2nthrt (problem 3.4.6)

Percentage Accurate: 58.1% → 92.6%

Time: 36.6s

Alternatives: 16

Speedup: 1.9×

• could not determine a ground truth

  1. x = -2.0868734146508698e-13
  2. n = -1.2394100938731442e+191

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: 58.1% 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: 92.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;x \leq -2.3 \cdot 10^{-56}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 1.2 \cdot 10^{-300}:\\ \;\;\;\;e^{\frac{x}{n}} - t\_0\\ \mathbf{elif}\;x \leq 20000:\\ \;\;\;\;\frac{\log \left(\frac{x}{e^{\mathsf{log1p}\left(x\right) + \frac{\mathsf{fma}\left(0.5, {\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}, 0.16666666666666666 \cdot \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{3} - {\log x}^{3}}{n}\right)}{n}}}\right)}{-n}\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \frac{\frac{1}{x}}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= x -2.3e-56)
     0.0
     (if (<= x 1.2e-300)
       (- (exp (/ x n)) t_0)
       (if (<= x 20000.0)
         (/
          (log
           (/
            x
            (exp
             (+
              (log1p x)
              (/
               (fma
                0.5
                (- (pow (log1p x) 2.0) (pow (log x) 2.0))
                (*
                 0.16666666666666666
                 (/ (- (pow (log1p x) 3.0) (pow (log x) 3.0)) n)))
               n)))))
          (- n))
         (* t_0 (/ (/ 1.0 x) n)))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double tmp;
	if (x <= -2.3e-56) {
		tmp = 0.0;
	} else if (x <= 1.2e-300) {
		tmp = exp((x / n)) - t_0;
	} else if (x <= 20000.0) {
		tmp = log((x / exp((log1p(x) + (fma(0.5, (pow(log1p(x), 2.0) - pow(log(x), 2.0)), (0.16666666666666666 * ((pow(log1p(x), 3.0) - pow(log(x), 3.0)) / n))) / n))))) / -n;
	} else {
		tmp = t_0 * ((1.0 / x) / n);
	}
	return tmp;
}

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (x <= -2.3e-56)
		tmp = 0.0;
	elseif (x <= 1.2e-300)
		tmp = Float64(exp(Float64(x / n)) - t_0);
	elseif (x <= 20000.0)
		tmp = Float64(log(Float64(x / exp(Float64(log1p(x) + Float64(fma(0.5, Float64((log1p(x) ^ 2.0) - (log(x) ^ 2.0)), Float64(0.16666666666666666 * Float64(Float64((log1p(x) ^ 3.0) - (log(x) ^ 3.0)) / n))) / n))))) / Float64(-n));
	else
		tmp = Float64(t_0 * Float64(Float64(1.0 / x) / n));
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, -2.3e-56], 0.0, If[LessEqual[x, 1.2e-300], N[(N[Exp[N[(x / n), $MachinePrecision]], $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[x, 20000.0], N[(N[Log[N[(x / N[Exp[N[(N[Log[1 + x], $MachinePrecision] + N[(N[(0.5 * N[(N[Power[N[Log[1 + x], $MachinePrecision], 2.0], $MachinePrecision] - N[Power[N[Log[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + N[(0.16666666666666666 * N[(N[(N[Power[N[Log[1 + x], $MachinePrecision], 3.0], $MachinePrecision] - N[Power[N[Log[x], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-n)), $MachinePrecision], N[(t$95$0 * N[(N[(1.0 / x), $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -2.30000000000000002e-56

   1. Initial program 72.5%

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

   3. Taylor expanded in x around inf 0.0%

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

      1. log-rec 0.0%

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

      2. mul-1-neg 0.0%

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

      3. neg-mul-1 0.0%

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

      4. mul-1-neg 0.0%

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

      5. distribute-frac-neg 0.0%

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

      6. remove-double-neg 0.0%

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

      7. *-rgt-identity 0.0%

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

      8. associate-/l* 0.0%

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

      9. exp-to-pow 88.0%

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

   5. Simplified 88.0%

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

   6. Taylor expanded in x around 0 96.1%

      \[\leadsto \color{blue}{0} \]

   if -2.30000000000000002e-56 < x < 1.2e-300

   1. Initial program 65.5%

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

   3. Taylor expanded in n around 0 15.2%

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

      1. log1p-define 15.2%

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

      2. *-rgt-identity 15.2%

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

      3. associate-*l/ 15.2%

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

      4. associate-/l* 15.2%

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

      5. exp-to-pow 95.2%

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

   5. Simplified 95.2%

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

   6. Taylor expanded in x around 0 95.2%

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

   if 1.2e-300 < x < 2e4

   1. Initial program 42.9%

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

   3. Taylor expanded in n around -inf 79.3%

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

   5. Simplified 79.3%

      \[\leadsto \color{blue}{\frac{\log x - \left(\mathsf{log1p}\left(x\right) + \frac{0.5 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}\right) + \frac{0.16666666666666666 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{3} - {\log x}^{3}\right)}{n}}{n}\right)}{-n}} \]
   6. Step-by-step derivation

      1. add-log-exp 87.1%

         \[\leadsto \frac{\log x - \color{blue}{\log \left(e^{\mathsf{log1p}\left(x\right) + \frac{0.5 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}\right) + \frac{0.16666666666666666 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{3} - {\log x}^{3}\right)}{n}}{n}}\right)}}{-n} \]

      2. diff-log 87.1%

         \[\leadsto \frac{\color{blue}{\log \left(\frac{x}{e^{\mathsf{log1p}\left(x\right) + \frac{0.5 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}\right) + \frac{0.16666666666666666 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{3} - {\log x}^{3}\right)}{n}}{n}}}\right)}}{-n} \]

   7. Applied egg-rr 87.1%

      \[\leadsto \frac{\color{blue}{\log \left(\frac{x}{e^{\mathsf{log1p}\left(x\right) + \frac{\mathsf{fma}\left(0.5, {\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}, 0.16666666666666666 \cdot \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{3} - {\log x}^{3}}{n}\right)}{n}}}\right)}}{-n} \]

   if 2e4 < x

   1. Initial program 69.8%

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

   3. Taylor expanded in x around inf 97.9%

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

      1. log-rec 97.9%

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

      2. mul-1-neg 97.9%

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

      3. neg-mul-1 97.9%

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

      4. mul-1-neg 97.9%

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

      5. distribute-frac-neg 97.9%

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

      6. remove-double-neg 97.9%

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

      7. *-rgt-identity 97.9%

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

      8. associate-/l* 97.9%

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

      9. exp-to-pow 97.9%

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

      10. *-commutative 97.9%

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

   5. Simplified 97.9%

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

      1. div-inv 97.9%

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

   7. Applied egg-rr 97.9%

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

      1. associate-/r* 99.5%

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

      2. *-commutative 99.5%

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

   9. Simplified 99.5%

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

4. Final simplification 93.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.3 \cdot 10^{-56}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 1.2 \cdot 10^{-300}:\\ \;\;\;\;e^{\frac{x}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;x \leq 20000:\\ \;\;\;\;\frac{\log \left(\frac{x}{e^{\mathsf{log1p}\left(x\right) + \frac{\mathsf{fma}\left(0.5, {\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}, 0.16666666666666666 \cdot \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{3} - {\log x}^{3}}{n}\right)}{n}}}\right)}{-n}\\ \mathbf{else}:\\ \;\;\;\;{x}^{\left(\frac{1}{n}\right)} \cdot \frac{\frac{1}{x}}{n}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 88.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;x \leq -1.95 \cdot 10^{-66}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 1.35 \cdot 10^{-300}:\\ \;\;\;\;e^{\frac{x}{n}} - t\_0\\ \mathbf{elif}\;x \leq 33000:\\ \;\;\;\;\frac{\left(\mathsf{log1p}\left(x\right) + \frac{0.5 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}\right) + \frac{0.16666666666666666 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{3} - {\log x}^{3}\right)}{n}}{n}\right) - \log x}{n}\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \frac{\frac{1}{x}}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= x -1.95e-66)
     0.0
     (if (<= x 1.35e-300)
       (- (exp (/ x n)) t_0)
       (if (<= x 33000.0)
         (/
          (-
           (+
            (log1p x)
            (/
             (+
              (* 0.5 (- (pow (log1p x) 2.0) (pow (log x) 2.0)))
              (/
               (*
                0.16666666666666666
                (- (pow (log1p x) 3.0) (pow (log x) 3.0)))
               n))
             n))
           (log x))
          n)
         (* t_0 (/ (/ 1.0 x) n)))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double tmp;
	if (x <= -1.95e-66) {
		tmp = 0.0;
	} else if (x <= 1.35e-300) {
		tmp = exp((x / n)) - t_0;
	} else if (x <= 33000.0) {
		tmp = ((log1p(x) + (((0.5 * (pow(log1p(x), 2.0) - pow(log(x), 2.0))) + ((0.16666666666666666 * (pow(log1p(x), 3.0) - pow(log(x), 3.0))) / n)) / n)) - log(x)) / n;
	} else {
		tmp = t_0 * ((1.0 / x) / n);
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, (1.0 / n));
	double tmp;
	if (x <= -1.95e-66) {
		tmp = 0.0;
	} else if (x <= 1.35e-300) {
		tmp = Math.exp((x / n)) - t_0;
	} else if (x <= 33000.0) {
		tmp = ((Math.log1p(x) + (((0.5 * (Math.pow(Math.log1p(x), 2.0) - Math.pow(Math.log(x), 2.0))) + ((0.16666666666666666 * (Math.pow(Math.log1p(x), 3.0) - Math.pow(Math.log(x), 3.0))) / n)) / n)) - Math.log(x)) / n;
	} else {
		tmp = t_0 * ((1.0 / x) / n);
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	tmp = 0
	if x <= -1.95e-66:
		tmp = 0.0
	elif x <= 1.35e-300:
		tmp = math.exp((x / n)) - t_0
	elif x <= 33000.0:
		tmp = ((math.log1p(x) + (((0.5 * (math.pow(math.log1p(x), 2.0) - math.pow(math.log(x), 2.0))) + ((0.16666666666666666 * (math.pow(math.log1p(x), 3.0) - math.pow(math.log(x), 3.0))) / n)) / n)) - math.log(x)) / n
	else:
		tmp = t_0 * ((1.0 / x) / n)
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (x <= -1.95e-66)
		tmp = 0.0;
	elseif (x <= 1.35e-300)
		tmp = Float64(exp(Float64(x / n)) - t_0);
	elseif (x <= 33000.0)
		tmp = Float64(Float64(Float64(log1p(x) + Float64(Float64(Float64(0.5 * Float64((log1p(x) ^ 2.0) - (log(x) ^ 2.0))) + Float64(Float64(0.16666666666666666 * Float64((log1p(x) ^ 3.0) - (log(x) ^ 3.0))) / n)) / n)) - log(x)) / n);
	else
		tmp = Float64(t_0 * Float64(Float64(1.0 / x) / n));
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, -1.95e-66], 0.0, If[LessEqual[x, 1.35e-300], N[(N[Exp[N[(x / n), $MachinePrecision]], $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[x, 33000.0], N[(N[(N[(N[Log[1 + x], $MachinePrecision] + N[(N[(N[(0.5 * N[(N[Power[N[Log[1 + x], $MachinePrecision], 2.0], $MachinePrecision] - N[Power[N[Log[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(0.16666666666666666 * N[(N[Power[N[Log[1 + x], $MachinePrecision], 3.0], $MachinePrecision] - N[Power[N[Log[x], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision] - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], N[(t$95$0 * N[(N[(1.0 / x), $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -1.94999999999999991e-66

   1. Initial program 72.5%

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

   3. Taylor expanded in x around inf 0.0%

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

      1. log-rec 0.0%

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

      2. mul-1-neg 0.0%

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

      3. neg-mul-1 0.0%

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

      4. mul-1-neg 0.0%

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

      5. distribute-frac-neg 0.0%

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

      6. remove-double-neg 0.0%

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

      7. *-rgt-identity 0.0%

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

      8. associate-/l* 0.0%

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

      9. exp-to-pow 88.0%

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

   5. Simplified 88.0%

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

   6. Taylor expanded in x around 0 96.1%

      \[\leadsto \color{blue}{0} \]

   if -1.94999999999999991e-66 < x < 1.34999999999999998e-300

   1. Initial program 65.5%

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

   3. Taylor expanded in n around 0 15.2%

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

      1. log1p-define 15.2%

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

      2. *-rgt-identity 15.2%

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

      3. associate-*l/ 15.2%

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

      4. associate-/l* 15.2%

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

      5. exp-to-pow 95.2%

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

   5. Simplified 95.2%

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

   6. Taylor expanded in x around 0 95.2%

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

   if 1.34999999999999998e-300 < x < 33000

   1. Initial program 42.9%

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

   3. Taylor expanded in n around -inf 79.3%

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

   5. Simplified 79.3%

      \[\leadsto \color{blue}{\frac{\log x - \left(\mathsf{log1p}\left(x\right) + \frac{0.5 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}\right) + \frac{0.16666666666666666 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{3} - {\log x}^{3}\right)}{n}}{n}\right)}{-n}} \]

   if 33000 < x

   1. Initial program 69.8%

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

   3. Taylor expanded in x around inf 97.9%

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

      1. log-rec 97.9%

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

      2. mul-1-neg 97.9%

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

      3. neg-mul-1 97.9%

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

      4. mul-1-neg 97.9%

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

      5. distribute-frac-neg 97.9%

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

      6. remove-double-neg 97.9%

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

      7. *-rgt-identity 97.9%

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

      8. associate-/l* 97.9%

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

      9. exp-to-pow 97.9%

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

      10. *-commutative 97.9%

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

   5. Simplified 97.9%

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

      1. div-inv 97.9%

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

   7. Applied egg-rr 97.9%

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

      1. associate-/r* 99.5%

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

      2. *-commutative 99.5%

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

   9. Simplified 99.5%

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

4. Final simplification 90.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.95 \cdot 10^{-66}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 1.35 \cdot 10^{-300}:\\ \;\;\;\;e^{\frac{x}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;x \leq 33000:\\ \;\;\;\;\frac{\left(\mathsf{log1p}\left(x\right) + \frac{0.5 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}\right) + \frac{0.16666666666666666 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{3} - {\log x}^{3}\right)}{n}}{n}\right) - \log x}{n}\\ \mathbf{else}:\\ \;\;\;\;{x}^{\left(\frac{1}{n}\right)} \cdot \frac{\frac{1}{x}}{n}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 88.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{-62}:\\ \;\;\;\;t\_0 \cdot \frac{\frac{1}{x}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-22}:\\ \;\;\;\;\frac{\log \left(\frac{x}{x + 1}\right)}{-n}\\ \mathbf{else}:\\ \;\;\;\;{\left({\left(e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - t\_0\right)}^{3}\right)}^{0.3333333333333333}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -1e-62)
     (* t_0 (/ (/ 1.0 x) n))
     (if (<= (/ 1.0 n) 1e-22)
       (/ (log (/ x (+ x 1.0))) (- n))
       (pow (pow (- (exp (/ (log1p x) n)) t_0) 3.0) 0.3333333333333333)))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -1e-62) {
		tmp = t_0 * ((1.0 / x) / n);
	} else if ((1.0 / n) <= 1e-22) {
		tmp = log((x / (x + 1.0))) / -n;
	} else {
		tmp = pow(pow((exp((log1p(x) / n)) - t_0), 3.0), 0.3333333333333333);
	}
	return tmp;
}

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-62) {
		tmp = t_0 * ((1.0 / x) / n);
	} else if ((1.0 / n) <= 1e-22) {
		tmp = Math.log((x / (x + 1.0))) / -n;
	} else {
		tmp = Math.pow(Math.pow((Math.exp((Math.log1p(x) / n)) - t_0), 3.0), 0.3333333333333333);
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	tmp = 0
	if (1.0 / n) <= -1e-62:
		tmp = t_0 * ((1.0 / x) / n)
	elif (1.0 / n) <= 1e-22:
		tmp = math.log((x / (x + 1.0))) / -n
	else:
		tmp = math.pow(math.pow((math.exp((math.log1p(x) / n)) - t_0), 3.0), 0.3333333333333333)
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -1e-62)
		tmp = Float64(t_0 * Float64(Float64(1.0 / x) / n));
	elseif (Float64(1.0 / n) <= 1e-22)
		tmp = Float64(log(Float64(x / Float64(x + 1.0))) / Float64(-n));
	else
		tmp = (Float64(exp(Float64(log1p(x) / n)) - t_0) ^ 3.0) ^ 0.3333333333333333;
	end
	return 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-62], N[(t$95$0 * N[(N[(1.0 / x), $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 1e-22], N[(N[Log[N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-n)), $MachinePrecision], N[Power[N[Power[N[(N[Exp[N[(N[Log[1 + x], $MachinePrecision] / n), $MachinePrecision]], $MachinePrecision] - t$95$0), $MachinePrecision], 3.0], $MachinePrecision], 0.3333333333333333], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -1e-62

   1. Initial program 86.6%

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

   3. Taylor expanded in x around inf 75.3%

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

      1. log-rec 75.3%

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

      2. mul-1-neg 75.3%

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

      3. neg-mul-1 75.3%

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

      4. mul-1-neg 75.3%

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

      5. distribute-frac-neg 75.3%

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

      6. remove-double-neg 75.3%

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

      7. *-rgt-identity 75.3%

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

      8. associate-/l* 75.3%

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

      9. exp-to-pow 94.2%

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

      10. *-commutative 94.2%

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

   5. Simplified 94.2%

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

      1. div-inv 94.2%

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

   7. Applied egg-rr 94.2%

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

      1. associate-/r* 95.0%

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

      2. *-commutative 95.0%

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

   9. Simplified 95.0%

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

   if -1e-62 < (/.f64 #s(literal 1 binary64) n) < 1e-22

   1. Initial program 40.1%

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

   3. Taylor expanded in n around -inf 79.9%

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

   5. Simplified 79.9%

      \[\leadsto \color{blue}{\frac{\log x - \left(\mathsf{log1p}\left(x\right) + \frac{0.5 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}\right) + \frac{0.16666666666666666 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{3} - {\log x}^{3}\right)}{n}}{n}\right)}{-n}} \]
   6. Step-by-step derivation

      1. add-log-exp 79.9%

         \[\leadsto \frac{\log x - \color{blue}{\log \left(e^{\mathsf{log1p}\left(x\right) + \frac{0.5 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}\right) + \frac{0.16666666666666666 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{3} - {\log x}^{3}\right)}{n}}{n}}\right)}}{-n} \]

      2. diff-log 47.9%

         \[\leadsto \frac{\color{blue}{\log \left(\frac{x}{e^{\mathsf{log1p}\left(x\right) + \frac{0.5 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}\right) + \frac{0.16666666666666666 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{3} - {\log x}^{3}\right)}{n}}{n}}}\right)}}{-n} \]

   7. Applied egg-rr 47.9%

      \[\leadsto \frac{\color{blue}{\log \left(\frac{x}{e^{\mathsf{log1p}\left(x\right) + \frac{\mathsf{fma}\left(0.5, {\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}, 0.16666666666666666 \cdot \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{3} - {\log x}^{3}}{n}\right)}{n}}}\right)}}{-n} \]

   8. Taylor expanded in n around inf 80.3%

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

      1. associate-*r/ 80.3%

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

      2. mul-1-neg 80.3%

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

      3. +-commutative 80.3%

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

   10. Simplified 80.3%

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

   if 1e-22 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 50.0%

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

      1. add-cbrt-cube 50.0%

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

      2. pow1/3 50.0%

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

      3. pow3 50.0%

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

      4. pow-to-exp 50.0%

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

      5. un-div-inv 50.0%

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

      6. +-commutative 50.0%

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

      7. log1p-define 91.5%

         \[\leadsto {\left({\left(e^{\frac{\color{blue}{\mathsf{log1p}\left(x\right)}}{n}} - {x}^{\left(\frac{1}{n}\right)}\right)}^{3}\right)}^{0.3333333333333333} \]

   4. Applied egg-rr 91.5%

      \[\leadsto \color{blue}{{\left({\left(e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{1}{n}\right)}\right)}^{3}\right)}^{0.3333333333333333}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 88.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{-62}:\\ \;\;\;\;{x}^{\left(\frac{1}{n}\right)} \cdot \frac{\frac{1}{x}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-22}:\\ \;\;\;\;\frac{\log \left(\frac{x}{x + 1}\right)}{-n}\\ \mathbf{else}:\\ \;\;\;\;{\left({\left(e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{1}{n}\right)}\right)}^{3}\right)}^{0.3333333333333333}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 88.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{-62}:\\ \;\;\;\;t\_0 \cdot \frac{\frac{1}{x}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-22}:\\ \;\;\;\;\frac{\log \left(\frac{x}{x + 1}\right)}{-n}\\ \mathbf{else}:\\ \;\;\;\;e^{\log \left(e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - t\_0\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -1e-62)
     (* t_0 (/ (/ 1.0 x) n))
     (if (<= (/ 1.0 n) 1e-22)
       (/ (log (/ x (+ x 1.0))) (- n))
       (exp (log (- (exp (/ (log1p x) n)) t_0)))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -1e-62) {
		tmp = t_0 * ((1.0 / x) / n);
	} else if ((1.0 / n) <= 1e-22) {
		tmp = log((x / (x + 1.0))) / -n;
	} else {
		tmp = exp(log((exp((log1p(x) / n)) - t_0)));
	}
	return tmp;
}

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-62) {
		tmp = t_0 * ((1.0 / x) / n);
	} else if ((1.0 / n) <= 1e-22) {
		tmp = Math.log((x / (x + 1.0))) / -n;
	} else {
		tmp = Math.exp(Math.log((Math.exp((Math.log1p(x) / n)) - t_0)));
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	tmp = 0
	if (1.0 / n) <= -1e-62:
		tmp = t_0 * ((1.0 / x) / n)
	elif (1.0 / n) <= 1e-22:
		tmp = math.log((x / (x + 1.0))) / -n
	else:
		tmp = math.exp(math.log((math.exp((math.log1p(x) / n)) - t_0)))
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -1e-62)
		tmp = Float64(t_0 * Float64(Float64(1.0 / x) / n));
	elseif (Float64(1.0 / n) <= 1e-22)
		tmp = Float64(log(Float64(x / Float64(x + 1.0))) / Float64(-n));
	else
		tmp = exp(log(Float64(exp(Float64(log1p(x) / n)) - t_0)));
	end
	return 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-62], N[(t$95$0 * N[(N[(1.0 / x), $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 1e-22], N[(N[Log[N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-n)), $MachinePrecision], N[Exp[N[Log[N[(N[Exp[N[(N[Log[1 + x], $MachinePrecision] / n), $MachinePrecision]], $MachinePrecision] - t$95$0), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -1e-62

   1. Initial program 86.6%

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

   3. Taylor expanded in x around inf 75.3%

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

      1. log-rec 75.3%

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

      2. mul-1-neg 75.3%

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

      3. neg-mul-1 75.3%

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

      4. mul-1-neg 75.3%

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

      5. distribute-frac-neg 75.3%

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

      6. remove-double-neg 75.3%

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

      7. *-rgt-identity 75.3%

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

      8. associate-/l* 75.3%

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

      9. exp-to-pow 94.2%

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

      10. *-commutative 94.2%

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

   5. Simplified 94.2%

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

      1. div-inv 94.2%

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

   7. Applied egg-rr 94.2%

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

      1. associate-/r* 95.0%

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

      2. *-commutative 95.0%

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

   9. Simplified 95.0%

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

   if -1e-62 < (/.f64 #s(literal 1 binary64) n) < 1e-22

   1. Initial program 40.1%

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

   3. Taylor expanded in n around -inf 79.9%

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

   5. Simplified 79.9%

      \[\leadsto \color{blue}{\frac{\log x - \left(\mathsf{log1p}\left(x\right) + \frac{0.5 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}\right) + \frac{0.16666666666666666 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{3} - {\log x}^{3}\right)}{n}}{n}\right)}{-n}} \]
   6. Step-by-step derivation

      1. add-log-exp 79.9%

         \[\leadsto \frac{\log x - \color{blue}{\log \left(e^{\mathsf{log1p}\left(x\right) + \frac{0.5 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}\right) + \frac{0.16666666666666666 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{3} - {\log x}^{3}\right)}{n}}{n}}\right)}}{-n} \]

      2. diff-log 47.9%

         \[\leadsto \frac{\color{blue}{\log \left(\frac{x}{e^{\mathsf{log1p}\left(x\right) + \frac{0.5 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}\right) + \frac{0.16666666666666666 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{3} - {\log x}^{3}\right)}{n}}{n}}}\right)}}{-n} \]

   7. Applied egg-rr 47.9%

      \[\leadsto \frac{\color{blue}{\log \left(\frac{x}{e^{\mathsf{log1p}\left(x\right) + \frac{\mathsf{fma}\left(0.5, {\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}, 0.16666666666666666 \cdot \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{3} - {\log x}^{3}}{n}\right)}{n}}}\right)}}{-n} \]

   8. Taylor expanded in n around inf 80.3%

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

      1. associate-*r/ 80.3%

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

      2. mul-1-neg 80.3%

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

      3. +-commutative 80.3%

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

   10. Simplified 80.3%

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

   if 1e-22 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 50.0%

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

      1. add-exp-log 50.0%

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

      2. pow-to-exp 50.0%

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

      3. un-div-inv 50.0%

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

      4. +-commutative 50.0%

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

      5. log1p-define 91.5%

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

   4. Applied egg-rr 91.5%

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

4. Final simplification 88.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{-62}:\\ \;\;\;\;{x}^{\left(\frac{1}{n}\right)} \cdot \frac{\frac{1}{x}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-22}:\\ \;\;\;\;\frac{\log \left(\frac{x}{x + 1}\right)}{-n}\\ \mathbf{else}:\\ \;\;\;\;e^{\log \left(e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{1}{n}\right)}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 88.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{-62}:\\ \;\;\;\;t\_0 \cdot \frac{\frac{1}{x}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-22}:\\ \;\;\;\;\frac{\log \left(\frac{x}{x + 1}\right)}{-n}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -1e-62)
     (* t_0 (/ (/ 1.0 x) n))
     (if (<= (/ 1.0 n) 1e-22)
       (/ (log (/ x (+ x 1.0))) (- n))
       (- (exp (/ (log1p x) n)) t_0)))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -1e-62) {
		tmp = t_0 * ((1.0 / x) / n);
	} else if ((1.0 / n) <= 1e-22) {
		tmp = log((x / (x + 1.0))) / -n;
	} else {
		tmp = exp((log1p(x) / n)) - t_0;
	}
	return tmp;
}

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-62) {
		tmp = t_0 * ((1.0 / x) / n);
	} else if ((1.0 / n) <= 1e-22) {
		tmp = Math.log((x / (x + 1.0))) / -n;
	} else {
		tmp = Math.exp((Math.log1p(x) / n)) - t_0;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	tmp = 0
	if (1.0 / n) <= -1e-62:
		tmp = t_0 * ((1.0 / x) / n)
	elif (1.0 / n) <= 1e-22:
		tmp = math.log((x / (x + 1.0))) / -n
	else:
		tmp = math.exp((math.log1p(x) / n)) - t_0
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -1e-62)
		tmp = Float64(t_0 * Float64(Float64(1.0 / x) / n));
	elseif (Float64(1.0 / n) <= 1e-22)
		tmp = Float64(log(Float64(x / Float64(x + 1.0))) / Float64(-n));
	else
		tmp = Float64(exp(Float64(log1p(x) / n)) - t_0);
	end
	return 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-62], N[(t$95$0 * N[(N[(1.0 / x), $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 1e-22], N[(N[Log[N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-n)), $MachinePrecision], N[(N[Exp[N[(N[Log[1 + x], $MachinePrecision] / n), $MachinePrecision]], $MachinePrecision] - t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -1e-62

   1. Initial program 86.6%

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

   3. Taylor expanded in x around inf 75.3%

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

      1. log-rec 75.3%

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

      2. mul-1-neg 75.3%

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

      3. neg-mul-1 75.3%

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

      4. mul-1-neg 75.3%

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

      5. distribute-frac-neg 75.3%

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

      6. remove-double-neg 75.3%

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

      7. *-rgt-identity 75.3%

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

      8. associate-/l* 75.3%

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

      9. exp-to-pow 94.2%

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

      10. *-commutative 94.2%

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

   5. Simplified 94.2%

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

      1. div-inv 94.2%

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

   7. Applied egg-rr 94.2%

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

      1. associate-/r* 95.0%

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

      2. *-commutative 95.0%

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

   9. Simplified 95.0%

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

   if -1e-62 < (/.f64 #s(literal 1 binary64) n) < 1e-22

   1. Initial program 40.1%

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

   3. Taylor expanded in n around -inf 79.9%

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

   5. Simplified 79.9%

      \[\leadsto \color{blue}{\frac{\log x - \left(\mathsf{log1p}\left(x\right) + \frac{0.5 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}\right) + \frac{0.16666666666666666 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{3} - {\log x}^{3}\right)}{n}}{n}\right)}{-n}} \]
   6. Step-by-step derivation

      1. add-log-exp 79.9%

         \[\leadsto \frac{\log x - \color{blue}{\log \left(e^{\mathsf{log1p}\left(x\right) + \frac{0.5 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}\right) + \frac{0.16666666666666666 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{3} - {\log x}^{3}\right)}{n}}{n}}\right)}}{-n} \]

      2. diff-log 47.9%

         \[\leadsto \frac{\color{blue}{\log \left(\frac{x}{e^{\mathsf{log1p}\left(x\right) + \frac{0.5 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}\right) + \frac{0.16666666666666666 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{3} - {\log x}^{3}\right)}{n}}{n}}}\right)}}{-n} \]

   7. Applied egg-rr 47.9%

      \[\leadsto \frac{\color{blue}{\log \left(\frac{x}{e^{\mathsf{log1p}\left(x\right) + \frac{\mathsf{fma}\left(0.5, {\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}, 0.16666666666666666 \cdot \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{3} - {\log x}^{3}}{n}\right)}{n}}}\right)}}{-n} \]

   8. Taylor expanded in n around inf 80.3%

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

      1. associate-*r/ 80.3%

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

      2. mul-1-neg 80.3%

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

      3. +-commutative 80.3%

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

   10. Simplified 80.3%

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

   if 1e-22 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 50.0%

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

   3. Taylor expanded in n around 0 31.7%

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

      1. log1p-define 57.1%

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

      2. *-rgt-identity 57.1%

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

      3. associate-*l/ 57.1%

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

      4. associate-/l* 57.1%

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

      5. exp-to-pow 91.5%

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

   5. Simplified 91.5%

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

4. Final simplification 88.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{-62}:\\ \;\;\;\;{x}^{\left(\frac{1}{n}\right)} \cdot \frac{\frac{1}{x}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-22}:\\ \;\;\;\;\frac{\log \left(\frac{x}{x + 1}\right)}{-n}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 79.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{-62}:\\ \;\;\;\;t\_0 \cdot \frac{\frac{1}{x}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-12}:\\ \;\;\;\;\frac{\log \left(\frac{x}{x + 1}\right)}{-n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{+179}:\\ \;\;\;\;\left(\frac{x}{n} + 1\right) - t\_0\\ \mathbf{else}:\\ \;\;\;\;\log \left({\left(\frac{1}{x}\right)}^{\left(\frac{1}{n}\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -1e-62)
     (* t_0 (/ (/ 1.0 x) n))
     (if (<= (/ 1.0 n) 2e-12)
       (/ (log (/ x (+ x 1.0))) (- n))
       (if (<= (/ 1.0 n) 5e+179)
         (- (+ (/ x n) 1.0) t_0)
         (log (pow (/ 1.0 x) (/ 1.0 n))))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -1e-62) {
		tmp = t_0 * ((1.0 / x) / n);
	} else if ((1.0 / n) <= 2e-12) {
		tmp = log((x / (x + 1.0))) / -n;
	} else if ((1.0 / n) <= 5e+179) {
		tmp = ((x / n) + 1.0) - t_0;
	} else {
		tmp = log(pow((1.0 / x), (1.0 / n)));
	}
	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-62)) then
        tmp = t_0 * ((1.0d0 / x) / n)
    else if ((1.0d0 / n) <= 2d-12) then
        tmp = log((x / (x + 1.0d0))) / -n
    else if ((1.0d0 / n) <= 5d+179) then
        tmp = ((x / n) + 1.0d0) - t_0
    else
        tmp = log(((1.0d0 / x) ** (1.0d0 / n)))
    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-62) {
		tmp = t_0 * ((1.0 / x) / n);
	} else if ((1.0 / n) <= 2e-12) {
		tmp = Math.log((x / (x + 1.0))) / -n;
	} else if ((1.0 / n) <= 5e+179) {
		tmp = ((x / n) + 1.0) - t_0;
	} else {
		tmp = Math.log(Math.pow((1.0 / x), (1.0 / n)));
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	tmp = 0
	if (1.0 / n) <= -1e-62:
		tmp = t_0 * ((1.0 / x) / n)
	elif (1.0 / n) <= 2e-12:
		tmp = math.log((x / (x + 1.0))) / -n
	elif (1.0 / n) <= 5e+179:
		tmp = ((x / n) + 1.0) - t_0
	else:
		tmp = math.log(math.pow((1.0 / x), (1.0 / n)))
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -1e-62)
		tmp = Float64(t_0 * Float64(Float64(1.0 / x) / n));
	elseif (Float64(1.0 / n) <= 2e-12)
		tmp = Float64(log(Float64(x / Float64(x + 1.0))) / Float64(-n));
	elseif (Float64(1.0 / n) <= 5e+179)
		tmp = Float64(Float64(Float64(x / n) + 1.0) - t_0);
	else
		tmp = log((Float64(1.0 / x) ^ Float64(1.0 / n)));
	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-62)
		tmp = t_0 * ((1.0 / x) / n);
	elseif ((1.0 / n) <= 2e-12)
		tmp = log((x / (x + 1.0))) / -n;
	elseif ((1.0 / n) <= 5e+179)
		tmp = ((x / n) + 1.0) - t_0;
	else
		tmp = log(((1.0 / x) ^ (1.0 / n)));
	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-62], N[(t$95$0 * N[(N[(1.0 / x), $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-12], N[(N[Log[N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-n)), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e+179], N[(N[(N[(x / n), $MachinePrecision] + 1.0), $MachinePrecision] - t$95$0), $MachinePrecision], N[Log[N[Power[N[(1.0 / x), $MachinePrecision], N[(1.0 / n), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -1e-62

   1. Initial program 86.6%

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

   3. Taylor expanded in x around inf 75.3%

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

      1. log-rec 75.3%

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

      2. mul-1-neg 75.3%

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

      3. neg-mul-1 75.3%

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

      4. mul-1-neg 75.3%

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

      5. distribute-frac-neg 75.3%

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

      6. remove-double-neg 75.3%

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

      7. *-rgt-identity 75.3%

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

      8. associate-/l* 75.3%

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

      9. exp-to-pow 94.2%

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

      10. *-commutative 94.2%

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

   5. Simplified 94.2%

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

      1. div-inv 94.2%

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

   7. Applied egg-rr 94.2%

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

      1. associate-/r* 95.0%

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

      2. *-commutative 95.0%

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

   9. Simplified 95.0%

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

   if -1e-62 < (/.f64 #s(literal 1 binary64) n) < 1.99999999999999996e-12

   1. Initial program 39.4%

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

   3. Taylor expanded in n around -inf 78.5%

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

   5. Simplified 78.5%

      \[\leadsto \color{blue}{\frac{\log x - \left(\mathsf{log1p}\left(x\right) + \frac{0.5 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}\right) + \frac{0.16666666666666666 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{3} - {\log x}^{3}\right)}{n}}{n}\right)}{-n}} \]
   6. Step-by-step derivation

      1. add-log-exp 78.5%

         \[\leadsto \frac{\log x - \color{blue}{\log \left(e^{\mathsf{log1p}\left(x\right) + \frac{0.5 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}\right) + \frac{0.16666666666666666 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{3} - {\log x}^{3}\right)}{n}}{n}}\right)}}{-n} \]

      2. diff-log 47.1%

         \[\leadsto \frac{\color{blue}{\log \left(\frac{x}{e^{\mathsf{log1p}\left(x\right) + \frac{0.5 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}\right) + \frac{0.16666666666666666 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{3} - {\log x}^{3}\right)}{n}}{n}}}\right)}}{-n} \]

   7. Applied egg-rr 47.1%

      \[\leadsto \frac{\color{blue}{\log \left(\frac{x}{e^{\mathsf{log1p}\left(x\right) + \frac{\mathsf{fma}\left(0.5, {\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}, 0.16666666666666666 \cdot \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{3} - {\log x}^{3}}{n}\right)}{n}}}\right)}}{-n} \]

   8. Taylor expanded in n around inf 78.8%

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

      1. associate-*r/ 78.8%

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

      2. mul-1-neg 78.8%

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

      3. +-commutative 78.8%

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

   10. Simplified 78.8%

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

   if 1.99999999999999996e-12 < (/.f64 #s(literal 1 binary64) n) < 5e179

   1. Initial program 72.9%

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

   3. Taylor expanded in x around 0 67.6%

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

   if 5e179 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 22.5%

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

   3. Taylor expanded in x around 0 5.6%

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

      1. *-rgt-identity 5.6%

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

      2. associate-*l/ 5.6%

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

      3. associate-/l* 5.6%

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

      4. exp-to-pow 18.6%

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

   5. Simplified 18.6%

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

   6. Taylor expanded in n around inf 8.0%

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

      1. associate-*r/ 8.0%

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

      2. neg-mul-1 8.0%

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

   8. Simplified 8.0%

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

      1. add-log-exp 52.1%

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

      2. div-inv 52.1%

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

      3. neg-log 52.1%

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

      4. exp-to-pow 53.0%

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

   10. Applied egg-rr 53.0%

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

4. Final simplification 80.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{-62}:\\ \;\;\;\;{x}^{\left(\frac{1}{n}\right)} \cdot \frac{\frac{1}{x}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-12}:\\ \;\;\;\;\frac{\log \left(\frac{x}{x + 1}\right)}{-n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{+179}:\\ \;\;\;\;\left(\frac{x}{n} + 1\right) - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\log \left({\left(\frac{1}{x}\right)}^{\left(\frac{1}{n}\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 88.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{-62}:\\ \;\;\;\;t\_0 \cdot \frac{\frac{1}{x}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-12}:\\ \;\;\;\;\frac{\log \left(\frac{x}{x + 1}\right)}{-n}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{x}{n}} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -1e-62)
     (* t_0 (/ (/ 1.0 x) n))
     (if (<= (/ 1.0 n) 2e-12)
       (/ (log (/ x (+ x 1.0))) (- n))
       (- (exp (/ x n)) t_0)))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -1e-62) {
		tmp = t_0 * ((1.0 / x) / n);
	} else if ((1.0 / n) <= 2e-12) {
		tmp = log((x / (x + 1.0))) / -n;
	} else {
		tmp = exp((x / n)) - t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x ** (1.0d0 / n)
    if ((1.0d0 / n) <= (-1d-62)) then
        tmp = t_0 * ((1.0d0 / x) / n)
    else if ((1.0d0 / n) <= 2d-12) then
        tmp = log((x / (x + 1.0d0))) / -n
    else
        tmp = exp((x / n)) - t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -1e-62) {
		tmp = t_0 * ((1.0 / x) / n);
	} else if ((1.0 / n) <= 2e-12) {
		tmp = Math.log((x / (x + 1.0))) / -n;
	} else {
		tmp = Math.exp((x / n)) - t_0;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	tmp = 0
	if (1.0 / n) <= -1e-62:
		tmp = t_0 * ((1.0 / x) / n)
	elif (1.0 / n) <= 2e-12:
		tmp = math.log((x / (x + 1.0))) / -n
	else:
		tmp = math.exp((x / n)) - t_0
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -1e-62)
		tmp = Float64(t_0 * Float64(Float64(1.0 / x) / n));
	elseif (Float64(1.0 / n) <= 2e-12)
		tmp = Float64(log(Float64(x / Float64(x + 1.0))) / Float64(-n));
	else
		tmp = Float64(exp(Float64(x / n)) - t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = x ^ (1.0 / n);
	tmp = 0.0;
	if ((1.0 / n) <= -1e-62)
		tmp = t_0 * ((1.0 / x) / n);
	elseif ((1.0 / n) <= 2e-12)
		tmp = log((x / (x + 1.0))) / -n;
	else
		tmp = exp((x / n)) - t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -1e-62], N[(t$95$0 * N[(N[(1.0 / x), $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-12], N[(N[Log[N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-n)), $MachinePrecision], N[(N[Exp[N[(x / n), $MachinePrecision]], $MachinePrecision] - t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -1e-62

   1. Initial program 86.6%

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

   3. Taylor expanded in x around inf 75.3%

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

      1. log-rec 75.3%

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

      2. mul-1-neg 75.3%

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

      3. neg-mul-1 75.3%

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

      4. mul-1-neg 75.3%

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

      5. distribute-frac-neg 75.3%

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

      6. remove-double-neg 75.3%

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

      7. *-rgt-identity 75.3%

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

      8. associate-/l* 75.3%

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

      9. exp-to-pow 94.2%

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

      10. *-commutative 94.2%

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

   5. Simplified 94.2%

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

      1. div-inv 94.2%

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

   7. Applied egg-rr 94.2%

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

      1. associate-/r* 95.0%

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

      2. *-commutative 95.0%

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

   9. Simplified 95.0%

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

   if -1e-62 < (/.f64 #s(literal 1 binary64) n) < 1.99999999999999996e-12

   1. Initial program 39.4%

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

   3. Taylor expanded in n around -inf 78.5%

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

   5. Simplified 78.5%

      \[\leadsto \color{blue}{\frac{\log x - \left(\mathsf{log1p}\left(x\right) + \frac{0.5 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}\right) + \frac{0.16666666666666666 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{3} - {\log x}^{3}\right)}{n}}{n}\right)}{-n}} \]
   6. Step-by-step derivation

      1. add-log-exp 78.5%

         \[\leadsto \frac{\log x - \color{blue}{\log \left(e^{\mathsf{log1p}\left(x\right) + \frac{0.5 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}\right) + \frac{0.16666666666666666 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{3} - {\log x}^{3}\right)}{n}}{n}}\right)}}{-n} \]

      2. diff-log 47.1%

         \[\leadsto \frac{\color{blue}{\log \left(\frac{x}{e^{\mathsf{log1p}\left(x\right) + \frac{0.5 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}\right) + \frac{0.16666666666666666 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{3} - {\log x}^{3}\right)}{n}}{n}}}\right)}}{-n} \]

   7. Applied egg-rr 47.1%

      \[\leadsto \frac{\color{blue}{\log \left(\frac{x}{e^{\mathsf{log1p}\left(x\right) + \frac{\mathsf{fma}\left(0.5, {\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}, 0.16666666666666666 \cdot \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{3} - {\log x}^{3}}{n}\right)}{n}}}\right)}}{-n} \]

   8. Taylor expanded in n around inf 78.8%

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

      1. associate-*r/ 78.8%

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

      2. mul-1-neg 78.8%

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

      3. +-commutative 78.8%

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

   10. Simplified 78.8%

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

   if 1.99999999999999996e-12 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 51.5%

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

   3. Taylor expanded in n around 0 32.6%

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

      1. log1p-define 58.9%

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

      2. *-rgt-identity 58.9%

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

      3. associate-*l/ 58.9%

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

      4. associate-/l* 58.9%

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

      5. exp-to-pow 94.5%

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

   5. Simplified 94.5%

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

   6. Taylor expanded in x around 0 94.4%

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

4. Final simplification 88.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{-62}:\\ \;\;\;\;{x}^{\left(\frac{1}{n}\right)} \cdot \frac{\frac{1}{x}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-12}:\\ \;\;\;\;\frac{\log \left(\frac{x}{x + 1}\right)}{-n}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{x}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 78.9% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{-62}:\\ \;\;\;\;t\_0 \cdot \frac{\frac{1}{x}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-12}:\\ \;\;\;\;\frac{\log \left(\frac{x}{x + 1}\right)}{-n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{+267}:\\ \;\;\;\;\left(\frac{x}{n} + 1\right) - t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x \cdot n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -1e-62)
     (* t_0 (/ (/ 1.0 x) n))
     (if (<= (/ 1.0 n) 2e-12)
       (/ (log (/ x (+ x 1.0))) (- n))
       (if (<= (/ 1.0 n) 2e+267) (- (+ (/ x n) 1.0) t_0) (/ 1.0 (* x n)))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -1e-62) {
		tmp = t_0 * ((1.0 / x) / n);
	} else if ((1.0 / n) <= 2e-12) {
		tmp = log((x / (x + 1.0))) / -n;
	} else if ((1.0 / n) <= 2e+267) {
		tmp = ((x / n) + 1.0) - t_0;
	} else {
		tmp = 1.0 / (x * n);
	}
	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-62)) then
        tmp = t_0 * ((1.0d0 / x) / n)
    else if ((1.0d0 / n) <= 2d-12) then
        tmp = log((x / (x + 1.0d0))) / -n
    else if ((1.0d0 / n) <= 2d+267) then
        tmp = ((x / n) + 1.0d0) - t_0
    else
        tmp = 1.0d0 / (x * n)
    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-62) {
		tmp = t_0 * ((1.0 / x) / n);
	} else if ((1.0 / n) <= 2e-12) {
		tmp = Math.log((x / (x + 1.0))) / -n;
	} else if ((1.0 / n) <= 2e+267) {
		tmp = ((x / n) + 1.0) - t_0;
	} else {
		tmp = 1.0 / (x * n);
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	tmp = 0
	if (1.0 / n) <= -1e-62:
		tmp = t_0 * ((1.0 / x) / n)
	elif (1.0 / n) <= 2e-12:
		tmp = math.log((x / (x + 1.0))) / -n
	elif (1.0 / n) <= 2e+267:
		tmp = ((x / n) + 1.0) - t_0
	else:
		tmp = 1.0 / (x * n)
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -1e-62)
		tmp = Float64(t_0 * Float64(Float64(1.0 / x) / n));
	elseif (Float64(1.0 / n) <= 2e-12)
		tmp = Float64(log(Float64(x / Float64(x + 1.0))) / Float64(-n));
	elseif (Float64(1.0 / n) <= 2e+267)
		tmp = Float64(Float64(Float64(x / n) + 1.0) - t_0);
	else
		tmp = Float64(1.0 / Float64(x * n));
	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-62)
		tmp = t_0 * ((1.0 / x) / n);
	elseif ((1.0 / n) <= 2e-12)
		tmp = log((x / (x + 1.0))) / -n;
	elseif ((1.0 / n) <= 2e+267)
		tmp = ((x / n) + 1.0) - t_0;
	else
		tmp = 1.0 / (x * n);
	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-62], N[(t$95$0 * N[(N[(1.0 / x), $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-12], N[(N[Log[N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-n)), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e+267], N[(N[(N[(x / n), $MachinePrecision] + 1.0), $MachinePrecision] - t$95$0), $MachinePrecision], N[(1.0 / N[(x * n), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -1e-62

   1. Initial program 86.6%

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

   3. Taylor expanded in x around inf 75.3%

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

      1. log-rec 75.3%

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

      2. mul-1-neg 75.3%

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

      3. neg-mul-1 75.3%

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

      4. mul-1-neg 75.3%

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

      5. distribute-frac-neg 75.3%

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

      6. remove-double-neg 75.3%

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

      7. *-rgt-identity 75.3%

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

      8. associate-/l* 75.3%

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

      9. exp-to-pow 94.2%

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

      10. *-commutative 94.2%

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

   5. Simplified 94.2%

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

      1. div-inv 94.2%

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

   7. Applied egg-rr 94.2%

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

      1. associate-/r* 95.0%

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

      2. *-commutative 95.0%

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

   9. Simplified 95.0%

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

   if -1e-62 < (/.f64 #s(literal 1 binary64) n) < 1.99999999999999996e-12

   1. Initial program 39.4%

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

   3. Taylor expanded in n around -inf 78.5%

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

   5. Simplified 78.5%

      \[\leadsto \color{blue}{\frac{\log x - \left(\mathsf{log1p}\left(x\right) + \frac{0.5 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}\right) + \frac{0.16666666666666666 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{3} - {\log x}^{3}\right)}{n}}{n}\right)}{-n}} \]
   6. Step-by-step derivation

      1. add-log-exp 78.5%

         \[\leadsto \frac{\log x - \color{blue}{\log \left(e^{\mathsf{log1p}\left(x\right) + \frac{0.5 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}\right) + \frac{0.16666666666666666 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{3} - {\log x}^{3}\right)}{n}}{n}}\right)}}{-n} \]

      2. diff-log 47.1%

         \[\leadsto \frac{\color{blue}{\log \left(\frac{x}{e^{\mathsf{log1p}\left(x\right) + \frac{0.5 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}\right) + \frac{0.16666666666666666 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{3} - {\log x}^{3}\right)}{n}}{n}}}\right)}}{-n} \]

   7. Applied egg-rr 47.1%

      \[\leadsto \frac{\color{blue}{\log \left(\frac{x}{e^{\mathsf{log1p}\left(x\right) + \frac{\mathsf{fma}\left(0.5, {\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}, 0.16666666666666666 \cdot \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{3} - {\log x}^{3}}{n}\right)}{n}}}\right)}}{-n} \]

   8. Taylor expanded in n around inf 78.8%

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

      1. associate-*r/ 78.8%

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

      2. mul-1-neg 78.8%

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

      3. +-commutative 78.8%

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

   10. Simplified 78.8%

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

   if 1.99999999999999996e-12 < (/.f64 #s(literal 1 binary64) n) < 1.9999999999999999e267

   1. Initial program 62.6%

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

   3. Taylor expanded in x around 0 57.0%

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

   if 1.9999999999999999e267 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 3.1%

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

   3. Taylor expanded in x around inf 0.0%

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

      1. log-rec 0.0%

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

      2. mul-1-neg 0.0%

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

      3. neg-mul-1 0.0%

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

      4. mul-1-neg 0.0%

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

      5. distribute-frac-neg 0.0%

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

      6. remove-double-neg 0.0%

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

      7. *-rgt-identity 0.0%

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

      8. associate-/l* 0.0%

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

      9. exp-to-pow 0.0%

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

      10. *-commutative 0.0%

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

   5. Simplified 0.0%

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

   6. Taylor expanded in n around inf 73.2%

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

4. Final simplification 80.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{-62}:\\ \;\;\;\;{x}^{\left(\frac{1}{n}\right)} \cdot \frac{\frac{1}{x}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-12}:\\ \;\;\;\;\frac{\log \left(\frac{x}{x + 1}\right)}{-n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{+267}:\\ \;\;\;\;\left(\frac{x}{n} + 1\right) - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x \cdot n}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 78.7% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{-62}:\\ \;\;\;\;t\_0 \cdot \frac{\frac{1}{x}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-12}:\\ \;\;\;\;\frac{\log \left(\frac{x}{x + 1}\right)}{-n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{+267}:\\ \;\;\;\;1 - t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x \cdot n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -1e-62)
     (* t_0 (/ (/ 1.0 x) n))
     (if (<= (/ 1.0 n) 2e-12)
       (/ (log (/ x (+ x 1.0))) (- n))
       (if (<= (/ 1.0 n) 2e+267) (- 1.0 t_0) (/ 1.0 (* x n)))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -1e-62) {
		tmp = t_0 * ((1.0 / x) / n);
	} else if ((1.0 / n) <= 2e-12) {
		tmp = log((x / (x + 1.0))) / -n;
	} else if ((1.0 / n) <= 2e+267) {
		tmp = 1.0 - t_0;
	} else {
		tmp = 1.0 / (x * n);
	}
	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-62)) then
        tmp = t_0 * ((1.0d0 / x) / n)
    else if ((1.0d0 / n) <= 2d-12) then
        tmp = log((x / (x + 1.0d0))) / -n
    else if ((1.0d0 / n) <= 2d+267) then
        tmp = 1.0d0 - t_0
    else
        tmp = 1.0d0 / (x * n)
    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-62) {
		tmp = t_0 * ((1.0 / x) / n);
	} else if ((1.0 / n) <= 2e-12) {
		tmp = Math.log((x / (x + 1.0))) / -n;
	} else if ((1.0 / n) <= 2e+267) {
		tmp = 1.0 - t_0;
	} else {
		tmp = 1.0 / (x * n);
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	tmp = 0
	if (1.0 / n) <= -1e-62:
		tmp = t_0 * ((1.0 / x) / n)
	elif (1.0 / n) <= 2e-12:
		tmp = math.log((x / (x + 1.0))) / -n
	elif (1.0 / n) <= 2e+267:
		tmp = 1.0 - t_0
	else:
		tmp = 1.0 / (x * n)
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -1e-62)
		tmp = Float64(t_0 * Float64(Float64(1.0 / x) / n));
	elseif (Float64(1.0 / n) <= 2e-12)
		tmp = Float64(log(Float64(x / Float64(x + 1.0))) / Float64(-n));
	elseif (Float64(1.0 / n) <= 2e+267)
		tmp = Float64(1.0 - t_0);
	else
		tmp = Float64(1.0 / Float64(x * n));
	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-62)
		tmp = t_0 * ((1.0 / x) / n);
	elseif ((1.0 / n) <= 2e-12)
		tmp = log((x / (x + 1.0))) / -n;
	elseif ((1.0 / n) <= 2e+267)
		tmp = 1.0 - t_0;
	else
		tmp = 1.0 / (x * n);
	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-62], N[(t$95$0 * N[(N[(1.0 / x), $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-12], N[(N[Log[N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-n)), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e+267], N[(1.0 - t$95$0), $MachinePrecision], N[(1.0 / N[(x * n), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -1e-62

   1. Initial program 86.6%

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

   3. Taylor expanded in x around inf 75.3%

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

      1. log-rec 75.3%

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

      2. mul-1-neg 75.3%

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

      3. neg-mul-1 75.3%

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

      4. mul-1-neg 75.3%

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

      5. distribute-frac-neg 75.3%

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

      6. remove-double-neg 75.3%

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

      7. *-rgt-identity 75.3%

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

      8. associate-/l* 75.3%

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

      9. exp-to-pow 94.2%

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

      10. *-commutative 94.2%

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

   5. Simplified 94.2%

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

      1. div-inv 94.2%

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

   7. Applied egg-rr 94.2%

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

      1. associate-/r* 95.0%

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

      2. *-commutative 95.0%

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

   9. Simplified 95.0%

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

   if -1e-62 < (/.f64 #s(literal 1 binary64) n) < 1.99999999999999996e-12

   1. Initial program 39.4%

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

   3. Taylor expanded in n around -inf 78.5%

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

   5. Simplified 78.5%

      \[\leadsto \color{blue}{\frac{\log x - \left(\mathsf{log1p}\left(x\right) + \frac{0.5 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}\right) + \frac{0.16666666666666666 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{3} - {\log x}^{3}\right)}{n}}{n}\right)}{-n}} \]
   6. Step-by-step derivation

      1. add-log-exp 78.5%

         \[\leadsto \frac{\log x - \color{blue}{\log \left(e^{\mathsf{log1p}\left(x\right) + \frac{0.5 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}\right) + \frac{0.16666666666666666 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{3} - {\log x}^{3}\right)}{n}}{n}}\right)}}{-n} \]

      2. diff-log 47.1%

         \[\leadsto \frac{\color{blue}{\log \left(\frac{x}{e^{\mathsf{log1p}\left(x\right) + \frac{0.5 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}\right) + \frac{0.16666666666666666 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{3} - {\log x}^{3}\right)}{n}}{n}}}\right)}}{-n} \]

   7. Applied egg-rr 47.1%

      \[\leadsto \frac{\color{blue}{\log \left(\frac{x}{e^{\mathsf{log1p}\left(x\right) + \frac{\mathsf{fma}\left(0.5, {\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}, 0.16666666666666666 \cdot \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{3} - {\log x}^{3}}{n}\right)}{n}}}\right)}}{-n} \]

   8. Taylor expanded in n around inf 78.8%

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

      1. associate-*r/ 78.8%

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

      2. mul-1-neg 78.8%

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

      3. +-commutative 78.8%

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

   10. Simplified 78.8%

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

   if 1.99999999999999996e-12 < (/.f64 #s(literal 1 binary64) n) < 1.9999999999999999e267

   1. Initial program 62.6%

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

   3. Taylor expanded in x around 0 35.4%

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

      1. *-rgt-identity 35.4%

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

      2. associate-*l/ 35.4%

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

      3. associate-/l* 35.4%

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

      4. exp-to-pow 56.5%

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

   5. Simplified 56.5%

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

   if 1.9999999999999999e267 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 3.1%

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

   3. Taylor expanded in x around inf 0.0%

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

      1. log-rec 0.0%

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

      2. mul-1-neg 0.0%

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

      3. neg-mul-1 0.0%

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

      4. mul-1-neg 0.0%

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

      5. distribute-frac-neg 0.0%

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

      6. remove-double-neg 0.0%

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

      7. *-rgt-identity 0.0%

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

      8. associate-/l* 0.0%

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

      9. exp-to-pow 0.0%

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

      10. *-commutative 0.0%

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

   5. Simplified 0.0%

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

   6. Taylor expanded in n around inf 73.2%

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

4. Final simplification 80.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{-62}:\\ \;\;\;\;{x}^{\left(\frac{1}{n}\right)} \cdot \frac{\frac{1}{x}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-12}:\\ \;\;\;\;\frac{\log \left(\frac{x}{x + 1}\right)}{-n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{+267}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x \cdot n}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 78.7% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{-62}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n} + -1\right)}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-12}:\\ \;\;\;\;\frac{\log \left(\frac{x}{x + 1}\right)}{-n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{+267}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x \cdot n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= (/ 1.0 n) -1e-62)
   (/ (pow x (+ (/ 1.0 n) -1.0)) n)
   (if (<= (/ 1.0 n) 2e-12)
     (/ (log (/ x (+ x 1.0))) (- n))
     (if (<= (/ 1.0 n) 2e+267) (- 1.0 (pow x (/ 1.0 n))) (/ 1.0 (* x n))))))

C

double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -1e-62) {
		tmp = pow(x, ((1.0 / n) + -1.0)) / n;
	} else if ((1.0 / n) <= 2e-12) {
		tmp = log((x / (x + 1.0))) / -n;
	} else if ((1.0 / n) <= 2e+267) {
		tmp = 1.0 - pow(x, (1.0 / n));
	} else {
		tmp = 1.0 / (x * n);
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if ((1.0d0 / n) <= (-1d-62)) then
        tmp = (x ** ((1.0d0 / n) + (-1.0d0))) / n
    else if ((1.0d0 / n) <= 2d-12) then
        tmp = log((x / (x + 1.0d0))) / -n
    else if ((1.0d0 / n) <= 2d+267) then
        tmp = 1.0d0 - (x ** (1.0d0 / n))
    else
        tmp = 1.0d0 / (x * n)
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -1e-62) {
		tmp = Math.pow(x, ((1.0 / n) + -1.0)) / n;
	} else if ((1.0 / n) <= 2e-12) {
		tmp = Math.log((x / (x + 1.0))) / -n;
	} else if ((1.0 / n) <= 2e+267) {
		tmp = 1.0 - Math.pow(x, (1.0 / n));
	} else {
		tmp = 1.0 / (x * n);
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if (1.0 / n) <= -1e-62:
		tmp = math.pow(x, ((1.0 / n) + -1.0)) / n
	elif (1.0 / n) <= 2e-12:
		tmp = math.log((x / (x + 1.0))) / -n
	elif (1.0 / n) <= 2e+267:
		tmp = 1.0 - math.pow(x, (1.0 / n))
	else:
		tmp = 1.0 / (x * n)
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -1e-62)
		tmp = Float64((x ^ Float64(Float64(1.0 / n) + -1.0)) / n);
	elseif (Float64(1.0 / n) <= 2e-12)
		tmp = Float64(log(Float64(x / Float64(x + 1.0))) / Float64(-n));
	elseif (Float64(1.0 / n) <= 2e+267)
		tmp = Float64(1.0 - (x ^ Float64(1.0 / n)));
	else
		tmp = Float64(1.0 / Float64(x * n));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if ((1.0 / n) <= -1e-62)
		tmp = (x ^ ((1.0 / n) + -1.0)) / n;
	elseif ((1.0 / n) <= 2e-12)
		tmp = log((x / (x + 1.0))) / -n;
	elseif ((1.0 / n) <= 2e+267)
		tmp = 1.0 - (x ^ (1.0 / n));
	else
		tmp = 1.0 / (x * n);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[N[(1.0 / n), $MachinePrecision], -1e-62], N[(N[Power[x, N[(N[(1.0 / n), $MachinePrecision] + -1.0), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-12], N[(N[Log[N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-n)), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e+267], N[(1.0 - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(x * n), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -1e-62

   1. Initial program 86.6%

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

   3. Taylor expanded in x around inf 75.3%

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

      1. log-rec 75.3%

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

      2. mul-1-neg 75.3%

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

      3. neg-mul-1 75.3%

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

      4. mul-1-neg 75.3%

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

      5. distribute-frac-neg 75.3%

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

      6. remove-double-neg 75.3%

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

      7. *-rgt-identity 75.3%

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

      8. associate-/l* 75.3%

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

      9. exp-to-pow 94.2%

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

      10. *-commutative 94.2%

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

   5. Simplified 94.2%

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

      1. *-un-lft-identity 94.2%

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

      2. associate-/r* 95.0%

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

      3. pow1 95.0%

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

      4. pow-div 94.9%

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

   7. Applied egg-rr 94.9%

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

      1. *-lft-identity 94.9%

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

      2. sub-neg 94.9%

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

      3. metadata-eval 94.9%

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

   9. Simplified 94.9%

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

   if -1e-62 < (/.f64 #s(literal 1 binary64) n) < 1.99999999999999996e-12

   1. Initial program 39.4%

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

   3. Taylor expanded in n around -inf 78.5%

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

   5. Simplified 78.5%

      \[\leadsto \color{blue}{\frac{\log x - \left(\mathsf{log1p}\left(x\right) + \frac{0.5 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}\right) + \frac{0.16666666666666666 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{3} - {\log x}^{3}\right)}{n}}{n}\right)}{-n}} \]
   6. Step-by-step derivation

      1. add-log-exp 78.5%

         \[\leadsto \frac{\log x - \color{blue}{\log \left(e^{\mathsf{log1p}\left(x\right) + \frac{0.5 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}\right) + \frac{0.16666666666666666 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{3} - {\log x}^{3}\right)}{n}}{n}}\right)}}{-n} \]

      2. diff-log 47.1%

         \[\leadsto \frac{\color{blue}{\log \left(\frac{x}{e^{\mathsf{log1p}\left(x\right) + \frac{0.5 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}\right) + \frac{0.16666666666666666 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{3} - {\log x}^{3}\right)}{n}}{n}}}\right)}}{-n} \]

   7. Applied egg-rr 47.1%

      \[\leadsto \frac{\color{blue}{\log \left(\frac{x}{e^{\mathsf{log1p}\left(x\right) + \frac{\mathsf{fma}\left(0.5, {\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}, 0.16666666666666666 \cdot \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{3} - {\log x}^{3}}{n}\right)}{n}}}\right)}}{-n} \]

   8. Taylor expanded in n around inf 78.8%

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

      1. associate-*r/ 78.8%

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

      2. mul-1-neg 78.8%

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

      3. +-commutative 78.8%

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

   10. Simplified 78.8%

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

   if 1.99999999999999996e-12 < (/.f64 #s(literal 1 binary64) n) < 1.9999999999999999e267

   1. Initial program 62.6%

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

   3. Taylor expanded in x around 0 35.4%

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

      1. *-rgt-identity 35.4%

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

      2. associate-*l/ 35.4%

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

      3. associate-/l* 35.4%

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

      4. exp-to-pow 56.5%

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

   5. Simplified 56.5%

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

   if 1.9999999999999999e267 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 3.1%

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

   3. Taylor expanded in x around inf 0.0%

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

      1. log-rec 0.0%

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

      2. mul-1-neg 0.0%

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

      3. neg-mul-1 0.0%

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

      4. mul-1-neg 0.0%

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

      5. distribute-frac-neg 0.0%

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

      6. remove-double-neg 0.0%

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

      7. *-rgt-identity 0.0%

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

      8. associate-/l* 0.0%

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

      9. exp-to-pow 0.0%

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

      10. *-commutative 0.0%

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

   5. Simplified 0.0%

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

   6. Taylor expanded in n around inf 73.2%

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

4. Final simplification 80.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{-62}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n} + -1\right)}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-12}:\\ \;\;\;\;\frac{\log \left(\frac{x}{x + 1}\right)}{-n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{+267}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x \cdot n}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 68.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq 2 \cdot 10^{-12}:\\ \;\;\;\;\frac{\log \left(\frac{x}{x + 1}\right)}{-n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{+267}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x \cdot n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= (/ 1.0 n) 2e-12)
   (/ (log (/ x (+ x 1.0))) (- n))
   (if (<= (/ 1.0 n) 2e+267) (- 1.0 (pow x (/ 1.0 n))) (/ 1.0 (* x n)))))

C

double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= 2e-12) {
		tmp = log((x / (x + 1.0))) / -n;
	} else if ((1.0 / n) <= 2e+267) {
		tmp = 1.0 - pow(x, (1.0 / n));
	} else {
		tmp = 1.0 / (x * n);
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if ((1.0d0 / n) <= 2d-12) then
        tmp = log((x / (x + 1.0d0))) / -n
    else if ((1.0d0 / n) <= 2d+267) then
        tmp = 1.0d0 - (x ** (1.0d0 / n))
    else
        tmp = 1.0d0 / (x * n)
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= 2e-12) {
		tmp = Math.log((x / (x + 1.0))) / -n;
	} else if ((1.0 / n) <= 2e+267) {
		tmp = 1.0 - Math.pow(x, (1.0 / n));
	} else {
		tmp = 1.0 / (x * n);
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if (1.0 / n) <= 2e-12:
		tmp = math.log((x / (x + 1.0))) / -n
	elif (1.0 / n) <= 2e+267:
		tmp = 1.0 - math.pow(x, (1.0 / n))
	else:
		tmp = 1.0 / (x * n)
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (Float64(1.0 / n) <= 2e-12)
		tmp = Float64(log(Float64(x / Float64(x + 1.0))) / Float64(-n));
	elseif (Float64(1.0 / n) <= 2e+267)
		tmp = Float64(1.0 - (x ^ Float64(1.0 / n)));
	else
		tmp = Float64(1.0 / Float64(x * n));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if ((1.0 / n) <= 2e-12)
		tmp = log((x / (x + 1.0))) / -n;
	elseif ((1.0 / n) <= 2e+267)
		tmp = 1.0 - (x ^ (1.0 / n));
	else
		tmp = 1.0 / (x * n);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-12], N[(N[Log[N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-n)), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e+267], N[(1.0 - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(x * n), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 #s(literal 1 binary64) n) < 1.99999999999999996e-12

   1. Initial program 61.0%

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

   3. Taylor expanded in n around -inf 71.6%

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

   5. Simplified 71.6%

      \[\leadsto \color{blue}{\frac{\log x - \left(\mathsf{log1p}\left(x\right) + \frac{0.5 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}\right) + \frac{0.16666666666666666 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{3} - {\log x}^{3}\right)}{n}}{n}\right)}{-n}} \]
   6. Step-by-step derivation

      1. add-log-exp 74.9%

         \[\leadsto \frac{\log x - \color{blue}{\log \left(e^{\mathsf{log1p}\left(x\right) + \frac{0.5 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}\right) + \frac{0.16666666666666666 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{3} - {\log x}^{3}\right)}{n}}{n}}\right)}}{-n} \]

      2. diff-log 40.8%

         \[\leadsto \frac{\color{blue}{\log \left(\frac{x}{e^{\mathsf{log1p}\left(x\right) + \frac{0.5 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}\right) + \frac{0.16666666666666666 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{3} - {\log x}^{3}\right)}{n}}{n}}}\right)}}{-n} \]

   7. Applied egg-rr 40.8%

      \[\leadsto \frac{\color{blue}{\log \left(\frac{x}{e^{\mathsf{log1p}\left(x\right) + \frac{\mathsf{fma}\left(0.5, {\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}, 0.16666666666666666 \cdot \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{3} - {\log x}^{3}}{n}\right)}{n}}}\right)}}{-n} \]

   8. Taylor expanded in n around inf 72.3%

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

      1. associate-*r/ 72.3%

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

      2. mul-1-neg 72.3%

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

      3. +-commutative 72.3%

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

   10. Simplified 72.3%

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

   if 1.99999999999999996e-12 < (/.f64 #s(literal 1 binary64) n) < 1.9999999999999999e267

   1. Initial program 62.6%

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

   3. Taylor expanded in x around 0 35.4%

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

      1. *-rgt-identity 35.4%

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

      2. associate-*l/ 35.4%

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

      3. associate-/l* 35.4%

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

      4. exp-to-pow 56.5%

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

   5. Simplified 56.5%

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

   if 1.9999999999999999e267 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 3.1%

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

   3. Taylor expanded in x around inf 0.0%

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

      1. log-rec 0.0%

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

      2. mul-1-neg 0.0%

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

      3. neg-mul-1 0.0%

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

      4. mul-1-neg 0.0%

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

      5. distribute-frac-neg 0.0%

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

      6. remove-double-neg 0.0%

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

      7. *-rgt-identity 0.0%

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

      8. associate-/l* 0.0%

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

      9. exp-to-pow 0.0%

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

      10. *-commutative 0.0%

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

   5. Simplified 0.0%

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

   6. Taylor expanded in n around inf 73.2%

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

4. Final simplification 69.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq 2 \cdot 10^{-12}:\\ \;\;\;\;\frac{\log \left(\frac{x}{x + 1}\right)}{-n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{+267}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x \cdot n}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 64.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{-208}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 4.6 \cdot 10^{-137}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;x \leq 0.55:\\ \;\;\;\;\frac{\log x}{-n}\\ \mathbf{elif}\;x \leq 2.9 \cdot 10^{+136}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x -2e-208)
   0.0
   (if (<= x 4.6e-137)
     (- 1.0 (pow x (/ 1.0 n)))
     (if (<= x 0.55)
       (/ (log x) (- n))
       (if (<= x 2.9e+136) (/ (/ 1.0 x) n) 0.0)))))

C

double code(double x, double n) {
	double tmp;
	if (x <= -2e-208) {
		tmp = 0.0;
	} else if (x <= 4.6e-137) {
		tmp = 1.0 - pow(x, (1.0 / n));
	} else if (x <= 0.55) {
		tmp = log(x) / -n;
	} else if (x <= 2.9e+136) {
		tmp = (1.0 / 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 <= (-2d-208)) then
        tmp = 0.0d0
    else if (x <= 4.6d-137) then
        tmp = 1.0d0 - (x ** (1.0d0 / n))
    else if (x <= 0.55d0) then
        tmp = log(x) / -n
    else if (x <= 2.9d+136) then
        tmp = (1.0d0 / 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 <= -2e-208) {
		tmp = 0.0;
	} else if (x <= 4.6e-137) {
		tmp = 1.0 - Math.pow(x, (1.0 / n));
	} else if (x <= 0.55) {
		tmp = Math.log(x) / -n;
	} else if (x <= 2.9e+136) {
		tmp = (1.0 / x) / n;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= -2e-208:
		tmp = 0.0
	elif x <= 4.6e-137:
		tmp = 1.0 - math.pow(x, (1.0 / n))
	elif x <= 0.55:
		tmp = math.log(x) / -n
	elif x <= 2.9e+136:
		tmp = (1.0 / x) / n
	else:
		tmp = 0.0
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= -2e-208)
		tmp = 0.0;
	elseif (x <= 4.6e-137)
		tmp = Float64(1.0 - (x ^ Float64(1.0 / n)));
	elseif (x <= 0.55)
		tmp = Float64(log(x) / Float64(-n));
	elseif (x <= 2.9e+136)
		tmp = Float64(Float64(1.0 / x) / n);
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (x <= -2e-208)
		tmp = 0.0;
	elseif (x <= 4.6e-137)
		tmp = 1.0 - (x ^ (1.0 / n));
	elseif (x <= 0.55)
		tmp = log(x) / -n;
	elseif (x <= 2.9e+136)
		tmp = (1.0 / x) / n;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, -2e-208], 0.0, If[LessEqual[x, 4.6e-137], N[(1.0 - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.55], N[(N[Log[x], $MachinePrecision] / (-n)), $MachinePrecision], If[LessEqual[x, 2.9e+136], N[(N[(1.0 / x), $MachinePrecision] / n), $MachinePrecision], 0.0]]]]

Derivation

1. Split input into 4 regimes

2. if x < -2.0000000000000002e-208 or 2.89999999999999974e136 < x

   1. Initial program 79.2%

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

   3. Taylor expanded in x around inf 56.0%

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

      1. log-rec 56.0%

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

      2. mul-1-neg 56.0%

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

      3. neg-mul-1 56.0%

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

      4. mul-1-neg 56.0%

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

      5. distribute-frac-neg 56.0%

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

      6. remove-double-neg 56.0%

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

      7. *-rgt-identity 56.0%

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

      8. associate-/l* 56.0%

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

      9. exp-to-pow 85.3%

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

   5. Simplified 85.3%

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

   6. Taylor expanded in x around 0 87.4%

      \[\leadsto \color{blue}{0} \]

   if -2.0000000000000002e-208 < x < 4.60000000000000016e-137

   1. Initial program 63.5%

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

   3. Taylor expanded in x around 0 54.0%

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

      1. *-rgt-identity 54.0%

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

      2. associate-*l/ 54.0%

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

      3. associate-/l* 54.0%

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

      4. exp-to-pow 63.5%

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

   5. Simplified 63.5%

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

   if 4.60000000000000016e-137 < x < 0.55000000000000004

   1. Initial program 28.6%

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

   3. Taylor expanded in x around 0 24.5%

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

      1. *-rgt-identity 24.5%

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

      2. associate-*l/ 24.5%

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

      3. associate-/l* 24.5%

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

      4. exp-to-pow 24.5%

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

   5. Simplified 24.5%

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

   6. Taylor expanded in n around inf 49.6%

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

      1. associate-*r/ 49.6%

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

      2. neg-mul-1 49.6%

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

   8. Simplified 49.6%

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

   if 0.55000000000000004 < x < 2.89999999999999974e136

   1. Initial program 41.4%

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

   3. Taylor expanded in x around inf 92.0%

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

      1. log-rec 92.0%

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

      2. mul-1-neg 92.0%

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

      3. neg-mul-1 92.0%

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

      4. mul-1-neg 92.0%

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

      5. distribute-frac-neg 92.0%

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

      6. remove-double-neg 92.0%

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

      7. *-rgt-identity 92.0%

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

      8. associate-/l* 92.0%

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

      9. exp-to-pow 92.1%

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

      10. *-commutative 92.1%

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

   5. Simplified 92.1%

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

   6. Taylor expanded in n around inf 68.6%

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

      1. *-commutative 68.6%

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

      2. associate-/r* 69.9%

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

   8. Simplified 69.9%

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

4. Final simplification 71.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{-208}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 4.6 \cdot 10^{-137}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;x \leq 0.55:\\ \;\;\;\;\frac{\log x}{-n}\\ \mathbf{elif}\;x \leq 2.9 \cdot 10^{+136}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 63.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4 \cdot 10^{-310}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 0.55:\\ \;\;\;\;\frac{\log x}{-n}\\ \mathbf{elif}\;x \leq 3.1 \cdot 10^{+136}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x -4e-310)
   0.0
   (if (<= x 0.55)
     (/ (log x) (- n))
     (if (<= x 3.1e+136) (/ (/ 1.0 x) n) 0.0))))

C

double code(double x, double n) {
	double tmp;
	if (x <= -4e-310) {
		tmp = 0.0;
	} else if (x <= 0.55) {
		tmp = log(x) / -n;
	} else if (x <= 3.1e+136) {
		tmp = (1.0 / 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 <= (-4d-310)) then
        tmp = 0.0d0
    else if (x <= 0.55d0) then
        tmp = log(x) / -n
    else if (x <= 3.1d+136) then
        tmp = (1.0d0 / 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 <= -4e-310) {
		tmp = 0.0;
	} else if (x <= 0.55) {
		tmp = Math.log(x) / -n;
	} else if (x <= 3.1e+136) {
		tmp = (1.0 / x) / n;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= -4e-310:
		tmp = 0.0
	elif x <= 0.55:
		tmp = math.log(x) / -n
	elif x <= 3.1e+136:
		tmp = (1.0 / x) / n
	else:
		tmp = 0.0
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= -4e-310)
		tmp = 0.0;
	elseif (x <= 0.55)
		tmp = Float64(log(x) / Float64(-n));
	elseif (x <= 3.1e+136)
		tmp = Float64(Float64(1.0 / x) / n);
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (x <= -4e-310)
		tmp = 0.0;
	elseif (x <= 0.55)
		tmp = log(x) / -n;
	elseif (x <= 3.1e+136)
		tmp = (1.0 / x) / n;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, -4e-310], 0.0, If[LessEqual[x, 0.55], N[(N[Log[x], $MachinePrecision] / (-n)), $MachinePrecision], If[LessEqual[x, 3.1e+136], N[(N[(1.0 / x), $MachinePrecision] / n), $MachinePrecision], 0.0]]]

Derivation

1. Split input into 3 regimes

2. if x < -3.999999999999988e-310 or 3.09999999999999983e136 < 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 inf 52.7%

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

      1. log-rec 52.7%

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

      2. mul-1-neg 52.7%

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

      3. neg-mul-1 52.7%

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

      4. mul-1-neg 52.7%

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

      5. distribute-frac-neg 52.7%

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

      6. remove-double-neg 52.7%

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

      7. *-rgt-identity 52.7%

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

      8. associate-/l* 52.7%

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

      9. exp-to-pow 80.5%

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

   5. Simplified 80.5%

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

   6. Taylor expanded in x around 0 82.5%

      \[\leadsto \color{blue}{0} \]

   if -3.999999999999988e-310 < x < 0.55000000000000004

   1. Initial program 45.6%

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

   3. Taylor expanded in x around 0 43.8%

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

      1. *-rgt-identity 43.8%

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

      2. associate-*l/ 43.8%

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

      3. associate-/l* 43.8%

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

      4. exp-to-pow 43.8%

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

   5. Simplified 43.8%

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

   6. Taylor expanded in n around inf 46.0%

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

      1. associate-*r/ 46.0%

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

      2. neg-mul-1 46.0%

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

   8. Simplified 46.0%

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

   if 0.55000000000000004 < x < 3.09999999999999983e136

   1. Initial program 41.4%

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

   3. Taylor expanded in x around inf 92.0%

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

      1. log-rec 92.0%

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

      2. mul-1-neg 92.0%

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

      3. neg-mul-1 92.0%

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

      4. mul-1-neg 92.0%

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

      5. distribute-frac-neg 92.0%

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

      6. remove-double-neg 92.0%

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

      7. *-rgt-identity 92.0%

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

      8. associate-/l* 92.0%

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

      9. exp-to-pow 92.1%

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

      10. *-commutative 92.1%

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

   5. Simplified 92.1%

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

   6. Taylor expanded in n around inf 68.6%

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

      1. *-commutative 68.6%

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

      2. associate-/r* 69.9%

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

   8. Simplified 69.9%

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

4. Final simplification 65.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4 \cdot 10^{-310}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 0.55:\\ \;\;\;\;\frac{\log x}{-n}\\ \mathbf{elif}\;x \leq 3.1 \cdot 10^{+136}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 51.1% accurate, 14.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 4.5 \cdot 10^{-304}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 3.6 \cdot 10^{+137}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 4.5e-304) 0.0 (if (<= x 3.6e+137) (/ (/ 1.0 x) n) 0.0)))

C

double code(double x, double n) {
	double tmp;
	if (x <= 4.5e-304) {
		tmp = 0.0;
	} else if (x <= 3.6e+137) {
		tmp = (1.0 / 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 <= 4.5d-304) then
        tmp = 0.0d0
    else if (x <= 3.6d+137) then
        tmp = (1.0d0 / 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 <= 4.5e-304) {
		tmp = 0.0;
	} else if (x <= 3.6e+137) {
		tmp = (1.0 / x) / n;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 4.5e-304:
		tmp = 0.0
	elif x <= 3.6e+137:
		tmp = (1.0 / x) / n
	else:
		tmp = 0.0
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 4.5e-304)
		tmp = 0.0;
	elseif (x <= 3.6e+137)
		tmp = Float64(Float64(1.0 / x) / n);
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (x <= 4.5e-304)
		tmp = 0.0;
	elseif (x <= 3.6e+137)
		tmp = (1.0 / x) / n;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 4.5e-304], 0.0, If[LessEqual[x, 3.6e+137], N[(N[(1.0 / x), $MachinePrecision] / n), $MachinePrecision], 0.0]]

Derivation

1. Split input into 2 regimes

2. if x < 4.4999999999999998e-304 or 3.6e137 < x

   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 inf 51.8%

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

      1. log-rec 51.8%

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

      2. mul-1-neg 51.8%

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

      3. neg-mul-1 51.8%

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

      4. mul-1-neg 51.8%

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

      5. distribute-frac-neg 51.8%

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

      6. remove-double-neg 51.8%

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

      7. *-rgt-identity 51.8%

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

      8. associate-/l* 51.8%

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

      9. exp-to-pow 79.1%

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

   5. Simplified 79.1%

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

   6. Taylor expanded in x around 0 81.0%

      \[\leadsto \color{blue}{0} \]

   if 4.4999999999999998e-304 < x < 3.6e137

   1. Initial program 44.2%

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

   3. Taylor expanded in x around inf 49.0%

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

      1. log-rec 49.0%

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

      2. mul-1-neg 49.0%

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

      3. neg-mul-1 49.0%

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

      4. mul-1-neg 49.0%

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

      5. distribute-frac-neg 49.0%

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

      6. remove-double-neg 49.0%

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

      7. *-rgt-identity 49.0%

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

      8. associate-/l* 49.0%

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

      9. exp-to-pow 49.0%

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

      10. *-commutative 49.0%

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

   5. Simplified 49.0%

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

   6. Taylor expanded in n around inf 42.8%

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

      1. *-commutative 42.8%

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

      2. associate-/r* 43.2%

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

   8. Simplified 43.2%

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

Alternative 15: 50.9% accurate, 14.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 4.5 \cdot 10^{-304}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 1.16 \cdot 10^{+137}:\\ \;\;\;\;\frac{1}{x \cdot n}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 4.5e-304) 0.0 (if (<= x 1.16e+137) (/ 1.0 (* x n)) 0.0)))

C

double code(double x, double n) {
	double tmp;
	if (x <= 4.5e-304) {
		tmp = 0.0;
	} else if (x <= 1.16e+137) {
		tmp = 1.0 / (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 <= 4.5d-304) then
        tmp = 0.0d0
    else if (x <= 1.16d+137) then
        tmp = 1.0d0 / (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 <= 4.5e-304) {
		tmp = 0.0;
	} else if (x <= 1.16e+137) {
		tmp = 1.0 / (x * n);
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 4.5e-304:
		tmp = 0.0
	elif x <= 1.16e+137:
		tmp = 1.0 / (x * n)
	else:
		tmp = 0.0
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 4.5e-304)
		tmp = 0.0;
	elseif (x <= 1.16e+137)
		tmp = Float64(1.0 / Float64(x * n));
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (x <= 4.5e-304)
		tmp = 0.0;
	elseif (x <= 1.16e+137)
		tmp = 1.0 / (x * n);
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 4.5e-304], 0.0, If[LessEqual[x, 1.16e+137], N[(1.0 / N[(x * n), $MachinePrecision]), $MachinePrecision], 0.0]]

Derivation

1. Split input into 2 regimes

2. if x < 4.4999999999999998e-304 or 1.1599999999999999e137 < x

   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 inf 51.8%

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

      1. log-rec 51.8%

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

      2. mul-1-neg 51.8%

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

      3. neg-mul-1 51.8%

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

      4. mul-1-neg 51.8%

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

      5. distribute-frac-neg 51.8%

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

      6. remove-double-neg 51.8%

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

      7. *-rgt-identity 51.8%

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

      8. associate-/l* 51.8%

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

      9. exp-to-pow 79.1%

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

   5. Simplified 79.1%

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

   6. Taylor expanded in x around 0 81.0%

      \[\leadsto \color{blue}{0} \]

   if 4.4999999999999998e-304 < x < 1.1599999999999999e137

   1. Initial program 44.2%

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

   3. Taylor expanded in x around inf 49.0%

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

      1. log-rec 49.0%

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

      2. mul-1-neg 49.0%

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

      3. neg-mul-1 49.0%

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

      4. mul-1-neg 49.0%

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

      5. distribute-frac-neg 49.0%

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

      6. remove-double-neg 49.0%

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

      7. *-rgt-identity 49.0%

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

      8. associate-/l* 49.0%

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

      9. exp-to-pow 49.0%

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

      10. *-commutative 49.0%

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

   5. Simplified 49.0%

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

   6. Taylor expanded in n around inf 42.8%

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

4. Final simplification 58.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 4.5 \cdot 10^{-304}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 1.16 \cdot 10^{+137}:\\ \;\;\;\;\frac{1}{x \cdot n}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 40.7% 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 58.8%

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

3. Taylor expanded in x around inf 30.4%

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

   1. log-rec 30.4%

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

   2. mul-1-neg 30.4%

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

   3. neg-mul-1 30.4%

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

   4. mul-1-neg 30.4%

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

   5. distribute-frac-neg 30.4%

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

   6. remove-double-neg 30.4%

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

   7. *-rgt-identity 30.4%

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

   8. associate-/l* 30.4%

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

   9. exp-to-pow 41.5%

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

5. Simplified 41.5%

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

6. Taylor expanded in x around 0 42.5%

   \[\leadsto \color{blue}{0} \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024180 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  :precision binary64
  (- (pow (+ x 1.0) (/ 1.0 n)) (pow x (/ 1.0 n))))