (x <= 2.8e+73)
		tmp = ((1.0 / n) + (0.5 / (n * x))) / x;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 2.8e+73], N[(N[(N[(1.0 / n), $MachinePrecision] + N[(0.5 / N[(n * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if x < 2.80000000000000008e73

   1. Initial program 41.3%

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

   3. Taylor expanded in n around inf 49.8%

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

   4. Taylor expanded in x around -inf 12.5%

      \[\leadsto \frac{\color{blue}{-1 \cdot \frac{-1 \cdot \frac{-1 \cdot \frac{0.25 \cdot \frac{1}{x} - 0.3333333333333333}{x} - 0.5}{x} - 1}{x}}}{n} \]
   5. Step-by-step derivation

      1. add-cube-cbrt 12.3%

         \[\leadsto \frac{-1 \cdot \color{blue}{\left(\left(\sqrt[3]{\frac{-1 \cdot \frac{-1 \cdot \frac{0.25 \cdot \frac{1}{x} - 0.3333333333333333}{x} - 0.5}{x} - 1}{x}} \cdot \sqrt[3]{\frac{-1 \cdot \frac{-1 \cdot \frac{0.25 \cdot \frac{1}{x} - 0.3333333333333333}{x} - 0.5}{x} - 1}{x}}\right) \cdot \sqrt[3]{\frac{-1 \cdot \frac{-1 \cdot \frac{0.25 \cdot \frac{1}{x} - 0.3333333333333333}{x} - 0.5}{x} - 1}{x}}\right)}}{n} \]

      2. pow3 12.3%

         \[\leadsto \frac{-1 \cdot \color{blue}{{\left(\sqrt[3]{\frac{-1 \cdot \frac{-1 \cdot \frac{0.25 \cdot \frac{1}{x} - 0.3333333333333333}{x} - 0.5}{x} - 1}{x}}\right)}^{3}}}{n} \]

   6. Applied egg-rr 12.0%

      \[\leadsto \frac{-1 \cdot \color{blue}{{\left(\sqrt[3]{\frac{\frac{\frac{\frac{0.25}{x} + -0.3333333333333333}{x} + -0.5}{x} + -1}{x}}\right)}^{3}}}{n} \]

   7. Taylor expanded in x around inf 34.1%

      \[\leadsto \color{blue}{\frac{\frac{1}{n} + 0.5 \cdot \frac{1}{n \cdot x}}{x}} \]
   8. Step-by-step derivation

      1. associate-*r/ 34.1%

         \[\leadsto \frac{\frac{1}{n} + \color{blue}{\frac{0.5 \cdot 1}{n \cdot x}}}{x} \]

      2. metadata-eval 34.1%

         \[\leadsto \frac{\frac{1}{n} + \frac{\color{blue}{0.5}}{n \cdot x}}{x} \]

      3. *-commutative 34.1%

         \[\leadsto \frac{\frac{1}{n} + \frac{0.5}{\color{blue}{x \cdot n}}}{x} \]

   9. Simplified 34.1%

      \[\leadsto \color{blue}{\frac{\frac{1}{n} + \frac{0.5}{x \cdot n}}{x}} \]

   if 2.80000000000000008e73 < x

   1. Initial program 80.4%

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

   3. Taylor expanded in x around 0 46.0%

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

   4. Taylor expanded in n around inf 80.4%

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

4. Final simplification 45.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.8 \cdot 10^{+73}:\\ \;\;\;\;\frac{\frac{1}{n} + \frac{0.5}{n \cdot x}}{x}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 49.5% accurate, 14.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.12 \cdot 10^{+74}:\\ \;\;\;\;\frac{\frac{-1 + \frac{-0.5}{x}}{x}}{-n}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 1.12e+74) (/ (/ (+ -1.0 (/ -0.5 x)) x) (- n)) 0.0))

C

double code(double x, double n) {
	double tmp;
	if (x <= 1.12e+74) {
		tmp = ((-1.0 + (-0.5 / x)) / x) / -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 <= 1.12d+74) then
        tmp = (((-1.0d0) + ((-0.5d0) / x)) / x) / -n
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (x <= 1.12e+74) {
		tmp = ((-1.0 + (-0.5 / x)) / x) / -n;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 1.12e+74:
		tmp = ((-1.0 + (-0.5 / x)) / x) / -n
	else:
		tmp = 0.0
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 1.12e+74)
		tmp = Float64(Float64(Float64(-1.0 + Float64(-0.5 / x)) / x) / Float64(-n));
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (x <= 1.12e+74)
		tmp = ((-1.0 + (-0.5 / x)) / x) / -n;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 1.12e+74], N[(N[(N[(-1.0 + N[(-0.5 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] / (-n)), $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if x < 1.12000000000000003e74

   1. Initial program 41.3%

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

   3. Taylor expanded in n around inf 49.8%

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

   4. Taylor expanded in x around -inf 12.5%

      \[\leadsto \frac{\color{blue}{-1 \cdot \frac{-1 \cdot \frac{-1 \cdot \frac{0.25 \cdot \frac{1}{x} - 0.3333333333333333}{x} - 0.5}{x} - 1}{x}}}{n} \]
   5. Step-by-step derivation

      1. add-cube-cbrt 12.3%

         \[\leadsto \frac{-1 \cdot \color{blue}{\left(\left(\sqrt[3]{\frac{-1 \cdot \frac{-1 \cdot \frac{0.25 \cdot \frac{1}{x} - 0.3333333333333333}{x} - 0.5}{x} - 1}{x}} \cdot \sqrt[3]{\frac{-1 \cdot \frac{-1 \cdot \frac{0.25 \cdot \frac{1}{x} - 0.3333333333333333}{x} - 0.5}{x} - 1}{x}}\right) \cdot \sqrt[3]{\frac{-1 \cdot \frac{-1 \cdot \frac{0.25 \cdot \frac{1}{x} - 0.3333333333333333}{x} - 0.5}{x} - 1}{x}}\right)}}{n} \]

      2. pow3 12.3%

         \[\leadsto \frac{-1 \cdot \color{blue}{{\left(\sqrt[3]{\frac{-1 \cdot \frac{-1 \cdot \frac{0.25 \cdot \frac{1}{x} - 0.3333333333333333}{x} - 0.5}{x} - 1}{x}}\right)}^{3}}}{n} \]

   6. Applied egg-rr 12.0%

      \[\leadsto \frac{-1 \cdot \color{blue}{{\left(\sqrt[3]{\frac{\frac{\frac{\frac{0.25}{x} + -0.3333333333333333}{x} + -0.5}{x} + -1}{x}}\right)}^{3}}}{n} \]

   7. Taylor expanded in x around inf 34.0%

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

      1. associate-*r/ 34.0%

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

      2. distribute-lft-in 34.0%

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

      3. metadata-eval 34.0%

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

      4. neg-mul-1 34.0%

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

      5. associate-*r/ 34.0%

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

      6. metadata-eval 34.0%

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

      7. distribute-neg-frac 34.0%

         \[\leadsto \frac{-1 \cdot \frac{-1 + \color{blue}{\frac{-0.5}{x}}}{x}}{n} \]

      8. metadata-eval 34.0%

         \[\leadsto \frac{-1 \cdot \frac{-1 + \frac{\color{blue}{-0.5}}{x}}{x}}{n} \]

   9. Simplified 34.0%

      \[\leadsto \frac{-1 \cdot \color{blue}{\frac{-1 + \frac{-0.5}{x}}{x}}}{n} \]

   if 1.12000000000000003e74 < x

   1. Initial program 80.4%

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

   3. Taylor expanded in x around 0 46.0%

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

   4. Taylor expanded in n around inf 80.4%

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

4. Final simplification 45.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.12 \cdot 10^{+74}:\\ \;\;\;\;\frac{\frac{-1 + \frac{-0.5}{x}}{x}}{-n}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 45.2% accurate, 21.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.15 \cdot 10^{+72}:\\ \;\;\;\;\frac{1}{n \cdot x}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n) :precision binary64 (if (<= x 1.15e+72) (/ 1.0 (* n x)) 0.0))

C

double code(double x, double n) {
	double tmp;
	if (x <= 1.15e+72) {
		tmp = 1.0 / (n * x);
	} 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 <= 1.15d+72) then
        tmp = 1.0d0 / (n * x)
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (x <= 1.15e+72) {
		tmp = 1.0 / (n * x);
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 1.15e+72:
		tmp = 1.0 / (n * x)
	else:
		tmp = 0.0
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 1.15e+72)
		tmp = Float64(1.0 / Float64(n * x));
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (x <= 1.15e+72)
		tmp = 1.0 / (n * x);
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 1.15e+72], N[(1.0 / N[(n * x), $MachinePrecision]), $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if x < 1.15e72

   1. Initial program 41.3%

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

   3. Taylor expanded in n around inf 49.8%

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

   4. Taylor expanded in x around inf 29.4%

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

      1. *-commutative 29.4%

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

   6. Simplified 29.4%

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

   if 1.15e72 < x

   1. Initial program 80.4%

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

   3. Taylor expanded in x around 0 46.0%

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

   4. Taylor expanded in n around inf 80.4%

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

4. Final simplification 42.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.15 \cdot 10^{+72}:\\ \;\;\;\;\frac{1}{n \cdot x}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 32.2% accurate, 211.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (x n) :precision binary64 0.0)

C

double code(double x, double n) {
	return 0.0;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    code = 0.0d0
end function

Java

public static double code(double x, double n) {
	return 0.0;
}

Python

def code(x, n):
	return 0.0

Julia

function code(x, n)
	return 0.0
end

MATLAB

function tmp = code(x, n)
	tmp = 0.0;
end

Wolfram

code[x_, n_] := 0.0

Derivation

1. Initial program 51.3%

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

3. Taylor expanded in x around 0 38.7%

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

4. Taylor expanded in n around inf 26.1%

   \[\leadsto 1 - \color{blue}{1} \]

5. Final simplification 26.1%

   \[\leadsto 0 \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024054 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  :precision binary64
  (- (pow (+ x 1.0) (/ 1.0 n)) (pow x (/ 1.0 n))))