2nthrt (problem 3.4.6)

Percentage Accurate: 53.6% → 84.5%

Time: 1.1min

Alternatives: 13

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: 53.6% 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, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1500000:\\ \;\;\;\;\frac{\log \left(\frac{e^{\mathsf{log1p}\left(x\right) + 0.5 \cdot \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}}{n}}}{x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x} \cdot \frac{{x}^{\left(\frac{1}{n}\right)}}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 1500000.0)
   (/
    (log
     (/
      (exp
       (+ (log1p x) (* 0.5 (/ (- (pow (log1p x) 2.0) (pow (log x) 2.0)) n))))
      x))
    n)
   (* (/ 1.0 x) (/ (pow x (/ 1.0 n)) n))))

C

double code(double x, double n) {
	double tmp;
	if (x <= 1500000.0) {
		tmp = log((exp((log1p(x) + (0.5 * ((pow(log1p(x), 2.0) - pow(log(x), 2.0)) / n)))) / x)) / n;
	} else {
		tmp = (1.0 / x) * (pow(x, (1.0 / n)) / n);
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double tmp;
	if (x <= 1500000.0) {
		tmp = Math.log((Math.exp((Math.log1p(x) + (0.5 * ((Math.pow(Math.log1p(x), 2.0) - Math.pow(Math.log(x), 2.0)) / n)))) / x)) / n;
	} else {
		tmp = (1.0 / x) * (Math.pow(x, (1.0 / n)) / n);
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 1500000.0:
		tmp = math.log((math.exp((math.log1p(x) + (0.5 * ((math.pow(math.log1p(x), 2.0) - math.pow(math.log(x), 2.0)) / n)))) / x)) / n
	else:
		tmp = (1.0 / x) * (math.pow(x, (1.0 / n)) / n)
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 1500000.0)
		tmp = Float64(log(Float64(exp(Float64(log1p(x) + Float64(0.5 * Float64(Float64((log1p(x) ^ 2.0) - (log(x) ^ 2.0)) / n)))) / x)) / n);
	else
		tmp = Float64(Float64(1.0 / x) * Float64((x ^ Float64(1.0 / n)) / n));
	end
	return tmp
end

Wolfram

code[x_, n_] := If[LessEqual[x, 1500000.0], N[(N[Log[N[(N[Exp[N[(N[Log[1 + x], $MachinePrecision] + N[(0.5 * N[(N[(N[Power[N[Log[1 + x], $MachinePrecision], 2.0], $MachinePrecision] - N[Power[N[Log[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], N[(N[(1.0 / x), $MachinePrecision] * N[(N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.5e6

   1. Initial program 35.3%

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

   3. Taylor expanded in n around inf 69.5%

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

   5. Simplified 69.5%

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

      1. associate-+r- 69.5%

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

      2. add-log-exp 79.7%

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

      3. exp-diff 79.7%

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

      4. add-exp-log 79.8%

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

   7. Applied egg-rr 79.8%

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

      1. +-commutative 79.8%

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

   9. Simplified 79.8%

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

   if 1.5e6 < x

   1. Initial program 61.9%

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

   3. Taylor expanded in x around inf 97.7%

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

      1. mul-1-neg 97.7%

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

      2. log-rec 97.7%

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

      3. mul-1-neg 97.7%

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

      4. distribute-neg-frac 97.7%

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

      5. mul-1-neg 97.7%

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

      6. remove-double-neg 97.7%

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

      7. *-commutative 97.7%

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

   5. Simplified 97.7%

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

      1. div-inv 97.7%

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

      2. pow-to-exp 97.7%

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

      3. *-un-lft-identity 97.7%

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

      4. times-frac 99.2%

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

   7. Applied egg-rr 99.2%

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

4. Final simplification 87.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1500000:\\ \;\;\;\;\frac{\log \left(\frac{e^{\mathsf{log1p}\left(x\right) + 0.5 \cdot \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}}{n}}}{x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x} \cdot \frac{{x}^{\left(\frac{1}{n}\right)}}{n}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 85.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{-147}:\\ \;\;\;\;\frac{1}{x} \cdot \frac{t\_0}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-6}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right) + \left(0.5 \cdot \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}}{n} - \log x\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;e^{\log \left(e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - t\_0\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -1e-147)
     (* (/ 1.0 x) (/ t_0 n))
     (if (<= (/ 1.0 n) 2e-6)
       (/
        (+
         (log1p x)
         (- (* 0.5 (/ (- (pow (log1p x) 2.0) (pow (log x) 2.0)) n)) (log x)))
        n)
       (exp (log (- (exp (/ (log1p x) n)) t_0)))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -1e-147) {
		tmp = (1.0 / x) * (t_0 / n);
	} else if ((1.0 / n) <= 2e-6) {
		tmp = (log1p(x) + ((0.5 * ((pow(log1p(x), 2.0) - pow(log(x), 2.0)) / n)) - log(x))) / n;
	} else {
		tmp = exp(log((exp((log1p(x) / n)) - t_0)));
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -1e-147) {
		tmp = (1.0 / x) * (t_0 / n);
	} else if ((1.0 / n) <= 2e-6) {
		tmp = (Math.log1p(x) + ((0.5 * ((Math.pow(Math.log1p(x), 2.0) - Math.pow(Math.log(x), 2.0)) / n)) - Math.log(x))) / n;
	} else {
		tmp = Math.exp(Math.log((Math.exp((Math.log1p(x) / n)) - t_0)));
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	tmp = 0
	if (1.0 / n) <= -1e-147:
		tmp = (1.0 / x) * (t_0 / n)
	elif (1.0 / n) <= 2e-6:
		tmp = (math.log1p(x) + ((0.5 * ((math.pow(math.log1p(x), 2.0) - math.pow(math.log(x), 2.0)) / n)) - math.log(x))) / n
	else:
		tmp = math.exp(math.log((math.exp((math.log1p(x) / n)) - t_0)))
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -1e-147)
		tmp = Float64(Float64(1.0 / x) * Float64(t_0 / n));
	elseif (Float64(1.0 / n) <= 2e-6)
		tmp = Float64(Float64(log1p(x) + Float64(Float64(0.5 * Float64(Float64((log1p(x) ^ 2.0) - (log(x) ^ 2.0)) / n)) - log(x))) / n);
	else
		tmp = exp(log(Float64(exp(Float64(log1p(x) / n)) - t_0)));
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -1e-147], N[(N[(1.0 / x), $MachinePrecision] * N[(t$95$0 / n), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-6], N[(N[(N[Log[1 + x], $MachinePrecision] + N[(N[(0.5 * N[(N[(N[Power[N[Log[1 + x], $MachinePrecision], 2.0], $MachinePrecision] - N[Power[N[Log[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision] - N[Log[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], N[Exp[N[Log[N[(N[Exp[N[(N[Log[1 + x], $MachinePrecision] / n), $MachinePrecision]], $MachinePrecision] - t$95$0), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -9.9999999999999997e-148

   1. Initial program 65.4%

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

   3. Taylor expanded in x around inf 84.6%

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

      1. mul-1-neg 84.6%

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

      2. log-rec 84.6%

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

      3. mul-1-neg 84.6%

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

      4. distribute-neg-frac 84.6%

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

      5. mul-1-neg 84.6%

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

      6. remove-double-neg 84.6%

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

      7. *-commutative 84.6%

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

   5. Simplified 84.6%

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

      1. div-inv 84.6%

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

      2. pow-to-exp 84.6%

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

      3. *-un-lft-identity 84.6%

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

      4. times-frac 85.2%

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

   7. Applied egg-rr 85.2%

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

   if -9.9999999999999997e-148 < (/.f64 #s(literal 1 binary64) n) < 1.99999999999999991e-6

   1. Initial program 27.4%

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

   3. Taylor expanded in n around inf 85.4%

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

   5. Simplified 85.4%

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

   if 1.99999999999999991e-6 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 53.3%

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

      1. add-exp-log 53.3%

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

      2. pow-to-exp 53.3%

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

      3. un-div-inv 53.3%

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

      4. +-commutative 53.3%

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

      5. log1p-define 99.7%

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

   4. Applied egg-rr 99.7%

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

4. Final simplification 87.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{-147}:\\ \;\;\;\;\frac{1}{x} \cdot \frac{{x}^{\left(\frac{1}{n}\right)}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-6}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right) + \left(0.5 \cdot \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}}{n} - \log x\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;e^{\log \left(e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{1}{n}\right)}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 85.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{-147}:\\ \;\;\;\;\frac{1}{x} \cdot \frac{t\_0}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-14}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right) - \log x}{n}\\ \mathbf{else}:\\ \;\;\;\;e^{\log \left(e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - t\_0\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -1e-147)
     (* (/ 1.0 x) (/ t_0 n))
     (if (<= (/ 1.0 n) 5e-14)
       (/ (- (log1p x) (log x)) n)
       (exp (log (- (exp (/ (log1p x) n)) t_0)))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -1e-147) {
		tmp = (1.0 / x) * (t_0 / n);
	} else if ((1.0 / n) <= 5e-14) {
		tmp = (log1p(x) - log(x)) / n;
	} else {
		tmp = exp(log((exp((log1p(x) / n)) - t_0)));
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -1e-147) {
		tmp = (1.0 / x) * (t_0 / n);
	} else if ((1.0 / n) <= 5e-14) {
		tmp = (Math.log1p(x) - Math.log(x)) / n;
	} else {
		tmp = Math.exp(Math.log((Math.exp((Math.log1p(x) / n)) - t_0)));
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	tmp = 0
	if (1.0 / n) <= -1e-147:
		tmp = (1.0 / x) * (t_0 / n)
	elif (1.0 / n) <= 5e-14:
		tmp = (math.log1p(x) - math.log(x)) / n
	else:
		tmp = math.exp(math.log((math.exp((math.log1p(x) / n)) - t_0)))
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -1e-147)
		tmp = Float64(Float64(1.0 / x) * Float64(t_0 / n));
	elseif (Float64(1.0 / n) <= 5e-14)
		tmp = Float64(Float64(log1p(x) - log(x)) / n);
	else
		tmp = exp(log(Float64(exp(Float64(log1p(x) / n)) - t_0)));
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -1e-147], N[(N[(1.0 / x), $MachinePrecision] * N[(t$95$0 / n), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e-14], N[(N[(N[Log[1 + x], $MachinePrecision] - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], N[Exp[N[Log[N[(N[Exp[N[(N[Log[1 + x], $MachinePrecision] / n), $MachinePrecision]], $MachinePrecision] - t$95$0), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -9.9999999999999997e-148

   1. Initial program 65.4%

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

   3. Taylor expanded in x around inf 84.6%

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

      1. mul-1-neg 84.6%

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

      2. log-rec 84.6%

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

      3. mul-1-neg 84.6%

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

      4. distribute-neg-frac 84.6%

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

      5. mul-1-neg 84.6%

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

      6. remove-double-neg 84.6%

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

      7. *-commutative 84.6%

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

   5. Simplified 84.6%

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

      1. div-inv 84.6%

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

      2. pow-to-exp 84.6%

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

      3. *-un-lft-identity 84.6%

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

      4. times-frac 85.2%

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

   7. Applied egg-rr 85.2%

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

   if -9.9999999999999997e-148 < (/.f64 #s(literal 1 binary64) n) < 5.0000000000000002e-14

   1. Initial program 28.0%

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

   3. Taylor expanded in n around inf 87.1%

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

      1. log1p-define 87.1%

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

   5. Simplified 87.1%

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

   if 5.0000000000000002e-14 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 49.2%

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

      1. add-exp-log 49.2%

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

      2. pow-to-exp 49.2%

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

      3. un-div-inv 49.2%

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

      4. +-commutative 49.2%

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

      5. log1p-define 91.5%

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

   4. Applied egg-rr 91.5%

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

4. Final simplification 86.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{-147}:\\ \;\;\;\;\frac{1}{x} \cdot \frac{{x}^{\left(\frac{1}{n}\right)}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-14}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right) - \log x}{n}\\ \mathbf{else}:\\ \;\;\;\;e^{\log \left(e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{1}{n}\right)}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 85.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{-147}:\\ \;\;\;\;\frac{1}{x} \cdot \frac{t\_0}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-14}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right) - \log x}{n}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -1e-147)
     (* (/ 1.0 x) (/ t_0 n))
     (if (<= (/ 1.0 n) 5e-14)
       (/ (- (log1p x) (log x)) n)
       (- (exp (/ (log1p x) n)) t_0)))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -1e-147) {
		tmp = (1.0 / x) * (t_0 / n);
	} else if ((1.0 / n) <= 5e-14) {
		tmp = (log1p(x) - log(x)) / n;
	} else {
		tmp = exp((log1p(x) / n)) - t_0;
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -1e-147) {
		tmp = (1.0 / x) * (t_0 / n);
	} else if ((1.0 / n) <= 5e-14) {
		tmp = (Math.log1p(x) - Math.log(x)) / n;
	} else {
		tmp = Math.exp((Math.log1p(x) / n)) - t_0;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	tmp = 0
	if (1.0 / n) <= -1e-147:
		tmp = (1.0 / x) * (t_0 / n)
	elif (1.0 / n) <= 5e-14:
		tmp = (math.log1p(x) - math.log(x)) / n
	else:
		tmp = math.exp((math.log1p(x) / n)) - t_0
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -1e-147)
		tmp = Float64(Float64(1.0 / x) * Float64(t_0 / n));
	elseif (Float64(1.0 / n) <= 5e-14)
		tmp = Float64(Float64(log1p(x) - log(x)) / n);
	else
		tmp = Float64(exp(Float64(log1p(x) / n)) - t_0);
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -1e-147], N[(N[(1.0 / x), $MachinePrecision] * N[(t$95$0 / n), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e-14], N[(N[(N[Log[1 + x], $MachinePrecision] - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], N[(N[Exp[N[(N[Log[1 + x], $MachinePrecision] / n), $MachinePrecision]], $MachinePrecision] - t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -9.9999999999999997e-148

   1. Initial program 65.4%

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

   3. Taylor expanded in x around inf 84.6%

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

      1. mul-1-neg 84.6%

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

      2. log-rec 84.6%

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

      3. mul-1-neg 84.6%

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

      4. distribute-neg-frac 84.6%

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

      5. mul-1-neg 84.6%

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

      6. remove-double-neg 84.6%

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

      7. *-commutative 84.6%

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

   5. Simplified 84.6%

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

      1. div-inv 84.6%

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

      2. pow-to-exp 84.6%

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

      3. *-un-lft-identity 84.6%

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

      4. times-frac 85.2%

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

   7. Applied egg-rr 85.2%

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

   if -9.9999999999999997e-148 < (/.f64 #s(literal 1 binary64) n) < 5.0000000000000002e-14

   1. Initial program 28.0%

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

   3. Taylor expanded in n around inf 87.1%

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

      1. log1p-define 87.1%

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

   5. Simplified 87.1%

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

   if 5.0000000000000002e-14 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 49.2%

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

   3. Taylor expanded in n around 0 49.2%

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

      1. log1p-define 91.5%

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

   5. Simplified 91.5%

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

4. Final simplification 86.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{-147}:\\ \;\;\;\;\frac{1}{x} \cdot \frac{{x}^{\left(\frac{1}{n}\right)}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-14}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right) - \log x}{n}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 81.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{-147}:\\ \;\;\;\;\frac{1}{x} \cdot \frac{t\_0}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-6}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right) - \log x}{n}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + x \cdot \left(\frac{1}{n} + x \cdot \left(0.5 \cdot \frac{1}{{n}^{2}} + 0.5 \cdot \frac{-1}{n}\right)\right)\right) - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -1e-147)
     (* (/ 1.0 x) (/ t_0 n))
     (if (<= (/ 1.0 n) 2e-6)
       (/ (- (log1p x) (log x)) n)
       (-
        (+
         1.0
         (*
          x
          (+
           (/ 1.0 n)
           (* x (+ (* 0.5 (/ 1.0 (pow n 2.0))) (* 0.5 (/ -1.0 n)))))))
        t_0)))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -1e-147) {
		tmp = (1.0 / x) * (t_0 / n);
	} else if ((1.0 / n) <= 2e-6) {
		tmp = (log1p(x) - log(x)) / n;
	} else {
		tmp = (1.0 + (x * ((1.0 / n) + (x * ((0.5 * (1.0 / pow(n, 2.0))) + (0.5 * (-1.0 / n))))))) - t_0;
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -1e-147) {
		tmp = (1.0 / x) * (t_0 / n);
	} else if ((1.0 / n) <= 2e-6) {
		tmp = (Math.log1p(x) - Math.log(x)) / n;
	} else {
		tmp = (1.0 + (x * ((1.0 / n) + (x * ((0.5 * (1.0 / Math.pow(n, 2.0))) + (0.5 * (-1.0 / n))))))) - t_0;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	tmp = 0
	if (1.0 / n) <= -1e-147:
		tmp = (1.0 / x) * (t_0 / n)
	elif (1.0 / n) <= 2e-6:
		tmp = (math.log1p(x) - math.log(x)) / n
	else:
		tmp = (1.0 + (x * ((1.0 / n) + (x * ((0.5 * (1.0 / math.pow(n, 2.0))) + (0.5 * (-1.0 / n))))))) - t_0
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -1e-147)
		tmp = Float64(Float64(1.0 / x) * Float64(t_0 / n));
	elseif (Float64(1.0 / n) <= 2e-6)
		tmp = Float64(Float64(log1p(x) - log(x)) / n);
	else
		tmp = Float64(Float64(1.0 + Float64(x * Float64(Float64(1.0 / n) + Float64(x * Float64(Float64(0.5 * Float64(1.0 / (n ^ 2.0))) + Float64(0.5 * Float64(-1.0 / n))))))) - t_0);
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -1e-147], N[(N[(1.0 / x), $MachinePrecision] * N[(t$95$0 / n), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-6], N[(N[(N[Log[1 + x], $MachinePrecision] - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], N[(N[(1.0 + N[(x * N[(N[(1.0 / n), $MachinePrecision] + N[(x * N[(N[(0.5 * N[(1.0 / N[Power[n, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(-1.0 / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -9.9999999999999997e-148

   1. Initial program 65.4%

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

   3. Taylor expanded in x around inf 84.6%

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

      1. mul-1-neg 84.6%

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

      2. log-rec 84.6%

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

      3. mul-1-neg 84.6%

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

      4. distribute-neg-frac 84.6%

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

      5. mul-1-neg 84.6%

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

      6. remove-double-neg 84.6%

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

      7. *-commutative 84.6%

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

   5. Simplified 84.6%

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

      1. div-inv 84.6%

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

      2. pow-to-exp 84.6%

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

      3. *-un-lft-identity 84.6%

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

      4. times-frac 85.2%

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

   7. Applied egg-rr 85.2%

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

   if -9.9999999999999997e-148 < (/.f64 #s(literal 1 binary64) n) < 1.99999999999999991e-6

   1. Initial program 27.4%

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

   3. Taylor expanded in n around inf 85.1%

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

      1. log1p-define 85.1%

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

   5. Simplified 85.1%

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

   if 1.99999999999999991e-6 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 53.3%

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

   3. Taylor expanded in x around 0 74.8%

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

4. Final simplification 83.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{-147}:\\ \;\;\;\;\frac{1}{x} \cdot \frac{{x}^{\left(\frac{1}{n}\right)}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-6}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right) - \log x}{n}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + x \cdot \left(\frac{1}{n} + x \cdot \left(0.5 \cdot \frac{1}{{n}^{2}} + 0.5 \cdot \frac{-1}{n}\right)\right)\right) - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 70.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\log x}{-n}\\ t_1 := {x}^{\left(\frac{1}{n}\right)}\\ t_2 := 1 - t\_1\\ \mathbf{if}\;x \leq 4 \cdot 10^{-282}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 1.45 \cdot 10^{-270}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;x \leq 5.5 \cdot 10^{-214}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 1.75 \cdot 10^{-199}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;x \leq 1.2 \cdot 10^{-78}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 1.9 \cdot 10^{-70}:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{\frac{1}{x}}{n}\right)\right)\\ \mathbf{elif}\;x \leq 5.6 \cdot 10^{-12}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x} \cdot \frac{t\_1}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (/ (log x) (- n))) (t_1 (pow x (/ 1.0 n))) (t_2 (- 1.0 t_1)))
   (if (<= x 4e-282)
     t_0
     (if (<= x 1.45e-270)
       t_2
       (if (<= x 5.5e-214)
         t_0
         (if (<= x 1.75e-199)
           t_2
           (if (<= x 1.2e-78)
             t_0
             (if (<= x 1.9e-70)
               (log1p (expm1 (/ (/ 1.0 x) n)))
               (if (<= x 5.6e-12) t_0 (* (/ 1.0 x) (/ t_1 n)))))))))))

C

double code(double x, double n) {
	double t_0 = log(x) / -n;
	double t_1 = pow(x, (1.0 / n));
	double t_2 = 1.0 - t_1;
	double tmp;
	if (x <= 4e-282) {
		tmp = t_0;
	} else if (x <= 1.45e-270) {
		tmp = t_2;
	} else if (x <= 5.5e-214) {
		tmp = t_0;
	} else if (x <= 1.75e-199) {
		tmp = t_2;
	} else if (x <= 1.2e-78) {
		tmp = t_0;
	} else if (x <= 1.9e-70) {
		tmp = log1p(expm1(((1.0 / x) / n)));
	} else if (x <= 5.6e-12) {
		tmp = t_0;
	} else {
		tmp = (1.0 / x) * (t_1 / n);
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double t_0 = Math.log(x) / -n;
	double t_1 = Math.pow(x, (1.0 / n));
	double t_2 = 1.0 - t_1;
	double tmp;
	if (x <= 4e-282) {
		tmp = t_0;
	} else if (x <= 1.45e-270) {
		tmp = t_2;
	} else if (x <= 5.5e-214) {
		tmp = t_0;
	} else if (x <= 1.75e-199) {
		tmp = t_2;
	} else if (x <= 1.2e-78) {
		tmp = t_0;
	} else if (x <= 1.9e-70) {
		tmp = Math.log1p(Math.expm1(((1.0 / x) / n)));
	} else if (x <= 5.6e-12) {
		tmp = t_0;
	} else {
		tmp = (1.0 / x) * (t_1 / n);
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.log(x) / -n
	t_1 = math.pow(x, (1.0 / n))
	t_2 = 1.0 - t_1
	tmp = 0
	if x <= 4e-282:
		tmp = t_0
	elif x <= 1.45e-270:
		tmp = t_2
	elif x <= 5.5e-214:
		tmp = t_0
	elif x <= 1.75e-199:
		tmp = t_2
	elif x <= 1.2e-78:
		tmp = t_0
	elif x <= 1.9e-70:
		tmp = math.log1p(math.expm1(((1.0 / x) / n)))
	elif x <= 5.6e-12:
		tmp = t_0
	else:
		tmp = (1.0 / x) * (t_1 / n)
	return tmp

Julia

function code(x, n)
	t_0 = Float64(log(x) / Float64(-n))
	t_1 = x ^ Float64(1.0 / n)
	t_2 = Float64(1.0 - t_1)
	tmp = 0.0
	if (x <= 4e-282)
		tmp = t_0;
	elseif (x <= 1.45e-270)
		tmp = t_2;
	elseif (x <= 5.5e-214)
		tmp = t_0;
	elseif (x <= 1.75e-199)
		tmp = t_2;
	elseif (x <= 1.2e-78)
		tmp = t_0;
	elseif (x <= 1.9e-70)
		tmp = log1p(expm1(Float64(Float64(1.0 / x) / n)));
	elseif (x <= 5.6e-12)
		tmp = t_0;
	else
		tmp = Float64(Float64(1.0 / x) * Float64(t_1 / n));
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(N[Log[x], $MachinePrecision] / (-n)), $MachinePrecision]}, Block[{t$95$1 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(1.0 - t$95$1), $MachinePrecision]}, If[LessEqual[x, 4e-282], t$95$0, If[LessEqual[x, 1.45e-270], t$95$2, If[LessEqual[x, 5.5e-214], t$95$0, If[LessEqual[x, 1.75e-199], t$95$2, If[LessEqual[x, 1.2e-78], t$95$0, If[LessEqual[x, 1.9e-70], N[Log[1 + N[(Exp[N[(N[(1.0 / x), $MachinePrecision] / n), $MachinePrecision]] - 1), $MachinePrecision]], $MachinePrecision], If[LessEqual[x, 5.6e-12], t$95$0, N[(N[(1.0 / x), $MachinePrecision] * N[(t$95$1 / n), $MachinePrecision]), $MachinePrecision]]]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if x < 4.0000000000000001e-282 or 1.44999999999999991e-270 < x < 5.50000000000000024e-214 or 1.7499999999999999e-199 < x < 1.2e-78 or 1.8999999999999999e-70 < x < 5.6000000000000004e-12

   1. Initial program 26.3%

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

   3. Taylor expanded in x around 0 25.6%

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

   4. Taylor expanded in n around inf 69.8%

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

      1. associate-*r/ 69.8%

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

      2. neg-mul-1 69.8%

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

   6. Simplified 69.8%

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

   if 4.0000000000000001e-282 < x < 1.44999999999999991e-270 or 5.50000000000000024e-214 < x < 1.7499999999999999e-199

   1. Initial program 92.6%

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

   3. Taylor expanded in x around 0 92.6%

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

   if 1.2e-78 < x < 1.8999999999999999e-70

   1. Initial program 71.1%

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

   3. Taylor expanded in x around inf 67.7%

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

      1. mul-1-neg 67.7%

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

      2. log-rec 67.7%

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

      3. mul-1-neg 67.7%

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

      4. distribute-neg-frac 67.7%

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

      5. mul-1-neg 67.7%

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

      6. remove-double-neg 67.7%

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

      7. *-commutative 67.7%

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

   5. Simplified 67.7%

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

   6. Taylor expanded in n around inf 16.5%

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

      1. *-commutative 16.5%

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

   8. Simplified 16.5%

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

      1. log1p-expm1-u 89.2%

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

      2. associate-/r* 89.2%

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

   10. Applied egg-rr 89.2%

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

   if 5.6000000000000004e-12 < x

   1. Initial program 60.5%

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

   3. Taylor expanded in x around inf 93.7%

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

      1. mul-1-neg 93.7%

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

      2. log-rec 93.7%

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

      3. mul-1-neg 93.7%

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

      4. distribute-neg-frac 93.7%

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

      5. mul-1-neg 93.7%

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

      6. remove-double-neg 93.7%

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

      7. *-commutative 93.7%

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

   5. Simplified 93.7%

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

      1. div-inv 93.7%

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

      2. pow-to-exp 93.7%

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

      3. *-un-lft-identity 93.7%

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

      4. times-frac 95.1%

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

   7. Applied egg-rr 95.1%

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

4. Final simplification 82.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 4 \cdot 10^{-282}:\\ \;\;\;\;\frac{\log x}{-n}\\ \mathbf{elif}\;x \leq 1.45 \cdot 10^{-270}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;x \leq 5.5 \cdot 10^{-214}:\\ \;\;\;\;\frac{\log x}{-n}\\ \mathbf{elif}\;x \leq 1.75 \cdot 10^{-199}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;x \leq 1.2 \cdot 10^{-78}:\\ \;\;\;\;\frac{\log x}{-n}\\ \mathbf{elif}\;x \leq 1.9 \cdot 10^{-70}:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{\frac{1}{x}}{n}\right)\right)\\ \mathbf{elif}\;x \leq 5.6 \cdot 10^{-12}:\\ \;\;\;\;\frac{\log x}{-n}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x} \cdot \frac{{x}^{\left(\frac{1}{n}\right)}}{n}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 81.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ t_1 := \frac{1}{x \cdot n}\\ \mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{-147}:\\ \;\;\;\;\frac{1}{x} \cdot \frac{t\_0}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-6}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right) - \log x}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{+131}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - t\_0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{t\_1 \cdot t\_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))) (t_1 (/ 1.0 (* x n))))
   (if (<= (/ 1.0 n) -1e-147)
     (* (/ 1.0 x) (/ t_0 n))
     (if (<= (/ 1.0 n) 2e-6)
       (/ (- (log1p x) (log x)) n)
       (if (<= (/ 1.0 n) 5e+131)
         (- (+ 1.0 (/ x n)) t_0)
         (sqrt (* t_1 t_1)))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double t_1 = 1.0 / (x * n);
	double tmp;
	if ((1.0 / n) <= -1e-147) {
		tmp = (1.0 / x) * (t_0 / n);
	} else if ((1.0 / n) <= 2e-6) {
		tmp = (log1p(x) - log(x)) / n;
	} else if ((1.0 / n) <= 5e+131) {
		tmp = (1.0 + (x / n)) - t_0;
	} else {
		tmp = sqrt((t_1 * t_1));
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, (1.0 / n));
	double t_1 = 1.0 / (x * n);
	double tmp;
	if ((1.0 / n) <= -1e-147) {
		tmp = (1.0 / x) * (t_0 / n);
	} else if ((1.0 / n) <= 2e-6) {
		tmp = (Math.log1p(x) - Math.log(x)) / n;
	} else if ((1.0 / n) <= 5e+131) {
		tmp = (1.0 + (x / n)) - t_0;
	} else {
		tmp = Math.sqrt((t_1 * t_1));
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	t_1 = 1.0 / (x * n)
	tmp = 0
	if (1.0 / n) <= -1e-147:
		tmp = (1.0 / x) * (t_0 / n)
	elif (1.0 / n) <= 2e-6:
		tmp = (math.log1p(x) - math.log(x)) / n
	elif (1.0 / n) <= 5e+131:
		tmp = (1.0 + (x / n)) - t_0
	else:
		tmp = math.sqrt((t_1 * t_1))
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	t_1 = Float64(1.0 / Float64(x * n))
	tmp = 0.0
	if (Float64(1.0 / n) <= -1e-147)
		tmp = Float64(Float64(1.0 / x) * Float64(t_0 / n));
	elseif (Float64(1.0 / n) <= 2e-6)
		tmp = Float64(Float64(log1p(x) - log(x)) / n);
	elseif (Float64(1.0 / n) <= 5e+131)
		tmp = Float64(Float64(1.0 + Float64(x / n)) - t_0);
	else
		tmp = sqrt(Float64(t_1 * t_1));
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(1.0 / N[(x * n), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -1e-147], N[(N[(1.0 / x), $MachinePrecision] * N[(t$95$0 / n), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-6], N[(N[(N[Log[1 + x], $MachinePrecision] - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e+131], N[(N[(1.0 + N[(x / n), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], N[Sqrt[N[(t$95$1 * t$95$1), $MachinePrecision]], $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -9.9999999999999997e-148

   1. Initial program 65.4%

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

   3. Taylor expanded in x around inf 84.6%

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

      1. mul-1-neg 84.6%

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

      2. log-rec 84.6%

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

      3. mul-1-neg 84.6%

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

      4. distribute-neg-frac 84.6%

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

      5. mul-1-neg 84.6%

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

      6. remove-double-neg 84.6%

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

      7. *-commutative 84.6%

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

   5. Simplified 84.6%

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

      1. div-inv 84.6%

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

      2. pow-to-exp 84.6%

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

      3. *-un-lft-identity 84.6%

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

      4. times-frac 85.2%

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

   7. Applied egg-rr 85.2%

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

   if -9.9999999999999997e-148 < (/.f64 #s(literal 1 binary64) n) < 1.99999999999999991e-6

   1. Initial program 27.4%

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

   3. Taylor expanded in n around inf 85.1%

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

      1. log1p-define 85.1%

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

   5. Simplified 85.1%

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

   if 1.99999999999999991e-6 < (/.f64 #s(literal 1 binary64) n) < 4.99999999999999995e131

   1. Initial program 78.3%

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

   3. Taylor expanded in x around 0 85.5%

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

   if 4.99999999999999995e131 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 32.7%

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

   3. Taylor expanded in x around inf 0.6%

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

      1. mul-1-neg 0.6%

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

      2. log-rec 0.6%

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

      3. mul-1-neg 0.6%

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

      4. distribute-neg-frac 0.6%

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

      5. mul-1-neg 0.6%

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

      6. remove-double-neg 0.6%

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

      7. *-commutative 0.6%

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

   5. Simplified 0.6%

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

   6. Taylor expanded in n around inf 39.4%

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

      1. *-commutative 39.4%

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

   8. Simplified 39.4%

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

      1. add-sqr-sqrt 39.4%

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

      2. sqrt-unprod 77.2%

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

      3. inv-pow 77.2%

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

      4. inv-pow 77.2%

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

      5. pow-prod-up 77.2%

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

      6. metadata-eval 77.2%

         \[\leadsto \sqrt{{\left(x \cdot n\right)}^{\color{blue}{-2}}} \]

   10. Applied egg-rr 77.2%

       \[\leadsto \color{blue}{\sqrt{{\left(x \cdot n\right)}^{-2}}} \]
   11. Step-by-step derivation

       1. metadata-eval 77.2%

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

       2. pow-prod-up 77.2%

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

       3. inv-pow 77.2%

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

       4. inv-pow 77.2%

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

   12. Applied egg-rr 77.2%

       \[\leadsto \sqrt{\color{blue}{\frac{1}{x \cdot n} \cdot \frac{1}{x \cdot n}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 84.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{-147}:\\ \;\;\;\;\frac{1}{x} \cdot \frac{{x}^{\left(\frac{1}{n}\right)}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-6}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right) - \log x}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{+131}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1}{x \cdot n} \cdot \frac{1}{x \cdot n}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 70.7% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\log x}{-n}\\ t_1 := {x}^{\left(\frac{1}{n}\right)}\\ t_2 := 1 - t\_1\\ \mathbf{if}\;x \leq 4.1 \cdot 10^{-282}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 4.8 \cdot 10^{-271}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;x \leq 5.5 \cdot 10^{-214}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 1.9 \cdot 10^{-199}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;x \leq 5.5 \cdot 10^{-12}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x} \cdot \frac{t\_1}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (/ (log x) (- n))) (t_1 (pow x (/ 1.0 n))) (t_2 (- 1.0 t_1)))
   (if (<= x 4.1e-282)
     t_0
     (if (<= x 4.8e-271)
       t_2
       (if (<= x 5.5e-214)
         t_0
         (if (<= x 1.9e-199)
           t_2
           (if (<= x 5.5e-12) t_0 (* (/ 1.0 x) (/ t_1 n)))))))))

C

double code(double x, double n) {
	double t_0 = log(x) / -n;
	double t_1 = pow(x, (1.0 / n));
	double t_2 = 1.0 - t_1;
	double tmp;
	if (x <= 4.1e-282) {
		tmp = t_0;
	} else if (x <= 4.8e-271) {
		tmp = t_2;
	} else if (x <= 5.5e-214) {
		tmp = t_0;
	} else if (x <= 1.9e-199) {
		tmp = t_2;
	} else if (x <= 5.5e-12) {
		tmp = t_0;
	} else {
		tmp = (1.0 / x) * (t_1 / n);
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = log(x) / -n
    t_1 = x ** (1.0d0 / n)
    t_2 = 1.0d0 - t_1
    if (x <= 4.1d-282) then
        tmp = t_0
    else if (x <= 4.8d-271) then
        tmp = t_2
    else if (x <= 5.5d-214) then
        tmp = t_0
    else if (x <= 1.9d-199) then
        tmp = t_2
    else if (x <= 5.5d-12) then
        tmp = t_0
    else
        tmp = (1.0d0 / x) * (t_1 / n)
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = Math.log(x) / -n;
	double t_1 = Math.pow(x, (1.0 / n));
	double t_2 = 1.0 - t_1;
	double tmp;
	if (x <= 4.1e-282) {
		tmp = t_0;
	} else if (x <= 4.8e-271) {
		tmp = t_2;
	} else if (x <= 5.5e-214) {
		tmp = t_0;
	} else if (x <= 1.9e-199) {
		tmp = t_2;
	} else if (x <= 5.5e-12) {
		tmp = t_0;
	} else {
		tmp = (1.0 / x) * (t_1 / n);
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.log(x) / -n
	t_1 = math.pow(x, (1.0 / n))
	t_2 = 1.0 - t_1
	tmp = 0
	if x <= 4.1e-282:
		tmp = t_0
	elif x <= 4.8e-271:
		tmp = t_2
	elif x <= 5.5e-214:
		tmp = t_0
	elif x <= 1.9e-199:
		tmp = t_2
	elif x <= 5.5e-12:
		tmp = t_0
	else:
		tmp = (1.0 / x) * (t_1 / n)
	return tmp

Julia

function code(x, n)
	t_0 = Float64(log(x) / Float64(-n))
	t_1 = x ^ Float64(1.0 / n)
	t_2 = Float64(1.0 - t_1)
	tmp = 0.0
	if (x <= 4.1e-282)
		tmp = t_0;
	elseif (x <= 4.8e-271)
		tmp = t_2;
	elseif (x <= 5.5e-214)
		tmp = t_0;
	elseif (x <= 1.9e-199)
		tmp = t_2;
	elseif (x <= 5.5e-12)
		tmp = t_0;
	else
		tmp = Float64(Float64(1.0 / x) * Float64(t_1 / n));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = log(x) / -n;
	t_1 = x ^ (1.0 / n);
	t_2 = 1.0 - t_1;
	tmp = 0.0;
	if (x <= 4.1e-282)
		tmp = t_0;
	elseif (x <= 4.8e-271)
		tmp = t_2;
	elseif (x <= 5.5e-214)
		tmp = t_0;
	elseif (x <= 1.9e-199)
		tmp = t_2;
	elseif (x <= 5.5e-12)
		tmp = t_0;
	else
		tmp = (1.0 / x) * (t_1 / n);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(N[Log[x], $MachinePrecision] / (-n)), $MachinePrecision]}, Block[{t$95$1 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(1.0 - t$95$1), $MachinePrecision]}, If[LessEqual[x, 4.1e-282], t$95$0, If[LessEqual[x, 4.8e-271], t$95$2, If[LessEqual[x, 5.5e-214], t$95$0, If[LessEqual[x, 1.9e-199], t$95$2, If[LessEqual[x, 5.5e-12], t$95$0, N[(N[(1.0 / x), $MachinePrecision] * N[(t$95$1 / n), $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < 4.09999999999999977e-282 or 4.8000000000000005e-271 < x < 5.50000000000000024e-214 or 1.8999999999999999e-199 < x < 5.5000000000000004e-12

   1. Initial program 29.3%

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

   3. Taylor expanded in x around 0 28.6%

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

   4. Taylor expanded in n around inf 66.1%

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

      1. associate-*r/ 66.1%

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

      2. neg-mul-1 66.1%

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

   6. Simplified 66.1%

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

   if 4.09999999999999977e-282 < x < 4.8000000000000005e-271 or 5.50000000000000024e-214 < x < 1.8999999999999999e-199

   1. Initial program 92.6%

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

   3. Taylor expanded in x around 0 92.6%

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

   if 5.5000000000000004e-12 < x

   1. Initial program 60.5%

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

   3. Taylor expanded in x around inf 93.7%

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

      1. mul-1-neg 93.7%

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

      2. log-rec 93.7%

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

      3. mul-1-neg 93.7%

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

      4. distribute-neg-frac 93.7%

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

      5. mul-1-neg 93.7%

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

      6. remove-double-neg 93.7%

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

      7. *-commutative 93.7%

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

   5. Simplified 93.7%

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

      1. div-inv 93.7%

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

      2. pow-to-exp 93.7%

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

      3. *-un-lft-identity 93.7%

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

      4. times-frac 95.1%

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

   7. Applied egg-rr 95.1%

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

4. Final simplification 79.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 4.1 \cdot 10^{-282}:\\ \;\;\;\;\frac{\log x}{-n}\\ \mathbf{elif}\;x \leq 4.8 \cdot 10^{-271}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;x \leq 5.5 \cdot 10^{-214}:\\ \;\;\;\;\frac{\log x}{-n}\\ \mathbf{elif}\;x \leq 1.9 \cdot 10^{-199}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;x \leq 5.5 \cdot 10^{-12}:\\ \;\;\;\;\frac{\log x}{-n}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x} \cdot \frac{{x}^{\left(\frac{1}{n}\right)}}{n}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 70.6% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\log x}{-n}\\ t_1 := 1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;x \leq 1.25 \cdot 10^{-282}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 8.4 \cdot 10^{-271}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 3.2 \cdot 10^{-215}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 9.5 \cdot 10^{-197}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 1.72 \cdot 10^{-12}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n} + -1\right)}}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (/ (log x) (- n))) (t_1 (- 1.0 (pow x (/ 1.0 n)))))
   (if (<= x 1.25e-282)
     t_0
     (if (<= x 8.4e-271)
       t_1
       (if (<= x 3.2e-215)
         t_0
         (if (<= x 9.5e-197)
           t_1
           (if (<= x 1.72e-12) t_0 (/ (pow x (+ (/ 1.0 n) -1.0)) n))))))))

C

double code(double x, double n) {
	double t_0 = log(x) / -n;
	double t_1 = 1.0 - pow(x, (1.0 / n));
	double tmp;
	if (x <= 1.25e-282) {
		tmp = t_0;
	} else if (x <= 8.4e-271) {
		tmp = t_1;
	} else if (x <= 3.2e-215) {
		tmp = t_0;
	} else if (x <= 9.5e-197) {
		tmp = t_1;
	} else if (x <= 1.72e-12) {
		tmp = t_0;
	} else {
		tmp = pow(x, ((1.0 / n) + -1.0)) / n;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = log(x) / -n
    t_1 = 1.0d0 - (x ** (1.0d0 / n))
    if (x <= 1.25d-282) then
        tmp = t_0
    else if (x <= 8.4d-271) then
        tmp = t_1
    else if (x <= 3.2d-215) then
        tmp = t_0
    else if (x <= 9.5d-197) then
        tmp = t_1
    else if (x <= 1.72d-12) then
        tmp = t_0
    else
        tmp = (x ** ((1.0d0 / n) + (-1.0d0))) / n
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = Math.log(x) / -n;
	double t_1 = 1.0 - Math.pow(x, (1.0 / n));
	double tmp;
	if (x <= 1.25e-282) {
		tmp = t_0;
	} else if (x <= 8.4e-271) {
		tmp = t_1;
	} else if (x <= 3.2e-215) {
		tmp = t_0;
	} else if (x <= 9.5e-197) {
		tmp = t_1;
	} else if (x <= 1.72e-12) {
		tmp = t_0;
	} else {
		tmp = Math.pow(x, ((1.0 / n) + -1.0)) / n;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.log(x) / -n
	t_1 = 1.0 - math.pow(x, (1.0 / n))
	tmp = 0
	if x <= 1.25e-282:
		tmp = t_0
	elif x <= 8.4e-271:
		tmp = t_1
	elif x <= 3.2e-215:
		tmp = t_0
	elif x <= 9.5e-197:
		tmp = t_1
	elif x <= 1.72e-12:
		tmp = t_0
	else:
		tmp = math.pow(x, ((1.0 / n) + -1.0)) / n
	return tmp

Julia

function code(x, n)
	t_0 = Float64(log(x) / Float64(-n))
	t_1 = Float64(1.0 - (x ^ Float64(1.0 / n)))
	tmp = 0.0
	if (x <= 1.25e-282)
		tmp = t_0;
	elseif (x <= 8.4e-271)
		tmp = t_1;
	elseif (x <= 3.2e-215)
		tmp = t_0;
	elseif (x <= 9.5e-197)
		tmp = t_1;
	elseif (x <= 1.72e-12)
		tmp = t_0;
	else
		tmp = Float64((x ^ Float64(Float64(1.0 / n) + -1.0)) / n);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = log(x) / -n;
	t_1 = 1.0 - (x ^ (1.0 / n));
	tmp = 0.0;
	if (x <= 1.25e-282)
		tmp = t_0;
	elseif (x <= 8.4e-271)
		tmp = t_1;
	elseif (x <= 3.2e-215)
		tmp = t_0;
	elseif (x <= 9.5e-197)
		tmp = t_1;
	elseif (x <= 1.72e-12)
		tmp = t_0;
	else
		tmp = (x ^ ((1.0 / n) + -1.0)) / n;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(N[Log[x], $MachinePrecision] / (-n)), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 1.25e-282], t$95$0, If[LessEqual[x, 8.4e-271], t$95$1, If[LessEqual[x, 3.2e-215], t$95$0, If[LessEqual[x, 9.5e-197], t$95$1, If[LessEqual[x, 1.72e-12], t$95$0, N[(N[Power[x, N[(N[(1.0 / n), $MachinePrecision] + -1.0), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < 1.25e-282 or 8.4000000000000003e-271 < x < 3.2000000000000001e-215 or 9.5000000000000003e-197 < x < 1.7199999999999999e-12

   1. Initial program 29.3%

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

   3. Taylor expanded in x around 0 28.6%

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

   4. Taylor expanded in n around inf 66.1%

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

      1. associate-*r/ 66.1%

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

      2. neg-mul-1 66.1%

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

   6. Simplified 66.1%

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

   if 1.25e-282 < x < 8.4000000000000003e-271 or 3.2000000000000001e-215 < x < 9.5000000000000003e-197

   1. Initial program 92.6%

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

   3. Taylor expanded in x around 0 92.6%

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

   if 1.7199999999999999e-12 < x

   1. Initial program 60.5%

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

   3. Taylor expanded in x around inf 93.7%

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

      1. mul-1-neg 93.7%

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

      2. log-rec 93.7%

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

      3. mul-1-neg 93.7%

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

      4. distribute-neg-frac 93.7%

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

      5. mul-1-neg 93.7%

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

      6. remove-double-neg 93.7%

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

      7. *-commutative 93.7%

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

   5. Simplified 93.7%

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

      1. *-un-lft-identity 93.7%

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

      2. associate-/r* 95.2%

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

      3. div-inv 95.2%

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

      4. pow-to-exp 95.2%

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

      5. pow1 95.2%

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

      6. pow-div 94.8%

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

   7. Applied egg-rr 94.8%

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

      1. *-lft-identity 94.8%

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

      2. sub-neg 94.8%

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

      3. metadata-eval 94.8%

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

   9. Simplified 94.8%

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

4. Final simplification 79.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.25 \cdot 10^{-282}:\\ \;\;\;\;\frac{\log x}{-n}\\ \mathbf{elif}\;x \leq 8.4 \cdot 10^{-271}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;x \leq 3.2 \cdot 10^{-215}:\\ \;\;\;\;\frac{\log x}{-n}\\ \mathbf{elif}\;x \leq 9.5 \cdot 10^{-197}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;x \leq 1.72 \cdot 10^{-12}:\\ \;\;\;\;\frac{\log x}{-n}\\ \mathbf{else}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n} + -1\right)}}{n}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 56.2% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\log x}{-n}\\ t_1 := 1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;x \leq 1.75 \cdot 10^{-282}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 6.2 \cdot 10^{-271}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 3.65 \cdot 10^{-214}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 1.76 \cdot 10^{-199}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 0.00145:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (/ (log x) (- n))) (t_1 (- 1.0 (pow x (/ 1.0 n)))))
   (if (<= x 1.75e-282)
     t_0
     (if (<= x 6.2e-271)
       t_1
       (if (<= x 3.65e-214)
         t_0
         (if (<= x 1.76e-199) t_1 (if (<= x 0.00145) t_0 (/ (/ 1.0 x) n))))))))

C

double code(double x, double n) {
	double t_0 = log(x) / -n;
	double t_1 = 1.0 - pow(x, (1.0 / n));
	double tmp;
	if (x <= 1.75e-282) {
		tmp = t_0;
	} else if (x <= 6.2e-271) {
		tmp = t_1;
	} else if (x <= 3.65e-214) {
		tmp = t_0;
	} else if (x <= 1.76e-199) {
		tmp = t_1;
	} else if (x <= 0.00145) {
		tmp = t_0;
	} else {
		tmp = (1.0 / x) / n;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = log(x) / -n
    t_1 = 1.0d0 - (x ** (1.0d0 / n))
    if (x <= 1.75d-282) then
        tmp = t_0
    else if (x <= 6.2d-271) then
        tmp = t_1
    else if (x <= 3.65d-214) then
        tmp = t_0
    else if (x <= 1.76d-199) then
        tmp = t_1
    else if (x <= 0.00145d0) then
        tmp = t_0
    else
        tmp = (1.0d0 / x) / n
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = Math.log(x) / -n;
	double t_1 = 1.0 - Math.pow(x, (1.0 / n));
	double tmp;
	if (x <= 1.75e-282) {
		tmp = t_0;
	} else if (x <= 6.2e-271) {
		tmp = t_1;
	} else if (x <= 3.65e-214) {
		tmp = t_0;
	} else if (x <= 1.76e-199) {
		tmp = t_1;
	} else if (x <= 0.00145) {
		tmp = t_0;
	} else {
		tmp = (1.0 / x) / n;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.log(x) / -n
	t_1 = 1.0 - math.pow(x, (1.0 / n))
	tmp = 0
	if x <= 1.75e-282:
		tmp = t_0
	elif x <= 6.2e-271:
		tmp = t_1
	elif x <= 3.65e-214:
		tmp = t_0
	elif x <= 1.76e-199:
		tmp = t_1
	elif x <= 0.00145:
		tmp = t_0
	else:
		tmp = (1.0 / x) / n
	return tmp

Julia

function code(x, n)
	t_0 = Float64(log(x) / Float64(-n))
	t_1 = Float64(1.0 - (x ^ Float64(1.0 / n)))
	tmp = 0.0
	if (x <= 1.75e-282)
		tmp = t_0;
	elseif (x <= 6.2e-271)
		tmp = t_1;
	elseif (x <= 3.65e-214)
		tmp = t_0;
	elseif (x <= 1.76e-199)
		tmp = t_1;
	elseif (x <= 0.00145)
		tmp = t_0;
	else
		tmp = Float64(Float64(1.0 / x) / n);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = log(x) / -n;
	t_1 = 1.0 - (x ^ (1.0 / n));
	tmp = 0.0;
	if (x <= 1.75e-282)
		tmp = t_0;
	elseif (x <= 6.2e-271)
		tmp = t_1;
	elseif (x <= 3.65e-214)
		tmp = t_0;
	elseif (x <= 1.76e-199)
		tmp = t_1;
	elseif (x <= 0.00145)
		tmp = t_0;
	else
		tmp = (1.0 / x) / n;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(N[Log[x], $MachinePrecision] / (-n)), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 1.75e-282], t$95$0, If[LessEqual[x, 6.2e-271], t$95$1, If[LessEqual[x, 3.65e-214], t$95$0, If[LessEqual[x, 1.76e-199], t$95$1, If[LessEqual[x, 0.00145], t$95$0, N[(N[(1.0 / x), $MachinePrecision] / n), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < 1.75000000000000003e-282 or 6.1999999999999998e-271 < x < 3.65000000000000015e-214 or 1.76000000000000011e-199 < x < 0.00145

   1. Initial program 30.1%

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

   3. Taylor expanded in x around 0 29.4%

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

   4. Taylor expanded in n around inf 65.0%

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

      1. associate-*r/ 65.0%

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

      2. neg-mul-1 65.0%

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

   6. Simplified 65.0%

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

   if 1.75000000000000003e-282 < x < 6.1999999999999998e-271 or 3.65000000000000015e-214 < x < 1.76000000000000011e-199

   1. Initial program 92.6%

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

   3. Taylor expanded in x around 0 92.6%

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

   if 0.00145 < x

   1. Initial program 60.7%

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

   3. Taylor expanded in x around inf 95.2%

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

      1. mul-1-neg 95.2%

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

      2. log-rec 95.2%

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

      3. mul-1-neg 95.2%

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

      4. distribute-neg-frac 95.2%

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

      5. mul-1-neg 95.2%

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

      6. remove-double-neg 95.2%

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

      7. *-commutative 95.2%

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

   5. Simplified 95.2%

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

   6. Taylor expanded in n around inf 64.0%

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

      1. *-commutative 64.0%

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

   8. Simplified 64.0%

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

      1. *-un-lft-identity 64.0%

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

      2. associate-/r* 65.5%

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

   10. Applied egg-rr 65.5%

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

4. Final simplification 66.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.75 \cdot 10^{-282}:\\ \;\;\;\;\frac{\log x}{-n}\\ \mathbf{elif}\;x \leq 6.2 \cdot 10^{-271}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;x \leq 3.65 \cdot 10^{-214}:\\ \;\;\;\;\frac{\log x}{-n}\\ \mathbf{elif}\;x \leq 1.76 \cdot 10^{-199}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;x \leq 0.00145:\\ \;\;\;\;\frac{\log x}{-n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 56.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.00145:\\ \;\;\;\;\frac{\log x}{-n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 0.00145) (/ (log x) (- n)) (/ (/ 1.0 x) n)))

C

double code(double x, double n) {
	double tmp;
	if (x <= 0.00145) {
		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 <= 0.00145d0) 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 <= 0.00145) {
		tmp = Math.log(x) / -n;
	} else {
		tmp = (1.0 / x) / n;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 0.00145:
		tmp = math.log(x) / -n
	else:
		tmp = (1.0 / x) / n
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 0.00145)
		tmp = Float64(log(x) / Float64(-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 <= 0.00145)
		tmp = log(x) / -n;
	else
		tmp = (1.0 / x) / n;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 0.00145], 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 < 0.00145

   1. Initial program 35.4%

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

   3. Taylor expanded in x around 0 34.8%

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

   4. Taylor expanded in n around inf 60.6%

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

      1. associate-*r/ 60.6%

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

      2. neg-mul-1 60.6%

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

   6. Simplified 60.6%

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

   if 0.00145 < x

   1. Initial program 60.7%

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

   3. Taylor expanded in x around inf 95.2%

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

      1. mul-1-neg 95.2%

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

      2. log-rec 95.2%

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

      3. mul-1-neg 95.2%

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

      4. distribute-neg-frac 95.2%

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

      5. mul-1-neg 95.2%

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

      6. remove-double-neg 95.2%

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

      7. *-commutative 95.2%

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

   5. Simplified 95.2%

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

   6. Taylor expanded in n around inf 64.0%

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

      1. *-commutative 64.0%

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

   8. Simplified 64.0%

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

      1. *-un-lft-identity 64.0%

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

      2. associate-/r* 65.5%

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

   10. Applied egg-rr 65.5%

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

4. Final simplification 62.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.00145:\\ \;\;\;\;\frac{\log x}{-n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 39.9% accurate, 42.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 45.7%

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

3. Taylor expanded in x around inf 53.1%

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

   1. mul-1-neg 53.1%

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

   2. log-rec 53.1%

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

   3. mul-1-neg 53.1%

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

   4. distribute-neg-frac 53.1%

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

   5. mul-1-neg 53.1%

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

   6. remove-double-neg 53.1%

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

   7. *-commutative 53.1%

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

5. Simplified 53.1%

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

6. Taylor expanded in n around inf 37.6%

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

   1. *-commutative 37.6%

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

8. Simplified 37.6%

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

9. Final simplification 37.6%

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

Alternative 13: 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 45.7%

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

3. Taylor expanded in x around inf 53.1%

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

   1. mul-1-neg 53.1%

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

   2. log-rec 53.1%

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

   3. mul-1-neg 53.1%

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

   4. distribute-neg-frac 53.1%

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

   5. mul-1-neg 53.1%

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

   6. remove-double-neg 53.1%

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

   7. *-commutative 53.1%

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

5. Simplified 53.1%

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

6. Taylor expanded in n around inf 37.6%

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

   1. *-commutative 37.6%

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

8. Simplified 37.6%

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

   1. *-un-lft-identity 37.6%

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

   2. associate-/r* 38.2%

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

10. Applied egg-rr 38.2%

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

11. Final simplification 38.2%

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

Reproduce

herbie shell --seed 2024071 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  :precision binary64
  (- (pow (+ x 1.0) (/ 1.0 n)) (pow x (/ 1.0 n))))