2nthrt (problem 3.4.6)

Percentage Accurate: 52.8% → 85.0%

Time: 40.5s

Alternatives: 23

Speedup: 1.9×

• 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: 52.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 85.0% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -4 \cdot 10^{-94}:\\ \;\;\;\;\frac{t\_0}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 10:\\ \;\;\;\;\frac{\left(\mathsf{log1p}\left(x\right) + \frac{0.5 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}\right) - \frac{-0.16666666666666666 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{3} - {\log x}^{3}\right) + -0.041666666666666664 \cdot \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{4} - {\log x}^{4}}{n}}{n}}{n}\right) - \log x}{n}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -4e-94)
     (/ t_0 (* n x))
     (if (<= (/ 1.0 n) 10.0)
       (/
        (-
         (+
          (log1p x)
          (/
           (-
            (* 0.5 (- (pow (log1p x) 2.0) (pow (log x) 2.0)))
            (/
             (+
              (*
               -0.16666666666666666
               (- (pow (log1p x) 3.0) (pow (log x) 3.0)))
              (*
               -0.041666666666666664
               (/ (- (pow (log1p x) 4.0) (pow (log x) 4.0)) n)))
             n))
           n))
         (log x))
        n)
       (- (exp (/ (log1p x) n)) t_0)))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -4e-94) {
		tmp = t_0 / (n * x);
	} else if ((1.0 / n) <= 10.0) {
		tmp = ((log1p(x) + (((0.5 * (pow(log1p(x), 2.0) - pow(log(x), 2.0))) - (((-0.16666666666666666 * (pow(log1p(x), 3.0) - pow(log(x), 3.0))) + (-0.041666666666666664 * ((pow(log1p(x), 4.0) - pow(log(x), 4.0)) / n))) / n)) / n)) - log(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 tmp;
	if ((1.0 / n) <= -4e-94) {
		tmp = t_0 / (n * x);
	} else if ((1.0 / n) <= 10.0) {
		tmp = ((Math.log1p(x) + (((0.5 * (Math.pow(Math.log1p(x), 2.0) - Math.pow(Math.log(x), 2.0))) - (((-0.16666666666666666 * (Math.pow(Math.log1p(x), 3.0) - Math.pow(Math.log(x), 3.0))) + (-0.041666666666666664 * ((Math.pow(Math.log1p(x), 4.0) - Math.pow(Math.log(x), 4.0)) / n))) / n)) / n)) - Math.log(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))
	tmp = 0
	if (1.0 / n) <= -4e-94:
		tmp = t_0 / (n * x)
	elif (1.0 / n) <= 10.0:
		tmp = ((math.log1p(x) + (((0.5 * (math.pow(math.log1p(x), 2.0) - math.pow(math.log(x), 2.0))) - (((-0.16666666666666666 * (math.pow(math.log1p(x), 3.0) - math.pow(math.log(x), 3.0))) + (-0.041666666666666664 * ((math.pow(math.log1p(x), 4.0) - math.pow(math.log(x), 4.0)) / n))) / n)) / n)) - math.log(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)
	tmp = 0.0
	if (Float64(1.0 / n) <= -4e-94)
		tmp = Float64(t_0 / Float64(n * x));
	elseif (Float64(1.0 / n) <= 10.0)
		tmp = Float64(Float64(Float64(log1p(x) + Float64(Float64(Float64(0.5 * Float64((log1p(x) ^ 2.0) - (log(x) ^ 2.0))) - Float64(Float64(Float64(-0.16666666666666666 * Float64((log1p(x) ^ 3.0) - (log(x) ^ 3.0))) + Float64(-0.041666666666666664 * Float64(Float64((log1p(x) ^ 4.0) - (log(x) ^ 4.0)) / n))) / n)) / n)) - log(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]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -4e-94], N[(t$95$0 / N[(n * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 10.0], N[(N[(N[(N[Log[1 + x], $MachinePrecision] + N[(N[(N[(0.5 * N[(N[Power[N[Log[1 + x], $MachinePrecision], 2.0], $MachinePrecision] - N[Power[N[Log[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(N[(-0.16666666666666666 * N[(N[Power[N[Log[1 + x], $MachinePrecision], 3.0], $MachinePrecision] - N[Power[N[Log[x], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.041666666666666664 * N[(N[(N[Power[N[Log[1 + x], $MachinePrecision], 4.0], $MachinePrecision] - N[Power[N[Log[x], $MachinePrecision], 4.0], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision] - N[Log[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 #s(literal 1 binary64) n) < -3.9999999999999998e-94

   1. Initial program 81.3%

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

   3. Taylor expanded in x around inf 93.4%

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

      1. mul-1-neg 93.4%

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

      2. log-rec 93.4%

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

      3. mul-1-neg 93.4%

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

      4. distribute-neg-frac 93.4%

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

      5. mul-1-neg 93.4%

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

      6. remove-double-neg 93.4%

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

      7. *-rgt-identity 93.4%

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

      8. associate-/l* 93.4%

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

      9. exp-to-pow 93.4%

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

      10. *-commutative 93.4%

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

   5. Simplified 93.4%

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

   if -3.9999999999999998e-94 < (/.f64 #s(literal 1 binary64) n) < 10

   1. Initial program 36.0%

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

   3. Taylor expanded in n around -inf 83.1%

      \[\leadsto \color{blue}{-1 \cdot \frac{\left(-1 \cdot \log \left(1 + x\right) + -1 \cdot \frac{\left(-1 \cdot \frac{\left(-1 \cdot \frac{0.041666666666666664 \cdot {\log \left(1 + x\right)}^{4} - 0.041666666666666664 \cdot {\log x}^{4}}{n} + -0.16666666666666666 \cdot {\log \left(1 + x\right)}^{3}\right) - -0.16666666666666666 \cdot {\log x}^{3}}{n} + 0.5 \cdot {\log \left(1 + x\right)}^{2}\right) - 0.5 \cdot {\log x}^{2}}{n}\right) - -1 \cdot \log x}{n}} \]

   5. Simplified 83.1%

      \[\leadsto \color{blue}{\frac{\log x - \left(\mathsf{log1p}\left(x\right) + \frac{0.5 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}\right) - \frac{-0.16666666666666666 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{3} - {\log x}^{3}\right) + -0.041666666666666664 \cdot \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{4} - {\log x}^{4}}{n}}{n}}{n}\right)}{-n}} \]

   if 10 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 63.0%

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

   3. Taylor expanded in n around 0 63.0%

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

      1. log1p-define 100.0%

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

      2. *-rgt-identity 100.0%

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

      3. associate-/l* 100.0%

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

      4. exp-to-pow 100.0%

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

   5. Simplified 100.0%

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

4. Final simplification 88.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -4 \cdot 10^{-94}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 10:\\ \;\;\;\;\frac{\left(\mathsf{log1p}\left(x\right) + \frac{0.5 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}\right) - \frac{-0.16666666666666666 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{3} - {\log x}^{3}\right) + -0.041666666666666664 \cdot \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{4} - {\log x}^{4}}{n}}{n}}{n}\right) - \log x}{n}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 85.0% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -4 \cdot 10^{-94}:\\ \;\;\;\;\frac{t\_0}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 10:\\ \;\;\;\;\frac{\left(\mathsf{log1p}\left(x\right) + \frac{0.5 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}\right) + 0.16666666666666666 \cdot \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{3} - {\log x}^{3}}{n}}{n}\right) - \log x}{n}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -4e-94)
     (/ t_0 (* n x))
     (if (<= (/ 1.0 n) 10.0)
       (/
        (-
         (+
          (log1p x)
          (/
           (+
            (* 0.5 (- (pow (log1p x) 2.0) (pow (log x) 2.0)))
            (*
             0.16666666666666666
             (/ (- (pow (log1p x) 3.0) (pow (log x) 3.0)) n)))
           n))
         (log x))
        n)
       (- (exp (/ (log1p x) n)) t_0)))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -4e-94) {
		tmp = t_0 / (n * x);
	} else if ((1.0 / n) <= 10.0) {
		tmp = ((log1p(x) + (((0.5 * (pow(log1p(x), 2.0) - pow(log(x), 2.0))) + (0.16666666666666666 * ((pow(log1p(x), 3.0) - pow(log(x), 3.0)) / n))) / n)) - log(x)) / n;
	} else {
		tmp = exp((log1p(x) / n)) - t_0;
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -4e-94) {
		tmp = t_0 / (n * x);
	} else if ((1.0 / n) <= 10.0) {
		tmp = ((Math.log1p(x) + (((0.5 * (Math.pow(Math.log1p(x), 2.0) - Math.pow(Math.log(x), 2.0))) + (0.16666666666666666 * ((Math.pow(Math.log1p(x), 3.0) - Math.pow(Math.log(x), 3.0)) / n))) / n)) - Math.log(x)) / n;
	} else {
		tmp = Math.exp((Math.log1p(x) / n)) - t_0;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	tmp = 0
	if (1.0 / n) <= -4e-94:
		tmp = t_0 / (n * x)
	elif (1.0 / n) <= 10.0:
		tmp = ((math.log1p(x) + (((0.5 * (math.pow(math.log1p(x), 2.0) - math.pow(math.log(x), 2.0))) + (0.16666666666666666 * ((math.pow(math.log1p(x), 3.0) - math.pow(math.log(x), 3.0)) / n))) / n)) - math.log(x)) / n
	else:
		tmp = math.exp((math.log1p(x) / n)) - t_0
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -4e-94)
		tmp = Float64(t_0 / Float64(n * x));
	elseif (Float64(1.0 / n) <= 10.0)
		tmp = Float64(Float64(Float64(log1p(x) + Float64(Float64(Float64(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)) - log(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]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -4e-94], N[(t$95$0 / N[(n * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 10.0], N[(N[(N[(N[Log[1 + x], $MachinePrecision] + N[(N[(N[(0.5 * N[(N[Power[N[Log[1 + x], $MachinePrecision], 2.0], $MachinePrecision] - N[Power[N[Log[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(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] - N[Log[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 #s(literal 1 binary64) n) < -3.9999999999999998e-94

   1. Initial program 81.3%

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

   3. Taylor expanded in x around inf 93.4%

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

      1. mul-1-neg 93.4%

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

      2. log-rec 93.4%

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

      3. mul-1-neg 93.4%

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

      4. distribute-neg-frac 93.4%

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

      5. mul-1-neg 93.4%

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

      6. remove-double-neg 93.4%

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

      7. *-rgt-identity 93.4%

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

      8. associate-/l* 93.4%

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

      9. exp-to-pow 93.4%

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

      10. *-commutative 93.4%

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

   5. Simplified 93.4%

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

   if -3.9999999999999998e-94 < (/.f64 #s(literal 1 binary64) n) < 10

   1. Initial program 36.0%

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

   3. Taylor expanded in n around -inf 83.0%

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

   5. Simplified 83.0%

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

   if 10 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 63.0%

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

   3. Taylor expanded in n around 0 63.0%

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

      1. log1p-define 100.0%

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

      2. *-rgt-identity 100.0%

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

      3. associate-/l* 100.0%

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

      4. exp-to-pow 100.0%

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

   5. Simplified 100.0%

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

4. Final simplification 88.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -4 \cdot 10^{-94}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 10:\\ \;\;\;\;\frac{\left(\mathsf{log1p}\left(x\right) + \frac{0.5 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}\right) + 0.16666666666666666 \cdot \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{3} - {\log x}^{3}}{n}}{n}\right) - \log x}{n}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 85.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -4 \cdot 10^{-94}:\\ \;\;\;\;\frac{t\_0}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-10}:\\ \;\;\;\;\frac{\log \left(\frac{\sqrt{e^{\frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}}{n}}} \cdot \left(1 + x\right)}{x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -4e-94)
     (/ t_0 (* n x))
     (if (<= (/ 1.0 n) 5e-10)
       (/
        (log
         (/
          (*
           (sqrt (exp (/ (- (pow (log1p x) 2.0) (pow (log x) 2.0)) n)))
           (+ 1.0 x))
          x))
        n)
       (- (exp (/ (log1p x) n)) t_0)))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -4e-94) {
		tmp = t_0 / (n * x);
	} else if ((1.0 / n) <= 5e-10) {
		tmp = log(((sqrt(exp(((pow(log1p(x), 2.0) - pow(log(x), 2.0)) / n))) * (1.0 + x)) / 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 tmp;
	if ((1.0 / n) <= -4e-94) {
		tmp = t_0 / (n * x);
	} else if ((1.0 / n) <= 5e-10) {
		tmp = Math.log(((Math.sqrt(Math.exp(((Math.pow(Math.log1p(x), 2.0) - Math.pow(Math.log(x), 2.0)) / n))) * (1.0 + x)) / 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))
	tmp = 0
	if (1.0 / n) <= -4e-94:
		tmp = t_0 / (n * x)
	elif (1.0 / n) <= 5e-10:
		tmp = math.log(((math.sqrt(math.exp(((math.pow(math.log1p(x), 2.0) - math.pow(math.log(x), 2.0)) / n))) * (1.0 + x)) / 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)
	tmp = 0.0
	if (Float64(1.0 / n) <= -4e-94)
		tmp = Float64(t_0 / Float64(n * x));
	elseif (Float64(1.0 / n) <= 5e-10)
		tmp = Float64(log(Float64(Float64(sqrt(exp(Float64(Float64((log1p(x) ^ 2.0) - (log(x) ^ 2.0)) / n))) * Float64(1.0 + x)) / 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]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -4e-94], N[(t$95$0 / N[(n * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e-10], N[(N[Log[N[(N[(N[Sqrt[N[Exp[N[(N[(N[Power[N[Log[1 + x], $MachinePrecision], 2.0], $MachinePrecision] - N[Power[N[Log[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * N[(1.0 + x), $MachinePrecision]), $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 #s(literal 1 binary64) n) < -3.9999999999999998e-94

   1. Initial program 81.3%

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

   3. Taylor expanded in x around inf 93.4%

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

      1. mul-1-neg 93.4%

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

      2. log-rec 93.4%

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

      3. mul-1-neg 93.4%

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

      4. distribute-neg-frac 93.4%

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

      5. mul-1-neg 93.4%

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

      6. remove-double-neg 93.4%

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

      7. *-rgt-identity 93.4%

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

      8. associate-/l* 93.4%

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

      9. exp-to-pow 93.4%

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

      10. *-commutative 93.4%

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

   5. Simplified 93.4%

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

   if -3.9999999999999998e-94 < (/.f64 #s(literal 1 binary64) n) < 5.00000000000000031e-10

   1. Initial program 35.9%

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

   3. Taylor expanded in n around inf 83.9%

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

   5. Simplified 83.9%

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

      1. associate-+r- 83.9%

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

      2. add-log-exp 83.9%

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

      3. exp-diff 83.9%

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

      4. add-exp-log 58.1%

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

   7. Applied egg-rr 58.1%

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

      1. +-commutative 58.1%

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

      2. exp-sum 58.1%

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

      3. *-commutative 58.1%

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

      4. exp-prod 58.1%

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

      5. unpow1/2 58.1%

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

      6. log1p-define 58.1%

         \[\leadsto \frac{\log \left(\frac{\sqrt{e^{\frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}}{n}}} \cdot e^{\color{blue}{\log \left(1 + x\right)}}}{x}\right)}{n} \]

      7. rem-exp-log 84.1%

         \[\leadsto \frac{\log \left(\frac{\sqrt{e^{\frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}}{n}}} \cdot \color{blue}{\left(1 + x\right)}}{x}\right)}{n} \]

   9. Simplified 84.1%

      \[\leadsto \frac{\color{blue}{\log \left(\frac{\sqrt{e^{\frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}}{n}}} \cdot \left(1 + x\right)}{x}\right)}}{n} \]

   if 5.00000000000000031e-10 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 61.0%

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

   3. Taylor expanded in n around 0 61.0%

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

      1. log1p-define 94.5%

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

      2. *-rgt-identity 94.5%

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

      3. associate-/l* 94.5%

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

      4. exp-to-pow 94.5%

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

   5. Simplified 94.5%

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

4. Final simplification 88.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -4 \cdot 10^{-94}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-10}:\\ \;\;\;\;\frac{\log \left(\frac{\sqrt{e^{\frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}}{n}}} \cdot \left(1 + x\right)}{x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 85.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -4 \cdot 10^{-94}:\\ \;\;\;\;\frac{t\_0}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-10}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right) - \left(\log x + 0.5 \cdot \frac{{\log x}^{2} - {\left(\mathsf{log1p}\left(x\right)\right)}^{2}}{n}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -4e-94)
     (/ t_0 (* n x))
     (if (<= (/ 1.0 n) 5e-10)
       (/
        (-
         (log1p x)
         (+ (log x) (* 0.5 (/ (- (pow (log x) 2.0) (pow (log1p x) 2.0)) n))))
        n)
       (- (exp (/ (log1p x) n)) t_0)))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -4e-94) {
		tmp = t_0 / (n * x);
	} else if ((1.0 / n) <= 5e-10) {
		tmp = (log1p(x) - (log(x) + (0.5 * ((pow(log(x), 2.0) - pow(log1p(x), 2.0)) / n)))) / n;
	} else {
		tmp = exp((log1p(x) / n)) - t_0;
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -4e-94) {
		tmp = t_0 / (n * x);
	} else if ((1.0 / n) <= 5e-10) {
		tmp = (Math.log1p(x) - (Math.log(x) + (0.5 * ((Math.pow(Math.log(x), 2.0) - Math.pow(Math.log1p(x), 2.0)) / n)))) / n;
	} else {
		tmp = Math.exp((Math.log1p(x) / n)) - t_0;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	tmp = 0
	if (1.0 / n) <= -4e-94:
		tmp = t_0 / (n * x)
	elif (1.0 / n) <= 5e-10:
		tmp = (math.log1p(x) - (math.log(x) + (0.5 * ((math.pow(math.log(x), 2.0) - math.pow(math.log1p(x), 2.0)) / n)))) / n
	else:
		tmp = math.exp((math.log1p(x) / n)) - t_0
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -4e-94)
		tmp = Float64(t_0 / Float64(n * x));
	elseif (Float64(1.0 / n) <= 5e-10)
		tmp = Float64(Float64(log1p(x) - Float64(log(x) + Float64(0.5 * Float64(Float64((log(x) ^ 2.0) - (log1p(x) ^ 2.0)) / n)))) / n);
	else
		tmp = Float64(exp(Float64(log1p(x) / n)) - t_0);
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -4e-94], N[(t$95$0 / N[(n * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e-10], N[(N[(N[Log[1 + x], $MachinePrecision] - N[(N[Log[x], $MachinePrecision] + N[(0.5 * N[(N[(N[Power[N[Log[x], $MachinePrecision], 2.0], $MachinePrecision] - N[Power[N[Log[1 + x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], N[(N[Exp[N[(N[Log[1 + x], $MachinePrecision] / n), $MachinePrecision]], $MachinePrecision] - t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -3.9999999999999998e-94

   1. Initial program 81.3%

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

   3. Taylor expanded in x around inf 93.4%

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

      1. mul-1-neg 93.4%

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

      2. log-rec 93.4%

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

      3. mul-1-neg 93.4%

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

      4. distribute-neg-frac 93.4%

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

      5. mul-1-neg 93.4%

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

      6. remove-double-neg 93.4%

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

      7. *-rgt-identity 93.4%

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

      8. associate-/l* 93.4%

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

      9. exp-to-pow 93.4%

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

      10. *-commutative 93.4%

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

   5. Simplified 93.4%

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

   if -3.9999999999999998e-94 < (/.f64 #s(literal 1 binary64) n) < 5.00000000000000031e-10

   1. Initial program 35.9%

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

   3. Taylor expanded in n around inf 83.9%

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

   5. Simplified 83.9%

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

   if 5.00000000000000031e-10 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 61.0%

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

   3. Taylor expanded in n around 0 61.0%

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

      1. log1p-define 94.5%

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

      2. *-rgt-identity 94.5%

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

      3. associate-/l* 94.5%

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

      4. exp-to-pow 94.5%

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

   5. Simplified 94.5%

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

4. Final simplification 88.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -4 \cdot 10^{-94}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-10}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right) - \left(\log x + 0.5 \cdot \frac{{\log x}^{2} - {\left(\mathsf{log1p}\left(x\right)\right)}^{2}}{n}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 85.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -4 \cdot 10^{-94}:\\ \;\;\;\;\frac{t\_0}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-12}:\\ \;\;\;\;\frac{\log \left(\frac{x}{1 + x}\right)}{-n}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -4e-94)
     (/ t_0 (* n x))
     (if (<= (/ 1.0 n) 2e-12)
       (/ (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 tmp;
	if ((1.0 / n) <= -4e-94) {
		tmp = t_0 / (n * x);
	} else if ((1.0 / n) <= 2e-12) {
		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 tmp;
	if ((1.0 / n) <= -4e-94) {
		tmp = t_0 / (n * x);
	} else if ((1.0 / n) <= 2e-12) {
		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))
	tmp = 0
	if (1.0 / n) <= -4e-94:
		tmp = t_0 / (n * x)
	elif (1.0 / n) <= 2e-12:
		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)
	tmp = 0.0
	if (Float64(1.0 / n) <= -4e-94)
		tmp = Float64(t_0 / Float64(n * x));
	elseif (Float64(1.0 / n) <= 2e-12)
		tmp = Float64(log(Float64(x / Float64(1.0 + x))) / Float64(-n));
	else
		tmp = Float64(exp(Float64(log1p(x) / n)) - t_0);
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -4e-94], N[(t$95$0 / N[(n * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-12], N[(N[Log[N[(x / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-n)), $MachinePrecision], N[(N[Exp[N[(N[Log[1 + x], $MachinePrecision] / n), $MachinePrecision]], $MachinePrecision] - t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -3.9999999999999998e-94

   1. Initial program 81.3%

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

   3. Taylor expanded in x around inf 93.4%

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

      1. mul-1-neg 93.4%

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

      2. log-rec 93.4%

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

      3. mul-1-neg 93.4%

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

      4. distribute-neg-frac 93.4%

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

      5. mul-1-neg 93.4%

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

      6. remove-double-neg 93.4%

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

      7. *-rgt-identity 93.4%

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

      8. associate-/l* 93.4%

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

      9. exp-to-pow 93.4%

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

      10. *-commutative 93.4%

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

   5. Simplified 93.4%

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

   if -3.9999999999999998e-94 < (/.f64 #s(literal 1 binary64) n) < 1.99999999999999996e-12

   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 n around inf 83.8%

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

   5. Simplified 83.8%

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

      1. associate-+r- 83.8%

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

      2. add-log-exp 83.8%

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

      3. exp-diff 83.8%

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

      4. add-exp-log 57.9%

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

   7. Applied egg-rr 57.9%

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

      1. +-commutative 57.9%

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

      2. exp-sum 57.9%

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

      3. *-commutative 57.9%

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

      4. exp-prod 57.9%

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

      5. unpow1/2 57.9%

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

      6. log1p-define 57.9%

         \[\leadsto \frac{\log \left(\frac{\sqrt{e^{\frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}}{n}}} \cdot e^{\color{blue}{\log \left(1 + x\right)}}}{x}\right)}{n} \]

      7. rem-exp-log 84.1%

         \[\leadsto \frac{\log \left(\frac{\sqrt{e^{\frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}}{n}}} \cdot \color{blue}{\left(1 + x\right)}}{x}\right)}{n} \]

   9. Simplified 84.1%

      \[\leadsto \frac{\color{blue}{\log \left(\frac{\sqrt{e^{\frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}}{n}}} \cdot \left(1 + x\right)}{x}\right)}}{n} \]

   10. Taylor expanded in n around inf 83.6%

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

       1. +-commutative 83.6%

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

   12. Simplified 83.6%

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

       1. clear-num 83.6%

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

       2. log-div 83.6%

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

       3. metadata-eval 83.6%

          \[\leadsto \frac{\color{blue}{0} - \log \left(\frac{x}{x + 1}\right)}{n} \]

   14. Applied egg-rr 83.6%

       \[\leadsto \frac{\color{blue}{0 - \log \left(\frac{x}{x + 1}\right)}}{n} \]
   15. Step-by-step derivation

       1. neg-sub0 83.6%

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

   16. Simplified 83.6%

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

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

   1. Initial program 61.2%

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

   3. Taylor expanded in n around 0 61.2%

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

      1. log1p-define 94.0%

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

      2. *-rgt-identity 94.0%

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

      3. associate-/l* 94.0%

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

      4. exp-to-pow 94.0%

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

   5. Simplified 94.0%

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

4. Final simplification 87.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -4 \cdot 10^{-94}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-12}:\\ \;\;\;\;\frac{\log \left(\frac{x}{1 + x}\right)}{-n}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 81.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -4 \cdot 10^{-94}:\\ \;\;\;\;\frac{t\_0}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-12}:\\ \;\;\;\;\frac{\log \left(\frac{x}{1 + x}\right)}{-n}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + x \cdot \left(\frac{1}{n} + x \cdot \left(0.5 \cdot \frac{1}{{n}^{2}} + 0.5 \cdot \frac{-1}{n}\right)\right)\right) - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -4e-94)
     (/ t_0 (* n x))
     (if (<= (/ 1.0 n) 2e-12)
       (/ (log (/ x (+ 1.0 x))) (- n))
       (-
        (+
         1.0
         (*
          x
          (+
           (/ 1.0 n)
           (* x (+ (* 0.5 (/ 1.0 (pow n 2.0))) (* 0.5 (/ -1.0 n)))))))
        t_0)))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -4e-94) {
		tmp = t_0 / (n * x);
	} else if ((1.0 / n) <= 2e-12) {
		tmp = log((x / (1.0 + x))) / -n;
	} else {
		tmp = (1.0 + (x * ((1.0 / n) + (x * ((0.5 * (1.0 / pow(n, 2.0))) + (0.5 * (-1.0 / n))))))) - t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x ** (1.0d0 / n)
    if ((1.0d0 / n) <= (-4d-94)) then
        tmp = t_0 / (n * x)
    else if ((1.0d0 / n) <= 2d-12) then
        tmp = log((x / (1.0d0 + x))) / -n
    else
        tmp = (1.0d0 + (x * ((1.0d0 / n) + (x * ((0.5d0 * (1.0d0 / (n ** 2.0d0))) + (0.5d0 * ((-1.0d0) / n))))))) - t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -4e-94) {
		tmp = t_0 / (n * x);
	} else if ((1.0 / n) <= 2e-12) {
		tmp = Math.log((x / (1.0 + x))) / -n;
	} else {
		tmp = (1.0 + (x * ((1.0 / n) + (x * ((0.5 * (1.0 / Math.pow(n, 2.0))) + (0.5 * (-1.0 / n))))))) - t_0;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	tmp = 0
	if (1.0 / n) <= -4e-94:
		tmp = t_0 / (n * x)
	elif (1.0 / n) <= 2e-12:
		tmp = math.log((x / (1.0 + x))) / -n
	else:
		tmp = (1.0 + (x * ((1.0 / n) + (x * ((0.5 * (1.0 / math.pow(n, 2.0))) + (0.5 * (-1.0 / n))))))) - t_0
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -4e-94)
		tmp = Float64(t_0 / Float64(n * x));
	elseif (Float64(1.0 / n) <= 2e-12)
		tmp = Float64(log(Float64(x / Float64(1.0 + x))) / Float64(-n));
	else
		tmp = Float64(Float64(1.0 + Float64(x * Float64(Float64(1.0 / n) + Float64(x * Float64(Float64(0.5 * Float64(1.0 / (n ^ 2.0))) + Float64(0.5 * Float64(-1.0 / n))))))) - t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = x ^ (1.0 / n);
	tmp = 0.0;
	if ((1.0 / n) <= -4e-94)
		tmp = t_0 / (n * x);
	elseif ((1.0 / n) <= 2e-12)
		tmp = log((x / (1.0 + x))) / -n;
	else
		tmp = (1.0 + (x * ((1.0 / n) + (x * ((0.5 * (1.0 / (n ^ 2.0))) + (0.5 * (-1.0 / n))))))) - t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -4e-94], N[(t$95$0 / N[(n * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-12], N[(N[Log[N[(x / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-n)), $MachinePrecision], N[(N[(1.0 + N[(x * N[(N[(1.0 / n), $MachinePrecision] + N[(x * N[(N[(0.5 * N[(1.0 / N[Power[n, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(-1.0 / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -3.9999999999999998e-94

   1. Initial program 81.3%

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

   3. Taylor expanded in x around inf 93.4%

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

      1. mul-1-neg 93.4%

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

      2. log-rec 93.4%

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

      3. mul-1-neg 93.4%

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

      4. distribute-neg-frac 93.4%

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

      5. mul-1-neg 93.4%

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

      6. remove-double-neg 93.4%

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

      7. *-rgt-identity 93.4%

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

      8. associate-/l* 93.4%

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

      9. exp-to-pow 93.4%

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

      10. *-commutative 93.4%

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

   5. Simplified 93.4%

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

   if -3.9999999999999998e-94 < (/.f64 #s(literal 1 binary64) n) < 1.99999999999999996e-12

   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 n around inf 83.8%

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

   5. Simplified 83.8%

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

      1. associate-+r- 83.8%

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

      2. add-log-exp 83.8%

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

      3. exp-diff 83.8%

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

      4. add-exp-log 57.9%

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

   7. Applied egg-rr 57.9%

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

      1. +-commutative 57.9%

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

      2. exp-sum 57.9%

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

      3. *-commutative 57.9%

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

      4. exp-prod 57.9%

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

      5. unpow1/2 57.9%

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

      6. log1p-define 57.9%

         \[\leadsto \frac{\log \left(\frac{\sqrt{e^{\frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}}{n}}} \cdot e^{\color{blue}{\log \left(1 + x\right)}}}{x}\right)}{n} \]

      7. rem-exp-log 84.1%

         \[\leadsto \frac{\log \left(\frac{\sqrt{e^{\frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}}{n}}} \cdot \color{blue}{\left(1 + x\right)}}{x}\right)}{n} \]

   9. Simplified 84.1%

      \[\leadsto \frac{\color{blue}{\log \left(\frac{\sqrt{e^{\frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}}{n}}} \cdot \left(1 + x\right)}{x}\right)}}{n} \]

   10. Taylor expanded in n around inf 83.6%

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

       1. +-commutative 83.6%

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

   12. Simplified 83.6%

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

       1. clear-num 83.6%

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

       2. log-div 83.6%

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

       3. metadata-eval 83.6%

          \[\leadsto \frac{\color{blue}{0} - \log \left(\frac{x}{x + 1}\right)}{n} \]

   14. Applied egg-rr 83.6%

       \[\leadsto \frac{\color{blue}{0 - \log \left(\frac{x}{x + 1}\right)}}{n} \]
   15. Step-by-step derivation

       1. neg-sub0 83.6%

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

   16. Simplified 83.6%

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

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

   1. Initial program 61.2%

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

   3. Taylor expanded in x around 0 74.6%

      \[\leadsto \color{blue}{\left(1 + x \cdot \left(x \cdot \left(0.5 \cdot \frac{1}{{n}^{2}} - 0.5 \cdot \frac{1}{n}\right) + \frac{1}{n}\right)\right)} - {x}^{\left(\frac{1}{n}\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 84.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -4 \cdot 10^{-94}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-12}:\\ \;\;\;\;\frac{\log \left(\frac{x}{1 + x}\right)}{-n}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + x \cdot \left(\frac{1}{n} + x \cdot \left(0.5 \cdot \frac{1}{{n}^{2}} + 0.5 \cdot \frac{-1}{n}\right)\right)\right) - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 82.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -4 \cdot 10^{-94}:\\ \;\;\;\;\frac{t\_0}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-12}:\\ \;\;\;\;\frac{\log \left(\frac{x}{1 + x}\right)}{-n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{+161}:\\ \;\;\;\;{\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - t\_0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{x}{n}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -4e-94)
     (/ t_0 (* n x))
     (if (<= (/ 1.0 n) 2e-12)
       (/ (log (/ x (+ 1.0 x))) (- n))
       (if (<= (/ 1.0 n) 2e+161)
         (- (pow (+ 1.0 x) (/ 1.0 n)) t_0)
         (log1p (expm1 (/ x n))))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -4e-94) {
		tmp = t_0 / (n * x);
	} else if ((1.0 / n) <= 2e-12) {
		tmp = log((x / (1.0 + x))) / -n;
	} else if ((1.0 / n) <= 2e+161) {
		tmp = pow((1.0 + x), (1.0 / n)) - t_0;
	} else {
		tmp = log1p(expm1((x / n)));
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -4e-94) {
		tmp = t_0 / (n * x);
	} else if ((1.0 / n) <= 2e-12) {
		tmp = Math.log((x / (1.0 + x))) / -n;
	} else if ((1.0 / n) <= 2e+161) {
		tmp = Math.pow((1.0 + x), (1.0 / n)) - t_0;
	} else {
		tmp = Math.log1p(Math.expm1((x / n)));
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	tmp = 0
	if (1.0 / n) <= -4e-94:
		tmp = t_0 / (n * x)
	elif (1.0 / n) <= 2e-12:
		tmp = math.log((x / (1.0 + x))) / -n
	elif (1.0 / n) <= 2e+161:
		tmp = math.pow((1.0 + x), (1.0 / n)) - t_0
	else:
		tmp = math.log1p(math.expm1((x / n)))
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -4e-94)
		tmp = Float64(t_0 / Float64(n * x));
	elseif (Float64(1.0 / n) <= 2e-12)
		tmp = Float64(log(Float64(x / Float64(1.0 + x))) / Float64(-n));
	elseif (Float64(1.0 / n) <= 2e+161)
		tmp = Float64((Float64(1.0 + x) ^ Float64(1.0 / n)) - t_0);
	else
		tmp = log1p(expm1(Float64(x / n)));
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -4e-94], N[(t$95$0 / N[(n * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-12], N[(N[Log[N[(x / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-n)), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e+161], N[(N[Power[N[(1.0 + x), $MachinePrecision], N[(1.0 / n), $MachinePrecision]], $MachinePrecision] - t$95$0), $MachinePrecision], N[Log[1 + N[(Exp[N[(x / n), $MachinePrecision]] - 1), $MachinePrecision]], $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -3.9999999999999998e-94

   1. Initial program 81.3%

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

   3. Taylor expanded in x around inf 93.4%

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

      1. mul-1-neg 93.4%

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

      2. log-rec 93.4%

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

      3. mul-1-neg 93.4%

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

      4. distribute-neg-frac 93.4%

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

      5. mul-1-neg 93.4%

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

      6. remove-double-neg 93.4%

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

      7. *-rgt-identity 93.4%

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

      8. associate-/l* 93.4%

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

      9. exp-to-pow 93.4%

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

      10. *-commutative 93.4%

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

   5. Simplified 93.4%

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

   if -3.9999999999999998e-94 < (/.f64 #s(literal 1 binary64) n) < 1.99999999999999996e-12

   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 n around inf 83.8%

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

   5. Simplified 83.8%

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

      1. associate-+r- 83.8%

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

      2. add-log-exp 83.8%

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

      3. exp-diff 83.8%

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

      4. add-exp-log 57.9%

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

   7. Applied egg-rr 57.9%

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

      1. +-commutative 57.9%

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

      2. exp-sum 57.9%

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

      3. *-commutative 57.9%

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

      4. exp-prod 57.9%

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

      5. unpow1/2 57.9%

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

      6. log1p-define 57.9%

         \[\leadsto \frac{\log \left(\frac{\sqrt{e^{\frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}}{n}}} \cdot e^{\color{blue}{\log \left(1 + x\right)}}}{x}\right)}{n} \]

      7. rem-exp-log 84.1%

         \[\leadsto \frac{\log \left(\frac{\sqrt{e^{\frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}}{n}}} \cdot \color{blue}{\left(1 + x\right)}}{x}\right)}{n} \]

   9. Simplified 84.1%

      \[\leadsto \frac{\color{blue}{\log \left(\frac{\sqrt{e^{\frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}}{n}}} \cdot \left(1 + x\right)}{x}\right)}}{n} \]

   10. Taylor expanded in n around inf 83.6%

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

       1. +-commutative 83.6%

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

   12. Simplified 83.6%

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

       1. clear-num 83.6%

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

       2. log-div 83.6%

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

       3. metadata-eval 83.6%

          \[\leadsto \frac{\color{blue}{0} - \log \left(\frac{x}{x + 1}\right)}{n} \]

   14. Applied egg-rr 83.6%

       \[\leadsto \frac{\color{blue}{0 - \log \left(\frac{x}{x + 1}\right)}}{n} \]
   15. Step-by-step derivation

       1. neg-sub0 83.6%

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

   16. Simplified 83.6%

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

   if 1.99999999999999996e-12 < (/.f64 #s(literal 1 binary64) n) < 2.0000000000000001e161

   1. Initial program 79.9%

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

   if 2.0000000000000001e161 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 31.6%

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

   3. Taylor expanded in n around inf 5.6%

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

      1. log1p-define 5.6%

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

   5. Simplified 5.6%

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

   6. Taylor expanded in x around 0 5.6%

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

   7. Taylor expanded in x around inf 5.3%

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

      1. log1p-expm1-u 78.4%

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

   9. Applied egg-rr 78.4%

      \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{x}{n}\right)\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 85.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -4 \cdot 10^{-94}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-12}:\\ \;\;\;\;\frac{\log \left(\frac{x}{1 + x}\right)}{-n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{+161}:\\ \;\;\;\;{\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{x}{n}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 81.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -4 \cdot 10^{-94}:\\ \;\;\;\;\frac{t\_0}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-12}:\\ \;\;\;\;\frac{\log \left(\frac{x}{1 + x}\right)}{-n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{+161}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - t\_0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{x}{n}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -4e-94)
     (/ t_0 (* n x))
     (if (<= (/ 1.0 n) 2e-12)
       (/ (log (/ x (+ 1.0 x))) (- n))
       (if (<= (/ 1.0 n) 2e+161)
         (- (+ 1.0 (/ x n)) t_0)
         (log1p (expm1 (/ x n))))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -4e-94) {
		tmp = t_0 / (n * x);
	} else if ((1.0 / n) <= 2e-12) {
		tmp = log((x / (1.0 + x))) / -n;
	} else if ((1.0 / n) <= 2e+161) {
		tmp = (1.0 + (x / n)) - t_0;
	} else {
		tmp = log1p(expm1((x / n)));
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -4e-94) {
		tmp = t_0 / (n * x);
	} else if ((1.0 / n) <= 2e-12) {
		tmp = Math.log((x / (1.0 + x))) / -n;
	} else if ((1.0 / n) <= 2e+161) {
		tmp = (1.0 + (x / n)) - t_0;
	} else {
		tmp = Math.log1p(Math.expm1((x / n)));
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	tmp = 0
	if (1.0 / n) <= -4e-94:
		tmp = t_0 / (n * x)
	elif (1.0 / n) <= 2e-12:
		tmp = math.log((x / (1.0 + x))) / -n
	elif (1.0 / n) <= 2e+161:
		tmp = (1.0 + (x / n)) - t_0
	else:
		tmp = math.log1p(math.expm1((x / n)))
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -4e-94)
		tmp = Float64(t_0 / Float64(n * x));
	elseif (Float64(1.0 / n) <= 2e-12)
		tmp = Float64(log(Float64(x / Float64(1.0 + x))) / Float64(-n));
	elseif (Float64(1.0 / n) <= 2e+161)
		tmp = Float64(Float64(1.0 + Float64(x / n)) - t_0);
	else
		tmp = log1p(expm1(Float64(x / n)));
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -4e-94], N[(t$95$0 / N[(n * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-12], N[(N[Log[N[(x / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-n)), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e+161], N[(N[(1.0 + N[(x / n), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], N[Log[1 + N[(Exp[N[(x / n), $MachinePrecision]] - 1), $MachinePrecision]], $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -3.9999999999999998e-94

   1. Initial program 81.3%

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

   3. Taylor expanded in x around inf 93.4%

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

      1. mul-1-neg 93.4%

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

      2. log-rec 93.4%

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

      3. mul-1-neg 93.4%

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

      4. distribute-neg-frac 93.4%

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

      5. mul-1-neg 93.4%

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

      6. remove-double-neg 93.4%

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

      7. *-rgt-identity 93.4%

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

      8. associate-/l* 93.4%

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

      9. exp-to-pow 93.4%

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

      10. *-commutative 93.4%

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

   5. Simplified 93.4%

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

   if -3.9999999999999998e-94 < (/.f64 #s(literal 1 binary64) n) < 1.99999999999999996e-12

   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 n around inf 83.8%

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

   5. Simplified 83.8%

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

      1. associate-+r- 83.8%

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

      2. add-log-exp 83.8%

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

      3. exp-diff 83.8%

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

      4. add-exp-log 57.9%

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

   7. Applied egg-rr 57.9%

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

      1. +-commutative 57.9%

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

      2. exp-sum 57.9%

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

      3. *-commutative 57.9%

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

      4. exp-prod 57.9%

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

      5. unpow1/2 57.9%

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

      6. log1p-define 57.9%

         \[\leadsto \frac{\log \left(\frac{\sqrt{e^{\frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}}{n}}} \cdot e^{\color{blue}{\log \left(1 + x\right)}}}{x}\right)}{n} \]

      7. rem-exp-log 84.1%

         \[\leadsto \frac{\log \left(\frac{\sqrt{e^{\frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}}{n}}} \cdot \color{blue}{\left(1 + x\right)}}{x}\right)}{n} \]

   9. Simplified 84.1%

      \[\leadsto \frac{\color{blue}{\log \left(\frac{\sqrt{e^{\frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}}{n}}} \cdot \left(1 + x\right)}{x}\right)}}{n} \]

   10. Taylor expanded in n around inf 83.6%

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

       1. +-commutative 83.6%

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

   12. Simplified 83.6%

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

       1. clear-num 83.6%

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

       2. log-div 83.6%

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

       3. metadata-eval 83.6%

          \[\leadsto \frac{\color{blue}{0} - \log \left(\frac{x}{x + 1}\right)}{n} \]

   14. Applied egg-rr 83.6%

       \[\leadsto \frac{\color{blue}{0 - \log \left(\frac{x}{x + 1}\right)}}{n} \]
   15. Step-by-step derivation

       1. neg-sub0 83.6%

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

   16. Simplified 83.6%

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

   if 1.99999999999999996e-12 < (/.f64 #s(literal 1 binary64) n) < 2.0000000000000001e161

   1. Initial program 79.9%

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

   3. Taylor expanded in x around 0 76.5%

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

   if 2.0000000000000001e161 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 31.6%

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

   3. Taylor expanded in n around inf 5.6%

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

      1. log1p-define 5.6%

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

   5. Simplified 5.6%

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

   6. Taylor expanded in x around 0 5.6%

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

   7. Taylor expanded in x around inf 5.3%

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

      1. log1p-expm1-u 78.4%

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

   9. Applied egg-rr 78.4%

      \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{x}{n}\right)\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 85.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -4 \cdot 10^{-94}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-12}:\\ \;\;\;\;\frac{\log \left(\frac{x}{1 + x}\right)}{-n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{+161}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{x}{n}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 81.3% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -4 \cdot 10^{-94}:\\ \;\;\;\;\frac{t\_0}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-12}:\\ \;\;\;\;\frac{\log \left(\frac{x}{1 + x}\right)}{-n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{+161}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x - \mathsf{log1p}\left(x + -1\right)}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -4e-94)
     (/ t_0 (* n x))
     (if (<= (/ 1.0 n) 2e-12)
       (/ (log (/ x (+ 1.0 x))) (- n))
       (if (<= (/ 1.0 n) 2e+161)
         (- (+ 1.0 (/ x n)) t_0)
         (/ (- x (log1p (+ x -1.0))) n))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -4e-94) {
		tmp = t_0 / (n * x);
	} else if ((1.0 / n) <= 2e-12) {
		tmp = log((x / (1.0 + x))) / -n;
	} else if ((1.0 / n) <= 2e+161) {
		tmp = (1.0 + (x / n)) - t_0;
	} else {
		tmp = (x - log1p((x + -1.0))) / n;
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -4e-94) {
		tmp = t_0 / (n * x);
	} else if ((1.0 / n) <= 2e-12) {
		tmp = Math.log((x / (1.0 + x))) / -n;
	} else if ((1.0 / n) <= 2e+161) {
		tmp = (1.0 + (x / n)) - t_0;
	} else {
		tmp = (x - Math.log1p((x + -1.0))) / n;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	tmp = 0
	if (1.0 / n) <= -4e-94:
		tmp = t_0 / (n * x)
	elif (1.0 / n) <= 2e-12:
		tmp = math.log((x / (1.0 + x))) / -n
	elif (1.0 / n) <= 2e+161:
		tmp = (1.0 + (x / n)) - t_0
	else:
		tmp = (x - math.log1p((x + -1.0))) / n
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -4e-94)
		tmp = Float64(t_0 / Float64(n * x));
	elseif (Float64(1.0 / n) <= 2e-12)
		tmp = Float64(log(Float64(x / Float64(1.0 + x))) / Float64(-n));
	elseif (Float64(1.0 / n) <= 2e+161)
		tmp = Float64(Float64(1.0 + Float64(x / n)) - t_0);
	else
		tmp = Float64(Float64(x - log1p(Float64(x + -1.0))) / n);
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -4e-94], N[(t$95$0 / N[(n * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-12], N[(N[Log[N[(x / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-n)), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e+161], N[(N[(1.0 + N[(x / n), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], N[(N[(x - N[Log[1 + N[(x + -1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -3.9999999999999998e-94

   1. Initial program 81.3%

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

   3. Taylor expanded in x around inf 93.4%

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

      1. mul-1-neg 93.4%

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

      2. log-rec 93.4%

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

      3. mul-1-neg 93.4%

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

      4. distribute-neg-frac 93.4%

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

      5. mul-1-neg 93.4%

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

      6. remove-double-neg 93.4%

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

      7. *-rgt-identity 93.4%

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

      8. associate-/l* 93.4%

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

      9. exp-to-pow 93.4%

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

      10. *-commutative 93.4%

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

   5. Simplified 93.4%

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

   if -3.9999999999999998e-94 < (/.f64 #s(literal 1 binary64) n) < 1.99999999999999996e-12

   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 n around inf 83.8%

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

   5. Simplified 83.8%

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

      1. associate-+r- 83.8%

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

      2. add-log-exp 83.8%

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

      3. exp-diff 83.8%

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

      4. add-exp-log 57.9%

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

   7. Applied egg-rr 57.9%

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

      1. +-commutative 57.9%

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

      2. exp-sum 57.9%

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

      3. *-commutative 57.9%

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

      4. exp-prod 57.9%

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

      5. unpow1/2 57.9%

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

      6. log1p-define 57.9%

         \[\leadsto \frac{\log \left(\frac{\sqrt{e^{\frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}}{n}}} \cdot e^{\color{blue}{\log \left(1 + x\right)}}}{x}\right)}{n} \]

      7. rem-exp-log 84.1%

         \[\leadsto \frac{\log \left(\frac{\sqrt{e^{\frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}}{n}}} \cdot \color{blue}{\left(1 + x\right)}}{x}\right)}{n} \]

   9. Simplified 84.1%

      \[\leadsto \frac{\color{blue}{\log \left(\frac{\sqrt{e^{\frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}}{n}}} \cdot \left(1 + x\right)}{x}\right)}}{n} \]

   10. Taylor expanded in n around inf 83.6%

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

       1. +-commutative 83.6%

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

   12. Simplified 83.6%

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

       1. clear-num 83.6%

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

       2. log-div 83.6%

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

       3. metadata-eval 83.6%

          \[\leadsto \frac{\color{blue}{0} - \log \left(\frac{x}{x + 1}\right)}{n} \]

   14. Applied egg-rr 83.6%

       \[\leadsto \frac{\color{blue}{0 - \log \left(\frac{x}{x + 1}\right)}}{n} \]
   15. Step-by-step derivation

       1. neg-sub0 83.6%

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

   16. Simplified 83.6%

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

   if 1.99999999999999996e-12 < (/.f64 #s(literal 1 binary64) n) < 2.0000000000000001e161

   1. Initial program 79.9%

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

   3. Taylor expanded in x around 0 76.5%

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

   if 2.0000000000000001e161 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 31.6%

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

   3. Taylor expanded in n around inf 5.6%

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

      1. log1p-define 5.6%

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

   5. Simplified 5.6%

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

   6. Taylor expanded in x around 0 5.6%

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

      1. log1p-expm1-u 71.6%

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

      2. expm1-undefine 71.6%

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

      3. add-exp-log 71.6%

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

   8. Applied egg-rr 71.6%

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

4. Final simplification 84.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -4 \cdot 10^{-94}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-12}:\\ \;\;\;\;\frac{\log \left(\frac{x}{1 + x}\right)}{-n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{+161}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x - \mathsf{log1p}\left(x + -1\right)}{n}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 81.2% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -4 \cdot 10^{-94}:\\ \;\;\;\;\frac{t\_0}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-12}:\\ \;\;\;\;\frac{\log \left(\frac{x}{1 + x}\right)}{-n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{+161}:\\ \;\;\;\;1 - t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x - \mathsf{log1p}\left(x + -1\right)}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -4e-94)
     (/ t_0 (* n x))
     (if (<= (/ 1.0 n) 2e-12)
       (/ (log (/ x (+ 1.0 x))) (- n))
       (if (<= (/ 1.0 n) 2e+161)
         (- 1.0 t_0)
         (/ (- x (log1p (+ x -1.0))) n))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -4e-94) {
		tmp = t_0 / (n * x);
	} else if ((1.0 / n) <= 2e-12) {
		tmp = log((x / (1.0 + x))) / -n;
	} else if ((1.0 / n) <= 2e+161) {
		tmp = 1.0 - t_0;
	} else {
		tmp = (x - log1p((x + -1.0))) / n;
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -4e-94) {
		tmp = t_0 / (n * x);
	} else if ((1.0 / n) <= 2e-12) {
		tmp = Math.log((x / (1.0 + x))) / -n;
	} else if ((1.0 / n) <= 2e+161) {
		tmp = 1.0 - t_0;
	} else {
		tmp = (x - Math.log1p((x + -1.0))) / n;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	tmp = 0
	if (1.0 / n) <= -4e-94:
		tmp = t_0 / (n * x)
	elif (1.0 / n) <= 2e-12:
		tmp = math.log((x / (1.0 + x))) / -n
	elif (1.0 / n) <= 2e+161:
		tmp = 1.0 - t_0
	else:
		tmp = (x - math.log1p((x + -1.0))) / n
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -4e-94)
		tmp = Float64(t_0 / Float64(n * x));
	elseif (Float64(1.0 / n) <= 2e-12)
		tmp = Float64(log(Float64(x / Float64(1.0 + x))) / Float64(-n));
	elseif (Float64(1.0 / n) <= 2e+161)
		tmp = Float64(1.0 - t_0);
	else
		tmp = Float64(Float64(x - log1p(Float64(x + -1.0))) / n);
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -4e-94], N[(t$95$0 / N[(n * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-12], N[(N[Log[N[(x / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-n)), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e+161], N[(1.0 - t$95$0), $MachinePrecision], N[(N[(x - N[Log[1 + N[(x + -1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -3.9999999999999998e-94

   1. Initial program 81.3%

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

   3. Taylor expanded in x around inf 93.4%

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

      1. mul-1-neg 93.4%

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

      2. log-rec 93.4%

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

      3. mul-1-neg 93.4%

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

      4. distribute-neg-frac 93.4%

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

      5. mul-1-neg 93.4%

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

      6. remove-double-neg 93.4%

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

      7. *-rgt-identity 93.4%

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

      8. associate-/l* 93.4%

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

      9. exp-to-pow 93.4%

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

      10. *-commutative 93.4%

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

   5. Simplified 93.4%

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

   if -3.9999999999999998e-94 < (/.f64 #s(literal 1 binary64) n) < 1.99999999999999996e-12

   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 n around inf 83.8%

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

   5. Simplified 83.8%

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

      1. associate-+r- 83.8%

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

      2. add-log-exp 83.8%

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

      3. exp-diff 83.8%

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

      4. add-exp-log 57.9%

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

   7. Applied egg-rr 57.9%

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

      1. +-commutative 57.9%

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

      2. exp-sum 57.9%

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

      3. *-commutative 57.9%

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

      4. exp-prod 57.9%

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

      5. unpow1/2 57.9%

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

      6. log1p-define 57.9%

         \[\leadsto \frac{\log \left(\frac{\sqrt{e^{\frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}}{n}}} \cdot e^{\color{blue}{\log \left(1 + x\right)}}}{x}\right)}{n} \]

      7. rem-exp-log 84.1%

         \[\leadsto \frac{\log \left(\frac{\sqrt{e^{\frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}}{n}}} \cdot \color{blue}{\left(1 + x\right)}}{x}\right)}{n} \]

   9. Simplified 84.1%

      \[\leadsto \frac{\color{blue}{\log \left(\frac{\sqrt{e^{\frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}}{n}}} \cdot \left(1 + x\right)}{x}\right)}}{n} \]

   10. Taylor expanded in n around inf 83.6%

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

       1. +-commutative 83.6%

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

   12. Simplified 83.6%

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

       1. clear-num 83.6%

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

       2. log-div 83.6%

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

       3. metadata-eval 83.6%

          \[\leadsto \frac{\color{blue}{0} - \log \left(\frac{x}{x + 1}\right)}{n} \]

   14. Applied egg-rr 83.6%

       \[\leadsto \frac{\color{blue}{0 - \log \left(\frac{x}{x + 1}\right)}}{n} \]
   15. Step-by-step derivation

       1. neg-sub0 83.6%

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

   16. Simplified 83.6%

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

   if 1.99999999999999996e-12 < (/.f64 #s(literal 1 binary64) n) < 2.0000000000000001e161

   1. Initial program 79.9%

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

   3. Taylor expanded in x around 0 76.2%

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

      1. *-rgt-identity 76.2%

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

      2. associate-/l* 76.2%

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

      3. exp-to-pow 76.2%

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

   5. Simplified 76.2%

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

   if 2.0000000000000001e161 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 31.6%

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

   3. Taylor expanded in n around inf 5.6%

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

      1. log1p-define 5.6%

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

   5. Simplified 5.6%

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

   6. Taylor expanded in x around 0 5.6%

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

      1. log1p-expm1-u 71.6%

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

      2. expm1-undefine 71.6%

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

      3. add-exp-log 71.6%

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

   8. Applied egg-rr 71.6%

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

4. Final simplification 84.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -4 \cdot 10^{-94}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-12}:\\ \;\;\;\;\frac{\log \left(\frac{x}{1 + x}\right)}{-n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{+161}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x - \mathsf{log1p}\left(x + -1\right)}{n}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 67.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq 2 \cdot 10^{-12}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{+161}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x - \mathsf{log1p}\left(x + -1\right)}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= (/ 1.0 n) 2e-12)
   (/ (log (/ (+ 1.0 x) x)) n)
   (if (<= (/ 1.0 n) 2e+161)
     (- 1.0 (pow x (/ 1.0 n)))
     (/ (- x (log1p (+ x -1.0))) n))))

C

double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= 2e-12) {
		tmp = log(((1.0 + x) / x)) / n;
	} else if ((1.0 / n) <= 2e+161) {
		tmp = 1.0 - pow(x, (1.0 / n));
	} else {
		tmp = (x - log1p((x + -1.0))) / n;
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= 2e-12) {
		tmp = Math.log(((1.0 + x) / x)) / n;
	} else if ((1.0 / n) <= 2e+161) {
		tmp = 1.0 - Math.pow(x, (1.0 / n));
	} else {
		tmp = (x - Math.log1p((x + -1.0))) / n;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if (1.0 / n) <= 2e-12:
		tmp = math.log(((1.0 + x) / x)) / n
	elif (1.0 / n) <= 2e+161:
		tmp = 1.0 - math.pow(x, (1.0 / n))
	else:
		tmp = (x - math.log1p((x + -1.0))) / n
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (Float64(1.0 / n) <= 2e-12)
		tmp = Float64(log(Float64(Float64(1.0 + x) / x)) / n);
	elseif (Float64(1.0 / n) <= 2e+161)
		tmp = Float64(1.0 - (x ^ Float64(1.0 / n)));
	else
		tmp = Float64(Float64(x - log1p(Float64(x + -1.0))) / n);
	end
	return tmp
end

Wolfram

code[x_, n_] := If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-12], N[(N[Log[N[(N[(1.0 + x), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e+161], N[(1.0 - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(x - N[Log[1 + N[(x + -1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 49.7%

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

   3. Taylor expanded in n around inf 73.8%

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

      1. log1p-define 73.8%

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

   5. Simplified 73.8%

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

      1. log1p-undefine 73.8%

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

      2. diff-log 73.9%

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

   7. Applied egg-rr 73.9%

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

   if 1.99999999999999996e-12 < (/.f64 #s(literal 1 binary64) n) < 2.0000000000000001e161

   1. Initial program 79.9%

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

   3. Taylor expanded in x around 0 76.2%

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

      1. *-rgt-identity 76.2%

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

      2. associate-/l* 76.2%

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

      3. exp-to-pow 76.2%

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

   5. Simplified 76.2%

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

   if 2.0000000000000001e161 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 31.6%

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

   3. Taylor expanded in n around inf 5.6%

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

      1. log1p-define 5.6%

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

   5. Simplified 5.6%

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

   6. Taylor expanded in x around 0 5.6%

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

      1. log1p-expm1-u 71.6%

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

      2. expm1-undefine 71.6%

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

      3. add-exp-log 71.6%

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

   8. Applied egg-rr 71.6%

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

4. Final simplification 74.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq 2 \cdot 10^{-12}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{+161}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x - \mathsf{log1p}\left(x + -1\right)}{n}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 59.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2.8 \cdot 10^{-204}:\\ \;\;\;\;\frac{-1}{\frac{n}{\log x}}\\ \mathbf{elif}\;x \leq 1.8 \cdot 10^{-143}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;x \leq 0.92:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{elif}\;x \leq 1.46 \cdot 10^{+94}:\\ \;\;\;\;\frac{\frac{1 - \frac{0.5 - \frac{0.3333333333333333 - \frac{0.25}{x}}{x}}{x}}{x}}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 2.8e-204)
   (/ -1.0 (/ n (log x)))
   (if (<= x 1.8e-143)
     (- 1.0 (pow x (/ 1.0 n)))
     (if (<= x 0.92)
       (/ (- x (log x)) n)
       (if (<= x 1.46e+94)
         (/
          (/ (- 1.0 (/ (- 0.5 (/ (- 0.3333333333333333 (/ 0.25 x)) x)) x)) x)
          n)
         (/ 0.0 n))))))

C

double code(double x, double n) {
	double tmp;
	if (x <= 2.8e-204) {
		tmp = -1.0 / (n / log(x));
	} else if (x <= 1.8e-143) {
		tmp = 1.0 - pow(x, (1.0 / n));
	} else if (x <= 0.92) {
		tmp = (x - log(x)) / n;
	} else if (x <= 1.46e+94) {
		tmp = ((1.0 - ((0.5 - ((0.3333333333333333 - (0.25 / x)) / x)) / x)) / x) / n;
	} else {
		tmp = 0.0 / n;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if (x <= 2.8d-204) then
        tmp = (-1.0d0) / (n / log(x))
    else if (x <= 1.8d-143) then
        tmp = 1.0d0 - (x ** (1.0d0 / n))
    else if (x <= 0.92d0) then
        tmp = (x - log(x)) / n
    else if (x <= 1.46d+94) then
        tmp = ((1.0d0 - ((0.5d0 - ((0.3333333333333333d0 - (0.25d0 / x)) / x)) / x)) / x) / n
    else
        tmp = 0.0d0 / n
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (x <= 2.8e-204) {
		tmp = -1.0 / (n / Math.log(x));
	} else if (x <= 1.8e-143) {
		tmp = 1.0 - Math.pow(x, (1.0 / n));
	} else if (x <= 0.92) {
		tmp = (x - Math.log(x)) / n;
	} else if (x <= 1.46e+94) {
		tmp = ((1.0 - ((0.5 - ((0.3333333333333333 - (0.25 / x)) / x)) / x)) / x) / n;
	} else {
		tmp = 0.0 / n;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 2.8e-204:
		tmp = -1.0 / (n / math.log(x))
	elif x <= 1.8e-143:
		tmp = 1.0 - math.pow(x, (1.0 / n))
	elif x <= 0.92:
		tmp = (x - math.log(x)) / n
	elif x <= 1.46e+94:
		tmp = ((1.0 - ((0.5 - ((0.3333333333333333 - (0.25 / x)) / x)) / x)) / x) / n
	else:
		tmp = 0.0 / n
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 2.8e-204)
		tmp = Float64(-1.0 / Float64(n / log(x)));
	elseif (x <= 1.8e-143)
		tmp = Float64(1.0 - (x ^ Float64(1.0 / n)));
	elseif (x <= 0.92)
		tmp = Float64(Float64(x - log(x)) / n);
	elseif (x <= 1.46e+94)
		tmp = Float64(Float64(Float64(1.0 - Float64(Float64(0.5 - Float64(Float64(0.3333333333333333 - Float64(0.25 / x)) / x)) / x)) / x) / n);
	else
		tmp = Float64(0.0 / n);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (x <= 2.8e-204)
		tmp = -1.0 / (n / log(x));
	elseif (x <= 1.8e-143)
		tmp = 1.0 - (x ^ (1.0 / n));
	elseif (x <= 0.92)
		tmp = (x - log(x)) / n;
	elseif (x <= 1.46e+94)
		tmp = ((1.0 - ((0.5 - ((0.3333333333333333 - (0.25 / x)) / x)) / x)) / x) / n;
	else
		tmp = 0.0 / n;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 2.8e-204], N[(-1.0 / N[(n / N[Log[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.8e-143], N[(1.0 - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.92], N[(N[(x - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[x, 1.46e+94], N[(N[(N[(1.0 - N[(N[(0.5 - N[(N[(0.3333333333333333 - N[(0.25 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] / n), $MachinePrecision], N[(0.0 / n), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if x < 2.8e-204

   1. Initial program 45.9%

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

   3. Taylor expanded in n around inf 58.9%

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

      1. log1p-define 58.9%

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

   5. Simplified 58.9%

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

      1. clear-num 58.9%

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

      2. inv-pow 58.9%

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

   7. Applied egg-rr 58.9%

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

      1. unpow-1 58.9%

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

   9. Simplified 58.9%

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

   10. Taylor expanded in x around 0 58.9%

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

       1. associate-*r/ 58.9%

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

       2. neg-mul-1 58.9%

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

   12. Simplified 58.9%

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

   if 2.8e-204 < x < 1.7999999999999999e-143

   1. Initial program 66.6%

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

   3. Taylor expanded in x around 0 66.6%

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

      1. *-rgt-identity 66.6%

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

      2. associate-/l* 66.6%

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

      3. exp-to-pow 66.6%

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

   5. Simplified 66.6%

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

   if 1.7999999999999999e-143 < x < 0.92000000000000004

   1. Initial program 21.1%

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

   3. Taylor expanded in n around inf 69.9%

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

      1. log1p-define 69.9%

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

   5. Simplified 69.9%

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

   6. Taylor expanded in x around 0 69.1%

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

   if 0.92000000000000004 < x < 1.46000000000000005e94

   1. Initial program 38.2%

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

   3. Taylor expanded in n around inf 39.1%

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

      1. log1p-define 39.1%

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

   5. Simplified 39.1%

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

   6. Taylor expanded in x around -inf 68.8%

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

   7. Taylor expanded in x around -inf 68.8%

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

      1. mul-1-neg 68.8%

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

      2. distribute-neg-frac2 68.8%

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

      3. mul-1-neg 68.8%

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

      4. unsub-neg 68.8%

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

      5. mul-1-neg 68.8%

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

      6. unsub-neg 68.8%

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

      7. associate-*r/ 68.8%

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

      8. metadata-eval 68.8%

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

   9. Simplified 68.8%

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

   if 1.46000000000000005e94 < x

   1. Initial program 81.7%

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

   3. Taylor expanded in n around inf 81.7%

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

   5. Simplified 81.7%

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

      1. associate-+r- 81.7%

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

      2. add-log-exp 81.7%

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

      3. exp-diff 81.7%

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

      4. add-exp-log 8.7%

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

   7. Applied egg-rr 8.7%

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

      1. +-commutative 8.7%

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

      2. exp-sum 8.7%

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

      3. *-commutative 8.7%

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

      4. exp-prod 8.7%

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

      5. unpow1/2 8.7%

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

      6. log1p-define 8.7%

         \[\leadsto \frac{\log \left(\frac{\sqrt{e^{\frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}}{n}}} \cdot e^{\color{blue}{\log \left(1 + x\right)}}}{x}\right)}{n} \]

      7. rem-exp-log 81.7%

         \[\leadsto \frac{\log \left(\frac{\sqrt{e^{\frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}}{n}}} \cdot \color{blue}{\left(1 + x\right)}}{x}\right)}{n} \]

   9. Simplified 81.7%

      \[\leadsto \frac{\color{blue}{\log \left(\frac{\sqrt{e^{\frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}}{n}}} \cdot \left(1 + x\right)}{x}\right)}}{n} \]

   10. Taylor expanded in x around inf 81.7%

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

4. Final simplification 70.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.8 \cdot 10^{-204}:\\ \;\;\;\;\frac{-1}{\frac{n}{\log x}}\\ \mathbf{elif}\;x \leq 1.8 \cdot 10^{-143}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;x \leq 0.92:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{elif}\;x \leq 1.46 \cdot 10^{+94}:\\ \;\;\;\;\frac{\frac{1 - \frac{0.5 - \frac{0.3333333333333333 - \frac{0.25}{x}}{x}}{x}}{x}}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{n}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 59.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2.3 \cdot 10^{-204}:\\ \;\;\;\;\frac{\log x}{-n}\\ \mathbf{elif}\;x \leq 3.6 \cdot 10^{-143}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;x \leq 0.88:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{elif}\;x \leq 1.6 \cdot 10^{+94}:\\ \;\;\;\;\frac{\frac{1 - \frac{0.5 - \frac{0.3333333333333333 - \frac{0.25}{x}}{x}}{x}}{x}}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 2.3e-204)
   (/ (log x) (- n))
   (if (<= x 3.6e-143)
     (- 1.0 (pow x (/ 1.0 n)))
     (if (<= x 0.88)
       (/ (- x (log x)) n)
       (if (<= x 1.6e+94)
         (/
          (/ (- 1.0 (/ (- 0.5 (/ (- 0.3333333333333333 (/ 0.25 x)) x)) x)) x)
          n)
         (/ 0.0 n))))))

C

double code(double x, double n) {
	double tmp;
	if (x <= 2.3e-204) {
		tmp = log(x) / -n;
	} else if (x <= 3.6e-143) {
		tmp = 1.0 - pow(x, (1.0 / n));
	} else if (x <= 0.88) {
		tmp = (x - log(x)) / n;
	} else if (x <= 1.6e+94) {
		tmp = ((1.0 - ((0.5 - ((0.3333333333333333 - (0.25 / x)) / x)) / x)) / x) / n;
	} else {
		tmp = 0.0 / n;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if (x <= 2.3d-204) then
        tmp = log(x) / -n
    else if (x <= 3.6d-143) then
        tmp = 1.0d0 - (x ** (1.0d0 / n))
    else if (x <= 0.88d0) then
        tmp = (x - log(x)) / n
    else if (x <= 1.6d+94) then
        tmp = ((1.0d0 - ((0.5d0 - ((0.3333333333333333d0 - (0.25d0 / x)) / x)) / x)) / x) / n
    else
        tmp = 0.0d0 / n
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (x <= 2.3e-204) {
		tmp = Math.log(x) / -n;
	} else if (x <= 3.6e-143) {
		tmp = 1.0 - Math.pow(x, (1.0 / n));
	} else if (x <= 0.88) {
		tmp = (x - Math.log(x)) / n;
	} else if (x <= 1.6e+94) {
		tmp = ((1.0 - ((0.5 - ((0.3333333333333333 - (0.25 / x)) / x)) / x)) / x) / n;
	} else {
		tmp = 0.0 / n;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 2.3e-204:
		tmp = math.log(x) / -n
	elif x <= 3.6e-143:
		tmp = 1.0 - math.pow(x, (1.0 / n))
	elif x <= 0.88:
		tmp = (x - math.log(x)) / n
	elif x <= 1.6e+94:
		tmp = ((1.0 - ((0.5 - ((0.3333333333333333 - (0.25 / x)) / x)) / x)) / x) / n
	else:
		tmp = 0.0 / n
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 2.3e-204)
		tmp = Float64(log(x) / Float64(-n));
	elseif (x <= 3.6e-143)
		tmp = Float64(1.0 - (x ^ Float64(1.0 / n)));
	elseif (x <= 0.88)
		tmp = Float64(Float64(x - log(x)) / n);
	elseif (x <= 1.6e+94)
		tmp = Float64(Float64(Float64(1.0 - Float64(Float64(0.5 - Float64(Float64(0.3333333333333333 - Float64(0.25 / x)) / x)) / x)) / x) / n);
	else
		tmp = Float64(0.0 / n);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (x <= 2.3e-204)
		tmp = log(x) / -n;
	elseif (x <= 3.6e-143)
		tmp = 1.0 - (x ^ (1.0 / n));
	elseif (x <= 0.88)
		tmp = (x - log(x)) / n;
	elseif (x <= 1.6e+94)
		tmp = ((1.0 - ((0.5 - ((0.3333333333333333 - (0.25 / x)) / x)) / x)) / x) / n;
	else
		tmp = 0.0 / n;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 2.3e-204], N[(N[Log[x], $MachinePrecision] / (-n)), $MachinePrecision], If[LessEqual[x, 3.6e-143], N[(1.0 - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.88], N[(N[(x - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[x, 1.6e+94], N[(N[(N[(1.0 - N[(N[(0.5 - N[(N[(0.3333333333333333 - N[(0.25 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] / n), $MachinePrecision], N[(0.0 / n), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if x < 2.2999999999999999e-204

   1. Initial program 45.9%

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

   3. Taylor expanded in n around inf 58.9%

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

      1. log1p-define 58.9%

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

   5. Simplified 58.9%

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

   6. Taylor expanded in x around 0 58.9%

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

      1. neg-mul-1 58.9%

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

   8. Simplified 58.9%

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

   if 2.2999999999999999e-204 < x < 3.5999999999999998e-143

   1. Initial program 66.6%

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

   3. Taylor expanded in x around 0 66.6%

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

      1. *-rgt-identity 66.6%

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

      2. associate-/l* 66.6%

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

      3. exp-to-pow 66.6%

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

   5. Simplified 66.6%

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

   if 3.5999999999999998e-143 < x < 0.880000000000000004

   1. Initial program 21.1%

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

   3. Taylor expanded in n around inf 69.9%

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

      1. log1p-define 69.9%

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

   5. Simplified 69.9%

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

   6. Taylor expanded in x around 0 69.1%

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

   if 0.880000000000000004 < x < 1.60000000000000007e94

   1. Initial program 38.2%

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

   3. Taylor expanded in n around inf 39.1%

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

      1. log1p-define 39.1%

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

   5. Simplified 39.1%

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

   6. Taylor expanded in x around -inf 68.8%

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

   7. Taylor expanded in x around -inf 68.8%

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

      1. mul-1-neg 68.8%

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

      2. distribute-neg-frac2 68.8%

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

      3. mul-1-neg 68.8%

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

      4. unsub-neg 68.8%

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

      5. mul-1-neg 68.8%

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

      6. unsub-neg 68.8%

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

      7. associate-*r/ 68.8%

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

      8. metadata-eval 68.8%

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

   9. Simplified 68.8%

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

   if 1.60000000000000007e94 < x

   1. Initial program 81.7%

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

   3. Taylor expanded in n around inf 81.7%

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

   5. Simplified 81.7%

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

      1. associate-+r- 81.7%

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

      2. add-log-exp 81.7%

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

      3. exp-diff 81.7%

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

      4. add-exp-log 8.7%

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

   7. Applied egg-rr 8.7%

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

      1. +-commutative 8.7%

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

      2. exp-sum 8.7%

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

      3. *-commutative 8.7%

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

      4. exp-prod 8.7%

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

      5. unpow1/2 8.7%

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

      6. log1p-define 8.7%

         \[\leadsto \frac{\log \left(\frac{\sqrt{e^{\frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}}{n}}} \cdot e^{\color{blue}{\log \left(1 + x\right)}}}{x}\right)}{n} \]

      7. rem-exp-log 81.7%

         \[\leadsto \frac{\log \left(\frac{\sqrt{e^{\frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}}{n}}} \cdot \color{blue}{\left(1 + x\right)}}{x}\right)}{n} \]

   9. Simplified 81.7%

      \[\leadsto \frac{\color{blue}{\log \left(\frac{\sqrt{e^{\frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}}{n}}} \cdot \left(1 + x\right)}{x}\right)}}{n} \]

   10. Taylor expanded in x around inf 81.7%

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

4. Final simplification 70.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.3 \cdot 10^{-204}:\\ \;\;\;\;\frac{\log x}{-n}\\ \mathbf{elif}\;x \leq 3.6 \cdot 10^{-143}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;x \leq 0.88:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{elif}\;x \leq 1.6 \cdot 10^{+94}:\\ \;\;\;\;\frac{\frac{1 - \frac{0.5 - \frac{0.3333333333333333 - \frac{0.25}{x}}{x}}{x}}{x}}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{n}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 66.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq 2 \cdot 10^{-12}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{+161}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{0.5 + \frac{-0.3333333333333333}{x}}{x} + -1}{-x}}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= (/ 1.0 n) 2e-12)
   (/ (log (/ (+ 1.0 x) x)) n)
   (if (<= (/ 1.0 n) 2e+161)
     (- 1.0 (pow x (/ 1.0 n)))
     (/ (/ (+ (/ (+ 0.5 (/ -0.3333333333333333 x)) x) -1.0) (- x)) n))))

C

double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= 2e-12) {
		tmp = log(((1.0 + x) / x)) / n;
	} else if ((1.0 / n) <= 2e+161) {
		tmp = 1.0 - pow(x, (1.0 / n));
	} else {
		tmp = ((((0.5 + (-0.3333333333333333 / x)) / x) + -1.0) / -x) / n;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if ((1.0d0 / n) <= 2d-12) then
        tmp = log(((1.0d0 + x) / x)) / n
    else if ((1.0d0 / n) <= 2d+161) then
        tmp = 1.0d0 - (x ** (1.0d0 / n))
    else
        tmp = ((((0.5d0 + ((-0.3333333333333333d0) / x)) / x) + (-1.0d0)) / -x) / n
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= 2e-12) {
		tmp = Math.log(((1.0 + x) / x)) / n;
	} else if ((1.0 / n) <= 2e+161) {
		tmp = 1.0 - Math.pow(x, (1.0 / n));
	} else {
		tmp = ((((0.5 + (-0.3333333333333333 / x)) / x) + -1.0) / -x) / n;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if (1.0 / n) <= 2e-12:
		tmp = math.log(((1.0 + x) / x)) / n
	elif (1.0 / n) <= 2e+161:
		tmp = 1.0 - math.pow(x, (1.0 / n))
	else:
		tmp = ((((0.5 + (-0.3333333333333333 / x)) / x) + -1.0) / -x) / n
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (Float64(1.0 / n) <= 2e-12)
		tmp = Float64(log(Float64(Float64(1.0 + x) / x)) / n);
	elseif (Float64(1.0 / n) <= 2e+161)
		tmp = Float64(1.0 - (x ^ Float64(1.0 / n)));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(0.5 + Float64(-0.3333333333333333 / x)) / x) + -1.0) / Float64(-x)) / n);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if ((1.0 / n) <= 2e-12)
		tmp = log(((1.0 + x) / x)) / n;
	elseif ((1.0 / n) <= 2e+161)
		tmp = 1.0 - (x ^ (1.0 / n));
	else
		tmp = ((((0.5 + (-0.3333333333333333 / x)) / x) + -1.0) / -x) / n;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-12], N[(N[Log[N[(N[(1.0 + x), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e+161], N[(1.0 - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(0.5 + N[(-0.3333333333333333 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] + -1.0), $MachinePrecision] / (-x)), $MachinePrecision] / n), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 49.7%

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

   3. Taylor expanded in n around inf 73.8%

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

      1. log1p-define 73.8%

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

   5. Simplified 73.8%

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

      1. log1p-undefine 73.8%

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

      2. diff-log 73.9%

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

   7. Applied egg-rr 73.9%

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

   if 1.99999999999999996e-12 < (/.f64 #s(literal 1 binary64) n) < 2.0000000000000001e161

   1. Initial program 79.9%

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

   3. Taylor expanded in x around 0 76.2%

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

      1. *-rgt-identity 76.2%

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

      2. associate-/l* 76.2%

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

      3. exp-to-pow 76.2%

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

   5. Simplified 76.2%

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

   if 2.0000000000000001e161 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 31.6%

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

   3. Taylor expanded in n around inf 0.2%

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

   5. Simplified 0.2%

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

      1. associate-+r- 0.2%

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

      2. add-log-exp 0.2%

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

      3. exp-diff 0.2%

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

      4. add-exp-log 0.2%

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

   7. Applied egg-rr 0.2%

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

      1. +-commutative 0.2%

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

      2. exp-sum 0.2%

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

      3. *-commutative 0.2%

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

      4. exp-prod 0.2%

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

      5. unpow1/2 0.2%

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

      6. log1p-define 0.2%

         \[\leadsto \frac{\log \left(\frac{\sqrt{e^{\frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}}{n}}} \cdot e^{\color{blue}{\log \left(1 + x\right)}}}{x}\right)}{n} \]

      7. rem-exp-log 0.2%

         \[\leadsto \frac{\log \left(\frac{\sqrt{e^{\frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}}{n}}} \cdot \color{blue}{\left(1 + x\right)}}{x}\right)}{n} \]

   9. Simplified 0.2%

      \[\leadsto \frac{\color{blue}{\log \left(\frac{\sqrt{e^{\frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}}{n}}} \cdot \left(1 + x\right)}{x}\right)}}{n} \]

   10. Taylor expanded in n around inf 5.6%

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

       1. +-commutative 5.6%

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

   12. Simplified 5.6%

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

   13. Taylor expanded in x around -inf 66.5%

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

       1. mul-1-neg 66.5%

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

       2. distribute-neg-frac2 66.5%

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

       3. sub-neg 66.5%

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

       4. associate-*r/ 66.5%

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

       5. sub-neg 66.5%

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

       6. metadata-eval 66.5%

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

       7. distribute-lft-in 66.5%

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

       8. neg-mul-1 66.5%

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

       9. associate-*r/ 66.5%

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

       10. metadata-eval 66.5%

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

       11. distribute-neg-frac 66.5%

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

       12. metadata-eval 66.5%

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

       13. metadata-eval 66.5%

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

       14. metadata-eval 66.5%

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

   15. Simplified 66.5%

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

4. Final simplification 73.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq 2 \cdot 10^{-12}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{+161}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{0.5 + \frac{-0.3333333333333333}{x}}{x} + -1}{-x}}{n}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 60.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.89:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{elif}\;x \leq 1.6 \cdot 10^{+94}:\\ \;\;\;\;\frac{\frac{1 - \frac{0.5 - \frac{0.3333333333333333 - \frac{0.25}{x}}{x}}{x}}{x}}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 0.89)
   (/ (- x (log x)) n)
   (if (<= x 1.6e+94)
     (/ (/ (- 1.0 (/ (- 0.5 (/ (- 0.3333333333333333 (/ 0.25 x)) x)) x)) x) n)
     (/ 0.0 n))))

C

double code(double x, double n) {
	double tmp;
	if (x <= 0.89) {
		tmp = (x - log(x)) / n;
	} else if (x <= 1.6e+94) {
		tmp = ((1.0 - ((0.5 - ((0.3333333333333333 - (0.25 / x)) / x)) / x)) / x) / n;
	} else {
		tmp = 0.0 / n;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if (x <= 0.89d0) then
        tmp = (x - log(x)) / n
    else if (x <= 1.6d+94) then
        tmp = ((1.0d0 - ((0.5d0 - ((0.3333333333333333d0 - (0.25d0 / x)) / x)) / x)) / x) / n
    else
        tmp = 0.0d0 / n
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (x <= 0.89) {
		tmp = (x - Math.log(x)) / n;
	} else if (x <= 1.6e+94) {
		tmp = ((1.0 - ((0.5 - ((0.3333333333333333 - (0.25 / x)) / x)) / x)) / x) / n;
	} else {
		tmp = 0.0 / n;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 0.89:
		tmp = (x - math.log(x)) / n
	elif x <= 1.6e+94:
		tmp = ((1.0 - ((0.5 - ((0.3333333333333333 - (0.25 / x)) / x)) / x)) / x) / n
	else:
		tmp = 0.0 / n
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 0.89)
		tmp = Float64(Float64(x - log(x)) / n);
	elseif (x <= 1.6e+94)
		tmp = Float64(Float64(Float64(1.0 - Float64(Float64(0.5 - Float64(Float64(0.3333333333333333 - Float64(0.25 / x)) / x)) / x)) / x) / n);
	else
		tmp = Float64(0.0 / n);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (x <= 0.89)
		tmp = (x - log(x)) / n;
	elseif (x <= 1.6e+94)
		tmp = ((1.0 - ((0.5 - ((0.3333333333333333 - (0.25 / x)) / x)) / x)) / x) / n;
	else
		tmp = 0.0 / n;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 0.89], N[(N[(x - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[x, 1.6e+94], N[(N[(N[(1.0 - N[(N[(0.5 - N[(N[(0.3333333333333333 - N[(0.25 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] / n), $MachinePrecision], N[(0.0 / n), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 0.890000000000000013

   1. Initial program 38.6%

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

   3. Taylor expanded in n around inf 57.9%

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

      1. log1p-define 57.9%

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

   5. Simplified 57.9%

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

   6. Taylor expanded in x around 0 57.5%

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

   if 0.890000000000000013 < x < 1.60000000000000007e94

   1. Initial program 38.2%

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

   3. Taylor expanded in n around inf 39.1%

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

      1. log1p-define 39.1%

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

   5. Simplified 39.1%

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

   6. Taylor expanded in x around -inf 68.8%

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

   7. Taylor expanded in x around -inf 68.8%

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

      1. mul-1-neg 68.8%

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

      2. distribute-neg-frac2 68.8%

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

      3. mul-1-neg 68.8%

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

      4. unsub-neg 68.8%

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

      5. mul-1-neg 68.8%

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

      6. unsub-neg 68.8%

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

      7. associate-*r/ 68.8%

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

      8. metadata-eval 68.8%

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

   9. Simplified 68.8%

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

   if 1.60000000000000007e94 < x

   1. Initial program 81.7%

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

   3. Taylor expanded in n around inf 81.7%

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

   5. Simplified 81.7%

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

      1. associate-+r- 81.7%

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

      2. add-log-exp 81.7%

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

      3. exp-diff 81.7%

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

      4. add-exp-log 8.7%

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

   7. Applied egg-rr 8.7%

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

      1. +-commutative 8.7%

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

      2. exp-sum 8.7%

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

      3. *-commutative 8.7%

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

      4. exp-prod 8.7%

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

      5. unpow1/2 8.7%

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

      6. log1p-define 8.7%

         \[\leadsto \frac{\log \left(\frac{\sqrt{e^{\frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}}{n}}} \cdot e^{\color{blue}{\log \left(1 + x\right)}}}{x}\right)}{n} \]

      7. rem-exp-log 81.7%

         \[\leadsto \frac{\log \left(\frac{\sqrt{e^{\frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}}{n}}} \cdot \color{blue}{\left(1 + x\right)}}{x}\right)}{n} \]

   9. Simplified 81.7%

      \[\leadsto \frac{\color{blue}{\log \left(\frac{\sqrt{e^{\frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}}{n}}} \cdot \left(1 + x\right)}{x}\right)}}{n} \]

   10. Taylor expanded in x around inf 81.7%

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

4. Final simplification 66.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.89:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{elif}\;x \leq 1.6 \cdot 10^{+94}:\\ \;\;\;\;\frac{\frac{1 - \frac{0.5 - \frac{0.3333333333333333 - \frac{0.25}{x}}{x}}{x}}{x}}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{n}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 60.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.7:\\ \;\;\;\;\frac{\log x}{-n}\\ \mathbf{elif}\;x \leq 1.46 \cdot 10^{+94}:\\ \;\;\;\;\frac{\frac{1 - \frac{0.5 - \frac{0.3333333333333333 - \frac{0.25}{x}}{x}}{x}}{x}}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 0.7)
   (/ (log x) (- n))
   (if (<= x 1.46e+94)
     (/ (/ (- 1.0 (/ (- 0.5 (/ (- 0.3333333333333333 (/ 0.25 x)) x)) x)) x) n)
     (/ 0.0 n))))

C

double code(double x, double n) {
	double tmp;
	if (x <= 0.7) {
		tmp = log(x) / -n;
	} else if (x <= 1.46e+94) {
		tmp = ((1.0 - ((0.5 - ((0.3333333333333333 - (0.25 / x)) / x)) / x)) / x) / n;
	} else {
		tmp = 0.0 / n;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if (x <= 0.7d0) then
        tmp = log(x) / -n
    else if (x <= 1.46d+94) then
        tmp = ((1.0d0 - ((0.5d0 - ((0.3333333333333333d0 - (0.25d0 / x)) / x)) / x)) / x) / n
    else
        tmp = 0.0d0 / n
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (x <= 0.7) {
		tmp = Math.log(x) / -n;
	} else if (x <= 1.46e+94) {
		tmp = ((1.0 - ((0.5 - ((0.3333333333333333 - (0.25 / x)) / x)) / x)) / x) / n;
	} else {
		tmp = 0.0 / n;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 0.7:
		tmp = math.log(x) / -n
	elif x <= 1.46e+94:
		tmp = ((1.0 - ((0.5 - ((0.3333333333333333 - (0.25 / x)) / x)) / x)) / x) / n
	else:
		tmp = 0.0 / n
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 0.7)
		tmp = Float64(log(x) / Float64(-n));
	elseif (x <= 1.46e+94)
		tmp = Float64(Float64(Float64(1.0 - Float64(Float64(0.5 - Float64(Float64(0.3333333333333333 - Float64(0.25 / x)) / x)) / x)) / x) / n);
	else
		tmp = Float64(0.0 / n);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (x <= 0.7)
		tmp = log(x) / -n;
	elseif (x <= 1.46e+94)
		tmp = ((1.0 - ((0.5 - ((0.3333333333333333 - (0.25 / x)) / x)) / x)) / x) / n;
	else
		tmp = 0.0 / n;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 0.7], N[(N[Log[x], $MachinePrecision] / (-n)), $MachinePrecision], If[LessEqual[x, 1.46e+94], N[(N[(N[(1.0 - N[(N[(0.5 - N[(N[(0.3333333333333333 - N[(0.25 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] / n), $MachinePrecision], N[(0.0 / n), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 0.69999999999999996

   1. Initial program 38.6%

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

   3. Taylor expanded in n around inf 57.9%

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

      1. log1p-define 57.9%

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

   5. Simplified 57.9%

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

   6. Taylor expanded in x around 0 56.7%

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

      1. neg-mul-1 56.7%

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

   8. Simplified 56.7%

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

   if 0.69999999999999996 < x < 1.46000000000000005e94

   1. Initial program 38.2%

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

   3. Taylor expanded in n around inf 39.1%

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

      1. log1p-define 39.1%

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

   5. Simplified 39.1%

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

   6. Taylor expanded in x around -inf 68.8%

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

   7. Taylor expanded in x around -inf 68.8%

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

      1. mul-1-neg 68.8%

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

      2. distribute-neg-frac2 68.8%

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

      3. mul-1-neg 68.8%

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

      4. unsub-neg 68.8%

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

      5. mul-1-neg 68.8%

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

      6. unsub-neg 68.8%

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

      7. associate-*r/ 68.8%

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

      8. metadata-eval 68.8%

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

   9. Simplified 68.8%

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

   if 1.46000000000000005e94 < x

   1. Initial program 81.7%

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

   3. Taylor expanded in n around inf 81.7%

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

   5. Simplified 81.7%

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

      1. associate-+r- 81.7%

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

      2. add-log-exp 81.7%

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

      3. exp-diff 81.7%

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

      4. add-exp-log 8.7%

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

   7. Applied egg-rr 8.7%

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

      1. +-commutative 8.7%

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

      2. exp-sum 8.7%

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

      3. *-commutative 8.7%

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

      4. exp-prod 8.7%

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

      5. unpow1/2 8.7%

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

      6. log1p-define 8.7%

         \[\leadsto \frac{\log \left(\frac{\sqrt{e^{\frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}}{n}}} \cdot e^{\color{blue}{\log \left(1 + x\right)}}}{x}\right)}{n} \]

      7. rem-exp-log 81.7%

         \[\leadsto \frac{\log \left(\frac{\sqrt{e^{\frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}}{n}}} \cdot \color{blue}{\left(1 + x\right)}}{x}\right)}{n} \]

   9. Simplified 81.7%

      \[\leadsto \frac{\color{blue}{\log \left(\frac{\sqrt{e^{\frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}}{n}}} \cdot \left(1 + x\right)}{x}\right)}}{n} \]

   10. Taylor expanded in x around inf 81.7%

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

4. Final simplification 66.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.7:\\ \;\;\;\;\frac{\log x}{-n}\\ \mathbf{elif}\;x \leq 1.46 \cdot 10^{+94}:\\ \;\;\;\;\frac{\frac{1 - \frac{0.5 - \frac{0.3333333333333333 - \frac{0.25}{x}}{x}}{x}}{x}}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{n}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 48.5% accurate, 8.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{+62}:\\ \;\;\;\;\frac{0}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{n} - \frac{\frac{0.5}{n} - \frac{0.3333333333333333}{n \cdot x}}{x}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= (/ 1.0 n) -1e+62)
   (/ 0.0 n)
   (/ (- (/ 1.0 n) (/ (- (/ 0.5 n) (/ 0.3333333333333333 (* n x))) x)) x)))

C

double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -1e+62) {
		tmp = 0.0 / n;
	} else {
		tmp = ((1.0 / n) - (((0.5 / n) - (0.3333333333333333 / (n * x))) / 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) <= (-1d+62)) then
        tmp = 0.0d0 / n
    else
        tmp = ((1.0d0 / n) - (((0.5d0 / n) - (0.3333333333333333d0 / (n * x))) / x)) / x
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -1e+62) {
		tmp = 0.0 / n;
	} else {
		tmp = ((1.0 / n) - (((0.5 / n) - (0.3333333333333333 / (n * x))) / x)) / x;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if (1.0 / n) <= -1e+62:
		tmp = 0.0 / n
	else:
		tmp = ((1.0 / n) - (((0.5 / n) - (0.3333333333333333 / (n * x))) / x)) / x
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -1e+62)
		tmp = Float64(0.0 / n);
	else
		tmp = Float64(Float64(Float64(1.0 / n) - Float64(Float64(Float64(0.5 / n) - Float64(0.3333333333333333 / Float64(n * x))) / x)) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if ((1.0 / n) <= -1e+62)
		tmp = 0.0 / n;
	else
		tmp = ((1.0 / n) - (((0.5 / n) - (0.3333333333333333 / (n * x))) / x)) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[N[(1.0 / n), $MachinePrecision], -1e+62], N[(0.0 / n), $MachinePrecision], N[(N[(N[(1.0 / n), $MachinePrecision] - N[(N[(N[(0.5 / n), $MachinePrecision] - N[(0.3333333333333333 / N[(n * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -1.00000000000000004e62

   1. Initial program 100.0%

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

   3. Taylor expanded in n around inf 82.9%

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

   5. Simplified 82.9%

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

      1. associate-+r- 82.9%

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

      2. add-log-exp 95.0%

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

      3. exp-diff 95.0%

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

      4. add-exp-log 37.3%

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

   7. Applied egg-rr 37.3%

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

      1. +-commutative 37.3%

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

      2. exp-sum 37.3%

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

      3. *-commutative 37.3%

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

      4. exp-prod 37.3%

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

      5. unpow1/2 37.3%

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

      6. log1p-define 37.3%

         \[\leadsto \frac{\log \left(\frac{\sqrt{e^{\frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}}{n}}} \cdot e^{\color{blue}{\log \left(1 + x\right)}}}{x}\right)}{n} \]

      7. rem-exp-log 95.0%

         \[\leadsto \frac{\log \left(\frac{\sqrt{e^{\frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}}{n}}} \cdot \color{blue}{\left(1 + x\right)}}{x}\right)}{n} \]

   9. Simplified 95.0%

      \[\leadsto \frac{\color{blue}{\log \left(\frac{\sqrt{e^{\frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}}{n}}} \cdot \left(1 + x\right)}{x}\right)}}{n} \]

   10. Taylor expanded in x around inf 64.7%

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

   if -1.00000000000000004e62 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 43.0%

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

   3. Taylor expanded in n around inf 62.8%

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

      1. log1p-define 62.8%

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

   5. Simplified 62.8%

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

   6. Taylor expanded in x around -inf 46.1%

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

      1. mul-1-neg 46.1%

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

      2. mul-1-neg 46.1%

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

      3. associate-*r/ 46.1%

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

      4. metadata-eval 46.1%

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

      5. *-commutative 46.1%

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

      6. associate-*r/ 46.1%

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

      7. metadata-eval 46.1%

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

   8. Simplified 46.1%

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

4. Final simplification 48.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{+62}:\\ \;\;\;\;\frac{0}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{n} - \frac{\frac{0.5}{n} - \frac{0.3333333333333333}{n \cdot x}}{x}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 47.0% accurate, 12.4× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{1}{n} - \frac{\frac{0.5}{n} - \frac{0.3333333333333333}{n \cdot x}}{x}}{x} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (/ (- (/ 1.0 n) (/ (- (/ 0.5 n) (/ 0.3333333333333333 (* n x))) x)) x))

C

double code(double x, double n) {
	return ((1.0 / n) - (((0.5 / n) - (0.3333333333333333 / (n * x))) / x)) / x;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    code = ((1.0d0 / n) - (((0.5d0 / n) - (0.3333333333333333d0 / (n * x))) / x)) / x
end function

Java

public static double code(double x, double n) {
	return ((1.0 / n) - (((0.5 / n) - (0.3333333333333333 / (n * x))) / x)) / x;
}

Python

def code(x, n):
	return ((1.0 / n) - (((0.5 / n) - (0.3333333333333333 / (n * x))) / x)) / x

Julia

function code(x, n)
	return Float64(Float64(Float64(1.0 / n) - Float64(Float64(Float64(0.5 / n) - Float64(0.3333333333333333 / Float64(n * x))) / x)) / x)
end

MATLAB

function tmp = code(x, n)
	tmp = ((1.0 / n) - (((0.5 / n) - (0.3333333333333333 / (n * x))) / x)) / x;
end

Wolfram

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

Derivation

1. Initial program 51.6%

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

3. Taylor expanded in n around inf 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 44.8%

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

   1. mul-1-neg 44.8%

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

   2. mul-1-neg 44.8%

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

   3. associate-*r/ 44.8%

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

   4. metadata-eval 44.8%

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

   5. *-commutative 44.8%

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

   6. associate-*r/ 44.8%

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

   7. metadata-eval 44.8%

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

8. Simplified 44.8%

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

9. Final simplification 44.8%

   \[\leadsto \frac{\frac{1}{n} - \frac{\frac{0.5}{n} - \frac{0.3333333333333333}{n \cdot x}}{x}}{x} \]
10. Add Preprocessing

Alternative 19: 47.0% accurate, 15.1× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{\frac{0.5 + \frac{-0.3333333333333333}{x}}{x} + -1}{-x}}{n} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (/ (/ (+ (/ (+ 0.5 (/ -0.3333333333333333 x)) x) -1.0) (- x)) n))

C

double code(double x, double n) {
	return ((((0.5 + (-0.3333333333333333 / x)) / x) + -1.0) / -x) / n;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    code = ((((0.5d0 + ((-0.3333333333333333d0) / x)) / x) + (-1.0d0)) / -x) / n
end function

Java

public static double code(double x, double n) {
	return ((((0.5 + (-0.3333333333333333 / x)) / x) + -1.0) / -x) / n;
}

Python

def code(x, n):
	return ((((0.5 + (-0.3333333333333333 / x)) / x) + -1.0) / -x) / n

Julia

function code(x, n)
	return Float64(Float64(Float64(Float64(Float64(0.5 + Float64(-0.3333333333333333 / x)) / x) + -1.0) / Float64(-x)) / n)
end

MATLAB

function tmp = code(x, n)
	tmp = ((((0.5 + (-0.3333333333333333 / x)) / x) + -1.0) / -x) / n;
end

Wolfram

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

Derivation

1. Initial program 51.6%

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

3. Taylor expanded in n around inf 65.7%

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

5. Simplified 65.7%

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

   1. associate-+r- 65.7%

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

   2. add-log-exp 70.5%

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

   3. exp-diff 70.5%

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

   4. add-exp-log 44.8%

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

7. Applied egg-rr 44.8%

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

   1. +-commutative 44.8%

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

   2. exp-sum 44.8%

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

   3. *-commutative 44.8%

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

   4. exp-prod 44.8%

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

   5. unpow1/2 44.8%

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

   6. log1p-define 44.8%

      \[\leadsto \frac{\log \left(\frac{\sqrt{e^{\frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}}{n}}} \cdot e^{\color{blue}{\log \left(1 + x\right)}}}{x}\right)}{n} \]

   7. rem-exp-log 70.7%

      \[\leadsto \frac{\log \left(\frac{\sqrt{e^{\frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}}{n}}} \cdot \color{blue}{\left(1 + x\right)}}{x}\right)}{n} \]

9. Simplified 70.7%

   \[\leadsto \frac{\color{blue}{\log \left(\frac{\sqrt{e^{\frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}}{n}}} \cdot \left(1 + x\right)}{x}\right)}}{n} \]

10. Taylor expanded in n around inf 62.7%

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

    1. +-commutative 62.7%

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

12. Simplified 62.7%

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

13. Taylor expanded in x around -inf 44.8%

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

    1. mul-1-neg 44.8%

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

    2. distribute-neg-frac2 44.8%

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

    3. sub-neg 44.8%

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

    4. associate-*r/ 44.8%

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

    5. sub-neg 44.8%

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

    6. metadata-eval 44.8%

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

    7. distribute-lft-in 44.8%

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

    8. neg-mul-1 44.8%

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

    9. associate-*r/ 44.8%

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

    10. metadata-eval 44.8%

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

    11. distribute-neg-frac 44.8%

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

    12. metadata-eval 44.8%

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

    13. metadata-eval 44.8%

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

    14. metadata-eval 44.8%

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

15. Simplified 44.8%

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

16. Final simplification 44.8%

    \[\leadsto \frac{\frac{\frac{0.5 + \frac{-0.3333333333333333}{x}}{x} + -1}{-x}}{n} \]
17. Add Preprocessing

Alternative 20: 41.0% accurate, 30.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 51.6%

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

3. Taylor expanded in n around inf 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 38.9%

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

   1. *-commutative 38.9%

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

8. Simplified 38.9%

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

   1. associate-/r* 39.6%

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

   2. div-inv 39.6%

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

10. Applied egg-rr 39.6%

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

11. Final simplification 39.6%

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

Alternative 21: 41.1% accurate, 42.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 51.6%

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

3. Taylor expanded in n around inf 65.7%

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

5. Simplified 65.7%

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

   1. associate-+r- 65.7%

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

   2. add-log-exp 70.5%

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

   3. exp-diff 70.5%

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

   4. add-exp-log 44.8%

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

7. Applied egg-rr 44.8%

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

   1. +-commutative 44.8%

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

   2. exp-sum 44.8%

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

   3. *-commutative 44.8%

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

   4. exp-prod 44.8%

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

   5. unpow1/2 44.8%

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

   6. log1p-define 44.8%

      \[\leadsto \frac{\log \left(\frac{\sqrt{e^{\frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}}{n}}} \cdot e^{\color{blue}{\log \left(1 + x\right)}}}{x}\right)}{n} \]

   7. rem-exp-log 70.7%

      \[\leadsto \frac{\log \left(\frac{\sqrt{e^{\frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}}{n}}} \cdot \color{blue}{\left(1 + x\right)}}{x}\right)}{n} \]

9. Simplified 70.7%

   \[\leadsto \frac{\color{blue}{\log \left(\frac{\sqrt{e^{\frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}}{n}}} \cdot \left(1 + x\right)}{x}\right)}}{n} \]

10. Taylor expanded in n around inf 62.7%

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

    1. +-commutative 62.7%

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

12. Simplified 62.7%

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

13. Taylor expanded in x around inf 38.9%

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

    1. associate-/r* 39.6%

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

15. Simplified 39.6%

    \[\leadsto \color{blue}{\frac{\frac{1}{n}}{x}} \]
16. Add Preprocessing

Alternative 22: 40.5% accurate, 42.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 51.6%

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

3. Taylor expanded in n around inf 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 38.9%

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

   1. *-commutative 38.9%

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

8. Simplified 38.9%

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

9. Final simplification 38.9%

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

Alternative 23: 4.5% accurate, 70.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 51.6%

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

3. Taylor expanded in n around inf 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 0 33.7%

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

7. Taylor expanded in x around inf 4.9%

   \[\leadsto \color{blue}{\frac{x}{n}} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024146 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  :precision binary64
  (- (pow (+ x 1.0) (/ 1.0 n)) (pow x (/ 1.0 n))))