2nthrt (problem 3.4.6)

Percentage Accurate: 53.2% → 85.0%

Time: 46.7s

Alternatives: 10

Speedup: 1.7×

• could not determine a ground truth

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 53.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 85.0% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ t_1 := {\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - t\_0\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;1 - t\_0\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;\frac{\log \left(\frac{x + 1}{x}\right)}{n} + \frac{0.5}{n} \cdot \left(\frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2}}{n} - \frac{{\log x}^{2}}{n}\right)\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))) (t_1 (- (pow (+ x 1.0) (/ 1.0 n)) t_0)))
   (if (<= t_1 (- INFINITY))
     (- 1.0 t_0)
     (if (<= t_1 0.0)
       (+
        (/ (log (/ (+ x 1.0) x)) n)
        (* (/ 0.5 n) (- (/ (pow (log1p x) 2.0) n) (/ (pow (log x) 2.0) n))))
       (- (exp (/ (log1p x) n)) t_0)))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double t_1 = pow((x + 1.0), (1.0 / n)) - t_0;
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = 1.0 - t_0;
	} else if (t_1 <= 0.0) {
		tmp = (log(((x + 1.0) / x)) / n) + ((0.5 / n) * ((pow(log1p(x), 2.0) / n) - (pow(log(x), 2.0) / n)));
	} else {
		tmp = exp((log1p(x) / n)) - t_0;
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, (1.0 / n));
	double t_1 = Math.pow((x + 1.0), (1.0 / n)) - t_0;
	double tmp;
	if (t_1 <= -Double.POSITIVE_INFINITY) {
		tmp = 1.0 - t_0;
	} else if (t_1 <= 0.0) {
		tmp = (Math.log(((x + 1.0) / x)) / n) + ((0.5 / n) * ((Math.pow(Math.log1p(x), 2.0) / n) - (Math.pow(Math.log(x), 2.0) / n)));
	} else {
		tmp = Math.exp((Math.log1p(x) / n)) - t_0;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	t_1 = math.pow((x + 1.0), (1.0 / n)) - t_0
	tmp = 0
	if t_1 <= -math.inf:
		tmp = 1.0 - t_0
	elif t_1 <= 0.0:
		tmp = (math.log(((x + 1.0) / x)) / n) + ((0.5 / n) * ((math.pow(math.log1p(x), 2.0) / n) - (math.pow(math.log(x), 2.0) / n)))
	else:
		tmp = math.exp((math.log1p(x) / n)) - t_0
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	t_1 = Float64((Float64(x + 1.0) ^ Float64(1.0 / n)) - t_0)
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(1.0 - t_0);
	elseif (t_1 <= 0.0)
		tmp = Float64(Float64(log(Float64(Float64(x + 1.0) / x)) / n) + Float64(Float64(0.5 / n) * Float64(Float64((log1p(x) ^ 2.0) / n) - Float64((log(x) ^ 2.0) / n))));
	else
		tmp = Float64(exp(Float64(log1p(x) / n)) - t_0);
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[Power[N[(x + 1.0), $MachinePrecision], N[(1.0 / n), $MachinePrecision]], $MachinePrecision] - t$95$0), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(1.0 - t$95$0), $MachinePrecision], If[LessEqual[t$95$1, 0.0], N[(N[(N[Log[N[(N[(x + 1.0), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision] + N[(N[(0.5 / n), $MachinePrecision] * N[(N[(N[Power[N[Log[1 + x], $MachinePrecision], 2.0], $MachinePrecision] / n), $MachinePrecision] - N[(N[Power[N[Log[x], $MachinePrecision], 2.0], $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Exp[N[(N[Log[1 + x], $MachinePrecision] / n), $MachinePrecision]], $MachinePrecision] - t$95$0), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (pow.f64 (+.f64 x 1) (/.f64 1 n)) (pow.f64 x (/.f64 1 n))) < -inf.0

   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 x around 0 100.0%

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

      1. *-rgt-identity 100.0%

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

      2. associate-*l/ 100.0%

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

      3. associate-/l* 100.0%

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

      4. exp-to-pow 100.0%

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

   5. Simplified 100.0%

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

   if -inf.0 < (-.f64 (pow.f64 (+.f64 x 1) (/.f64 1 n)) (pow.f64 x (/.f64 1 n))) < 0.0

   1. Initial program 40.3%

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

   3. Taylor expanded in n around inf 71.8%

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

   5. Simplified 79.8%

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

      1. log1p-undefine 79.8%

         \[\leadsto \frac{\color{blue}{\log \left(1 + x\right)} - \log x}{n} + \frac{0.5}{n} \cdot \left(\frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2}}{n} - \frac{{\log x}^{2}}{n}\right) \]

      2. diff-log 79.9%

         \[\leadsto \frac{\color{blue}{\log \left(\frac{1 + x}{x}\right)}}{n} + \frac{0.5}{n} \cdot \left(\frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2}}{n} - \frac{{\log x}^{2}}{n}\right) \]

   7. Applied egg-rr 79.9%

      \[\leadsto \frac{\color{blue}{\log \left(\frac{1 + x}{x}\right)}}{n} + \frac{0.5}{n} \cdot \left(\frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2}}{n} - \frac{{\log x}^{2}}{n}\right) \]
   8. Step-by-step derivation

      1. +-commutative 79.9%

         \[\leadsto \frac{\log \left(\frac{\color{blue}{x + 1}}{x}\right)}{n} + \frac{0.5}{n} \cdot \left(\frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2}}{n} - \frac{{\log x}^{2}}{n}\right) \]

   9. Simplified 79.9%

      \[\leadsto \frac{\color{blue}{\log \left(\frac{x + 1}{x}\right)}}{n} + \frac{0.5}{n} \cdot \left(\frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2}}{n} - \frac{{\log x}^{2}}{n}\right) \]

   if 0.0 < (-.f64 (pow.f64 (+.f64 x 1) (/.f64 1 n)) (pow.f64 x (/.f64 1 n)))

   1. Initial program 51.3%

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

   3. Taylor expanded in n around 0 51.3%

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

      1. log1p-define 97.4%

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

      2. *-rgt-identity 97.4%

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

      3. associate-*l/ 97.4%

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

      4. associate-/l* 97.4%

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

      5. exp-to-pow 97.4%

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

   5. Simplified 97.4%

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

4. Final simplification 84.8%

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

Alternative 2: 85.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ t_1 := {\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - t\_0\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;1 - t\_0\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right) - \log x}{n}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))) (t_1 (- (pow (+ x 1.0) (/ 1.0 n)) t_0)))
   (if (<= t_1 (- INFINITY))
     (- 1.0 t_0)
     (if (<= t_1 0.0)
       (/ (- (log1p x) (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 t_1 = pow((x + 1.0), (1.0 / n)) - t_0;
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = 1.0 - t_0;
	} else if (t_1 <= 0.0) {
		tmp = (log1p(x) - 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 t_1 = Math.pow((x + 1.0), (1.0 / n)) - t_0;
	double tmp;
	if (t_1 <= -Double.POSITIVE_INFINITY) {
		tmp = 1.0 - t_0;
	} else if (t_1 <= 0.0) {
		tmp = (Math.log1p(x) - 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))
	t_1 = math.pow((x + 1.0), (1.0 / n)) - t_0
	tmp = 0
	if t_1 <= -math.inf:
		tmp = 1.0 - t_0
	elif t_1 <= 0.0:
		tmp = (math.log1p(x) - 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)
	t_1 = Float64((Float64(x + 1.0) ^ Float64(1.0 / n)) - t_0)
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(1.0 - t_0);
	elseif (t_1 <= 0.0)
		tmp = Float64(Float64(log1p(x) - 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]}, 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, (-Infinity)], N[(1.0 - t$95$0), $MachinePrecision], If[LessEqual[t$95$1, 0.0], N[(N[(N[Log[1 + x], $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 (pow.f64 (+.f64 x 1) (/.f64 1 n)) (pow.f64 x (/.f64 1 n))) < -inf.0

   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 x around 0 100.0%

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

      1. *-rgt-identity 100.0%

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

      2. associate-*l/ 100.0%

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

      3. associate-/l* 100.0%

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

      4. exp-to-pow 100.0%

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

   5. Simplified 100.0%

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

   if -inf.0 < (-.f64 (pow.f64 (+.f64 x 1) (/.f64 1 n)) (pow.f64 x (/.f64 1 n))) < 0.0

   1. Initial program 40.3%

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

   3. Taylor expanded in n around inf 79.4%

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

      1. log1p-define 79.4%

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

   5. Simplified 79.4%

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

   if 0.0 < (-.f64 (pow.f64 (+.f64 x 1) (/.f64 1 n)) (pow.f64 x (/.f64 1 n)))

   1. Initial program 51.3%

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

   3. Taylor expanded in n around 0 51.3%

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

      1. log1p-define 97.4%

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

      2. *-rgt-identity 97.4%

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

      3. associate-*l/ 97.4%

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

      4. associate-/l* 97.4%

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

      5. exp-to-pow 97.4%

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

   5. Simplified 97.4%

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

4. Final simplification 84.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \leq -\infty:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \leq 0:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right) - \log x}{n}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 78.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ t_1 := {\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - t\_0\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;1 - t\_0\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right) - \log x}{n}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

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 (- INFINITY))
     (- 1.0 t_0)
     (if (<= t_1 0.0) (/ (- (log1p x) (log x)) n) t_1))))

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 <= -((double) INFINITY)) {
		tmp = 1.0 - t_0;
	} else if (t_1 <= 0.0) {
		tmp = (log1p(x) - log(x)) / n;
	} else {
		tmp = t_1;
	}
	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 <= -Double.POSITIVE_INFINITY) {
		tmp = 1.0 - t_0;
	} else if (t_1 <= 0.0) {
		tmp = (Math.log1p(x) - Math.log(x)) / n;
	} else {
		tmp = t_1;
	}
	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 <= -math.inf:
		tmp = 1.0 - t_0
	elif t_1 <= 0.0:
		tmp = (math.log1p(x) - math.log(x)) / n
	else:
		tmp = t_1
	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 <= Float64(-Inf))
		tmp = Float64(1.0 - t_0);
	elseif (t_1 <= 0.0)
		tmp = Float64(Float64(log1p(x) - log(x)) / n);
	else
		tmp = t_1;
	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, (-Infinity)], N[(1.0 - t$95$0), $MachinePrecision], If[LessEqual[t$95$1, 0.0], N[(N[(N[Log[1 + x], $MachinePrecision] - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (pow.f64 (+.f64 x 1) (/.f64 1 n)) (pow.f64 x (/.f64 1 n))) < -inf.0

   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 x around 0 100.0%

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

      1. *-rgt-identity 100.0%

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

      2. associate-*l/ 100.0%

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

      3. associate-/l* 100.0%

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

      4. exp-to-pow 100.0%

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

   5. Simplified 100.0%

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

   if -inf.0 < (-.f64 (pow.f64 (+.f64 x 1) (/.f64 1 n)) (pow.f64 x (/.f64 1 n))) < 0.0

   1. Initial program 40.3%

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

   3. Taylor expanded in n around inf 79.4%

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

      1. log1p-define 79.4%

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

   5. Simplified 79.4%

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

   if 0.0 < (-.f64 (pow.f64 (+.f64 x 1) (/.f64 1 n)) (pow.f64 x (/.f64 1 n)))

   1. Initial program 51.3%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   2. Add Preprocessing
3. Recombined 3 regimes into one program.

4. Final simplification 77.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \leq -\infty:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \leq 0:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right) - \log x}{n}\\ \mathbf{else}:\\ \;\;\;\;{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 80.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ t_1 := \frac{\mathsf{log1p}\left(x\right) - \log x}{n}\\ \mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{-10}:\\ \;\;\;\;\frac{t\_0}{x \cdot n}\\ \mathbf{elif}\;\frac{1}{n} \leq -5 \cdot 10^{-59}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\frac{1}{n} \leq -1 \cdot 10^{-119}:\\ \;\;\;\;\frac{1}{x \cdot n} + \frac{\frac{0.3333333333333333}{x} + -0.5}{n \cdot {x}^{2}}\\ \mathbf{elif}\;\frac{1}{n} \leq 0.001:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\frac{1}{n} \leq 1.03 \cdot 10^{+214}:\\ \;\;\;\;\left(1 - t\_0\right) + \frac{x}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x} + \frac{0.3333333333333333}{{x}^{3}}}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))) (t_1 (/ (- (log1p x) (log x)) n)))
   (if (<= (/ 1.0 n) -1e-10)
     (/ t_0 (* x n))
     (if (<= (/ 1.0 n) -5e-59)
       t_1
       (if (<= (/ 1.0 n) -1e-119)
         (+
          (/ 1.0 (* x n))
          (/ (+ (/ 0.3333333333333333 x) -0.5) (* n (pow x 2.0))))
         (if (<= (/ 1.0 n) 0.001)
           t_1
           (if (<= (/ 1.0 n) 1.03e+214)
             (+ (- 1.0 t_0) (/ x n))
             (/ (+ (/ 1.0 x) (/ 0.3333333333333333 (pow x 3.0))) n))))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double t_1 = (log1p(x) - log(x)) / n;
	double tmp;
	if ((1.0 / n) <= -1e-10) {
		tmp = t_0 / (x * n);
	} else if ((1.0 / n) <= -5e-59) {
		tmp = t_1;
	} else if ((1.0 / n) <= -1e-119) {
		tmp = (1.0 / (x * n)) + (((0.3333333333333333 / x) + -0.5) / (n * pow(x, 2.0)));
	} else if ((1.0 / n) <= 0.001) {
		tmp = t_1;
	} else if ((1.0 / n) <= 1.03e+214) {
		tmp = (1.0 - t_0) + (x / n);
	} else {
		tmp = ((1.0 / x) + (0.3333333333333333 / pow(x, 3.0))) / n;
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, (1.0 / n));
	double t_1 = (Math.log1p(x) - Math.log(x)) / n;
	double tmp;
	if ((1.0 / n) <= -1e-10) {
		tmp = t_0 / (x * n);
	} else if ((1.0 / n) <= -5e-59) {
		tmp = t_1;
	} else if ((1.0 / n) <= -1e-119) {
		tmp = (1.0 / (x * n)) + (((0.3333333333333333 / x) + -0.5) / (n * Math.pow(x, 2.0)));
	} else if ((1.0 / n) <= 0.001) {
		tmp = t_1;
	} else if ((1.0 / n) <= 1.03e+214) {
		tmp = (1.0 - t_0) + (x / n);
	} else {
		tmp = ((1.0 / x) + (0.3333333333333333 / Math.pow(x, 3.0))) / n;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	t_1 = (math.log1p(x) - math.log(x)) / n
	tmp = 0
	if (1.0 / n) <= -1e-10:
		tmp = t_0 / (x * n)
	elif (1.0 / n) <= -5e-59:
		tmp = t_1
	elif (1.0 / n) <= -1e-119:
		tmp = (1.0 / (x * n)) + (((0.3333333333333333 / x) + -0.5) / (n * math.pow(x, 2.0)))
	elif (1.0 / n) <= 0.001:
		tmp = t_1
	elif (1.0 / n) <= 1.03e+214:
		tmp = (1.0 - t_0) + (x / n)
	else:
		tmp = ((1.0 / x) + (0.3333333333333333 / math.pow(x, 3.0))) / n
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	t_1 = Float64(Float64(log1p(x) - log(x)) / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -1e-10)
		tmp = Float64(t_0 / Float64(x * n));
	elseif (Float64(1.0 / n) <= -5e-59)
		tmp = t_1;
	elseif (Float64(1.0 / n) <= -1e-119)
		tmp = Float64(Float64(1.0 / Float64(x * n)) + Float64(Float64(Float64(0.3333333333333333 / x) + -0.5) / Float64(n * (x ^ 2.0))));
	elseif (Float64(1.0 / n) <= 0.001)
		tmp = t_1;
	elseif (Float64(1.0 / n) <= 1.03e+214)
		tmp = Float64(Float64(1.0 - t_0) + Float64(x / n));
	else
		tmp = Float64(Float64(Float64(1.0 / x) + Float64(0.3333333333333333 / (x ^ 3.0))) / n);
	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[(N[Log[1 + x], $MachinePrecision] - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -1e-10], N[(t$95$0 / N[(x * n), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], -5e-59], t$95$1, If[LessEqual[N[(1.0 / n), $MachinePrecision], -1e-119], N[(N[(1.0 / N[(x * n), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(0.3333333333333333 / x), $MachinePrecision] + -0.5), $MachinePrecision] / N[(n * N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 0.001], t$95$1, If[LessEqual[N[(1.0 / n), $MachinePrecision], 1.03e+214], N[(N[(1.0 - t$95$0), $MachinePrecision] + N[(x / n), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 / x), $MachinePrecision] + N[(0.3333333333333333 / N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if (/.f64 1 n) < -1.00000000000000004e-10

   1. Initial program 96.8%

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

   3. Taylor expanded in x around inf 100.0%

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

      1. log-rec 100.0%

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

      2. mul-1-neg 100.0%

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

      3. associate-*r/ 100.0%

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

      4. associate-*r* 100.0%

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

      5. metadata-eval 100.0%

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

      6. *-commutative 100.0%

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

      7. associate-/l* 100.0%

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

      8. exp-to-pow 100.0%

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

      9. *-commutative 100.0%

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

   5. Simplified 100.0%

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

   if -1.00000000000000004e-10 < (/.f64 1 n) < -5.0000000000000001e-59 or -1.00000000000000001e-119 < (/.f64 1 n) < 1e-3

   1. Initial program 31.4%

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

   3. Taylor expanded in n around inf 79.9%

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

      1. log1p-define 79.9%

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

   5. Simplified 79.9%

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

   if -5.0000000000000001e-59 < (/.f64 1 n) < -1.00000000000000001e-119

   1. Initial program 12.8%

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

   3. Taylor expanded in x around inf 71.2%

      \[\leadsto \color{blue}{\frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}}{n \cdot x} + \left(\frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}} \cdot \left(0.5 \cdot \frac{1}{{n}^{2}} - 0.5 \cdot \frac{1}{n}\right)}{{x}^{2}} + \frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}} \cdot \left(\left(0.16666666666666666 \cdot \frac{1}{{n}^{3}} + 0.3333333333333333 \cdot \frac{1}{n}\right) - 0.5 \cdot \frac{1}{{n}^{2}}\right)}{{x}^{3}}\right)} \]

   5. Simplified 71.7%

      \[\leadsto \color{blue}{\frac{{x}^{\left(\frac{1}{n}\right)}}{x \cdot n} + \frac{{x}^{\left(\frac{1}{n}\right)}}{{x}^{2}} \cdot \left(\left(\frac{0.5}{{n}^{2}} + \frac{-0.5}{n}\right) + \frac{\frac{0.16666666666666666}{{n}^{3}} + \left(\frac{0.3333333333333333}{n} + \frac{-0.5}{{n}^{2}}\right)}{x}\right)} \]

   6. Taylor expanded in n around inf 71.7%

      \[\leadsto \frac{{x}^{\left(\frac{1}{n}\right)}}{x \cdot n} + \color{blue}{\frac{0.3333333333333333 \cdot \frac{1}{x} - 0.5}{n \cdot {x}^{2}}} \]
   7. Step-by-step derivation

      1. sub-neg 71.7%

         \[\leadsto \frac{{x}^{\left(\frac{1}{n}\right)}}{x \cdot n} + \frac{\color{blue}{0.3333333333333333 \cdot \frac{1}{x} + \left(-0.5\right)}}{n \cdot {x}^{2}} \]

      2. associate-*r/ 71.7%

         \[\leadsto \frac{{x}^{\left(\frac{1}{n}\right)}}{x \cdot n} + \frac{\color{blue}{\frac{0.3333333333333333 \cdot 1}{x}} + \left(-0.5\right)}{n \cdot {x}^{2}} \]

      3. metadata-eval 71.7%

         \[\leadsto \frac{{x}^{\left(\frac{1}{n}\right)}}{x \cdot n} + \frac{\frac{\color{blue}{0.3333333333333333}}{x} + \left(-0.5\right)}{n \cdot {x}^{2}} \]

      4. metadata-eval 71.7%

         \[\leadsto \frac{{x}^{\left(\frac{1}{n}\right)}}{x \cdot n} + \frac{\frac{0.3333333333333333}{x} + \color{blue}{-0.5}}{n \cdot {x}^{2}} \]

      5. *-commutative 71.7%

         \[\leadsto \frac{{x}^{\left(\frac{1}{n}\right)}}{x \cdot n} + \frac{\frac{0.3333333333333333}{x} + -0.5}{\color{blue}{{x}^{2} \cdot n}} \]

   8. Simplified 71.7%

      \[\leadsto \frac{{x}^{\left(\frac{1}{n}\right)}}{x \cdot n} + \color{blue}{\frac{\frac{0.3333333333333333}{x} + -0.5}{{x}^{2} \cdot n}} \]

   9. Taylor expanded in n around inf 71.7%

      \[\leadsto \color{blue}{\frac{1}{n \cdot x}} + \frac{\frac{0.3333333333333333}{x} + -0.5}{{x}^{2} \cdot n} \]
   10. Step-by-step derivation

       1. *-commutative 71.7%

          \[\leadsto \frac{1}{\color{blue}{x \cdot n}} + \frac{\frac{0.3333333333333333}{x} + -0.5}{{x}^{2} \cdot n} \]

   11. Simplified 71.7%

       \[\leadsto \color{blue}{\frac{1}{x \cdot n}} + \frac{\frac{0.3333333333333333}{x} + -0.5}{{x}^{2} \cdot n} \]

   if 1e-3 < (/.f64 1 n) < 1.03e214

   1. Initial program 63.8%

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

   3. Taylor expanded in x around 0 61.9%

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

      1. +-commutative 61.9%

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

      2. *-rgt-identity 61.9%

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

      3. associate-*l/ 61.9%

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

      4. associate-/l* 61.9%

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

      5. exp-to-pow 61.9%

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

      6. associate-+r- 61.9%

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

   5. Simplified 61.9%

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

   if 1.03e214 < (/.f64 1 n)

   1. Initial program 20.7%

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

   3. Taylor expanded in x around inf 0.0%

      \[\leadsto \color{blue}{\frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}}{n \cdot x} + \left(\frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}} \cdot \left(0.5 \cdot \frac{1}{{n}^{2}} - 0.5 \cdot \frac{1}{n}\right)}{{x}^{2}} + \frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}} \cdot \left(\left(0.16666666666666666 \cdot \frac{1}{{n}^{3}} + 0.3333333333333333 \cdot \frac{1}{n}\right) - 0.5 \cdot \frac{1}{{n}^{2}}\right)}{{x}^{3}}\right)} \]

   5. Simplified 0.0%

      \[\leadsto \color{blue}{\frac{{x}^{\left(\frac{1}{n}\right)}}{x \cdot n} + \frac{{x}^{\left(\frac{1}{n}\right)}}{{x}^{2}} \cdot \left(\left(\frac{0.5}{{n}^{2}} + \frac{-0.5}{n}\right) + \frac{\frac{0.16666666666666666}{{n}^{3}} + \left(\frac{0.3333333333333333}{n} + \frac{-0.5}{{n}^{2}}\right)}{x}\right)} \]

   6. Taylor expanded in n around inf 19.1%

      \[\leadsto \frac{{x}^{\left(\frac{1}{n}\right)}}{x \cdot n} + \color{blue}{\frac{0.3333333333333333 \cdot \frac{1}{x} - 0.5}{n \cdot {x}^{2}}} \]
   7. Step-by-step derivation

      1. sub-neg 19.1%

         \[\leadsto \frac{{x}^{\left(\frac{1}{n}\right)}}{x \cdot n} + \frac{\color{blue}{0.3333333333333333 \cdot \frac{1}{x} + \left(-0.5\right)}}{n \cdot {x}^{2}} \]

      2. associate-*r/ 19.1%

         \[\leadsto \frac{{x}^{\left(\frac{1}{n}\right)}}{x \cdot n} + \frac{\color{blue}{\frac{0.3333333333333333 \cdot 1}{x}} + \left(-0.5\right)}{n \cdot {x}^{2}} \]

      3. metadata-eval 19.1%

         \[\leadsto \frac{{x}^{\left(\frac{1}{n}\right)}}{x \cdot n} + \frac{\frac{\color{blue}{0.3333333333333333}}{x} + \left(-0.5\right)}{n \cdot {x}^{2}} \]

      4. metadata-eval 19.1%

         \[\leadsto \frac{{x}^{\left(\frac{1}{n}\right)}}{x \cdot n} + \frac{\frac{0.3333333333333333}{x} + \color{blue}{-0.5}}{n \cdot {x}^{2}} \]

      5. *-commutative 19.1%

         \[\leadsto \frac{{x}^{\left(\frac{1}{n}\right)}}{x \cdot n} + \frac{\frac{0.3333333333333333}{x} + -0.5}{\color{blue}{{x}^{2} \cdot n}} \]

   8. Simplified 19.1%

      \[\leadsto \frac{{x}^{\left(\frac{1}{n}\right)}}{x \cdot n} + \color{blue}{\frac{\frac{0.3333333333333333}{x} + -0.5}{{x}^{2} \cdot n}} \]

   9. Taylor expanded in x around 0 19.1%

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

       1. associate-/r* 19.1%

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

   11. Simplified 19.1%

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

   12. Taylor expanded in n around inf 74.2%

       \[\leadsto \color{blue}{\frac{\frac{1}{x} + 0.3333333333333333 \cdot \frac{1}{{x}^{3}}}{n}} \]
   13. Step-by-step derivation

       1. associate-*r/ 74.2%

          \[\leadsto \frac{\frac{1}{x} + \color{blue}{\frac{0.3333333333333333 \cdot 1}{{x}^{3}}}}{n} \]

       2. metadata-eval 74.2%

          \[\leadsto \frac{\frac{1}{x} + \frac{\color{blue}{0.3333333333333333}}{{x}^{3}}}{n} \]

   14. Simplified 74.2%

       \[\leadsto \color{blue}{\frac{\frac{1}{x} + \frac{0.3333333333333333}{{x}^{3}}}{n}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 82.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{-10}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{x \cdot n}\\ \mathbf{elif}\;\frac{1}{n} \leq -5 \cdot 10^{-59}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right) - \log x}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq -1 \cdot 10^{-119}:\\ \;\;\;\;\frac{1}{x \cdot n} + \frac{\frac{0.3333333333333333}{x} + -0.5}{n \cdot {x}^{2}}\\ \mathbf{elif}\;\frac{1}{n} \leq 0.001:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right) - \log x}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 1.03 \cdot 10^{+214}:\\ \;\;\;\;\left(1 - {x}^{\left(\frac{1}{n}\right)}\right) + \frac{x}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x} + \frac{0.3333333333333333}{{x}^{3}}}{n}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 50.3% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - {x}^{\left(\frac{1}{n}\right)}\\ t_1 := \frac{x - \log x}{n}\\ \mathbf{if}\;n \leq -2.2 \cdot 10^{+195}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n \leq -60000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;n \leq 1400:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n \leq 10^{+280}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;n \leq 8 \cdot 10^{+295}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\log x}{-n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (- 1.0 (pow x (/ 1.0 n)))) (t_1 (/ (- x (log x)) n)))
   (if (<= n -2.2e+195)
     t_0
     (if (<= n -60000.0)
       t_1
       (if (<= n 1400.0)
         t_0
         (if (<= n 1e+280) t_1 (if (<= n 8e+295) t_0 (/ (log x) (- n)))))))))

C

double code(double x, double n) {
	double t_0 = 1.0 - pow(x, (1.0 / n));
	double t_1 = (x - log(x)) / n;
	double tmp;
	if (n <= -2.2e+195) {
		tmp = t_0;
	} else if (n <= -60000.0) {
		tmp = t_1;
	} else if (n <= 1400.0) {
		tmp = t_0;
	} else if (n <= 1e+280) {
		tmp = t_1;
	} else if (n <= 8e+295) {
		tmp = t_0;
	} else {
		tmp = log(x) / -n;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = 1.0d0 - (x ** (1.0d0 / n))
    t_1 = (x - log(x)) / n
    if (n <= (-2.2d+195)) then
        tmp = t_0
    else if (n <= (-60000.0d0)) then
        tmp = t_1
    else if (n <= 1400.0d0) then
        tmp = t_0
    else if (n <= 1d+280) then
        tmp = t_1
    else if (n <= 8d+295) then
        tmp = t_0
    else
        tmp = log(x) / -n
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = 1.0 - Math.pow(x, (1.0 / n));
	double t_1 = (x - Math.log(x)) / n;
	double tmp;
	if (n <= -2.2e+195) {
		tmp = t_0;
	} else if (n <= -60000.0) {
		tmp = t_1;
	} else if (n <= 1400.0) {
		tmp = t_0;
	} else if (n <= 1e+280) {
		tmp = t_1;
	} else if (n <= 8e+295) {
		tmp = t_0;
	} else {
		tmp = Math.log(x) / -n;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = 1.0 - math.pow(x, (1.0 / n))
	t_1 = (x - math.log(x)) / n
	tmp = 0
	if n <= -2.2e+195:
		tmp = t_0
	elif n <= -60000.0:
		tmp = t_1
	elif n <= 1400.0:
		tmp = t_0
	elif n <= 1e+280:
		tmp = t_1
	elif n <= 8e+295:
		tmp = t_0
	else:
		tmp = math.log(x) / -n
	return tmp

Julia

function code(x, n)
	t_0 = Float64(1.0 - (x ^ Float64(1.0 / n)))
	t_1 = Float64(Float64(x - log(x)) / n)
	tmp = 0.0
	if (n <= -2.2e+195)
		tmp = t_0;
	elseif (n <= -60000.0)
		tmp = t_1;
	elseif (n <= 1400.0)
		tmp = t_0;
	elseif (n <= 1e+280)
		tmp = t_1;
	elseif (n <= 8e+295)
		tmp = t_0;
	else
		tmp = Float64(log(x) / Float64(-n));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = 1.0 - (x ^ (1.0 / n));
	t_1 = (x - log(x)) / n;
	tmp = 0.0;
	if (n <= -2.2e+195)
		tmp = t_0;
	elseif (n <= -60000.0)
		tmp = t_1;
	elseif (n <= 1400.0)
		tmp = t_0;
	elseif (n <= 1e+280)
		tmp = t_1;
	elseif (n <= 8e+295)
		tmp = t_0;
	else
		tmp = log(x) / -n;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(1.0 - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision]}, If[LessEqual[n, -2.2e+195], t$95$0, If[LessEqual[n, -60000.0], t$95$1, If[LessEqual[n, 1400.0], t$95$0, If[LessEqual[n, 1e+280], t$95$1, If[LessEqual[n, 8e+295], t$95$0, N[(N[Log[x], $MachinePrecision] / (-n)), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if n < -2.2e195 or -6e4 < n < 1400 or 1e280 < n < 7.9999999999999999e295

   1. Initial program 75.9%

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

   3. Taylor expanded in x around 0 53.6%

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

      1. *-rgt-identity 53.6%

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

      2. associate-*l/ 53.6%

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

      3. associate-/l* 53.6%

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

      4. exp-to-pow 53.6%

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

   5. Simplified 53.6%

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

   if -2.2e195 < n < -6e4 or 1400 < n < 1e280

   1. Initial program 20.1%

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

   3. Taylor expanded in x around 0 7.0%

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

      1. +-commutative 7.0%

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

      2. *-rgt-identity 7.0%

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

      3. associate-*l/ 7.0%

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

      4. associate-/l* 7.0%

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

      5. exp-to-pow 7.0%

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

      6. associate-+r- 6.1%

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

   5. Simplified 6.1%

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

   6. Taylor expanded in n around inf 52.5%

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

   if 7.9999999999999999e295 < n

   1. Initial program 33.4%

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

   3. Taylor expanded in x around 0 33.4%

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

      1. *-rgt-identity 33.4%

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

      2. associate-*l/ 33.4%

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

      3. associate-/l* 33.4%

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

      4. exp-to-pow 33.4%

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

   5. Simplified 33.4%

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

   6. Taylor expanded in n around inf 77.5%

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

      1. associate-*r/ 77.5%

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

      2. neg-mul-1 77.5%

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

   8. Simplified 77.5%

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

4. Final simplification 53.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -2.2 \cdot 10^{+195}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;n \leq -60000:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{elif}\;n \leq 1400:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;n \leq 10^{+280}:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{elif}\;n \leq 8 \cdot 10^{+295}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log x}{-n}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 71.2% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;x \leq 8.8 \cdot 10^{-243}:\\ \;\;\;\;\frac{\log x}{-n}\\ \mathbf{elif}\;x \leq 1.8 \cdot 10^{-202}:\\ \;\;\;\;1 - t\_0\\ \mathbf{elif}\;x \leq 1.55 \cdot 10^{-5}:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0}{x \cdot n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= x 8.8e-243)
     (/ (log x) (- n))
     (if (<= x 1.8e-202)
       (- 1.0 t_0)
       (if (<= x 1.55e-5) (/ (- x (log x)) n) (/ t_0 (* x n)))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double tmp;
	if (x <= 8.8e-243) {
		tmp = log(x) / -n;
	} else if (x <= 1.8e-202) {
		tmp = 1.0 - t_0;
	} else if (x <= 1.55e-5) {
		tmp = (x - log(x)) / n;
	} else {
		tmp = t_0 / (x * n);
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x ** (1.0d0 / n)
    if (x <= 8.8d-243) then
        tmp = log(x) / -n
    else if (x <= 1.8d-202) then
        tmp = 1.0d0 - t_0
    else if (x <= 1.55d-5) then
        tmp = (x - log(x)) / n
    else
        tmp = t_0 / (x * n)
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, (1.0 / n));
	double tmp;
	if (x <= 8.8e-243) {
		tmp = Math.log(x) / -n;
	} else if (x <= 1.8e-202) {
		tmp = 1.0 - t_0;
	} else if (x <= 1.55e-5) {
		tmp = (x - Math.log(x)) / n;
	} else {
		tmp = t_0 / (x * n);
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	tmp = 0
	if x <= 8.8e-243:
		tmp = math.log(x) / -n
	elif x <= 1.8e-202:
		tmp = 1.0 - t_0
	elif x <= 1.55e-5:
		tmp = (x - math.log(x)) / n
	else:
		tmp = t_0 / (x * n)
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (x <= 8.8e-243)
		tmp = Float64(log(x) / Float64(-n));
	elseif (x <= 1.8e-202)
		tmp = Float64(1.0 - t_0);
	elseif (x <= 1.55e-5)
		tmp = Float64(Float64(x - log(x)) / n);
	else
		tmp = Float64(t_0 / Float64(x * n));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = x ^ (1.0 / n);
	tmp = 0.0;
	if (x <= 8.8e-243)
		tmp = log(x) / -n;
	elseif (x <= 1.8e-202)
		tmp = 1.0 - t_0;
	elseif (x <= 1.55e-5)
		tmp = (x - log(x)) / n;
	else
		tmp = t_0 / (x * n);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, 8.8e-243], N[(N[Log[x], $MachinePrecision] / (-n)), $MachinePrecision], If[LessEqual[x, 1.8e-202], N[(1.0 - t$95$0), $MachinePrecision], If[LessEqual[x, 1.55e-5], N[(N[(x - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], N[(t$95$0 / N[(x * n), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if x < 8.7999999999999996e-243

   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 x around 0 35.9%

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

      1. *-rgt-identity 35.9%

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

      2. associate-*l/ 35.9%

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

      3. associate-/l* 35.9%

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

      4. exp-to-pow 35.9%

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

   5. Simplified 35.9%

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

   6. Taylor expanded in n around inf 69.0%

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

      1. associate-*r/ 69.0%

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

      2. neg-mul-1 69.0%

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

   8. Simplified 69.0%

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

   if 8.7999999999999996e-243 < x < 1.8000000000000001e-202

   1. Initial program 61.3%

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

   3. Taylor expanded in x around 0 61.3%

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

      1. *-rgt-identity 61.3%

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

      2. associate-*l/ 61.3%

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

      3. associate-/l* 61.3%

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

      4. exp-to-pow 61.3%

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

   5. Simplified 61.3%

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

   if 1.8000000000000001e-202 < x < 1.55000000000000007e-5

   1. Initial program 32.1%

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

   3. Taylor expanded in x around 0 31.7%

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

      1. +-commutative 31.7%

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

      2. *-rgt-identity 31.7%

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

      3. associate-*l/ 31.7%

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

      4. associate-/l* 31.7%

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

      5. exp-to-pow 31.7%

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

      6. associate-+r- 32.6%

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

   5. Simplified 32.6%

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

   6. Taylor expanded in n around inf 57.3%

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

   if 1.55000000000000007e-5 < x

   1. Initial program 62.5%

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

   3. Taylor expanded in x around inf 95.9%

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

      1. log-rec 95.9%

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

      2. mul-1-neg 95.9%

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

      3. associate-*r/ 95.9%

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

      4. associate-*r* 95.9%

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

      5. metadata-eval 95.9%

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

      6. *-commutative 95.9%

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

      7. associate-/l* 95.9%

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

      8. exp-to-pow 95.9%

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

      9. *-commutative 95.9%

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

   5. Simplified 95.9%

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

4. Final simplification 76.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 8.8 \cdot 10^{-243}:\\ \;\;\;\;\frac{\log x}{-n}\\ \mathbf{elif}\;x \leq 1.8 \cdot 10^{-202}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;x \leq 1.55 \cdot 10^{-5}:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{x \cdot n}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 71.2% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;x \leq 8 \cdot 10^{-242}:\\ \;\;\;\;\frac{\log x}{-n}\\ \mathbf{elif}\;x \leq 1.5 \cdot 10^{-202}:\\ \;\;\;\;\left(1 - t\_0\right) + \frac{x}{n}\\ \mathbf{elif}\;x \leq 1.36 \cdot 10^{-5}:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0}{x \cdot n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= x 8e-242)
     (/ (log x) (- n))
     (if (<= x 1.5e-202)
       (+ (- 1.0 t_0) (/ x n))
       (if (<= x 1.36e-5) (/ (- x (log x)) n) (/ t_0 (* x n)))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double tmp;
	if (x <= 8e-242) {
		tmp = log(x) / -n;
	} else if (x <= 1.5e-202) {
		tmp = (1.0 - t_0) + (x / n);
	} else if (x <= 1.36e-5) {
		tmp = (x - log(x)) / n;
	} else {
		tmp = t_0 / (x * n);
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x ** (1.0d0 / n)
    if (x <= 8d-242) then
        tmp = log(x) / -n
    else if (x <= 1.5d-202) then
        tmp = (1.0d0 - t_0) + (x / n)
    else if (x <= 1.36d-5) then
        tmp = (x - log(x)) / n
    else
        tmp = t_0 / (x * n)
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, (1.0 / n));
	double tmp;
	if (x <= 8e-242) {
		tmp = Math.log(x) / -n;
	} else if (x <= 1.5e-202) {
		tmp = (1.0 - t_0) + (x / n);
	} else if (x <= 1.36e-5) {
		tmp = (x - Math.log(x)) / n;
	} else {
		tmp = t_0 / (x * n);
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	tmp = 0
	if x <= 8e-242:
		tmp = math.log(x) / -n
	elif x <= 1.5e-202:
		tmp = (1.0 - t_0) + (x / n)
	elif x <= 1.36e-5:
		tmp = (x - math.log(x)) / n
	else:
		tmp = t_0 / (x * n)
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (x <= 8e-242)
		tmp = Float64(log(x) / Float64(-n));
	elseif (x <= 1.5e-202)
		tmp = Float64(Float64(1.0 - t_0) + Float64(x / n));
	elseif (x <= 1.36e-5)
		tmp = Float64(Float64(x - log(x)) / n);
	else
		tmp = Float64(t_0 / Float64(x * n));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = x ^ (1.0 / n);
	tmp = 0.0;
	if (x <= 8e-242)
		tmp = log(x) / -n;
	elseif (x <= 1.5e-202)
		tmp = (1.0 - t_0) + (x / n);
	elseif (x <= 1.36e-5)
		tmp = (x - log(x)) / n;
	else
		tmp = t_0 / (x * n);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, 8e-242], N[(N[Log[x], $MachinePrecision] / (-n)), $MachinePrecision], If[LessEqual[x, 1.5e-202], N[(N[(1.0 - t$95$0), $MachinePrecision] + N[(x / n), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.36e-5], N[(N[(x - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], N[(t$95$0 / N[(x * n), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if x < 8e-242

   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 x around 0 35.9%

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

      1. *-rgt-identity 35.9%

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

      2. associate-*l/ 35.9%

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

      3. associate-/l* 35.9%

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

      4. exp-to-pow 35.9%

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

   5. Simplified 35.9%

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

   6. Taylor expanded in n around inf 69.0%

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

      1. associate-*r/ 69.0%

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

      2. neg-mul-1 69.0%

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

   8. Simplified 69.0%

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

   if 8e-242 < x < 1.50000000000000005e-202

   1. Initial program 61.3%

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

   3. Taylor expanded in x around 0 61.4%

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

      1. +-commutative 61.4%

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

      2. *-rgt-identity 61.4%

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

      3. associate-*l/ 61.4%

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

      4. associate-/l* 61.4%

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

      5. exp-to-pow 61.4%

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

      6. associate-+r- 61.5%

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

   5. Simplified 61.5%

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

   if 1.50000000000000005e-202 < x < 1.36000000000000002e-5

   1. Initial program 32.1%

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

   3. Taylor expanded in x around 0 31.7%

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

      1. +-commutative 31.7%

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

      2. *-rgt-identity 31.7%

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

      3. associate-*l/ 31.7%

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

      4. associate-/l* 31.7%

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

      5. exp-to-pow 31.7%

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

      6. associate-+r- 32.6%

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

   5. Simplified 32.6%

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

   6. Taylor expanded in n around inf 57.3%

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

   if 1.36000000000000002e-5 < x

   1. Initial program 62.5%

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

   3. Taylor expanded in x around inf 95.9%

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

      1. log-rec 95.9%

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

      2. mul-1-neg 95.9%

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

      3. associate-*r/ 95.9%

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

      4. associate-*r* 95.9%

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

      5. metadata-eval 95.9%

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

      6. *-commutative 95.9%

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

      7. associate-/l* 95.9%

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

      8. exp-to-pow 95.9%

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

      9. *-commutative 95.9%

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

   5. Simplified 95.9%

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

4. Final simplification 76.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 8 \cdot 10^{-242}:\\ \;\;\;\;\frac{\log x}{-n}\\ \mathbf{elif}\;x \leq 1.5 \cdot 10^{-202}:\\ \;\;\;\;\left(1 - {x}^{\left(\frac{1}{n}\right)}\right) + \frac{x}{n}\\ \mathbf{elif}\;x \leq 1.36 \cdot 10^{-5}:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{x \cdot n}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 31.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \frac{x - \log x}{n} \end{array} \]

FPCore

(FPCore (x n) :precision binary64 (/ (- x (log x)) n))

C

double code(double x, double n) {
	return (x - log(x)) / n;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    code = (x - log(x)) / n
end function

Java

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

Python

def code(x, n):
	return (x - math.log(x)) / n

Julia

function code(x, n)
	return Float64(Float64(x - log(x)) / n)
end

MATLAB

function tmp = code(x, n)
	tmp = (x - log(x)) / n;
end

Wolfram

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

Derivation

1. Initial program 48.7%

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

3. Taylor expanded in x around 0 29.3%

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

   1. +-commutative 29.3%

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

   2. *-rgt-identity 29.3%

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

   3. associate-*l/ 29.3%

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

   4. associate-/l* 29.3%

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

   5. exp-to-pow 29.3%

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

   6. associate-+r- 22.9%

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

5. Simplified 22.9%

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

6. Taylor expanded in n around inf 33.2%

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

7. Final simplification 33.2%

   \[\leadsto \frac{x - \log x}{n} \]
8. Add Preprocessing

Alternative 9: 31.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \frac{\log x}{-n} \end{array} \]

FPCore

(FPCore (x n) :precision binary64 (/ (log x) (- n)))

C

double code(double x, double n) {
	return log(x) / -n;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    code = log(x) / -n
end function

Java

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

Python

def code(x, n):
	return math.log(x) / -n

Julia

function code(x, n)
	return Float64(log(x) / Float64(-n))
end

MATLAB

function tmp = code(x, n)
	tmp = log(x) / -n;
end

Wolfram

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

Derivation

1. Initial program 48.7%

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

3. Taylor expanded in x around 0 37.3%

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

   1. *-rgt-identity 37.3%

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

   2. associate-*l/ 37.3%

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

   3. associate-/l* 37.3%

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

   4. exp-to-pow 37.3%

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

5. Simplified 37.3%

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

6. Taylor expanded in n around inf 32.8%

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

   1. associate-*r/ 32.8%

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

   2. neg-mul-1 32.8%

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

8. Simplified 32.8%

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

9. Final simplification 32.8%

   \[\leadsto \frac{\log x}{-n} \]
10. Add Preprocessing

Alternative 10: 4.6% 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 48.7%

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

3. Taylor expanded in x around 0 29.3%

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

   1. +-commutative 29.3%

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

   2. *-rgt-identity 29.3%

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

   3. associate-*l/ 29.3%

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

   4. associate-/l* 29.3%

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

   5. exp-to-pow 29.3%

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

   6. associate-+r- 22.9%

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

5. Simplified 22.9%

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

6. Taylor expanded in x around inf 4.8%

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

7. Final simplification 4.8%

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

Reproduce

herbie shell --seed 2024046 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  :precision binary64
  (- (pow (+ x 1.0) (/ 1.0 n)) (pow x (/ 1.0 n))))