2nthrt (problem 3.4.6)

Percentage Accurate: 54.3% → 98.3%

Time: 18.0s

Alternatives: 14

Speedup: 1.9×

• could not determine a ground truth

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 54.3% 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: 98.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{-8}:\\ \;\;\;\;\frac{\frac{t_0}{n}}{x}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-10}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(\frac{1}{x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{x}{n}} - t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -1e-8)
     (/ (/ t_0 n) x)
     (if (<= (/ 1.0 n) 1e-10) (/ (log1p (/ 1.0 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) <= -1e-8) {
		tmp = (t_0 / n) / x;
	} else if ((1.0 / n) <= 1e-10) {
		tmp = log1p((1.0 / 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) <= -1e-8) {
		tmp = (t_0 / n) / x;
	} else if ((1.0 / n) <= 1e-10) {
		tmp = Math.log1p((1.0 / 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) <= -1e-8:
		tmp = (t_0 / n) / x
	elif (1.0 / n) <= 1e-10:
		tmp = math.log1p((1.0 / 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) <= -1e-8)
		tmp = Float64(Float64(t_0 / n) / x);
	elseif (Float64(1.0 / n) <= 1e-10)
		tmp = Float64(log1p(Float64(1.0 / 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], -1e-8], N[(N[(t$95$0 / n), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 1e-10], N[(N[Log[1 + N[(1.0 / 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 1 n) < -1e-8

   1. Initial program 99.0%

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

   2. Taylor expanded in x around inf 100.0%

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

      1. mul-1-neg 100.0%

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

      2. log-rec 100.0%

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

      3. mul-1-neg 100.0%

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

      4. distribute-neg-frac 100.0%

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

      5. mul-1-neg 100.0%

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

      6. remove-double-neg 100.0%

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

      7. *-commutative 100.0%

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

   4. Simplified 100.0%

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

      1. div-inv 100.0%

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

      2. pow-to-exp 100.0%

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

      3. *-un-lft-identity 100.0%

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

      4. times-frac 100.0%

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

   6. Applied egg-rr 100.0%

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

      1. associate-*l/ 100.0%

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

      2. *-un-lft-identity 100.0%

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

   8. Applied egg-rr 100.0%

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

   if -1e-8 < (/.f64 1 n) < 1.00000000000000004e-10

   1. Initial program 30.8%

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

   2. Taylor expanded in n around inf 72.7%

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

      1. +-rgt-identity 72.7%

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

      2. +-rgt-identity 72.7%

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

      3. log1p-def 72.7%

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

   4. Simplified 72.7%

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

      1. log1p-udef 72.7%

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

      2. diff-log 72.7%

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

      3. +-commutative 72.7%

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

   6. Applied egg-rr 72.7%

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

      1. log-div 72.7%

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

      2. +-commutative 72.7%

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

      3. log1p-udef 72.7%

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

      4. log1p-expm1-u 72.7%

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

      5. expm1-udef 72.7%

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

      6. log1p-udef 72.7%

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

      7. +-commutative 72.7%

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

      8. log-div 72.7%

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

      9. add-exp-log 72.7%

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

      10. +-commutative 72.7%

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

   8. Applied egg-rr 72.7%

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

      1. *-lft-identity 72.7%

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

      2. associate-*l/ 70.8%

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

      3. distribute-rgt-in 70.8%

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

      4. *-lft-identity 70.8%

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

      5. rgt-mult-inverse 72.7%

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

      6. associate--l+ 99.1%

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

      7. metadata-eval 99.1%

         \[\leadsto \frac{\mathsf{log1p}\left(\frac{1}{x} + \color{blue}{0}\right)}{n} \]

   10. Simplified 99.1%

       \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(\frac{1}{x} + 0\right)}}{n} \]

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

   1. Initial program 52.6%

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

   2. Taylor expanded in n around 0 52.6%

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

      1. log1p-def 100.0%

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

   4. Simplified 100.0%

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

   5. Taylor expanded in x around 0 100.0%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{-8}:\\ \;\;\;\;\frac{\frac{{x}^{\left(\frac{1}{n}\right)}}{n}}{x}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-10}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(\frac{1}{x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{x}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \]

Alternative 2: 94.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{-8}:\\ \;\;\;\;\frac{\frac{t_0}{n}}{x}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-10}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(\frac{1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{+190}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - t_0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{x}{n}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -1e-8)
     (/ (/ t_0 n) x)
     (if (<= (/ 1.0 n) 1e-10)
       (/ (log1p (/ 1.0 x)) n)
       (if (<= (/ 1.0 n) 1e+190)
         (- (+ 1.0 (/ x n)) t_0)
         (log1p (expm1 (/ x n))))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -1e-8) {
		tmp = (t_0 / n) / x;
	} else if ((1.0 / n) <= 1e-10) {
		tmp = log1p((1.0 / x)) / n;
	} else if ((1.0 / n) <= 1e+190) {
		tmp = (1.0 + (x / n)) - t_0;
	} else {
		tmp = log1p(expm1((x / n)));
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -1e-8) {
		tmp = (t_0 / n) / x;
	} else if ((1.0 / n) <= 1e-10) {
		tmp = Math.log1p((1.0 / x)) / n;
	} else if ((1.0 / n) <= 1e+190) {
		tmp = (1.0 + (x / n)) - t_0;
	} else {
		tmp = Math.log1p(Math.expm1((x / n)));
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	tmp = 0
	if (1.0 / n) <= -1e-8:
		tmp = (t_0 / n) / x
	elif (1.0 / n) <= 1e-10:
		tmp = math.log1p((1.0 / x)) / n
	elif (1.0 / n) <= 1e+190:
		tmp = (1.0 + (x / n)) - t_0
	else:
		tmp = math.log1p(math.expm1((x / n)))
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -1e-8)
		tmp = Float64(Float64(t_0 / n) / x);
	elseif (Float64(1.0 / n) <= 1e-10)
		tmp = Float64(log1p(Float64(1.0 / x)) / n);
	elseif (Float64(1.0 / n) <= 1e+190)
		tmp = Float64(Float64(1.0 + Float64(x / n)) - t_0);
	else
		tmp = log1p(expm1(Float64(x / n)));
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -1e-8], N[(N[(t$95$0 / n), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 1e-10], N[(N[Log[1 + N[(1.0 / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 1e+190], N[(N[(1.0 + N[(x / n), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], N[Log[1 + N[(Exp[N[(x / n), $MachinePrecision]] - 1), $MachinePrecision]], $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 1 n) < -1e-8

   1. Initial program 99.0%

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

   2. Taylor expanded in x around inf 100.0%

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

      1. mul-1-neg 100.0%

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

      2. log-rec 100.0%

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

      3. mul-1-neg 100.0%

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

      4. distribute-neg-frac 100.0%

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

      5. mul-1-neg 100.0%

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

      6. remove-double-neg 100.0%

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

      7. *-commutative 100.0%

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

   4. Simplified 100.0%

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

      1. div-inv 100.0%

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

      2. pow-to-exp 100.0%

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

      3. *-un-lft-identity 100.0%

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

      4. times-frac 100.0%

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

   6. Applied egg-rr 100.0%

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

      1. associate-*l/ 100.0%

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

      2. *-un-lft-identity 100.0%

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

   8. Applied egg-rr 100.0%

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

   if -1e-8 < (/.f64 1 n) < 1.00000000000000004e-10

   1. Initial program 30.8%

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

   2. Taylor expanded in n around inf 72.7%

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

      1. +-rgt-identity 72.7%

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

      2. +-rgt-identity 72.7%

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

      3. log1p-def 72.7%

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

   4. Simplified 72.7%

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

      1. log1p-udef 72.7%

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

      2. diff-log 72.7%

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

      3. +-commutative 72.7%

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

   6. Applied egg-rr 72.7%

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

      1. log-div 72.7%

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

      2. +-commutative 72.7%

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

      3. log1p-udef 72.7%

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

      4. log1p-expm1-u 72.7%

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

      5. expm1-udef 72.7%

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

      6. log1p-udef 72.7%

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

      7. +-commutative 72.7%

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

      8. log-div 72.7%

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

      9. add-exp-log 72.7%

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

      10. +-commutative 72.7%

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

   8. Applied egg-rr 72.7%

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

      1. *-lft-identity 72.7%

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

      2. associate-*l/ 70.8%

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

      3. distribute-rgt-in 70.8%

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

      4. *-lft-identity 70.8%

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

      5. rgt-mult-inverse 72.7%

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

      6. associate--l+ 99.1%

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

      7. metadata-eval 99.1%

         \[\leadsto \frac{\mathsf{log1p}\left(\frac{1}{x} + \color{blue}{0}\right)}{n} \]

   10. Simplified 99.1%

       \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(\frac{1}{x} + 0\right)}}{n} \]

   if 1.00000000000000004e-10 < (/.f64 1 n) < 1.0000000000000001e190

   1. Initial program 86.6%

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

   2. Taylor expanded in x around 0 88.9%

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

   if 1.0000000000000001e190 < (/.f64 1 n)

   1. Initial program 30.8%

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

   2. Taylor expanded in x around inf 0.3%

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

      1. mul-1-neg 0.3%

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

      2. log-rec 0.3%

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

      3. mul-1-neg 0.3%

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

      4. distribute-neg-frac 0.3%

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

      5. mul-1-neg 0.3%

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

      6. remove-double-neg 0.3%

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

      7. *-commutative 0.3%

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

   4. Simplified 0.3%

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

      1. div-inv 0.3%

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

      2. pow-to-exp 0.3%

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

      3. *-un-lft-identity 0.3%

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

      4. times-frac 1.8%

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

   6. Applied egg-rr 1.8%

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

   7. Taylor expanded in n around inf 63.1%

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

      1. associate-/r* 63.1%

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

   9. Simplified 63.1%

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

       1. associate-/l/ 63.1%

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

       2. inv-pow 63.1%

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

       3. metadata-eval 63.1%

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

       4. sqrt-pow2 63.1%

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

       5. log1p-expm1-u 82.7%

          \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left({\left(\sqrt{x \cdot n}\right)}^{-2}\right)\right)} \]

       6. sqrt-pow2 82.7%

          \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\color{blue}{{\left(x \cdot n\right)}^{\left(\frac{-2}{2}\right)}}\right)\right) \]

       7. metadata-eval 82.7%

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

       8. inv-pow 82.7%

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

       9. associate-/r* 82.7%

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

       10. add-exp-log 82.7%

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

       11. neg-log 82.7%

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

       12. add-sqr-sqrt 82.7%

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

       13. sqrt-unprod 82.7%

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

       14. sqr-neg 82.7%

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

       15. sqrt-unprod 0.0%

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

       16. add-sqr-sqrt 83.5%

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

       17. add-exp-log 83.5%

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

   11. Applied egg-rr 83.5%

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

4. Final simplification 97.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{-8}:\\ \;\;\;\;\frac{\frac{{x}^{\left(\frac{1}{n}\right)}}{n}}{x}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-10}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(\frac{1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{+190}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{x}{n}\right)\right)\\ \end{array} \]

Alternative 3: 66.5% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ t_1 := \frac{1}{n \cdot x}\\ t_2 := 1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{+93}:\\ \;\;\;\;\frac{0}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq -5 \cdot 10^{+75}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\frac{1}{n} \leq -5 \cdot 10^{-47}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\frac{1}{n} \leq -4 \cdot 10^{-119}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-10}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{+219}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (/ (log (/ (+ 1.0 x) x)) n))
        (t_1 (/ 1.0 (* n x)))
        (t_2 (- 1.0 (pow x (/ 1.0 n)))))
   (if (<= (/ 1.0 n) -2e+93)
     (/ 0.0 n)
     (if (<= (/ 1.0 n) -5e+75)
       t_2
       (if (<= (/ 1.0 n) -5e-47)
         t_0
         (if (<= (/ 1.0 n) -4e-119)
           t_1
           (if (<= (/ 1.0 n) 1e-10)
             t_0
             (if (<= (/ 1.0 n) 2e+219) t_2 t_1))))))))

C

double code(double x, double n) {
	double t_0 = log(((1.0 + x) / x)) / n;
	double t_1 = 1.0 / (n * x);
	double t_2 = 1.0 - pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -2e+93) {
		tmp = 0.0 / n;
	} else if ((1.0 / n) <= -5e+75) {
		tmp = t_2;
	} else if ((1.0 / n) <= -5e-47) {
		tmp = t_0;
	} else if ((1.0 / n) <= -4e-119) {
		tmp = t_1;
	} else if ((1.0 / n) <= 1e-10) {
		tmp = t_0;
	} else if ((1.0 / n) <= 2e+219) {
		tmp = t_2;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = log(((1.0d0 + x) / x)) / n
    t_1 = 1.0d0 / (n * x)
    t_2 = 1.0d0 - (x ** (1.0d0 / n))
    if ((1.0d0 / n) <= (-2d+93)) then
        tmp = 0.0d0 / n
    else if ((1.0d0 / n) <= (-5d+75)) then
        tmp = t_2
    else if ((1.0d0 / n) <= (-5d-47)) then
        tmp = t_0
    else if ((1.0d0 / n) <= (-4d-119)) then
        tmp = t_1
    else if ((1.0d0 / n) <= 1d-10) then
        tmp = t_0
    else if ((1.0d0 / n) <= 2d+219) then
        tmp = t_2
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = Math.log(((1.0 + x) / x)) / n;
	double t_1 = 1.0 / (n * x);
	double t_2 = 1.0 - Math.pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -2e+93) {
		tmp = 0.0 / n;
	} else if ((1.0 / n) <= -5e+75) {
		tmp = t_2;
	} else if ((1.0 / n) <= -5e-47) {
		tmp = t_0;
	} else if ((1.0 / n) <= -4e-119) {
		tmp = t_1;
	} else if ((1.0 / n) <= 1e-10) {
		tmp = t_0;
	} else if ((1.0 / n) <= 2e+219) {
		tmp = t_2;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.log(((1.0 + x) / x)) / n
	t_1 = 1.0 / (n * x)
	t_2 = 1.0 - math.pow(x, (1.0 / n))
	tmp = 0
	if (1.0 / n) <= -2e+93:
		tmp = 0.0 / n
	elif (1.0 / n) <= -5e+75:
		tmp = t_2
	elif (1.0 / n) <= -5e-47:
		tmp = t_0
	elif (1.0 / n) <= -4e-119:
		tmp = t_1
	elif (1.0 / n) <= 1e-10:
		tmp = t_0
	elif (1.0 / n) <= 2e+219:
		tmp = t_2
	else:
		tmp = t_1
	return tmp

Julia

function code(x, n)
	t_0 = Float64(log(Float64(Float64(1.0 + x) / x)) / n)
	t_1 = Float64(1.0 / Float64(n * x))
	t_2 = Float64(1.0 - (x ^ Float64(1.0 / n)))
	tmp = 0.0
	if (Float64(1.0 / n) <= -2e+93)
		tmp = Float64(0.0 / n);
	elseif (Float64(1.0 / n) <= -5e+75)
		tmp = t_2;
	elseif (Float64(1.0 / n) <= -5e-47)
		tmp = t_0;
	elseif (Float64(1.0 / n) <= -4e-119)
		tmp = t_1;
	elseif (Float64(1.0 / n) <= 1e-10)
		tmp = t_0;
	elseif (Float64(1.0 / n) <= 2e+219)
		tmp = t_2;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = log(((1.0 + x) / x)) / n;
	t_1 = 1.0 / (n * x);
	t_2 = 1.0 - (x ^ (1.0 / n));
	tmp = 0.0;
	if ((1.0 / n) <= -2e+93)
		tmp = 0.0 / n;
	elseif ((1.0 / n) <= -5e+75)
		tmp = t_2;
	elseif ((1.0 / n) <= -5e-47)
		tmp = t_0;
	elseif ((1.0 / n) <= -4e-119)
		tmp = t_1;
	elseif ((1.0 / n) <= 1e-10)
		tmp = t_0;
	elseif ((1.0 / n) <= 2e+219)
		tmp = t_2;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(N[Log[N[(N[(1.0 + x), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 / N[(n * x), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(1.0 - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -2e+93], N[(0.0 / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], -5e+75], t$95$2, If[LessEqual[N[(1.0 / n), $MachinePrecision], -5e-47], t$95$0, If[LessEqual[N[(1.0 / n), $MachinePrecision], -4e-119], t$95$1, If[LessEqual[N[(1.0 / n), $MachinePrecision], 1e-10], t$95$0, If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e+219], t$95$2, t$95$1]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 1 n) < -2.00000000000000009e93

   1. Initial program 100.0%

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

   2. Taylor expanded in n around inf 63.6%

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

      1. +-rgt-identity 63.6%

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

      2. +-rgt-identity 63.6%

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

      3. log1p-def 63.6%

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

   4. Simplified 63.6%

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

      1. log1p-udef 63.6%

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

      2. diff-log 62.2%

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

      3. +-commutative 62.2%

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

   6. Applied egg-rr 62.2%

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

   7. Taylor expanded in x around inf 63.3%

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

   if -2.00000000000000009e93 < (/.f64 1 n) < -5.0000000000000002e75 or 1.00000000000000004e-10 < (/.f64 1 n) < 1.99999999999999993e219

   1. Initial program 85.0%

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

   2. Taylor expanded in x around 0 81.6%

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

   if -5.0000000000000002e75 < (/.f64 1 n) < -5.00000000000000011e-47 or -4.00000000000000005e-119 < (/.f64 1 n) < 1.00000000000000004e-10

   1. Initial program 43.7%

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

   2. Taylor expanded in n around inf 75.1%

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

      1. +-rgt-identity 75.1%

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

      2. +-rgt-identity 75.1%

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

      3. log1p-def 75.1%

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

   4. Simplified 75.1%

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

      1. log1p-udef 75.1%

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

      2. diff-log 75.1%

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

      3. +-commutative 75.1%

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

   6. Applied egg-rr 75.1%

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

   if -5.00000000000000011e-47 < (/.f64 1 n) < -4.00000000000000005e-119 or 1.99999999999999993e219 < (/.f64 1 n)

   1. Initial program 19.9%

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

   2. Taylor expanded in n around inf 20.2%

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

      1. +-rgt-identity 20.2%

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

      2. +-rgt-identity 20.2%

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

      3. log1p-def 20.2%

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

   4. Simplified 20.2%

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

   5. Taylor expanded in x around inf 76.7%

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

      1. *-commutative 76.7%

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

   7. Simplified 76.7%

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

4. Final simplification 73.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{+93}:\\ \;\;\;\;\frac{0}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq -5 \cdot 10^{+75}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;\frac{1}{n} \leq -5 \cdot 10^{-47}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq -4 \cdot 10^{-119}:\\ \;\;\;\;\frac{1}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-10}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{+219}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{n \cdot x}\\ \end{array} \]

Alternative 4: 79.6% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{+93}:\\ \;\;\;\;\frac{0}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq -5 \cdot 10^{+75}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\frac{1}{n} \leq -100:\\ \;\;\;\;\frac{0}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-10}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(\frac{1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{+219}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{n \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (- 1.0 (pow x (/ 1.0 n)))))
   (if (<= (/ 1.0 n) -2e+93)
     (/ 0.0 n)
     (if (<= (/ 1.0 n) -5e+75)
       t_0
       (if (<= (/ 1.0 n) -100.0)
         (/ 0.0 n)
         (if (<= (/ 1.0 n) 1e-10)
           (/ (log1p (/ 1.0 x)) n)
           (if (<= (/ 1.0 n) 2e+219) t_0 (/ 1.0 (* n x)))))))))

C

double code(double x, double n) {
	double t_0 = 1.0 - pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -2e+93) {
		tmp = 0.0 / n;
	} else if ((1.0 / n) <= -5e+75) {
		tmp = t_0;
	} else if ((1.0 / n) <= -100.0) {
		tmp = 0.0 / n;
	} else if ((1.0 / n) <= 1e-10) {
		tmp = log1p((1.0 / x)) / n;
	} else if ((1.0 / n) <= 2e+219) {
		tmp = t_0;
	} else {
		tmp = 1.0 / (n * x);
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double t_0 = 1.0 - Math.pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -2e+93) {
		tmp = 0.0 / n;
	} else if ((1.0 / n) <= -5e+75) {
		tmp = t_0;
	} else if ((1.0 / n) <= -100.0) {
		tmp = 0.0 / n;
	} else if ((1.0 / n) <= 1e-10) {
		tmp = Math.log1p((1.0 / x)) / n;
	} else if ((1.0 / n) <= 2e+219) {
		tmp = t_0;
	} else {
		tmp = 1.0 / (n * x);
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = 1.0 - math.pow(x, (1.0 / n))
	tmp = 0
	if (1.0 / n) <= -2e+93:
		tmp = 0.0 / n
	elif (1.0 / n) <= -5e+75:
		tmp = t_0
	elif (1.0 / n) <= -100.0:
		tmp = 0.0 / n
	elif (1.0 / n) <= 1e-10:
		tmp = math.log1p((1.0 / x)) / n
	elif (1.0 / n) <= 2e+219:
		tmp = t_0
	else:
		tmp = 1.0 / (n * x)
	return tmp

Julia

function code(x, n)
	t_0 = Float64(1.0 - (x ^ Float64(1.0 / n)))
	tmp = 0.0
	if (Float64(1.0 / n) <= -2e+93)
		tmp = Float64(0.0 / n);
	elseif (Float64(1.0 / n) <= -5e+75)
		tmp = t_0;
	elseif (Float64(1.0 / n) <= -100.0)
		tmp = Float64(0.0 / n);
	elseif (Float64(1.0 / n) <= 1e-10)
		tmp = Float64(log1p(Float64(1.0 / x)) / n);
	elseif (Float64(1.0 / n) <= 2e+219)
		tmp = t_0;
	else
		tmp = Float64(1.0 / Float64(n * x));
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(1.0 - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -2e+93], N[(0.0 / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], -5e+75], t$95$0, If[LessEqual[N[(1.0 / n), $MachinePrecision], -100.0], N[(0.0 / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 1e-10], N[(N[Log[1 + N[(1.0 / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e+219], t$95$0, N[(1.0 / N[(n * x), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 1 n) < -2.00000000000000009e93 or -5.0000000000000002e75 < (/.f64 1 n) < -100

   1. Initial program 100.0%

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

   2. Taylor expanded in n around inf 64.6%

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

      1. +-rgt-identity 64.6%

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

      2. +-rgt-identity 64.6%

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

      3. log1p-def 64.6%

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

   4. Simplified 64.6%

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

      1. log1p-udef 64.6%

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

      2. diff-log 63.5%

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

      3. +-commutative 63.5%

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

   6. Applied egg-rr 63.5%

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

   7. Taylor expanded in x around inf 65.3%

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

   if -2.00000000000000009e93 < (/.f64 1 n) < -5.0000000000000002e75 or 1.00000000000000004e-10 < (/.f64 1 n) < 1.99999999999999993e219

   1. Initial program 85.0%

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

   2. Taylor expanded in x around 0 81.6%

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

   if -100 < (/.f64 1 n) < 1.00000000000000004e-10

   1. Initial program 30.6%

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

   2. Taylor expanded in n around inf 72.1%

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

      1. +-rgt-identity 72.1%

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

      2. +-rgt-identity 72.1%

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

      3. log1p-def 72.1%

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

   4. Simplified 72.1%

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

      1. log1p-udef 72.1%

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

      2. diff-log 72.2%

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

      3. +-commutative 72.2%

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

   6. Applied egg-rr 72.2%

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

      1. log-div 72.1%

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

      2. +-commutative 72.1%

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

      3. log1p-udef 72.1%

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

      4. log1p-expm1-u 72.1%

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

      5. expm1-udef 72.1%

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

      6. log1p-udef 72.1%

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

      7. +-commutative 72.1%

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

      8. log-div 72.2%

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

      9. add-exp-log 72.2%

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

      10. +-commutative 72.2%

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

   8. Applied egg-rr 72.2%

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

      1. *-lft-identity 72.2%

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

      2. associate-*l/ 70.3%

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

      3. distribute-rgt-in 70.3%

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

      4. *-lft-identity 70.3%

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

      5. rgt-mult-inverse 72.2%

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

      6. associate--l+ 98.7%

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

      7. metadata-eval 98.7%

         \[\leadsto \frac{\mathsf{log1p}\left(\frac{1}{x} + \color{blue}{0}\right)}{n} \]

   10. Simplified 98.7%

       \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(\frac{1}{x} + 0\right)}}{n} \]

   if 1.99999999999999993e219 < (/.f64 1 n)

   1. Initial program 20.7%

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

   2. Taylor expanded in n around inf 8.1%

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

      1. +-rgt-identity 8.1%

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

      2. +-rgt-identity 8.1%

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

      3. log1p-def 8.1%

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

   4. Simplified 8.1%

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

   5. Taylor expanded in x around inf 74.6%

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

      1. *-commutative 74.6%

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

   7. Simplified 74.6%

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

4. Final simplification 83.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{+93}:\\ \;\;\;\;\frac{0}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq -5 \cdot 10^{+75}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;\frac{1}{n} \leq -100:\\ \;\;\;\;\frac{0}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-10}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(\frac{1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{+219}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{n \cdot x}\\ \end{array} \]

Alternative 5: 93.3% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{-8}:\\ \;\;\;\;\frac{\frac{t_0}{n}}{x}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-10}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(\frac{1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{+219}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{n \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -1e-8)
     (/ (/ t_0 n) x)
     (if (<= (/ 1.0 n) 1e-10)
       (/ (log1p (/ 1.0 x)) n)
       (if (<= (/ 1.0 n) 2e+219) (- (+ 1.0 (/ x n)) t_0) (/ 1.0 (* n x)))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -1e-8) {
		tmp = (t_0 / n) / x;
	} else if ((1.0 / n) <= 1e-10) {
		tmp = log1p((1.0 / x)) / n;
	} else if ((1.0 / n) <= 2e+219) {
		tmp = (1.0 + (x / n)) - t_0;
	} else {
		tmp = 1.0 / (n * x);
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -1e-8) {
		tmp = (t_0 / n) / x;
	} else if ((1.0 / n) <= 1e-10) {
		tmp = Math.log1p((1.0 / x)) / n;
	} else if ((1.0 / n) <= 2e+219) {
		tmp = (1.0 + (x / n)) - t_0;
	} else {
		tmp = 1.0 / (n * x);
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	tmp = 0
	if (1.0 / n) <= -1e-8:
		tmp = (t_0 / n) / x
	elif (1.0 / n) <= 1e-10:
		tmp = math.log1p((1.0 / x)) / n
	elif (1.0 / n) <= 2e+219:
		tmp = (1.0 + (x / n)) - t_0
	else:
		tmp = 1.0 / (n * x)
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -1e-8)
		tmp = Float64(Float64(t_0 / n) / x);
	elseif (Float64(1.0 / n) <= 1e-10)
		tmp = Float64(log1p(Float64(1.0 / x)) / n);
	elseif (Float64(1.0 / n) <= 2e+219)
		tmp = Float64(Float64(1.0 + Float64(x / n)) - t_0);
	else
		tmp = Float64(1.0 / Float64(n * x));
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -1e-8], N[(N[(t$95$0 / n), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 1e-10], N[(N[Log[1 + N[(1.0 / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e+219], N[(N[(1.0 + N[(x / n), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], N[(1.0 / N[(n * x), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 1 n) < -1e-8

   1. Initial program 99.0%

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

   2. Taylor expanded in x around inf 100.0%

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

      1. mul-1-neg 100.0%

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

      2. log-rec 100.0%

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

      3. mul-1-neg 100.0%

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

      4. distribute-neg-frac 100.0%

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

      5. mul-1-neg 100.0%

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

      6. remove-double-neg 100.0%

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

      7. *-commutative 100.0%

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

   4. Simplified 100.0%

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

      1. div-inv 100.0%

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

      2. pow-to-exp 100.0%

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

      3. *-un-lft-identity 100.0%

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

      4. times-frac 100.0%

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

   6. Applied egg-rr 100.0%

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

      1. associate-*l/ 100.0%

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

      2. *-un-lft-identity 100.0%

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

   8. Applied egg-rr 100.0%

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

   if -1e-8 < (/.f64 1 n) < 1.00000000000000004e-10

   1. Initial program 30.8%

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

   2. Taylor expanded in n around inf 72.7%

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

      1. +-rgt-identity 72.7%

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

      2. +-rgt-identity 72.7%

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

      3. log1p-def 72.7%

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

   4. Simplified 72.7%

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

      1. log1p-udef 72.7%

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

      2. diff-log 72.7%

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

      3. +-commutative 72.7%

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

   6. Applied egg-rr 72.7%

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

      1. log-div 72.7%

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

      2. +-commutative 72.7%

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

      3. log1p-udef 72.7%

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

      4. log1p-expm1-u 72.7%

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

      5. expm1-udef 72.7%

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

      6. log1p-udef 72.7%

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

      7. +-commutative 72.7%

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

      8. log-div 72.7%

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

      9. add-exp-log 72.7%

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

      10. +-commutative 72.7%

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

   8. Applied egg-rr 72.7%

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

      1. *-lft-identity 72.7%

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

      2. associate-*l/ 70.8%

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

      3. distribute-rgt-in 70.8%

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

      4. *-lft-identity 70.8%

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

      5. rgt-mult-inverse 72.7%

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

      6. associate--l+ 99.1%

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

      7. metadata-eval 99.1%

         \[\leadsto \frac{\mathsf{log1p}\left(\frac{1}{x} + \color{blue}{0}\right)}{n} \]

   10. Simplified 99.1%

       \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(\frac{1}{x} + 0\right)}}{n} \]

   if 1.00000000000000004e-10 < (/.f64 1 n) < 1.99999999999999993e219

   1. Initial program 81.8%

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

   2. Taylor expanded in x around 0 79.7%

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

   if 1.99999999999999993e219 < (/.f64 1 n)

   1. Initial program 20.7%

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

   2. Taylor expanded in n around inf 8.1%

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

      1. +-rgt-identity 8.1%

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

      2. +-rgt-identity 8.1%

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

      3. log1p-def 8.1%

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

   4. Simplified 8.1%

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

   5. Taylor expanded in x around inf 74.6%

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

      1. *-commutative 74.6%

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

   7. Simplified 74.6%

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

4. Final simplification 95.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{-8}:\\ \;\;\;\;\frac{\frac{{x}^{\left(\frac{1}{n}\right)}}{n}}{x}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-10}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(\frac{1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{+219}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{n \cdot x}\\ \end{array} \]

Alternative 6: 93.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{-8}:\\ \;\;\;\;\frac{\frac{t_0}{n}}{x}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-10}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(\frac{1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{+219}:\\ \;\;\;\;1 - t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{n \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -1e-8)
     (/ (/ t_0 n) x)
     (if (<= (/ 1.0 n) 1e-10)
       (/ (log1p (/ 1.0 x)) n)
       (if (<= (/ 1.0 n) 2e+219) (- 1.0 t_0) (/ 1.0 (* n x)))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -1e-8) {
		tmp = (t_0 / n) / x;
	} else if ((1.0 / n) <= 1e-10) {
		tmp = log1p((1.0 / x)) / n;
	} else if ((1.0 / n) <= 2e+219) {
		tmp = 1.0 - t_0;
	} else {
		tmp = 1.0 / (n * x);
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -1e-8) {
		tmp = (t_0 / n) / x;
	} else if ((1.0 / n) <= 1e-10) {
		tmp = Math.log1p((1.0 / x)) / n;
	} else if ((1.0 / n) <= 2e+219) {
		tmp = 1.0 - t_0;
	} else {
		tmp = 1.0 / (n * x);
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	tmp = 0
	if (1.0 / n) <= -1e-8:
		tmp = (t_0 / n) / x
	elif (1.0 / n) <= 1e-10:
		tmp = math.log1p((1.0 / x)) / n
	elif (1.0 / n) <= 2e+219:
		tmp = 1.0 - t_0
	else:
		tmp = 1.0 / (n * x)
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -1e-8)
		tmp = Float64(Float64(t_0 / n) / x);
	elseif (Float64(1.0 / n) <= 1e-10)
		tmp = Float64(log1p(Float64(1.0 / x)) / n);
	elseif (Float64(1.0 / n) <= 2e+219)
		tmp = Float64(1.0 - t_0);
	else
		tmp = Float64(1.0 / Float64(n * x));
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -1e-8], N[(N[(t$95$0 / n), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 1e-10], N[(N[Log[1 + N[(1.0 / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e+219], N[(1.0 - t$95$0), $MachinePrecision], N[(1.0 / N[(n * x), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 1 n) < -1e-8

   1. Initial program 99.0%

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

   2. Taylor expanded in x around inf 100.0%

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

      1. mul-1-neg 100.0%

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

      2. log-rec 100.0%

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

      3. mul-1-neg 100.0%

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

      4. distribute-neg-frac 100.0%

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

      5. mul-1-neg 100.0%

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

      6. remove-double-neg 100.0%

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

      7. *-commutative 100.0%

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

   4. Simplified 100.0%

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

      1. div-inv 100.0%

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

      2. pow-to-exp 100.0%

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

      3. *-un-lft-identity 100.0%

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

      4. times-frac 100.0%

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

   6. Applied egg-rr 100.0%

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

      1. associate-*l/ 100.0%

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

      2. *-un-lft-identity 100.0%

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

   8. Applied egg-rr 100.0%

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

   if -1e-8 < (/.f64 1 n) < 1.00000000000000004e-10

   1. Initial program 30.8%

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

   2. Taylor expanded in n around inf 72.7%

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

      1. +-rgt-identity 72.7%

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

      2. +-rgt-identity 72.7%

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

      3. log1p-def 72.7%

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

   4. Simplified 72.7%

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

      1. log1p-udef 72.7%

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

      2. diff-log 72.7%

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

      3. +-commutative 72.7%

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

   6. Applied egg-rr 72.7%

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

      1. log-div 72.7%

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

      2. +-commutative 72.7%

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

      3. log1p-udef 72.7%

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

      4. log1p-expm1-u 72.7%

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

      5. expm1-udef 72.7%

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

      6. log1p-udef 72.7%

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

      7. +-commutative 72.7%

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

      8. log-div 72.7%

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

      9. add-exp-log 72.7%

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

      10. +-commutative 72.7%

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

   8. Applied egg-rr 72.7%

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

      1. *-lft-identity 72.7%

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

      2. associate-*l/ 70.8%

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

      3. distribute-rgt-in 70.8%

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

      4. *-lft-identity 70.8%

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

      5. rgt-mult-inverse 72.7%

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

      6. associate--l+ 99.1%

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

      7. metadata-eval 99.1%

         \[\leadsto \frac{\mathsf{log1p}\left(\frac{1}{x} + \color{blue}{0}\right)}{n} \]

   10. Simplified 99.1%

       \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(\frac{1}{x} + 0\right)}}{n} \]

   if 1.00000000000000004e-10 < (/.f64 1 n) < 1.99999999999999993e219

   1. Initial program 81.8%

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

   2. Taylor expanded in x around 0 77.8%

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

   if 1.99999999999999993e219 < (/.f64 1 n)

   1. Initial program 20.7%

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

   2. Taylor expanded in n around inf 8.1%

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

      1. +-rgt-identity 8.1%

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

      2. +-rgt-identity 8.1%

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

      3. log1p-def 8.1%

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

   4. Simplified 8.1%

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

   5. Taylor expanded in x around inf 74.6%

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

      1. *-commutative 74.6%

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

   7. Simplified 74.6%

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

4. Final simplification 95.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{-8}:\\ \;\;\;\;\frac{\frac{{x}^{\left(\frac{1}{n}\right)}}{n}}{x}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-10}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(\frac{1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{+219}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{n \cdot x}\\ \end{array} \]

Alternative 7: 56.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - {x}^{\left(\frac{1}{n}\right)}\\ t_1 := \frac{-\log x}{n}\\ \mathbf{if}\;x \leq 1.05 \cdot 10^{-236}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 3.5 \cdot 10^{-184}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 1.5 \cdot 10^{-136}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 2.3 \cdot 10^{-90}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 6.8 \cdot 10^{-19}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 1.25 \cdot 10^{-5}:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{elif}\;x \leq 2.7 \cdot 10^{+115}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (- 1.0 (pow x (/ 1.0 n)))) (t_1 (/ (- (log x)) n)))
   (if (<= x 1.05e-236)
     t_0
     (if (<= x 3.5e-184)
       t_1
       (if (<= x 1.5e-136)
         t_0
         (if (<= x 2.3e-90)
           t_1
           (if (<= x 6.8e-19)
             t_0
             (if (<= x 1.25e-5)
               (/ (- x (log x)) n)
               (if (<= x 2.7e+115) (/ (/ 1.0 x) n) (/ 0.0 n))))))))))

C

double code(double x, double n) {
	double t_0 = 1.0 - pow(x, (1.0 / n));
	double t_1 = -log(x) / n;
	double tmp;
	if (x <= 1.05e-236) {
		tmp = t_0;
	} else if (x <= 3.5e-184) {
		tmp = t_1;
	} else if (x <= 1.5e-136) {
		tmp = t_0;
	} else if (x <= 2.3e-90) {
		tmp = t_1;
	} else if (x <= 6.8e-19) {
		tmp = t_0;
	} else if (x <= 1.25e-5) {
		tmp = (x - log(x)) / n;
	} else if (x <= 2.7e+115) {
		tmp = (1.0 / x) / n;
	} else {
		tmp = 0.0 / n;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = 1.0d0 - (x ** (1.0d0 / n))
    t_1 = -log(x) / n
    if (x <= 1.05d-236) then
        tmp = t_0
    else if (x <= 3.5d-184) then
        tmp = t_1
    else if (x <= 1.5d-136) then
        tmp = t_0
    else if (x <= 2.3d-90) then
        tmp = t_1
    else if (x <= 6.8d-19) then
        tmp = t_0
    else if (x <= 1.25d-5) then
        tmp = (x - log(x)) / n
    else if (x <= 2.7d+115) then
        tmp = (1.0d0 / x) / n
    else
        tmp = 0.0d0 / n
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = 1.0 - Math.pow(x, (1.0 / n));
	double t_1 = -Math.log(x) / n;
	double tmp;
	if (x <= 1.05e-236) {
		tmp = t_0;
	} else if (x <= 3.5e-184) {
		tmp = t_1;
	} else if (x <= 1.5e-136) {
		tmp = t_0;
	} else if (x <= 2.3e-90) {
		tmp = t_1;
	} else if (x <= 6.8e-19) {
		tmp = t_0;
	} else if (x <= 1.25e-5) {
		tmp = (x - Math.log(x)) / n;
	} else if (x <= 2.7e+115) {
		tmp = (1.0 / x) / n;
	} else {
		tmp = 0.0 / n;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = 1.0 - math.pow(x, (1.0 / n))
	t_1 = -math.log(x) / n
	tmp = 0
	if x <= 1.05e-236:
		tmp = t_0
	elif x <= 3.5e-184:
		tmp = t_1
	elif x <= 1.5e-136:
		tmp = t_0
	elif x <= 2.3e-90:
		tmp = t_1
	elif x <= 6.8e-19:
		tmp = t_0
	elif x <= 1.25e-5:
		tmp = (x - math.log(x)) / n
	elif x <= 2.7e+115:
		tmp = (1.0 / x) / n
	else:
		tmp = 0.0 / n
	return tmp

Julia

function code(x, n)
	t_0 = Float64(1.0 - (x ^ Float64(1.0 / n)))
	t_1 = Float64(Float64(-log(x)) / n)
	tmp = 0.0
	if (x <= 1.05e-236)
		tmp = t_0;
	elseif (x <= 3.5e-184)
		tmp = t_1;
	elseif (x <= 1.5e-136)
		tmp = t_0;
	elseif (x <= 2.3e-90)
		tmp = t_1;
	elseif (x <= 6.8e-19)
		tmp = t_0;
	elseif (x <= 1.25e-5)
		tmp = Float64(Float64(x - log(x)) / n);
	elseif (x <= 2.7e+115)
		tmp = Float64(Float64(1.0 / x) / n);
	else
		tmp = Float64(0.0 / n);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = 1.0 - (x ^ (1.0 / n));
	t_1 = -log(x) / n;
	tmp = 0.0;
	if (x <= 1.05e-236)
		tmp = t_0;
	elseif (x <= 3.5e-184)
		tmp = t_1;
	elseif (x <= 1.5e-136)
		tmp = t_0;
	elseif (x <= 2.3e-90)
		tmp = t_1;
	elseif (x <= 6.8e-19)
		tmp = t_0;
	elseif (x <= 1.25e-5)
		tmp = (x - log(x)) / n;
	elseif (x <= 2.7e+115)
		tmp = (1.0 / x) / n;
	else
		tmp = 0.0 / n;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(1.0 - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[((-N[Log[x], $MachinePrecision]) / n), $MachinePrecision]}, If[LessEqual[x, 1.05e-236], t$95$0, If[LessEqual[x, 3.5e-184], t$95$1, If[LessEqual[x, 1.5e-136], t$95$0, If[LessEqual[x, 2.3e-90], t$95$1, If[LessEqual[x, 6.8e-19], t$95$0, If[LessEqual[x, 1.25e-5], N[(N[(x - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[x, 2.7e+115], N[(N[(1.0 / x), $MachinePrecision] / n), $MachinePrecision], N[(0.0 / n), $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if x < 1.04999999999999989e-236 or 3.49999999999999981e-184 < x < 1.4999999999999999e-136 or 2.2999999999999998e-90 < x < 6.8000000000000004e-19

   1. Initial program 59.3%

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

   2. Taylor expanded in x around 0 59.3%

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

   if 1.04999999999999989e-236 < x < 3.49999999999999981e-184 or 1.4999999999999999e-136 < x < 2.2999999999999998e-90

   1. Initial program 24.4%

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

   2. Taylor expanded in x around 0 24.4%

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

   3. Taylor expanded in n around inf 60.9%

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

      1. neg-mul-1 60.9%

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

      2. distribute-neg-frac 60.9%

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

   5. Simplified 60.9%

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

   if 6.8000000000000004e-19 < x < 1.25000000000000006e-5

   1. Initial program 5.5%

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

   2. Taylor expanded in x around 0 5.5%

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

   3. Taylor expanded in n around inf 99.5%

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

   if 1.25000000000000006e-5 < x < 2.70000000000000004e115

   1. Initial program 55.8%

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

   2. Taylor expanded in n around inf 46.0%

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

      1. +-rgt-identity 46.0%

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

      2. +-rgt-identity 46.0%

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

      3. log1p-def 46.0%

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

   4. Simplified 46.0%

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

   5. Taylor expanded in x around inf 58.3%

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

   if 2.70000000000000004e115 < x

   1. Initial program 88.6%

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

   2. Taylor expanded in n around inf 88.6%

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

      1. +-rgt-identity 88.6%

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

      2. +-rgt-identity 88.6%

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

      3. log1p-def 88.6%

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

   4. Simplified 88.6%

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

      1. log1p-udef 88.6%

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

      2. diff-log 88.6%

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

      3. +-commutative 88.6%

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

   6. Applied egg-rr 88.6%

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

   7. Taylor expanded in x around inf 88.6%

      \[\leadsto \frac{\log \color{blue}{1}}{n} \]
3. Recombined 5 regimes into one program.

4. Final simplification 68.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.05 \cdot 10^{-236}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;x \leq 3.5 \cdot 10^{-184}:\\ \;\;\;\;\frac{-\log x}{n}\\ \mathbf{elif}\;x \leq 1.5 \cdot 10^{-136}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;x \leq 2.3 \cdot 10^{-90}:\\ \;\;\;\;\frac{-\log x}{n}\\ \mathbf{elif}\;x \leq 6.8 \cdot 10^{-19}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;x \leq 1.25 \cdot 10^{-5}:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{elif}\;x \leq 2.7 \cdot 10^{+115}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{n}\\ \end{array} \]

Alternative 8: 59.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 4.8 \cdot 10^{-184}:\\ \;\;\;\;\frac{-\log x}{n}\\ \mathbf{elif}\;x \leq 5.2 \cdot 10^{-169}:\\ \;\;\;\;\frac{1}{n \cdot x}\\ \mathbf{elif}\;x \leq 1.25 \cdot 10^{-5}:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{elif}\;x \leq 2.4 \cdot 10^{+115}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 4.8e-184)
   (/ (- (log x)) n)
   (if (<= x 5.2e-169)
     (/ 1.0 (* n x))
     (if (<= x 1.25e-5)
       (/ (- x (log x)) n)
       (if (<= x 2.4e+115) (/ (/ 1.0 x) n) (/ 0.0 n))))))

C

double code(double x, double n) {
	double tmp;
	if (x <= 4.8e-184) {
		tmp = -log(x) / n;
	} else if (x <= 5.2e-169) {
		tmp = 1.0 / (n * x);
	} else if (x <= 1.25e-5) {
		tmp = (x - log(x)) / n;
	} else if (x <= 2.4e+115) {
		tmp = (1.0 / x) / n;
	} else {
		tmp = 0.0 / n;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if (x <= 4.8d-184) then
        tmp = -log(x) / n
    else if (x <= 5.2d-169) then
        tmp = 1.0d0 / (n * x)
    else if (x <= 1.25d-5) then
        tmp = (x - log(x)) / n
    else if (x <= 2.4d+115) then
        tmp = (1.0d0 / x) / n
    else
        tmp = 0.0d0 / n
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (x <= 4.8e-184) {
		tmp = -Math.log(x) / n;
	} else if (x <= 5.2e-169) {
		tmp = 1.0 / (n * x);
	} else if (x <= 1.25e-5) {
		tmp = (x - Math.log(x)) / n;
	} else if (x <= 2.4e+115) {
		tmp = (1.0 / x) / n;
	} else {
		tmp = 0.0 / n;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 4.8e-184:
		tmp = -math.log(x) / n
	elif x <= 5.2e-169:
		tmp = 1.0 / (n * x)
	elif x <= 1.25e-5:
		tmp = (x - math.log(x)) / n
	elif x <= 2.4e+115:
		tmp = (1.0 / x) / n
	else:
		tmp = 0.0 / n
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 4.8e-184)
		tmp = Float64(Float64(-log(x)) / n);
	elseif (x <= 5.2e-169)
		tmp = Float64(1.0 / Float64(n * x));
	elseif (x <= 1.25e-5)
		tmp = Float64(Float64(x - log(x)) / n);
	elseif (x <= 2.4e+115)
		tmp = Float64(Float64(1.0 / x) / n);
	else
		tmp = Float64(0.0 / n);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (x <= 4.8e-184)
		tmp = -log(x) / n;
	elseif (x <= 5.2e-169)
		tmp = 1.0 / (n * x);
	elseif (x <= 1.25e-5)
		tmp = (x - log(x)) / n;
	elseif (x <= 2.4e+115)
		tmp = (1.0 / x) / n;
	else
		tmp = 0.0 / n;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 4.8e-184], N[((-N[Log[x], $MachinePrecision]) / n), $MachinePrecision], If[LessEqual[x, 5.2e-169], N[(1.0 / N[(n * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.25e-5], N[(N[(x - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[x, 2.4e+115], N[(N[(1.0 / x), $MachinePrecision] / n), $MachinePrecision], N[(0.0 / n), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if x < 4.80000000000000049e-184

   1. Initial program 47.3%

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

   2. Taylor expanded in x around 0 47.3%

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

   3. Taylor expanded in n around inf 44.7%

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

      1. neg-mul-1 44.7%

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

      2. distribute-neg-frac 44.7%

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

   5. Simplified 44.7%

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

   if 4.80000000000000049e-184 < x < 5.20000000000000028e-169

   1. Initial program 48.1%

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

   2. Taylor expanded in n around inf 23.9%

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

      1. +-rgt-identity 23.9%

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

      2. +-rgt-identity 23.9%

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

      3. log1p-def 23.9%

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

   4. Simplified 23.9%

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

   5. Taylor expanded in x around inf 65.3%

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

      1. *-commutative 65.3%

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

   7. Simplified 65.3%

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

   if 5.20000000000000028e-169 < x < 1.25000000000000006e-5

   1. Initial program 43.8%

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

   2. Taylor expanded in x around 0 44.8%

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

   3. Taylor expanded in n around inf 45.8%

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

   if 1.25000000000000006e-5 < x < 2.4e115

   1. Initial program 55.8%

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

   2. Taylor expanded in n around inf 46.0%

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

      1. +-rgt-identity 46.0%

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

      2. +-rgt-identity 46.0%

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

      3. log1p-def 46.0%

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

   4. Simplified 46.0%

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

   5. Taylor expanded in x around inf 58.3%

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

   if 2.4e115 < x

   1. Initial program 88.6%

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

   2. Taylor expanded in n around inf 88.6%

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

      1. +-rgt-identity 88.6%

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

      2. +-rgt-identity 88.6%

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

      3. log1p-def 88.6%

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

   4. Simplified 88.6%

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

      1. log1p-udef 88.6%

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

      2. diff-log 88.6%

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

      3. +-commutative 88.6%

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

   6. Applied egg-rr 88.6%

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

   7. Taylor expanded in x around inf 88.6%

      \[\leadsto \frac{\log \color{blue}{1}}{n} \]
3. Recombined 5 regimes into one program.

4. Final simplification 61.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 4.8 \cdot 10^{-184}:\\ \;\;\;\;\frac{-\log x}{n}\\ \mathbf{elif}\;x \leq 5.2 \cdot 10^{-169}:\\ \;\;\;\;\frac{1}{n \cdot x}\\ \mathbf{elif}\;x \leq 1.25 \cdot 10^{-5}:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{elif}\;x \leq 2.4 \cdot 10^{+115}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{n}\\ \end{array} \]

Alternative 9: 59.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-\log x}{n}\\ \mathbf{if}\;x \leq 4.8 \cdot 10^{-184}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 4.2 \cdot 10^{-171}:\\ \;\;\;\;\frac{1}{n \cdot x}\\ \mathbf{elif}\;x \leq 1.25 \cdot 10^{-5}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 7.5 \cdot 10^{+114}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (/ (- (log x)) n)))
   (if (<= x 4.8e-184)
     t_0
     (if (<= x 4.2e-171)
       (/ 1.0 (* n x))
       (if (<= x 1.25e-5)
         t_0
         (if (<= x 7.5e+114) (/ (/ 1.0 x) n) (/ 0.0 n)))))))

C

double code(double x, double n) {
	double t_0 = -log(x) / n;
	double tmp;
	if (x <= 4.8e-184) {
		tmp = t_0;
	} else if (x <= 4.2e-171) {
		tmp = 1.0 / (n * x);
	} else if (x <= 1.25e-5) {
		tmp = t_0;
	} else if (x <= 7.5e+114) {
		tmp = (1.0 / x) / n;
	} else {
		tmp = 0.0 / n;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: tmp
    t_0 = -log(x) / n
    if (x <= 4.8d-184) then
        tmp = t_0
    else if (x <= 4.2d-171) then
        tmp = 1.0d0 / (n * x)
    else if (x <= 1.25d-5) then
        tmp = t_0
    else if (x <= 7.5d+114) then
        tmp = (1.0d0 / x) / n
    else
        tmp = 0.0d0 / n
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = -Math.log(x) / n;
	double tmp;
	if (x <= 4.8e-184) {
		tmp = t_0;
	} else if (x <= 4.2e-171) {
		tmp = 1.0 / (n * x);
	} else if (x <= 1.25e-5) {
		tmp = t_0;
	} else if (x <= 7.5e+114) {
		tmp = (1.0 / x) / n;
	} else {
		tmp = 0.0 / n;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = -math.log(x) / n
	tmp = 0
	if x <= 4.8e-184:
		tmp = t_0
	elif x <= 4.2e-171:
		tmp = 1.0 / (n * x)
	elif x <= 1.25e-5:
		tmp = t_0
	elif x <= 7.5e+114:
		tmp = (1.0 / x) / n
	else:
		tmp = 0.0 / n
	return tmp

Julia

function code(x, n)
	t_0 = Float64(Float64(-log(x)) / n)
	tmp = 0.0
	if (x <= 4.8e-184)
		tmp = t_0;
	elseif (x <= 4.2e-171)
		tmp = Float64(1.0 / Float64(n * x));
	elseif (x <= 1.25e-5)
		tmp = t_0;
	elseif (x <= 7.5e+114)
		tmp = Float64(Float64(1.0 / x) / n);
	else
		tmp = Float64(0.0 / n);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = -log(x) / n;
	tmp = 0.0;
	if (x <= 4.8e-184)
		tmp = t_0;
	elseif (x <= 4.2e-171)
		tmp = 1.0 / (n * x);
	elseif (x <= 1.25e-5)
		tmp = t_0;
	elseif (x <= 7.5e+114)
		tmp = (1.0 / x) / n;
	else
		tmp = 0.0 / n;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[((-N[Log[x], $MachinePrecision]) / n), $MachinePrecision]}, If[LessEqual[x, 4.8e-184], t$95$0, If[LessEqual[x, 4.2e-171], N[(1.0 / N[(n * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.25e-5], t$95$0, If[LessEqual[x, 7.5e+114], N[(N[(1.0 / x), $MachinePrecision] / n), $MachinePrecision], N[(0.0 / n), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if x < 4.80000000000000049e-184 or 4.2e-171 < x < 1.25000000000000006e-5

   1. Initial program 45.5%

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

   2. Taylor expanded in x around 0 45.5%

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

   3. Taylor expanded in n around inf 44.6%

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

      1. neg-mul-1 44.6%

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

      2. distribute-neg-frac 44.6%

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

   5. Simplified 44.6%

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

   if 4.80000000000000049e-184 < x < 4.2e-171

   1. Initial program 48.1%

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

   2. Taylor expanded in n around inf 23.9%

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

      1. +-rgt-identity 23.9%

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

      2. +-rgt-identity 23.9%

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

      3. log1p-def 23.9%

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

   4. Simplified 23.9%

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

   5. Taylor expanded in x around inf 65.3%

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

      1. *-commutative 65.3%

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

   7. Simplified 65.3%

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

   if 1.25000000000000006e-5 < x < 7.5000000000000001e114

   1. Initial program 55.8%

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

   2. Taylor expanded in n around inf 46.0%

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

      1. +-rgt-identity 46.0%

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

      2. +-rgt-identity 46.0%

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

      3. log1p-def 46.0%

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

   4. Simplified 46.0%

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

   5. Taylor expanded in x around inf 58.3%

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

   if 7.5000000000000001e114 < x

   1. Initial program 88.6%

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

   2. Taylor expanded in n around inf 88.6%

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

      1. +-rgt-identity 88.6%

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

      2. +-rgt-identity 88.6%

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

      3. log1p-def 88.6%

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

   4. Simplified 88.6%

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

      1. log1p-udef 88.6%

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

      2. diff-log 88.6%

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

      3. +-commutative 88.6%

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

   6. Applied egg-rr 88.6%

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

   7. Taylor expanded in x around inf 88.6%

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

4. Final simplification 60.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 4.8 \cdot 10^{-184}:\\ \;\;\;\;\frac{-\log x}{n}\\ \mathbf{elif}\;x \leq 4.2 \cdot 10^{-171}:\\ \;\;\;\;\frac{1}{n \cdot x}\\ \mathbf{elif}\;x \leq 1.25 \cdot 10^{-5}:\\ \;\;\;\;\frac{-\log x}{n}\\ \mathbf{elif}\;x \leq 7.5 \cdot 10^{+114}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{n}\\ \end{array} \]

Alternative 10: 48.0% accurate, 23.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -100:\\ \;\;\;\;\frac{0}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= (/ 1.0 n) -100.0) (/ 0.0 n) (/ (/ 1.0 x) n)))

C

double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -100.0) {
		tmp = 0.0 / n;
	} else {
		tmp = (1.0 / x) / n;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if ((1.0d0 / n) <= (-100.0d0)) then
        tmp = 0.0d0 / n
    else
        tmp = (1.0d0 / x) / n
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -100.0) {
		tmp = 0.0 / n;
	} else {
		tmp = (1.0 / x) / n;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if (1.0 / n) <= -100.0:
		tmp = 0.0 / n
	else:
		tmp = (1.0 / x) / n
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -100.0)
		tmp = Float64(0.0 / n);
	else
		tmp = Float64(Float64(1.0 / x) / n);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if ((1.0 / n) <= -100.0)
		tmp = 0.0 / n;
	else
		tmp = (1.0 / x) / n;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[N[(1.0 / n), $MachinePrecision], -100.0], N[(0.0 / n), $MachinePrecision], N[(N[(1.0 / x), $MachinePrecision] / n), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 1 n) < -100

   1. Initial program 100.0%

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

   2. Taylor expanded in n around inf 61.4%

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

      1. +-rgt-identity 61.4%

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

      2. +-rgt-identity 61.4%

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

      3. log1p-def 61.4%

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

   4. Simplified 61.4%

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

      1. log1p-udef 61.4%

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

      2. diff-log 60.3%

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

      3. +-commutative 60.3%

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

   6. Applied egg-rr 60.3%

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

   7. Taylor expanded in x around inf 61.9%

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

   if -100 < (/.f64 1 n)

   1. Initial program 36.8%

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

   2. Taylor expanded in n around inf 53.8%

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

      1. +-rgt-identity 53.8%

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

      2. +-rgt-identity 53.8%

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

      3. log1p-def 53.8%

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

   4. Simplified 53.8%

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

   5. Taylor expanded in x around inf 52.1%

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

4. Final simplification 55.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -100:\\ \;\;\;\;\frac{0}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \end{array} \]

Alternative 11: 40.4% accurate, 42.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 59.8%

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

2. Taylor expanded in n around inf 56.5%

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

   1. +-rgt-identity 56.5%

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

   2. +-rgt-identity 56.5%

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

   3. log1p-def 56.5%

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

4. Simplified 56.5%

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

5. Taylor expanded in x around inf 41.5%

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

   1. *-commutative 41.5%

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

7. Simplified 41.5%

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

8. Final simplification 41.5%

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

Alternative 12: 41.0% accurate, 42.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 59.8%

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

2. Taylor expanded in x around inf 62.5%

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

   1. mul-1-neg 62.5%

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

   2. log-rec 62.5%

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

   3. mul-1-neg 62.5%

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

   4. distribute-neg-frac 62.5%

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

   5. mul-1-neg 62.5%

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

   6. remove-double-neg 62.5%

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

   7. *-commutative 62.5%

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

4. Simplified 62.5%

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

   1. div-inv 62.5%

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

   2. pow-to-exp 62.5%

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

   3. *-un-lft-identity 62.5%

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

   4. times-frac 63.1%

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

6. Applied egg-rr 63.1%

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

7. Taylor expanded in n around inf 41.5%

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

   1. associate-/r* 41.8%

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

9. Simplified 41.8%

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

10. Final simplification 41.8%

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

Alternative 13: 41.0% accurate, 42.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 59.8%

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

2. Taylor expanded in n around inf 56.5%

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

   1. +-rgt-identity 56.5%

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

   2. +-rgt-identity 56.5%

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

   3. log1p-def 56.5%

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

4. Simplified 56.5%

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

5. Taylor expanded in x around inf 41.8%

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

6. Final simplification 41.8%

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

Alternative 14: 4.5% accurate, 70.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 59.8%

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

2. Taylor expanded in x around inf 62.5%

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

   1. mul-1-neg 62.5%

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

   2. log-rec 62.5%

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

   3. mul-1-neg 62.5%

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

   4. distribute-neg-frac 62.5%

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

   5. mul-1-neg 62.5%

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

   6. remove-double-neg 62.5%

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

   7. *-commutative 62.5%

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

4. Simplified 62.5%

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

   1. div-inv 62.5%

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

   2. pow-to-exp 62.5%

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

   3. *-un-lft-identity 62.5%

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

   4. times-frac 63.1%

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

6. Applied egg-rr 63.1%

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

7. Taylor expanded in n around inf 41.5%

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

   1. associate-/r* 41.8%

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

9. Simplified 41.8%

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

    1. expm1-log1p-u 33.5%

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

    2. expm1-udef 31.4%

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

    3. associate-/l/ 31.4%

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

    4. associate-/r* 31.4%

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

    5. add-exp-log 31.4%

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

    6. neg-log 31.4%

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

    7. add-sqr-sqrt 8.0%

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

    8. sqrt-unprod 14.6%

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

    9. sqr-neg 14.6%

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

    10. sqrt-unprod 6.5%

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

    11. add-sqr-sqrt 8.5%

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

    12. add-exp-log 8.5%

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

11. Applied egg-rr 8.5%

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

    1. expm1-def 3.3%

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

    2. expm1-log1p 4.2%

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

13. Simplified 4.2%

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

14. Final simplification 4.2%

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

Reproduce

herbie shell --seed 2023334 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  :precision binary64
  (- (pow (+ x 1.0) (/ 1.0 n)) (pow x (/ 1.0 n))))