2nthrt (problem 3.4.6)

Percentage Accurate: 57.4% → 87.6%

Time: 38.1s

Alternatives: 16

Speedup: 1.9×

• could not determine a ground truth

  1. x = -1.8063241171927504e+284
  2. n = -2.7809054161984717e+212

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: 57.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 87.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{-75}:\\ \;\;\;\;\frac{\frac{t\_0}{n}}{x}\\ \mathbf{elif}\;\frac{1}{n} \leq 4 \cdot 10^{-6}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right) + \left(0.5 \cdot \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}}{n} - \log x\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -2e-75)
     (/ (/ t_0 n) x)
     (if (<= (/ 1.0 n) 4e-6)
       (/
        (+
         (log1p x)
         (- (* 0.5 (/ (- (pow (log1p x) 2.0) (pow (log x) 2.0)) n)) (log x)))
        n)
       (- (exp (/ (log1p x) n)) t_0)))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -2e-75) {
		tmp = (t_0 / n) / x;
	} else if ((1.0 / n) <= 4e-6) {
		tmp = (log1p(x) + ((0.5 * ((pow(log1p(x), 2.0) - pow(log(x), 2.0)) / n)) - log(x))) / n;
	} else {
		tmp = exp((log1p(x) / n)) - t_0;
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -2e-75) {
		tmp = (t_0 / n) / x;
	} else if ((1.0 / n) <= 4e-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.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) <= -2e-75:
		tmp = (t_0 / n) / x
	elif (1.0 / n) <= 4e-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.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) <= -2e-75)
		tmp = Float64(Float64(t_0 / n) / x);
	elseif (Float64(1.0 / n) <= 4e-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 = 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], -2e-75], N[(N[(t$95$0 / n), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 4e-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[(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) < -1.9999999999999999e-75

   1. Initial program 85.0%

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

   3. Taylor expanded in x around inf 74.0%

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

      1. log-rec 74.0%

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

      2. mul-1-neg 74.0%

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

      3. neg-mul-1 74.0%

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

      4. mul-1-neg 74.0%

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

      5. distribute-frac-neg 74.0%

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

      6. remove-double-neg 74.0%

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

      7. *-rgt-identity 74.0%

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

      8. associate-/l* 74.0%

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

      9. exp-to-pow 96.5%

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

      10. *-commutative 96.5%

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

   5. Simplified 96.5%

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

      1. *-un-lft-identity 96.5%

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

      2. times-frac 96.6%

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

   7. Applied egg-rr 96.6%

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

      1. associate-*l/ 96.6%

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

      2. *-un-lft-identity 96.6%

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

   9. Applied egg-rr 96.6%

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

   if -1.9999999999999999e-75 < (/.f64 #s(literal 1 binary64) n) < 3.99999999999999982e-6

   1. Initial program 29.2%

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

   3. Taylor expanded in n around inf 79.9%

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

      1. associate--l+ 79.9%

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

      2. log1p-define 79.9%

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

      3. +-commutative 79.9%

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

      4. associate--r+ 79.9%

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

      5. distribute-lft-out-- 79.9%

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

      6. div-sub 79.9%

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

      7. log1p-define 79.9%

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

   5. Simplified 79.9%

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

   if 3.99999999999999982e-6 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 51.1%

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

   3. Taylor expanded in n around 0 34.6%

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

      1. log1p-define 56.1%

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

      2. *-rgt-identity 56.1%

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

      3. associate-*l/ 56.1%

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

      4. associate-/l* 56.1%

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

      5. exp-to-pow 93.0%

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

   5. Simplified 93.0%

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

Alternative 2: 87.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{-75}:\\ \;\;\;\;\frac{\frac{t\_0}{n}}{x}\\ \mathbf{elif}\;\frac{1}{n} \leq 4 \cdot 10^{-6}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right) - \log x}{n}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -2e-75)
     (/ (/ t_0 n) x)
     (if (<= (/ 1.0 n) 4e-6)
       (/ (- (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) <= -2e-75) {
		tmp = (t_0 / n) / x;
	} else if ((1.0 / n) <= 4e-6) {
		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) <= -2e-75) {
		tmp = (t_0 / n) / x;
	} else if ((1.0 / n) <= 4e-6) {
		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) <= -2e-75:
		tmp = (t_0 / n) / x
	elif (1.0 / n) <= 4e-6:
		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) <= -2e-75)
		tmp = Float64(Float64(t_0 / n) / x);
	elseif (Float64(1.0 / n) <= 4e-6)
		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], -2e-75], N[(N[(t$95$0 / n), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 4e-6], 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) < -1.9999999999999999e-75

   1. Initial program 85.0%

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

   3. Taylor expanded in x around inf 74.0%

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

      1. log-rec 74.0%

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

      2. mul-1-neg 74.0%

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

      3. neg-mul-1 74.0%

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

      4. mul-1-neg 74.0%

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

      5. distribute-frac-neg 74.0%

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

      6. remove-double-neg 74.0%

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

      7. *-rgt-identity 74.0%

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

      8. associate-/l* 74.0%

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

      9. exp-to-pow 96.5%

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

      10. *-commutative 96.5%

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

   5. Simplified 96.5%

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

      1. *-un-lft-identity 96.5%

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

      2. times-frac 96.6%

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

   7. Applied egg-rr 96.6%

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

      1. associate-*l/ 96.6%

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

      2. *-un-lft-identity 96.6%

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

   9. Applied egg-rr 96.6%

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

   if -1.9999999999999999e-75 < (/.f64 #s(literal 1 binary64) n) < 3.99999999999999982e-6

   1. Initial program 29.2%

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

   3. Taylor expanded in n around inf 79.4%

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

      1. log1p-define 79.4%

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

   5. Simplified 79.4%

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

   if 3.99999999999999982e-6 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 51.1%

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

   3. Taylor expanded in n around 0 34.6%

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

      1. log1p-define 56.1%

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

      2. *-rgt-identity 56.1%

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

      3. associate-*l/ 56.1%

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

      4. associate-/l* 56.1%

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

      5. exp-to-pow 93.0%

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

   5. Simplified 93.0%

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

Alternative 3: 87.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{-75}:\\ \;\;\;\;\frac{\frac{t\_0}{n}}{x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-5}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right) - \log x}{n}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{x}{n}} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -2e-75)
     (/ (/ t_0 n) x)
     (if (<= (/ 1.0 n) 2e-5)
       (/ (- (log1p x) (log x)) n)
       (- (exp (/ x n)) t_0)))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -2e-75) {
		tmp = (t_0 / n) / x;
	} else if ((1.0 / n) <= 2e-5) {
		tmp = (log1p(x) - log(x)) / n;
	} else {
		tmp = exp((x / n)) - t_0;
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -2e-75) {
		tmp = (t_0 / n) / x;
	} else if ((1.0 / n) <= 2e-5) {
		tmp = (Math.log1p(x) - Math.log(x)) / n;
	} else {
		tmp = Math.exp((x / n)) - t_0;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	tmp = 0
	if (1.0 / n) <= -2e-75:
		tmp = (t_0 / n) / x
	elif (1.0 / n) <= 2e-5:
		tmp = (math.log1p(x) - math.log(x)) / n
	else:
		tmp = math.exp((x / n)) - t_0
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -2e-75)
		tmp = Float64(Float64(t_0 / n) / x);
	elseif (Float64(1.0 / n) <= 2e-5)
		tmp = Float64(Float64(log1p(x) - log(x)) / n);
	else
		tmp = Float64(exp(Float64(x / n)) - t_0);
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -2e-75], N[(N[(t$95$0 / n), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-5], N[(N[(N[Log[1 + x], $MachinePrecision] - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], N[(N[Exp[N[(x / n), $MachinePrecision]], $MachinePrecision] - t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -1.9999999999999999e-75

   1. Initial program 85.0%

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

   3. Taylor expanded in x around inf 74.0%

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

      1. log-rec 74.0%

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

      2. mul-1-neg 74.0%

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

      3. neg-mul-1 74.0%

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

      4. mul-1-neg 74.0%

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

      5. distribute-frac-neg 74.0%

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

      6. remove-double-neg 74.0%

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

      7. *-rgt-identity 74.0%

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

      8. associate-/l* 74.0%

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

      9. exp-to-pow 96.5%

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

      10. *-commutative 96.5%

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

   5. Simplified 96.5%

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

      1. *-un-lft-identity 96.5%

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

      2. times-frac 96.6%

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

   7. Applied egg-rr 96.6%

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

      1. associate-*l/ 96.6%

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

      2. *-un-lft-identity 96.6%

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

   9. Applied egg-rr 96.6%

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

   if -1.9999999999999999e-75 < (/.f64 #s(literal 1 binary64) n) < 2.00000000000000016e-5

   1. Initial program 29.0%

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

   3. Taylor expanded in n around inf 78.8%

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

      1. log1p-define 78.8%

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

   5. Simplified 78.8%

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

   if 2.00000000000000016e-5 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 51.8%

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

   3. Taylor expanded in n around 0 35.1%

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

      1. log1p-define 56.9%

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

      2. *-rgt-identity 56.9%

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

      3. associate-*l/ 56.9%

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

      4. associate-/l* 56.9%

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

      5. exp-to-pow 94.4%

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

   5. Simplified 94.4%

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

   6. Taylor expanded in x around 0 94.4%

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

Alternative 4: 76.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{-75}:\\ \;\;\;\;\frac{\frac{t\_0}{n}}{x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-5}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right) - \log x}{n}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{x}{n} + 1\right) - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -2e-75)
     (/ (/ t_0 n) x)
     (if (<= (/ 1.0 n) 2e-5)
       (/ (- (log1p x) (log x)) n)
       (- (+ (/ x n) 1.0) t_0)))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -2e-75) {
		tmp = (t_0 / n) / x;
	} else if ((1.0 / n) <= 2e-5) {
		tmp = (log1p(x) - log(x)) / n;
	} else {
		tmp = ((x / n) + 1.0) - 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) <= -2e-75) {
		tmp = (t_0 / n) / x;
	} else if ((1.0 / n) <= 2e-5) {
		tmp = (Math.log1p(x) - Math.log(x)) / n;
	} else {
		tmp = ((x / n) + 1.0) - t_0;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	tmp = 0
	if (1.0 / n) <= -2e-75:
		tmp = (t_0 / n) / x
	elif (1.0 / n) <= 2e-5:
		tmp = (math.log1p(x) - math.log(x)) / n
	else:
		tmp = ((x / n) + 1.0) - t_0
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -2e-75)
		tmp = Float64(Float64(t_0 / n) / x);
	elseif (Float64(1.0 / n) <= 2e-5)
		tmp = Float64(Float64(log1p(x) - log(x)) / n);
	else
		tmp = Float64(Float64(Float64(x / n) + 1.0) - 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], -2e-75], N[(N[(t$95$0 / n), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-5], N[(N[(N[Log[1 + x], $MachinePrecision] - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], N[(N[(N[(x / n), $MachinePrecision] + 1.0), $MachinePrecision] - t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -1.9999999999999999e-75

   1. Initial program 85.0%

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

   3. Taylor expanded in x around inf 74.0%

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

      1. log-rec 74.0%

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

      2. mul-1-neg 74.0%

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

      3. neg-mul-1 74.0%

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

      4. mul-1-neg 74.0%

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

      5. distribute-frac-neg 74.0%

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

      6. remove-double-neg 74.0%

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

      7. *-rgt-identity 74.0%

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

      8. associate-/l* 74.0%

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

      9. exp-to-pow 96.5%

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

      10. *-commutative 96.5%

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

   5. Simplified 96.5%

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

      1. *-un-lft-identity 96.5%

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

      2. times-frac 96.6%

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

   7. Applied egg-rr 96.6%

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

      1. associate-*l/ 96.6%

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

      2. *-un-lft-identity 96.6%

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

   9. Applied egg-rr 96.6%

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

   if -1.9999999999999999e-75 < (/.f64 #s(literal 1 binary64) n) < 2.00000000000000016e-5

   1. Initial program 29.0%

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

   3. Taylor expanded in n around inf 78.8%

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

      1. log1p-define 78.8%

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

   5. Simplified 78.8%

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

   if 2.00000000000000016e-5 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 51.8%

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

   3. Taylor expanded in x around 0 50.2%

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

4. Final simplification 78.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{-75}:\\ \;\;\;\;\frac{\frac{{x}^{\left(\frac{1}{n}\right)}}{n}}{x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-5}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right) - \log x}{n}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{x}{n} + 1\right) - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 74.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;x \leq -3.1 \cdot 10^{-192}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 2.3 \cdot 10^{-247}:\\ \;\;\;\;\left(\frac{x}{n} + 1\right) - t\_0\\ \mathbf{elif}\;x \leq 0.25:\\ \;\;\;\;x \cdot \left(\frac{1}{n} + x \cdot \left(\frac{x}{n} \cdot 0.3333333333333333 + 0.5 \cdot \frac{-1}{n}\right)\right) - \frac{\log x}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{t\_0}{n}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= x -3.1e-192)
     0.0
     (if (<= x 2.3e-247)
       (- (+ (/ x n) 1.0) t_0)
       (if (<= x 0.25)
         (-
          (*
           x
           (+
            (/ 1.0 n)
            (* x (+ (* (/ x n) 0.3333333333333333) (* 0.5 (/ -1.0 n))))))
          (/ (log x) n))
         (/ (/ t_0 n) x))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double tmp;
	if (x <= -3.1e-192) {
		tmp = 0.0;
	} else if (x <= 2.3e-247) {
		tmp = ((x / n) + 1.0) - t_0;
	} else if (x <= 0.25) {
		tmp = (x * ((1.0 / n) + (x * (((x / n) * 0.3333333333333333) + (0.5 * (-1.0 / n)))))) - (log(x) / n);
	} else {
		tmp = (t_0 / n) / x;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x ** (1.0d0 / n)
    if (x <= (-3.1d-192)) then
        tmp = 0.0d0
    else if (x <= 2.3d-247) then
        tmp = ((x / n) + 1.0d0) - t_0
    else if (x <= 0.25d0) then
        tmp = (x * ((1.0d0 / n) + (x * (((x / n) * 0.3333333333333333d0) + (0.5d0 * ((-1.0d0) / n)))))) - (log(x) / n)
    else
        tmp = (t_0 / n) / x
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, (1.0 / n));
	double tmp;
	if (x <= -3.1e-192) {
		tmp = 0.0;
	} else if (x <= 2.3e-247) {
		tmp = ((x / n) + 1.0) - t_0;
	} else if (x <= 0.25) {
		tmp = (x * ((1.0 / n) + (x * (((x / n) * 0.3333333333333333) + (0.5 * (-1.0 / n)))))) - (Math.log(x) / n);
	} else {
		tmp = (t_0 / n) / x;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	tmp = 0
	if x <= -3.1e-192:
		tmp = 0.0
	elif x <= 2.3e-247:
		tmp = ((x / n) + 1.0) - t_0
	elif x <= 0.25:
		tmp = (x * ((1.0 / n) + (x * (((x / n) * 0.3333333333333333) + (0.5 * (-1.0 / n)))))) - (math.log(x) / n)
	else:
		tmp = (t_0 / n) / x
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (x <= -3.1e-192)
		tmp = 0.0;
	elseif (x <= 2.3e-247)
		tmp = Float64(Float64(Float64(x / n) + 1.0) - t_0);
	elseif (x <= 0.25)
		tmp = Float64(Float64(x * Float64(Float64(1.0 / n) + Float64(x * Float64(Float64(Float64(x / n) * 0.3333333333333333) + Float64(0.5 * Float64(-1.0 / n)))))) - Float64(log(x) / n));
	else
		tmp = Float64(Float64(t_0 / n) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = x ^ (1.0 / n);
	tmp = 0.0;
	if (x <= -3.1e-192)
		tmp = 0.0;
	elseif (x <= 2.3e-247)
		tmp = ((x / n) + 1.0) - t_0;
	elseif (x <= 0.25)
		tmp = (x * ((1.0 / n) + (x * (((x / n) * 0.3333333333333333) + (0.5 * (-1.0 / n)))))) - (log(x) / n);
	else
		tmp = (t_0 / n) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, -3.1e-192], 0.0, If[LessEqual[x, 2.3e-247], N[(N[(N[(x / n), $MachinePrecision] + 1.0), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[x, 0.25], N[(N[(x * N[(N[(1.0 / n), $MachinePrecision] + N[(x * N[(N[(N[(x / n), $MachinePrecision] * 0.3333333333333333), $MachinePrecision] + N[(0.5 * N[(-1.0 / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[Log[x], $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$0 / n), $MachinePrecision] / x), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -3.1e-192

   1. Initial program 65.8%

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

   3. Taylor expanded in x around inf 0.0%

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

      1. log-rec 0.0%

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

      2. mul-1-neg 0.0%

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

      3. neg-mul-1 0.0%

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

      4. mul-1-neg 0.0%

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

      5. distribute-frac-neg 0.0%

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

      6. remove-double-neg 0.0%

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

      7. *-rgt-identity 0.0%

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

      8. associate-/l* 0.0%

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

      9. exp-to-pow 85.8%

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

   5. Simplified 85.8%

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

   6. Taylor expanded in x around 0 85.9%

      \[\leadsto \color{blue}{0} \]

   if -3.1e-192 < x < 2.3e-247

   1. Initial program 64.7%

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

   3. Taylor expanded in x around 0 65.9%

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

   if 2.3e-247 < x < 0.25

   1. Initial program 37.1%

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

      1. add-sqr-sqrt 14.5%

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

      2. pow2 14.5%

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

      3. pow-to-exp 14.5%

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

      4. un-div-inv 14.5%

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

      5. +-commutative 14.5%

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

      6. log1p-define 26.3%

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

   4. Applied egg-rr 26.3%

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

   5. Taylor expanded in n around inf 30.9%

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

      1. log1p-define 30.9%

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

   7. Simplified 30.9%

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

   8. Taylor expanded in x around 0 56.1%

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

   if 0.25 < x

   1. Initial program 61.2%

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

   3. Taylor expanded in x around inf 97.9%

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

      1. log-rec 97.9%

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

      2. mul-1-neg 97.9%

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

      3. neg-mul-1 97.9%

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

      4. mul-1-neg 97.9%

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

      5. distribute-frac-neg 97.9%

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

      6. remove-double-neg 97.9%

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

      7. *-rgt-identity 97.9%

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

      8. associate-/l* 97.9%

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

      9. exp-to-pow 97.9%

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

      10. *-commutative 97.9%

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

   5. Simplified 97.9%

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

      1. *-un-lft-identity 97.9%

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

      2. times-frac 98.6%

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

   7. Applied egg-rr 98.6%

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

      1. associate-*l/ 98.6%

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

      2. *-un-lft-identity 98.6%

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

   9. Applied egg-rr 98.6%

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

4. Final simplification 76.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.1 \cdot 10^{-192}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 2.3 \cdot 10^{-247}:\\ \;\;\;\;\left(\frac{x}{n} + 1\right) - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;x \leq 0.25:\\ \;\;\;\;x \cdot \left(\frac{1}{n} + x \cdot \left(\frac{x}{n} \cdot 0.3333333333333333 + 0.5 \cdot \frac{-1}{n}\right)\right) - \frac{\log x}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{{x}^{\left(\frac{1}{n}\right)}}{n}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 74.4% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;x \leq -4.9 \cdot 10^{-206}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 2.3 \cdot 10^{-247}:\\ \;\;\;\;\left(\frac{x}{n} + 1\right) - t\_0\\ \mathbf{elif}\;x \leq 0.22:\\ \;\;\;\;x \cdot \left(\frac{1}{n} + \frac{x}{n} \cdot -0.5\right) - \frac{\log x}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{t\_0}{n}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= x -4.9e-206)
     0.0
     (if (<= x 2.3e-247)
       (- (+ (/ x n) 1.0) t_0)
       (if (<= x 0.22)
         (- (* x (+ (/ 1.0 n) (* (/ x n) -0.5))) (/ (log x) n))
         (/ (/ t_0 n) x))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double tmp;
	if (x <= -4.9e-206) {
		tmp = 0.0;
	} else if (x <= 2.3e-247) {
		tmp = ((x / n) + 1.0) - t_0;
	} else if (x <= 0.22) {
		tmp = (x * ((1.0 / n) + ((x / n) * -0.5))) - (log(x) / n);
	} else {
		tmp = (t_0 / n) / x;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x ** (1.0d0 / n)
    if (x <= (-4.9d-206)) then
        tmp = 0.0d0
    else if (x <= 2.3d-247) then
        tmp = ((x / n) + 1.0d0) - t_0
    else if (x <= 0.22d0) then
        tmp = (x * ((1.0d0 / n) + ((x / n) * (-0.5d0)))) - (log(x) / n)
    else
        tmp = (t_0 / n) / x
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, (1.0 / n));
	double tmp;
	if (x <= -4.9e-206) {
		tmp = 0.0;
	} else if (x <= 2.3e-247) {
		tmp = ((x / n) + 1.0) - t_0;
	} else if (x <= 0.22) {
		tmp = (x * ((1.0 / n) + ((x / n) * -0.5))) - (Math.log(x) / n);
	} else {
		tmp = (t_0 / n) / x;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	tmp = 0
	if x <= -4.9e-206:
		tmp = 0.0
	elif x <= 2.3e-247:
		tmp = ((x / n) + 1.0) - t_0
	elif x <= 0.22:
		tmp = (x * ((1.0 / n) + ((x / n) * -0.5))) - (math.log(x) / n)
	else:
		tmp = (t_0 / n) / x
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (x <= -4.9e-206)
		tmp = 0.0;
	elseif (x <= 2.3e-247)
		tmp = Float64(Float64(Float64(x / n) + 1.0) - t_0);
	elseif (x <= 0.22)
		tmp = Float64(Float64(x * Float64(Float64(1.0 / n) + Float64(Float64(x / n) * -0.5))) - Float64(log(x) / n));
	else
		tmp = Float64(Float64(t_0 / n) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = x ^ (1.0 / n);
	tmp = 0.0;
	if (x <= -4.9e-206)
		tmp = 0.0;
	elseif (x <= 2.3e-247)
		tmp = ((x / n) + 1.0) - t_0;
	elseif (x <= 0.22)
		tmp = (x * ((1.0 / n) + ((x / n) * -0.5))) - (log(x) / n);
	else
		tmp = (t_0 / n) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, -4.9e-206], 0.0, If[LessEqual[x, 2.3e-247], N[(N[(N[(x / n), $MachinePrecision] + 1.0), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[x, 0.22], N[(N[(x * N[(N[(1.0 / n), $MachinePrecision] + N[(N[(x / n), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[Log[x], $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$0 / n), $MachinePrecision] / x), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -4.9e-206

   1. Initial program 65.8%

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

   3. Taylor expanded in x around inf 0.0%

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

      1. log-rec 0.0%

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

      2. mul-1-neg 0.0%

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

      3. neg-mul-1 0.0%

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

      4. mul-1-neg 0.0%

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

      5. distribute-frac-neg 0.0%

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

      6. remove-double-neg 0.0%

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

      7. *-rgt-identity 0.0%

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

      8. associate-/l* 0.0%

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

      9. exp-to-pow 85.8%

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

   5. Simplified 85.8%

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

   6. Taylor expanded in x around 0 85.9%

      \[\leadsto \color{blue}{0} \]

   if -4.9e-206 < x < 2.3e-247

   1. Initial program 64.7%

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

   3. Taylor expanded in x around 0 65.9%

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

   if 2.3e-247 < x < 0.220000000000000001

   1. Initial program 37.1%

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

      1. add-sqr-sqrt 14.5%

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

      2. pow2 14.5%

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

      3. pow-to-exp 14.5%

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

      4. un-div-inv 14.5%

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

      5. +-commutative 14.5%

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

      6. log1p-define 26.3%

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

   4. Applied egg-rr 26.3%

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

   5. Taylor expanded in n around inf 30.9%

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

      1. log1p-define 30.9%

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

   7. Simplified 30.9%

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

   8. Taylor expanded in x around 0 56.0%

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

   if 0.220000000000000001 < x

   1. Initial program 61.2%

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

   3. Taylor expanded in x around inf 97.9%

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

      1. log-rec 97.9%

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

      2. mul-1-neg 97.9%

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

      3. neg-mul-1 97.9%

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

      4. mul-1-neg 97.9%

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

      5. distribute-frac-neg 97.9%

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

      6. remove-double-neg 97.9%

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

      7. *-rgt-identity 97.9%

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

      8. associate-/l* 97.9%

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

      9. exp-to-pow 97.9%

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

      10. *-commutative 97.9%

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

   5. Simplified 97.9%

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

      1. *-un-lft-identity 97.9%

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

      2. times-frac 98.6%

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

   7. Applied egg-rr 98.6%

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

      1. associate-*l/ 98.6%

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

      2. *-un-lft-identity 98.6%

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

   9. Applied egg-rr 98.6%

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

4. Final simplification 76.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.9 \cdot 10^{-206}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 2.3 \cdot 10^{-247}:\\ \;\;\;\;\left(\frac{x}{n} + 1\right) - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;x \leq 0.22:\\ \;\;\;\;x \cdot \left(\frac{1}{n} + \frac{x}{n} \cdot -0.5\right) - \frac{\log x}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{{x}^{\left(\frac{1}{n}\right)}}{n}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 74.4% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;x \leq -2.95 \cdot 10^{-192}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 1.4 \cdot 10^{-247}:\\ \;\;\;\;\left(\frac{x}{n} + 1\right) - t\_0\\ \mathbf{elif}\;x \leq 0.22:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{t\_0}{n}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= x -2.95e-192)
     0.0
     (if (<= x 1.4e-247)
       (- (+ (/ x n) 1.0) t_0)
       (if (<= x 0.22) (/ (- x (log x)) n) (/ (/ t_0 n) x))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double tmp;
	if (x <= -2.95e-192) {
		tmp = 0.0;
	} else if (x <= 1.4e-247) {
		tmp = ((x / n) + 1.0) - t_0;
	} else if (x <= 0.22) {
		tmp = (x - log(x)) / n;
	} else {
		tmp = (t_0 / n) / x;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x ** (1.0d0 / n)
    if (x <= (-2.95d-192)) then
        tmp = 0.0d0
    else if (x <= 1.4d-247) then
        tmp = ((x / n) + 1.0d0) - t_0
    else if (x <= 0.22d0) then
        tmp = (x - log(x)) / n
    else
        tmp = (t_0 / n) / x
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, (1.0 / n));
	double tmp;
	if (x <= -2.95e-192) {
		tmp = 0.0;
	} else if (x <= 1.4e-247) {
		tmp = ((x / n) + 1.0) - t_0;
	} else if (x <= 0.22) {
		tmp = (x - Math.log(x)) / n;
	} else {
		tmp = (t_0 / n) / x;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	tmp = 0
	if x <= -2.95e-192:
		tmp = 0.0
	elif x <= 1.4e-247:
		tmp = ((x / n) + 1.0) - t_0
	elif x <= 0.22:
		tmp = (x - math.log(x)) / n
	else:
		tmp = (t_0 / n) / x
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (x <= -2.95e-192)
		tmp = 0.0;
	elseif (x <= 1.4e-247)
		tmp = Float64(Float64(Float64(x / n) + 1.0) - t_0);
	elseif (x <= 0.22)
		tmp = Float64(Float64(x - log(x)) / n);
	else
		tmp = Float64(Float64(t_0 / n) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = x ^ (1.0 / n);
	tmp = 0.0;
	if (x <= -2.95e-192)
		tmp = 0.0;
	elseif (x <= 1.4e-247)
		tmp = ((x / n) + 1.0) - t_0;
	elseif (x <= 0.22)
		tmp = (x - log(x)) / n;
	else
		tmp = (t_0 / n) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, -2.95e-192], 0.0, If[LessEqual[x, 1.4e-247], N[(N[(N[(x / n), $MachinePrecision] + 1.0), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[x, 0.22], N[(N[(x - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], N[(N[(t$95$0 / n), $MachinePrecision] / x), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -2.9499999999999998e-192

   1. Initial program 65.8%

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

   3. Taylor expanded in x around inf 0.0%

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

      1. log-rec 0.0%

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

      2. mul-1-neg 0.0%

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

      3. neg-mul-1 0.0%

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

      4. mul-1-neg 0.0%

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

      5. distribute-frac-neg 0.0%

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

      6. remove-double-neg 0.0%

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

      7. *-rgt-identity 0.0%

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

      8. associate-/l* 0.0%

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

      9. exp-to-pow 85.8%

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

   5. Simplified 85.8%

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

   6. Taylor expanded in x around 0 85.9%

      \[\leadsto \color{blue}{0} \]

   if -2.9499999999999998e-192 < x < 1.39999999999999993e-247

   1. Initial program 64.7%

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

   3. Taylor expanded in x around 0 65.9%

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

   if 1.39999999999999993e-247 < x < 0.220000000000000001

   1. Initial program 37.1%

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

   3. Taylor expanded in x around 0 35.7%

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

   4. Taylor expanded in n around inf 55.5%

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

   if 0.220000000000000001 < x

   1. Initial program 61.2%

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

   3. Taylor expanded in x around inf 97.9%

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

      1. log-rec 97.9%

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

      2. mul-1-neg 97.9%

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

      3. neg-mul-1 97.9%

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

      4. mul-1-neg 97.9%

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

      5. distribute-frac-neg 97.9%

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

      6. remove-double-neg 97.9%

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

      7. *-rgt-identity 97.9%

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

      8. associate-/l* 97.9%

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

      9. exp-to-pow 97.9%

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

      10. *-commutative 97.9%

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

   5. Simplified 97.9%

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

      1. *-un-lft-identity 97.9%

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

      2. times-frac 98.6%

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

   7. Applied egg-rr 98.6%

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

      1. associate-*l/ 98.6%

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

      2. *-un-lft-identity 98.6%

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

   9. Applied egg-rr 98.6%

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

4. Final simplification 75.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.95 \cdot 10^{-192}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 1.4 \cdot 10^{-247}:\\ \;\;\;\;\left(\frac{x}{n} + 1\right) - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;x \leq 0.22:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{{x}^{\left(\frac{1}{n}\right)}}{n}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 74.3% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;x \leq -2.4 \cdot 10^{-198}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 1.55 \cdot 10^{-247}:\\ \;\;\;\;1 - t\_0\\ \mathbf{elif}\;x \leq 0.22:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{t\_0}{n}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= x -2.4e-198)
     0.0
     (if (<= x 1.55e-247)
       (- 1.0 t_0)
       (if (<= x 0.22) (/ (- x (log x)) n) (/ (/ t_0 n) x))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double tmp;
	if (x <= -2.4e-198) {
		tmp = 0.0;
	} else if (x <= 1.55e-247) {
		tmp = 1.0 - t_0;
	} else if (x <= 0.22) {
		tmp = (x - log(x)) / n;
	} else {
		tmp = (t_0 / n) / x;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x ** (1.0d0 / n)
    if (x <= (-2.4d-198)) then
        tmp = 0.0d0
    else if (x <= 1.55d-247) then
        tmp = 1.0d0 - t_0
    else if (x <= 0.22d0) then
        tmp = (x - log(x)) / n
    else
        tmp = (t_0 / n) / x
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, (1.0 / n));
	double tmp;
	if (x <= -2.4e-198) {
		tmp = 0.0;
	} else if (x <= 1.55e-247) {
		tmp = 1.0 - t_0;
	} else if (x <= 0.22) {
		tmp = (x - Math.log(x)) / n;
	} else {
		tmp = (t_0 / n) / x;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	tmp = 0
	if x <= -2.4e-198:
		tmp = 0.0
	elif x <= 1.55e-247:
		tmp = 1.0 - t_0
	elif x <= 0.22:
		tmp = (x - math.log(x)) / n
	else:
		tmp = (t_0 / n) / x
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (x <= -2.4e-198)
		tmp = 0.0;
	elseif (x <= 1.55e-247)
		tmp = Float64(1.0 - t_0);
	elseif (x <= 0.22)
		tmp = Float64(Float64(x - log(x)) / n);
	else
		tmp = Float64(Float64(t_0 / n) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = x ^ (1.0 / n);
	tmp = 0.0;
	if (x <= -2.4e-198)
		tmp = 0.0;
	elseif (x <= 1.55e-247)
		tmp = 1.0 - t_0;
	elseif (x <= 0.22)
		tmp = (x - log(x)) / n;
	else
		tmp = (t_0 / n) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, -2.4e-198], 0.0, If[LessEqual[x, 1.55e-247], N[(1.0 - t$95$0), $MachinePrecision], If[LessEqual[x, 0.22], N[(N[(x - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], N[(N[(t$95$0 / n), $MachinePrecision] / x), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -2.39999999999999986e-198

   1. Initial program 65.8%

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

   3. Taylor expanded in x around inf 0.0%

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

      1. log-rec 0.0%

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

      2. mul-1-neg 0.0%

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

      3. neg-mul-1 0.0%

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

      4. mul-1-neg 0.0%

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

      5. distribute-frac-neg 0.0%

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

      6. remove-double-neg 0.0%

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

      7. *-rgt-identity 0.0%

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

      8. associate-/l* 0.0%

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

      9. exp-to-pow 85.8%

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

   5. Simplified 85.8%

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

   6. Taylor expanded in x around 0 85.9%

      \[\leadsto \color{blue}{0} \]

   if -2.39999999999999986e-198 < x < 1.55000000000000008e-247

   1. Initial program 64.7%

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

   3. Taylor expanded in x around 0 47.3%

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

      1. *-rgt-identity 47.3%

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

      2. associate-*l/ 47.3%

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

      3. associate-/l* 47.3%

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

      4. exp-to-pow 64.7%

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

   5. Simplified 64.7%

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

   if 1.55000000000000008e-247 < x < 0.220000000000000001

   1. Initial program 37.1%

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

   3. Taylor expanded in x around 0 35.7%

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

   4. Taylor expanded in n around inf 55.5%

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

   if 0.220000000000000001 < x

   1. Initial program 61.2%

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

   3. Taylor expanded in x around inf 97.9%

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

      1. log-rec 97.9%

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

      2. mul-1-neg 97.9%

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

      3. neg-mul-1 97.9%

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

      4. mul-1-neg 97.9%

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

      5. distribute-frac-neg 97.9%

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

      6. remove-double-neg 97.9%

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

      7. *-rgt-identity 97.9%

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

      8. associate-/l* 97.9%

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

      9. exp-to-pow 97.9%

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

      10. *-commutative 97.9%

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

   5. Simplified 97.9%

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

      1. *-un-lft-identity 97.9%

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

      2. times-frac 98.6%

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

   7. Applied egg-rr 98.6%

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

      1. associate-*l/ 98.6%

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

      2. *-un-lft-identity 98.6%

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

   9. Applied egg-rr 98.6%

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

Alternative 9: 66.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{-192}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 1.5 \cdot 10^{-247}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;x \leq 0.85:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{elif}\;x \leq 2.1 \cdot 10^{+160}:\\ \;\;\;\;\frac{\frac{1}{n} + \frac{\frac{0.3333333333333333}{n \cdot x} - \frac{0.5}{n}}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x -2e-192)
   0.0
   (if (<= x 1.5e-247)
     (- 1.0 (pow x (/ 1.0 n)))
     (if (<= x 0.85)
       (/ (- x (log x)) n)
       (if (<= x 2.1e+160)
         (/ (+ (/ 1.0 n) (/ (- (/ 0.3333333333333333 (* n x)) (/ 0.5 n)) x)) x)
         0.0)))))

C

double code(double x, double n) {
	double tmp;
	if (x <= -2e-192) {
		tmp = 0.0;
	} else if (x <= 1.5e-247) {
		tmp = 1.0 - pow(x, (1.0 / n));
	} else if (x <= 0.85) {
		tmp = (x - log(x)) / n;
	} else if (x <= 2.1e+160) {
		tmp = ((1.0 / n) + (((0.3333333333333333 / (n * x)) - (0.5 / n)) / x)) / x;
	} else {
		tmp = 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 <= (-2d-192)) then
        tmp = 0.0d0
    else if (x <= 1.5d-247) then
        tmp = 1.0d0 - (x ** (1.0d0 / n))
    else if (x <= 0.85d0) then
        tmp = (x - log(x)) / n
    else if (x <= 2.1d+160) then
        tmp = ((1.0d0 / n) + (((0.3333333333333333d0 / (n * x)) - (0.5d0 / n)) / x)) / x
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (x <= -2e-192) {
		tmp = 0.0;
	} else if (x <= 1.5e-247) {
		tmp = 1.0 - Math.pow(x, (1.0 / n));
	} else if (x <= 0.85) {
		tmp = (x - Math.log(x)) / n;
	} else if (x <= 2.1e+160) {
		tmp = ((1.0 / n) + (((0.3333333333333333 / (n * x)) - (0.5 / n)) / x)) / x;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= -2e-192:
		tmp = 0.0
	elif x <= 1.5e-247:
		tmp = 1.0 - math.pow(x, (1.0 / n))
	elif x <= 0.85:
		tmp = (x - math.log(x)) / n
	elif x <= 2.1e+160:
		tmp = ((1.0 / n) + (((0.3333333333333333 / (n * x)) - (0.5 / n)) / x)) / x
	else:
		tmp = 0.0
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= -2e-192)
		tmp = 0.0;
	elseif (x <= 1.5e-247)
		tmp = Float64(1.0 - (x ^ Float64(1.0 / n)));
	elseif (x <= 0.85)
		tmp = Float64(Float64(x - log(x)) / n);
	elseif (x <= 2.1e+160)
		tmp = Float64(Float64(Float64(1.0 / n) + Float64(Float64(Float64(0.3333333333333333 / Float64(n * x)) - Float64(0.5 / n)) / x)) / x);
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (x <= -2e-192)
		tmp = 0.0;
	elseif (x <= 1.5e-247)
		tmp = 1.0 - (x ^ (1.0 / n));
	elseif (x <= 0.85)
		tmp = (x - log(x)) / n;
	elseif (x <= 2.1e+160)
		tmp = ((1.0 / n) + (((0.3333333333333333 / (n * x)) - (0.5 / n)) / x)) / x;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, -2e-192], 0.0, If[LessEqual[x, 1.5e-247], N[(1.0 - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.85], N[(N[(x - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[x, 2.1e+160], N[(N[(N[(1.0 / n), $MachinePrecision] + N[(N[(N[(0.3333333333333333 / N[(n * x), $MachinePrecision]), $MachinePrecision] - N[(0.5 / n), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], 0.0]]]]

Derivation

1. Split input into 4 regimes

2. if x < -2.0000000000000002e-192 or 2.09999999999999997e160 < x

   1. Initial program 73.7%

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

   3. Taylor expanded in x around inf 46.1%

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

      1. log-rec 46.1%

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

      2. mul-1-neg 46.1%

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

      3. neg-mul-1 46.1%

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

      4. mul-1-neg 46.1%

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

      5. distribute-frac-neg 46.1%

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

      6. remove-double-neg 46.1%

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

      7. *-rgt-identity 46.1%

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

      8. associate-/l* 46.1%

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

      9. exp-to-pow 82.1%

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

   5. Simplified 82.1%

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

   6. Taylor expanded in x around 0 82.1%

      \[\leadsto \color{blue}{0} \]

   if -2.0000000000000002e-192 < x < 1.4999999999999999e-247

   1. Initial program 64.7%

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

   3. Taylor expanded in x around 0 47.3%

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

      1. *-rgt-identity 47.3%

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

      2. associate-*l/ 47.3%

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

      3. associate-/l* 47.3%

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

      4. exp-to-pow 64.7%

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

   5. Simplified 64.7%

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

   if 1.4999999999999999e-247 < x < 0.849999999999999978

   1. Initial program 37.1%

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

   3. Taylor expanded in x around 0 35.7%

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

   4. Taylor expanded in n around inf 55.5%

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

   if 0.849999999999999978 < x < 2.09999999999999997e160

   1. Initial program 39.8%

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

      1. add-sqr-sqrt 39.8%

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

      2. pow2 39.8%

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

      3. pow-to-exp 39.8%

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

      4. un-div-inv 39.8%

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

      5. +-commutative 39.8%

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

      6. log1p-define 39.8%

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

   4. Applied egg-rr 39.8%

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

   5. Taylor expanded in n around inf 38.7%

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

      1. log1p-define 38.7%

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

   7. Simplified 38.7%

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

   8. Taylor expanded in x around -inf 74.1%

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

      1. mul-1-neg 74.1%

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

      2. mul-1-neg 74.1%

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

      3. associate-*r/ 74.1%

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

      4. metadata-eval 74.1%

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

      5. *-commutative 74.1%

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

      6. associate-*r/ 74.1%

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

      7. metadata-eval 74.1%

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

   10. Simplified 74.1%

       \[\leadsto \color{blue}{-\frac{\left(-\frac{\frac{0.3333333333333333}{x \cdot n} - \frac{0.5}{n}}{x}\right) - \frac{1}{n}}{x}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 68.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{-192}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 1.5 \cdot 10^{-247}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;x \leq 0.85:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{elif}\;x \leq 2.1 \cdot 10^{+160}:\\ \;\;\;\;\frac{\frac{1}{n} + \frac{\frac{0.3333333333333333}{n \cdot x} - \frac{0.5}{n}}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 64.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4 \cdot 10^{-310}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 0.85:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{elif}\;x \leq 1.95 \cdot 10^{+160}:\\ \;\;\;\;\frac{\frac{1}{n} + \frac{\frac{0.3333333333333333}{n \cdot x} - \frac{0.5}{n}}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x -4e-310)
   0.0
   (if (<= x 0.85)
     (/ (- x (log x)) n)
     (if (<= x 1.95e+160)
       (/ (+ (/ 1.0 n) (/ (- (/ 0.3333333333333333 (* n x)) (/ 0.5 n)) x)) x)
       0.0))))

C

double code(double x, double n) {
	double tmp;
	if (x <= -4e-310) {
		tmp = 0.0;
	} else if (x <= 0.85) {
		tmp = (x - log(x)) / n;
	} else if (x <= 1.95e+160) {
		tmp = ((1.0 / n) + (((0.3333333333333333 / (n * x)) - (0.5 / n)) / x)) / x;
	} else {
		tmp = 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 <= (-4d-310)) then
        tmp = 0.0d0
    else if (x <= 0.85d0) then
        tmp = (x - log(x)) / n
    else if (x <= 1.95d+160) then
        tmp = ((1.0d0 / n) + (((0.3333333333333333d0 / (n * x)) - (0.5d0 / n)) / x)) / x
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (x <= -4e-310) {
		tmp = 0.0;
	} else if (x <= 0.85) {
		tmp = (x - Math.log(x)) / n;
	} else if (x <= 1.95e+160) {
		tmp = ((1.0 / n) + (((0.3333333333333333 / (n * x)) - (0.5 / n)) / x)) / x;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= -4e-310:
		tmp = 0.0
	elif x <= 0.85:
		tmp = (x - math.log(x)) / n
	elif x <= 1.95e+160:
		tmp = ((1.0 / n) + (((0.3333333333333333 / (n * x)) - (0.5 / n)) / x)) / x
	else:
		tmp = 0.0
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= -4e-310)
		tmp = 0.0;
	elseif (x <= 0.85)
		tmp = Float64(Float64(x - log(x)) / n);
	elseif (x <= 1.95e+160)
		tmp = Float64(Float64(Float64(1.0 / n) + Float64(Float64(Float64(0.3333333333333333 / Float64(n * x)) - Float64(0.5 / n)) / x)) / x);
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (x <= -4e-310)
		tmp = 0.0;
	elseif (x <= 0.85)
		tmp = (x - log(x)) / n;
	elseif (x <= 1.95e+160)
		tmp = ((1.0 / n) + (((0.3333333333333333 / (n * x)) - (0.5 / n)) / x)) / x;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, -4e-310], 0.0, If[LessEqual[x, 0.85], N[(N[(x - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[x, 1.95e+160], N[(N[(N[(1.0 / n), $MachinePrecision] + N[(N[(N[(0.3333333333333333 / N[(n * x), $MachinePrecision]), $MachinePrecision] - N[(0.5 / n), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], 0.0]]]

Derivation

1. Split input into 3 regimes

2. if x < -3.999999999999988e-310 or 1.95000000000000004e160 < x

   1. Initial program 72.8%

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

   3. Taylor expanded in x around inf 40.6%

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

      1. log-rec 40.6%

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

      2. mul-1-neg 40.6%

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

      3. neg-mul-1 40.6%

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

      4. mul-1-neg 40.6%

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

      5. distribute-frac-neg 40.6%

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

      6. remove-double-neg 40.6%

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

      7. *-rgt-identity 40.6%

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

      8. associate-/l* 40.6%

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

      9. exp-to-pow 73.6%

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

   5. Simplified 73.6%

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

   6. Taylor expanded in x around 0 73.7%

      \[\leadsto \color{blue}{0} \]

   if -3.999999999999988e-310 < x < 0.849999999999999978

   1. Initial program 43.9%

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

   3. Taylor expanded in x around 0 43.1%

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

   4. Taylor expanded in n around inf 50.6%

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

   if 0.849999999999999978 < x < 1.95000000000000004e160

   1. Initial program 39.8%

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

      1. add-sqr-sqrt 39.8%

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

      2. pow2 39.8%

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

      3. pow-to-exp 39.8%

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

      4. un-div-inv 39.8%

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

      5. +-commutative 39.8%

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

      6. log1p-define 39.8%

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

   4. Applied egg-rr 39.8%

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

   5. Taylor expanded in n around inf 38.7%

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

      1. log1p-define 38.7%

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

   7. Simplified 38.7%

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

   8. Taylor expanded in x around -inf 74.1%

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

      1. mul-1-neg 74.1%

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

      2. mul-1-neg 74.1%

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

      3. associate-*r/ 74.1%

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

      4. metadata-eval 74.1%

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

      5. *-commutative 74.1%

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

      6. associate-*r/ 74.1%

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

      7. metadata-eval 74.1%

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

   10. Simplified 74.1%

       \[\leadsto \color{blue}{-\frac{\left(-\frac{\frac{0.3333333333333333}{x \cdot n} - \frac{0.5}{n}}{x}\right) - \frac{1}{n}}{x}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 62.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4 \cdot 10^{-310}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 0.85:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{elif}\;x \leq 1.95 \cdot 10^{+160}:\\ \;\;\;\;\frac{\frac{1}{n} + \frac{\frac{0.3333333333333333}{n \cdot x} - \frac{0.5}{n}}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 64.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4 \cdot 10^{-310}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 0.6:\\ \;\;\;\;\frac{\log x}{-n}\\ \mathbf{elif}\;x \leq 2 \cdot 10^{+160}:\\ \;\;\;\;\frac{\frac{1}{n} + \frac{\frac{0.3333333333333333}{n \cdot x} - \frac{0.5}{n}}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x -4e-310)
   0.0
   (if (<= x 0.6)
     (/ (log x) (- n))
     (if (<= x 2e+160)
       (/ (+ (/ 1.0 n) (/ (- (/ 0.3333333333333333 (* n x)) (/ 0.5 n)) x)) x)
       0.0))))

C

double code(double x, double n) {
	double tmp;
	if (x <= -4e-310) {
		tmp = 0.0;
	} else if (x <= 0.6) {
		tmp = log(x) / -n;
	} else if (x <= 2e+160) {
		tmp = ((1.0 / n) + (((0.3333333333333333 / (n * x)) - (0.5 / n)) / x)) / x;
	} else {
		tmp = 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 <= (-4d-310)) then
        tmp = 0.0d0
    else if (x <= 0.6d0) then
        tmp = log(x) / -n
    else if (x <= 2d+160) then
        tmp = ((1.0d0 / n) + (((0.3333333333333333d0 / (n * x)) - (0.5d0 / n)) / x)) / x
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (x <= -4e-310) {
		tmp = 0.0;
	} else if (x <= 0.6) {
		tmp = Math.log(x) / -n;
	} else if (x <= 2e+160) {
		tmp = ((1.0 / n) + (((0.3333333333333333 / (n * x)) - (0.5 / n)) / x)) / x;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= -4e-310:
		tmp = 0.0
	elif x <= 0.6:
		tmp = math.log(x) / -n
	elif x <= 2e+160:
		tmp = ((1.0 / n) + (((0.3333333333333333 / (n * x)) - (0.5 / n)) / x)) / x
	else:
		tmp = 0.0
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= -4e-310)
		tmp = 0.0;
	elseif (x <= 0.6)
		tmp = Float64(log(x) / Float64(-n));
	elseif (x <= 2e+160)
		tmp = Float64(Float64(Float64(1.0 / n) + Float64(Float64(Float64(0.3333333333333333 / Float64(n * x)) - Float64(0.5 / n)) / x)) / x);
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (x <= -4e-310)
		tmp = 0.0;
	elseif (x <= 0.6)
		tmp = log(x) / -n;
	elseif (x <= 2e+160)
		tmp = ((1.0 / n) + (((0.3333333333333333 / (n * x)) - (0.5 / n)) / x)) / x;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, -4e-310], 0.0, If[LessEqual[x, 0.6], N[(N[Log[x], $MachinePrecision] / (-n)), $MachinePrecision], If[LessEqual[x, 2e+160], N[(N[(N[(1.0 / n), $MachinePrecision] + N[(N[(N[(0.3333333333333333 / N[(n * x), $MachinePrecision]), $MachinePrecision] - N[(0.5 / n), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], 0.0]]]

Derivation

1. Split input into 3 regimes

2. if x < -3.999999999999988e-310 or 2.00000000000000001e160 < x

   1. Initial program 72.8%

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

   3. Taylor expanded in x around inf 40.6%

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

      1. log-rec 40.6%

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

      2. mul-1-neg 40.6%

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

      3. neg-mul-1 40.6%

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

      4. mul-1-neg 40.6%

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

      5. distribute-frac-neg 40.6%

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

      6. remove-double-neg 40.6%

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

      7. *-rgt-identity 40.6%

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

      8. associate-/l* 40.6%

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

      9. exp-to-pow 73.6%

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

   5. Simplified 73.6%

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

   6. Taylor expanded in x around 0 73.7%

      \[\leadsto \color{blue}{0} \]

   if -3.999999999999988e-310 < x < 0.599999999999999978

   1. Initial program 43.9%

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

   3. Taylor expanded in x around 0 42.3%

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

      1. *-rgt-identity 42.3%

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

      2. associate-*l/ 42.3%

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

      3. associate-/l* 42.3%

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

      4. exp-to-pow 42.3%

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

   5. Simplified 42.3%

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

   6. Taylor expanded in n around inf 50.2%

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

      1. associate-*r/ 50.2%

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

      2. mul-1-neg 50.2%

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

   8. Simplified 50.2%

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

   if 0.599999999999999978 < x < 2.00000000000000001e160

   1. Initial program 39.8%

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

      1. add-sqr-sqrt 39.8%

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

      2. pow2 39.8%

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

      3. pow-to-exp 39.8%

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

      4. un-div-inv 39.8%

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

      5. +-commutative 39.8%

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

      6. log1p-define 39.8%

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

   4. Applied egg-rr 39.8%

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

   5. Taylor expanded in n around inf 38.7%

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

      1. log1p-define 38.7%

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

   7. Simplified 38.7%

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

   8. Taylor expanded in x around -inf 74.1%

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

      1. mul-1-neg 74.1%

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

      2. mul-1-neg 74.1%

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

      3. associate-*r/ 74.1%

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

      4. metadata-eval 74.1%

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

      5. *-commutative 74.1%

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

      6. associate-*r/ 74.1%

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

      7. metadata-eval 74.1%

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

   10. Simplified 74.1%

       \[\leadsto \color{blue}{-\frac{\left(-\frac{\frac{0.3333333333333333}{x \cdot n} - \frac{0.5}{n}}{x}\right) - \frac{1}{n}}{x}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 62.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4 \cdot 10^{-310}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 0.6:\\ \;\;\;\;\frac{\log x}{-n}\\ \mathbf{elif}\;x \leq 2 \cdot 10^{+160}:\\ \;\;\;\;\frac{\frac{1}{n} + \frac{\frac{0.3333333333333333}{n \cdot x} - \frac{0.5}{n}}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 56.0% accurate, 7.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.16 \cdot 10^{-292}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 1.95 \cdot 10^{+160}:\\ \;\;\;\;\frac{\frac{1}{n} + \frac{\frac{0.3333333333333333}{n \cdot x} - \frac{0.5}{n}}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 1.16e-292)
   0.0
   (if (<= x 1.95e+160)
     (/ (+ (/ 1.0 n) (/ (- (/ 0.3333333333333333 (* n x)) (/ 0.5 n)) x)) x)
     0.0)))

C

double code(double x, double n) {
	double tmp;
	if (x <= 1.16e-292) {
		tmp = 0.0;
	} else if (x <= 1.95e+160) {
		tmp = ((1.0 / n) + (((0.3333333333333333 / (n * x)) - (0.5 / n)) / x)) / x;
	} else {
		tmp = 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 <= 1.16d-292) then
        tmp = 0.0d0
    else if (x <= 1.95d+160) then
        tmp = ((1.0d0 / n) + (((0.3333333333333333d0 / (n * x)) - (0.5d0 / n)) / x)) / x
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (x <= 1.16e-292) {
		tmp = 0.0;
	} else if (x <= 1.95e+160) {
		tmp = ((1.0 / n) + (((0.3333333333333333 / (n * x)) - (0.5 / n)) / x)) / x;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 1.16e-292:
		tmp = 0.0
	elif x <= 1.95e+160:
		tmp = ((1.0 / n) + (((0.3333333333333333 / (n * x)) - (0.5 / n)) / x)) / x
	else:
		tmp = 0.0
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 1.16e-292)
		tmp = 0.0;
	elseif (x <= 1.95e+160)
		tmp = Float64(Float64(Float64(1.0 / n) + Float64(Float64(Float64(0.3333333333333333 / Float64(n * x)) - Float64(0.5 / n)) / x)) / x);
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (x <= 1.16e-292)
		tmp = 0.0;
	elseif (x <= 1.95e+160)
		tmp = ((1.0 / n) + (((0.3333333333333333 / (n * x)) - (0.5 / n)) / x)) / x;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 1.16e-292], 0.0, If[LessEqual[x, 1.95e+160], N[(N[(N[(1.0 / n), $MachinePrecision] + N[(N[(N[(0.3333333333333333 / N[(n * x), $MachinePrecision]), $MachinePrecision] - N[(0.5 / n), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], 0.0]]

Derivation

1. Split input into 2 regimes

2. if x < 1.16e-292 or 1.95000000000000004e160 < x

   1. Initial program 70.5%

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

   3. Taylor expanded in x around inf 38.7%

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

      1. log-rec 38.7%

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

      2. mul-1-neg 38.7%

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

      3. neg-mul-1 38.7%

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

      4. mul-1-neg 38.7%

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

      5. distribute-frac-neg 38.7%

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

      6. remove-double-neg 38.7%

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

      7. *-rgt-identity 38.7%

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

      8. associate-/l* 38.7%

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

      9. exp-to-pow 70.1%

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

   5. Simplified 70.1%

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

   6. Taylor expanded in x around 0 70.2%

      \[\leadsto \color{blue}{0} \]

   if 1.16e-292 < x < 1.95000000000000004e160

   1. Initial program 43.3%

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

      1. add-sqr-sqrt 24.5%

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

      2. pow2 24.5%

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

      3. pow-to-exp 24.5%

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

      4. un-div-inv 24.5%

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

      5. +-commutative 24.5%

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

      6. log1p-define 31.9%

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

   4. Applied egg-rr 31.9%

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

   5. Taylor expanded in n around inf 28.8%

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

      1. log1p-define 28.8%

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

   7. Simplified 28.8%

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

   8. Taylor expanded in x around -inf 40.9%

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

      1. mul-1-neg 40.9%

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

      2. mul-1-neg 40.9%

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

      3. associate-*r/ 40.9%

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

      4. metadata-eval 40.9%

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

      5. *-commutative 40.9%

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

      6. associate-*r/ 40.9%

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

      7. metadata-eval 40.9%

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

   10. Simplified 40.9%

       \[\leadsto \color{blue}{-\frac{\left(-\frac{\frac{0.3333333333333333}{x \cdot n} - \frac{0.5}{n}}{x}\right) - \frac{1}{n}}{x}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 52.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.16 \cdot 10^{-292}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 1.95 \cdot 10^{+160}:\\ \;\;\;\;\frac{\frac{1}{n} + \frac{\frac{0.3333333333333333}{n \cdot x} - \frac{0.5}{n}}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 51.5% accurate, 14.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.16 \cdot 10^{-292}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 1.95 \cdot 10^{+160}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 1.16e-292) 0.0 (if (<= x 1.95e+160) (/ (/ 1.0 x) n) 0.0)))

C

double code(double x, double n) {
	double tmp;
	if (x <= 1.16e-292) {
		tmp = 0.0;
	} else if (x <= 1.95e+160) {
		tmp = (1.0 / x) / n;
	} else {
		tmp = 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 <= 1.16d-292) then
        tmp = 0.0d0
    else if (x <= 1.95d+160) then
        tmp = (1.0d0 / x) / n
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (x <= 1.16e-292) {
		tmp = 0.0;
	} else if (x <= 1.95e+160) {
		tmp = (1.0 / x) / n;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 1.16e-292:
		tmp = 0.0
	elif x <= 1.95e+160:
		tmp = (1.0 / x) / n
	else:
		tmp = 0.0
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 1.16e-292)
		tmp = 0.0;
	elseif (x <= 1.95e+160)
		tmp = Float64(Float64(1.0 / x) / n);
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (x <= 1.16e-292)
		tmp = 0.0;
	elseif (x <= 1.95e+160)
		tmp = (1.0 / x) / n;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 1.16e-292], 0.0, If[LessEqual[x, 1.95e+160], N[(N[(1.0 / x), $MachinePrecision] / n), $MachinePrecision], 0.0]]

Derivation

1. Split input into 2 regimes

2. if x < 1.16e-292 or 1.95000000000000004e160 < x

   1. Initial program 70.5%

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

   3. Taylor expanded in x around inf 38.7%

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

      1. log-rec 38.7%

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

      2. mul-1-neg 38.7%

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

      3. neg-mul-1 38.7%

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

      4. mul-1-neg 38.7%

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

      5. distribute-frac-neg 38.7%

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

      6. remove-double-neg 38.7%

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

      7. *-rgt-identity 38.7%

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

      8. associate-/l* 38.7%

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

      9. exp-to-pow 70.1%

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

   5. Simplified 70.1%

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

   6. Taylor expanded in x around 0 70.2%

      \[\leadsto \color{blue}{0} \]

   if 1.16e-292 < x < 1.95000000000000004e160

   1. Initial program 43.3%

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

   3. Taylor expanded in x around inf 45.6%

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

      1. log-rec 45.6%

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

      2. mul-1-neg 45.6%

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

      3. neg-mul-1 45.6%

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

      4. mul-1-neg 45.6%

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

      5. distribute-frac-neg 45.6%

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

      6. remove-double-neg 45.6%

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

      7. *-rgt-identity 45.6%

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

      8. associate-/l* 45.6%

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

      9. exp-to-pow 45.6%

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

      10. *-commutative 45.6%

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

   5. Simplified 45.6%

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

      1. *-un-lft-identity 45.6%

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

      2. times-frac 45.9%

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

   7. Applied egg-rr 45.9%

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

      1. associate-*l/ 45.9%

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

      2. *-un-lft-identity 45.9%

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

   9. Applied egg-rr 45.9%

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

   10. Taylor expanded in n around inf 32.5%

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

       1. *-commutative 32.5%

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

       2. associate-/r* 32.5%

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

   12. Simplified 32.5%

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

Alternative 14: 51.5% accurate, 14.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.16 \cdot 10^{-292}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 1.95 \cdot 10^{+160}:\\ \;\;\;\;\frac{\frac{1}{n}}{x}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 1.16e-292) 0.0 (if (<= x 1.95e+160) (/ (/ 1.0 n) x) 0.0)))

C

double code(double x, double n) {
	double tmp;
	if (x <= 1.16e-292) {
		tmp = 0.0;
	} else if (x <= 1.95e+160) {
		tmp = (1.0 / n) / x;
	} else {
		tmp = 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 <= 1.16d-292) then
        tmp = 0.0d0
    else if (x <= 1.95d+160) then
        tmp = (1.0d0 / n) / x
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (x <= 1.16e-292) {
		tmp = 0.0;
	} else if (x <= 1.95e+160) {
		tmp = (1.0 / n) / x;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 1.16e-292:
		tmp = 0.0
	elif x <= 1.95e+160:
		tmp = (1.0 / n) / x
	else:
		tmp = 0.0
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 1.16e-292)
		tmp = 0.0;
	elseif (x <= 1.95e+160)
		tmp = Float64(Float64(1.0 / n) / x);
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (x <= 1.16e-292)
		tmp = 0.0;
	elseif (x <= 1.95e+160)
		tmp = (1.0 / n) / x;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 1.16e-292], 0.0, If[LessEqual[x, 1.95e+160], N[(N[(1.0 / n), $MachinePrecision] / x), $MachinePrecision], 0.0]]

Derivation

1. Split input into 2 regimes

2. if x < 1.16e-292 or 1.95000000000000004e160 < x

   1. Initial program 70.5%

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

   3. Taylor expanded in x around inf 38.7%

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

      1. log-rec 38.7%

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

      2. mul-1-neg 38.7%

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

      3. neg-mul-1 38.7%

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

      4. mul-1-neg 38.7%

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

      5. distribute-frac-neg 38.7%

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

      6. remove-double-neg 38.7%

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

      7. *-rgt-identity 38.7%

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

      8. associate-/l* 38.7%

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

      9. exp-to-pow 70.1%

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

   5. Simplified 70.1%

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

   6. Taylor expanded in x around 0 70.2%

      \[\leadsto \color{blue}{0} \]

   if 1.16e-292 < x < 1.95000000000000004e160

   1. Initial program 43.3%

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

   3. Taylor expanded in x around inf 45.6%

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

      1. log-rec 45.6%

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

      2. mul-1-neg 45.6%

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

      3. neg-mul-1 45.6%

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

      4. mul-1-neg 45.6%

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

      5. distribute-frac-neg 45.6%

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

      6. remove-double-neg 45.6%

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

      7. *-rgt-identity 45.6%

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

      8. associate-/l* 45.6%

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

      9. exp-to-pow 45.6%

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

      10. *-commutative 45.6%

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

   5. Simplified 45.6%

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

      1. *-un-lft-identity 45.6%

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

      2. times-frac 45.9%

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

   7. Applied egg-rr 45.9%

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

      1. associate-*l/ 45.9%

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

      2. *-un-lft-identity 45.9%

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

   9. Applied egg-rr 45.9%

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

   10. Taylor expanded in n around inf 32.5%

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

Alternative 15: 51.4% accurate, 14.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.16 \cdot 10^{-292}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 1.95 \cdot 10^{+160}:\\ \;\;\;\;\frac{1}{n \cdot x}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 1.16e-292) 0.0 (if (<= x 1.95e+160) (/ 1.0 (* n x)) 0.0)))

C

double code(double x, double n) {
	double tmp;
	if (x <= 1.16e-292) {
		tmp = 0.0;
	} else if (x <= 1.95e+160) {
		tmp = 1.0 / (n * x);
	} else {
		tmp = 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 <= 1.16d-292) then
        tmp = 0.0d0
    else if (x <= 1.95d+160) then
        tmp = 1.0d0 / (n * x)
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (x <= 1.16e-292) {
		tmp = 0.0;
	} else if (x <= 1.95e+160) {
		tmp = 1.0 / (n * x);
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 1.16e-292:
		tmp = 0.0
	elif x <= 1.95e+160:
		tmp = 1.0 / (n * x)
	else:
		tmp = 0.0
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 1.16e-292)
		tmp = 0.0;
	elseif (x <= 1.95e+160)
		tmp = Float64(1.0 / Float64(n * x));
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (x <= 1.16e-292)
		tmp = 0.0;
	elseif (x <= 1.95e+160)
		tmp = 1.0 / (n * x);
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 1.16e-292], 0.0, If[LessEqual[x, 1.95e+160], N[(1.0 / N[(n * x), $MachinePrecision]), $MachinePrecision], 0.0]]

Derivation

1. Split input into 2 regimes

2. if x < 1.16e-292 or 1.95000000000000004e160 < x

   1. Initial program 70.5%

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

   3. Taylor expanded in x around inf 38.7%

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

      1. log-rec 38.7%

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

      2. mul-1-neg 38.7%

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

      3. neg-mul-1 38.7%

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

      4. mul-1-neg 38.7%

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

      5. distribute-frac-neg 38.7%

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

      6. remove-double-neg 38.7%

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

      7. *-rgt-identity 38.7%

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

      8. associate-/l* 38.7%

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

      9. exp-to-pow 70.1%

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

   5. Simplified 70.1%

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

   6. Taylor expanded in x around 0 70.2%

      \[\leadsto \color{blue}{0} \]

   if 1.16e-292 < x < 1.95000000000000004e160

   1. Initial program 43.3%

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

   3. Taylor expanded in x around inf 45.6%

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

      1. log-rec 45.6%

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

      2. mul-1-neg 45.6%

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

      3. neg-mul-1 45.6%

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

      4. mul-1-neg 45.6%

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

      5. distribute-frac-neg 45.6%

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

      6. remove-double-neg 45.6%

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

      7. *-rgt-identity 45.6%

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

      8. associate-/l* 45.6%

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

      9. exp-to-pow 45.6%

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

      10. *-commutative 45.6%

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

   5. Simplified 45.6%

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

   6. Taylor expanded in n around inf 32.5%

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

      1. *-commutative 32.5%

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

   8. Simplified 32.5%

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

4. Final simplification 46.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.16 \cdot 10^{-292}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 1.95 \cdot 10^{+160}:\\ \;\;\;\;\frac{1}{n \cdot x}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 41.2% accurate, 211.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (x n) :precision binary64 0.0)

C

double code(double x, double n) {
	return 0.0;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    code = 0.0d0
end function

Java

public static double code(double x, double n) {
	return 0.0;
}

Python

def code(x, n):
	return 0.0

Julia

function code(x, n)
	return 0.0
end

MATLAB

function tmp = code(x, n)
	tmp = 0.0;
end

Wolfram

code[x_, n_] := 0.0

Derivation

1. Initial program 53.6%

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

3. Taylor expanded in x around inf 22.3%

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

   1. log-rec 22.3%

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

   2. mul-1-neg 22.3%

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

   3. neg-mul-1 22.3%

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

   4. mul-1-neg 22.3%

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

   5. distribute-frac-neg 22.3%

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

   6. remove-double-neg 22.3%

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

   7. *-rgt-identity 22.3%

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

   8. associate-/l* 22.3%

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

   9. exp-to-pow 34.2%

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

5. Simplified 34.2%

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

6. Taylor expanded in x around 0 34.4%

   \[\leadsto \color{blue}{0} \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024172 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  :precision binary64
  (- (pow (+ x 1.0) (/ 1.0 n)) (pow x (/ 1.0 n))))