2nthrt (problem 3.4.6)

Percentage Accurate: 53.4% → 86.2%

Time: 44.1s

Alternatives: 12

Speedup: 1.8×

• could not determine a ground truth

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 53.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 86.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ t_1 := {\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - t\_0\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{-7}:\\ \;\;\;\;1 - \sqrt[3]{\mathsf{expm1}\left(\mathsf{log1p}\left({x}^{\left(\frac{3}{n}\right)}\right)\right)}\\ \mathbf{elif}\;t\_1 \leq 10^{-13}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right) - \log x}{n}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{x}{n}} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))) (t_1 (- (pow (+ x 1.0) (/ 1.0 n)) t_0)))
   (if (<= t_1 -1e-7)
     (- 1.0 (cbrt (expm1 (log1p (pow x (/ 3.0 n))))))
     (if (<= t_1 1e-13) (/ (- (log1p x) (log x)) n) (- (exp (/ x n)) t_0)))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double t_1 = pow((x + 1.0), (1.0 / n)) - t_0;
	double tmp;
	if (t_1 <= -1e-7) {
		tmp = 1.0 - cbrt(expm1(log1p(pow(x, (3.0 / n)))));
	} else if (t_1 <= 1e-13) {
		tmp = (log1p(x) - log(x)) / n;
	} else {
		tmp = exp((x / n)) - t_0;
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, (1.0 / n));
	double t_1 = Math.pow((x + 1.0), (1.0 / n)) - t_0;
	double tmp;
	if (t_1 <= -1e-7) {
		tmp = 1.0 - Math.cbrt(Math.expm1(Math.log1p(Math.pow(x, (3.0 / n)))));
	} else if (t_1 <= 1e-13) {
		tmp = (Math.log1p(x) - Math.log(x)) / n;
	} else {
		tmp = Math.exp((x / n)) - t_0;
	}
	return tmp;
}

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	t_1 = Float64((Float64(x + 1.0) ^ Float64(1.0 / n)) - t_0)
	tmp = 0.0
	if (t_1 <= -1e-7)
		tmp = Float64(1.0 - cbrt(expm1(log1p((x ^ Float64(3.0 / n))))));
	elseif (t_1 <= 1e-13)
		tmp = Float64(Float64(log1p(x) - log(x)) / n);
	else
		tmp = Float64(exp(Float64(x / n)) - t_0);
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[Power[N[(x + 1.0), $MachinePrecision], N[(1.0 / n), $MachinePrecision]], $MachinePrecision] - t$95$0), $MachinePrecision]}, If[LessEqual[t$95$1, -1e-7], N[(1.0 - N[Power[N[(Exp[N[Log[1 + N[Power[x, N[(3.0 / n), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]] - 1), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e-13], N[(N[(N[Log[1 + x], $MachinePrecision] - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], N[(N[Exp[N[(x / n), $MachinePrecision]], $MachinePrecision] - t$95$0), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (pow.f64 (+.f64 x 1) (/.f64 1 n)) (pow.f64 x (/.f64 1 n))) < -9.9999999999999995e-8

   1. Initial program 99.2%

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

      1. add-cbrt-cube 99.2%

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

      2. pow3 99.2%

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

      3. pow-pow 99.2%

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

   4. Applied egg-rr 99.2%

      \[\leadsto {\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - \color{blue}{\sqrt[3]{{x}^{\left(\frac{1}{n} \cdot 3\right)}}} \]
   5. Step-by-step derivation

      1. associate-*l/ 99.2%

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

      2. metadata-eval 99.2%

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

   6. Simplified 99.2%

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

   7. Taylor expanded in x around 0 99.2%

      \[\leadsto \color{blue}{1 - {\left(e^{3 \cdot \frac{\log x}{n}}\right)}^{0.3333333333333333}} \]
   8. Step-by-step derivation

      1. associate-*r/ 99.2%

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

      2. associate-*l/ 99.1%

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

      3. *-commutative 99.1%

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

      4. exp-to-pow 99.1%

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

      5. unpow1/3 99.2%

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

   9. Simplified 99.2%

      \[\leadsto \color{blue}{1 - \sqrt[3]{{x}^{\left(\frac{3}{n}\right)}}} \]
   10. Step-by-step derivation

       1. expm1-log1p-u 99.3%

          \[\leadsto 1 - \sqrt[3]{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({x}^{\left(\frac{3}{n}\right)}\right)\right)}} \]

       2. expm1-undefine 99.3%

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

   11. Applied egg-rr 99.3%

       \[\leadsto 1 - \sqrt[3]{\color{blue}{e^{\mathsf{log1p}\left({x}^{\left(\frac{3}{n}\right)}\right)} - 1}} \]
   12. Step-by-step derivation

       1. expm1-define 99.3%

          \[\leadsto 1 - \sqrt[3]{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({x}^{\left(\frac{3}{n}\right)}\right)\right)}} \]

   13. Simplified 99.3%

       \[\leadsto 1 - \sqrt[3]{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({x}^{\left(\frac{3}{n}\right)}\right)\right)}} \]

   if -9.9999999999999995e-8 < (-.f64 (pow.f64 (+.f64 x 1) (/.f64 1 n)) (pow.f64 x (/.f64 1 n))) < 1e-13

   1. Initial program 40.8%

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

   3. Taylor expanded in n around inf 82.3%

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

      1. +-rgt-identity 82.3%

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

      2. +-rgt-identity 82.3%

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

      3. log1p-define 82.3%

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

   5. Simplified 82.3%

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

   if 1e-13 < (-.f64 (pow.f64 (+.f64 x 1) (/.f64 1 n)) (pow.f64 x (/.f64 1 n)))

   1. Initial program 44.5%

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

   3. Taylor expanded in n around 0 44.5%

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

      1. log1p-define 99.6%

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

      2. *-rgt-identity 99.6%

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

      3. associate-/l* 99.6%

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

      4. exp-to-pow 99.6%

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

   5. Simplified 99.6%

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

   6. Taylor expanded in x around 0 99.6%

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

4. Final simplification 87.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \leq -1 \cdot 10^{-7}:\\ \;\;\;\;1 - \sqrt[3]{\mathsf{expm1}\left(\mathsf{log1p}\left({x}^{\left(\frac{3}{n}\right)}\right)\right)}\\ \mathbf{elif}\;{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \leq 10^{-13}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right) - \log x}{n}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{x}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 86.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ t_1 := {\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - t\_0\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{-7}:\\ \;\;\;\;1 - t\_0\\ \mathbf{elif}\;t\_1 \leq 10^{-13}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right) - \log x}{n}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{x}{n}} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))) (t_1 (- (pow (+ x 1.0) (/ 1.0 n)) t_0)))
   (if (<= t_1 -1e-7)
     (- 1.0 t_0)
     (if (<= t_1 1e-13) (/ (- (log1p x) (log x)) n) (- (exp (/ x n)) t_0)))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double t_1 = pow((x + 1.0), (1.0 / n)) - t_0;
	double tmp;
	if (t_1 <= -1e-7) {
		tmp = 1.0 - t_0;
	} else if (t_1 <= 1e-13) {
		tmp = (log1p(x) - log(x)) / n;
	} else {
		tmp = exp((x / n)) - t_0;
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, (1.0 / n));
	double t_1 = Math.pow((x + 1.0), (1.0 / n)) - t_0;
	double tmp;
	if (t_1 <= -1e-7) {
		tmp = 1.0 - t_0;
	} else if (t_1 <= 1e-13) {
		tmp = (Math.log1p(x) - Math.log(x)) / n;
	} else {
		tmp = Math.exp((x / n)) - t_0;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	t_1 = math.pow((x + 1.0), (1.0 / n)) - t_0
	tmp = 0
	if t_1 <= -1e-7:
		tmp = 1.0 - t_0
	elif t_1 <= 1e-13:
		tmp = (math.log1p(x) - math.log(x)) / n
	else:
		tmp = math.exp((x / n)) - t_0
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	t_1 = Float64((Float64(x + 1.0) ^ Float64(1.0 / n)) - t_0)
	tmp = 0.0
	if (t_1 <= -1e-7)
		tmp = Float64(1.0 - t_0);
	elseif (t_1 <= 1e-13)
		tmp = Float64(Float64(log1p(x) - log(x)) / n);
	else
		tmp = Float64(exp(Float64(x / n)) - t_0);
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[Power[N[(x + 1.0), $MachinePrecision], N[(1.0 / n), $MachinePrecision]], $MachinePrecision] - t$95$0), $MachinePrecision]}, If[LessEqual[t$95$1, -1e-7], N[(1.0 - t$95$0), $MachinePrecision], If[LessEqual[t$95$1, 1e-13], N[(N[(N[Log[1 + x], $MachinePrecision] - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], N[(N[Exp[N[(x / n), $MachinePrecision]], $MachinePrecision] - t$95$0), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (pow.f64 (+.f64 x 1) (/.f64 1 n)) (pow.f64 x (/.f64 1 n))) < -9.9999999999999995e-8

   1. Initial program 99.2%

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

   3. Taylor expanded in x around 0 99.2%

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

      1. *-rgt-identity 99.2%

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

      2. associate-/l* 99.2%

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

      3. exp-to-pow 99.2%

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

   5. Simplified 99.2%

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

   if -9.9999999999999995e-8 < (-.f64 (pow.f64 (+.f64 x 1) (/.f64 1 n)) (pow.f64 x (/.f64 1 n))) < 1e-13

   1. Initial program 40.8%

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

   3. Taylor expanded in n around inf 82.3%

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

      1. +-rgt-identity 82.3%

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

      2. +-rgt-identity 82.3%

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

      3. log1p-define 82.3%

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

   5. Simplified 82.3%

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

   if 1e-13 < (-.f64 (pow.f64 (+.f64 x 1) (/.f64 1 n)) (pow.f64 x (/.f64 1 n)))

   1. Initial program 44.5%

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

   3. Taylor expanded in n around 0 44.5%

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

      1. log1p-define 99.6%

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

      2. *-rgt-identity 99.6%

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

      3. associate-/l* 99.6%

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

      4. exp-to-pow 99.6%

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

   5. Simplified 99.6%

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

   6. Taylor expanded in x around 0 99.6%

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

4. Final simplification 87.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \leq -1 \cdot 10^{-7}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \leq 10^{-13}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right) - \log x}{n}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{x}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 86.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ t_1 := {\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - t\_0\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{-7}:\\ \;\;\;\;1 - \mathsf{expm1}\left(\mathsf{log1p}\left(t\_0\right)\right)\\ \mathbf{elif}\;t\_1 \leq 10^{-13}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right) - \log x}{n}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{x}{n}} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))) (t_1 (- (pow (+ x 1.0) (/ 1.0 n)) t_0)))
   (if (<= t_1 -1e-7)
     (- 1.0 (expm1 (log1p t_0)))
     (if (<= t_1 1e-13) (/ (- (log1p x) (log x)) n) (- (exp (/ x n)) t_0)))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double t_1 = pow((x + 1.0), (1.0 / n)) - t_0;
	double tmp;
	if (t_1 <= -1e-7) {
		tmp = 1.0 - expm1(log1p(t_0));
	} else if (t_1 <= 1e-13) {
		tmp = (log1p(x) - log(x)) / n;
	} else {
		tmp = exp((x / n)) - t_0;
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, (1.0 / n));
	double t_1 = Math.pow((x + 1.0), (1.0 / n)) - t_0;
	double tmp;
	if (t_1 <= -1e-7) {
		tmp = 1.0 - Math.expm1(Math.log1p(t_0));
	} else if (t_1 <= 1e-13) {
		tmp = (Math.log1p(x) - Math.log(x)) / n;
	} else {
		tmp = Math.exp((x / n)) - t_0;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	t_1 = math.pow((x + 1.0), (1.0 / n)) - t_0
	tmp = 0
	if t_1 <= -1e-7:
		tmp = 1.0 - math.expm1(math.log1p(t_0))
	elif t_1 <= 1e-13:
		tmp = (math.log1p(x) - math.log(x)) / n
	else:
		tmp = math.exp((x / n)) - t_0
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	t_1 = Float64((Float64(x + 1.0) ^ Float64(1.0 / n)) - t_0)
	tmp = 0.0
	if (t_1 <= -1e-7)
		tmp = Float64(1.0 - expm1(log1p(t_0)));
	elseif (t_1 <= 1e-13)
		tmp = Float64(Float64(log1p(x) - log(x)) / n);
	else
		tmp = Float64(exp(Float64(x / n)) - t_0);
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[Power[N[(x + 1.0), $MachinePrecision], N[(1.0 / n), $MachinePrecision]], $MachinePrecision] - t$95$0), $MachinePrecision]}, If[LessEqual[t$95$1, -1e-7], N[(1.0 - N[(Exp[N[Log[1 + t$95$0], $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e-13], N[(N[(N[Log[1 + x], $MachinePrecision] - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], N[(N[Exp[N[(x / n), $MachinePrecision]], $MachinePrecision] - t$95$0), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (pow.f64 (+.f64 x 1) (/.f64 1 n)) (pow.f64 x (/.f64 1 n))) < -9.9999999999999995e-8

   1. Initial program 99.2%

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

   3. Taylor expanded in x around 0 99.2%

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

      1. *-rgt-identity 99.2%

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

      2. associate-/l* 99.2%

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

      3. exp-to-pow 99.2%

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

   5. Simplified 99.2%

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

      1. expm1-log1p-u 99.2%

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

      2. expm1-undefine 99.2%

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

   7. Applied egg-rr 99.2%

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

      1. expm1-define 99.2%

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

   9. Simplified 99.2%

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

   if -9.9999999999999995e-8 < (-.f64 (pow.f64 (+.f64 x 1) (/.f64 1 n)) (pow.f64 x (/.f64 1 n))) < 1e-13

   1. Initial program 40.8%

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

   3. Taylor expanded in n around inf 82.3%

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

      1. +-rgt-identity 82.3%

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

      2. +-rgt-identity 82.3%

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

      3. log1p-define 82.3%

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

   5. Simplified 82.3%

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

   if 1e-13 < (-.f64 (pow.f64 (+.f64 x 1) (/.f64 1 n)) (pow.f64 x (/.f64 1 n)))

   1. Initial program 44.5%

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

   3. Taylor expanded in n around 0 44.5%

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

      1. log1p-define 99.6%

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

      2. *-rgt-identity 99.6%

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

      3. associate-/l* 99.6%

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

      4. exp-to-pow 99.6%

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

   5. Simplified 99.6%

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

   6. Taylor expanded in x around 0 99.6%

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

4. Final simplification 87.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \leq -1 \cdot 10^{-7}:\\ \;\;\;\;1 - \mathsf{expm1}\left(\mathsf{log1p}\left({x}^{\left(\frac{1}{n}\right)}\right)\right)\\ \mathbf{elif}\;{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \leq 10^{-13}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right) - \log x}{n}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{x}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 83.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{-7}:\\ \;\;\;\;\frac{t\_0}{x \cdot n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-15}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right) - \log x}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{+121}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - t\_0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{x}{n}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -5e-7)
     (/ t_0 (* x n))
     (if (<= (/ 1.0 n) 5e-15)
       (/ (- (log1p x) (log x)) n)
       (if (<= (/ 1.0 n) 2e+121)
         (- (+ 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) <= -5e-7) {
		tmp = t_0 / (x * n);
	} else if ((1.0 / n) <= 5e-15) {
		tmp = (log1p(x) - log(x)) / n;
	} else if ((1.0 / n) <= 2e+121) {
		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) <= -5e-7) {
		tmp = t_0 / (x * n);
	} else if ((1.0 / n) <= 5e-15) {
		tmp = (Math.log1p(x) - Math.log(x)) / n;
	} else if ((1.0 / n) <= 2e+121) {
		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) <= -5e-7:
		tmp = t_0 / (x * n)
	elif (1.0 / n) <= 5e-15:
		tmp = (math.log1p(x) - math.log(x)) / n
	elif (1.0 / n) <= 2e+121:
		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) <= -5e-7)
		tmp = Float64(t_0 / Float64(x * n));
	elseif (Float64(1.0 / n) <= 5e-15)
		tmp = Float64(Float64(log1p(x) - log(x)) / n);
	elseif (Float64(1.0 / n) <= 2e+121)
		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], -5e-7], N[(t$95$0 / N[(x * n), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e-15], N[(N[(N[Log[1 + x], $MachinePrecision] - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e+121], 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 1 n) < -4.99999999999999977e-7

   1. Initial program 97.6%

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

   3. Taylor expanded in x around inf 97.7%

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

      1. log-rec 97.7%

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

      2. mul-1-neg 97.7%

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

      3. associate-*r/ 97.7%

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

      4. associate-*r* 97.7%

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

      5. metadata-eval 97.7%

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

      6. *-commutative 97.7%

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

      7. associate-/l* 97.7%

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

      8. exp-to-pow 97.7%

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

      9. *-commutative 97.7%

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

   5. Simplified 97.7%

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

   if -4.99999999999999977e-7 < (/.f64 1 n) < 4.99999999999999999e-15

   1. Initial program 30.1%

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

   3. Taylor expanded in n around inf 81.7%

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

      1. +-rgt-identity 81.7%

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

      2. +-rgt-identity 81.7%

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

      3. log1p-define 81.7%

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

   5. Simplified 81.7%

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

   if 4.99999999999999999e-15 < (/.f64 1 n) < 2.00000000000000007e121

   1. Initial program 65.8%

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

   3. Taylor expanded in x around 0 66.2%

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

   if 2.00000000000000007e121 < (/.f64 1 n)

   1. Initial program 30.0%

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

   3. Taylor expanded in n around inf 0.0%

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

   4. Taylor expanded in x around inf 0.1%

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

      1. +-commutative 0.1%

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

      2. mul-1-neg 0.1%

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

      3. distribute-frac-neg 0.1%

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

      4. log-rec 0.1%

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

      5. remove-double-neg 0.1%

         \[\leadsto \frac{\frac{1}{n} + \frac{\color{blue}{\log x}}{{n}^{2}}}{x} \]

   6. Simplified 0.1%

      \[\leadsto \color{blue}{\frac{\frac{1}{n} + \frac{\log x}{{n}^{2}}}{x}} \]

   7. Taylor expanded in n around inf 55.0%

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

      1. *-commutative 55.0%

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

   9. Simplified 55.0%

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

       1. log1p-expm1-u 79.6%

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

       2. associate-/r* 79.6%

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

       3. rem-exp-log 79.6%

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

       4. neg-log 79.6%

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

       5. add-sqr-sqrt 79.6%

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

       6. sqrt-unprod 79.6%

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

       7. sqr-neg 79.6%

          \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{e^{\sqrt{\color{blue}{\log x \cdot \log x}}}}{n}\right)\right) \]

       8. sqrt-unprod 0.0%

          \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{e^{\color{blue}{\sqrt{\log x} \cdot \sqrt{\log x}}}}{n}\right)\right) \]

       9. add-sqr-sqrt 80.8%

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

       10. add-exp-log 80.8%

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

   11. Applied egg-rr 80.8%

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

4. Final simplification 86.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{-7}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{x \cdot n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-15}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right) - \log x}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{+121}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{x}{n}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 67.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\log x}{-n}\\ \mathbf{if}\;x \leq 8.5 \cdot 10^{-190}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 4.2 \cdot 10^{-172}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;x \leq 3.1 \cdot 10^{-136}:\\ \;\;\;\;\log x \cdot \frac{-1}{n}\\ \mathbf{elif}\;x \leq 2.8 \cdot 10^{-73}:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{x}{n}\right)\right)\\ \mathbf{elif}\;x \leq 2.2 \cdot 10^{-60}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n} + -1\right)}}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (/ (log x) (- n))))
   (if (<= x 8.5e-190)
     t_0
     (if (<= x 4.2e-172)
       (- (+ 1.0 (/ x n)) (pow x (/ 1.0 n)))
       (if (<= x 3.1e-136)
         (* (log x) (/ -1.0 n))
         (if (<= x 2.8e-73)
           (log1p (expm1 (/ x n)))
           (if (<= x 2.2e-60) t_0 (/ (pow x (+ (/ 1.0 n) -1.0)) n))))))))

C

double code(double x, double n) {
	double t_0 = log(x) / -n;
	double tmp;
	if (x <= 8.5e-190) {
		tmp = t_0;
	} else if (x <= 4.2e-172) {
		tmp = (1.0 + (x / n)) - pow(x, (1.0 / n));
	} else if (x <= 3.1e-136) {
		tmp = log(x) * (-1.0 / n);
	} else if (x <= 2.8e-73) {
		tmp = log1p(expm1((x / n)));
	} else if (x <= 2.2e-60) {
		tmp = t_0;
	} else {
		tmp = pow(x, ((1.0 / n) + -1.0)) / n;
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double t_0 = Math.log(x) / -n;
	double tmp;
	if (x <= 8.5e-190) {
		tmp = t_0;
	} else if (x <= 4.2e-172) {
		tmp = (1.0 + (x / n)) - Math.pow(x, (1.0 / n));
	} else if (x <= 3.1e-136) {
		tmp = Math.log(x) * (-1.0 / n);
	} else if (x <= 2.8e-73) {
		tmp = Math.log1p(Math.expm1((x / n)));
	} else if (x <= 2.2e-60) {
		tmp = t_0;
	} else {
		tmp = Math.pow(x, ((1.0 / n) + -1.0)) / n;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.log(x) / -n
	tmp = 0
	if x <= 8.5e-190:
		tmp = t_0
	elif x <= 4.2e-172:
		tmp = (1.0 + (x / n)) - math.pow(x, (1.0 / n))
	elif x <= 3.1e-136:
		tmp = math.log(x) * (-1.0 / n)
	elif x <= 2.8e-73:
		tmp = math.log1p(math.expm1((x / n)))
	elif x <= 2.2e-60:
		tmp = t_0
	else:
		tmp = math.pow(x, ((1.0 / n) + -1.0)) / n
	return tmp

Julia

function code(x, n)
	t_0 = Float64(log(x) / Float64(-n))
	tmp = 0.0
	if (x <= 8.5e-190)
		tmp = t_0;
	elseif (x <= 4.2e-172)
		tmp = Float64(Float64(1.0 + Float64(x / n)) - (x ^ Float64(1.0 / n)));
	elseif (x <= 3.1e-136)
		tmp = Float64(log(x) * Float64(-1.0 / n));
	elseif (x <= 2.8e-73)
		tmp = log1p(expm1(Float64(x / n)));
	elseif (x <= 2.2e-60)
		tmp = t_0;
	else
		tmp = Float64((x ^ Float64(Float64(1.0 / n) + -1.0)) / n);
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(N[Log[x], $MachinePrecision] / (-n)), $MachinePrecision]}, If[LessEqual[x, 8.5e-190], t$95$0, If[LessEqual[x, 4.2e-172], N[(N[(1.0 + N[(x / n), $MachinePrecision]), $MachinePrecision] - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 3.1e-136], N[(N[Log[x], $MachinePrecision] * N[(-1.0 / n), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 2.8e-73], N[Log[1 + N[(Exp[N[(x / n), $MachinePrecision]] - 1), $MachinePrecision]], $MachinePrecision], If[LessEqual[x, 2.2e-60], t$95$0, N[(N[Power[x, N[(N[(1.0 / n), $MachinePrecision] + -1.0), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision]]]]]]]

Derivation

1. Split input into 5 regimes

2. if x < 8.5000000000000003e-190 or 2.80000000000000012e-73 < x < 2.1999999999999999e-60

   1. Initial program 39.4%

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

   3. Taylor expanded in x around 0 39.4%

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

      1. *-rgt-identity 39.4%

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

      2. associate-/l* 39.4%

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

      3. exp-to-pow 39.4%

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

   5. Simplified 39.4%

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

   6. Taylor expanded in n around inf 61.6%

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

      1. associate-*r/ 61.6%

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

      2. mul-1-neg 61.6%

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

   8. Simplified 61.6%

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

   if 8.5000000000000003e-190 < x < 4.1999999999999999e-172

   1. Initial program 83.9%

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

   3. Taylor expanded in x around 0 84.0%

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

   if 4.1999999999999999e-172 < x < 3.1e-136

   1. Initial program 36.9%

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

   3. Taylor expanded in x around 0 36.9%

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

      1. *-rgt-identity 36.9%

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

      2. associate-/l* 36.9%

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

      3. exp-to-pow 36.9%

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

   5. Simplified 36.9%

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

   6. Taylor expanded in n around inf 63.1%

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

      1. associate-*r/ 63.1%

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

      2. mul-1-neg 63.1%

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

   8. Simplified 63.1%

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

      1. div-inv 63.3%

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

   10. Applied egg-rr 63.3%

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

   if 3.1e-136 < x < 2.80000000000000012e-73

   1. Initial program 39.8%

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

   3. Taylor expanded in n around inf 34.1%

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

   4. Taylor expanded in x around inf 27.4%

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

      1. +-commutative 27.4%

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

      2. mul-1-neg 27.4%

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

      3. distribute-frac-neg 27.4%

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

      4. log-rec 27.4%

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

      5. remove-double-neg 27.4%

         \[\leadsto \frac{\frac{1}{n} + \frac{\color{blue}{\log x}}{{n}^{2}}}{x} \]

   6. Simplified 27.4%

      \[\leadsto \color{blue}{\frac{\frac{1}{n} + \frac{\log x}{{n}^{2}}}{x}} \]

   7. Taylor expanded in n around inf 37.6%

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

      1. *-commutative 37.6%

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

   9. Simplified 37.6%

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

       1. log1p-expm1-u 60.1%

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

       2. associate-/r* 60.1%

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

       3. rem-exp-log 60.1%

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

       4. neg-log 60.1%

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

       5. add-sqr-sqrt 60.1%

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

       6. sqrt-unprod 60.1%

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

       7. sqr-neg 60.1%

          \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{e^{\sqrt{\color{blue}{\log x \cdot \log x}}}}{n}\right)\right) \]

       8. sqrt-unprod 0.0%

          \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{e^{\color{blue}{\sqrt{\log x} \cdot \sqrt{\log x}}}}{n}\right)\right) \]

       9. add-sqr-sqrt 57.2%

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

       10. add-exp-log 57.2%

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

   11. Applied egg-rr 57.2%

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

   if 2.1999999999999999e-60 < x

   1. Initial program 65.7%

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

   3. Taylor expanded in x around inf 84.3%

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

      1. log-rec 84.3%

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

      2. mul-1-neg 84.3%

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

      3. associate-*r/ 84.3%

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

      4. associate-*r* 84.3%

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

      5. metadata-eval 84.3%

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

      6. *-commutative 84.3%

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

      7. associate-/l* 84.3%

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

      8. exp-to-pow 84.3%

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

      9. *-commutative 84.3%

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

   5. Simplified 84.3%

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

      1. *-un-lft-identity 84.3%

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

      2. associate-/r* 84.7%

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

      3. pow1 84.7%

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

      4. pow-div 84.4%

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

   7. Applied egg-rr 84.4%

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

      1. *-lft-identity 84.4%

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

      2. sub-neg 84.4%

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

      3. metadata-eval 84.4%

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

   9. Simplified 84.4%

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

4. Final simplification 73.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 8.5 \cdot 10^{-190}:\\ \;\;\;\;\frac{\log x}{-n}\\ \mathbf{elif}\;x \leq 4.2 \cdot 10^{-172}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;x \leq 3.1 \cdot 10^{-136}:\\ \;\;\;\;\log x \cdot \frac{-1}{n}\\ \mathbf{elif}\;x \leq 2.8 \cdot 10^{-73}:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{x}{n}\right)\right)\\ \mathbf{elif}\;x \leq 2.2 \cdot 10^{-60}:\\ \;\;\;\;\frac{\log x}{-n}\\ \mathbf{else}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n} + -1\right)}}{n}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 54.6% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\log x}{-n}\\ t_1 := 1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;x \leq 10^{-189}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 2.4 \cdot 10^{-171}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 2.3 \cdot 10^{-60}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 435:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{n}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (/ (log x) (- n))) (t_1 (- 1.0 (pow x (/ 1.0 n)))))
   (if (<= x 1e-189)
     t_0
     (if (<= x 2.4e-171)
       t_1
       (if (<= x 2.3e-60) t_0 (if (<= x 435.0) t_1 (/ (/ 1.0 n) x)))))))

C

double code(double x, double n) {
	double t_0 = log(x) / -n;
	double t_1 = 1.0 - pow(x, (1.0 / n));
	double tmp;
	if (x <= 1e-189) {
		tmp = t_0;
	} else if (x <= 2.4e-171) {
		tmp = t_1;
	} else if (x <= 2.3e-60) {
		tmp = t_0;
	} else if (x <= 435.0) {
		tmp = t_1;
	} else {
		tmp = (1.0 / n) / x;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = log(x) / -n
    t_1 = 1.0d0 - (x ** (1.0d0 / n))
    if (x <= 1d-189) then
        tmp = t_0
    else if (x <= 2.4d-171) then
        tmp = t_1
    else if (x <= 2.3d-60) then
        tmp = t_0
    else if (x <= 435.0d0) then
        tmp = t_1
    else
        tmp = (1.0d0 / n) / x
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = Math.log(x) / -n;
	double t_1 = 1.0 - Math.pow(x, (1.0 / n));
	double tmp;
	if (x <= 1e-189) {
		tmp = t_0;
	} else if (x <= 2.4e-171) {
		tmp = t_1;
	} else if (x <= 2.3e-60) {
		tmp = t_0;
	} else if (x <= 435.0) {
		tmp = t_1;
	} else {
		tmp = (1.0 / n) / x;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.log(x) / -n
	t_1 = 1.0 - math.pow(x, (1.0 / n))
	tmp = 0
	if x <= 1e-189:
		tmp = t_0
	elif x <= 2.4e-171:
		tmp = t_1
	elif x <= 2.3e-60:
		tmp = t_0
	elif x <= 435.0:
		tmp = t_1
	else:
		tmp = (1.0 / n) / x
	return tmp

Julia

function code(x, n)
	t_0 = Float64(log(x) / Float64(-n))
	t_1 = Float64(1.0 - (x ^ Float64(1.0 / n)))
	tmp = 0.0
	if (x <= 1e-189)
		tmp = t_0;
	elseif (x <= 2.4e-171)
		tmp = t_1;
	elseif (x <= 2.3e-60)
		tmp = t_0;
	elseif (x <= 435.0)
		tmp = t_1;
	else
		tmp = Float64(Float64(1.0 / n) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = log(x) / -n;
	t_1 = 1.0 - (x ^ (1.0 / n));
	tmp = 0.0;
	if (x <= 1e-189)
		tmp = t_0;
	elseif (x <= 2.4e-171)
		tmp = t_1;
	elseif (x <= 2.3e-60)
		tmp = t_0;
	elseif (x <= 435.0)
		tmp = t_1;
	else
		tmp = (1.0 / n) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(N[Log[x], $MachinePrecision] / (-n)), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 1e-189], t$95$0, If[LessEqual[x, 2.4e-171], t$95$1, If[LessEqual[x, 2.3e-60], t$95$0, If[LessEqual[x, 435.0], t$95$1, N[(N[(1.0 / n), $MachinePrecision] / x), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < 1.00000000000000007e-189 or 2.39999999999999987e-171 < x < 2.3000000000000001e-60

   1. Initial program 39.1%

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

   3. Taylor expanded in x around 0 39.1%

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

      1. *-rgt-identity 39.1%

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

      2. associate-/l* 39.1%

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

      3. exp-to-pow 39.1%

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

   5. Simplified 39.1%

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

   6. Taylor expanded in n around inf 56.6%

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

      1. associate-*r/ 56.6%

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

      2. mul-1-neg 56.6%

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

   8. Simplified 56.6%

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

   if 1.00000000000000007e-189 < x < 2.39999999999999987e-171 or 2.3000000000000001e-60 < x < 435

   1. Initial program 62.6%

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

   3. Taylor expanded in x around 0 56.0%

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

      1. *-rgt-identity 56.0%

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

      2. associate-/l* 56.0%

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

      3. exp-to-pow 56.0%

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

   5. Simplified 56.0%

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

   if 435 < x

   1. Initial program 68.4%

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

   3. Taylor expanded in n around inf 54.3%

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

   4. Taylor expanded in x around inf 68.9%

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

      1. +-commutative 68.9%

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

      2. mul-1-neg 68.9%

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

      3. distribute-frac-neg 68.9%

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

      4. log-rec 68.9%

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

      5. remove-double-neg 68.9%

         \[\leadsto \frac{\frac{1}{n} + \frac{\color{blue}{\log x}}{{n}^{2}}}{x} \]

   6. Simplified 68.9%

      \[\leadsto \color{blue}{\frac{\frac{1}{n} + \frac{\log x}{{n}^{2}}}{x}} \]

   7. Taylor expanded in n around inf 67.9%

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

      1. associate-/r* 68.4%

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

   9. Simplified 68.4%

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

4. Final simplification 60.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 10^{-189}:\\ \;\;\;\;\frac{\log x}{-n}\\ \mathbf{elif}\;x \leq 2.4 \cdot 10^{-171}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;x \leq 2.3 \cdot 10^{-60}:\\ \;\;\;\;\frac{\log x}{-n}\\ \mathbf{elif}\;x \leq 435:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{n}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 68.8% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\log x}{-n}\\ \mathbf{if}\;x \leq 2.6 \cdot 10^{-190}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 4.1 \cdot 10^{-171}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;x \leq 2.3 \cdot 10^{-60}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n} + -1\right)}}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (/ (log x) (- n))))
   (if (<= x 2.6e-190)
     t_0
     (if (<= x 4.1e-171)
       (- 1.0 (pow x (/ 1.0 n)))
       (if (<= x 2.3e-60) t_0 (/ (pow x (+ (/ 1.0 n) -1.0)) n))))))

C

double code(double x, double n) {
	double t_0 = log(x) / -n;
	double tmp;
	if (x <= 2.6e-190) {
		tmp = t_0;
	} else if (x <= 4.1e-171) {
		tmp = 1.0 - pow(x, (1.0 / n));
	} else if (x <= 2.3e-60) {
		tmp = t_0;
	} else {
		tmp = pow(x, ((1.0 / n) + -1.0)) / n;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: tmp
    t_0 = log(x) / -n
    if (x <= 2.6d-190) then
        tmp = t_0
    else if (x <= 4.1d-171) then
        tmp = 1.0d0 - (x ** (1.0d0 / n))
    else if (x <= 2.3d-60) then
        tmp = t_0
    else
        tmp = (x ** ((1.0d0 / n) + (-1.0d0))) / n
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = Math.log(x) / -n;
	double tmp;
	if (x <= 2.6e-190) {
		tmp = t_0;
	} else if (x <= 4.1e-171) {
		tmp = 1.0 - Math.pow(x, (1.0 / n));
	} else if (x <= 2.3e-60) {
		tmp = t_0;
	} else {
		tmp = Math.pow(x, ((1.0 / n) + -1.0)) / n;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.log(x) / -n
	tmp = 0
	if x <= 2.6e-190:
		tmp = t_0
	elif x <= 4.1e-171:
		tmp = 1.0 - math.pow(x, (1.0 / n))
	elif x <= 2.3e-60:
		tmp = t_0
	else:
		tmp = math.pow(x, ((1.0 / n) + -1.0)) / n
	return tmp

Julia

function code(x, n)
	t_0 = Float64(log(x) / Float64(-n))
	tmp = 0.0
	if (x <= 2.6e-190)
		tmp = t_0;
	elseif (x <= 4.1e-171)
		tmp = Float64(1.0 - (x ^ Float64(1.0 / n)));
	elseif (x <= 2.3e-60)
		tmp = t_0;
	else
		tmp = Float64((x ^ Float64(Float64(1.0 / n) + -1.0)) / n);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = log(x) / -n;
	tmp = 0.0;
	if (x <= 2.6e-190)
		tmp = t_0;
	elseif (x <= 4.1e-171)
		tmp = 1.0 - (x ^ (1.0 / n));
	elseif (x <= 2.3e-60)
		tmp = t_0;
	else
		tmp = (x ^ ((1.0 / n) + -1.0)) / n;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(N[Log[x], $MachinePrecision] / (-n)), $MachinePrecision]}, If[LessEqual[x, 2.6e-190], t$95$0, If[LessEqual[x, 4.1e-171], N[(1.0 - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 2.3e-60], t$95$0, N[(N[Power[x, N[(N[(1.0 / n), $MachinePrecision] + -1.0), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if x < 2.5999999999999998e-190 or 4.1e-171 < x < 2.3000000000000001e-60

   1. Initial program 39.1%

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

   3. Taylor expanded in x around 0 39.1%

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

      1. *-rgt-identity 39.1%

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

      2. associate-/l* 39.1%

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

      3. exp-to-pow 39.1%

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

   5. Simplified 39.1%

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

   6. Taylor expanded in n around inf 56.6%

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

      1. associate-*r/ 56.6%

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

      2. mul-1-neg 56.6%

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

   8. Simplified 56.6%

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

   if 2.5999999999999998e-190 < x < 4.1e-171

   1. Initial program 83.9%

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

   3. Taylor expanded in x around 0 83.9%

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

      1. *-rgt-identity 83.9%

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

      2. associate-/l* 83.9%

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

      3. exp-to-pow 83.9%

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

   5. Simplified 83.9%

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

   if 2.3000000000000001e-60 < x

   1. Initial program 65.7%

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

   3. Taylor expanded in x around inf 84.3%

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

      1. log-rec 84.3%

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

      2. mul-1-neg 84.3%

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

      3. associate-*r/ 84.3%

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

      4. associate-*r* 84.3%

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

      5. metadata-eval 84.3%

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

      6. *-commutative 84.3%

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

      7. associate-/l* 84.3%

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

      8. exp-to-pow 84.3%

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

      9. *-commutative 84.3%

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

   5. Simplified 84.3%

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

      1. *-un-lft-identity 84.3%

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

      2. associate-/r* 84.7%

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

      3. pow1 84.7%

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

      4. pow-div 84.4%

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

   7. Applied egg-rr 84.4%

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

      1. *-lft-identity 84.4%

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

      2. sub-neg 84.4%

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

      3. metadata-eval 84.4%

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

   9. Simplified 84.4%

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

4. Final simplification 71.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.6 \cdot 10^{-190}:\\ \;\;\;\;\frac{\log x}{-n}\\ \mathbf{elif}\;x \leq 4.1 \cdot 10^{-171}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;x \leq 2.3 \cdot 10^{-60}:\\ \;\;\;\;\frac{\log x}{-n}\\ \mathbf{else}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n} + -1\right)}}{n}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 69.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\log x}{-n}\\ \mathbf{if}\;x \leq 6.4 \cdot 10^{-190}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 4.3 \cdot 10^{-172}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;x \leq 2.3 \cdot 10^{-60}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n} + -1\right)}}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (/ (log x) (- n))))
   (if (<= x 6.4e-190)
     t_0
     (if (<= x 4.3e-172)
       (- (+ 1.0 (/ x n)) (pow x (/ 1.0 n)))
       (if (<= x 2.3e-60) t_0 (/ (pow x (+ (/ 1.0 n) -1.0)) n))))))

C

double code(double x, double n) {
	double t_0 = log(x) / -n;
	double tmp;
	if (x <= 6.4e-190) {
		tmp = t_0;
	} else if (x <= 4.3e-172) {
		tmp = (1.0 + (x / n)) - pow(x, (1.0 / n));
	} else if (x <= 2.3e-60) {
		tmp = t_0;
	} else {
		tmp = pow(x, ((1.0 / n) + -1.0)) / n;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: tmp
    t_0 = log(x) / -n
    if (x <= 6.4d-190) then
        tmp = t_0
    else if (x <= 4.3d-172) then
        tmp = (1.0d0 + (x / n)) - (x ** (1.0d0 / n))
    else if (x <= 2.3d-60) then
        tmp = t_0
    else
        tmp = (x ** ((1.0d0 / n) + (-1.0d0))) / n
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = Math.log(x) / -n;
	double tmp;
	if (x <= 6.4e-190) {
		tmp = t_0;
	} else if (x <= 4.3e-172) {
		tmp = (1.0 + (x / n)) - Math.pow(x, (1.0 / n));
	} else if (x <= 2.3e-60) {
		tmp = t_0;
	} else {
		tmp = Math.pow(x, ((1.0 / n) + -1.0)) / n;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.log(x) / -n
	tmp = 0
	if x <= 6.4e-190:
		tmp = t_0
	elif x <= 4.3e-172:
		tmp = (1.0 + (x / n)) - math.pow(x, (1.0 / n))
	elif x <= 2.3e-60:
		tmp = t_0
	else:
		tmp = math.pow(x, ((1.0 / n) + -1.0)) / n
	return tmp

Julia

function code(x, n)
	t_0 = Float64(log(x) / Float64(-n))
	tmp = 0.0
	if (x <= 6.4e-190)
		tmp = t_0;
	elseif (x <= 4.3e-172)
		tmp = Float64(Float64(1.0 + Float64(x / n)) - (x ^ Float64(1.0 / n)));
	elseif (x <= 2.3e-60)
		tmp = t_0;
	else
		tmp = Float64((x ^ Float64(Float64(1.0 / n) + -1.0)) / n);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = log(x) / -n;
	tmp = 0.0;
	if (x <= 6.4e-190)
		tmp = t_0;
	elseif (x <= 4.3e-172)
		tmp = (1.0 + (x / n)) - (x ^ (1.0 / n));
	elseif (x <= 2.3e-60)
		tmp = t_0;
	else
		tmp = (x ^ ((1.0 / n) + -1.0)) / n;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(N[Log[x], $MachinePrecision] / (-n)), $MachinePrecision]}, If[LessEqual[x, 6.4e-190], t$95$0, If[LessEqual[x, 4.3e-172], N[(N[(1.0 + N[(x / n), $MachinePrecision]), $MachinePrecision] - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 2.3e-60], t$95$0, N[(N[Power[x, N[(N[(1.0 / n), $MachinePrecision] + -1.0), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if x < 6.4000000000000001e-190 or 4.2999999999999997e-172 < x < 2.3000000000000001e-60

   1. Initial program 39.1%

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

   3. Taylor expanded in x around 0 39.1%

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

      1. *-rgt-identity 39.1%

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

      2. associate-/l* 39.1%

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

      3. exp-to-pow 39.1%

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

   5. Simplified 39.1%

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

   6. Taylor expanded in n around inf 56.6%

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

      1. associate-*r/ 56.6%

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

      2. mul-1-neg 56.6%

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

   8. Simplified 56.6%

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

   if 6.4000000000000001e-190 < x < 4.2999999999999997e-172

   1. Initial program 83.9%

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

   3. Taylor expanded in x around 0 84.0%

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

   if 2.3000000000000001e-60 < x

   1. Initial program 65.7%

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

   3. Taylor expanded in x around inf 84.3%

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

      1. log-rec 84.3%

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

      2. mul-1-neg 84.3%

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

      3. associate-*r/ 84.3%

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

      4. associate-*r* 84.3%

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

      5. metadata-eval 84.3%

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

      6. *-commutative 84.3%

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

      7. associate-/l* 84.3%

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

      8. exp-to-pow 84.3%

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

      9. *-commutative 84.3%

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

   5. Simplified 84.3%

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

      1. *-un-lft-identity 84.3%

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

      2. associate-/r* 84.7%

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

      3. pow1 84.7%

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

      4. pow-div 84.4%

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

   7. Applied egg-rr 84.4%

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

      1. *-lft-identity 84.4%

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

      2. sub-neg 84.4%

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

      3. metadata-eval 84.4%

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

   9. Simplified 84.4%

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

4. Final simplification 71.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 6.4 \cdot 10^{-190}:\\ \;\;\;\;\frac{\log x}{-n}\\ \mathbf{elif}\;x \leq 4.3 \cdot 10^{-172}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;x \leq 2.3 \cdot 10^{-60}:\\ \;\;\;\;\frac{\log x}{-n}\\ \mathbf{else}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n} + -1\right)}}{n}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 56.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\log x}{-n}\\ \mathbf{if}\;x \leq 4.7 \cdot 10^{-237}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 5.4 \cdot 10^{-229}:\\ \;\;\;\;\frac{1}{x \cdot n}\\ \mathbf{elif}\;x \leq 7.8 \cdot 10^{-7}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{n}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (/ (log x) (- n))))
   (if (<= x 4.7e-237)
     t_0
     (if (<= x 5.4e-229)
       (/ 1.0 (* x n))
       (if (<= x 7.8e-7) t_0 (/ (/ 1.0 n) x))))))

C

double code(double x, double n) {
	double t_0 = log(x) / -n;
	double tmp;
	if (x <= 4.7e-237) {
		tmp = t_0;
	} else if (x <= 5.4e-229) {
		tmp = 1.0 / (x * n);
	} else if (x <= 7.8e-7) {
		tmp = t_0;
	} else {
		tmp = (1.0 / n) / x;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: tmp
    t_0 = log(x) / -n
    if (x <= 4.7d-237) then
        tmp = t_0
    else if (x <= 5.4d-229) then
        tmp = 1.0d0 / (x * n)
    else if (x <= 7.8d-7) then
        tmp = t_0
    else
        tmp = (1.0d0 / n) / x
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = Math.log(x) / -n;
	double tmp;
	if (x <= 4.7e-237) {
		tmp = t_0;
	} else if (x <= 5.4e-229) {
		tmp = 1.0 / (x * n);
	} else if (x <= 7.8e-7) {
		tmp = t_0;
	} else {
		tmp = (1.0 / n) / x;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.log(x) / -n
	tmp = 0
	if x <= 4.7e-237:
		tmp = t_0
	elif x <= 5.4e-229:
		tmp = 1.0 / (x * n)
	elif x <= 7.8e-7:
		tmp = t_0
	else:
		tmp = (1.0 / n) / x
	return tmp

Julia

function code(x, n)
	t_0 = Float64(log(x) / Float64(-n))
	tmp = 0.0
	if (x <= 4.7e-237)
		tmp = t_0;
	elseif (x <= 5.4e-229)
		tmp = Float64(1.0 / Float64(x * n));
	elseif (x <= 7.8e-7)
		tmp = t_0;
	else
		tmp = Float64(Float64(1.0 / n) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = log(x) / -n;
	tmp = 0.0;
	if (x <= 4.7e-237)
		tmp = t_0;
	elseif (x <= 5.4e-229)
		tmp = 1.0 / (x * n);
	elseif (x <= 7.8e-7)
		tmp = t_0;
	else
		tmp = (1.0 / n) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(N[Log[x], $MachinePrecision] / (-n)), $MachinePrecision]}, If[LessEqual[x, 4.7e-237], t$95$0, If[LessEqual[x, 5.4e-229], N[(1.0 / N[(x * n), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 7.8e-7], t$95$0, N[(N[(1.0 / n), $MachinePrecision] / x), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if x < 4.6999999999999998e-237 or 5.3999999999999997e-229 < x < 7.80000000000000049e-7

   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 x around 0 41.7%

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

      1. *-rgt-identity 41.7%

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

      2. associate-/l* 41.7%

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

      3. exp-to-pow 41.7%

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

   5. Simplified 41.7%

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

   6. Taylor expanded in n around inf 53.5%

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

      1. associate-*r/ 53.5%

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

      2. mul-1-neg 53.5%

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

   8. Simplified 53.5%

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

   if 4.6999999999999998e-237 < x < 5.3999999999999997e-229

   1. Initial program 76.1%

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

   3. Taylor expanded in n around inf 1.3%

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

   4. Taylor expanded in x around inf 58.7%

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

      1. +-commutative 58.7%

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

      2. mul-1-neg 58.7%

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

      3. distribute-frac-neg 58.7%

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

      4. log-rec 58.7%

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

      5. remove-double-neg 58.7%

         \[\leadsto \frac{\frac{1}{n} + \frac{\color{blue}{\log x}}{{n}^{2}}}{x} \]

   6. Simplified 58.7%

      \[\leadsto \color{blue}{\frac{\frac{1}{n} + \frac{\log x}{{n}^{2}}}{x}} \]

   7. Taylor expanded in n around inf 72.9%

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

      1. *-commutative 72.9%

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

   9. Simplified 72.9%

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

   if 7.80000000000000049e-7 < x

   1. Initial program 69.4%

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

   3. Taylor expanded in n around inf 52.7%

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

   4. Taylor expanded in x around inf 67.8%

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

      1. +-commutative 67.8%

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

      2. mul-1-neg 67.8%

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

      3. distribute-frac-neg 67.8%

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

      4. log-rec 67.8%

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

      5. remove-double-neg 67.8%

         \[\leadsto \frac{\frac{1}{n} + \frac{\color{blue}{\log x}}{{n}^{2}}}{x} \]

   6. Simplified 67.8%

      \[\leadsto \color{blue}{\frac{\frac{1}{n} + \frac{\log x}{{n}^{2}}}{x}} \]

   7. Taylor expanded in n around inf 65.9%

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

      1. associate-/r* 66.4%

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

   9. Simplified 66.4%

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

4. Final simplification 58.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 4.7 \cdot 10^{-237}:\\ \;\;\;\;\frac{\log x}{-n}\\ \mathbf{elif}\;x \leq 5.4 \cdot 10^{-229}:\\ \;\;\;\;\frac{1}{x \cdot n}\\ \mathbf{elif}\;x \leq 7.8 \cdot 10^{-7}:\\ \;\;\;\;\frac{\log x}{-n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{n}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 40.1% accurate, 42.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 53.8%

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

3. Taylor expanded in n around inf 51.7%

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

4. Taylor expanded in x around inf 41.0%

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

   1. +-commutative 41.0%

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

   2. mul-1-neg 41.0%

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

   3. distribute-frac-neg 41.0%

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

   4. log-rec 41.0%

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

   5. remove-double-neg 41.0%

      \[\leadsto \frac{\frac{1}{n} + \frac{\color{blue}{\log x}}{{n}^{2}}}{x} \]

6. Simplified 41.0%

   \[\leadsto \color{blue}{\frac{\frac{1}{n} + \frac{\log x}{{n}^{2}}}{x}} \]

7. Taylor expanded in n around inf 39.6%

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

   1. *-commutative 39.6%

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

9. Simplified 39.6%

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

10. Final simplification 39.6%

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

Alternative 11: 40.5% 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 53.8%

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

3. Taylor expanded in n around inf 51.7%

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

4. Taylor expanded in x around inf 41.0%

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

   1. +-commutative 41.0%

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

   2. mul-1-neg 41.0%

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

   3. distribute-frac-neg 41.0%

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

   4. log-rec 41.0%

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

   5. remove-double-neg 41.0%

      \[\leadsto \frac{\frac{1}{n} + \frac{\color{blue}{\log x}}{{n}^{2}}}{x} \]

6. Simplified 41.0%

   \[\leadsto \color{blue}{\frac{\frac{1}{n} + \frac{\log x}{{n}^{2}}}{x}} \]

7. Taylor expanded in n around inf 39.6%

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

   1. associate-/r* 39.8%

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

9. Simplified 39.8%

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

10. Final simplification 39.8%

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

Alternative 12: 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 53.8%

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

3. Taylor expanded in n around inf 51.7%

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

4. Taylor expanded in x around inf 41.0%

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

   1. +-commutative 41.0%

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

   2. mul-1-neg 41.0%

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

   3. distribute-frac-neg 41.0%

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

   4. log-rec 41.0%

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

   5. remove-double-neg 41.0%

      \[\leadsto \frac{\frac{1}{n} + \frac{\color{blue}{\log x}}{{n}^{2}}}{x} \]

6. Simplified 41.0%

   \[\leadsto \color{blue}{\frac{\frac{1}{n} + \frac{\log x}{{n}^{2}}}{x}} \]

7. Taylor expanded in n around inf 39.6%

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

   1. *-commutative 39.6%

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

9. Simplified 39.6%

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

    1. inv-pow 39.6%

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

    2. unpow-prod-down 39.8%

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

    3. pow-to-exp 39.1%

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

    4. *-commutative 39.1%

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

    5. neg-mul-1 39.1%

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

    6. add-sqr-sqrt 15.0%

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

    7. sqrt-unprod 16.3%

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

    8. sqr-neg 16.3%

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

    9. sqrt-unprod 1.3%

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

    10. add-sqr-sqrt 4.6%

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

    11. add-exp-log 4.6%

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

    12. inv-pow 4.6%

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

11. Applied egg-rr 4.6%

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

    1. associate-*r/ 4.6%

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

    2. *-rgt-identity 4.6%

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

13. Simplified 4.6%

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

14. Final simplification 4.6%

    \[\leadsto \frac{x}{n} \]
15. Add Preprocessing

Reproduce

herbie shell --seed 2024039 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  :precision binary64
  (- (pow (+ x 1.0) (/ 1.0 n)) (pow x (/ 1.0 n))))