2nthrt (problem 3.4.6)

Percentage Accurate: 52.9% → 84.5%

Time: 28.7s

Alternatives: 12

Speedup: 1.9×

• could not determine a ground truth

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 52.9% 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: 84.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{log1p}\left(x\right) - \log x\\ t_1 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{-19}:\\ \;\;\;\;t_1 \cdot \frac{\frac{1}{x}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq -1 \cdot 10^{-137}:\\ \;\;\;\;\frac{1}{n} \cdot t_0\\ \mathbf{elif}\;\frac{1}{n} \leq -2 \cdot 10^{-154}:\\ \;\;\;\;\frac{\left(\frac{1}{x} + \frac{0.3333333333333333}{{x}^{3}}\right) - \left(\frac{0.5}{{x}^{2}} + \frac{0.25}{{x}^{4}}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-150}:\\ \;\;\;\;\frac{t_0}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 100000:\\ \;\;\;\;\frac{\frac{t_1}{n}}{x}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{x}{n}} - t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (- (log1p x) (log x))) (t_1 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -5e-19)
     (* t_1 (/ (/ 1.0 x) n))
     (if (<= (/ 1.0 n) -1e-137)
       (* (/ 1.0 n) t_0)
       (if (<= (/ 1.0 n) -2e-154)
         (/
          (-
           (+ (/ 1.0 x) (/ 0.3333333333333333 (pow x 3.0)))
           (+ (/ 0.5 (pow x 2.0)) (/ 0.25 (pow x 4.0))))
          n)
         (if (<= (/ 1.0 n) 2e-150)
           (/ t_0 n)
           (if (<= (/ 1.0 n) 100000.0)
             (/ (/ t_1 n) x)
             (- (exp (/ x n)) t_1))))))))

C

double code(double x, double n) {
	double t_0 = log1p(x) - log(x);
	double t_1 = pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -5e-19) {
		tmp = t_1 * ((1.0 / x) / n);
	} else if ((1.0 / n) <= -1e-137) {
		tmp = (1.0 / n) * t_0;
	} else if ((1.0 / n) <= -2e-154) {
		tmp = (((1.0 / x) + (0.3333333333333333 / pow(x, 3.0))) - ((0.5 / pow(x, 2.0)) + (0.25 / pow(x, 4.0)))) / n;
	} else if ((1.0 / n) <= 2e-150) {
		tmp = t_0 / n;
	} else if ((1.0 / n) <= 100000.0) {
		tmp = (t_1 / n) / x;
	} else {
		tmp = exp((x / n)) - t_1;
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double t_0 = Math.log1p(x) - Math.log(x);
	double t_1 = Math.pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -5e-19) {
		tmp = t_1 * ((1.0 / x) / n);
	} else if ((1.0 / n) <= -1e-137) {
		tmp = (1.0 / n) * t_0;
	} else if ((1.0 / n) <= -2e-154) {
		tmp = (((1.0 / x) + (0.3333333333333333 / Math.pow(x, 3.0))) - ((0.5 / Math.pow(x, 2.0)) + (0.25 / Math.pow(x, 4.0)))) / n;
	} else if ((1.0 / n) <= 2e-150) {
		tmp = t_0 / n;
	} else if ((1.0 / n) <= 100000.0) {
		tmp = (t_1 / n) / x;
	} else {
		tmp = Math.exp((x / n)) - t_1;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.log1p(x) - math.log(x)
	t_1 = math.pow(x, (1.0 / n))
	tmp = 0
	if (1.0 / n) <= -5e-19:
		tmp = t_1 * ((1.0 / x) / n)
	elif (1.0 / n) <= -1e-137:
		tmp = (1.0 / n) * t_0
	elif (1.0 / n) <= -2e-154:
		tmp = (((1.0 / x) + (0.3333333333333333 / math.pow(x, 3.0))) - ((0.5 / math.pow(x, 2.0)) + (0.25 / math.pow(x, 4.0)))) / n
	elif (1.0 / n) <= 2e-150:
		tmp = t_0 / n
	elif (1.0 / n) <= 100000.0:
		tmp = (t_1 / n) / x
	else:
		tmp = math.exp((x / n)) - t_1
	return tmp

Julia

function code(x, n)
	t_0 = Float64(log1p(x) - log(x))
	t_1 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -5e-19)
		tmp = Float64(t_1 * Float64(Float64(1.0 / x) / n));
	elseif (Float64(1.0 / n) <= -1e-137)
		tmp = Float64(Float64(1.0 / n) * t_0);
	elseif (Float64(1.0 / n) <= -2e-154)
		tmp = Float64(Float64(Float64(Float64(1.0 / x) + Float64(0.3333333333333333 / (x ^ 3.0))) - Float64(Float64(0.5 / (x ^ 2.0)) + Float64(0.25 / (x ^ 4.0)))) / n);
	elseif (Float64(1.0 / n) <= 2e-150)
		tmp = Float64(t_0 / n);
	elseif (Float64(1.0 / n) <= 100000.0)
		tmp = Float64(Float64(t_1 / n) / x);
	else
		tmp = Float64(exp(Float64(x / n)) - t_1);
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(N[Log[1 + x], $MachinePrecision] - N[Log[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -5e-19], N[(t$95$1 * N[(N[(1.0 / x), $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], -1e-137], N[(N[(1.0 / n), $MachinePrecision] * t$95$0), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], -2e-154], N[(N[(N[(N[(1.0 / x), $MachinePrecision] + N[(0.3333333333333333 / N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(0.5 / N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision] + N[(0.25 / N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-150], N[(t$95$0 / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 100000.0], N[(N[(t$95$1 / n), $MachinePrecision] / x), $MachinePrecision], N[(N[Exp[N[(x / n), $MachinePrecision]], $MachinePrecision] - t$95$1), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 6 regimes

2. if (/.f64 1 n) < -5.0000000000000004e-19

   1. Initial program 97.2%

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

   2. Taylor expanded in x around inf 98.8%

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

      1. mul-1-neg 98.8%

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

      2. log-rec 98.8%

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

      3. mul-1-neg 98.8%

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

      4. distribute-neg-frac 98.8%

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

      5. mul-1-neg 98.8%

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

      6. remove-double-neg 98.8%

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

      7. *-commutative 98.8%

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

   4. Simplified 98.8%

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

      1. div-inv 98.8%

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

      2. associate-/r* 98.8%

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

      3. div-inv 98.8%

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

      4. exp-to-pow 98.8%

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

   6. Applied egg-rr 98.8%

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

   if -5.0000000000000004e-19 < (/.f64 1 n) < -9.99999999999999978e-138

   1. Initial program 14.1%

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

   2. Taylor expanded in n around inf 76.8%

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

      1. log1p-def 76.8%

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

   4. Simplified 76.8%

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

      1. div-sub 76.8%

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

   6. Applied egg-rr 76.8%

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

      1. div-inv 73.9%

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

      2. div-inv 76.8%

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

      3. distribute-rgt-out-- 76.8%

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

   8. Applied egg-rr 76.8%

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

   if -9.99999999999999978e-138 < (/.f64 1 n) < -1.9999999999999999e-154

   1. Initial program 22.0%

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

   2. Taylor expanded in n around inf 39.7%

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

      1. log1p-def 39.7%

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

   4. Simplified 39.7%

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

   5. Taylor expanded in x around inf 86.8%

      \[\leadsto \frac{\color{blue}{\left(0.3333333333333333 \cdot \frac{1}{{x}^{3}} + \frac{1}{x}\right) - \left(0.25 \cdot \frac{1}{{x}^{4}} + 0.5 \cdot \frac{1}{{x}^{2}}\right)}}{n} \]
   6. Step-by-step derivation

      1. +-commutative 86.8%

         \[\leadsto \frac{\color{blue}{\left(\frac{1}{x} + 0.3333333333333333 \cdot \frac{1}{{x}^{3}}\right)} - \left(0.25 \cdot \frac{1}{{x}^{4}} + 0.5 \cdot \frac{1}{{x}^{2}}\right)}{n} \]

      2. associate-*r/ 86.8%

         \[\leadsto \frac{\left(\frac{1}{x} + \color{blue}{\frac{0.3333333333333333 \cdot 1}{{x}^{3}}}\right) - \left(0.25 \cdot \frac{1}{{x}^{4}} + 0.5 \cdot \frac{1}{{x}^{2}}\right)}{n} \]

      3. metadata-eval 86.8%

         \[\leadsto \frac{\left(\frac{1}{x} + \frac{\color{blue}{0.3333333333333333}}{{x}^{3}}\right) - \left(0.25 \cdot \frac{1}{{x}^{4}} + 0.5 \cdot \frac{1}{{x}^{2}}\right)}{n} \]

      4. +-commutative 86.8%

         \[\leadsto \frac{\left(\frac{1}{x} + \frac{0.3333333333333333}{{x}^{3}}\right) - \color{blue}{\left(0.5 \cdot \frac{1}{{x}^{2}} + 0.25 \cdot \frac{1}{{x}^{4}}\right)}}{n} \]

      5. associate-*r/ 86.8%

         \[\leadsto \frac{\left(\frac{1}{x} + \frac{0.3333333333333333}{{x}^{3}}\right) - \left(\color{blue}{\frac{0.5 \cdot 1}{{x}^{2}}} + 0.25 \cdot \frac{1}{{x}^{4}}\right)}{n} \]

      6. metadata-eval 86.8%

         \[\leadsto \frac{\left(\frac{1}{x} + \frac{0.3333333333333333}{{x}^{3}}\right) - \left(\frac{\color{blue}{0.5}}{{x}^{2}} + 0.25 \cdot \frac{1}{{x}^{4}}\right)}{n} \]

      7. associate-*r/ 86.8%

         \[\leadsto \frac{\left(\frac{1}{x} + \frac{0.3333333333333333}{{x}^{3}}\right) - \left(\frac{0.5}{{x}^{2}} + \color{blue}{\frac{0.25 \cdot 1}{{x}^{4}}}\right)}{n} \]

      8. metadata-eval 86.8%

         \[\leadsto \frac{\left(\frac{1}{x} + \frac{0.3333333333333333}{{x}^{3}}\right) - \left(\frac{0.5}{{x}^{2}} + \frac{\color{blue}{0.25}}{{x}^{4}}\right)}{n} \]

   7. Simplified 86.8%

      \[\leadsto \frac{\color{blue}{\left(\frac{1}{x} + \frac{0.3333333333333333}{{x}^{3}}\right) - \left(\frac{0.5}{{x}^{2}} + \frac{0.25}{{x}^{4}}\right)}}{n} \]

   if -1.9999999999999999e-154 < (/.f64 1 n) < 2.00000000000000001e-150

   1. Initial program 40.0%

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

   2. Taylor expanded in n around inf 96.5%

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

      1. log1p-def 96.5%

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

   4. Simplified 96.5%

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

   if 2.00000000000000001e-150 < (/.f64 1 n) < 1e5

   1. Initial program 18.2%

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

   2. Taylor expanded in x around inf 76.7%

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

      1. mul-1-neg 76.7%

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

      2. log-rec 76.7%

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

      3. mul-1-neg 76.7%

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

      4. distribute-neg-frac 76.7%

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

      5. mul-1-neg 76.7%

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

      6. remove-double-neg 76.7%

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

      7. *-commutative 76.7%

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

   4. Simplified 76.7%

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

      1. *-un-lft-identity 76.7%

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

      2. times-frac 76.7%

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

      3. div-inv 76.7%

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

      4. exp-to-pow 76.7%

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

   6. Applied egg-rr 76.7%

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

      1. associate-*l/ 76.9%

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

      2. *-un-lft-identity 76.9%

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

   8. Applied egg-rr 76.9%

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

   if 1e5 < (/.f64 1 n)

   1. Initial program 60.0%

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

   2. Taylor expanded in n around 0 60.0%

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

      1. log1p-def 100.0%

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

   4. Simplified 100.0%

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

   5. Taylor expanded in x around 0 100.0%

      \[\leadsto e^{\color{blue}{\frac{x}{n}}} - {x}^{\left(\frac{1}{n}\right)} \]
3. Recombined 6 regimes into one program.

4. Final simplification 92.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{-19}:\\ \;\;\;\;{x}^{\left(\frac{1}{n}\right)} \cdot \frac{\frac{1}{x}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq -1 \cdot 10^{-137}:\\ \;\;\;\;\frac{1}{n} \cdot \left(\mathsf{log1p}\left(x\right) - \log x\right)\\ \mathbf{elif}\;\frac{1}{n} \leq -2 \cdot 10^{-154}:\\ \;\;\;\;\frac{\left(\frac{1}{x} + \frac{0.3333333333333333}{{x}^{3}}\right) - \left(\frac{0.5}{{x}^{2}} + \frac{0.25}{{x}^{4}}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-150}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right) - \log x}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 100000:\\ \;\;\;\;\frac{\frac{{x}^{\left(\frac{1}{n}\right)}}{n}}{x}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{x}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \]

Alternative 2: 84.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{log1p}\left(x\right) - \log x\\ t_1 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{-19}:\\ \;\;\;\;t_1 \cdot \frac{\frac{1}{x}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq -1 \cdot 10^{-137}:\\ \;\;\;\;\frac{1}{n} \cdot t_0\\ \mathbf{elif}\;\frac{1}{n} \leq -2 \cdot 10^{-154}:\\ \;\;\;\;\frac{\frac{1}{x} - \frac{0.5}{{x}^{2}}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-150}:\\ \;\;\;\;\frac{t_0}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 100000:\\ \;\;\;\;\frac{\frac{t_1}{n}}{x}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{x}{n}} - t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (- (log1p x) (log x))) (t_1 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -5e-19)
     (* t_1 (/ (/ 1.0 x) n))
     (if (<= (/ 1.0 n) -1e-137)
       (* (/ 1.0 n) t_0)
       (if (<= (/ 1.0 n) -2e-154)
         (/ (- (/ 1.0 x) (/ 0.5 (pow x 2.0))) n)
         (if (<= (/ 1.0 n) 2e-150)
           (/ t_0 n)
           (if (<= (/ 1.0 n) 100000.0)
             (/ (/ t_1 n) x)
             (- (exp (/ x n)) t_1))))))))

C

double code(double x, double n) {
	double t_0 = log1p(x) - log(x);
	double t_1 = pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -5e-19) {
		tmp = t_1 * ((1.0 / x) / n);
	} else if ((1.0 / n) <= -1e-137) {
		tmp = (1.0 / n) * t_0;
	} else if ((1.0 / n) <= -2e-154) {
		tmp = ((1.0 / x) - (0.5 / pow(x, 2.0))) / n;
	} else if ((1.0 / n) <= 2e-150) {
		tmp = t_0 / n;
	} else if ((1.0 / n) <= 100000.0) {
		tmp = (t_1 / n) / x;
	} else {
		tmp = exp((x / n)) - t_1;
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double t_0 = Math.log1p(x) - Math.log(x);
	double t_1 = Math.pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -5e-19) {
		tmp = t_1 * ((1.0 / x) / n);
	} else if ((1.0 / n) <= -1e-137) {
		tmp = (1.0 / n) * t_0;
	} else if ((1.0 / n) <= -2e-154) {
		tmp = ((1.0 / x) - (0.5 / Math.pow(x, 2.0))) / n;
	} else if ((1.0 / n) <= 2e-150) {
		tmp = t_0 / n;
	} else if ((1.0 / n) <= 100000.0) {
		tmp = (t_1 / n) / x;
	} else {
		tmp = Math.exp((x / n)) - t_1;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.log1p(x) - math.log(x)
	t_1 = math.pow(x, (1.0 / n))
	tmp = 0
	if (1.0 / n) <= -5e-19:
		tmp = t_1 * ((1.0 / x) / n)
	elif (1.0 / n) <= -1e-137:
		tmp = (1.0 / n) * t_0
	elif (1.0 / n) <= -2e-154:
		tmp = ((1.0 / x) - (0.5 / math.pow(x, 2.0))) / n
	elif (1.0 / n) <= 2e-150:
		tmp = t_0 / n
	elif (1.0 / n) <= 100000.0:
		tmp = (t_1 / n) / x
	else:
		tmp = math.exp((x / n)) - t_1
	return tmp

Julia

function code(x, n)
	t_0 = Float64(log1p(x) - log(x))
	t_1 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -5e-19)
		tmp = Float64(t_1 * Float64(Float64(1.0 / x) / n));
	elseif (Float64(1.0 / n) <= -1e-137)
		tmp = Float64(Float64(1.0 / n) * t_0);
	elseif (Float64(1.0 / n) <= -2e-154)
		tmp = Float64(Float64(Float64(1.0 / x) - Float64(0.5 / (x ^ 2.0))) / n);
	elseif (Float64(1.0 / n) <= 2e-150)
		tmp = Float64(t_0 / n);
	elseif (Float64(1.0 / n) <= 100000.0)
		tmp = Float64(Float64(t_1 / n) / x);
	else
		tmp = Float64(exp(Float64(x / n)) - t_1);
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(N[Log[1 + x], $MachinePrecision] - N[Log[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -5e-19], N[(t$95$1 * N[(N[(1.0 / x), $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], -1e-137], N[(N[(1.0 / n), $MachinePrecision] * t$95$0), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], -2e-154], N[(N[(N[(1.0 / x), $MachinePrecision] - N[(0.5 / N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-150], N[(t$95$0 / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 100000.0], N[(N[(t$95$1 / n), $MachinePrecision] / x), $MachinePrecision], N[(N[Exp[N[(x / n), $MachinePrecision]], $MachinePrecision] - t$95$1), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 6 regimes

2. if (/.f64 1 n) < -5.0000000000000004e-19

   1. Initial program 97.2%

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

   2. Taylor expanded in x around inf 98.8%

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

      1. mul-1-neg 98.8%

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

      2. log-rec 98.8%

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

      3. mul-1-neg 98.8%

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

      4. distribute-neg-frac 98.8%

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

      5. mul-1-neg 98.8%

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

      6. remove-double-neg 98.8%

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

      7. *-commutative 98.8%

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

   4. Simplified 98.8%

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

      1. div-inv 98.8%

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

      2. associate-/r* 98.8%

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

      3. div-inv 98.8%

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

      4. exp-to-pow 98.8%

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

   6. Applied egg-rr 98.8%

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

   if -5.0000000000000004e-19 < (/.f64 1 n) < -9.99999999999999978e-138

   1. Initial program 14.1%

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

   2. Taylor expanded in n around inf 76.8%

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

      1. log1p-def 76.8%

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

   4. Simplified 76.8%

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

      1. div-sub 76.8%

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

   6. Applied egg-rr 76.8%

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

      1. div-inv 73.9%

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

      2. div-inv 76.8%

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

      3. distribute-rgt-out-- 76.8%

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

   8. Applied egg-rr 76.8%

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

   if -9.99999999999999978e-138 < (/.f64 1 n) < -1.9999999999999999e-154

   1. Initial program 22.0%

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

   2. Taylor expanded in n around inf 39.7%

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

      1. log1p-def 39.7%

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

   4. Simplified 39.7%

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

   5. Taylor expanded in x around inf 84.4%

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

      1. associate-*r/ 84.4%

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

      2. metadata-eval 84.4%

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

   7. Simplified 84.4%

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

   if -1.9999999999999999e-154 < (/.f64 1 n) < 2.00000000000000001e-150

   1. Initial program 40.0%

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

   2. Taylor expanded in n around inf 96.5%

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

      1. log1p-def 96.5%

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

   4. Simplified 96.5%

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

   if 2.00000000000000001e-150 < (/.f64 1 n) < 1e5

   1. Initial program 18.2%

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

   2. Taylor expanded in x around inf 76.7%

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

      1. mul-1-neg 76.7%

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

      2. log-rec 76.7%

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

      3. mul-1-neg 76.7%

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

      4. distribute-neg-frac 76.7%

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

      5. mul-1-neg 76.7%

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

      6. remove-double-neg 76.7%

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

      7. *-commutative 76.7%

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

   4. Simplified 76.7%

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

      1. *-un-lft-identity 76.7%

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

      2. times-frac 76.7%

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

      3. div-inv 76.7%

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

      4. exp-to-pow 76.7%

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

   6. Applied egg-rr 76.7%

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

      1. associate-*l/ 76.9%

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

      2. *-un-lft-identity 76.9%

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

   8. Applied egg-rr 76.9%

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

   if 1e5 < (/.f64 1 n)

   1. Initial program 60.0%

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

   2. Taylor expanded in n around 0 60.0%

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

      1. log1p-def 100.0%

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

   4. Simplified 100.0%

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

   5. Taylor expanded in x around 0 100.0%

      \[\leadsto e^{\color{blue}{\frac{x}{n}}} - {x}^{\left(\frac{1}{n}\right)} \]
3. Recombined 6 regimes into one program.

4. Final simplification 92.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{-19}:\\ \;\;\;\;{x}^{\left(\frac{1}{n}\right)} \cdot \frac{\frac{1}{x}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq -1 \cdot 10^{-137}:\\ \;\;\;\;\frac{1}{n} \cdot \left(\mathsf{log1p}\left(x\right) - \log x\right)\\ \mathbf{elif}\;\frac{1}{n} \leq -2 \cdot 10^{-154}:\\ \;\;\;\;\frac{\frac{1}{x} - \frac{0.5}{{x}^{2}}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-150}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right) - \log x}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 100000:\\ \;\;\;\;\frac{\frac{{x}^{\left(\frac{1}{n}\right)}}{n}}{x}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{x}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \]

Alternative 3: 84.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{log1p}\left(x\right) - \log x\\ t_1 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{-19}:\\ \;\;\;\;t_1 \cdot \frac{\frac{1}{x}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq -1 \cdot 10^{-137}:\\ \;\;\;\;\frac{1}{n} \cdot t_0\\ \mathbf{elif}\;\frac{1}{n} \leq -2 \cdot 10^{-154}:\\ \;\;\;\;\frac{\frac{0.3333333333333333}{{x}^{3}} + \left(\frac{1}{x} - \frac{0.5}{{x}^{2}}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-150}:\\ \;\;\;\;\frac{t_0}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 100000:\\ \;\;\;\;\frac{\frac{t_1}{n}}{x}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{x}{n}} - t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (- (log1p x) (log x))) (t_1 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -5e-19)
     (* t_1 (/ (/ 1.0 x) n))
     (if (<= (/ 1.0 n) -1e-137)
       (* (/ 1.0 n) t_0)
       (if (<= (/ 1.0 n) -2e-154)
         (/
          (+
           (/ 0.3333333333333333 (pow x 3.0))
           (- (/ 1.0 x) (/ 0.5 (pow x 2.0))))
          n)
         (if (<= (/ 1.0 n) 2e-150)
           (/ t_0 n)
           (if (<= (/ 1.0 n) 100000.0)
             (/ (/ t_1 n) x)
             (- (exp (/ x n)) t_1))))))))

C

double code(double x, double n) {
	double t_0 = log1p(x) - log(x);
	double t_1 = pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -5e-19) {
		tmp = t_1 * ((1.0 / x) / n);
	} else if ((1.0 / n) <= -1e-137) {
		tmp = (1.0 / n) * t_0;
	} else if ((1.0 / n) <= -2e-154) {
		tmp = ((0.3333333333333333 / pow(x, 3.0)) + ((1.0 / x) - (0.5 / pow(x, 2.0)))) / n;
	} else if ((1.0 / n) <= 2e-150) {
		tmp = t_0 / n;
	} else if ((1.0 / n) <= 100000.0) {
		tmp = (t_1 / n) / x;
	} else {
		tmp = exp((x / n)) - t_1;
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double t_0 = Math.log1p(x) - Math.log(x);
	double t_1 = Math.pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -5e-19) {
		tmp = t_1 * ((1.0 / x) / n);
	} else if ((1.0 / n) <= -1e-137) {
		tmp = (1.0 / n) * t_0;
	} else if ((1.0 / n) <= -2e-154) {
		tmp = ((0.3333333333333333 / Math.pow(x, 3.0)) + ((1.0 / x) - (0.5 / Math.pow(x, 2.0)))) / n;
	} else if ((1.0 / n) <= 2e-150) {
		tmp = t_0 / n;
	} else if ((1.0 / n) <= 100000.0) {
		tmp = (t_1 / n) / x;
	} else {
		tmp = Math.exp((x / n)) - t_1;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.log1p(x) - math.log(x)
	t_1 = math.pow(x, (1.0 / n))
	tmp = 0
	if (1.0 / n) <= -5e-19:
		tmp = t_1 * ((1.0 / x) / n)
	elif (1.0 / n) <= -1e-137:
		tmp = (1.0 / n) * t_0
	elif (1.0 / n) <= -2e-154:
		tmp = ((0.3333333333333333 / math.pow(x, 3.0)) + ((1.0 / x) - (0.5 / math.pow(x, 2.0)))) / n
	elif (1.0 / n) <= 2e-150:
		tmp = t_0 / n
	elif (1.0 / n) <= 100000.0:
		tmp = (t_1 / n) / x
	else:
		tmp = math.exp((x / n)) - t_1
	return tmp

Julia

function code(x, n)
	t_0 = Float64(log1p(x) - log(x))
	t_1 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -5e-19)
		tmp = Float64(t_1 * Float64(Float64(1.0 / x) / n));
	elseif (Float64(1.0 / n) <= -1e-137)
		tmp = Float64(Float64(1.0 / n) * t_0);
	elseif (Float64(1.0 / n) <= -2e-154)
		tmp = Float64(Float64(Float64(0.3333333333333333 / (x ^ 3.0)) + Float64(Float64(1.0 / x) - Float64(0.5 / (x ^ 2.0)))) / n);
	elseif (Float64(1.0 / n) <= 2e-150)
		tmp = Float64(t_0 / n);
	elseif (Float64(1.0 / n) <= 100000.0)
		tmp = Float64(Float64(t_1 / n) / x);
	else
		tmp = Float64(exp(Float64(x / n)) - t_1);
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(N[Log[1 + x], $MachinePrecision] - N[Log[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -5e-19], N[(t$95$1 * N[(N[(1.0 / x), $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], -1e-137], N[(N[(1.0 / n), $MachinePrecision] * t$95$0), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], -2e-154], N[(N[(N[(0.3333333333333333 / N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 / x), $MachinePrecision] - N[(0.5 / N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-150], N[(t$95$0 / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 100000.0], N[(N[(t$95$1 / n), $MachinePrecision] / x), $MachinePrecision], N[(N[Exp[N[(x / n), $MachinePrecision]], $MachinePrecision] - t$95$1), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 6 regimes

2. if (/.f64 1 n) < -5.0000000000000004e-19

   1. Initial program 97.2%

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

   2. Taylor expanded in x around inf 98.8%

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

      1. mul-1-neg 98.8%

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

      2. log-rec 98.8%

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

      3. mul-1-neg 98.8%

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

      4. distribute-neg-frac 98.8%

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

      5. mul-1-neg 98.8%

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

      6. remove-double-neg 98.8%

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

      7. *-commutative 98.8%

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

   4. Simplified 98.8%

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

      1. div-inv 98.8%

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

      2. associate-/r* 98.8%

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

      3. div-inv 98.8%

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

      4. exp-to-pow 98.8%

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

   6. Applied egg-rr 98.8%

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

   if -5.0000000000000004e-19 < (/.f64 1 n) < -9.99999999999999978e-138

   1. Initial program 14.1%

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

   2. Taylor expanded in n around inf 76.8%

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

      1. log1p-def 76.8%

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

   4. Simplified 76.8%

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

      1. div-sub 76.8%

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

   6. Applied egg-rr 76.8%

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

      1. div-inv 73.9%

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

      2. div-inv 76.8%

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

      3. distribute-rgt-out-- 76.8%

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

   8. Applied egg-rr 76.8%

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

   if -9.99999999999999978e-138 < (/.f64 1 n) < -1.9999999999999999e-154

   1. Initial program 22.0%

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

   2. Taylor expanded in n around inf 39.7%

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

      1. log1p-def 39.7%

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

   4. Simplified 39.7%

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

   5. Taylor expanded in x around inf 85.8%

      \[\leadsto \frac{\color{blue}{\left(0.3333333333333333 \cdot \frac{1}{{x}^{3}} + \frac{1}{x}\right) - 0.5 \cdot \frac{1}{{x}^{2}}}}{n} \]
   6. Step-by-step derivation

      1. associate--l+ 85.8%

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

      2. associate-*r/ 85.8%

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

      3. metadata-eval 85.8%

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

      4. associate-*r/ 85.8%

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

      5. metadata-eval 85.8%

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

   7. Simplified 85.8%

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

   if -1.9999999999999999e-154 < (/.f64 1 n) < 2.00000000000000001e-150

   1. Initial program 40.0%

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

   2. Taylor expanded in n around inf 96.5%

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

      1. log1p-def 96.5%

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

   4. Simplified 96.5%

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

   if 2.00000000000000001e-150 < (/.f64 1 n) < 1e5

   1. Initial program 18.2%

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

   2. Taylor expanded in x around inf 76.7%

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

      1. mul-1-neg 76.7%

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

      2. log-rec 76.7%

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

      3. mul-1-neg 76.7%

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

      4. distribute-neg-frac 76.7%

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

      5. mul-1-neg 76.7%

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

      6. remove-double-neg 76.7%

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

      7. *-commutative 76.7%

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

   4. Simplified 76.7%

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

      1. *-un-lft-identity 76.7%

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

      2. times-frac 76.7%

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

      3. div-inv 76.7%

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

      4. exp-to-pow 76.7%

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

   6. Applied egg-rr 76.7%

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

      1. associate-*l/ 76.9%

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

      2. *-un-lft-identity 76.9%

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

   8. Applied egg-rr 76.9%

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

   if 1e5 < (/.f64 1 n)

   1. Initial program 60.0%

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

   2. Taylor expanded in n around 0 60.0%

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

      1. log1p-def 100.0%

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

   4. Simplified 100.0%

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

   5. Taylor expanded in x around 0 100.0%

      \[\leadsto e^{\color{blue}{\frac{x}{n}}} - {x}^{\left(\frac{1}{n}\right)} \]
3. Recombined 6 regimes into one program.

4. Final simplification 92.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{-19}:\\ \;\;\;\;{x}^{\left(\frac{1}{n}\right)} \cdot \frac{\frac{1}{x}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq -1 \cdot 10^{-137}:\\ \;\;\;\;\frac{1}{n} \cdot \left(\mathsf{log1p}\left(x\right) - \log x\right)\\ \mathbf{elif}\;\frac{1}{n} \leq -2 \cdot 10^{-154}:\\ \;\;\;\;\frac{\frac{0.3333333333333333}{{x}^{3}} + \left(\frac{1}{x} - \frac{0.5}{{x}^{2}}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-150}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right) - \log x}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 100000:\\ \;\;\;\;\frac{\frac{{x}^{\left(\frac{1}{n}\right)}}{n}}{x}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{x}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \]

Alternative 4: 80.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{log1p}\left(x\right) - \log x}{n}\\ t_1 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{-19}:\\ \;\;\;\;t_1 \cdot \frac{\frac{1}{x}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq -1 \cdot 10^{-137}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\frac{1}{n} \leq -2 \cdot 10^{-154}:\\ \;\;\;\;\frac{\frac{1}{x} - \frac{0.5}{{x}^{2}}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-150}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\frac{1}{n} \leq 100000:\\ \;\;\;\;\frac{\frac{t_1}{n}}{x}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{+204}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{0.3333333333333333}{n \cdot {x}^{3}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (/ (- (log1p x) (log x)) n)) (t_1 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -5e-19)
     (* t_1 (/ (/ 1.0 x) n))
     (if (<= (/ 1.0 n) -1e-137)
       t_0
       (if (<= (/ 1.0 n) -2e-154)
         (/ (- (/ 1.0 x) (/ 0.5 (pow x 2.0))) n)
         (if (<= (/ 1.0 n) 2e-150)
           t_0
           (if (<= (/ 1.0 n) 100000.0)
             (/ (/ t_1 n) x)
             (if (<= (/ 1.0 n) 1e+204)
               (- (+ 1.0 (/ x n)) t_1)
               (/ 0.3333333333333333 (* n (pow x 3.0)))))))))))

C

double code(double x, double n) {
	double t_0 = (log1p(x) - log(x)) / n;
	double t_1 = pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -5e-19) {
		tmp = t_1 * ((1.0 / x) / n);
	} else if ((1.0 / n) <= -1e-137) {
		tmp = t_0;
	} else if ((1.0 / n) <= -2e-154) {
		tmp = ((1.0 / x) - (0.5 / pow(x, 2.0))) / n;
	} else if ((1.0 / n) <= 2e-150) {
		tmp = t_0;
	} else if ((1.0 / n) <= 100000.0) {
		tmp = (t_1 / n) / x;
	} else if ((1.0 / n) <= 1e+204) {
		tmp = (1.0 + (x / n)) - t_1;
	} else {
		tmp = 0.3333333333333333 / (n * pow(x, 3.0));
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double t_0 = (Math.log1p(x) - Math.log(x)) / n;
	double t_1 = Math.pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -5e-19) {
		tmp = t_1 * ((1.0 / x) / n);
	} else if ((1.0 / n) <= -1e-137) {
		tmp = t_0;
	} else if ((1.0 / n) <= -2e-154) {
		tmp = ((1.0 / x) - (0.5 / Math.pow(x, 2.0))) / n;
	} else if ((1.0 / n) <= 2e-150) {
		tmp = t_0;
	} else if ((1.0 / n) <= 100000.0) {
		tmp = (t_1 / n) / x;
	} else if ((1.0 / n) <= 1e+204) {
		tmp = (1.0 + (x / n)) - t_1;
	} else {
		tmp = 0.3333333333333333 / (n * Math.pow(x, 3.0));
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = (math.log1p(x) - math.log(x)) / n
	t_1 = math.pow(x, (1.0 / n))
	tmp = 0
	if (1.0 / n) <= -5e-19:
		tmp = t_1 * ((1.0 / x) / n)
	elif (1.0 / n) <= -1e-137:
		tmp = t_0
	elif (1.0 / n) <= -2e-154:
		tmp = ((1.0 / x) - (0.5 / math.pow(x, 2.0))) / n
	elif (1.0 / n) <= 2e-150:
		tmp = t_0
	elif (1.0 / n) <= 100000.0:
		tmp = (t_1 / n) / x
	elif (1.0 / n) <= 1e+204:
		tmp = (1.0 + (x / n)) - t_1
	else:
		tmp = 0.3333333333333333 / (n * math.pow(x, 3.0))
	return tmp

Julia

function code(x, n)
	t_0 = Float64(Float64(log1p(x) - log(x)) / n)
	t_1 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -5e-19)
		tmp = Float64(t_1 * Float64(Float64(1.0 / x) / n));
	elseif (Float64(1.0 / n) <= -1e-137)
		tmp = t_0;
	elseif (Float64(1.0 / n) <= -2e-154)
		tmp = Float64(Float64(Float64(1.0 / x) - Float64(0.5 / (x ^ 2.0))) / n);
	elseif (Float64(1.0 / n) <= 2e-150)
		tmp = t_0;
	elseif (Float64(1.0 / n) <= 100000.0)
		tmp = Float64(Float64(t_1 / n) / x);
	elseif (Float64(1.0 / n) <= 1e+204)
		tmp = Float64(Float64(1.0 + Float64(x / n)) - t_1);
	else
		tmp = Float64(0.3333333333333333 / Float64(n * (x ^ 3.0)));
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(N[(N[Log[1 + x], $MachinePrecision] - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision]}, Block[{t$95$1 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -5e-19], N[(t$95$1 * N[(N[(1.0 / x), $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], -1e-137], t$95$0, If[LessEqual[N[(1.0 / n), $MachinePrecision], -2e-154], N[(N[(N[(1.0 / x), $MachinePrecision] - N[(0.5 / N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-150], t$95$0, If[LessEqual[N[(1.0 / n), $MachinePrecision], 100000.0], N[(N[(t$95$1 / n), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 1e+204], N[(N[(1.0 + N[(x / n), $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision], N[(0.3333333333333333 / N[(n * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 6 regimes

2. if (/.f64 1 n) < -5.0000000000000004e-19

   1. Initial program 97.2%

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

   2. Taylor expanded in x around inf 98.8%

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

      1. mul-1-neg 98.8%

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

      2. log-rec 98.8%

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

      3. mul-1-neg 98.8%

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

      4. distribute-neg-frac 98.8%

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

      5. mul-1-neg 98.8%

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

      6. remove-double-neg 98.8%

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

      7. *-commutative 98.8%

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

   4. Simplified 98.8%

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

      1. div-inv 98.8%

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

      2. associate-/r* 98.8%

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

      3. div-inv 98.8%

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

      4. exp-to-pow 98.8%

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

   6. Applied egg-rr 98.8%

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

   if -5.0000000000000004e-19 < (/.f64 1 n) < -9.99999999999999978e-138 or -1.9999999999999999e-154 < (/.f64 1 n) < 2.00000000000000001e-150

   1. Initial program 33.0%

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

   2. Taylor expanded in n around inf 91.1%

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

      1. log1p-def 91.1%

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

   4. Simplified 91.1%

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

   if -9.99999999999999978e-138 < (/.f64 1 n) < -1.9999999999999999e-154

   1. Initial program 22.0%

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

   2. Taylor expanded in n around inf 39.7%

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

      1. log1p-def 39.7%

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

   4. Simplified 39.7%

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

   5. Taylor expanded in x around inf 84.4%

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

      1. associate-*r/ 84.4%

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

      2. metadata-eval 84.4%

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

   7. Simplified 84.4%

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

   if 2.00000000000000001e-150 < (/.f64 1 n) < 1e5

   1. Initial program 18.2%

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

   2. Taylor expanded in x around inf 76.7%

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

      1. mul-1-neg 76.7%

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

      2. log-rec 76.7%

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

      3. mul-1-neg 76.7%

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

      4. distribute-neg-frac 76.7%

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

      5. mul-1-neg 76.7%

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

      6. remove-double-neg 76.7%

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

      7. *-commutative 76.7%

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

   4. Simplified 76.7%

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

      1. *-un-lft-identity 76.7%

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

      2. times-frac 76.7%

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

      3. div-inv 76.7%

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

      4. exp-to-pow 76.7%

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

   6. Applied egg-rr 76.7%

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

      1. associate-*l/ 76.9%

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

      2. *-un-lft-identity 76.9%

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

   8. Applied egg-rr 76.9%

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

   if 1e5 < (/.f64 1 n) < 9.99999999999999989e203

   1. Initial program 84.4%

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

   2. Taylor expanded in x around 0 86.3%

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

   if 9.99999999999999989e203 < (/.f64 1 n)

   1. Initial program 3.1%

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

   2. Taylor expanded in n around inf 7.4%

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

      1. log1p-def 7.4%

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

   4. Simplified 7.4%

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

   5. Taylor expanded in x around inf 11.1%

      \[\leadsto \color{blue}{\left(0.3333333333333333 \cdot \frac{1}{n \cdot {x}^{3}} + \frac{1}{n \cdot x}\right) - 0.5 \cdot \frac{1}{n \cdot {x}^{2}}} \]
   6. Step-by-step derivation

      1. associate--l+ 11.1%

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

      2. associate-*r/ 11.1%

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

      3. metadata-eval 11.1%

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

      4. *-commutative 11.1%

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

      5. *-commutative 11.1%

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

      6. associate-*r/ 11.1%

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

      7. metadata-eval 11.1%

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

      8. *-commutative 11.1%

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

   7. Simplified 11.1%

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

   8. Taylor expanded in x around 0 100.0%

      \[\leadsto \color{blue}{\frac{0.3333333333333333}{n \cdot {x}^{3}}} \]
3. Recombined 6 regimes into one program.

4. Final simplification 91.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{-19}:\\ \;\;\;\;{x}^{\left(\frac{1}{n}\right)} \cdot \frac{\frac{1}{x}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq -1 \cdot 10^{-137}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right) - \log x}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq -2 \cdot 10^{-154}:\\ \;\;\;\;\frac{\frac{1}{x} - \frac{0.5}{{x}^{2}}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-150}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right) - \log x}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 100000:\\ \;\;\;\;\frac{\frac{{x}^{\left(\frac{1}{n}\right)}}{n}}{x}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{+204}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.3333333333333333}{n \cdot {x}^{3}}\\ \end{array} \]

Alternative 5: 80.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{log1p}\left(x\right) - \log x\\ t_1 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{-19}:\\ \;\;\;\;t_1 \cdot \frac{\frac{1}{x}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq -1 \cdot 10^{-137}:\\ \;\;\;\;\frac{1}{n} \cdot t_0\\ \mathbf{elif}\;\frac{1}{n} \leq -2 \cdot 10^{-154}:\\ \;\;\;\;\frac{\frac{1}{x} - \frac{0.5}{{x}^{2}}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-150}:\\ \;\;\;\;\frac{t_0}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 100000:\\ \;\;\;\;\frac{\frac{t_1}{n}}{x}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{+204}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{0.3333333333333333}{n \cdot {x}^{3}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (- (log1p x) (log x))) (t_1 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -5e-19)
     (* t_1 (/ (/ 1.0 x) n))
     (if (<= (/ 1.0 n) -1e-137)
       (* (/ 1.0 n) t_0)
       (if (<= (/ 1.0 n) -2e-154)
         (/ (- (/ 1.0 x) (/ 0.5 (pow x 2.0))) n)
         (if (<= (/ 1.0 n) 2e-150)
           (/ t_0 n)
           (if (<= (/ 1.0 n) 100000.0)
             (/ (/ t_1 n) x)
             (if (<= (/ 1.0 n) 1e+204)
               (- (+ 1.0 (/ x n)) t_1)
               (/ 0.3333333333333333 (* n (pow x 3.0)))))))))))

C

double code(double x, double n) {
	double t_0 = log1p(x) - log(x);
	double t_1 = pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -5e-19) {
		tmp = t_1 * ((1.0 / x) / n);
	} else if ((1.0 / n) <= -1e-137) {
		tmp = (1.0 / n) * t_0;
	} else if ((1.0 / n) <= -2e-154) {
		tmp = ((1.0 / x) - (0.5 / pow(x, 2.0))) / n;
	} else if ((1.0 / n) <= 2e-150) {
		tmp = t_0 / n;
	} else if ((1.0 / n) <= 100000.0) {
		tmp = (t_1 / n) / x;
	} else if ((1.0 / n) <= 1e+204) {
		tmp = (1.0 + (x / n)) - t_1;
	} else {
		tmp = 0.3333333333333333 / (n * pow(x, 3.0));
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double t_0 = Math.log1p(x) - Math.log(x);
	double t_1 = Math.pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -5e-19) {
		tmp = t_1 * ((1.0 / x) / n);
	} else if ((1.0 / n) <= -1e-137) {
		tmp = (1.0 / n) * t_0;
	} else if ((1.0 / n) <= -2e-154) {
		tmp = ((1.0 / x) - (0.5 / Math.pow(x, 2.0))) / n;
	} else if ((1.0 / n) <= 2e-150) {
		tmp = t_0 / n;
	} else if ((1.0 / n) <= 100000.0) {
		tmp = (t_1 / n) / x;
	} else if ((1.0 / n) <= 1e+204) {
		tmp = (1.0 + (x / n)) - t_1;
	} else {
		tmp = 0.3333333333333333 / (n * Math.pow(x, 3.0));
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.log1p(x) - math.log(x)
	t_1 = math.pow(x, (1.0 / n))
	tmp = 0
	if (1.0 / n) <= -5e-19:
		tmp = t_1 * ((1.0 / x) / n)
	elif (1.0 / n) <= -1e-137:
		tmp = (1.0 / n) * t_0
	elif (1.0 / n) <= -2e-154:
		tmp = ((1.0 / x) - (0.5 / math.pow(x, 2.0))) / n
	elif (1.0 / n) <= 2e-150:
		tmp = t_0 / n
	elif (1.0 / n) <= 100000.0:
		tmp = (t_1 / n) / x
	elif (1.0 / n) <= 1e+204:
		tmp = (1.0 + (x / n)) - t_1
	else:
		tmp = 0.3333333333333333 / (n * math.pow(x, 3.0))
	return tmp

Julia

function code(x, n)
	t_0 = Float64(log1p(x) - log(x))
	t_1 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -5e-19)
		tmp = Float64(t_1 * Float64(Float64(1.0 / x) / n));
	elseif (Float64(1.0 / n) <= -1e-137)
		tmp = Float64(Float64(1.0 / n) * t_0);
	elseif (Float64(1.0 / n) <= -2e-154)
		tmp = Float64(Float64(Float64(1.0 / x) - Float64(0.5 / (x ^ 2.0))) / n);
	elseif (Float64(1.0 / n) <= 2e-150)
		tmp = Float64(t_0 / n);
	elseif (Float64(1.0 / n) <= 100000.0)
		tmp = Float64(Float64(t_1 / n) / x);
	elseif (Float64(1.0 / n) <= 1e+204)
		tmp = Float64(Float64(1.0 + Float64(x / n)) - t_1);
	else
		tmp = Float64(0.3333333333333333 / Float64(n * (x ^ 3.0)));
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(N[Log[1 + x], $MachinePrecision] - N[Log[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -5e-19], N[(t$95$1 * N[(N[(1.0 / x), $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], -1e-137], N[(N[(1.0 / n), $MachinePrecision] * t$95$0), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], -2e-154], N[(N[(N[(1.0 / x), $MachinePrecision] - N[(0.5 / N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-150], N[(t$95$0 / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 100000.0], N[(N[(t$95$1 / n), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 1e+204], N[(N[(1.0 + N[(x / n), $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision], N[(0.3333333333333333 / N[(n * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 7 regimes

2. if (/.f64 1 n) < -5.0000000000000004e-19

   1. Initial program 97.2%

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

   2. Taylor expanded in x around inf 98.8%

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

      1. mul-1-neg 98.8%

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

      2. log-rec 98.8%

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

      3. mul-1-neg 98.8%

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

      4. distribute-neg-frac 98.8%

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

      5. mul-1-neg 98.8%

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

      6. remove-double-neg 98.8%

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

      7. *-commutative 98.8%

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

   4. Simplified 98.8%

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

      1. div-inv 98.8%

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

      2. associate-/r* 98.8%

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

      3. div-inv 98.8%

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

      4. exp-to-pow 98.8%

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

   6. Applied egg-rr 98.8%

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

   if -5.0000000000000004e-19 < (/.f64 1 n) < -9.99999999999999978e-138

   1. Initial program 14.1%

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

   2. Taylor expanded in n around inf 76.8%

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

      1. log1p-def 76.8%

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

   4. Simplified 76.8%

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

      1. div-sub 76.8%

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

   6. Applied egg-rr 76.8%

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

      1. div-inv 73.9%

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

      2. div-inv 76.8%

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

      3. distribute-rgt-out-- 76.8%

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

   8. Applied egg-rr 76.8%

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

   if -9.99999999999999978e-138 < (/.f64 1 n) < -1.9999999999999999e-154

   1. Initial program 22.0%

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

   2. Taylor expanded in n around inf 39.7%

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

      1. log1p-def 39.7%

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

   4. Simplified 39.7%

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

   5. Taylor expanded in x around inf 84.4%

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

      1. associate-*r/ 84.4%

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

      2. metadata-eval 84.4%

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

   7. Simplified 84.4%

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

   if -1.9999999999999999e-154 < (/.f64 1 n) < 2.00000000000000001e-150

   1. Initial program 40.0%

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

   2. Taylor expanded in n around inf 96.5%

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

      1. log1p-def 96.5%

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

   4. Simplified 96.5%

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

   if 2.00000000000000001e-150 < (/.f64 1 n) < 1e5

   1. Initial program 18.2%

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

   2. Taylor expanded in x around inf 76.7%

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

      1. mul-1-neg 76.7%

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

      2. log-rec 76.7%

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

      3. mul-1-neg 76.7%

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

      4. distribute-neg-frac 76.7%

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

      5. mul-1-neg 76.7%

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

      6. remove-double-neg 76.7%

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

      7. *-commutative 76.7%

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

   4. Simplified 76.7%

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

      1. *-un-lft-identity 76.7%

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

      2. times-frac 76.7%

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

      3. div-inv 76.7%

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

      4. exp-to-pow 76.7%

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

   6. Applied egg-rr 76.7%

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

      1. associate-*l/ 76.9%

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

      2. *-un-lft-identity 76.9%

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

   8. Applied egg-rr 76.9%

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

   if 1e5 < (/.f64 1 n) < 9.99999999999999989e203

   1. Initial program 84.4%

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

   2. Taylor expanded in x around 0 86.3%

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

   if 9.99999999999999989e203 < (/.f64 1 n)

   1. Initial program 3.1%

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

   2. Taylor expanded in n around inf 7.4%

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

      1. log1p-def 7.4%

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

   4. Simplified 7.4%

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

   5. Taylor expanded in x around inf 11.1%

      \[\leadsto \color{blue}{\left(0.3333333333333333 \cdot \frac{1}{n \cdot {x}^{3}} + \frac{1}{n \cdot x}\right) - 0.5 \cdot \frac{1}{n \cdot {x}^{2}}} \]
   6. Step-by-step derivation

      1. associate--l+ 11.1%

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

      2. associate-*r/ 11.1%

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

      3. metadata-eval 11.1%

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

      4. *-commutative 11.1%

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

      5. *-commutative 11.1%

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

      6. associate-*r/ 11.1%

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

      7. metadata-eval 11.1%

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

      8. *-commutative 11.1%

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

   7. Simplified 11.1%

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

   8. Taylor expanded in x around 0 100.0%

      \[\leadsto \color{blue}{\frac{0.3333333333333333}{n \cdot {x}^{3}}} \]
3. Recombined 7 regimes into one program.

4. Final simplification 91.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{-19}:\\ \;\;\;\;{x}^{\left(\frac{1}{n}\right)} \cdot \frac{\frac{1}{x}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq -1 \cdot 10^{-137}:\\ \;\;\;\;\frac{1}{n} \cdot \left(\mathsf{log1p}\left(x\right) - \log x\right)\\ \mathbf{elif}\;\frac{1}{n} \leq -2 \cdot 10^{-154}:\\ \;\;\;\;\frac{\frac{1}{x} - \frac{0.5}{{x}^{2}}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-150}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right) - \log x}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 100000:\\ \;\;\;\;\frac{\frac{{x}^{\left(\frac{1}{n}\right)}}{n}}{x}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{+204}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.3333333333333333}{n \cdot {x}^{3}}\\ \end{array} \]

Alternative 6: 60.2% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\frac{1}{x}}{n}\\ t_1 := \frac{-\log x}{n}\\ \mathbf{if}\;n \leq -5.8 \cdot 10^{+139}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;n \leq -18:\\ \;\;\;\;t_1\\ \mathbf{elif}\;n \leq 1.2 \cdot 10^{-204}:\\ \;\;\;\;\frac{0.3333333333333333}{n \cdot {x}^{3}}\\ \mathbf{elif}\;n \leq 6800000000:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;n \leq 4.6 \cdot 10^{+149}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (/ (/ 1.0 x) n)) (t_1 (/ (- (log x)) n)))
   (if (<= n -5.8e+139)
     t_0
     (if (<= n -18.0)
       t_1
       (if (<= n 1.2e-204)
         (/ 0.3333333333333333 (* n (pow x 3.0)))
         (if (<= n 6800000000.0)
           (- 1.0 (pow x (/ 1.0 n)))
           (if (<= n 4.6e+149) t_0 t_1)))))))

C

double code(double x, double n) {
	double t_0 = (1.0 / x) / n;
	double t_1 = -log(x) / n;
	double tmp;
	if (n <= -5.8e+139) {
		tmp = t_0;
	} else if (n <= -18.0) {
		tmp = t_1;
	} else if (n <= 1.2e-204) {
		tmp = 0.3333333333333333 / (n * pow(x, 3.0));
	} else if (n <= 6800000000.0) {
		tmp = 1.0 - pow(x, (1.0 / n));
	} else if (n <= 4.6e+149) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	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) / n
    t_1 = -log(x) / n
    if (n <= (-5.8d+139)) then
        tmp = t_0
    else if (n <= (-18.0d0)) then
        tmp = t_1
    else if (n <= 1.2d-204) then
        tmp = 0.3333333333333333d0 / (n * (x ** 3.0d0))
    else if (n <= 6800000000.0d0) then
        tmp = 1.0d0 - (x ** (1.0d0 / n))
    else if (n <= 4.6d+149) then
        tmp = t_0
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = (1.0 / x) / n;
	double t_1 = -Math.log(x) / n;
	double tmp;
	if (n <= -5.8e+139) {
		tmp = t_0;
	} else if (n <= -18.0) {
		tmp = t_1;
	} else if (n <= 1.2e-204) {
		tmp = 0.3333333333333333 / (n * Math.pow(x, 3.0));
	} else if (n <= 6800000000.0) {
		tmp = 1.0 - Math.pow(x, (1.0 / n));
	} else if (n <= 4.6e+149) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = (1.0 / x) / n
	t_1 = -math.log(x) / n
	tmp = 0
	if n <= -5.8e+139:
		tmp = t_0
	elif n <= -18.0:
		tmp = t_1
	elif n <= 1.2e-204:
		tmp = 0.3333333333333333 / (n * math.pow(x, 3.0))
	elif n <= 6800000000.0:
		tmp = 1.0 - math.pow(x, (1.0 / n))
	elif n <= 4.6e+149:
		tmp = t_0
	else:
		tmp = t_1
	return tmp

Julia

function code(x, n)
	t_0 = Float64(Float64(1.0 / x) / n)
	t_1 = Float64(Float64(-log(x)) / n)
	tmp = 0.0
	if (n <= -5.8e+139)
		tmp = t_0;
	elseif (n <= -18.0)
		tmp = t_1;
	elseif (n <= 1.2e-204)
		tmp = Float64(0.3333333333333333 / Float64(n * (x ^ 3.0)));
	elseif (n <= 6800000000.0)
		tmp = Float64(1.0 - (x ^ Float64(1.0 / n)));
	elseif (n <= 4.6e+149)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = (1.0 / x) / n;
	t_1 = -log(x) / n;
	tmp = 0.0;
	if (n <= -5.8e+139)
		tmp = t_0;
	elseif (n <= -18.0)
		tmp = t_1;
	elseif (n <= 1.2e-204)
		tmp = 0.3333333333333333 / (n * (x ^ 3.0));
	elseif (n <= 6800000000.0)
		tmp = 1.0 - (x ^ (1.0 / n));
	elseif (n <= 4.6e+149)
		tmp = t_0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(N[(1.0 / x), $MachinePrecision] / n), $MachinePrecision]}, Block[{t$95$1 = N[((-N[Log[x], $MachinePrecision]) / n), $MachinePrecision]}, If[LessEqual[n, -5.8e+139], t$95$0, If[LessEqual[n, -18.0], t$95$1, If[LessEqual[n, 1.2e-204], N[(0.3333333333333333 / N[(n * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 6800000000.0], N[(1.0 - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 4.6e+149], t$95$0, t$95$1]]]]]]]

Derivation

1. Split input into 4 regimes

2. if n < -5.7999999999999998e139 or 6.8e9 < n < 4.5999999999999997e149

   1. Initial program 33.0%

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

   2. Taylor expanded in n around inf 68.9%

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

      1. log1p-def 68.9%

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

   4. Simplified 68.9%

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

   5. Taylor expanded in x around inf 63.3%

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

   if -5.7999999999999998e139 < n < -18 or 4.5999999999999997e149 < n

   1. Initial program 24.6%

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

   2. Taylor expanded in n around inf 84.0%

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

      1. log1p-def 84.0%

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

   4. Simplified 84.0%

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

   5. Taylor expanded in x around 0 63.7%

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

      1. mul-1-neg 63.7%

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

   7. Simplified 63.7%

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

   if -18 < n < 1.2e-204

   1. Initial program 89.9%

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

   2. Taylor expanded in n around inf 48.6%

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

      1. log1p-def 48.6%

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

   4. Simplified 48.6%

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

   5. Taylor expanded in x around inf 6.7%

      \[\leadsto \color{blue}{\left(0.3333333333333333 \cdot \frac{1}{n \cdot {x}^{3}} + \frac{1}{n \cdot x}\right) - 0.5 \cdot \frac{1}{n \cdot {x}^{2}}} \]
   6. Step-by-step derivation

      1. associate--l+ 6.7%

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

      2. associate-*r/ 6.7%

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

      3. metadata-eval 6.7%

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

      4. *-commutative 6.7%

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

      5. *-commutative 6.7%

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

      6. associate-*r/ 6.7%

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

      7. metadata-eval 6.7%

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

      8. *-commutative 6.7%

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

   7. Simplified 6.7%

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

   8. Taylor expanded in x around 0 72.2%

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

   if 1.2e-204 < n < 6.8e9

   1. Initial program 76.0%

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

   2. Taylor expanded in x around 0 75.9%

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

4. Final simplification 67.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -5.8 \cdot 10^{+139}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \mathbf{elif}\;n \leq -18:\\ \;\;\;\;\frac{-\log x}{n}\\ \mathbf{elif}\;n \leq 1.2 \cdot 10^{-204}:\\ \;\;\;\;\frac{0.3333333333333333}{n \cdot {x}^{3}}\\ \mathbf{elif}\;n \leq 6800000000:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;n \leq 4.6 \cdot 10^{+149}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\log x}{n}\\ \end{array} \]

Alternative 7: 60.4% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\frac{1}{x}}{n}\\ \mathbf{if}\;n \leq -1.02 \cdot 10^{+139}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;n \leq -7.1:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{elif}\;n \leq 1.35 \cdot 10^{-204}:\\ \;\;\;\;\frac{0.3333333333333333}{n \cdot {x}^{3}}\\ \mathbf{elif}\;n \leq 1300000000:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;n \leq 6.8 \cdot 10^{+149}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{-\log x}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (/ (/ 1.0 x) n)))
   (if (<= n -1.02e+139)
     t_0
     (if (<= n -7.1)
       (/ (- x (log x)) n)
       (if (<= n 1.35e-204)
         (/ 0.3333333333333333 (* n (pow x 3.0)))
         (if (<= n 1300000000.0)
           (- 1.0 (pow x (/ 1.0 n)))
           (if (<= n 6.8e+149) t_0 (/ (- (log x)) n))))))))

C

double code(double x, double n) {
	double t_0 = (1.0 / x) / n;
	double tmp;
	if (n <= -1.02e+139) {
		tmp = t_0;
	} else if (n <= -7.1) {
		tmp = (x - log(x)) / n;
	} else if (n <= 1.35e-204) {
		tmp = 0.3333333333333333 / (n * pow(x, 3.0));
	} else if (n <= 1300000000.0) {
		tmp = 1.0 - pow(x, (1.0 / n));
	} else if (n <= 6.8e+149) {
		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) :: tmp
    t_0 = (1.0d0 / x) / n
    if (n <= (-1.02d+139)) then
        tmp = t_0
    else if (n <= (-7.1d0)) then
        tmp = (x - log(x)) / n
    else if (n <= 1.35d-204) then
        tmp = 0.3333333333333333d0 / (n * (x ** 3.0d0))
    else if (n <= 1300000000.0d0) then
        tmp = 1.0d0 - (x ** (1.0d0 / n))
    else if (n <= 6.8d+149) 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 / x) / n;
	double tmp;
	if (n <= -1.02e+139) {
		tmp = t_0;
	} else if (n <= -7.1) {
		tmp = (x - Math.log(x)) / n;
	} else if (n <= 1.35e-204) {
		tmp = 0.3333333333333333 / (n * Math.pow(x, 3.0));
	} else if (n <= 1300000000.0) {
		tmp = 1.0 - Math.pow(x, (1.0 / n));
	} else if (n <= 6.8e+149) {
		tmp = t_0;
	} else {
		tmp = -Math.log(x) / n;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = (1.0 / x) / n
	tmp = 0
	if n <= -1.02e+139:
		tmp = t_0
	elif n <= -7.1:
		tmp = (x - math.log(x)) / n
	elif n <= 1.35e-204:
		tmp = 0.3333333333333333 / (n * math.pow(x, 3.0))
	elif n <= 1300000000.0:
		tmp = 1.0 - math.pow(x, (1.0 / n))
	elif n <= 6.8e+149:
		tmp = t_0
	else:
		tmp = -math.log(x) / n
	return tmp

Julia

function code(x, n)
	t_0 = Float64(Float64(1.0 / x) / n)
	tmp = 0.0
	if (n <= -1.02e+139)
		tmp = t_0;
	elseif (n <= -7.1)
		tmp = Float64(Float64(x - log(x)) / n);
	elseif (n <= 1.35e-204)
		tmp = Float64(0.3333333333333333 / Float64(n * (x ^ 3.0)));
	elseif (n <= 1300000000.0)
		tmp = Float64(1.0 - (x ^ Float64(1.0 / n)));
	elseif (n <= 6.8e+149)
		tmp = t_0;
	else
		tmp = Float64(Float64(-log(x)) / n);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = (1.0 / x) / n;
	tmp = 0.0;
	if (n <= -1.02e+139)
		tmp = t_0;
	elseif (n <= -7.1)
		tmp = (x - log(x)) / n;
	elseif (n <= 1.35e-204)
		tmp = 0.3333333333333333 / (n * (x ^ 3.0));
	elseif (n <= 1300000000.0)
		tmp = 1.0 - (x ^ (1.0 / n));
	elseif (n <= 6.8e+149)
		tmp = t_0;
	else
		tmp = -log(x) / n;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(N[(1.0 / x), $MachinePrecision] / n), $MachinePrecision]}, If[LessEqual[n, -1.02e+139], t$95$0, If[LessEqual[n, -7.1], N[(N[(x - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[n, 1.35e-204], N[(0.3333333333333333 / N[(n * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 1300000000.0], N[(1.0 - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 6.8e+149], t$95$0, N[((-N[Log[x], $MachinePrecision]) / n), $MachinePrecision]]]]]]]

Derivation

1. Split input into 5 regimes

2. if n < -1.02e139 or 1.3e9 < n < 6.7999999999999997e149

   1. Initial program 33.0%

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

   2. Taylor expanded in n around inf 68.9%

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

      1. log1p-def 68.9%

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

   4. Simplified 68.9%

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

   5. Taylor expanded in x around inf 63.3%

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

   if -1.02e139 < n < -7.0999999999999996

   1. Initial program 15.3%

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

   2. Taylor expanded in n around inf 71.4%

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

      1. log1p-def 71.4%

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

   4. Simplified 71.4%

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

   5. Taylor expanded in x around 0 62.2%

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

      1. mul-1-neg 62.2%

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

   7. Simplified 62.2%

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

   if -7.0999999999999996 < n < 1.34999999999999996e-204

   1. Initial program 89.9%

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

   2. Taylor expanded in n around inf 48.6%

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

      1. log1p-def 48.6%

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

   4. Simplified 48.6%

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

   5. Taylor expanded in x around inf 6.7%

      \[\leadsto \color{blue}{\left(0.3333333333333333 \cdot \frac{1}{n \cdot {x}^{3}} + \frac{1}{n \cdot x}\right) - 0.5 \cdot \frac{1}{n \cdot {x}^{2}}} \]
   6. Step-by-step derivation

      1. associate--l+ 6.7%

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

      2. associate-*r/ 6.7%

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

      3. metadata-eval 6.7%

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

      4. *-commutative 6.7%

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

      5. *-commutative 6.7%

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

      6. associate-*r/ 6.7%

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

      7. metadata-eval 6.7%

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

      8. *-commutative 6.7%

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

   7. Simplified 6.7%

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

   8. Taylor expanded in x around 0 72.2%

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

   if 1.34999999999999996e-204 < n < 1.3e9

   1. Initial program 76.0%

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

   2. Taylor expanded in x around 0 75.9%

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

   if 6.7999999999999997e149 < n

   1. Initial program 32.1%

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

   2. Taylor expanded in n around inf 94.1%

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

      1. log1p-def 94.1%

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

   4. Simplified 94.1%

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

   5. Taylor expanded in x around 0 65.4%

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

      1. mul-1-neg 65.4%

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

   7. Simplified 65.4%

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

4. Final simplification 67.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -1.02 \cdot 10^{+139}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \mathbf{elif}\;n \leq -7.1:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{elif}\;n \leq 1.35 \cdot 10^{-204}:\\ \;\;\;\;\frac{0.3333333333333333}{n \cdot {x}^{3}}\\ \mathbf{elif}\;n \leq 1300000000:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;n \leq 6.8 \cdot 10^{+149}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\log x}{n}\\ \end{array} \]

Alternative 8: 60.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.55:\\ \;\;\;\;\frac{-\log x}{n}\\ \mathbf{elif}\;x \leq 6.5 \cdot 10^{+185}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log x}{n} \cdot 0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 0.55)
   (/ (- (log x)) n)
   (if (<= x 6.5e+185) (/ (/ 1.0 x) n) (* (/ (log x) n) 0.0))))

C

double code(double x, double n) {
	double tmp;
	if (x <= 0.55) {
		tmp = -log(x) / n;
	} else if (x <= 6.5e+185) {
		tmp = (1.0 / x) / n;
	} else {
		tmp = (log(x) / n) * 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if (x <= 0.55d0) then
        tmp = -log(x) / n
    else if (x <= 6.5d+185) then
        tmp = (1.0d0 / x) / n
    else
        tmp = (log(x) / n) * 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (x <= 0.55) {
		tmp = -Math.log(x) / n;
	} else if (x <= 6.5e+185) {
		tmp = (1.0 / x) / n;
	} else {
		tmp = (Math.log(x) / n) * 0.0;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 0.55:
		tmp = -math.log(x) / n
	elif x <= 6.5e+185:
		tmp = (1.0 / x) / n
	else:
		tmp = (math.log(x) / n) * 0.0
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 0.55)
		tmp = Float64(Float64(-log(x)) / n);
	elseif (x <= 6.5e+185)
		tmp = Float64(Float64(1.0 / x) / n);
	else
		tmp = Float64(Float64(log(x) / n) * 0.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (x <= 0.55)
		tmp = -log(x) / n;
	elseif (x <= 6.5e+185)
		tmp = (1.0 / x) / n;
	else
		tmp = (log(x) / n) * 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 0.55], N[((-N[Log[x], $MachinePrecision]) / n), $MachinePrecision], If[LessEqual[x, 6.5e+185], N[(N[(1.0 / x), $MachinePrecision] / n), $MachinePrecision], N[(N[(N[Log[x], $MachinePrecision] / n), $MachinePrecision] * 0.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 0.55000000000000004

   1. Initial program 43.8%

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

   2. Taylor expanded in n around inf 53.7%

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

      1. log1p-def 53.7%

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

   4. Simplified 53.7%

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

   5. Taylor expanded in x around 0 52.9%

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

      1. mul-1-neg 52.9%

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

   7. Simplified 52.9%

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

   if 0.55000000000000004 < x < 6.5000000000000002e185

   1. Initial program 53.9%

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

   2. Taylor expanded in n around inf 55.6%

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

      1. log1p-def 55.6%

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

   4. Simplified 55.6%

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

   5. Taylor expanded in x around inf 61.1%

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

   if 6.5000000000000002e185 < x

   1. Initial program 89.1%

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

   2. Taylor expanded in n around inf 89.1%

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

      1. log1p-def 89.1%

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

   4. Simplified 89.1%

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

      1. log1p-def 89.1%

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

      2. div-sub 86.7%

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

      3. add-sqr-sqrt 10.8%

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

      4. cancel-sign-sub-inv 10.8%

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

      5. log1p-def 10.8%

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

   6. Applied egg-rr 10.8%

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

      1. *-commutative 10.8%

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

   8. Simplified 10.8%

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

   9. Taylor expanded in x around inf 86.7%

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

       1. associate-*r/ 86.7%

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

       2. log-rec 86.7%

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

       3. neg-mul-1 86.7%

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

       4. remove-double-neg 86.7%

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

       5. log-rec 86.7%

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

       6. distribute-frac-neg 86.7%

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

       7. mul-1-neg 86.7%

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

       8. *-lft-identity 86.7%

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

       9. distribute-rgt-out 86.7%

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

       10. metadata-eval 86.7%

           \[\leadsto \frac{\log x}{n} \cdot \color{blue}{0} \]

   11. Simplified 86.7%

       \[\leadsto \color{blue}{\frac{\log x}{n} \cdot 0} \]
3. Recombined 3 regimes into one program.

4. Final simplification 60.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.55:\\ \;\;\;\;\frac{-\log x}{n}\\ \mathbf{elif}\;x \leq 6.5 \cdot 10^{+185}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log x}{n} \cdot 0\\ \end{array} \]

Alternative 9: 71.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 5 \cdot 10^{-32}:\\ \;\;\;\;\frac{-\log x}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{{x}^{\left(\frac{1}{n}\right)}}{n}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 5e-32) (/ (- (log x)) n) (/ (/ (pow x (/ 1.0 n)) n) x)))

C

double code(double x, double n) {
	double tmp;
	if (x <= 5e-32) {
		tmp = -log(x) / n;
	} else {
		tmp = (pow(x, (1.0 / n)) / n) / x;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if (x <= 5d-32) then
        tmp = -log(x) / n
    else
        tmp = ((x ** (1.0d0 / n)) / n) / x
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (x <= 5e-32) {
		tmp = -Math.log(x) / n;
	} else {
		tmp = (Math.pow(x, (1.0 / n)) / n) / x;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 5e-32:
		tmp = -math.log(x) / n
	else:
		tmp = (math.pow(x, (1.0 / n)) / n) / x
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 5e-32)
		tmp = Float64(Float64(-log(x)) / n);
	else
		tmp = Float64(Float64((x ^ Float64(1.0 / n)) / n) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (x <= 5e-32)
		tmp = -log(x) / n;
	else
		tmp = ((x ^ (1.0 / n)) / n) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 5e-32], N[((-N[Log[x], $MachinePrecision]) / n), $MachinePrecision], N[(N[(N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 5e-32

   1. Initial program 42.8%

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

   2. Taylor expanded in n around inf 54.8%

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

      1. log1p-def 54.8%

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

   4. Simplified 54.8%

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

   5. Taylor expanded in x around 0 54.8%

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

      1. mul-1-neg 54.8%

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

   7. Simplified 54.8%

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

   if 5e-32 < x

   1. Initial program 66.2%

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

   2. Taylor expanded in x around inf 94.3%

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

      1. mul-1-neg 94.3%

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

      2. log-rec 94.3%

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

      3. mul-1-neg 94.3%

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

      4. distribute-neg-frac 94.3%

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

      5. mul-1-neg 94.3%

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

      6. remove-double-neg 94.3%

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

      7. *-commutative 94.3%

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

   4. Simplified 94.3%

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

      1. *-un-lft-identity 94.3%

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

      2. times-frac 95.4%

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

      3. div-inv 95.4%

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

      4. exp-to-pow 95.4%

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

   6. Applied egg-rr 95.4%

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

      1. associate-*l/ 95.5%

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

      2. *-un-lft-identity 95.5%

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

   8. Applied egg-rr 95.5%

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

4. Final simplification 74.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 5 \cdot 10^{-32}:\\ \;\;\;\;\frac{-\log x}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{{x}^{\left(\frac{1}{n}\right)}}{n}}{x}\\ \end{array} \]

Alternative 10: 56.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.55:\\ \;\;\;\;\frac{-\log x}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 0.55) (/ (- (log x)) n) (/ (/ 1.0 x) n)))

C

double code(double x, double n) {
	double tmp;
	if (x <= 0.55) {
		tmp = -log(x) / n;
	} else {
		tmp = (1.0 / x) / n;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if (x <= 0.55d0) then
        tmp = -log(x) / n
    else
        tmp = (1.0d0 / x) / n
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (x <= 0.55) {
		tmp = -Math.log(x) / n;
	} else {
		tmp = (1.0 / x) / n;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 0.55:
		tmp = -math.log(x) / n
	else:
		tmp = (1.0 / x) / n
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 0.55)
		tmp = Float64(Float64(-log(x)) / n);
	else
		tmp = Float64(Float64(1.0 / x) / n);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (x <= 0.55)
		tmp = -log(x) / n;
	else
		tmp = (1.0 / x) / n;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 0.55], N[((-N[Log[x], $MachinePrecision]) / n), $MachinePrecision], N[(N[(1.0 / x), $MachinePrecision] / n), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.55000000000000004

   1. Initial program 43.8%

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

   2. Taylor expanded in n around inf 53.7%

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

      1. log1p-def 53.7%

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

   4. Simplified 53.7%

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

   5. Taylor expanded in x around 0 52.9%

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

      1. mul-1-neg 52.9%

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

   7. Simplified 52.9%

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

   if 0.55000000000000004 < x

   1. Initial program 66.8%

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

   2. Taylor expanded in n around inf 67.8%

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

      1. log1p-def 67.8%

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

   4. Simplified 67.8%

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

   5. Taylor expanded in x around inf 61.9%

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

4. Final simplification 57.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.55:\\ \;\;\;\;\frac{-\log x}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \end{array} \]

Alternative 11: 39.8% accurate, 42.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 54.1%

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

2. Taylor expanded in n around inf 60.0%

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

   1. log1p-def 60.0%

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

4. Simplified 60.0%

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

5. Taylor expanded in x around inf 40.4%

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

   1. *-commutative 40.4%

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

7. Simplified 40.4%

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

8. Final simplification 40.4%

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

Alternative 12: 40.3% accurate, 42.2× speedup

Math

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

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(Float64(1.0 / x) / n)
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 54.1%

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

2. Taylor expanded in n around inf 60.0%

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

   1. log1p-def 60.0%

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

4. Simplified 60.0%

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

5. Taylor expanded in x around inf 40.9%

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

6. Final simplification 40.9%

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

Reproduce

herbie shell --seed 2023336 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  :precision binary64
  (- (pow (+ x 1.0) (/ 1.0 n)) (pow x (/ 1.0 n))))