2nthrt (problem 3.4.6)

Percentage Accurate: 53.5% → 85.7%

Time: 22.1s

Alternatives: 13

Speedup: 2.0×

• could not determine a ground truth

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 53.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 85.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ t_1 := \frac{t_0}{n \cdot x}\\ \mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{-20}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-82}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 100000:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{x}{n} + -0.5 \cdot \frac{x \cdot x}{n}} - t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))) (t_1 (/ t_0 (* n x))))
   (if (<= (/ 1.0 n) -1e-20)
     t_1
     (if (<= (/ 1.0 n) 5e-82)
       (/ (log (/ (+ 1.0 x) x)) n)
       (if (<= (/ 1.0 n) 100000.0)
         t_1
         (- (exp (+ (/ x n) (* -0.5 (/ (* x x) n)))) t_0))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double t_1 = t_0 / (n * x);
	double tmp;
	if ((1.0 / n) <= -1e-20) {
		tmp = t_1;
	} else if ((1.0 / n) <= 5e-82) {
		tmp = log(((1.0 + x) / x)) / n;
	} else if ((1.0 / n) <= 100000.0) {
		tmp = t_1;
	} else {
		tmp = exp(((x / n) + (-0.5 * ((x * x) / n)))) - t_0;
	}
	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 = x ** (1.0d0 / n)
    t_1 = t_0 / (n * x)
    if ((1.0d0 / n) <= (-1d-20)) then
        tmp = t_1
    else if ((1.0d0 / n) <= 5d-82) then
        tmp = log(((1.0d0 + x) / x)) / n
    else if ((1.0d0 / n) <= 100000.0d0) then
        tmp = t_1
    else
        tmp = exp(((x / n) + ((-0.5d0) * ((x * x) / n)))) - t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, (1.0 / n));
	double t_1 = t_0 / (n * x);
	double tmp;
	if ((1.0 / n) <= -1e-20) {
		tmp = t_1;
	} else if ((1.0 / n) <= 5e-82) {
		tmp = Math.log(((1.0 + x) / x)) / n;
	} else if ((1.0 / n) <= 100000.0) {
		tmp = t_1;
	} else {
		tmp = Math.exp(((x / n) + (-0.5 * ((x * x) / n)))) - t_0;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	t_1 = t_0 / (n * x)
	tmp = 0
	if (1.0 / n) <= -1e-20:
		tmp = t_1
	elif (1.0 / n) <= 5e-82:
		tmp = math.log(((1.0 + x) / x)) / n
	elif (1.0 / n) <= 100000.0:
		tmp = t_1
	else:
		tmp = math.exp(((x / n) + (-0.5 * ((x * x) / n)))) - t_0
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	t_1 = Float64(t_0 / Float64(n * x))
	tmp = 0.0
	if (Float64(1.0 / n) <= -1e-20)
		tmp = t_1;
	elseif (Float64(1.0 / n) <= 5e-82)
		tmp = Float64(log(Float64(Float64(1.0 + x) / x)) / n);
	elseif (Float64(1.0 / n) <= 100000.0)
		tmp = t_1;
	else
		tmp = Float64(exp(Float64(Float64(x / n) + Float64(-0.5 * Float64(Float64(x * x) / n)))) - t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = x ^ (1.0 / n);
	t_1 = t_0 / (n * x);
	tmp = 0.0;
	if ((1.0 / n) <= -1e-20)
		tmp = t_1;
	elseif ((1.0 / n) <= 5e-82)
		tmp = log(((1.0 + x) / x)) / n;
	elseif ((1.0 / n) <= 100000.0)
		tmp = t_1;
	else
		tmp = exp(((x / n) + (-0.5 * ((x * x) / n)))) - t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 / N[(n * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -1e-20], t$95$1, If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e-82], N[(N[Log[N[(N[(1.0 + x), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 100000.0], t$95$1, N[(N[Exp[N[(N[(x / n), $MachinePrecision] + N[(-0.5 * N[(N[(x * x), $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - t$95$0), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 1 n) < -9.99999999999999945e-21 or 4.9999999999999998e-82 < (/.f64 1 n) < 1e5

   1. Initial program 73.2%

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

   2. Taylor expanded in x around inf 93.4%

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

      1. log-rec 93.4%

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

      2. mul-1-neg 93.4%

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

      3. mul-1-neg 93.4%

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

      4. distribute-frac-neg 93.4%

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

      5. neg-mul-1 93.4%

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

      6. remove-double-neg 93.4%

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

      7. *-rgt-identity 93.4%

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

      8. associate-*r/ 93.4%

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

      9. unpow-1 93.4%

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

      10. exp-to-pow 93.4%

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

      11. unpow-1 93.4%

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

      12. *-commutative 93.4%

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

   4. Simplified 93.4%

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

   if -9.99999999999999945e-21 < (/.f64 1 n) < 4.9999999999999998e-82

   1. Initial program 33.5%

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

   2. Taylor expanded in n around inf 83.6%

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

      1. log1p-def 83.6%

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

   4. Simplified 83.6%

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

      1. log1p-udef 83.6%

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

      2. diff-log 83.6%

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

   6. Applied egg-rr 83.6%

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

   if 1e5 < (/.f64 1 n)

   1. Initial program 46.8%

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

   2. Taylor expanded in n around 0 46.8%

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

      1. log1p-def 100.0%

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

      2. *-rgt-identity 100.0%

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

      3. associate-*r/ 100.0%

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

      4. unpow-1 100.0%

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

      5. exp-to-pow 100.0%

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

      6. /-rgt-identity 100.0%

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

      7. metadata-eval 100.0%

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

      8. associate-/l* 100.0%

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

      9. *-commutative 100.0%

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

      10. *-commutative 100.0%

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

      11. associate-/l* 100.0%

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

      12. metadata-eval 100.0%

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

      13. /-rgt-identity 100.0%

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

      14. unpow-1 100.0%

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

   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%

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

      1. unpow2 100.0%

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

   7. Simplified 100.0%

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

4. Final simplification 89.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{-20}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-82}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 100000:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{n \cdot x}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{x}{n} + -0.5 \cdot \frac{x \cdot x}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \]

Alternative 2: 82.0% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ t_1 := \frac{t_0}{n \cdot x}\\ t_2 := \frac{1}{n \cdot x}\\ \mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{-20}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-82}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 100000:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\frac{1}{n} \leq 4 \cdot 10^{+195}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - t_0\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{t_2 \cdot \frac{t_2}{n \cdot x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))) (t_1 (/ t_0 (* n x))) (t_2 (/ 1.0 (* n x))))
   (if (<= (/ 1.0 n) -1e-20)
     t_1
     (if (<= (/ 1.0 n) 5e-82)
       (/ (log (/ (+ 1.0 x) x)) n)
       (if (<= (/ 1.0 n) 100000.0)
         t_1
         (if (<= (/ 1.0 n) 4e+195)
           (- (+ 1.0 (/ x n)) t_0)
           (cbrt (* t_2 (/ t_2 (* n x))))))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double t_1 = t_0 / (n * x);
	double t_2 = 1.0 / (n * x);
	double tmp;
	if ((1.0 / n) <= -1e-20) {
		tmp = t_1;
	} else if ((1.0 / n) <= 5e-82) {
		tmp = log(((1.0 + x) / x)) / n;
	} else if ((1.0 / n) <= 100000.0) {
		tmp = t_1;
	} else if ((1.0 / n) <= 4e+195) {
		tmp = (1.0 + (x / n)) - t_0;
	} else {
		tmp = cbrt((t_2 * (t_2 / (n * x))));
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, (1.0 / n));
	double t_1 = t_0 / (n * x);
	double t_2 = 1.0 / (n * x);
	double tmp;
	if ((1.0 / n) <= -1e-20) {
		tmp = t_1;
	} else if ((1.0 / n) <= 5e-82) {
		tmp = Math.log(((1.0 + x) / x)) / n;
	} else if ((1.0 / n) <= 100000.0) {
		tmp = t_1;
	} else if ((1.0 / n) <= 4e+195) {
		tmp = (1.0 + (x / n)) - t_0;
	} else {
		tmp = Math.cbrt((t_2 * (t_2 / (n * x))));
	}
	return tmp;
}

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	t_1 = Float64(t_0 / Float64(n * x))
	t_2 = Float64(1.0 / Float64(n * x))
	tmp = 0.0
	if (Float64(1.0 / n) <= -1e-20)
		tmp = t_1;
	elseif (Float64(1.0 / n) <= 5e-82)
		tmp = Float64(log(Float64(Float64(1.0 + x) / x)) / n);
	elseif (Float64(1.0 / n) <= 100000.0)
		tmp = t_1;
	elseif (Float64(1.0 / n) <= 4e+195)
		tmp = Float64(Float64(1.0 + Float64(x / n)) - t_0);
	else
		tmp = cbrt(Float64(t_2 * Float64(t_2 / Float64(n * x))));
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 / N[(n * x), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(1.0 / N[(n * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -1e-20], t$95$1, If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e-82], N[(N[Log[N[(N[(1.0 + x), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 100000.0], t$95$1, If[LessEqual[N[(1.0 / n), $MachinePrecision], 4e+195], N[(N[(1.0 + N[(x / n), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], N[Power[N[(t$95$2 * N[(t$95$2 / N[(n * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 1 n) < -9.99999999999999945e-21 or 4.9999999999999998e-82 < (/.f64 1 n) < 1e5

   1. Initial program 73.2%

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

   2. Taylor expanded in x around inf 93.4%

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

      1. log-rec 93.4%

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

      2. mul-1-neg 93.4%

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

      3. mul-1-neg 93.4%

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

      4. distribute-frac-neg 93.4%

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

      5. neg-mul-1 93.4%

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

      6. remove-double-neg 93.4%

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

      7. *-rgt-identity 93.4%

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

      8. associate-*r/ 93.4%

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

      9. unpow-1 93.4%

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

      10. exp-to-pow 93.4%

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

      11. unpow-1 93.4%

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

      12. *-commutative 93.4%

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

   4. Simplified 93.4%

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

   if -9.99999999999999945e-21 < (/.f64 1 n) < 4.9999999999999998e-82

   1. Initial program 33.5%

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

   2. Taylor expanded in n around inf 83.6%

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

      1. log1p-def 83.6%

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

   4. Simplified 83.6%

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

      1. log1p-udef 83.6%

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

      2. diff-log 83.6%

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

   6. Applied egg-rr 83.6%

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

   if 1e5 < (/.f64 1 n) < 3.99999999999999991e195

   1. Initial program 67.8%

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

   2. Taylor expanded in x around 0 70.2%

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

   if 3.99999999999999991e195 < (/.f64 1 n)

   1. Initial program 20.3%

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

   2. Taylor expanded in n around inf 6.3%

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

      1. log1p-def 6.3%

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

   4. Simplified 6.3%

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

   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%

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

      1. add-cbrt-cube 80.6%

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

   9. Applied egg-rr 80.6%

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

       1. associate-*r/ 80.6%

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

       2. *-rgt-identity 80.6%

          \[\leadsto \sqrt[3]{\frac{\color{blue}{\frac{1}{x \cdot n} \cdot \frac{1}{x \cdot n}}}{x \cdot n}} \]

       3. associate-*r/ 80.6%

          \[\leadsto \sqrt[3]{\color{blue}{\frac{1}{x \cdot n} \cdot \frac{\frac{1}{x \cdot n}}{x \cdot n}}} \]

   11. Simplified 80.6%

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

4. Final simplification 85.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{-20}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-82}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 100000:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 4 \cdot 10^{+195}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\frac{1}{n \cdot x} \cdot \frac{\frac{1}{n \cdot x}}{n \cdot x}}\\ \end{array} \]

Alternative 3: 81.1% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ t_1 := \frac{t_0}{n \cdot x}\\ \mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{-20}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-82}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 100000:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\frac{1}{n} \leq 4 \cdot 10^{+195}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{n \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))) (t_1 (/ t_0 (* n x))))
   (if (<= (/ 1.0 n) -1e-20)
     t_1
     (if (<= (/ 1.0 n) 5e-82)
       (/ (log (/ (+ 1.0 x) x)) n)
       (if (<= (/ 1.0 n) 100000.0)
         t_1
         (if (<= (/ 1.0 n) 4e+195)
           (- (+ 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 t_1 = t_0 / (n * x);
	double tmp;
	if ((1.0 / n) <= -1e-20) {
		tmp = t_1;
	} else if ((1.0 / n) <= 5e-82) {
		tmp = log(((1.0 + x) / x)) / n;
	} else if ((1.0 / n) <= 100000.0) {
		tmp = t_1;
	} else if ((1.0 / n) <= 4e+195) {
		tmp = (1.0 + (x / n)) - t_0;
	} else {
		tmp = 1.0 / (n * x);
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = x ** (1.0d0 / n)
    t_1 = t_0 / (n * x)
    if ((1.0d0 / n) <= (-1d-20)) then
        tmp = t_1
    else if ((1.0d0 / n) <= 5d-82) then
        tmp = log(((1.0d0 + x) / x)) / n
    else if ((1.0d0 / n) <= 100000.0d0) then
        tmp = t_1
    else if ((1.0d0 / n) <= 4d+195) then
        tmp = (1.0d0 + (x / n)) - t_0
    else
        tmp = 1.0d0 / (n * x)
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, (1.0 / n));
	double t_1 = t_0 / (n * x);
	double tmp;
	if ((1.0 / n) <= -1e-20) {
		tmp = t_1;
	} else if ((1.0 / n) <= 5e-82) {
		tmp = Math.log(((1.0 + x) / x)) / n;
	} else if ((1.0 / n) <= 100000.0) {
		tmp = t_1;
	} else if ((1.0 / n) <= 4e+195) {
		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))
	t_1 = t_0 / (n * x)
	tmp = 0
	if (1.0 / n) <= -1e-20:
		tmp = t_1
	elif (1.0 / n) <= 5e-82:
		tmp = math.log(((1.0 + x) / x)) / n
	elif (1.0 / n) <= 100000.0:
		tmp = t_1
	elif (1.0 / n) <= 4e+195:
		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)
	t_1 = Float64(t_0 / Float64(n * x))
	tmp = 0.0
	if (Float64(1.0 / n) <= -1e-20)
		tmp = t_1;
	elseif (Float64(1.0 / n) <= 5e-82)
		tmp = Float64(log(Float64(Float64(1.0 + x) / x)) / n);
	elseif (Float64(1.0 / n) <= 100000.0)
		tmp = t_1;
	elseif (Float64(1.0 / n) <= 4e+195)
		tmp = Float64(Float64(1.0 + Float64(x / n)) - t_0);
	else
		tmp = Float64(1.0 / Float64(n * x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = x ^ (1.0 / n);
	t_1 = t_0 / (n * x);
	tmp = 0.0;
	if ((1.0 / n) <= -1e-20)
		tmp = t_1;
	elseif ((1.0 / n) <= 5e-82)
		tmp = log(((1.0 + x) / x)) / n;
	elseif ((1.0 / n) <= 100000.0)
		tmp = t_1;
	elseif ((1.0 / n) <= 4e+195)
		tmp = (1.0 + (x / n)) - t_0;
	else
		tmp = 1.0 / (n * x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 / N[(n * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -1e-20], t$95$1, If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e-82], N[(N[Log[N[(N[(1.0 + x), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 100000.0], t$95$1, If[LessEqual[N[(1.0 / n), $MachinePrecision], 4e+195], 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) < -9.99999999999999945e-21 or 4.9999999999999998e-82 < (/.f64 1 n) < 1e5

   1. Initial program 73.2%

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

   2. Taylor expanded in x around inf 93.4%

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

      1. log-rec 93.4%

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

      2. mul-1-neg 93.4%

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

      3. mul-1-neg 93.4%

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

      4. distribute-frac-neg 93.4%

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

      5. neg-mul-1 93.4%

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

      6. remove-double-neg 93.4%

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

      7. *-rgt-identity 93.4%

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

      8. associate-*r/ 93.4%

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

      9. unpow-1 93.4%

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

      10. exp-to-pow 93.4%

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

      11. unpow-1 93.4%

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

      12. *-commutative 93.4%

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

   4. Simplified 93.4%

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

   if -9.99999999999999945e-21 < (/.f64 1 n) < 4.9999999999999998e-82

   1. Initial program 33.5%

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

   2. Taylor expanded in n around inf 83.6%

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

      1. log1p-def 83.6%

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

   4. Simplified 83.6%

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

      1. log1p-udef 83.6%

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

      2. diff-log 83.6%

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

   6. Applied egg-rr 83.6%

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

   if 1e5 < (/.f64 1 n) < 3.99999999999999991e195

   1. Initial program 67.8%

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

   2. Taylor expanded in x around 0 70.2%

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

   if 3.99999999999999991e195 < (/.f64 1 n)

   1. Initial program 20.3%

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

   2. Taylor expanded in n around inf 6.3%

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

      1. log1p-def 6.3%

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

   4. Simplified 6.3%

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

   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 85.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{-20}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-82}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 100000:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 4 \cdot 10^{+195}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{n \cdot x}\\ \end{array} \]

Alternative 4: 81.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ t_1 := \frac{t_0}{n \cdot x}\\ \mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{-20}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-82}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 100000:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\frac{1}{n} \leq 4 \cdot 10^{+195}:\\ \;\;\;\;1 - t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{n \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))) (t_1 (/ t_0 (* n x))))
   (if (<= (/ 1.0 n) -1e-20)
     t_1
     (if (<= (/ 1.0 n) 5e-82)
       (/ (log (/ (+ 1.0 x) x)) n)
       (if (<= (/ 1.0 n) 100000.0)
         t_1
         (if (<= (/ 1.0 n) 4e+195) (- 1.0 t_0) (/ 1.0 (* n x))))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double t_1 = t_0 / (n * x);
	double tmp;
	if ((1.0 / n) <= -1e-20) {
		tmp = t_1;
	} else if ((1.0 / n) <= 5e-82) {
		tmp = log(((1.0 + x) / x)) / n;
	} else if ((1.0 / n) <= 100000.0) {
		tmp = t_1;
	} else if ((1.0 / n) <= 4e+195) {
		tmp = 1.0 - t_0;
	} else {
		tmp = 1.0 / (n * x);
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = x ** (1.0d0 / n)
    t_1 = t_0 / (n * x)
    if ((1.0d0 / n) <= (-1d-20)) then
        tmp = t_1
    else if ((1.0d0 / n) <= 5d-82) then
        tmp = log(((1.0d0 + x) / x)) / n
    else if ((1.0d0 / n) <= 100000.0d0) then
        tmp = t_1
    else if ((1.0d0 / n) <= 4d+195) then
        tmp = 1.0d0 - t_0
    else
        tmp = 1.0d0 / (n * x)
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, (1.0 / n));
	double t_1 = t_0 / (n * x);
	double tmp;
	if ((1.0 / n) <= -1e-20) {
		tmp = t_1;
	} else if ((1.0 / n) <= 5e-82) {
		tmp = Math.log(((1.0 + x) / x)) / n;
	} else if ((1.0 / n) <= 100000.0) {
		tmp = t_1;
	} else if ((1.0 / n) <= 4e+195) {
		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))
	t_1 = t_0 / (n * x)
	tmp = 0
	if (1.0 / n) <= -1e-20:
		tmp = t_1
	elif (1.0 / n) <= 5e-82:
		tmp = math.log(((1.0 + x) / x)) / n
	elif (1.0 / n) <= 100000.0:
		tmp = t_1
	elif (1.0 / n) <= 4e+195:
		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)
	t_1 = Float64(t_0 / Float64(n * x))
	tmp = 0.0
	if (Float64(1.0 / n) <= -1e-20)
		tmp = t_1;
	elseif (Float64(1.0 / n) <= 5e-82)
		tmp = Float64(log(Float64(Float64(1.0 + x) / x)) / n);
	elseif (Float64(1.0 / n) <= 100000.0)
		tmp = t_1;
	elseif (Float64(1.0 / n) <= 4e+195)
		tmp = Float64(1.0 - t_0);
	else
		tmp = Float64(1.0 / Float64(n * x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = x ^ (1.0 / n);
	t_1 = t_0 / (n * x);
	tmp = 0.0;
	if ((1.0 / n) <= -1e-20)
		tmp = t_1;
	elseif ((1.0 / n) <= 5e-82)
		tmp = log(((1.0 + x) / x)) / n;
	elseif ((1.0 / n) <= 100000.0)
		tmp = t_1;
	elseif ((1.0 / n) <= 4e+195)
		tmp = 1.0 - t_0;
	else
		tmp = 1.0 / (n * x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 / N[(n * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -1e-20], t$95$1, If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e-82], N[(N[Log[N[(N[(1.0 + x), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 100000.0], t$95$1, If[LessEqual[N[(1.0 / n), $MachinePrecision], 4e+195], 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) < -9.99999999999999945e-21 or 4.9999999999999998e-82 < (/.f64 1 n) < 1e5

   1. Initial program 73.2%

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

   2. Taylor expanded in x around inf 93.4%

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

      1. log-rec 93.4%

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

      2. mul-1-neg 93.4%

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

      3. mul-1-neg 93.4%

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

      4. distribute-frac-neg 93.4%

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

      5. neg-mul-1 93.4%

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

      6. remove-double-neg 93.4%

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

      7. *-rgt-identity 93.4%

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

      8. associate-*r/ 93.4%

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

      9. unpow-1 93.4%

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

      10. exp-to-pow 93.4%

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

      11. unpow-1 93.4%

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

      12. *-commutative 93.4%

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

   4. Simplified 93.4%

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

   if -9.99999999999999945e-21 < (/.f64 1 n) < 4.9999999999999998e-82

   1. Initial program 33.5%

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

   2. Taylor expanded in n around inf 83.6%

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

      1. log1p-def 83.6%

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

   4. Simplified 83.6%

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

      1. log1p-udef 83.6%

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

      2. diff-log 83.6%

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

   6. Applied egg-rr 83.6%

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

   if 1e5 < (/.f64 1 n) < 3.99999999999999991e195

   1. Initial program 67.8%

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

   2. Taylor expanded in x around 0 67.8%

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

      1. *-rgt-identity 67.8%

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

      2. associate-*r/ 67.8%

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

      3. unpow-1 67.8%

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

      4. exp-to-pow 67.8%

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

      5. unpow-1 67.8%

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

   4. Simplified 67.8%

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

   if 3.99999999999999991e195 < (/.f64 1 n)

   1. Initial program 20.3%

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

   2. Taylor expanded in n around inf 6.3%

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

      1. log1p-def 6.3%

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

   4. Simplified 6.3%

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

   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 85.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{-20}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-82}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 100000:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 4 \cdot 10^{+195}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{n \cdot x}\\ \end{array} \]

Alternative 5: 66.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq 5 \cdot 10^{-82}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 0.002:\\ \;\;\;\;\frac{\frac{1}{n}}{x}\\ \mathbf{elif}\;\frac{1}{n} \leq 4 \cdot 10^{+195}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{n \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= (/ 1.0 n) 5e-82)
   (/ (log (/ (+ 1.0 x) x)) n)
   (if (<= (/ 1.0 n) 0.002)
     (/ (/ 1.0 n) x)
     (if (<= (/ 1.0 n) 4e+195) (- 1.0 (pow x (/ 1.0 n))) (/ 1.0 (* n x))))))

C

double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= 5e-82) {
		tmp = log(((1.0 + x) / x)) / n;
	} else if ((1.0 / n) <= 0.002) {
		tmp = (1.0 / n) / x;
	} else if ((1.0 / n) <= 4e+195) {
		tmp = 1.0 - pow(x, (1.0 / n));
	} else {
		tmp = 1.0 / (n * x);
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if ((1.0d0 / n) <= 5d-82) then
        tmp = log(((1.0d0 + x) / x)) / n
    else if ((1.0d0 / n) <= 0.002d0) then
        tmp = (1.0d0 / n) / x
    else if ((1.0d0 / n) <= 4d+195) then
        tmp = 1.0d0 - (x ** (1.0d0 / n))
    else
        tmp = 1.0d0 / (n * x)
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= 5e-82) {
		tmp = Math.log(((1.0 + x) / x)) / n;
	} else if ((1.0 / n) <= 0.002) {
		tmp = (1.0 / n) / x;
	} else if ((1.0 / n) <= 4e+195) {
		tmp = 1.0 - Math.pow(x, (1.0 / n));
	} else {
		tmp = 1.0 / (n * x);
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if (1.0 / n) <= 5e-82:
		tmp = math.log(((1.0 + x) / x)) / n
	elif (1.0 / n) <= 0.002:
		tmp = (1.0 / n) / x
	elif (1.0 / n) <= 4e+195:
		tmp = 1.0 - math.pow(x, (1.0 / n))
	else:
		tmp = 1.0 / (n * x)
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (Float64(1.0 / n) <= 5e-82)
		tmp = Float64(log(Float64(Float64(1.0 + x) / x)) / n);
	elseif (Float64(1.0 / n) <= 0.002)
		tmp = Float64(Float64(1.0 / n) / x);
	elseif (Float64(1.0 / n) <= 4e+195)
		tmp = Float64(1.0 - (x ^ Float64(1.0 / n)));
	else
		tmp = Float64(1.0 / Float64(n * x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if ((1.0 / n) <= 5e-82)
		tmp = log(((1.0 + x) / x)) / n;
	elseif ((1.0 / n) <= 0.002)
		tmp = (1.0 / n) / x;
	elseif ((1.0 / n) <= 4e+195)
		tmp = 1.0 - (x ^ (1.0 / n));
	else
		tmp = 1.0 / (n * x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e-82], N[(N[Log[N[(N[(1.0 + x), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 0.002], N[(N[(1.0 / n), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 4e+195], N[(1.0 - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(n * x), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 1 n) < 4.9999999999999998e-82

   1. Initial program 53.2%

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

   2. Taylor expanded in n around inf 74.5%

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

      1. log1p-def 74.5%

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

   4. Simplified 74.5%

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

      1. log1p-udef 74.5%

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

      2. diff-log 74.6%

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

   6. Applied egg-rr 74.6%

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

   if 4.9999999999999998e-82 < (/.f64 1 n) < 2e-3

   1. Initial program 4.9%

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

   2. Taylor expanded in n around inf 26.2%

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

      1. log1p-def 26.2%

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

   4. Simplified 26.2%

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

   5. Taylor expanded in x around inf 68.4%

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

      1. *-commutative 68.4%

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

   7. Simplified 68.4%

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

   8. Taylor expanded in x around 0 68.4%

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

      1. associate-/r* 68.6%

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

   10. Simplified 68.6%

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

   if 2e-3 < (/.f64 1 n) < 3.99999999999999991e195

   1. Initial program 66.1%

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

   2. Taylor expanded in x around 0 66.1%

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

      1. *-rgt-identity 66.1%

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

      2. associate-*r/ 66.1%

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

      3. unpow-1 66.1%

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

      4. exp-to-pow 66.1%

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

      5. unpow-1 66.1%

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

   4. Simplified 66.1%

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

   if 3.99999999999999991e195 < (/.f64 1 n)

   1. Initial program 20.3%

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

   2. Taylor expanded in n around inf 6.3%

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

      1. log1p-def 6.3%

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

   4. Simplified 6.3%

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

   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 73.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq 5 \cdot 10^{-82}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 0.002:\\ \;\;\;\;\frac{\frac{1}{n}}{x}\\ \mathbf{elif}\;\frac{1}{n} \leq 4 \cdot 10^{+195}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{n \cdot x}\\ \end{array} \]

Alternative 6: 60.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.96:\\ \;\;\;\;\frac{x}{n} - \frac{\log x}{n}\\ \mathbf{elif}\;x \leq 4.2 \cdot 10^{+123}:\\ \;\;\;\;\frac{\frac{1}{x} - \frac{0.5}{x \cdot x}}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 0.96)
   (- (/ x n) (/ (log x) n))
   (if (<= x 4.2e+123) (/ (- (/ 1.0 x) (/ 0.5 (* x x))) n) (/ 0.0 n))))

C

double code(double x, double n) {
	double tmp;
	if (x <= 0.96) {
		tmp = (x / n) - (log(x) / n);
	} else if (x <= 4.2e+123) {
		tmp = ((1.0 / x) - (0.5 / (x * 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 <= 0.96d0) then
        tmp = (x / n) - (log(x) / n)
    else if (x <= 4.2d+123) then
        tmp = ((1.0d0 / x) - (0.5d0 / (x * 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 <= 0.96) {
		tmp = (x / n) - (Math.log(x) / n);
	} else if (x <= 4.2e+123) {
		tmp = ((1.0 / x) - (0.5 / (x * x))) / n;
	} else {
		tmp = 0.0 / n;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 0.96:
		tmp = (x / n) - (math.log(x) / n)
	elif x <= 4.2e+123:
		tmp = ((1.0 / x) - (0.5 / (x * x))) / n
	else:
		tmp = 0.0 / n
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 0.96)
		tmp = Float64(Float64(x / n) - Float64(log(x) / n));
	elseif (x <= 4.2e+123)
		tmp = Float64(Float64(Float64(1.0 / x) - Float64(0.5 / Float64(x * x))) / n);
	else
		tmp = Float64(0.0 / n);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (x <= 0.96)
		tmp = (x / n) - (log(x) / n);
	elseif (x <= 4.2e+123)
		tmp = ((1.0 / x) - (0.5 / (x * x))) / n;
	else
		tmp = 0.0 / n;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 0.96], N[(N[(x / n), $MachinePrecision] - N[(N[Log[x], $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 4.2e+123], N[(N[(N[(1.0 / x), $MachinePrecision] - N[(0.5 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], N[(0.0 / n), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 0.95999999999999996

   1. Initial program 34.5%

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

   2. Taylor expanded in n around inf 58.5%

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

      1. log1p-def 58.5%

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

   4. Simplified 58.5%

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

   5. Taylor expanded in x around 0 58.4%

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

      1. neg-mul-1 58.4%

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

      2. sub-neg 58.4%

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

   7. Simplified 58.4%

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

      1. div-sub 58.4%

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

   9. Applied egg-rr 58.4%

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

   if 0.95999999999999996 < x < 4.19999999999999988e123

   1. Initial program 42.3%

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

   2. Taylor expanded in n around inf 40.9%

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

      1. log1p-def 40.9%

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

   4. Simplified 40.9%

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

   5. Taylor expanded in x around inf 70.1%

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

      1. associate-*r/ 70.1%

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

      2. metadata-eval 70.1%

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

      3. unpow2 70.1%

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

   7. Simplified 70.1%

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

   if 4.19999999999999988e123 < x

   1. Initial program 80.7%

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

   2. Taylor expanded in n around inf 80.7%

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

      1. log1p-def 80.7%

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

   4. Simplified 80.7%

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

      1. log1p-udef 80.7%

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

      2. diff-log 80.7%

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

   6. Applied egg-rr 80.7%

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

   7. Taylor expanded in x around inf 80.7%

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

4. Final simplification 66.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.96:\\ \;\;\;\;\frac{x}{n} - \frac{\log x}{n}\\ \mathbf{elif}\;x \leq 4.2 \cdot 10^{+123}:\\ \;\;\;\;\frac{\frac{1}{x} - \frac{0.5}{x \cdot x}}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{n}\\ \end{array} \]

Alternative 7: 60.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{elif}\;x \leq 1.7 \cdot 10^{+124}:\\ \;\;\;\;\frac{\frac{1}{x} - \frac{0.5}{x \cdot x}}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 1.0)
   (/ (- x (log x)) n)
   (if (<= x 1.7e+124) (/ (- (/ 1.0 x) (/ 0.5 (* x x))) n) (/ 0.0 n))))

C

double code(double x, double n) {
	double tmp;
	if (x <= 1.0) {
		tmp = (x - log(x)) / n;
	} else if (x <= 1.7e+124) {
		tmp = ((1.0 / x) - (0.5 / (x * 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 <= 1.0d0) then
        tmp = (x - log(x)) / n
    else if (x <= 1.7d+124) then
        tmp = ((1.0d0 / x) - (0.5d0 / (x * 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 <= 1.0) {
		tmp = (x - Math.log(x)) / n;
	} else if (x <= 1.7e+124) {
		tmp = ((1.0 / x) - (0.5 / (x * x))) / n;
	} else {
		tmp = 0.0 / n;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 1.0:
		tmp = (x - math.log(x)) / n
	elif x <= 1.7e+124:
		tmp = ((1.0 / x) - (0.5 / (x * x))) / n
	else:
		tmp = 0.0 / n
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 1.0)
		tmp = Float64(Float64(x - log(x)) / n);
	elseif (x <= 1.7e+124)
		tmp = Float64(Float64(Float64(1.0 / x) - Float64(0.5 / Float64(x * x))) / n);
	else
		tmp = Float64(0.0 / n);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (x <= 1.0)
		tmp = (x - log(x)) / n;
	elseif (x <= 1.7e+124)
		tmp = ((1.0 / x) - (0.5 / (x * x))) / n;
	else
		tmp = 0.0 / n;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 1.0], N[(N[(x - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[x, 1.7e+124], N[(N[(N[(1.0 / x), $MachinePrecision] - N[(0.5 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], N[(0.0 / n), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 1

   1. Initial program 34.5%

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

   2. Taylor expanded in n around inf 58.5%

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

      1. log1p-def 58.5%

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

   4. Simplified 58.5%

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

   5. Taylor expanded in x around 0 58.4%

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

      1. neg-mul-1 58.4%

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

      2. sub-neg 58.4%

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

   7. Simplified 58.4%

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

   if 1 < x < 1.7e124

   1. Initial program 42.3%

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

   2. Taylor expanded in n around inf 40.9%

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

      1. log1p-def 40.9%

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

   4. Simplified 40.9%

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

   5. Taylor expanded in x around inf 70.1%

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

      1. associate-*r/ 70.1%

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

      2. metadata-eval 70.1%

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

      3. unpow2 70.1%

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

   7. Simplified 70.1%

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

   if 1.7e124 < x

   1. Initial program 80.7%

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

   2. Taylor expanded in n around inf 80.7%

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

      1. log1p-def 80.7%

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

   4. Simplified 80.7%

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

      1. log1p-udef 80.7%

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

      2. diff-log 80.7%

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

   6. Applied egg-rr 80.7%

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

   7. Taylor expanded in x around inf 80.7%

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

4. Final simplification 66.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{elif}\;x \leq 1.7 \cdot 10^{+124}:\\ \;\;\;\;\frac{\frac{1}{x} - \frac{0.5}{x \cdot x}}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{n}\\ \end{array} \]

Alternative 8: 60.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.7:\\ \;\;\;\;\frac{-\log x}{n}\\ \mathbf{elif}\;x \leq 4.8 \cdot 10^{+123}:\\ \;\;\;\;\frac{\frac{1}{x} - \frac{0.5}{x \cdot x}}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 0.7)
   (/ (- (log x)) n)
   (if (<= x 4.8e+123) (/ (- (/ 1.0 x) (/ 0.5 (* x x))) n) (/ 0.0 n))))

C

double code(double x, double n) {
	double tmp;
	if (x <= 0.7) {
		tmp = -log(x) / n;
	} else if (x <= 4.8e+123) {
		tmp = ((1.0 / x) - (0.5 / (x * 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 <= 0.7d0) then
        tmp = -log(x) / n
    else if (x <= 4.8d+123) then
        tmp = ((1.0d0 / x) - (0.5d0 / (x * 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 <= 0.7) {
		tmp = -Math.log(x) / n;
	} else if (x <= 4.8e+123) {
		tmp = ((1.0 / x) - (0.5 / (x * x))) / n;
	} else {
		tmp = 0.0 / n;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 0.7:
		tmp = -math.log(x) / n
	elif x <= 4.8e+123:
		tmp = ((1.0 / x) - (0.5 / (x * x))) / n
	else:
		tmp = 0.0 / n
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 0.7)
		tmp = Float64(Float64(-log(x)) / n);
	elseif (x <= 4.8e+123)
		tmp = Float64(Float64(Float64(1.0 / x) - Float64(0.5 / Float64(x * x))) / n);
	else
		tmp = Float64(0.0 / n);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (x <= 0.7)
		tmp = -log(x) / n;
	elseif (x <= 4.8e+123)
		tmp = ((1.0 / x) - (0.5 / (x * x))) / n;
	else
		tmp = 0.0 / n;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 0.7], N[((-N[Log[x], $MachinePrecision]) / n), $MachinePrecision], If[LessEqual[x, 4.8e+123], N[(N[(N[(1.0 / x), $MachinePrecision] - N[(0.5 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], N[(0.0 / n), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 0.69999999999999996

   1. Initial program 34.5%

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

   2. Taylor expanded in n around inf 58.5%

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

      1. log1p-def 58.5%

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

   4. Simplified 58.5%

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

   5. Taylor expanded in x around 0 58.2%

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

      1. neg-mul-1 58.2%

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

   7. Simplified 58.2%

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

   if 0.69999999999999996 < x < 4.79999999999999978e123

   1. Initial program 42.3%

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

   2. Taylor expanded in n around inf 40.9%

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

      1. log1p-def 40.9%

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

   4. Simplified 40.9%

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

   5. Taylor expanded in x around inf 70.1%

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

      1. associate-*r/ 70.1%

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

      2. metadata-eval 70.1%

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

      3. unpow2 70.1%

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

   7. Simplified 70.1%

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

   if 4.79999999999999978e123 < x

   1. Initial program 80.7%

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

   2. Taylor expanded in n around inf 80.7%

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

      1. log1p-def 80.7%

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

   4. Simplified 80.7%

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

      1. log1p-udef 80.7%

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

      2. diff-log 80.7%

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

   6. Applied egg-rr 80.7%

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

   7. Taylor expanded in x around inf 80.7%

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

4. Final simplification 66.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.7:\\ \;\;\;\;\frac{-\log x}{n}\\ \mathbf{elif}\;x \leq 4.8 \cdot 10^{+123}:\\ \;\;\;\;\frac{\frac{1}{x} - \frac{0.5}{x \cdot x}}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{n}\\ \end{array} \]

Alternative 9: 46.8% accurate, 23.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -2000:\\ \;\;\;\;\frac{0}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= (/ 1.0 n) -2000.0) (/ 0.0 n) (/ (/ 1.0 x) n)))

C

double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -2000.0) {
		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) <= (-2000.0d0)) 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) <= -2000.0) {
		tmp = 0.0 / n;
	} else {
		tmp = (1.0 / x) / n;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if (1.0 / n) <= -2000.0:
		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) <= -2000.0)
		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) <= -2000.0)
		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], -2000.0], 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) < -2e3

   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 60.2%

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

      1. log1p-def 60.2%

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

   4. Simplified 60.2%

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

      1. log1p-udef 60.2%

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

      2. diff-log 60.2%

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

   6. Applied egg-rr 60.2%

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

   7. Taylor expanded in x around inf 58.7%

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

   if -2e3 < (/.f64 1 n)

   1. Initial program 32.6%

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

   2. Taylor expanded in n around inf 61.9%

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

      1. log1p-def 61.9%

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

   4. Simplified 61.9%

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

   5. Taylor expanded in x around inf 48.5%

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

4. Final simplification 51.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -2000:\\ \;\;\;\;\frac{0}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \end{array} \]

Alternative 10: 39.2% 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 48.9%

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

2. Taylor expanded in n around inf 61.5%

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

   1. log1p-def 61.5%

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

4. Simplified 61.5%

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

5. Taylor expanded in x around inf 42.7%

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

   1. *-commutative 42.7%

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

7. Simplified 42.7%

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

8. Final simplification 42.7%

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

Alternative 11: 39.6% 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 48.9%

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

2. Taylor expanded in n around inf 61.5%

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

   1. log1p-def 61.5%

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

4. Simplified 61.5%

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

5. Taylor expanded in x around inf 42.7%

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

   1. *-commutative 42.7%

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

7. Simplified 42.7%

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

8. Taylor expanded in x around 0 42.7%

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

   1. associate-/r* 42.9%

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

10. Simplified 42.9%

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

11. Final simplification 42.9%

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

Alternative 12: 39.6% 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 48.9%

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

2. Taylor expanded in n around inf 61.5%

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

   1. log1p-def 61.5%

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

4. Simplified 61.5%

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

5. Taylor expanded in x around inf 43.0%

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

6. Final simplification 43.0%

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

Alternative 13: 4.6% accurate, 70.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 48.9%

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

2. Taylor expanded in x around 0 27.7%

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

3. Taylor expanded in x around inf 4.6%

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

4. Final simplification 4.6%

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

Reproduce

herbie shell --seed 2023279 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  :precision binary64
  (- (pow (+ x 1.0) (/ 1.0 n)) (pow x (/ 1.0 n))))