2nthrt (problem 3.4.6)

Percentage Accurate: 53.2% → 91.8%

Time: 1.0min

Alternatives: 21

Speedup: 1.8×

• could not determine a ground truth

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 53.2% 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: 91.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 35000:\\ \;\;\;\;\frac{\log \left(\frac{x}{e^{\mathsf{log1p}\left(x\right) + \frac{\mathsf{fma}\left(0.5, {\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}, 0.16666666666666666 \cdot \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{3} - {\log x}^{3}}{n}\right)}{n}}}\right)}{-n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{{x}^{\left(\frac{1}{n}\right)}}{n}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 35000.0)
   (/
    (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))
   (/ (/ (pow x (/ 1.0 n)) n) x)))

C

double code(double x, double n) {
	double tmp;
	if (x <= 35000.0) {
		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 = (pow(x, (1.0 / n)) / n) / x;
	}
	return tmp;
}

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 35000.0)
		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(Float64((x ^ Float64(1.0 / n)) / n) / x);
	end
	return tmp
end

Wolfram

code[x_, n_] := If[LessEqual[x, 35000.0], 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[(N[(N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 35000

   1. Initial program 36.2%

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

   3. Taylor expanded in n around -inf 84.5%

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

   5. Simplified 84.5%

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

      1. add-log-exp 89.2%

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

      2. diff-log 89.3%

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

      3. fma-define 89.3%

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

   7. Applied egg-rr 89.3%

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

   if 35000 < x

   1. Initial program 68.0%

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

   3. Taylor expanded in x around inf 96.9%

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

      1. associate-/r* 99.0%

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

      2. mul-1-neg 99.0%

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

      3. log-rec 99.0%

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

      4. mul-1-neg 99.0%

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

      5. distribute-neg-frac 99.0%

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

      6. mul-1-neg 99.0%

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

      7. remove-double-neg 99.0%

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

      8. *-rgt-identity 99.0%

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

      9. associate-/l* 99.0%

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

      10. exp-to-pow 99.0%

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

   5. Simplified 99.0%

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

4. Final simplification 93.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 35000:\\ \;\;\;\;\frac{\log \left(\frac{x}{e^{\mathsf{log1p}\left(x\right) + \frac{\mathsf{fma}\left(0.5, {\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}, 0.16666666666666666 \cdot \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{3} - {\log x}^{3}}{n}\right)}{n}}}\right)}{-n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{{x}^{\left(\frac{1}{n}\right)}}{n}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 86.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 28000:\\ \;\;\;\;\frac{\left(\mathsf{log1p}\left(x\right) + \frac{0.16666666666666666 \cdot \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{3} - {\log x}^{3}}{n} + 0.5 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}\right)}{n}\right) - \log x}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{{x}^{\left(\frac{1}{n}\right)}}{n}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 28000.0)
   (/
    (-
     (+
      (log1p x)
      (/
       (+
        (* 0.16666666666666666 (/ (- (pow (log1p x) 3.0) (pow (log x) 3.0)) n))
        (* 0.5 (- (pow (log1p x) 2.0) (pow (log x) 2.0))))
       n))
     (log x))
    n)
   (/ (/ (pow x (/ 1.0 n)) n) x)))

C

double code(double x, double n) {
	double tmp;
	if (x <= 28000.0) {
		tmp = ((log1p(x) + (((0.16666666666666666 * ((pow(log1p(x), 3.0) - pow(log(x), 3.0)) / n)) + (0.5 * (pow(log1p(x), 2.0) - pow(log(x), 2.0)))) / n)) - log(x)) / n;
	} else {
		tmp = (pow(x, (1.0 / n)) / n) / x;
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double tmp;
	if (x <= 28000.0) {
		tmp = ((Math.log1p(x) + (((0.16666666666666666 * ((Math.pow(Math.log1p(x), 3.0) - Math.pow(Math.log(x), 3.0)) / n)) + (0.5 * (Math.pow(Math.log1p(x), 2.0) - Math.pow(Math.log(x), 2.0)))) / n)) - Math.log(x)) / n;
	} else {
		tmp = (Math.pow(x, (1.0 / n)) / n) / x;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 28000.0:
		tmp = ((math.log1p(x) + (((0.16666666666666666 * ((math.pow(math.log1p(x), 3.0) - math.pow(math.log(x), 3.0)) / n)) + (0.5 * (math.pow(math.log1p(x), 2.0) - math.pow(math.log(x), 2.0)))) / n)) - math.log(x)) / n
	else:
		tmp = (math.pow(x, (1.0 / n)) / n) / x
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 28000.0)
		tmp = Float64(Float64(Float64(log1p(x) + Float64(Float64(Float64(0.16666666666666666 * Float64(Float64((log1p(x) ^ 3.0) - (log(x) ^ 3.0)) / n)) + Float64(0.5 * Float64((log1p(x) ^ 2.0) - (log(x) ^ 2.0)))) / n)) - log(x)) / n);
	else
		tmp = Float64(Float64((x ^ Float64(1.0 / n)) / n) / x);
	end
	return tmp
end

Wolfram

code[x_, n_] := If[LessEqual[x, 28000.0], N[(N[(N[(N[Log[1 + x], $MachinePrecision] + N[(N[(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] + 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]), $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision] - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], N[(N[(N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 28000

   1. Initial program 36.2%

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

   3. Taylor expanded in n around -inf 84.5%

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

   5. Simplified 84.5%

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

   if 28000 < x

   1. Initial program 68.0%

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

   3. Taylor expanded in x around inf 96.9%

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

      1. associate-/r* 99.0%

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

      2. mul-1-neg 99.0%

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

      3. log-rec 99.0%

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

      4. mul-1-neg 99.0%

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

      5. distribute-neg-frac 99.0%

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

      6. mul-1-neg 99.0%

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

      7. remove-double-neg 99.0%

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

      8. *-rgt-identity 99.0%

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

      9. associate-/l* 99.0%

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

      10. exp-to-pow 99.0%

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

   5. Simplified 99.0%

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

4. Final simplification 91.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 28000:\\ \;\;\;\;\frac{\left(\mathsf{log1p}\left(x\right) + \frac{0.16666666666666666 \cdot \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{3} - {\log x}^{3}}{n} + 0.5 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}\right)}{n}\right) - \log x}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{{x}^{\left(\frac{1}{n}\right)}}{n}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 85.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ t_1 := {\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - t\_0\\ \mathbf{if}\;t\_1 \leq -4 \cdot 10^{-6}:\\ \;\;\;\;\log \left(e \cdot e^{-e^{\frac{\log x}{n}}}\right)\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{-16}:\\ \;\;\;\;\frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;{\left({\left(e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - t\_0\right)}^{3}\right)}^{0.3333333333333333}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))) (t_1 (- (pow (+ x 1.0) (/ 1.0 n)) t_0)))
   (if (<= t_1 -4e-6)
     (log (* E (exp (- (exp (/ (log x) n))))))
     (if (<= t_1 2e-16)
       (/ (log (/ (+ x 1.0) x)) 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 t_1 = pow((x + 1.0), (1.0 / n)) - t_0;
	double tmp;
	if (t_1 <= -4e-6) {
		tmp = log((((double) M_E) * exp(-exp((log(x) / n)))));
	} else if (t_1 <= 2e-16) {
		tmp = log(((x + 1.0) / x)) / 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 t_1 = Math.pow((x + 1.0), (1.0 / n)) - t_0;
	double tmp;
	if (t_1 <= -4e-6) {
		tmp = Math.log((Math.E * Math.exp(-Math.exp((Math.log(x) / n)))));
	} else if (t_1 <= 2e-16) {
		tmp = Math.log(((x + 1.0) / x)) / 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))
	t_1 = math.pow((x + 1.0), (1.0 / n)) - t_0
	tmp = 0
	if t_1 <= -4e-6:
		tmp = math.log((math.e * math.exp(-math.exp((math.log(x) / n)))))
	elif t_1 <= 2e-16:
		tmp = math.log(((x + 1.0) / x)) / 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)
	t_1 = Float64((Float64(x + 1.0) ^ Float64(1.0 / n)) - t_0)
	tmp = 0.0
	if (t_1 <= -4e-6)
		tmp = log(Float64(exp(1) * exp(Float64(-exp(Float64(log(x) / n))))));
	elseif (t_1 <= 2e-16)
		tmp = Float64(log(Float64(Float64(x + 1.0) / x)) / 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]}, Block[{t$95$1 = N[(N[Power[N[(x + 1.0), $MachinePrecision], N[(1.0 / n), $MachinePrecision]], $MachinePrecision] - t$95$0), $MachinePrecision]}, If[LessEqual[t$95$1, -4e-6], N[Log[N[(E * N[Exp[(-N[Exp[N[(N[Log[x], $MachinePrecision] / n), $MachinePrecision]], $MachinePrecision])], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$1, 2e-16], N[(N[Log[N[(N[(x + 1.0), $MachinePrecision] / x), $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 (pow.f64 (+.f64 x #s(literal 1 binary64)) (/.f64 #s(literal 1 binary64) n)) (pow.f64 x (/.f64 #s(literal 1 binary64) n))) < -3.99999999999999982e-6

   1. Initial program 98.5%

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

      1. sub-neg 98.5%

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

      2. +-commutative 98.5%

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

      3. add-log-exp 98.5%

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

      4. add-log-exp 98.5%

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

      5. sum-log 98.7%

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

      6. pow-to-exp 98.7%

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

      7. un-div-inv 98.7%

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

      8. +-commutative 98.7%

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

      9. log1p-define 98.7%

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

   4. Applied egg-rr 98.7%

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

   5. Taylor expanded in x around 0 98.7%

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

   if -3.99999999999999982e-6 < (-.f64 (pow.f64 (+.f64 x #s(literal 1 binary64)) (/.f64 #s(literal 1 binary64) n)) (pow.f64 x (/.f64 #s(literal 1 binary64) n))) < 2e-16

   1. Initial program 45.7%

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

   3. Taylor expanded in n around inf 79.3%

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

      1. log1p-define 79.3%

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

   5. Simplified 79.3%

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

      1. log1p-undefine 79.3%

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

      2. diff-log 79.4%

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

   7. Applied egg-rr 79.4%

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

      1. +-commutative 79.4%

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

   9. Simplified 79.4%

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

   if 2e-16 < (-.f64 (pow.f64 (+.f64 x #s(literal 1 binary64)) (/.f64 #s(literal 1 binary64) n)) (pow.f64 x (/.f64 #s(literal 1 binary64) n)))

   1. Initial program 46.1%

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

      1. add-cbrt-cube 46.1%

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

      2. pow1/3 46.1%

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

      3. pow3 46.1%

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

      4. pow-to-exp 46.1%

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

      5. un-div-inv 46.1%

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

      6. +-commutative 46.1%

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

      7. log1p-define 95.9%

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

   4. Applied egg-rr 95.9%

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

4. Final simplification 83.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \leq -4 \cdot 10^{-6}:\\ \;\;\;\;\log \left(e \cdot e^{-e^{\frac{\log x}{n}}}\right)\\ \mathbf{elif}\;{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \leq 2 \cdot 10^{-16}:\\ \;\;\;\;\frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;{\left({\left(e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{1}{n}\right)}\right)}^{3}\right)}^{0.3333333333333333}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 85.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ t_1 := {\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - t\_0\\ \mathbf{if}\;t\_1 \leq -4 \cdot 10^{-6}:\\ \;\;\;\;\log \left(e \cdot e^{-e^{\frac{\log x}{n}}}\right)\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{-16}:\\ \;\;\;\;\frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;e^{\log \left(e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - t\_0\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))) (t_1 (- (pow (+ x 1.0) (/ 1.0 n)) t_0)))
   (if (<= t_1 -4e-6)
     (log (* E (exp (- (exp (/ (log x) n))))))
     (if (<= t_1 2e-16)
       (/ (log (/ (+ x 1.0) x)) 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 t_1 = pow((x + 1.0), (1.0 / n)) - t_0;
	double tmp;
	if (t_1 <= -4e-6) {
		tmp = log((((double) M_E) * exp(-exp((log(x) / n)))));
	} else if (t_1 <= 2e-16) {
		tmp = log(((x + 1.0) / x)) / 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 t_1 = Math.pow((x + 1.0), (1.0 / n)) - t_0;
	double tmp;
	if (t_1 <= -4e-6) {
		tmp = Math.log((Math.E * Math.exp(-Math.exp((Math.log(x) / n)))));
	} else if (t_1 <= 2e-16) {
		tmp = Math.log(((x + 1.0) / x)) / 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))
	t_1 = math.pow((x + 1.0), (1.0 / n)) - t_0
	tmp = 0
	if t_1 <= -4e-6:
		tmp = math.log((math.e * math.exp(-math.exp((math.log(x) / n)))))
	elif t_1 <= 2e-16:
		tmp = math.log(((x + 1.0) / x)) / 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)
	t_1 = Float64((Float64(x + 1.0) ^ Float64(1.0 / n)) - t_0)
	tmp = 0.0
	if (t_1 <= -4e-6)
		tmp = log(Float64(exp(1) * exp(Float64(-exp(Float64(log(x) / n))))));
	elseif (t_1 <= 2e-16)
		tmp = Float64(log(Float64(Float64(x + 1.0) / x)) / 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]}, Block[{t$95$1 = N[(N[Power[N[(x + 1.0), $MachinePrecision], N[(1.0 / n), $MachinePrecision]], $MachinePrecision] - t$95$0), $MachinePrecision]}, If[LessEqual[t$95$1, -4e-6], N[Log[N[(E * N[Exp[(-N[Exp[N[(N[Log[x], $MachinePrecision] / n), $MachinePrecision]], $MachinePrecision])], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$1, 2e-16], N[(N[Log[N[(N[(x + 1.0), $MachinePrecision] / x), $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 (pow.f64 (+.f64 x #s(literal 1 binary64)) (/.f64 #s(literal 1 binary64) n)) (pow.f64 x (/.f64 #s(literal 1 binary64) n))) < -3.99999999999999982e-6

   1. Initial program 98.5%

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

      1. sub-neg 98.5%

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

      2. +-commutative 98.5%

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

      3. add-log-exp 98.5%

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

      4. add-log-exp 98.5%

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

      5. sum-log 98.7%

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

      6. pow-to-exp 98.7%

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

      7. un-div-inv 98.7%

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

      8. +-commutative 98.7%

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

      9. log1p-define 98.7%

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

   4. Applied egg-rr 98.7%

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

   5. Taylor expanded in x around 0 98.7%

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

   if -3.99999999999999982e-6 < (-.f64 (pow.f64 (+.f64 x #s(literal 1 binary64)) (/.f64 #s(literal 1 binary64) n)) (pow.f64 x (/.f64 #s(literal 1 binary64) n))) < 2e-16

   1. Initial program 45.7%

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

   3. Taylor expanded in n around inf 79.3%

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

      1. log1p-define 79.3%

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

   5. Simplified 79.3%

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

      1. log1p-undefine 79.3%

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

      2. diff-log 79.4%

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

   7. Applied egg-rr 79.4%

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

      1. +-commutative 79.4%

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

   9. Simplified 79.4%

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

   if 2e-16 < (-.f64 (pow.f64 (+.f64 x #s(literal 1 binary64)) (/.f64 #s(literal 1 binary64) n)) (pow.f64 x (/.f64 #s(literal 1 binary64) n)))

   1. Initial program 46.1%

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

      1. add-exp-log 46.1%

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

      2. pow-to-exp 46.1%

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

      3. un-div-inv 46.1%

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

      4. +-commutative 46.1%

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

      5. log1p-define 95.9%

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

   4. Applied egg-rr 95.9%

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

4. Final simplification 83.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \leq -4 \cdot 10^{-6}:\\ \;\;\;\;\log \left(e \cdot e^{-e^{\frac{\log x}{n}}}\right)\\ \mathbf{elif}\;{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \leq 2 \cdot 10^{-16}:\\ \;\;\;\;\frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;e^{\log \left(e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{1}{n}\right)}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 85.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ t_1 := {\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - t\_0\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{-8}:\\ \;\;\;\;\log \left(e^{1 - t\_0}\right)\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{-16}:\\ \;\;\;\;\frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))) (t_1 (- (pow (+ x 1.0) (/ 1.0 n)) t_0)))
   (if (<= t_1 -1e-8)
     (log (exp (- 1.0 t_0)))
     (if (<= t_1 2e-16)
       (/ (log (/ (+ x 1.0) x)) n)
       (- (exp (/ (log1p x) n)) t_0)))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double t_1 = pow((x + 1.0), (1.0 / n)) - t_0;
	double tmp;
	if (t_1 <= -1e-8) {
		tmp = log(exp((1.0 - t_0)));
	} else if (t_1 <= 2e-16) {
		tmp = log(((x + 1.0) / x)) / 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 t_1 = Math.pow((x + 1.0), (1.0 / n)) - t_0;
	double tmp;
	if (t_1 <= -1e-8) {
		tmp = Math.log(Math.exp((1.0 - t_0)));
	} else if (t_1 <= 2e-16) {
		tmp = Math.log(((x + 1.0) / x)) / 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))
	t_1 = math.pow((x + 1.0), (1.0 / n)) - t_0
	tmp = 0
	if t_1 <= -1e-8:
		tmp = math.log(math.exp((1.0 - t_0)))
	elif t_1 <= 2e-16:
		tmp = math.log(((x + 1.0) / x)) / 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)
	t_1 = Float64((Float64(x + 1.0) ^ Float64(1.0 / n)) - t_0)
	tmp = 0.0
	if (t_1 <= -1e-8)
		tmp = log(exp(Float64(1.0 - t_0)));
	elseif (t_1 <= 2e-16)
		tmp = Float64(log(Float64(Float64(x + 1.0) / x)) / 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]}, Block[{t$95$1 = N[(N[Power[N[(x + 1.0), $MachinePrecision], N[(1.0 / n), $MachinePrecision]], $MachinePrecision] - t$95$0), $MachinePrecision]}, If[LessEqual[t$95$1, -1e-8], N[Log[N[Exp[N[(1.0 - t$95$0), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$1, 2e-16], N[(N[Log[N[(N[(x + 1.0), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], N[(N[Exp[N[(N[Log[1 + x], $MachinePrecision] / n), $MachinePrecision]], $MachinePrecision] - t$95$0), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (pow.f64 (+.f64 x #s(literal 1 binary64)) (/.f64 #s(literal 1 binary64) n)) (pow.f64 x (/.f64 #s(literal 1 binary64) n))) < -1e-8

   1. Initial program 97.2%

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

      1. sub-neg 97.2%

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

      2. +-commutative 97.2%

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

      3. add-log-exp 97.2%

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

      4. add-log-exp 97.2%

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

      5. sum-log 97.3%

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

      6. pow-to-exp 97.3%

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

      7. un-div-inv 97.3%

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

      8. +-commutative 97.3%

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

      9. log1p-define 97.3%

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

   4. Applied egg-rr 97.3%

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

   5. Taylor expanded in x around 0 97.3%

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

      1. *-commutative 97.3%

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

      2. prod-exp 97.3%

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

      3. *-rgt-identity 97.3%

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

      4. associate-*r/ 97.3%

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

      5. exp-to-pow 97.3%

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

      6. +-commutative 97.3%

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

      7. sub-neg 97.3%

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

   7. Simplified 97.3%

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

   if -1e-8 < (-.f64 (pow.f64 (+.f64 x #s(literal 1 binary64)) (/.f64 #s(literal 1 binary64) n)) (pow.f64 x (/.f64 #s(literal 1 binary64) n))) < 2e-16

   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 n around inf 79.4%

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

      1. log1p-define 79.4%

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

   5. Simplified 79.4%

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

      1. log1p-undefine 79.4%

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

      2. diff-log 79.5%

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

   7. Applied egg-rr 79.5%

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

      1. +-commutative 79.5%

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

   9. Simplified 79.5%

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

   if 2e-16 < (-.f64 (pow.f64 (+.f64 x #s(literal 1 binary64)) (/.f64 #s(literal 1 binary64) n)) (pow.f64 x (/.f64 #s(literal 1 binary64) n)))

   1. Initial program 46.1%

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

   3. Taylor expanded in n around 0 46.1%

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

      1. log1p-define 95.9%

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

      2. *-rgt-identity 95.9%

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

      3. associate-*l/ 95.9%

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

      4. associate-/l* 95.9%

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

      5. exp-to-pow 95.9%

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

   5. Simplified 95.9%

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

4. Final simplification 83.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \leq -1 \cdot 10^{-8}:\\ \;\;\;\;\log \left(e^{1 - {x}^{\left(\frac{1}{n}\right)}}\right)\\ \mathbf{elif}\;{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \leq 2 \cdot 10^{-16}:\\ \;\;\;\;\frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 85.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ t_1 := {\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - t\_0\\ \mathbf{if}\;t\_1 \leq -4 \cdot 10^{-6}:\\ \;\;\;\;\log \left(e \cdot e^{-e^{\frac{\log x}{n}}}\right)\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{-16}:\\ \;\;\;\;\frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))) (t_1 (- (pow (+ x 1.0) (/ 1.0 n)) t_0)))
   (if (<= t_1 -4e-6)
     (log (* E (exp (- (exp (/ (log x) n))))))
     (if (<= t_1 2e-16)
       (/ (log (/ (+ x 1.0) x)) n)
       (- (exp (/ (log1p x) n)) t_0)))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double t_1 = pow((x + 1.0), (1.0 / n)) - t_0;
	double tmp;
	if (t_1 <= -4e-6) {
		tmp = log((((double) M_E) * exp(-exp((log(x) / n)))));
	} else if (t_1 <= 2e-16) {
		tmp = log(((x + 1.0) / x)) / 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 t_1 = Math.pow((x + 1.0), (1.0 / n)) - t_0;
	double tmp;
	if (t_1 <= -4e-6) {
		tmp = Math.log((Math.E * Math.exp(-Math.exp((Math.log(x) / n)))));
	} else if (t_1 <= 2e-16) {
		tmp = Math.log(((x + 1.0) / x)) / 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))
	t_1 = math.pow((x + 1.0), (1.0 / n)) - t_0
	tmp = 0
	if t_1 <= -4e-6:
		tmp = math.log((math.e * math.exp(-math.exp((math.log(x) / n)))))
	elif t_1 <= 2e-16:
		tmp = math.log(((x + 1.0) / x)) / 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)
	t_1 = Float64((Float64(x + 1.0) ^ Float64(1.0 / n)) - t_0)
	tmp = 0.0
	if (t_1 <= -4e-6)
		tmp = log(Float64(exp(1) * exp(Float64(-exp(Float64(log(x) / n))))));
	elseif (t_1 <= 2e-16)
		tmp = Float64(log(Float64(Float64(x + 1.0) / x)) / 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]}, Block[{t$95$1 = N[(N[Power[N[(x + 1.0), $MachinePrecision], N[(1.0 / n), $MachinePrecision]], $MachinePrecision] - t$95$0), $MachinePrecision]}, If[LessEqual[t$95$1, -4e-6], N[Log[N[(E * N[Exp[(-N[Exp[N[(N[Log[x], $MachinePrecision] / n), $MachinePrecision]], $MachinePrecision])], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$1, 2e-16], N[(N[Log[N[(N[(x + 1.0), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], N[(N[Exp[N[(N[Log[1 + x], $MachinePrecision] / n), $MachinePrecision]], $MachinePrecision] - t$95$0), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (pow.f64 (+.f64 x #s(literal 1 binary64)) (/.f64 #s(literal 1 binary64) n)) (pow.f64 x (/.f64 #s(literal 1 binary64) n))) < -3.99999999999999982e-6

   1. Initial program 98.5%

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

      1. sub-neg 98.5%

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

      2. +-commutative 98.5%

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

      3. add-log-exp 98.5%

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

      4. add-log-exp 98.5%

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

      5. sum-log 98.7%

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

      6. pow-to-exp 98.7%

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

      7. un-div-inv 98.7%

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

      8. +-commutative 98.7%

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

      9. log1p-define 98.7%

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

   4. Applied egg-rr 98.7%

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

   5. Taylor expanded in x around 0 98.7%

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

   if -3.99999999999999982e-6 < (-.f64 (pow.f64 (+.f64 x #s(literal 1 binary64)) (/.f64 #s(literal 1 binary64) n)) (pow.f64 x (/.f64 #s(literal 1 binary64) n))) < 2e-16

   1. Initial program 45.7%

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

   3. Taylor expanded in n around inf 79.3%

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

      1. log1p-define 79.3%

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

   5. Simplified 79.3%

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

      1. log1p-undefine 79.3%

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

      2. diff-log 79.4%

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

   7. Applied egg-rr 79.4%

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

      1. +-commutative 79.4%

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

   9. Simplified 79.4%

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

   if 2e-16 < (-.f64 (pow.f64 (+.f64 x #s(literal 1 binary64)) (/.f64 #s(literal 1 binary64) n)) (pow.f64 x (/.f64 #s(literal 1 binary64) n)))

   1. Initial program 46.1%

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

   3. Taylor expanded in n around 0 46.1%

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

      1. log1p-define 95.9%

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

      2. *-rgt-identity 95.9%

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

      3. associate-*l/ 95.9%

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

      4. associate-/l* 95.9%

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

      5. exp-to-pow 95.9%

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

   5. Simplified 95.9%

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

4. Final simplification 83.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \leq -4 \cdot 10^{-6}:\\ \;\;\;\;\log \left(e \cdot e^{-e^{\frac{\log x}{n}}}\right)\\ \mathbf{elif}\;{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \leq 2 \cdot 10^{-16}:\\ \;\;\;\;\frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 82.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ t_1 := {\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - t\_0\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{-8}:\\ \;\;\;\;1 - t\_0\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{-16}:\\ \;\;\;\;\frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + x \cdot \left(\frac{1}{n} + \frac{x \cdot -0.5 + 0.5 \cdot \frac{x}{n}}{n}\right)\right) - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))) (t_1 (- (pow (+ x 1.0) (/ 1.0 n)) t_0)))
   (if (<= t_1 -1e-8)
     (- 1.0 t_0)
     (if (<= t_1 2e-16)
       (/ (log (/ (+ x 1.0) x)) n)
       (-
        (+ 1.0 (* x (+ (/ 1.0 n) (/ (+ (* x -0.5) (* 0.5 (/ x n))) n))))
        t_0)))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double t_1 = pow((x + 1.0), (1.0 / n)) - t_0;
	double tmp;
	if (t_1 <= -1e-8) {
		tmp = 1.0 - t_0;
	} else if (t_1 <= 2e-16) {
		tmp = log(((x + 1.0) / x)) / n;
	} else {
		tmp = (1.0 + (x * ((1.0 / n) + (((x * -0.5) + (0.5 * (x / n))) / n)))) - t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = x ** (1.0d0 / n)
    t_1 = ((x + 1.0d0) ** (1.0d0 / n)) - t_0
    if (t_1 <= (-1d-8)) then
        tmp = 1.0d0 - t_0
    else if (t_1 <= 2d-16) then
        tmp = log(((x + 1.0d0) / x)) / n
    else
        tmp = (1.0d0 + (x * ((1.0d0 / n) + (((x * (-0.5d0)) + (0.5d0 * (x / n))) / n)))) - t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, (1.0 / n));
	double t_1 = Math.pow((x + 1.0), (1.0 / n)) - t_0;
	double tmp;
	if (t_1 <= -1e-8) {
		tmp = 1.0 - t_0;
	} else if (t_1 <= 2e-16) {
		tmp = Math.log(((x + 1.0) / x)) / n;
	} else {
		tmp = (1.0 + (x * ((1.0 / n) + (((x * -0.5) + (0.5 * (x / n))) / n)))) - t_0;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	t_1 = math.pow((x + 1.0), (1.0 / n)) - t_0
	tmp = 0
	if t_1 <= -1e-8:
		tmp = 1.0 - t_0
	elif t_1 <= 2e-16:
		tmp = math.log(((x + 1.0) / x)) / n
	else:
		tmp = (1.0 + (x * ((1.0 / n) + (((x * -0.5) + (0.5 * (x / n))) / n)))) - t_0
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	t_1 = Float64((Float64(x + 1.0) ^ Float64(1.0 / n)) - t_0)
	tmp = 0.0
	if (t_1 <= -1e-8)
		tmp = Float64(1.0 - t_0);
	elseif (t_1 <= 2e-16)
		tmp = Float64(log(Float64(Float64(x + 1.0) / x)) / n);
	else
		tmp = Float64(Float64(1.0 + Float64(x * Float64(Float64(1.0 / n) + Float64(Float64(Float64(x * -0.5) + Float64(0.5 * Float64(x / n))) / n)))) - t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = x ^ (1.0 / n);
	t_1 = ((x + 1.0) ^ (1.0 / n)) - t_0;
	tmp = 0.0;
	if (t_1 <= -1e-8)
		tmp = 1.0 - t_0;
	elseif (t_1 <= 2e-16)
		tmp = log(((x + 1.0) / x)) / n;
	else
		tmp = (1.0 + (x * ((1.0 / n) + (((x * -0.5) + (0.5 * (x / n))) / n)))) - t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[Power[N[(x + 1.0), $MachinePrecision], N[(1.0 / n), $MachinePrecision]], $MachinePrecision] - t$95$0), $MachinePrecision]}, If[LessEqual[t$95$1, -1e-8], N[(1.0 - t$95$0), $MachinePrecision], If[LessEqual[t$95$1, 2e-16], N[(N[Log[N[(N[(x + 1.0), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], N[(N[(1.0 + N[(x * N[(N[(1.0 / n), $MachinePrecision] + N[(N[(N[(x * -0.5), $MachinePrecision] + N[(0.5 * N[(x / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (pow.f64 (+.f64 x #s(literal 1 binary64)) (/.f64 #s(literal 1 binary64) n)) (pow.f64 x (/.f64 #s(literal 1 binary64) n))) < -1e-8

   1. Initial program 97.2%

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

   3. Taylor expanded in x around 0 97.2%

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

      1. *-rgt-identity 97.2%

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

      2. associate-*l/ 97.2%

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

      3. associate-/l* 97.2%

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

      4. exp-to-pow 97.2%

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

   5. Simplified 97.2%

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

   if -1e-8 < (-.f64 (pow.f64 (+.f64 x #s(literal 1 binary64)) (/.f64 #s(literal 1 binary64) n)) (pow.f64 x (/.f64 #s(literal 1 binary64) n))) < 2e-16

   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 n around inf 79.4%

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

      1. log1p-define 79.4%

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

   5. Simplified 79.4%

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

      1. log1p-undefine 79.4%

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

      2. diff-log 79.5%

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

   7. Applied egg-rr 79.5%

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

      1. +-commutative 79.5%

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

   9. Simplified 79.5%

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

   if 2e-16 < (-.f64 (pow.f64 (+.f64 x #s(literal 1 binary64)) (/.f64 #s(literal 1 binary64) n)) (pow.f64 x (/.f64 #s(literal 1 binary64) n)))

   1. Initial program 46.1%

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

   3. Taylor expanded in x around 0 72.5%

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

   4. Taylor expanded in n around inf 72.5%

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

4. Final simplification 80.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \leq -1 \cdot 10^{-8}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \leq 2 \cdot 10^{-16}:\\ \;\;\;\;\frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + x \cdot \left(\frac{1}{n} + \frac{x \cdot -0.5 + 0.5 \cdot \frac{x}{n}}{n}\right)\right) - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 82.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ t_1 := {\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - t\_0\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{-8}:\\ \;\;\;\;\log \left(e^{1 - t\_0}\right)\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{-16}:\\ \;\;\;\;\frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + x \cdot \left(\frac{1}{n} + \frac{x \cdot -0.5 + 0.5 \cdot \frac{x}{n}}{n}\right)\right) - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))) (t_1 (- (pow (+ x 1.0) (/ 1.0 n)) t_0)))
   (if (<= t_1 -1e-8)
     (log (exp (- 1.0 t_0)))
     (if (<= t_1 2e-16)
       (/ (log (/ (+ x 1.0) x)) n)
       (-
        (+ 1.0 (* x (+ (/ 1.0 n) (/ (+ (* x -0.5) (* 0.5 (/ x n))) n))))
        t_0)))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double t_1 = pow((x + 1.0), (1.0 / n)) - t_0;
	double tmp;
	if (t_1 <= -1e-8) {
		tmp = log(exp((1.0 - t_0)));
	} else if (t_1 <= 2e-16) {
		tmp = log(((x + 1.0) / x)) / n;
	} else {
		tmp = (1.0 + (x * ((1.0 / n) + (((x * -0.5) + (0.5 * (x / n))) / n)))) - t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = x ** (1.0d0 / n)
    t_1 = ((x + 1.0d0) ** (1.0d0 / n)) - t_0
    if (t_1 <= (-1d-8)) then
        tmp = log(exp((1.0d0 - t_0)))
    else if (t_1 <= 2d-16) then
        tmp = log(((x + 1.0d0) / x)) / n
    else
        tmp = (1.0d0 + (x * ((1.0d0 / n) + (((x * (-0.5d0)) + (0.5d0 * (x / n))) / n)))) - t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, (1.0 / n));
	double t_1 = Math.pow((x + 1.0), (1.0 / n)) - t_0;
	double tmp;
	if (t_1 <= -1e-8) {
		tmp = Math.log(Math.exp((1.0 - t_0)));
	} else if (t_1 <= 2e-16) {
		tmp = Math.log(((x + 1.0) / x)) / n;
	} else {
		tmp = (1.0 + (x * ((1.0 / n) + (((x * -0.5) + (0.5 * (x / n))) / n)))) - t_0;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	t_1 = math.pow((x + 1.0), (1.0 / n)) - t_0
	tmp = 0
	if t_1 <= -1e-8:
		tmp = math.log(math.exp((1.0 - t_0)))
	elif t_1 <= 2e-16:
		tmp = math.log(((x + 1.0) / x)) / n
	else:
		tmp = (1.0 + (x * ((1.0 / n) + (((x * -0.5) + (0.5 * (x / n))) / n)))) - t_0
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	t_1 = Float64((Float64(x + 1.0) ^ Float64(1.0 / n)) - t_0)
	tmp = 0.0
	if (t_1 <= -1e-8)
		tmp = log(exp(Float64(1.0 - t_0)));
	elseif (t_1 <= 2e-16)
		tmp = Float64(log(Float64(Float64(x + 1.0) / x)) / n);
	else
		tmp = Float64(Float64(1.0 + Float64(x * Float64(Float64(1.0 / n) + Float64(Float64(Float64(x * -0.5) + Float64(0.5 * Float64(x / n))) / n)))) - t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = x ^ (1.0 / n);
	t_1 = ((x + 1.0) ^ (1.0 / n)) - t_0;
	tmp = 0.0;
	if (t_1 <= -1e-8)
		tmp = log(exp((1.0 - t_0)));
	elseif (t_1 <= 2e-16)
		tmp = log(((x + 1.0) / x)) / n;
	else
		tmp = (1.0 + (x * ((1.0 / n) + (((x * -0.5) + (0.5 * (x / n))) / n)))) - t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[Power[N[(x + 1.0), $MachinePrecision], N[(1.0 / n), $MachinePrecision]], $MachinePrecision] - t$95$0), $MachinePrecision]}, If[LessEqual[t$95$1, -1e-8], N[Log[N[Exp[N[(1.0 - t$95$0), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$1, 2e-16], N[(N[Log[N[(N[(x + 1.0), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], N[(N[(1.0 + N[(x * N[(N[(1.0 / n), $MachinePrecision] + N[(N[(N[(x * -0.5), $MachinePrecision] + N[(0.5 * N[(x / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (pow.f64 (+.f64 x #s(literal 1 binary64)) (/.f64 #s(literal 1 binary64) n)) (pow.f64 x (/.f64 #s(literal 1 binary64) n))) < -1e-8

   1. Initial program 97.2%

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

      1. sub-neg 97.2%

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

      2. +-commutative 97.2%

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

      3. add-log-exp 97.2%

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

      4. add-log-exp 97.2%

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

      5. sum-log 97.3%

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

      6. pow-to-exp 97.3%

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

      7. un-div-inv 97.3%

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

      8. +-commutative 97.3%

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

      9. log1p-define 97.3%

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

   4. Applied egg-rr 97.3%

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

   5. Taylor expanded in x around 0 97.3%

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

      1. *-commutative 97.3%

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

      2. prod-exp 97.3%

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

      3. *-rgt-identity 97.3%

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

      4. associate-*r/ 97.3%

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

      5. exp-to-pow 97.3%

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

      6. +-commutative 97.3%

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

      7. sub-neg 97.3%

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

   7. Simplified 97.3%

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

   if -1e-8 < (-.f64 (pow.f64 (+.f64 x #s(literal 1 binary64)) (/.f64 #s(literal 1 binary64) n)) (pow.f64 x (/.f64 #s(literal 1 binary64) n))) < 2e-16

   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 n around inf 79.4%

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

      1. log1p-define 79.4%

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

   5. Simplified 79.4%

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

      1. log1p-undefine 79.4%

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

      2. diff-log 79.5%

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

   7. Applied egg-rr 79.5%

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

      1. +-commutative 79.5%

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

   9. Simplified 79.5%

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

   if 2e-16 < (-.f64 (pow.f64 (+.f64 x #s(literal 1 binary64)) (/.f64 #s(literal 1 binary64) n)) (pow.f64 x (/.f64 #s(literal 1 binary64) n)))

   1. Initial program 46.1%

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

   3. Taylor expanded in x around 0 72.5%

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

   4. Taylor expanded in n around inf 72.5%

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

4. Final simplification 80.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \leq -1 \cdot 10^{-8}:\\ \;\;\;\;\log \left(e^{1 - {x}^{\left(\frac{1}{n}\right)}}\right)\\ \mathbf{elif}\;{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \leq 2 \cdot 10^{-16}:\\ \;\;\;\;\frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + x \cdot \left(\frac{1}{n} + \frac{x \cdot -0.5 + 0.5 \cdot \frac{x}{n}}{n}\right)\right) - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 86.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.0003:\\ \;\;\;\;\frac{\frac{-0.16666666666666666 \cdot \frac{{\log x}^{3}}{n} + {\log x}^{2} \cdot -0.5}{n} - \log x}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{{x}^{\left(\frac{1}{n}\right)}}{n}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 0.0003)
   (/
    (-
     (/
      (+
       (* -0.16666666666666666 (/ (pow (log x) 3.0) n))
       (* (pow (log x) 2.0) -0.5))
      n)
     (log x))
    n)
   (/ (/ (pow x (/ 1.0 n)) n) x)))

C

double code(double x, double n) {
	double tmp;
	if (x <= 0.0003) {
		tmp = ((((-0.16666666666666666 * (pow(log(x), 3.0) / n)) + (pow(log(x), 2.0) * -0.5)) / n) - log(x)) / n;
	} else {
		tmp = (pow(x, (1.0 / n)) / n) / x;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if (x <= 0.0003d0) then
        tmp = (((((-0.16666666666666666d0) * ((log(x) ** 3.0d0) / n)) + ((log(x) ** 2.0d0) * (-0.5d0))) / n) - log(x)) / n
    else
        tmp = ((x ** (1.0d0 / n)) / n) / x
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (x <= 0.0003) {
		tmp = ((((-0.16666666666666666 * (Math.pow(Math.log(x), 3.0) / n)) + (Math.pow(Math.log(x), 2.0) * -0.5)) / n) - Math.log(x)) / n;
	} else {
		tmp = (Math.pow(x, (1.0 / n)) / n) / x;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 0.0003:
		tmp = ((((-0.16666666666666666 * (math.pow(math.log(x), 3.0) / n)) + (math.pow(math.log(x), 2.0) * -0.5)) / n) - math.log(x)) / n
	else:
		tmp = (math.pow(x, (1.0 / n)) / n) / x
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 0.0003)
		tmp = Float64(Float64(Float64(Float64(Float64(-0.16666666666666666 * Float64((log(x) ^ 3.0) / n)) + Float64((log(x) ^ 2.0) * -0.5)) / n) - log(x)) / n);
	else
		tmp = Float64(Float64((x ^ Float64(1.0 / n)) / n) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (x <= 0.0003)
		tmp = ((((-0.16666666666666666 * ((log(x) ^ 3.0) / n)) + ((log(x) ^ 2.0) * -0.5)) / n) - log(x)) / n;
	else
		tmp = ((x ^ (1.0 / n)) / n) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 0.0003], N[(N[(N[(N[(N[(-0.16666666666666666 * N[(N[Power[N[Log[x], $MachinePrecision], 3.0], $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision] + N[(N[Power[N[Log[x], $MachinePrecision], 2.0], $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision] - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], N[(N[(N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.99999999999999974e-4

   1. Initial program 35.4%

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

   3. Taylor expanded in x around 0 34.7%

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

      1. *-rgt-identity 34.7%

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

      2. associate-*l/ 34.7%

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

      3. associate-/l* 34.7%

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

      4. exp-to-pow 34.7%

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

   5. Simplified 34.7%

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

   6. Taylor expanded in n around -inf 84.0%

      \[\leadsto \color{blue}{-1 \cdot \frac{-1 \cdot \frac{-0.16666666666666666 \cdot \frac{{\log x}^{3}}{n} - 0.5 \cdot {\log x}^{2}}{n} - -1 \cdot \log x}{n}} \]
   7. Step-by-step derivation

      1. mul-1-neg 84.0%

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

   8. Simplified 84.0%

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

   if 2.99999999999999974e-4 < x

   1. Initial program 67.3%

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

   3. Taylor expanded in x around inf 95.2%

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

      1. associate-/r* 97.2%

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

      2. mul-1-neg 97.2%

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

      3. log-rec 97.2%

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

      4. mul-1-neg 97.2%

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

      5. distribute-neg-frac 97.2%

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

      6. mul-1-neg 97.2%

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

      7. remove-double-neg 97.2%

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

      8. *-rgt-identity 97.2%

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

      9. associate-/l* 97.2%

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

      10. exp-to-pow 97.2%

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

   5. Simplified 97.2%

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

4. Final simplification 90.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.0003:\\ \;\;\;\;\frac{\frac{-0.16666666666666666 \cdot \frac{{\log x}^{3}}{n} + {\log x}^{2} \cdot -0.5}{n} - \log x}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{{x}^{\left(\frac{1}{n}\right)}}{n}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 80.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\frac{1 + \frac{0.3333333333333333 \cdot \frac{1}{x} - 0.5}{x}}{x}}{n}\\ t_1 := {x}^{\left(\frac{1}{n}\right)}\\ t_2 := \frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \mathbf{if}\;\frac{1}{n} \leq -0.2:\\ \;\;\;\;\frac{\frac{t\_1}{n}}{x}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-142}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-78}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-19}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{+131}:\\ \;\;\;\;\left(1 - t\_1\right) + \frac{x}{n}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0
         (/ (/ (+ 1.0 (/ (- (* 0.3333333333333333 (/ 1.0 x)) 0.5) x)) x) n))
        (t_1 (pow x (/ 1.0 n)))
        (t_2 (/ (log (/ (+ x 1.0) x)) n)))
   (if (<= (/ 1.0 n) -0.2)
     (/ (/ t_1 n) x)
     (if (<= (/ 1.0 n) 1e-142)
       t_2
       (if (<= (/ 1.0 n) 1e-78)
         t_0
         (if (<= (/ 1.0 n) 5e-19)
           t_2
           (if (<= (/ 1.0 n) 5e+131) (+ (- 1.0 t_1) (/ x n)) t_0)))))))

C

double code(double x, double n) {
	double t_0 = ((1.0 + (((0.3333333333333333 * (1.0 / x)) - 0.5) / x)) / x) / n;
	double t_1 = pow(x, (1.0 / n));
	double t_2 = log(((x + 1.0) / x)) / n;
	double tmp;
	if ((1.0 / n) <= -0.2) {
		tmp = (t_1 / n) / x;
	} else if ((1.0 / n) <= 1e-142) {
		tmp = t_2;
	} else if ((1.0 / n) <= 1e-78) {
		tmp = t_0;
	} else if ((1.0 / n) <= 5e-19) {
		tmp = t_2;
	} else if ((1.0 / n) <= 5e+131) {
		tmp = (1.0 - t_1) + (x / n);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = ((1.0d0 + (((0.3333333333333333d0 * (1.0d0 / x)) - 0.5d0) / x)) / x) / n
    t_1 = x ** (1.0d0 / n)
    t_2 = log(((x + 1.0d0) / x)) / n
    if ((1.0d0 / n) <= (-0.2d0)) then
        tmp = (t_1 / n) / x
    else if ((1.0d0 / n) <= 1d-142) then
        tmp = t_2
    else if ((1.0d0 / n) <= 1d-78) then
        tmp = t_0
    else if ((1.0d0 / n) <= 5d-19) then
        tmp = t_2
    else if ((1.0d0 / n) <= 5d+131) then
        tmp = (1.0d0 - t_1) + (x / n)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = ((1.0 + (((0.3333333333333333 * (1.0 / x)) - 0.5) / x)) / x) / n;
	double t_1 = Math.pow(x, (1.0 / n));
	double t_2 = Math.log(((x + 1.0) / x)) / n;
	double tmp;
	if ((1.0 / n) <= -0.2) {
		tmp = (t_1 / n) / x;
	} else if ((1.0 / n) <= 1e-142) {
		tmp = t_2;
	} else if ((1.0 / n) <= 1e-78) {
		tmp = t_0;
	} else if ((1.0 / n) <= 5e-19) {
		tmp = t_2;
	} else if ((1.0 / n) <= 5e+131) {
		tmp = (1.0 - t_1) + (x / n);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = ((1.0 + (((0.3333333333333333 * (1.0 / x)) - 0.5) / x)) / x) / n
	t_1 = math.pow(x, (1.0 / n))
	t_2 = math.log(((x + 1.0) / x)) / n
	tmp = 0
	if (1.0 / n) <= -0.2:
		tmp = (t_1 / n) / x
	elif (1.0 / n) <= 1e-142:
		tmp = t_2
	elif (1.0 / n) <= 1e-78:
		tmp = t_0
	elif (1.0 / n) <= 5e-19:
		tmp = t_2
	elif (1.0 / n) <= 5e+131:
		tmp = (1.0 - t_1) + (x / n)
	else:
		tmp = t_0
	return tmp

Julia

function code(x, n)
	t_0 = Float64(Float64(Float64(1.0 + Float64(Float64(Float64(0.3333333333333333 * Float64(1.0 / x)) - 0.5) / x)) / x) / n)
	t_1 = x ^ Float64(1.0 / n)
	t_2 = Float64(log(Float64(Float64(x + 1.0) / x)) / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -0.2)
		tmp = Float64(Float64(t_1 / n) / x);
	elseif (Float64(1.0 / n) <= 1e-142)
		tmp = t_2;
	elseif (Float64(1.0 / n) <= 1e-78)
		tmp = t_0;
	elseif (Float64(1.0 / n) <= 5e-19)
		tmp = t_2;
	elseif (Float64(1.0 / n) <= 5e+131)
		tmp = Float64(Float64(1.0 - t_1) + Float64(x / n));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = ((1.0 + (((0.3333333333333333 * (1.0 / x)) - 0.5) / x)) / x) / n;
	t_1 = x ^ (1.0 / n);
	t_2 = log(((x + 1.0) / x)) / n;
	tmp = 0.0;
	if ((1.0 / n) <= -0.2)
		tmp = (t_1 / n) / x;
	elseif ((1.0 / n) <= 1e-142)
		tmp = t_2;
	elseif ((1.0 / n) <= 1e-78)
		tmp = t_0;
	elseif ((1.0 / n) <= 5e-19)
		tmp = t_2;
	elseif ((1.0 / n) <= 5e+131)
		tmp = (1.0 - t_1) + (x / n);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(N[(N[(1.0 + N[(N[(N[(0.3333333333333333 * N[(1.0 / x), $MachinePrecision]), $MachinePrecision] - 0.5), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] / n), $MachinePrecision]}, Block[{t$95$1 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[Log[N[(N[(x + 1.0), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -0.2], N[(N[(t$95$1 / n), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 1e-142], t$95$2, If[LessEqual[N[(1.0 / n), $MachinePrecision], 1e-78], t$95$0, If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e-19], t$95$2, If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e+131], N[(N[(1.0 - t$95$1), $MachinePrecision] + N[(x / n), $MachinePrecision]), $MachinePrecision], t$95$0]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -0.20000000000000001

   1. Initial program 98.5%

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

   3. Taylor expanded in x around inf 99.9%

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

      1. associate-/r* 99.9%

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

      2. mul-1-neg 99.9%

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

      3. log-rec 99.9%

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

      4. mul-1-neg 99.9%

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

      5. distribute-neg-frac 99.9%

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

      6. mul-1-neg 99.9%

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

      7. remove-double-neg 99.9%

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

      8. *-rgt-identity 99.9%

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

      9. associate-/l* 99.9%

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

      10. exp-to-pow 99.9%

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

   5. Simplified 99.9%

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

   if -0.20000000000000001 < (/.f64 #s(literal 1 binary64) n) < 1e-142 or 9.99999999999999999e-79 < (/.f64 #s(literal 1 binary64) n) < 5.0000000000000004e-19

   1. Initial program 34.1%

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

   3. Taylor expanded in n around inf 79.4%

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

      1. log1p-define 79.4%

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

   5. Simplified 79.4%

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

      1. log1p-undefine 79.4%

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

      2. diff-log 79.5%

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

   7. Applied egg-rr 79.5%

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

      1. +-commutative 79.5%

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

   9. Simplified 79.5%

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

   if 1e-142 < (/.f64 #s(literal 1 binary64) n) < 9.99999999999999999e-79 or 4.99999999999999995e131 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 26.0%

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

   3. Taylor expanded in n around inf 30.0%

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

      1. log1p-define 30.0%

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

   5. Simplified 30.0%

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

   6. Taylor expanded in x around -inf 77.6%

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

   if 5.0000000000000004e-19 < (/.f64 #s(literal 1 binary64) n) < 4.99999999999999995e131

   1. Initial program 78.4%

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

      1. sub-neg 78.4%

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

      2. +-commutative 78.4%

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

      3. add-log-exp 78.0%

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

      4. add-log-exp 78.0%

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

      5. sum-log 77.8%

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

      6. pow-to-exp 77.8%

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

      7. un-div-inv 77.8%

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

      8. +-commutative 77.8%

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

      9. log1p-define 83.8%

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

   4. Applied egg-rr 83.8%

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

   5. Taylor expanded in x around 0 72.0%

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

      1. prod-exp 72.0%

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

      2. sub-neg 72.0%

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

      3. rem-log-exp 72.5%

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

      4. *-rgt-identity 72.5%

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

      5. associate-*r/ 72.5%

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

      6. exp-to-pow 72.5%

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

   7. Simplified 72.5%

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

4. Final simplification 83.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -0.2:\\ \;\;\;\;\frac{\frac{{x}^{\left(\frac{1}{n}\right)}}{n}}{x}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-142}:\\ \;\;\;\;\frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-78}:\\ \;\;\;\;\frac{\frac{1 + \frac{0.3333333333333333 \cdot \frac{1}{x} - 0.5}{x}}{x}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-19}:\\ \;\;\;\;\frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{+131}:\\ \;\;\;\;\left(1 - {x}^{\left(\frac{1}{n}\right)}\right) + \frac{x}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \frac{0.3333333333333333 \cdot \frac{1}{x} - 0.5}{x}}{x}}{n}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 80.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\frac{1 + \frac{0.3333333333333333 \cdot \frac{1}{x} - 0.5}{x}}{x}}{n}\\ t_1 := {x}^{\left(\frac{1}{n}\right)}\\ t_2 := \frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \mathbf{if}\;\frac{1}{n} \leq -0.2:\\ \;\;\;\;\frac{\frac{t\_1}{n}}{x}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-142}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-78}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-8}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{+131}:\\ \;\;\;\;1 - t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0
         (/ (/ (+ 1.0 (/ (- (* 0.3333333333333333 (/ 1.0 x)) 0.5) x)) x) n))
        (t_1 (pow x (/ 1.0 n)))
        (t_2 (/ (log (/ (+ x 1.0) x)) n)))
   (if (<= (/ 1.0 n) -0.2)
     (/ (/ t_1 n) x)
     (if (<= (/ 1.0 n) 1e-142)
       t_2
       (if (<= (/ 1.0 n) 1e-78)
         t_0
         (if (<= (/ 1.0 n) 2e-8)
           t_2
           (if (<= (/ 1.0 n) 5e+131) (- 1.0 t_1) t_0)))))))

C

double code(double x, double n) {
	double t_0 = ((1.0 + (((0.3333333333333333 * (1.0 / x)) - 0.5) / x)) / x) / n;
	double t_1 = pow(x, (1.0 / n));
	double t_2 = log(((x + 1.0) / x)) / n;
	double tmp;
	if ((1.0 / n) <= -0.2) {
		tmp = (t_1 / n) / x;
	} else if ((1.0 / n) <= 1e-142) {
		tmp = t_2;
	} else if ((1.0 / n) <= 1e-78) {
		tmp = t_0;
	} else if ((1.0 / n) <= 2e-8) {
		tmp = t_2;
	} else if ((1.0 / n) <= 5e+131) {
		tmp = 1.0 - t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = ((1.0d0 + (((0.3333333333333333d0 * (1.0d0 / x)) - 0.5d0) / x)) / x) / n
    t_1 = x ** (1.0d0 / n)
    t_2 = log(((x + 1.0d0) / x)) / n
    if ((1.0d0 / n) <= (-0.2d0)) then
        tmp = (t_1 / n) / x
    else if ((1.0d0 / n) <= 1d-142) then
        tmp = t_2
    else if ((1.0d0 / n) <= 1d-78) then
        tmp = t_0
    else if ((1.0d0 / n) <= 2d-8) then
        tmp = t_2
    else if ((1.0d0 / n) <= 5d+131) then
        tmp = 1.0d0 - t_1
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = ((1.0 + (((0.3333333333333333 * (1.0 / x)) - 0.5) / x)) / x) / n;
	double t_1 = Math.pow(x, (1.0 / n));
	double t_2 = Math.log(((x + 1.0) / x)) / n;
	double tmp;
	if ((1.0 / n) <= -0.2) {
		tmp = (t_1 / n) / x;
	} else if ((1.0 / n) <= 1e-142) {
		tmp = t_2;
	} else if ((1.0 / n) <= 1e-78) {
		tmp = t_0;
	} else if ((1.0 / n) <= 2e-8) {
		tmp = t_2;
	} else if ((1.0 / n) <= 5e+131) {
		tmp = 1.0 - t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = ((1.0 + (((0.3333333333333333 * (1.0 / x)) - 0.5) / x)) / x) / n
	t_1 = math.pow(x, (1.0 / n))
	t_2 = math.log(((x + 1.0) / x)) / n
	tmp = 0
	if (1.0 / n) <= -0.2:
		tmp = (t_1 / n) / x
	elif (1.0 / n) <= 1e-142:
		tmp = t_2
	elif (1.0 / n) <= 1e-78:
		tmp = t_0
	elif (1.0 / n) <= 2e-8:
		tmp = t_2
	elif (1.0 / n) <= 5e+131:
		tmp = 1.0 - t_1
	else:
		tmp = t_0
	return tmp

Julia

function code(x, n)
	t_0 = Float64(Float64(Float64(1.0 + Float64(Float64(Float64(0.3333333333333333 * Float64(1.0 / x)) - 0.5) / x)) / x) / n)
	t_1 = x ^ Float64(1.0 / n)
	t_2 = Float64(log(Float64(Float64(x + 1.0) / x)) / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -0.2)
		tmp = Float64(Float64(t_1 / n) / x);
	elseif (Float64(1.0 / n) <= 1e-142)
		tmp = t_2;
	elseif (Float64(1.0 / n) <= 1e-78)
		tmp = t_0;
	elseif (Float64(1.0 / n) <= 2e-8)
		tmp = t_2;
	elseif (Float64(1.0 / n) <= 5e+131)
		tmp = Float64(1.0 - t_1);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = ((1.0 + (((0.3333333333333333 * (1.0 / x)) - 0.5) / x)) / x) / n;
	t_1 = x ^ (1.0 / n);
	t_2 = log(((x + 1.0) / x)) / n;
	tmp = 0.0;
	if ((1.0 / n) <= -0.2)
		tmp = (t_1 / n) / x;
	elseif ((1.0 / n) <= 1e-142)
		tmp = t_2;
	elseif ((1.0 / n) <= 1e-78)
		tmp = t_0;
	elseif ((1.0 / n) <= 2e-8)
		tmp = t_2;
	elseif ((1.0 / n) <= 5e+131)
		tmp = 1.0 - t_1;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(N[(N[(1.0 + N[(N[(N[(0.3333333333333333 * N[(1.0 / x), $MachinePrecision]), $MachinePrecision] - 0.5), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] / n), $MachinePrecision]}, Block[{t$95$1 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[Log[N[(N[(x + 1.0), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -0.2], N[(N[(t$95$1 / n), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 1e-142], t$95$2, If[LessEqual[N[(1.0 / n), $MachinePrecision], 1e-78], t$95$0, If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-8], t$95$2, If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e+131], N[(1.0 - t$95$1), $MachinePrecision], t$95$0]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -0.20000000000000001

   1. Initial program 98.5%

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

   3. Taylor expanded in x around inf 99.9%

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

      1. associate-/r* 99.9%

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

      2. mul-1-neg 99.9%

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

      3. log-rec 99.9%

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

      4. mul-1-neg 99.9%

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

      5. distribute-neg-frac 99.9%

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

      6. mul-1-neg 99.9%

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

      7. remove-double-neg 99.9%

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

      8. *-rgt-identity 99.9%

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

      9. associate-/l* 99.9%

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

      10. exp-to-pow 99.9%

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

   5. Simplified 99.9%

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

   if -0.20000000000000001 < (/.f64 #s(literal 1 binary64) n) < 1e-142 or 9.99999999999999999e-79 < (/.f64 #s(literal 1 binary64) n) < 2e-8

   1. Initial program 34.1%

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

   3. Taylor expanded in n around inf 78.7%

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

      1. log1p-define 78.7%

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

   5. Simplified 78.7%

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

      1. log1p-undefine 78.7%

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

      2. diff-log 78.9%

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

   7. Applied egg-rr 78.9%

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

      1. +-commutative 78.9%

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

   9. Simplified 78.9%

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

   if 1e-142 < (/.f64 #s(literal 1 binary64) n) < 9.99999999999999999e-79 or 4.99999999999999995e131 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 26.0%

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

   3. Taylor expanded in n around inf 30.0%

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

      1. log1p-define 30.0%

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

   5. Simplified 30.0%

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

   6. Taylor expanded in x around -inf 77.6%

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

   if 2e-8 < (/.f64 #s(literal 1 binary64) n) < 4.99999999999999995e131

   1. Initial program 84.9%

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

   3. Taylor expanded in x around 0 78.0%

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

      1. *-rgt-identity 78.0%

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

      2. associate-*l/ 78.0%

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

      3. associate-/l* 78.0%

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

      4. exp-to-pow 78.0%

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

   5. Simplified 78.0%

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

4. Final simplification 83.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -0.2:\\ \;\;\;\;\frac{\frac{{x}^{\left(\frac{1}{n}\right)}}{n}}{x}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-142}:\\ \;\;\;\;\frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-78}:\\ \;\;\;\;\frac{\frac{1 + \frac{0.3333333333333333 \cdot \frac{1}{x} - 0.5}{x}}{x}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-8}:\\ \;\;\;\;\frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{+131}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \frac{0.3333333333333333 \cdot \frac{1}{x} - 0.5}{x}}{x}}{n}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 67.0% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\frac{1 + \frac{0.3333333333333333 \cdot \frac{1}{x} - 0.5}{x}}{x}}{n}\\ t_1 := \frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \mathbf{if}\;\frac{1}{n} \leq 10^{-142}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-78}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-8}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{+131}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0
         (/ (/ (+ 1.0 (/ (- (* 0.3333333333333333 (/ 1.0 x)) 0.5) x)) x) n))
        (t_1 (/ (log (/ (+ x 1.0) x)) n)))
   (if (<= (/ 1.0 n) 1e-142)
     t_1
     (if (<= (/ 1.0 n) 1e-78)
       t_0
       (if (<= (/ 1.0 n) 2e-8)
         t_1
         (if (<= (/ 1.0 n) 5e+131) (- 1.0 (pow x (/ 1.0 n))) t_0))))))

C

double code(double x, double n) {
	double t_0 = ((1.0 + (((0.3333333333333333 * (1.0 / x)) - 0.5) / x)) / x) / n;
	double t_1 = log(((x + 1.0) / x)) / n;
	double tmp;
	if ((1.0 / n) <= 1e-142) {
		tmp = t_1;
	} else if ((1.0 / n) <= 1e-78) {
		tmp = t_0;
	} else if ((1.0 / n) <= 2e-8) {
		tmp = t_1;
	} else if ((1.0 / n) <= 5e+131) {
		tmp = 1.0 - pow(x, (1.0 / n));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = ((1.0d0 + (((0.3333333333333333d0 * (1.0d0 / x)) - 0.5d0) / x)) / x) / n
    t_1 = log(((x + 1.0d0) / x)) / n
    if ((1.0d0 / n) <= 1d-142) then
        tmp = t_1
    else if ((1.0d0 / n) <= 1d-78) then
        tmp = t_0
    else if ((1.0d0 / n) <= 2d-8) then
        tmp = t_1
    else if ((1.0d0 / n) <= 5d+131) then
        tmp = 1.0d0 - (x ** (1.0d0 / n))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = ((1.0 + (((0.3333333333333333 * (1.0 / x)) - 0.5) / x)) / x) / n;
	double t_1 = Math.log(((x + 1.0) / x)) / n;
	double tmp;
	if ((1.0 / n) <= 1e-142) {
		tmp = t_1;
	} else if ((1.0 / n) <= 1e-78) {
		tmp = t_0;
	} else if ((1.0 / n) <= 2e-8) {
		tmp = t_1;
	} else if ((1.0 / n) <= 5e+131) {
		tmp = 1.0 - Math.pow(x, (1.0 / n));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = ((1.0 + (((0.3333333333333333 * (1.0 / x)) - 0.5) / x)) / x) / n
	t_1 = math.log(((x + 1.0) / x)) / n
	tmp = 0
	if (1.0 / n) <= 1e-142:
		tmp = t_1
	elif (1.0 / n) <= 1e-78:
		tmp = t_0
	elif (1.0 / n) <= 2e-8:
		tmp = t_1
	elif (1.0 / n) <= 5e+131:
		tmp = 1.0 - math.pow(x, (1.0 / n))
	else:
		tmp = t_0
	return tmp

Julia

function code(x, n)
	t_0 = Float64(Float64(Float64(1.0 + Float64(Float64(Float64(0.3333333333333333 * Float64(1.0 / x)) - 0.5) / x)) / x) / n)
	t_1 = Float64(log(Float64(Float64(x + 1.0) / x)) / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= 1e-142)
		tmp = t_1;
	elseif (Float64(1.0 / n) <= 1e-78)
		tmp = t_0;
	elseif (Float64(1.0 / n) <= 2e-8)
		tmp = t_1;
	elseif (Float64(1.0 / n) <= 5e+131)
		tmp = Float64(1.0 - (x ^ Float64(1.0 / n)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = ((1.0 + (((0.3333333333333333 * (1.0 / x)) - 0.5) / x)) / x) / n;
	t_1 = log(((x + 1.0) / x)) / n;
	tmp = 0.0;
	if ((1.0 / n) <= 1e-142)
		tmp = t_1;
	elseif ((1.0 / n) <= 1e-78)
		tmp = t_0;
	elseif ((1.0 / n) <= 2e-8)
		tmp = t_1;
	elseif ((1.0 / n) <= 5e+131)
		tmp = 1.0 - (x ^ (1.0 / n));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(N[(N[(1.0 + N[(N[(N[(0.3333333333333333 * N[(1.0 / x), $MachinePrecision]), $MachinePrecision] - 0.5), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] / n), $MachinePrecision]}, Block[{t$95$1 = N[(N[Log[N[(N[(x + 1.0), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], 1e-142], t$95$1, If[LessEqual[N[(1.0 / n), $MachinePrecision], 1e-78], t$95$0, If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-8], t$95$1, If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e+131], N[(1.0 - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$0]]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 #s(literal 1 binary64) n) < 1e-142 or 9.99999999999999999e-79 < (/.f64 #s(literal 1 binary64) n) < 2e-8

   1. Initial program 54.5%

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

   3. Taylor expanded in n around inf 72.2%

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

      1. log1p-define 72.2%

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

   5. Simplified 72.2%

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

      1. log1p-undefine 72.2%

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

      2. diff-log 72.3%

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

   7. Applied egg-rr 72.3%

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

      1. +-commutative 72.3%

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

   9. Simplified 72.3%

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

   if 1e-142 < (/.f64 #s(literal 1 binary64) n) < 9.99999999999999999e-79 or 4.99999999999999995e131 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 26.0%

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

   3. Taylor expanded in n around inf 30.0%

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

      1. log1p-define 30.0%

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

   5. Simplified 30.0%

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

   6. Taylor expanded in x around -inf 77.6%

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

   if 2e-8 < (/.f64 #s(literal 1 binary64) n) < 4.99999999999999995e131

   1. Initial program 84.9%

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

   3. Taylor expanded in x around 0 78.0%

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

      1. *-rgt-identity 78.0%

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

      2. associate-*l/ 78.0%

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

      3. associate-/l* 78.0%

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

      4. exp-to-pow 78.0%

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

   5. Simplified 78.0%

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

4. Final simplification 73.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq 10^{-142}:\\ \;\;\;\;\frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-78}:\\ \;\;\;\;\frac{\frac{1 + \frac{0.3333333333333333 \cdot \frac{1}{x} - 0.5}{x}}{x}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-8}:\\ \;\;\;\;\frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{+131}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \frac{0.3333333333333333 \cdot \frac{1}{x} - 0.5}{x}}{x}}{n}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 60.9% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\log x}{-n}\\ t_1 := 1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;x \leq 1.85 \cdot 10^{-266}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 5.4 \cdot 10^{-217}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 1.4 \cdot 10^{-176}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 9 \cdot 10^{-152}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 0.88:\\ \;\;\;\;\frac{x}{n} - \frac{\log x}{n}\\ \mathbf{elif}\;x \leq 7.4 \cdot 10^{+135}:\\ \;\;\;\;\frac{\frac{1 + \frac{\frac{0.3333333333333333 + 0.25 \cdot \frac{-1}{x}}{x} - 0.5}{x}}{x}}{n}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (/ (log x) (- n))) (t_1 (- 1.0 (pow x (/ 1.0 n)))))
   (if (<= x 1.85e-266)
     t_0
     (if (<= x 5.4e-217)
       t_1
       (if (<= x 1.4e-176)
         t_0
         (if (<= x 9e-152)
           t_1
           (if (<= x 0.88)
             (- (/ x n) (/ (log x) n))
             (if (<= x 7.4e+135)
               (/
                (/
                 (+
                  1.0
                  (/
                   (- (/ (+ 0.3333333333333333 (* 0.25 (/ -1.0 x))) x) 0.5)
                   x))
                 x)
                n)
               0.0))))))))

C

double code(double x, double n) {
	double t_0 = log(x) / -n;
	double t_1 = 1.0 - pow(x, (1.0 / n));
	double tmp;
	if (x <= 1.85e-266) {
		tmp = t_0;
	} else if (x <= 5.4e-217) {
		tmp = t_1;
	} else if (x <= 1.4e-176) {
		tmp = t_0;
	} else if (x <= 9e-152) {
		tmp = t_1;
	} else if (x <= 0.88) {
		tmp = (x / n) - (log(x) / n);
	} else if (x <= 7.4e+135) {
		tmp = ((1.0 + ((((0.3333333333333333 + (0.25 * (-1.0 / x))) / x) - 0.5) / x)) / x) / n;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = log(x) / -n
    t_1 = 1.0d0 - (x ** (1.0d0 / n))
    if (x <= 1.85d-266) then
        tmp = t_0
    else if (x <= 5.4d-217) then
        tmp = t_1
    else if (x <= 1.4d-176) then
        tmp = t_0
    else if (x <= 9d-152) then
        tmp = t_1
    else if (x <= 0.88d0) then
        tmp = (x / n) - (log(x) / n)
    else if (x <= 7.4d+135) then
        tmp = ((1.0d0 + ((((0.3333333333333333d0 + (0.25d0 * ((-1.0d0) / x))) / x) - 0.5d0) / x)) / x) / n
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = Math.log(x) / -n;
	double t_1 = 1.0 - Math.pow(x, (1.0 / n));
	double tmp;
	if (x <= 1.85e-266) {
		tmp = t_0;
	} else if (x <= 5.4e-217) {
		tmp = t_1;
	} else if (x <= 1.4e-176) {
		tmp = t_0;
	} else if (x <= 9e-152) {
		tmp = t_1;
	} else if (x <= 0.88) {
		tmp = (x / n) - (Math.log(x) / n);
	} else if (x <= 7.4e+135) {
		tmp = ((1.0 + ((((0.3333333333333333 + (0.25 * (-1.0 / x))) / x) - 0.5) / x)) / x) / n;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.log(x) / -n
	t_1 = 1.0 - math.pow(x, (1.0 / n))
	tmp = 0
	if x <= 1.85e-266:
		tmp = t_0
	elif x <= 5.4e-217:
		tmp = t_1
	elif x <= 1.4e-176:
		tmp = t_0
	elif x <= 9e-152:
		tmp = t_1
	elif x <= 0.88:
		tmp = (x / n) - (math.log(x) / n)
	elif x <= 7.4e+135:
		tmp = ((1.0 + ((((0.3333333333333333 + (0.25 * (-1.0 / x))) / x) - 0.5) / x)) / x) / n
	else:
		tmp = 0.0
	return tmp

Julia

function code(x, n)
	t_0 = Float64(log(x) / Float64(-n))
	t_1 = Float64(1.0 - (x ^ Float64(1.0 / n)))
	tmp = 0.0
	if (x <= 1.85e-266)
		tmp = t_0;
	elseif (x <= 5.4e-217)
		tmp = t_1;
	elseif (x <= 1.4e-176)
		tmp = t_0;
	elseif (x <= 9e-152)
		tmp = t_1;
	elseif (x <= 0.88)
		tmp = Float64(Float64(x / n) - Float64(log(x) / n));
	elseif (x <= 7.4e+135)
		tmp = Float64(Float64(Float64(1.0 + Float64(Float64(Float64(Float64(0.3333333333333333 + Float64(0.25 * Float64(-1.0 / x))) / x) - 0.5) / x)) / x) / n);
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = log(x) / -n;
	t_1 = 1.0 - (x ^ (1.0 / n));
	tmp = 0.0;
	if (x <= 1.85e-266)
		tmp = t_0;
	elseif (x <= 5.4e-217)
		tmp = t_1;
	elseif (x <= 1.4e-176)
		tmp = t_0;
	elseif (x <= 9e-152)
		tmp = t_1;
	elseif (x <= 0.88)
		tmp = (x / n) - (log(x) / n);
	elseif (x <= 7.4e+135)
		tmp = ((1.0 + ((((0.3333333333333333 + (0.25 * (-1.0 / x))) / x) - 0.5) / x)) / x) / n;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(N[Log[x], $MachinePrecision] / (-n)), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 1.85e-266], t$95$0, If[LessEqual[x, 5.4e-217], t$95$1, If[LessEqual[x, 1.4e-176], t$95$0, If[LessEqual[x, 9e-152], t$95$1, If[LessEqual[x, 0.88], N[(N[(x / n), $MachinePrecision] - N[(N[Log[x], $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 7.4e+135], N[(N[(N[(1.0 + N[(N[(N[(N[(0.3333333333333333 + N[(0.25 * N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] - 0.5), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] / n), $MachinePrecision], 0.0]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if x < 1.8500000000000001e-266 or 5.40000000000000032e-217 < x < 1.4000000000000001e-176

   1. Initial program 20.8%

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

   3. Taylor expanded in x around 0 20.8%

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

      1. *-rgt-identity 20.8%

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

      2. associate-*l/ 20.8%

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

      3. associate-/l* 20.8%

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

      4. exp-to-pow 20.8%

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

   5. Simplified 20.8%

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

   6. Taylor expanded in n around inf 72.3%

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

      1. mul-1-neg 72.3%

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

   8. Simplified 72.3%

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

   if 1.8500000000000001e-266 < x < 5.40000000000000032e-217 or 1.4000000000000001e-176 < x < 9.0000000000000008e-152

   1. Initial program 64.6%

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

   3. Taylor expanded in x around 0 64.6%

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

      1. *-rgt-identity 64.6%

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

      2. associate-*l/ 64.6%

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

      3. associate-/l* 64.6%

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

      4. exp-to-pow 64.6%

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

   5. Simplified 64.6%

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

   if 9.0000000000000008e-152 < x < 0.880000000000000004

   1. Initial program 30.3%

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

   3. Taylor expanded in n around inf 59.2%

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

      1. log1p-define 59.2%

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

   5. Simplified 59.2%

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

   6. Taylor expanded in x around 0 58.2%

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

      1. +-commutative 58.2%

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

      2. mul-1-neg 58.2%

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

      3. unsub-neg 58.2%

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

   8. Simplified 58.2%

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

   if 0.880000000000000004 < x < 7.39999999999999994e135

   1. Initial program 46.7%

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

   3. Taylor expanded in n around inf 49.3%

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

      1. log1p-define 49.3%

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

   5. Simplified 49.3%

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

   6. Taylor expanded in x around -inf 71.5%

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

   if 7.39999999999999994e135 < x

   1. Initial program 81.6%

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

      1. sub-neg 81.6%

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

      2. +-commutative 81.6%

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

      3. add-log-exp 81.6%

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

      4. add-log-exp 81.6%

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

      5. sum-log 81.6%

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

      6. pow-to-exp 81.6%

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

      7. un-div-inv 81.6%

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

      8. +-commutative 81.6%

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

      9. log1p-define 81.6%

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

   4. Applied egg-rr 81.6%

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

   5. Taylor expanded in x around inf 81.6%

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

      1. exp-neg 81.6%

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

      2. rgt-mult-inverse 81.6%

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

      3. metadata-eval 81.6%

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

   7. Simplified 81.6%

      \[\leadsto \color{blue}{0} \]
3. Recombined 5 regimes into one program.

4. Final simplification 70.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.85 \cdot 10^{-266}:\\ \;\;\;\;\frac{\log x}{-n}\\ \mathbf{elif}\;x \leq 5.4 \cdot 10^{-217}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;x \leq 1.4 \cdot 10^{-176}:\\ \;\;\;\;\frac{\log x}{-n}\\ \mathbf{elif}\;x \leq 9 \cdot 10^{-152}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;x \leq 0.88:\\ \;\;\;\;\frac{x}{n} - \frac{\log x}{n}\\ \mathbf{elif}\;x \leq 7.4 \cdot 10^{+135}:\\ \;\;\;\;\frac{\frac{1 + \frac{\frac{0.3333333333333333 + 0.25 \cdot \frac{-1}{x}}{x} - 0.5}{x}}{x}}{n}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 61.0% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\log x}{-n}\\ t_1 := 1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;x \leq 5.2 \cdot 10^{-267}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 1.32 \cdot 10^{-216}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 1.15 \cdot 10^{-176}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 9 \cdot 10^{-152}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 0.88:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{elif}\;x \leq 4.9 \cdot 10^{+135}:\\ \;\;\;\;\frac{\frac{1 + \frac{\frac{0.3333333333333333 + 0.25 \cdot \frac{-1}{x}}{x} - 0.5}{x}}{x}}{n}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (/ (log x) (- n))) (t_1 (- 1.0 (pow x (/ 1.0 n)))))
   (if (<= x 5.2e-267)
     t_0
     (if (<= x 1.32e-216)
       t_1
       (if (<= x 1.15e-176)
         t_0
         (if (<= x 9e-152)
           t_1
           (if (<= x 0.88)
             (/ (- x (log x)) n)
             (if (<= x 4.9e+135)
               (/
                (/
                 (+
                  1.0
                  (/
                   (- (/ (+ 0.3333333333333333 (* 0.25 (/ -1.0 x))) x) 0.5)
                   x))
                 x)
                n)
               0.0))))))))

C

double code(double x, double n) {
	double t_0 = log(x) / -n;
	double t_1 = 1.0 - pow(x, (1.0 / n));
	double tmp;
	if (x <= 5.2e-267) {
		tmp = t_0;
	} else if (x <= 1.32e-216) {
		tmp = t_1;
	} else if (x <= 1.15e-176) {
		tmp = t_0;
	} else if (x <= 9e-152) {
		tmp = t_1;
	} else if (x <= 0.88) {
		tmp = (x - log(x)) / n;
	} else if (x <= 4.9e+135) {
		tmp = ((1.0 + ((((0.3333333333333333 + (0.25 * (-1.0 / x))) / x) - 0.5) / x)) / x) / n;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = log(x) / -n
    t_1 = 1.0d0 - (x ** (1.0d0 / n))
    if (x <= 5.2d-267) then
        tmp = t_0
    else if (x <= 1.32d-216) then
        tmp = t_1
    else if (x <= 1.15d-176) then
        tmp = t_0
    else if (x <= 9d-152) then
        tmp = t_1
    else if (x <= 0.88d0) then
        tmp = (x - log(x)) / n
    else if (x <= 4.9d+135) then
        tmp = ((1.0d0 + ((((0.3333333333333333d0 + (0.25d0 * ((-1.0d0) / x))) / x) - 0.5d0) / x)) / x) / n
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = Math.log(x) / -n;
	double t_1 = 1.0 - Math.pow(x, (1.0 / n));
	double tmp;
	if (x <= 5.2e-267) {
		tmp = t_0;
	} else if (x <= 1.32e-216) {
		tmp = t_1;
	} else if (x <= 1.15e-176) {
		tmp = t_0;
	} else if (x <= 9e-152) {
		tmp = t_1;
	} else if (x <= 0.88) {
		tmp = (x - Math.log(x)) / n;
	} else if (x <= 4.9e+135) {
		tmp = ((1.0 + ((((0.3333333333333333 + (0.25 * (-1.0 / x))) / x) - 0.5) / x)) / x) / n;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.log(x) / -n
	t_1 = 1.0 - math.pow(x, (1.0 / n))
	tmp = 0
	if x <= 5.2e-267:
		tmp = t_0
	elif x <= 1.32e-216:
		tmp = t_1
	elif x <= 1.15e-176:
		tmp = t_0
	elif x <= 9e-152:
		tmp = t_1
	elif x <= 0.88:
		tmp = (x - math.log(x)) / n
	elif x <= 4.9e+135:
		tmp = ((1.0 + ((((0.3333333333333333 + (0.25 * (-1.0 / x))) / x) - 0.5) / x)) / x) / n
	else:
		tmp = 0.0
	return tmp

Julia

function code(x, n)
	t_0 = Float64(log(x) / Float64(-n))
	t_1 = Float64(1.0 - (x ^ Float64(1.0 / n)))
	tmp = 0.0
	if (x <= 5.2e-267)
		tmp = t_0;
	elseif (x <= 1.32e-216)
		tmp = t_1;
	elseif (x <= 1.15e-176)
		tmp = t_0;
	elseif (x <= 9e-152)
		tmp = t_1;
	elseif (x <= 0.88)
		tmp = Float64(Float64(x - log(x)) / n);
	elseif (x <= 4.9e+135)
		tmp = Float64(Float64(Float64(1.0 + Float64(Float64(Float64(Float64(0.3333333333333333 + Float64(0.25 * Float64(-1.0 / x))) / x) - 0.5) / x)) / x) / n);
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = log(x) / -n;
	t_1 = 1.0 - (x ^ (1.0 / n));
	tmp = 0.0;
	if (x <= 5.2e-267)
		tmp = t_0;
	elseif (x <= 1.32e-216)
		tmp = t_1;
	elseif (x <= 1.15e-176)
		tmp = t_0;
	elseif (x <= 9e-152)
		tmp = t_1;
	elseif (x <= 0.88)
		tmp = (x - log(x)) / n;
	elseif (x <= 4.9e+135)
		tmp = ((1.0 + ((((0.3333333333333333 + (0.25 * (-1.0 / x))) / x) - 0.5) / x)) / x) / n;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(N[Log[x], $MachinePrecision] / (-n)), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 5.2e-267], t$95$0, If[LessEqual[x, 1.32e-216], t$95$1, If[LessEqual[x, 1.15e-176], t$95$0, If[LessEqual[x, 9e-152], t$95$1, If[LessEqual[x, 0.88], N[(N[(x - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[x, 4.9e+135], N[(N[(N[(1.0 + N[(N[(N[(N[(0.3333333333333333 + N[(0.25 * N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] - 0.5), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] / n), $MachinePrecision], 0.0]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if x < 5.2000000000000003e-267 or 1.31999999999999997e-216 < x < 1.1500000000000001e-176

   1. Initial program 20.8%

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

   3. Taylor expanded in x around 0 20.8%

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

      1. *-rgt-identity 20.8%

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

      2. associate-*l/ 20.8%

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

      3. associate-/l* 20.8%

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

      4. exp-to-pow 20.8%

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

   5. Simplified 20.8%

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

   6. Taylor expanded in n around inf 72.3%

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

      1. mul-1-neg 72.3%

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

   8. Simplified 72.3%

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

   if 5.2000000000000003e-267 < x < 1.31999999999999997e-216 or 1.1500000000000001e-176 < x < 9.0000000000000008e-152

   1. Initial program 64.6%

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

   3. Taylor expanded in x around 0 64.6%

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

      1. *-rgt-identity 64.6%

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

      2. associate-*l/ 64.6%

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

      3. associate-/l* 64.6%

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

      4. exp-to-pow 64.6%

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

   5. Simplified 64.6%

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

   if 9.0000000000000008e-152 < x < 0.880000000000000004

   1. Initial program 30.3%

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

   3. Taylor expanded in n around inf 59.2%

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

      1. log1p-define 59.2%

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

   5. Simplified 59.2%

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

   6. Taylor expanded in x around 0 58.2%

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

   if 0.880000000000000004 < x < 4.9000000000000001e135

   1. Initial program 46.7%

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

   3. Taylor expanded in n around inf 49.3%

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

      1. log1p-define 49.3%

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

   5. Simplified 49.3%

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

   6. Taylor expanded in x around -inf 71.5%

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

   if 4.9000000000000001e135 < x

   1. Initial program 81.6%

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

      1. sub-neg 81.6%

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

      2. +-commutative 81.6%

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

      3. add-log-exp 81.6%

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

      4. add-log-exp 81.6%

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

      5. sum-log 81.6%

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

      6. pow-to-exp 81.6%

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

      7. un-div-inv 81.6%

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

      8. +-commutative 81.6%

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

      9. log1p-define 81.6%

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

   4. Applied egg-rr 81.6%

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

   5. Taylor expanded in x around inf 81.6%

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

      1. exp-neg 81.6%

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

      2. rgt-mult-inverse 81.6%

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

      3. metadata-eval 81.6%

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

   7. Simplified 81.6%

      \[\leadsto \color{blue}{0} \]
3. Recombined 5 regimes into one program.

4. Final simplification 70.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 5.2 \cdot 10^{-267}:\\ \;\;\;\;\frac{\log x}{-n}\\ \mathbf{elif}\;x \leq 1.32 \cdot 10^{-216}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;x \leq 1.15 \cdot 10^{-176}:\\ \;\;\;\;\frac{\log x}{-n}\\ \mathbf{elif}\;x \leq 9 \cdot 10^{-152}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;x \leq 0.88:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{elif}\;x \leq 4.9 \cdot 10^{+135}:\\ \;\;\;\;\frac{\frac{1 + \frac{\frac{0.3333333333333333 + 0.25 \cdot \frac{-1}{x}}{x} - 0.5}{x}}{x}}{n}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 60.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2.9 \cdot 10^{-183}:\\ \;\;\;\;\frac{\log x}{-n}\\ \mathbf{elif}\;x \leq 3.4 \cdot 10^{-160}:\\ \;\;\;\;\frac{\frac{1}{n} + \frac{0.3333333333333333 \cdot \frac{1}{x \cdot n} + 0.5 \cdot \frac{-1}{n}}{x}}{x}\\ \mathbf{elif}\;x \leq 0.88:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{elif}\;x \leq 8.4 \cdot 10^{+135}:\\ \;\;\;\;\frac{\frac{1 + \frac{\frac{0.3333333333333333 + 0.25 \cdot \frac{-1}{x}}{x} - 0.5}{x}}{x}}{n}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 2.9e-183)
   (/ (log x) (- n))
   (if (<= x 3.4e-160)
     (/
      (+
       (/ 1.0 n)
       (/ (+ (* 0.3333333333333333 (/ 1.0 (* x n))) (* 0.5 (/ -1.0 n))) x))
      x)
     (if (<= x 0.88)
       (/ (- x (log x)) n)
       (if (<= x 8.4e+135)
         (/
          (/
           (+
            1.0
            (/ (- (/ (+ 0.3333333333333333 (* 0.25 (/ -1.0 x))) x) 0.5) x))
           x)
          n)
         0.0)))))

C

double code(double x, double n) {
	double tmp;
	if (x <= 2.9e-183) {
		tmp = log(x) / -n;
	} else if (x <= 3.4e-160) {
		tmp = ((1.0 / n) + (((0.3333333333333333 * (1.0 / (x * n))) + (0.5 * (-1.0 / n))) / x)) / x;
	} else if (x <= 0.88) {
		tmp = (x - log(x)) / n;
	} else if (x <= 8.4e+135) {
		tmp = ((1.0 + ((((0.3333333333333333 + (0.25 * (-1.0 / x))) / x) - 0.5) / x)) / x) / n;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if (x <= 2.9d-183) then
        tmp = log(x) / -n
    else if (x <= 3.4d-160) then
        tmp = ((1.0d0 / n) + (((0.3333333333333333d0 * (1.0d0 / (x * n))) + (0.5d0 * ((-1.0d0) / n))) / x)) / x
    else if (x <= 0.88d0) then
        tmp = (x - log(x)) / n
    else if (x <= 8.4d+135) then
        tmp = ((1.0d0 + ((((0.3333333333333333d0 + (0.25d0 * ((-1.0d0) / x))) / x) - 0.5d0) / x)) / x) / n
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (x <= 2.9e-183) {
		tmp = Math.log(x) / -n;
	} else if (x <= 3.4e-160) {
		tmp = ((1.0 / n) + (((0.3333333333333333 * (1.0 / (x * n))) + (0.5 * (-1.0 / n))) / x)) / x;
	} else if (x <= 0.88) {
		tmp = (x - Math.log(x)) / n;
	} else if (x <= 8.4e+135) {
		tmp = ((1.0 + ((((0.3333333333333333 + (0.25 * (-1.0 / x))) / x) - 0.5) / x)) / x) / n;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 2.9e-183:
		tmp = math.log(x) / -n
	elif x <= 3.4e-160:
		tmp = ((1.0 / n) + (((0.3333333333333333 * (1.0 / (x * n))) + (0.5 * (-1.0 / n))) / x)) / x
	elif x <= 0.88:
		tmp = (x - math.log(x)) / n
	elif x <= 8.4e+135:
		tmp = ((1.0 + ((((0.3333333333333333 + (0.25 * (-1.0 / x))) / x) - 0.5) / x)) / x) / n
	else:
		tmp = 0.0
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 2.9e-183)
		tmp = Float64(log(x) / Float64(-n));
	elseif (x <= 3.4e-160)
		tmp = Float64(Float64(Float64(1.0 / n) + Float64(Float64(Float64(0.3333333333333333 * Float64(1.0 / Float64(x * n))) + Float64(0.5 * Float64(-1.0 / n))) / x)) / x);
	elseif (x <= 0.88)
		tmp = Float64(Float64(x - log(x)) / n);
	elseif (x <= 8.4e+135)
		tmp = Float64(Float64(Float64(1.0 + Float64(Float64(Float64(Float64(0.3333333333333333 + Float64(0.25 * Float64(-1.0 / x))) / x) - 0.5) / x)) / x) / n);
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (x <= 2.9e-183)
		tmp = log(x) / -n;
	elseif (x <= 3.4e-160)
		tmp = ((1.0 / n) + (((0.3333333333333333 * (1.0 / (x * n))) + (0.5 * (-1.0 / n))) / x)) / x;
	elseif (x <= 0.88)
		tmp = (x - log(x)) / n;
	elseif (x <= 8.4e+135)
		tmp = ((1.0 + ((((0.3333333333333333 + (0.25 * (-1.0 / x))) / x) - 0.5) / x)) / x) / n;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 2.9e-183], N[(N[Log[x], $MachinePrecision] / (-n)), $MachinePrecision], If[LessEqual[x, 3.4e-160], N[(N[(N[(1.0 / n), $MachinePrecision] + N[(N[(N[(0.3333333333333333 * N[(1.0 / N[(x * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(-1.0 / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[x, 0.88], N[(N[(x - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[x, 8.4e+135], N[(N[(N[(1.0 + N[(N[(N[(N[(0.3333333333333333 + N[(0.25 * N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] - 0.5), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] / n), $MachinePrecision], 0.0]]]]

Derivation

1. Split input into 5 regimes

2. if x < 2.9e-183

   1. Initial program 39.6%

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

   3. Taylor expanded in x around 0 39.6%

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

      1. *-rgt-identity 39.6%

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

      2. associate-*l/ 39.6%

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

      3. associate-/l* 39.6%

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

      4. exp-to-pow 39.6%

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

   5. Simplified 39.6%

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

   6. Taylor expanded in n around inf 61.3%

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

      1. mul-1-neg 61.3%

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

   8. Simplified 61.3%

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

   if 2.9e-183 < x < 3.40000000000000021e-160

   1. Initial program 44.8%

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

   3. Taylor expanded in n around inf 26.3%

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

      1. log1p-define 26.3%

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

   5. Simplified 26.3%

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

   6. Taylor expanded in x around -inf 65.2%

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

   if 3.40000000000000021e-160 < x < 0.880000000000000004

   1. Initial program 33.1%

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

   3. Taylor expanded in n around inf 57.4%

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

      1. log1p-define 57.4%

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

   5. Simplified 57.4%

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

   6. Taylor expanded in x around 0 56.5%

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

   if 0.880000000000000004 < x < 8.40000000000000039e135

   1. Initial program 46.7%

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

   3. Taylor expanded in n around inf 49.3%

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

      1. log1p-define 49.3%

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

   5. Simplified 49.3%

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

   6. Taylor expanded in x around -inf 71.5%

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

   if 8.40000000000000039e135 < x

   1. Initial program 81.6%

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

      1. sub-neg 81.6%

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

      2. +-commutative 81.6%

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

      3. add-log-exp 81.6%

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

      4. add-log-exp 81.6%

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

      5. sum-log 81.6%

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

      6. pow-to-exp 81.6%

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

      7. un-div-inv 81.6%

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

      8. +-commutative 81.6%

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

      9. log1p-define 81.6%

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

   4. Applied egg-rr 81.6%

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

   5. Taylor expanded in x around inf 81.6%

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

      1. exp-neg 81.6%

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

      2. rgt-mult-inverse 81.6%

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

      3. metadata-eval 81.6%

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

   7. Simplified 81.6%

      \[\leadsto \color{blue}{0} \]
3. Recombined 5 regimes into one program.

4. Final simplification 68.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.9 \cdot 10^{-183}:\\ \;\;\;\;\frac{\log x}{-n}\\ \mathbf{elif}\;x \leq 3.4 \cdot 10^{-160}:\\ \;\;\;\;\frac{\frac{1}{n} + \frac{0.3333333333333333 \cdot \frac{1}{x \cdot n} + 0.5 \cdot \frac{-1}{n}}{x}}{x}\\ \mathbf{elif}\;x \leq 0.88:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{elif}\;x \leq 8.4 \cdot 10^{+135}:\\ \;\;\;\;\frac{\frac{1 + \frac{\frac{0.3333333333333333 + 0.25 \cdot \frac{-1}{x}}{x} - 0.5}{x}}{x}}{n}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 60.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\log x}{-n}\\ \mathbf{if}\;x \leq 2.9 \cdot 10^{-183}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 2.05 \cdot 10^{-160}:\\ \;\;\;\;\frac{\frac{1}{n} + \frac{0.3333333333333333 \cdot \frac{1}{x \cdot n} + 0.5 \cdot \frac{-1}{n}}{x}}{x}\\ \mathbf{elif}\;x \leq 0.7:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 7 \cdot 10^{+135}:\\ \;\;\;\;\frac{\frac{1 + \frac{\frac{0.3333333333333333 + 0.25 \cdot \frac{-1}{x}}{x} - 0.5}{x}}{x}}{n}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (/ (log x) (- n))))
   (if (<= x 2.9e-183)
     t_0
     (if (<= x 2.05e-160)
       (/
        (+
         (/ 1.0 n)
         (/ (+ (* 0.3333333333333333 (/ 1.0 (* x n))) (* 0.5 (/ -1.0 n))) x))
        x)
       (if (<= x 0.7)
         t_0
         (if (<= x 7e+135)
           (/
            (/
             (+
              1.0
              (/ (- (/ (+ 0.3333333333333333 (* 0.25 (/ -1.0 x))) x) 0.5) x))
             x)
            n)
           0.0))))))

C

double code(double x, double n) {
	double t_0 = log(x) / -n;
	double tmp;
	if (x <= 2.9e-183) {
		tmp = t_0;
	} else if (x <= 2.05e-160) {
		tmp = ((1.0 / n) + (((0.3333333333333333 * (1.0 / (x * n))) + (0.5 * (-1.0 / n))) / x)) / x;
	} else if (x <= 0.7) {
		tmp = t_0;
	} else if (x <= 7e+135) {
		tmp = ((1.0 + ((((0.3333333333333333 + (0.25 * (-1.0 / x))) / x) - 0.5) / x)) / x) / n;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: tmp
    t_0 = log(x) / -n
    if (x <= 2.9d-183) then
        tmp = t_0
    else if (x <= 2.05d-160) then
        tmp = ((1.0d0 / n) + (((0.3333333333333333d0 * (1.0d0 / (x * n))) + (0.5d0 * ((-1.0d0) / n))) / x)) / x
    else if (x <= 0.7d0) then
        tmp = t_0
    else if (x <= 7d+135) then
        tmp = ((1.0d0 + ((((0.3333333333333333d0 + (0.25d0 * ((-1.0d0) / x))) / x) - 0.5d0) / x)) / x) / n
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = Math.log(x) / -n;
	double tmp;
	if (x <= 2.9e-183) {
		tmp = t_0;
	} else if (x <= 2.05e-160) {
		tmp = ((1.0 / n) + (((0.3333333333333333 * (1.0 / (x * n))) + (0.5 * (-1.0 / n))) / x)) / x;
	} else if (x <= 0.7) {
		tmp = t_0;
	} else if (x <= 7e+135) {
		tmp = ((1.0 + ((((0.3333333333333333 + (0.25 * (-1.0 / x))) / x) - 0.5) / x)) / x) / n;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.log(x) / -n
	tmp = 0
	if x <= 2.9e-183:
		tmp = t_0
	elif x <= 2.05e-160:
		tmp = ((1.0 / n) + (((0.3333333333333333 * (1.0 / (x * n))) + (0.5 * (-1.0 / n))) / x)) / x
	elif x <= 0.7:
		tmp = t_0
	elif x <= 7e+135:
		tmp = ((1.0 + ((((0.3333333333333333 + (0.25 * (-1.0 / x))) / x) - 0.5) / x)) / x) / n
	else:
		tmp = 0.0
	return tmp

Julia

function code(x, n)
	t_0 = Float64(log(x) / Float64(-n))
	tmp = 0.0
	if (x <= 2.9e-183)
		tmp = t_0;
	elseif (x <= 2.05e-160)
		tmp = Float64(Float64(Float64(1.0 / n) + Float64(Float64(Float64(0.3333333333333333 * Float64(1.0 / Float64(x * n))) + Float64(0.5 * Float64(-1.0 / n))) / x)) / x);
	elseif (x <= 0.7)
		tmp = t_0;
	elseif (x <= 7e+135)
		tmp = Float64(Float64(Float64(1.0 + Float64(Float64(Float64(Float64(0.3333333333333333 + Float64(0.25 * Float64(-1.0 / x))) / x) - 0.5) / x)) / x) / n);
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = log(x) / -n;
	tmp = 0.0;
	if (x <= 2.9e-183)
		tmp = t_0;
	elseif (x <= 2.05e-160)
		tmp = ((1.0 / n) + (((0.3333333333333333 * (1.0 / (x * n))) + (0.5 * (-1.0 / n))) / x)) / x;
	elseif (x <= 0.7)
		tmp = t_0;
	elseif (x <= 7e+135)
		tmp = ((1.0 + ((((0.3333333333333333 + (0.25 * (-1.0 / x))) / x) - 0.5) / x)) / x) / n;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(N[Log[x], $MachinePrecision] / (-n)), $MachinePrecision]}, If[LessEqual[x, 2.9e-183], t$95$0, If[LessEqual[x, 2.05e-160], N[(N[(N[(1.0 / n), $MachinePrecision] + N[(N[(N[(0.3333333333333333 * N[(1.0 / N[(x * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(-1.0 / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[x, 0.7], t$95$0, If[LessEqual[x, 7e+135], N[(N[(N[(1.0 + N[(N[(N[(N[(0.3333333333333333 + N[(0.25 * N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] - 0.5), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] / n), $MachinePrecision], 0.0]]]]]

Derivation

1. Split input into 4 regimes

2. if x < 2.9e-183 or 2.05000000000000001e-160 < x < 0.69999999999999996

   1. Initial program 35.7%

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

   3. Taylor expanded in x around 0 34.9%

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

      1. *-rgt-identity 34.9%

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

      2. associate-*l/ 34.9%

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

      3. associate-/l* 34.9%

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

      4. exp-to-pow 34.9%

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

   5. Simplified 34.9%

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

   6. Taylor expanded in n around inf 58.0%

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

      1. mul-1-neg 58.0%

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

   8. Simplified 58.0%

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

   if 2.9e-183 < x < 2.05000000000000001e-160

   1. Initial program 44.8%

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

   3. Taylor expanded in n around inf 26.3%

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

      1. log1p-define 26.3%

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

   5. Simplified 26.3%

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

   6. Taylor expanded in x around -inf 65.2%

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

   if 0.69999999999999996 < x < 7.0000000000000005e135

   1. Initial program 46.7%

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

   3. Taylor expanded in n around inf 49.3%

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

      1. log1p-define 49.3%

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

   5. Simplified 49.3%

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

   6. Taylor expanded in x around -inf 71.5%

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

   if 7.0000000000000005e135 < x

   1. Initial program 81.6%

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

      1. sub-neg 81.6%

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

      2. +-commutative 81.6%

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

      3. add-log-exp 81.6%

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

      4. add-log-exp 81.6%

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

      5. sum-log 81.6%

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

      6. pow-to-exp 81.6%

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

      7. un-div-inv 81.6%

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

      8. +-commutative 81.6%

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

      9. log1p-define 81.6%

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

   4. Applied egg-rr 81.6%

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

   5. Taylor expanded in x around inf 81.6%

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

      1. exp-neg 81.6%

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

      2. rgt-mult-inverse 81.6%

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

      3. metadata-eval 81.6%

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

   7. Simplified 81.6%

      \[\leadsto \color{blue}{0} \]
3. Recombined 4 regimes into one program.

4. Final simplification 67.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.9 \cdot 10^{-183}:\\ \;\;\;\;\frac{\log x}{-n}\\ \mathbf{elif}\;x \leq 2.05 \cdot 10^{-160}:\\ \;\;\;\;\frac{\frac{1}{n} + \frac{0.3333333333333333 \cdot \frac{1}{x \cdot n} + 0.5 \cdot \frac{-1}{n}}{x}}{x}\\ \mathbf{elif}\;x \leq 0.7:\\ \;\;\;\;\frac{\log x}{-n}\\ \mathbf{elif}\;x \leq 7 \cdot 10^{+135}:\\ \;\;\;\;\frac{\frac{1 + \frac{\frac{0.3333333333333333 + 0.25 \cdot \frac{-1}{x}}{x} - 0.5}{x}}{x}}{n}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 48.2% accurate, 7.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -50000000:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{n} + \frac{0.3333333333333333 \cdot \frac{1}{x \cdot n} + 0.5 \cdot \frac{-1}{n}}{x}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= (/ 1.0 n) -50000000.0)
   0.0
   (/
    (+
     (/ 1.0 n)
     (/ (+ (* 0.3333333333333333 (/ 1.0 (* x n))) (* 0.5 (/ -1.0 n))) x))
    x)))

C

double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -50000000.0) {
		tmp = 0.0;
	} else {
		tmp = ((1.0 / n) + (((0.3333333333333333 * (1.0 / (x * n))) + (0.5 * (-1.0 / n))) / x)) / x;
	}
	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) <= (-50000000.0d0)) then
        tmp = 0.0d0
    else
        tmp = ((1.0d0 / n) + (((0.3333333333333333d0 * (1.0d0 / (x * n))) + (0.5d0 * ((-1.0d0) / n))) / x)) / x
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -50000000.0) {
		tmp = 0.0;
	} else {
		tmp = ((1.0 / n) + (((0.3333333333333333 * (1.0 / (x * n))) + (0.5 * (-1.0 / n))) / x)) / x;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if (1.0 / n) <= -50000000.0:
		tmp = 0.0
	else:
		tmp = ((1.0 / n) + (((0.3333333333333333 * (1.0 / (x * n))) + (0.5 * (-1.0 / n))) / x)) / x
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -50000000.0)
		tmp = 0.0;
	else
		tmp = Float64(Float64(Float64(1.0 / n) + Float64(Float64(Float64(0.3333333333333333 * Float64(1.0 / Float64(x * n))) + Float64(0.5 * Float64(-1.0 / n))) / x)) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if ((1.0 / n) <= -50000000.0)
		tmp = 0.0;
	else
		tmp = ((1.0 / n) + (((0.3333333333333333 * (1.0 / (x * n))) + (0.5 * (-1.0 / n))) / x)) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[N[(1.0 / n), $MachinePrecision], -50000000.0], 0.0, N[(N[(N[(1.0 / n), $MachinePrecision] + N[(N[(N[(0.3333333333333333 * N[(1.0 / N[(x * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(-1.0 / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -5e7

   1. Initial program 100.0%

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

      1. sub-neg 100.0%

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

      2. +-commutative 100.0%

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

      3. add-log-exp 100.0%

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

      4. add-log-exp 100.0%

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

      5. sum-log 100.0%

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

      6. pow-to-exp 100.0%

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

      7. un-div-inv 100.0%

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

      8. +-commutative 100.0%

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

      9. log1p-define 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in x around inf 59.7%

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

      1. exp-neg 59.7%

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

      2. rgt-mult-inverse 60.3%

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

      3. metadata-eval 60.3%

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

   7. Simplified 60.3%

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

   if -5e7 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 35.8%

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

   3. Taylor expanded in n around inf 62.5%

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

      1. log1p-define 62.5%

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

   5. Simplified 62.5%

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

   6. Taylor expanded in x around -inf 53.2%

      \[\leadsto \color{blue}{-1 \cdot \frac{-1 \cdot \frac{0.3333333333333333 \cdot \frac{1}{n \cdot x} - 0.5 \cdot \frac{1}{n}}{x} - \frac{1}{n}}{x}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 54.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -50000000:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{n} + \frac{0.3333333333333333 \cdot \frac{1}{x \cdot n} + 0.5 \cdot \frac{-1}{n}}{x}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 48.2% accurate, 9.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -50000000:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \frac{0.3333333333333333 \cdot \frac{1}{x} - 0.5}{x}}{x}}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= (/ 1.0 n) -50000000.0)
   0.0
   (/ (/ (+ 1.0 (/ (- (* 0.3333333333333333 (/ 1.0 x)) 0.5) x)) x) n)))

C

double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -50000000.0) {
		tmp = 0.0;
	} else {
		tmp = ((1.0 + (((0.3333333333333333 * (1.0 / x)) - 0.5) / x)) / 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) <= (-50000000.0d0)) then
        tmp = 0.0d0
    else
        tmp = ((1.0d0 + (((0.3333333333333333d0 * (1.0d0 / x)) - 0.5d0) / x)) / x) / n
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -50000000.0) {
		tmp = 0.0;
	} else {
		tmp = ((1.0 + (((0.3333333333333333 * (1.0 / x)) - 0.5) / x)) / x) / n;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if (1.0 / n) <= -50000000.0:
		tmp = 0.0
	else:
		tmp = ((1.0 + (((0.3333333333333333 * (1.0 / x)) - 0.5) / x)) / x) / n
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -50000000.0)
		tmp = 0.0;
	else
		tmp = Float64(Float64(Float64(1.0 + Float64(Float64(Float64(0.3333333333333333 * Float64(1.0 / x)) - 0.5) / x)) / x) / n);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if ((1.0 / n) <= -50000000.0)
		tmp = 0.0;
	else
		tmp = ((1.0 + (((0.3333333333333333 * (1.0 / x)) - 0.5) / x)) / x) / n;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[N[(1.0 / n), $MachinePrecision], -50000000.0], 0.0, N[(N[(N[(1.0 + N[(N[(N[(0.3333333333333333 * N[(1.0 / x), $MachinePrecision]), $MachinePrecision] - 0.5), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] / n), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -5e7

   1. Initial program 100.0%

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

      1. sub-neg 100.0%

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

      2. +-commutative 100.0%

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

      3. add-log-exp 100.0%

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

      4. add-log-exp 100.0%

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

      5. sum-log 100.0%

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

      6. pow-to-exp 100.0%

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

      7. un-div-inv 100.0%

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

      8. +-commutative 100.0%

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

      9. log1p-define 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in x around inf 59.7%

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

      1. exp-neg 59.7%

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

      2. rgt-mult-inverse 60.3%

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

      3. metadata-eval 60.3%

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

   7. Simplified 60.3%

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

   if -5e7 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 35.8%

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

   3. Taylor expanded in n around inf 62.5%

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

      1. log1p-define 62.5%

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

   5. Simplified 62.5%

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

   6. Taylor expanded in x around -inf 53.2%

      \[\leadsto \frac{\color{blue}{-1 \cdot \frac{-1 \cdot \frac{0.3333333333333333 \cdot \frac{1}{x} - 0.5}{x} - 1}{x}}}{n} \]
3. Recombined 2 regimes into one program.

4. Final simplification 54.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -50000000:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \frac{0.3333333333333333 \cdot \frac{1}{x} - 0.5}{x}}{x}}{n}\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 46.3% accurate, 14.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -2.4 \lor \neg \left(n \leq -2.8 \cdot 10^{-293}\right):\\ \;\;\;\;\frac{1}{x \cdot n}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (or (<= n -2.4) (not (<= n -2.8e-293))) (/ 1.0 (* x n)) 0.0))

C

double code(double x, double n) {
	double tmp;
	if ((n <= -2.4) || !(n <= -2.8e-293)) {
		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 ((n <= (-2.4d0)) .or. (.not. (n <= (-2.8d-293)))) 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 ((n <= -2.4) || !(n <= -2.8e-293)) {
		tmp = 1.0 / (x * n);
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if (n <= -2.4) or not (n <= -2.8e-293):
		tmp = 1.0 / (x * n)
	else:
		tmp = 0.0
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if ((n <= -2.4) || !(n <= -2.8e-293))
		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 ((n <= -2.4) || ~((n <= -2.8e-293)))
		tmp = 1.0 / (x * n);
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[Or[LessEqual[n, -2.4], N[Not[LessEqual[n, -2.8e-293]], $MachinePrecision]], N[(1.0 / N[(x * n), $MachinePrecision]), $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if n < -2.39999999999999991 or -2.80000000000000025e-293 < n

   1. Initial program 36.8%

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

   3. Taylor expanded in x around inf 45.3%

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

      1. mul-1-neg 45.3%

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

      2. log-rec 45.3%

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

      3. mul-1-neg 45.3%

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

      4. distribute-neg-frac 45.3%

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

      5. mul-1-neg 45.3%

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

      6. remove-double-neg 45.3%

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

      7. *-commutative 45.3%

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

   5. Simplified 45.3%

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

   6. Taylor expanded in n around inf 49.0%

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

      1. *-commutative 49.0%

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

   8. Simplified 49.0%

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

   if -2.39999999999999991 < n < -2.80000000000000025e-293

   1. Initial program 100.0%

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

      1. sub-neg 100.0%

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

      2. +-commutative 100.0%

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

      3. add-log-exp 100.0%

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

      4. add-log-exp 100.0%

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

      5. sum-log 100.0%

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

      6. pow-to-exp 100.0%

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

      7. un-div-inv 100.0%

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

      8. +-commutative 100.0%

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

      9. log1p-define 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in x around inf 61.0%

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

      1. exp-neg 61.0%

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

      2. rgt-mult-inverse 61.6%

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

      3. metadata-eval 61.6%

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

   7. Simplified 61.6%

      \[\leadsto \color{blue}{0} \]
3. Recombined 2 regimes into one program.

4. Final simplification 51.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -2.4 \lor \neg \left(n \leq -2.8 \cdot 10^{-293}\right):\\ \;\;\;\;\frac{1}{x \cdot n}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 46.6% accurate, 17.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -50000000:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= (/ 1.0 n) -50000000.0) 0.0 (/ (/ 1.0 x) n)))

C

double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -50000000.0) {
		tmp = 0.0;
	} 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) :: tmp
    if ((1.0d0 / n) <= (-50000000.0d0)) then
        tmp = 0.0d0
    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) <= -50000000.0) {
		tmp = 0.0;
	} else {
		tmp = (1.0 / x) / n;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if (1.0 / n) <= -50000000.0:
		tmp = 0.0
	else:
		tmp = (1.0 / x) / n
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -50000000.0)
		tmp = 0.0;
	else
		tmp = Float64(Float64(1.0 / x) / n);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if ((1.0 / n) <= -50000000.0)
		tmp = 0.0;
	else
		tmp = (1.0 / x) / n;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[N[(1.0 / n), $MachinePrecision], -50000000.0], 0.0, N[(N[(1.0 / x), $MachinePrecision] / n), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -5e7

   1. Initial program 100.0%

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

      1. sub-neg 100.0%

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

      2. +-commutative 100.0%

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

      3. add-log-exp 100.0%

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

      4. add-log-exp 100.0%

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

      5. sum-log 100.0%

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

      6. pow-to-exp 100.0%

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

      7. un-div-inv 100.0%

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

      8. +-commutative 100.0%

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

      9. log1p-define 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in x around inf 59.7%

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

      1. exp-neg 59.7%

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

      2. rgt-mult-inverse 60.3%

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

      3. metadata-eval 60.3%

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

   7. Simplified 60.3%

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

   if -5e7 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 35.8%

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

   3. Taylor expanded in n around inf 62.5%

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

      1. log1p-define 62.5%

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

   5. Simplified 62.5%

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

   6. Taylor expanded in x around inf 50.1%

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

4. Final simplification 52.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -50000000:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 31.0% accurate, 211.0× speedup

Math

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

FPCore

(FPCore (x n) :precision binary64 0.0)

C

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

Fortran

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

Java

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

Python

def code(x, n):
	return 0.0

Julia

function code(x, n)
	return 0.0
end

MATLAB

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

Wolfram

code[x_, n_] := 0.0

Derivation

1. Initial program 51.3%

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

   1. sub-neg 51.3%

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

   2. +-commutative 51.3%

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

   3. add-log-exp 51.3%

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

   4. add-log-exp 51.3%

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

   5. sum-log 51.3%

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

   6. pow-to-exp 51.3%

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

   7. un-div-inv 51.3%

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

   8. +-commutative 51.3%

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

   9. log1p-define 58.5%

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

4. Applied egg-rr 58.5%

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

5. Taylor expanded in x around inf 34.2%

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

   1. exp-neg 34.1%

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

   2. rgt-mult-inverse 34.3%

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

   3. metadata-eval 34.3%

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

7. Simplified 34.3%

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

8. Final simplification 34.3%

   \[\leadsto 0 \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024075 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  :precision binary64
  (- (pow (+ x 1.0) (/ 1.0 n)) (pow x (/ 1.0 n))))