2nthrt (problem 3.4.6)

Percentage Accurate: 53.5% → 85.7%

Time: 59.3s

Alternatives: 16

Speedup: 1.8×

• 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, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -4 \cdot 10^{-68}:\\ \;\;\;\;\frac{\frac{t\_0}{n}}{x}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-13}:\\ \;\;\;\;\frac{\left(\mathsf{log1p}\left(x\right) + 0.5 \cdot \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}}{n}\right) - \log x}{n}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{x}{n}} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -4e-68)
     (/ (/ t_0 n) x)
     (if (<= (/ 1.0 n) 5e-13)
       (/
        (-
         (+ (log1p x) (* 0.5 (/ (- (pow (log1p x) 2.0) (pow (log x) 2.0)) n)))
         (log x))
        n)
       (- (exp (/ x n)) t_0)))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -4e-68) {
		tmp = (t_0 / n) / x;
	} else if ((1.0 / n) <= 5e-13) {
		tmp = ((log1p(x) + (0.5 * ((pow(log1p(x), 2.0) - pow(log(x), 2.0)) / n))) - log(x)) / n;
	} else {
		tmp = exp((x / n)) - t_0;
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -4e-68) {
		tmp = (t_0 / n) / x;
	} else if ((1.0 / n) <= 5e-13) {
		tmp = ((Math.log1p(x) + (0.5 * ((Math.pow(Math.log1p(x), 2.0) - Math.pow(Math.log(x), 2.0)) / n))) - Math.log(x)) / n;
	} else {
		tmp = Math.exp((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) <= -4e-68:
		tmp = (t_0 / n) / x
	elif (1.0 / n) <= 5e-13:
		tmp = ((math.log1p(x) + (0.5 * ((math.pow(math.log1p(x), 2.0) - math.pow(math.log(x), 2.0)) / n))) - math.log(x)) / n
	else:
		tmp = math.exp((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) <= -4e-68)
		tmp = Float64(Float64(t_0 / n) / x);
	elseif (Float64(1.0 / n) <= 5e-13)
		tmp = Float64(Float64(Float64(log1p(x) + Float64(0.5 * Float64(Float64((log1p(x) ^ 2.0) - (log(x) ^ 2.0)) / n))) - log(x)) / n);
	else
		tmp = Float64(exp(Float64(x / n)) - t_0);
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -4e-68], N[(N[(t$95$0 / n), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e-13], N[(N[(N[(N[Log[1 + x], $MachinePrecision] + N[(0.5 * N[(N[(N[Power[N[Log[1 + x], $MachinePrecision], 2.0], $MachinePrecision] - N[Power[N[Log[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], N[(N[Exp[N[(x / n), $MachinePrecision]], $MachinePrecision] - t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -4.00000000000000027e-68

   1. Initial program 84.2%

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

   3. Taylor expanded in x around inf 93.6%

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

      1. associate-/r* 93.6%

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

      2. mul-1-neg 93.6%

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

      3. log-rec 93.6%

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

      4. mul-1-neg 93.6%

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

      5. distribute-neg-frac 93.6%

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

      6. mul-1-neg 93.6%

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

      7. remove-double-neg 93.6%

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

      8. *-rgt-identity 93.6%

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

      9. associate-/l* 93.6%

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

      10. exp-to-pow 93.6%

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

   5. Simplified 93.6%

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

   if -4.00000000000000027e-68 < (/.f64 #s(literal 1 binary64) n) < 4.9999999999999999e-13

   1. Initial program 34.8%

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

   3. Taylor expanded in n around inf 81.0%

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

   5. Simplified 81.0%

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

   if 4.9999999999999999e-13 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 67.5%

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

   3. Taylor expanded in n around 0 67.5%

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

      1. log1p-define 99.8%

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

      2. *-rgt-identity 99.8%

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

      3. associate-/l* 99.8%

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

      4. exp-to-pow 99.8%

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

   5. Simplified 99.8%

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

   6. Taylor expanded in x around 0 99.8%

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

4. Final simplification 88.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -4 \cdot 10^{-68}:\\ \;\;\;\;\frac{\frac{{x}^{\left(\frac{1}{n}\right)}}{n}}{x}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-13}:\\ \;\;\;\;\frac{\left(\mathsf{log1p}\left(x\right) + 0.5 \cdot \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}}{n}\right) - \log x}{n}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{x}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 85.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -4 \cdot 10^{-68}:\\ \;\;\;\;\frac{\frac{t\_0}{n}}{x}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-13}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{x}{n}} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -4e-68)
     (/ (/ t_0 n) x)
     (if (<= (/ 1.0 n) 5e-13)
       (/ (log (/ (+ 1.0 x) 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) <= -4e-68) {
		tmp = (t_0 / n) / x;
	} else if ((1.0 / n) <= 5e-13) {
		tmp = log(((1.0 + x) / x)) / n;
	} else {
		tmp = exp((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) :: tmp
    t_0 = x ** (1.0d0 / n)
    if ((1.0d0 / n) <= (-4d-68)) then
        tmp = (t_0 / n) / x
    else if ((1.0d0 / n) <= 5d-13) then
        tmp = log(((1.0d0 + x) / x)) / n
    else
        tmp = exp((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 tmp;
	if ((1.0 / n) <= -4e-68) {
		tmp = (t_0 / n) / x;
	} else if ((1.0 / n) <= 5e-13) {
		tmp = Math.log(((1.0 + x) / 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) <= -4e-68:
		tmp = (t_0 / n) / x
	elif (1.0 / n) <= 5e-13:
		tmp = math.log(((1.0 + x) / 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) <= -4e-68)
		tmp = Float64(Float64(t_0 / n) / x);
	elseif (Float64(1.0 / n) <= 5e-13)
		tmp = Float64(log(Float64(Float64(1.0 + x) / x)) / n);
	else
		tmp = Float64(exp(Float64(x / n)) - t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = x ^ (1.0 / n);
	tmp = 0.0;
	if ((1.0 / n) <= -4e-68)
		tmp = (t_0 / n) / x;
	elseif ((1.0 / n) <= 5e-13)
		tmp = log(((1.0 + x) / x)) / n;
	else
		tmp = exp((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]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -4e-68], N[(N[(t$95$0 / n), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e-13], N[(N[Log[N[(N[(1.0 + x), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], N[(N[Exp[N[(x / n), $MachinePrecision]], $MachinePrecision] - t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -4.00000000000000027e-68

   1. Initial program 84.2%

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

   3. Taylor expanded in x around inf 93.6%

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

      1. associate-/r* 93.6%

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

      2. mul-1-neg 93.6%

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

      3. log-rec 93.6%

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

      4. mul-1-neg 93.6%

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

      5. distribute-neg-frac 93.6%

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

      6. mul-1-neg 93.6%

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

      7. remove-double-neg 93.6%

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

      8. *-rgt-identity 93.6%

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

      9. associate-/l* 93.6%

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

      10. exp-to-pow 93.6%

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

   5. Simplified 93.6%

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

   if -4.00000000000000027e-68 < (/.f64 #s(literal 1 binary64) n) < 4.9999999999999999e-13

   1. Initial program 34.8%

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

   3. Taylor expanded in n around inf 80.7%

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

      1. log1p-define 80.7%

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

   5. Simplified 80.7%

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

      1. log1p-undefine 80.7%

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

      2. diff-log 80.9%

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

   7. Applied egg-rr 80.9%

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

      1. +-commutative 80.9%

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

   9. Simplified 80.9%

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

   if 4.9999999999999999e-13 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 67.5%

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

   3. Taylor expanded in n around 0 67.5%

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

      1. log1p-define 99.8%

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

      2. *-rgt-identity 99.8%

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

      3. associate-/l* 99.8%

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

      4. exp-to-pow 99.8%

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

   5. Simplified 99.8%

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

   6. Taylor expanded in x around 0 99.8%

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

4. Final simplification 88.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -4 \cdot 10^{-68}:\\ \;\;\;\;\frac{\frac{{x}^{\left(\frac{1}{n}\right)}}{n}}{x}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-13}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{x}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 82.1% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -4 \cdot 10^{-68}:\\ \;\;\;\;\frac{\frac{t\_0}{n}}{x}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-13}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{+197}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{expm1}\left(\frac{1 + \frac{-1 + 1.1666666666666667 \cdot \frac{1}{x}}{x}}{x}\right)}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -4e-68)
     (/ (/ t_0 n) x)
     (if (<= (/ 1.0 n) 5e-13)
       (/ (log (/ (+ 1.0 x) x)) n)
       (if (<= (/ 1.0 n) 2e+197)
         (- (+ 1.0 (/ x n)) t_0)
         (/
          (expm1 (/ (+ 1.0 (/ (+ -1.0 (* 1.1666666666666667 (/ 1.0 x))) x)) x))
          n))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -4e-68) {
		tmp = (t_0 / n) / x;
	} else if ((1.0 / n) <= 5e-13) {
		tmp = log(((1.0 + x) / x)) / n;
	} else if ((1.0 / n) <= 2e+197) {
		tmp = (1.0 + (x / n)) - t_0;
	} else {
		tmp = expm1(((1.0 + ((-1.0 + (1.1666666666666667 * (1.0 / x))) / x)) / 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) <= -4e-68) {
		tmp = (t_0 / n) / x;
	} else if ((1.0 / n) <= 5e-13) {
		tmp = Math.log(((1.0 + x) / x)) / n;
	} else if ((1.0 / n) <= 2e+197) {
		tmp = (1.0 + (x / n)) - t_0;
	} else {
		tmp = Math.expm1(((1.0 + ((-1.0 + (1.1666666666666667 * (1.0 / x))) / x)) / x)) / n;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	tmp = 0
	if (1.0 / n) <= -4e-68:
		tmp = (t_0 / n) / x
	elif (1.0 / n) <= 5e-13:
		tmp = math.log(((1.0 + x) / x)) / n
	elif (1.0 / n) <= 2e+197:
		tmp = (1.0 + (x / n)) - t_0
	else:
		tmp = math.expm1(((1.0 + ((-1.0 + (1.1666666666666667 * (1.0 / x))) / x)) / x)) / n
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -4e-68)
		tmp = Float64(Float64(t_0 / n) / x);
	elseif (Float64(1.0 / n) <= 5e-13)
		tmp = Float64(log(Float64(Float64(1.0 + x) / x)) / n);
	elseif (Float64(1.0 / n) <= 2e+197)
		tmp = Float64(Float64(1.0 + Float64(x / n)) - t_0);
	else
		tmp = Float64(expm1(Float64(Float64(1.0 + Float64(Float64(-1.0 + Float64(1.1666666666666667 * Float64(1.0 / x))) / x)) / 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], -4e-68], N[(N[(t$95$0 / n), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e-13], N[(N[Log[N[(N[(1.0 + x), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e+197], N[(N[(1.0 + N[(x / n), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], N[(N[(Exp[N[(N[(1.0 + N[(N[(-1.0 + N[(1.1666666666666667 * N[(1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]] - 1), $MachinePrecision] / n), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -4.00000000000000027e-68

   1. Initial program 84.2%

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

   3. Taylor expanded in x around inf 93.6%

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

      1. associate-/r* 93.6%

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

      2. mul-1-neg 93.6%

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

      3. log-rec 93.6%

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

      4. mul-1-neg 93.6%

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

      5. distribute-neg-frac 93.6%

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

      6. mul-1-neg 93.6%

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

      7. remove-double-neg 93.6%

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

      8. *-rgt-identity 93.6%

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

      9. associate-/l* 93.6%

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

      10. exp-to-pow 93.6%

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

   5. Simplified 93.6%

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

   if -4.00000000000000027e-68 < (/.f64 #s(literal 1 binary64) n) < 4.9999999999999999e-13

   1. Initial program 34.8%

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

   3. Taylor expanded in n around inf 80.7%

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

      1. log1p-define 80.7%

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

   5. Simplified 80.7%

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

      1. log1p-undefine 80.7%

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

      2. diff-log 80.9%

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

   7. Applied egg-rr 80.9%

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

      1. +-commutative 80.9%

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

   9. Simplified 80.9%

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

   if 4.9999999999999999e-13 < (/.f64 #s(literal 1 binary64) n) < 1.9999999999999999e197

   1. Initial program 78.9%

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

   3. Taylor expanded in x around 0 75.5%

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

      1. *-rgt-identity 75.5%

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

      2. associate-/l* 75.5%

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

      3. exp-to-pow 75.5%

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

   5. Simplified 75.5%

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

   if 1.9999999999999999e197 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 27.3%

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

   3. Taylor expanded in n around inf 18.9%

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

      1. log1p-define 18.9%

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

   5. Simplified 18.9%

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

      1. expm1-log1p-u 18.9%

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

   7. Applied egg-rr 18.9%

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

   8. Taylor expanded in x around -inf 87.9%

      \[\leadsto \frac{\mathsf{expm1}\left(\color{blue}{-1 \cdot \frac{-1 \cdot \frac{1.1666666666666667 \cdot \frac{1}{x} - 1}{x} - 1}{x}}\right)}{n} \]
3. Recombined 4 regimes into one program.

4. Final simplification 85.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -4 \cdot 10^{-68}:\\ \;\;\;\;\frac{\frac{{x}^{\left(\frac{1}{n}\right)}}{n}}{x}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-13}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{+197}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{expm1}\left(\frac{1 + \frac{-1 + 1.1666666666666667 \cdot \frac{1}{x}}{x}}{x}\right)}{n}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 81.1% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -4 \cdot 10^{-68}:\\ \;\;\;\;\frac{\frac{t\_0}{n}}{x}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-13}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{+244}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-1 + \frac{0.5 + \frac{-0.3333333333333333}{x}}{x}}{-x}}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -4e-68)
     (/ (/ t_0 n) x)
     (if (<= (/ 1.0 n) 5e-13)
       (/ (log (/ (+ 1.0 x) x)) n)
       (if (<= (/ 1.0 n) 2e+244)
         (- (+ 1.0 (/ x n)) t_0)
         (/ (/ (+ -1.0 (/ (+ 0.5 (/ -0.3333333333333333 x)) x)) (- x)) n))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -4e-68) {
		tmp = (t_0 / n) / x;
	} else if ((1.0 / n) <= 5e-13) {
		tmp = log(((1.0 + x) / x)) / n;
	} else if ((1.0 / n) <= 2e+244) {
		tmp = (1.0 + (x / n)) - t_0;
	} else {
		tmp = ((-1.0 + ((0.5 + (-0.3333333333333333 / x)) / x)) / -x) / n;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x ** (1.0d0 / n)
    if ((1.0d0 / n) <= (-4d-68)) then
        tmp = (t_0 / n) / x
    else if ((1.0d0 / n) <= 5d-13) then
        tmp = log(((1.0d0 + x) / x)) / n
    else if ((1.0d0 / n) <= 2d+244) then
        tmp = (1.0d0 + (x / n)) - t_0
    else
        tmp = (((-1.0d0) + ((0.5d0 + ((-0.3333333333333333d0) / x)) / x)) / -x) / n
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -4e-68) {
		tmp = (t_0 / n) / x;
	} else if ((1.0 / n) <= 5e-13) {
		tmp = Math.log(((1.0 + x) / x)) / n;
	} else if ((1.0 / n) <= 2e+244) {
		tmp = (1.0 + (x / n)) - t_0;
	} else {
		tmp = ((-1.0 + ((0.5 + (-0.3333333333333333 / x)) / x)) / -x) / n;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	tmp = 0
	if (1.0 / n) <= -4e-68:
		tmp = (t_0 / n) / x
	elif (1.0 / n) <= 5e-13:
		tmp = math.log(((1.0 + x) / x)) / n
	elif (1.0 / n) <= 2e+244:
		tmp = (1.0 + (x / n)) - t_0
	else:
		tmp = ((-1.0 + ((0.5 + (-0.3333333333333333 / x)) / x)) / -x) / n
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -4e-68)
		tmp = Float64(Float64(t_0 / n) / x);
	elseif (Float64(1.0 / n) <= 5e-13)
		tmp = Float64(log(Float64(Float64(1.0 + x) / x)) / n);
	elseif (Float64(1.0 / n) <= 2e+244)
		tmp = Float64(Float64(1.0 + Float64(x / n)) - t_0);
	else
		tmp = Float64(Float64(Float64(-1.0 + Float64(Float64(0.5 + Float64(-0.3333333333333333 / x)) / x)) / Float64(-x)) / n);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = x ^ (1.0 / n);
	tmp = 0.0;
	if ((1.0 / n) <= -4e-68)
		tmp = (t_0 / n) / x;
	elseif ((1.0 / n) <= 5e-13)
		tmp = log(((1.0 + x) / x)) / n;
	elseif ((1.0 / n) <= 2e+244)
		tmp = (1.0 + (x / n)) - t_0;
	else
		tmp = ((-1.0 + ((0.5 + (-0.3333333333333333 / x)) / x)) / -x) / n;
	end
	tmp_2 = 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], -4e-68], N[(N[(t$95$0 / n), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e-13], N[(N[Log[N[(N[(1.0 + x), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e+244], N[(N[(1.0 + N[(x / n), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], N[(N[(N[(-1.0 + N[(N[(0.5 + N[(-0.3333333333333333 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / (-x)), $MachinePrecision] / n), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -4.00000000000000027e-68

   1. Initial program 84.2%

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

   3. Taylor expanded in x around inf 93.6%

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

      1. associate-/r* 93.6%

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

      2. mul-1-neg 93.6%

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

      3. log-rec 93.6%

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

      4. mul-1-neg 93.6%

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

      5. distribute-neg-frac 93.6%

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

      6. mul-1-neg 93.6%

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

      7. remove-double-neg 93.6%

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

      8. *-rgt-identity 93.6%

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

      9. associate-/l* 93.6%

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

      10. exp-to-pow 93.6%

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

   5. Simplified 93.6%

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

   if -4.00000000000000027e-68 < (/.f64 #s(literal 1 binary64) n) < 4.9999999999999999e-13

   1. Initial program 34.8%

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

   3. Taylor expanded in n around inf 80.7%

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

      1. log1p-define 80.7%

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

   5. Simplified 80.7%

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

      1. log1p-undefine 80.7%

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

      2. diff-log 80.9%

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

   7. Applied egg-rr 80.9%

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

      1. +-commutative 80.9%

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

   9. Simplified 80.9%

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

   if 4.9999999999999999e-13 < (/.f64 #s(literal 1 binary64) n) < 2.00000000000000015e244

   1. Initial program 77.8%

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

   3. Taylor expanded in x around 0 71.8%

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

      1. *-rgt-identity 71.8%

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

      2. associate-/l* 71.8%

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

      3. exp-to-pow 71.8%

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

   5. Simplified 71.8%

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

   if 2.00000000000000015e244 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 3.1%

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

   3. Taylor expanded in n around inf 27.0%

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

      1. log1p-define 27.0%

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

   5. Simplified 27.0%

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

      1. expm1-log1p-u 27.0%

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

   7. Applied egg-rr 27.0%

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

   8. Taylor expanded in x around -inf 100.0%

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

      1. mul-1-neg 100.0%

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

      2. distribute-neg-frac2 100.0%

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

      3. sub-neg 100.0%

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

      4. associate-*r/ 100.0%

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

      5. sub-neg 100.0%

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

      6. metadata-eval 100.0%

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

      7. distribute-lft-in 100.0%

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

      8. neg-mul-1 100.0%

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

      9. associate-*r/ 100.0%

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

      10. metadata-eval 100.0%

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

      11. distribute-neg-frac 100.0%

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

      12. metadata-eval 100.0%

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

      13. metadata-eval 100.0%

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

      14. metadata-eval 100.0%

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

   10. Simplified 100.0%

       \[\leadsto \frac{\color{blue}{\frac{\frac{\frac{-0.3333333333333333}{x} + 0.5}{x} + -1}{-x}}}{n} \]
3. Recombined 4 regimes into one program.

4. Final simplification 85.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -4 \cdot 10^{-68}:\\ \;\;\;\;\frac{\frac{{x}^{\left(\frac{1}{n}\right)}}{n}}{x}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-13}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{+244}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-1 + \frac{0.5 + \frac{-0.3333333333333333}{x}}{x}}{-x}}{n}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 62.1% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -9.8 \cdot 10^{+253}:\\ \;\;\;\;\frac{\log x}{-n}\\ \mathbf{elif}\;n \leq -4.6:\\ \;\;\;\;\frac{1}{n} \cdot \frac{1}{x}\\ \mathbf{elif}\;n \leq 2.05 \cdot 10^{-198}:\\ \;\;\;\;\frac{\frac{-0.25}{n}}{-{x}^{4}}\\ \mathbf{elif}\;n \leq 16200000000:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;n \leq 3.3 \cdot 10^{+93}:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{n} + \frac{\frac{0.3333333333333333}{n \cdot x} - \frac{0.5}{n}}{x}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= n -9.8e+253)
   (/ (log x) (- n))
   (if (<= n -4.6)
     (* (/ 1.0 n) (/ 1.0 x))
     (if (<= n 2.05e-198)
       (/ (/ -0.25 n) (- (pow x 4.0)))
       (if (<= n 16200000000.0)
         (- 1.0 (pow x (/ 1.0 n)))
         (if (<= n 3.3e+93)
           (/ (- x (log x)) n)
           (/
            (+ (/ 1.0 n) (/ (- (/ 0.3333333333333333 (* n x)) (/ 0.5 n)) x))
            x)))))))

C

double code(double x, double n) {
	double tmp;
	if (n <= -9.8e+253) {
		tmp = log(x) / -n;
	} else if (n <= -4.6) {
		tmp = (1.0 / n) * (1.0 / x);
	} else if (n <= 2.05e-198) {
		tmp = (-0.25 / n) / -pow(x, 4.0);
	} else if (n <= 16200000000.0) {
		tmp = 1.0 - pow(x, (1.0 / n));
	} else if (n <= 3.3e+93) {
		tmp = (x - log(x)) / n;
	} else {
		tmp = ((1.0 / n) + (((0.3333333333333333 / (n * x)) - (0.5 / n)) / x)) / x;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if (n <= (-9.8d+253)) then
        tmp = log(x) / -n
    else if (n <= (-4.6d0)) then
        tmp = (1.0d0 / n) * (1.0d0 / x)
    else if (n <= 2.05d-198) then
        tmp = ((-0.25d0) / n) / -(x ** 4.0d0)
    else if (n <= 16200000000.0d0) then
        tmp = 1.0d0 - (x ** (1.0d0 / n))
    else if (n <= 3.3d+93) then
        tmp = (x - log(x)) / n
    else
        tmp = ((1.0d0 / n) + (((0.3333333333333333d0 / (n * x)) - (0.5d0 / n)) / x)) / x
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (n <= -9.8e+253) {
		tmp = Math.log(x) / -n;
	} else if (n <= -4.6) {
		tmp = (1.0 / n) * (1.0 / x);
	} else if (n <= 2.05e-198) {
		tmp = (-0.25 / n) / -Math.pow(x, 4.0);
	} else if (n <= 16200000000.0) {
		tmp = 1.0 - Math.pow(x, (1.0 / n));
	} else if (n <= 3.3e+93) {
		tmp = (x - Math.log(x)) / n;
	} else {
		tmp = ((1.0 / n) + (((0.3333333333333333 / (n * x)) - (0.5 / n)) / x)) / x;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if n <= -9.8e+253:
		tmp = math.log(x) / -n
	elif n <= -4.6:
		tmp = (1.0 / n) * (1.0 / x)
	elif n <= 2.05e-198:
		tmp = (-0.25 / n) / -math.pow(x, 4.0)
	elif n <= 16200000000.0:
		tmp = 1.0 - math.pow(x, (1.0 / n))
	elif n <= 3.3e+93:
		tmp = (x - math.log(x)) / n
	else:
		tmp = ((1.0 / n) + (((0.3333333333333333 / (n * x)) - (0.5 / n)) / x)) / x
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (n <= -9.8e+253)
		tmp = Float64(log(x) / Float64(-n));
	elseif (n <= -4.6)
		tmp = Float64(Float64(1.0 / n) * Float64(1.0 / x));
	elseif (n <= 2.05e-198)
		tmp = Float64(Float64(-0.25 / n) / Float64(-(x ^ 4.0)));
	elseif (n <= 16200000000.0)
		tmp = Float64(1.0 - (x ^ Float64(1.0 / n)));
	elseif (n <= 3.3e+93)
		tmp = Float64(Float64(x - log(x)) / n);
	else
		tmp = Float64(Float64(Float64(1.0 / n) + Float64(Float64(Float64(0.3333333333333333 / Float64(n * x)) - Float64(0.5 / n)) / x)) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (n <= -9.8e+253)
		tmp = log(x) / -n;
	elseif (n <= -4.6)
		tmp = (1.0 / n) * (1.0 / x);
	elseif (n <= 2.05e-198)
		tmp = (-0.25 / n) / -(x ^ 4.0);
	elseif (n <= 16200000000.0)
		tmp = 1.0 - (x ^ (1.0 / n));
	elseif (n <= 3.3e+93)
		tmp = (x - log(x)) / n;
	else
		tmp = ((1.0 / n) + (((0.3333333333333333 / (n * x)) - (0.5 / n)) / x)) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[n, -9.8e+253], N[(N[Log[x], $MachinePrecision] / (-n)), $MachinePrecision], If[LessEqual[n, -4.6], N[(N[(1.0 / n), $MachinePrecision] * N[(1.0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 2.05e-198], N[(N[(-0.25 / n), $MachinePrecision] / (-N[Power[x, 4.0], $MachinePrecision])), $MachinePrecision], If[LessEqual[n, 16200000000.0], N[(1.0 - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 3.3e+93], N[(N[(x - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], N[(N[(N[(1.0 / n), $MachinePrecision] + N[(N[(N[(0.3333333333333333 / N[(n * x), $MachinePrecision]), $MachinePrecision] - N[(0.5 / n), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]]]]]]

Derivation

1. Split input into 6 regimes

2. if n < -9.8000000000000002e253

   1. Initial program 29.1%

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

   3. Taylor expanded in n around inf 99.5%

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

      1. log1p-define 99.5%

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

   5. Simplified 99.5%

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

   6. Taylor expanded in x around 0 79.0%

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

      1. neg-mul-1 79.0%

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

   8. Simplified 79.0%

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

   if -9.8000000000000002e253 < n < -4.5999999999999996

   1. Initial program 33.6%

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

   3. Taylor expanded in n around inf 71.9%

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

      1. log1p-define 71.9%

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

   5. Simplified 71.9%

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

   6. Taylor expanded in x around inf 57.8%

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

      1. *-commutative 57.8%

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

   8. Simplified 57.8%

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

      1. inv-pow 57.8%

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

      2. unpow-prod-down 60.4%

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

      3. inv-pow 60.4%

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

      4. inv-pow 60.4%

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

   10. Applied egg-rr 60.4%

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

   if -4.5999999999999996 < n < 2.05000000000000006e-198

   1. Initial program 93.3%

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

   3. Taylor expanded in n around inf 47.8%

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

      1. log1p-define 47.8%

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

   5. Simplified 47.8%

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

   6. Taylor expanded in x around -inf 1.5%

      \[\leadsto \color{blue}{-1 \cdot \frac{-1 \cdot \frac{-1 \cdot \frac{0.25 \cdot \frac{1}{n \cdot x} - 0.3333333333333333 \cdot \frac{1}{n}}{x} - 0.5 \cdot \frac{1}{n}}{x} - \frac{1}{n}}{x}} \]
   7. Step-by-step derivation

      1. mul-1-neg 1.5%

         \[\leadsto \color{blue}{-\frac{-1 \cdot \frac{-1 \cdot \frac{0.25 \cdot \frac{1}{n \cdot x} - 0.3333333333333333 \cdot \frac{1}{n}}{x} - 0.5 \cdot \frac{1}{n}}{x} - \frac{1}{n}}{x}} \]

   8. Simplified 1.5%

      \[\leadsto \color{blue}{-\frac{\left(-\frac{\left(-\frac{\frac{0.25}{x \cdot n} - \frac{0.3333333333333333}{n}}{x}\right) - \frac{0.5}{n}}{x}\right) - \frac{1}{n}}{x}} \]
   9. Step-by-step derivation

      1. add-sqr-sqrt 1.5%

         \[\leadsto -\frac{\left(-\frac{\left(-\color{blue}{\sqrt{\frac{\frac{0.25}{x \cdot n} - \frac{0.3333333333333333}{n}}{x}} \cdot \sqrt{\frac{\frac{0.25}{x \cdot n} - \frac{0.3333333333333333}{n}}{x}}}\right) - \frac{0.5}{n}}{x}\right) - \frac{1}{n}}{x} \]

      2. sqrt-unprod 46.6%

         \[\leadsto -\frac{\left(-\frac{\left(-\color{blue}{\sqrt{\frac{\frac{0.25}{x \cdot n} - \frac{0.3333333333333333}{n}}{x} \cdot \frac{\frac{0.25}{x \cdot n} - \frac{0.3333333333333333}{n}}{x}}}\right) - \frac{0.5}{n}}{x}\right) - \frac{1}{n}}{x} \]

      3. sqr-neg 46.6%

         \[\leadsto -\frac{\left(-\frac{\left(-\sqrt{\color{blue}{\left(-\frac{\frac{0.25}{x \cdot n} - \frac{0.3333333333333333}{n}}{x}\right) \cdot \left(-\frac{\frac{0.25}{x \cdot n} - \frac{0.3333333333333333}{n}}{x}\right)}}\right) - \frac{0.5}{n}}{x}\right) - \frac{1}{n}}{x} \]

      4. sqrt-unprod 42.9%

         \[\leadsto -\frac{\left(-\frac{\left(-\color{blue}{\sqrt{-\frac{\frac{0.25}{x \cdot n} - \frac{0.3333333333333333}{n}}{x}} \cdot \sqrt{-\frac{\frac{0.25}{x \cdot n} - \frac{0.3333333333333333}{n}}{x}}}\right) - \frac{0.5}{n}}{x}\right) - \frac{1}{n}}{x} \]

      5. add-sqr-sqrt 51.4%

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

      6. neg-sub0 51.4%

         \[\leadsto -\frac{\left(-\frac{\left(-\color{blue}{\left(0 - \frac{\frac{0.25}{x \cdot n} - \frac{0.3333333333333333}{n}}{x}\right)}\right) - \frac{0.5}{n}}{x}\right) - \frac{1}{n}}{x} \]

      7. associate-/r* 51.4%

         \[\leadsto -\frac{\left(-\frac{\left(-\left(0 - \frac{\color{blue}{\frac{\frac{0.25}{x}}{n}} - \frac{0.3333333333333333}{n}}{x}\right)\right) - \frac{0.5}{n}}{x}\right) - \frac{1}{n}}{x} \]

      8. sub-div 51.4%

         \[\leadsto -\frac{\left(-\frac{\left(-\left(0 - \frac{\color{blue}{\frac{\frac{0.25}{x} - 0.3333333333333333}{n}}}{x}\right)\right) - \frac{0.5}{n}}{x}\right) - \frac{1}{n}}{x} \]

   10. Applied egg-rr 51.4%

       \[\leadsto -\frac{\left(-\frac{\left(-\color{blue}{\left(0 - \frac{\frac{\frac{0.25}{x} - 0.3333333333333333}{n}}{x}\right)}\right) - \frac{0.5}{n}}{x}\right) - \frac{1}{n}}{x} \]
   11. Step-by-step derivation

       1. neg-sub0 51.4%

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

       2. distribute-neg-frac 51.4%

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

       3. distribute-neg-frac 51.4%

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

       4. mul-1-neg 51.4%

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

       5. sub-neg 51.4%

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

       6. metadata-eval 51.4%

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

       7. associate-*r/ 51.4%

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

       8. metadata-eval 51.4%

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

       9. distribute-lft-in 51.4%

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

       10. neg-mul-1 51.4%

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

       11. associate-*r/ 51.4%

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

       12. metadata-eval 51.4%

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

       13. distribute-neg-frac 51.4%

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

       14. metadata-eval 51.4%

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

       15. metadata-eval 51.4%

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

   12. Simplified 51.4%

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

   13. Taylor expanded in x around 0 85.8%

       \[\leadsto -\color{blue}{\frac{-0.25}{n \cdot {x}^{4}}} \]
   14. Step-by-step derivation

       1. associate-/r* 85.8%

          \[\leadsto -\color{blue}{\frac{\frac{-0.25}{n}}{{x}^{4}}} \]

   15. Simplified 85.8%

       \[\leadsto -\color{blue}{\frac{\frac{-0.25}{n}}{{x}^{4}}} \]

   if 2.05000000000000006e-198 < n < 1.62e10

   1. Initial program 78.9%

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

   3. Taylor expanded in x around 0 75.5%

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

      1. *-rgt-identity 75.5%

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

      2. associate-/l* 75.5%

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

      3. exp-to-pow 75.5%

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

   5. Simplified 75.5%

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

   if 1.62e10 < n < 3.30000000000000009e93

   1. Initial program 6.7%

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

   3. Taylor expanded in n around inf 73.2%

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

      1. log1p-define 73.2%

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

   5. Simplified 73.2%

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

   6. Taylor expanded in x around 0 68.3%

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

   if 3.30000000000000009e93 < n

   1. Initial program 40.6%

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

   3. Taylor expanded in n around inf 78.1%

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

      1. log1p-define 78.1%

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

   5. Simplified 78.1%

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

   6. Taylor expanded in x around -inf 63.1%

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

      1. mul-1-neg 63.1%

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

      2. mul-1-neg 63.1%

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

      3. associate-*r/ 63.1%

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

      4. metadata-eval 63.1%

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

      5. *-commutative 63.1%

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

      6. associate-*r/ 63.1%

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

      7. metadata-eval 63.1%

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

   8. Simplified 63.1%

      \[\leadsto \color{blue}{-\frac{\left(-\frac{\frac{0.3333333333333333}{x \cdot n} - \frac{0.5}{n}}{x}\right) - \frac{1}{n}}{x}} \]
3. Recombined 6 regimes into one program.

4. Final simplification 72.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -9.8 \cdot 10^{+253}:\\ \;\;\;\;\frac{\log x}{-n}\\ \mathbf{elif}\;n \leq -4.6:\\ \;\;\;\;\frac{1}{n} \cdot \frac{1}{x}\\ \mathbf{elif}\;n \leq 2.05 \cdot 10^{-198}:\\ \;\;\;\;\frac{\frac{-0.25}{n}}{-{x}^{4}}\\ \mathbf{elif}\;n \leq 16200000000:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;n \leq 3.3 \cdot 10^{+93}:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{n} + \frac{\frac{0.3333333333333333}{n \cdot x} - \frac{0.5}{n}}{x}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 76.7% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -20:\\ \;\;\;\;\frac{\frac{-0.25}{n}}{-{x}^{4}}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-13}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{+197}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{n} - \frac{\frac{\frac{-0.25}{n \cdot x}}{x} + \frac{0.5}{n}}{x}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= (/ 1.0 n) -20.0)
   (/ (/ -0.25 n) (- (pow x 4.0)))
   (if (<= (/ 1.0 n) 5e-13)
     (/ (log (/ (+ 1.0 x) x)) n)
     (if (<= (/ 1.0 n) 2e+197)
       (- 1.0 (pow x (/ 1.0 n)))
       (/ (- (/ 1.0 n) (/ (+ (/ (/ -0.25 (* n x)) x) (/ 0.5 n)) x)) x)))))

C

double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -20.0) {
		tmp = (-0.25 / n) / -pow(x, 4.0);
	} else if ((1.0 / n) <= 5e-13) {
		tmp = log(((1.0 + x) / x)) / n;
	} else if ((1.0 / n) <= 2e+197) {
		tmp = 1.0 - pow(x, (1.0 / n));
	} else {
		tmp = ((1.0 / n) - ((((-0.25 / (n * x)) / x) + (0.5 / n)) / x)) / 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) <= (-20.0d0)) then
        tmp = ((-0.25d0) / n) / -(x ** 4.0d0)
    else if ((1.0d0 / n) <= 5d-13) then
        tmp = log(((1.0d0 + x) / x)) / n
    else if ((1.0d0 / n) <= 2d+197) then
        tmp = 1.0d0 - (x ** (1.0d0 / n))
    else
        tmp = ((1.0d0 / n) - (((((-0.25d0) / (n * x)) / x) + (0.5d0 / n)) / x)) / x
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -20.0) {
		tmp = (-0.25 / n) / -Math.pow(x, 4.0);
	} else if ((1.0 / n) <= 5e-13) {
		tmp = Math.log(((1.0 + x) / x)) / n;
	} else if ((1.0 / n) <= 2e+197) {
		tmp = 1.0 - Math.pow(x, (1.0 / n));
	} else {
		tmp = ((1.0 / n) - ((((-0.25 / (n * x)) / x) + (0.5 / n)) / x)) / x;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if (1.0 / n) <= -20.0:
		tmp = (-0.25 / n) / -math.pow(x, 4.0)
	elif (1.0 / n) <= 5e-13:
		tmp = math.log(((1.0 + x) / x)) / n
	elif (1.0 / n) <= 2e+197:
		tmp = 1.0 - math.pow(x, (1.0 / n))
	else:
		tmp = ((1.0 / n) - ((((-0.25 / (n * x)) / x) + (0.5 / n)) / x)) / x
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -20.0)
		tmp = Float64(Float64(-0.25 / n) / Float64(-(x ^ 4.0)));
	elseif (Float64(1.0 / n) <= 5e-13)
		tmp = Float64(log(Float64(Float64(1.0 + x) / x)) / n);
	elseif (Float64(1.0 / n) <= 2e+197)
		tmp = Float64(1.0 - (x ^ Float64(1.0 / n)));
	else
		tmp = Float64(Float64(Float64(1.0 / n) - Float64(Float64(Float64(Float64(-0.25 / Float64(n * x)) / x) + Float64(0.5 / n)) / x)) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if ((1.0 / n) <= -20.0)
		tmp = (-0.25 / n) / -(x ^ 4.0);
	elseif ((1.0 / n) <= 5e-13)
		tmp = log(((1.0 + x) / x)) / n;
	elseif ((1.0 / n) <= 2e+197)
		tmp = 1.0 - (x ^ (1.0 / n));
	else
		tmp = ((1.0 / n) - ((((-0.25 / (n * x)) / x) + (0.5 / n)) / x)) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[N[(1.0 / n), $MachinePrecision], -20.0], N[(N[(-0.25 / n), $MachinePrecision] / (-N[Power[x, 4.0], $MachinePrecision])), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e-13], N[(N[Log[N[(N[(1.0 + x), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e+197], N[(1.0 - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 / n), $MachinePrecision] - N[(N[(N[(N[(-0.25 / N[(n * x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] + N[(0.5 / n), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -20

   1. Initial program 100.0%

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

   3. Taylor expanded in n around inf 50.8%

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

      1. log1p-define 50.8%

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

   5. Simplified 50.8%

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

   6. Taylor expanded in x around -inf 1.7%

      \[\leadsto \color{blue}{-1 \cdot \frac{-1 \cdot \frac{-1 \cdot \frac{0.25 \cdot \frac{1}{n \cdot x} - 0.3333333333333333 \cdot \frac{1}{n}}{x} - 0.5 \cdot \frac{1}{n}}{x} - \frac{1}{n}}{x}} \]
   7. Step-by-step derivation

      1. mul-1-neg 1.7%

         \[\leadsto \color{blue}{-\frac{-1 \cdot \frac{-1 \cdot \frac{0.25 \cdot \frac{1}{n \cdot x} - 0.3333333333333333 \cdot \frac{1}{n}}{x} - 0.5 \cdot \frac{1}{n}}{x} - \frac{1}{n}}{x}} \]

   8. Simplified 1.7%

      \[\leadsto \color{blue}{-\frac{\left(-\frac{\left(-\frac{\frac{0.25}{x \cdot n} - \frac{0.3333333333333333}{n}}{x}\right) - \frac{0.5}{n}}{x}\right) - \frac{1}{n}}{x}} \]
   9. Step-by-step derivation

      1. add-sqr-sqrt 1.7%

         \[\leadsto -\frac{\left(-\frac{\left(-\color{blue}{\sqrt{\frac{\frac{0.25}{x \cdot n} - \frac{0.3333333333333333}{n}}{x}} \cdot \sqrt{\frac{\frac{0.25}{x \cdot n} - \frac{0.3333333333333333}{n}}{x}}}\right) - \frac{0.5}{n}}{x}\right) - \frac{1}{n}}{x} \]

      2. sqrt-unprod 51.3%

         \[\leadsto -\frac{\left(-\frac{\left(-\color{blue}{\sqrt{\frac{\frac{0.25}{x \cdot n} - \frac{0.3333333333333333}{n}}{x} \cdot \frac{\frac{0.25}{x \cdot n} - \frac{0.3333333333333333}{n}}{x}}}\right) - \frac{0.5}{n}}{x}\right) - \frac{1}{n}}{x} \]

      3. sqr-neg 51.3%

         \[\leadsto -\frac{\left(-\frac{\left(-\sqrt{\color{blue}{\left(-\frac{\frac{0.25}{x \cdot n} - \frac{0.3333333333333333}{n}}{x}\right) \cdot \left(-\frac{\frac{0.25}{x \cdot n} - \frac{0.3333333333333333}{n}}{x}\right)}}\right) - \frac{0.5}{n}}{x}\right) - \frac{1}{n}}{x} \]

      4. sqrt-unprod 47.2%

         \[\leadsto -\frac{\left(-\frac{\left(-\color{blue}{\sqrt{-\frac{\frac{0.25}{x \cdot n} - \frac{0.3333333333333333}{n}}{x}} \cdot \sqrt{-\frac{\frac{0.25}{x \cdot n} - \frac{0.3333333333333333}{n}}{x}}}\right) - \frac{0.5}{n}}{x}\right) - \frac{1}{n}}{x} \]

      5. add-sqr-sqrt 48.9%

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

      6. neg-sub0 48.9%

         \[\leadsto -\frac{\left(-\frac{\left(-\color{blue}{\left(0 - \frac{\frac{0.25}{x \cdot n} - \frac{0.3333333333333333}{n}}{x}\right)}\right) - \frac{0.5}{n}}{x}\right) - \frac{1}{n}}{x} \]

      7. associate-/r* 48.9%

         \[\leadsto -\frac{\left(-\frac{\left(-\left(0 - \frac{\color{blue}{\frac{\frac{0.25}{x}}{n}} - \frac{0.3333333333333333}{n}}{x}\right)\right) - \frac{0.5}{n}}{x}\right) - \frac{1}{n}}{x} \]

      8. sub-div 48.9%

         \[\leadsto -\frac{\left(-\frac{\left(-\left(0 - \frac{\color{blue}{\frac{\frac{0.25}{x} - 0.3333333333333333}{n}}}{x}\right)\right) - \frac{0.5}{n}}{x}\right) - \frac{1}{n}}{x} \]

   10. Applied egg-rr 48.9%

       \[\leadsto -\frac{\left(-\frac{\left(-\color{blue}{\left(0 - \frac{\frac{\frac{0.25}{x} - 0.3333333333333333}{n}}{x}\right)}\right) - \frac{0.5}{n}}{x}\right) - \frac{1}{n}}{x} \]
   11. Step-by-step derivation

       1. neg-sub0 48.9%

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

       2. distribute-neg-frac 48.9%

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

       3. distribute-neg-frac 48.9%

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

       4. mul-1-neg 48.9%

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

       5. sub-neg 48.9%

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

       6. metadata-eval 48.9%

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

       7. associate-*r/ 48.9%

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

       8. metadata-eval 48.9%

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

       9. distribute-lft-in 48.9%

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

       10. neg-mul-1 48.9%

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

       11. associate-*r/ 48.9%

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

       12. metadata-eval 48.9%

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

       13. distribute-neg-frac 48.9%

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

       14. metadata-eval 48.9%

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

       15. metadata-eval 48.9%

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

   12. Simplified 48.9%

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

   13. Taylor expanded in x around 0 86.7%

       \[\leadsto -\color{blue}{\frac{-0.25}{n \cdot {x}^{4}}} \]
   14. Step-by-step derivation

       1. associate-/r* 86.7%

          \[\leadsto -\color{blue}{\frac{\frac{-0.25}{n}}{{x}^{4}}} \]

   15. Simplified 86.7%

       \[\leadsto -\color{blue}{\frac{\frac{-0.25}{n}}{{x}^{4}}} \]

   if -20 < (/.f64 #s(literal 1 binary64) n) < 4.9999999999999999e-13

   1. Initial program 32.3%

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

   3. Taylor expanded in n around inf 76.1%

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

      1. log1p-define 76.1%

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

   5. Simplified 76.1%

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

      1. log1p-undefine 76.1%

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

      2. diff-log 76.4%

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

   7. Applied egg-rr 76.4%

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

      1. +-commutative 76.4%

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

   9. Simplified 76.4%

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

   if 4.9999999999999999e-13 < (/.f64 #s(literal 1 binary64) n) < 1.9999999999999999e197

   1. Initial program 78.9%

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

   3. Taylor expanded in x around 0 75.5%

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

      1. *-rgt-identity 75.5%

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

      2. associate-/l* 75.5%

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

      3. exp-to-pow 75.5%

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

   5. Simplified 75.5%

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

   if 1.9999999999999999e197 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 27.3%

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

   3. Taylor expanded in n around inf 18.9%

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

      1. log1p-define 18.9%

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

   5. Simplified 18.9%

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

   6. Taylor expanded in x around -inf 0.1%

      \[\leadsto \color{blue}{-1 \cdot \frac{-1 \cdot \frac{-1 \cdot \frac{0.25 \cdot \frac{1}{n \cdot x} - 0.3333333333333333 \cdot \frac{1}{n}}{x} - 0.5 \cdot \frac{1}{n}}{x} - \frac{1}{n}}{x}} \]
   7. Step-by-step derivation

      1. mul-1-neg 0.1%

         \[\leadsto \color{blue}{-\frac{-1 \cdot \frac{-1 \cdot \frac{0.25 \cdot \frac{1}{n \cdot x} - 0.3333333333333333 \cdot \frac{1}{n}}{x} - 0.5 \cdot \frac{1}{n}}{x} - \frac{1}{n}}{x}} \]

   8. Simplified 0.1%

      \[\leadsto \color{blue}{-\frac{\left(-\frac{\left(-\frac{\frac{0.25}{x \cdot n} - \frac{0.3333333333333333}{n}}{x}\right) - \frac{0.5}{n}}{x}\right) - \frac{1}{n}}{x}} \]
   9. Step-by-step derivation

      1. add-sqr-sqrt 0.1%

         \[\leadsto -\frac{\left(-\frac{\left(-\color{blue}{\sqrt{\frac{\frac{0.25}{x \cdot n} - \frac{0.3333333333333333}{n}}{x}} \cdot \sqrt{\frac{\frac{0.25}{x \cdot n} - \frac{0.3333333333333333}{n}}{x}}}\right) - \frac{0.5}{n}}{x}\right) - \frac{1}{n}}{x} \]

      2. sqrt-unprod 0.1%

         \[\leadsto -\frac{\left(-\frac{\left(-\color{blue}{\sqrt{\frac{\frac{0.25}{x \cdot n} - \frac{0.3333333333333333}{n}}{x} \cdot \frac{\frac{0.25}{x \cdot n} - \frac{0.3333333333333333}{n}}{x}}}\right) - \frac{0.5}{n}}{x}\right) - \frac{1}{n}}{x} \]

      3. sqr-neg 0.1%

         \[\leadsto -\frac{\left(-\frac{\left(-\sqrt{\color{blue}{\left(-\frac{\frac{0.25}{x \cdot n} - \frac{0.3333333333333333}{n}}{x}\right) \cdot \left(-\frac{\frac{0.25}{x \cdot n} - \frac{0.3333333333333333}{n}}{x}\right)}}\right) - \frac{0.5}{n}}{x}\right) - \frac{1}{n}}{x} \]

      4. sqrt-unprod 0.0%

         \[\leadsto -\frac{\left(-\frac{\left(-\color{blue}{\sqrt{-\frac{\frac{0.25}{x \cdot n} - \frac{0.3333333333333333}{n}}{x}} \cdot \sqrt{-\frac{\frac{0.25}{x \cdot n} - \frac{0.3333333333333333}{n}}{x}}}\right) - \frac{0.5}{n}}{x}\right) - \frac{1}{n}}{x} \]

      5. add-sqr-sqrt 76.4%

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

      6. neg-sub0 76.4%

         \[\leadsto -\frac{\left(-\frac{\left(-\color{blue}{\left(0 - \frac{\frac{0.25}{x \cdot n} - \frac{0.3333333333333333}{n}}{x}\right)}\right) - \frac{0.5}{n}}{x}\right) - \frac{1}{n}}{x} \]

      7. associate-/r* 76.4%

         \[\leadsto -\frac{\left(-\frac{\left(-\left(0 - \frac{\color{blue}{\frac{\frac{0.25}{x}}{n}} - \frac{0.3333333333333333}{n}}{x}\right)\right) - \frac{0.5}{n}}{x}\right) - \frac{1}{n}}{x} \]

      8. sub-div 76.4%

         \[\leadsto -\frac{\left(-\frac{\left(-\left(0 - \frac{\color{blue}{\frac{\frac{0.25}{x} - 0.3333333333333333}{n}}}{x}\right)\right) - \frac{0.5}{n}}{x}\right) - \frac{1}{n}}{x} \]

   10. Applied egg-rr 76.4%

       \[\leadsto -\frac{\left(-\frac{\left(-\color{blue}{\left(0 - \frac{\frac{\frac{0.25}{x} - 0.3333333333333333}{n}}{x}\right)}\right) - \frac{0.5}{n}}{x}\right) - \frac{1}{n}}{x} \]
   11. Step-by-step derivation

       1. neg-sub0 76.4%

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

       2. distribute-neg-frac 76.4%

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

       3. distribute-neg-frac 76.4%

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

       4. mul-1-neg 76.4%

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

       5. sub-neg 76.4%

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

       6. metadata-eval 76.4%

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

       7. associate-*r/ 76.4%

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

       8. metadata-eval 76.4%

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

       9. distribute-lft-in 76.4%

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

       10. neg-mul-1 76.4%

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

       11. associate-*r/ 76.4%

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

       12. metadata-eval 76.4%

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

       13. distribute-neg-frac 76.4%

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

       14. metadata-eval 76.4%

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

       15. metadata-eval 76.4%

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

   12. Simplified 76.4%

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

   13. Taylor expanded in x around 0 76.4%

       \[\leadsto -\frac{\left(-\frac{\left(-\frac{\color{blue}{\frac{-0.25}{n \cdot x}}}{x}\right) - \frac{0.5}{n}}{x}\right) - \frac{1}{n}}{x} \]
   14. Step-by-step derivation

       1. *-commutative 76.4%

          \[\leadsto -\frac{\left(-\frac{\left(-\frac{\frac{-0.25}{\color{blue}{x \cdot n}}}{x}\right) - \frac{0.5}{n}}{x}\right) - \frac{1}{n}}{x} \]

   15. Simplified 76.4%

       \[\leadsto -\frac{\left(-\frac{\left(-\frac{\color{blue}{\frac{-0.25}{x \cdot n}}}{x}\right) - \frac{0.5}{n}}{x}\right) - \frac{1}{n}}{x} \]
3. Recombined 4 regimes into one program.

4. Final simplification 79.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -20:\\ \;\;\;\;\frac{\frac{-0.25}{n}}{-{x}^{4}}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-13}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{+197}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{n} - \frac{\frac{\frac{-0.25}{n \cdot x}}{x} + \frac{0.5}{n}}{x}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 81.8% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -4 \cdot 10^{-68}:\\ \;\;\;\;\frac{\frac{t\_0}{n}}{x}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-13}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{+197}:\\ \;\;\;\;1 - t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{n} - \frac{\frac{\frac{-0.25}{n \cdot x}}{x} + \frac{0.5}{n}}{x}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -4e-68)
     (/ (/ t_0 n) x)
     (if (<= (/ 1.0 n) 5e-13)
       (/ (log (/ (+ 1.0 x) x)) n)
       (if (<= (/ 1.0 n) 2e+197)
         (- 1.0 t_0)
         (/ (- (/ 1.0 n) (/ (+ (/ (/ -0.25 (* n x)) x) (/ 0.5 n)) x)) x))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -4e-68) {
		tmp = (t_0 / n) / x;
	} else if ((1.0 / n) <= 5e-13) {
		tmp = log(((1.0 + x) / x)) / n;
	} else if ((1.0 / n) <= 2e+197) {
		tmp = 1.0 - t_0;
	} else {
		tmp = ((1.0 / n) - ((((-0.25 / (n * x)) / x) + (0.5 / n)) / x)) / x;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x ** (1.0d0 / n)
    if ((1.0d0 / n) <= (-4d-68)) then
        tmp = (t_0 / n) / x
    else if ((1.0d0 / n) <= 5d-13) then
        tmp = log(((1.0d0 + x) / x)) / n
    else if ((1.0d0 / n) <= 2d+197) then
        tmp = 1.0d0 - t_0
    else
        tmp = ((1.0d0 / n) - (((((-0.25d0) / (n * x)) / x) + (0.5d0 / n)) / x)) / x
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -4e-68) {
		tmp = (t_0 / n) / x;
	} else if ((1.0 / n) <= 5e-13) {
		tmp = Math.log(((1.0 + x) / x)) / n;
	} else if ((1.0 / n) <= 2e+197) {
		tmp = 1.0 - t_0;
	} else {
		tmp = ((1.0 / n) - ((((-0.25 / (n * x)) / x) + (0.5 / n)) / x)) / x;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	tmp = 0
	if (1.0 / n) <= -4e-68:
		tmp = (t_0 / n) / x
	elif (1.0 / n) <= 5e-13:
		tmp = math.log(((1.0 + x) / x)) / n
	elif (1.0 / n) <= 2e+197:
		tmp = 1.0 - t_0
	else:
		tmp = ((1.0 / n) - ((((-0.25 / (n * x)) / x) + (0.5 / n)) / x)) / x
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -4e-68)
		tmp = Float64(Float64(t_0 / n) / x);
	elseif (Float64(1.0 / n) <= 5e-13)
		tmp = Float64(log(Float64(Float64(1.0 + x) / x)) / n);
	elseif (Float64(1.0 / n) <= 2e+197)
		tmp = Float64(1.0 - t_0);
	else
		tmp = Float64(Float64(Float64(1.0 / n) - Float64(Float64(Float64(Float64(-0.25 / Float64(n * x)) / x) + Float64(0.5 / n)) / x)) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = x ^ (1.0 / n);
	tmp = 0.0;
	if ((1.0 / n) <= -4e-68)
		tmp = (t_0 / n) / x;
	elseif ((1.0 / n) <= 5e-13)
		tmp = log(((1.0 + x) / x)) / n;
	elseif ((1.0 / n) <= 2e+197)
		tmp = 1.0 - t_0;
	else
		tmp = ((1.0 / n) - ((((-0.25 / (n * x)) / x) + (0.5 / n)) / x)) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -4e-68], N[(N[(t$95$0 / n), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e-13], N[(N[Log[N[(N[(1.0 + x), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e+197], N[(1.0 - t$95$0), $MachinePrecision], N[(N[(N[(1.0 / n), $MachinePrecision] - N[(N[(N[(N[(-0.25 / N[(n * x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] + N[(0.5 / n), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -4.00000000000000027e-68

   1. Initial program 84.2%

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

   3. Taylor expanded in x around inf 93.6%

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

      1. associate-/r* 93.6%

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

      2. mul-1-neg 93.6%

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

      3. log-rec 93.6%

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

      4. mul-1-neg 93.6%

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

      5. distribute-neg-frac 93.6%

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

      6. mul-1-neg 93.6%

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

      7. remove-double-neg 93.6%

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

      8. *-rgt-identity 93.6%

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

      9. associate-/l* 93.6%

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

      10. exp-to-pow 93.6%

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

   5. Simplified 93.6%

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

   if -4.00000000000000027e-68 < (/.f64 #s(literal 1 binary64) n) < 4.9999999999999999e-13

   1. Initial program 34.8%

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

   3. Taylor expanded in n around inf 80.7%

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

      1. log1p-define 80.7%

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

   5. Simplified 80.7%

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

      1. log1p-undefine 80.7%

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

      2. diff-log 80.9%

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

   7. Applied egg-rr 80.9%

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

      1. +-commutative 80.9%

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

   9. Simplified 80.9%

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

   if 4.9999999999999999e-13 < (/.f64 #s(literal 1 binary64) n) < 1.9999999999999999e197

   1. Initial program 78.9%

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

   3. Taylor expanded in x around 0 75.5%

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

      1. *-rgt-identity 75.5%

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

      2. associate-/l* 75.5%

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

      3. exp-to-pow 75.5%

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

   5. Simplified 75.5%

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

   if 1.9999999999999999e197 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 27.3%

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

   3. Taylor expanded in n around inf 18.9%

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

      1. log1p-define 18.9%

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

   5. Simplified 18.9%

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

   6. Taylor expanded in x around -inf 0.1%

      \[\leadsto \color{blue}{-1 \cdot \frac{-1 \cdot \frac{-1 \cdot \frac{0.25 \cdot \frac{1}{n \cdot x} - 0.3333333333333333 \cdot \frac{1}{n}}{x} - 0.5 \cdot \frac{1}{n}}{x} - \frac{1}{n}}{x}} \]
   7. Step-by-step derivation

      1. mul-1-neg 0.1%

         \[\leadsto \color{blue}{-\frac{-1 \cdot \frac{-1 \cdot \frac{0.25 \cdot \frac{1}{n \cdot x} - 0.3333333333333333 \cdot \frac{1}{n}}{x} - 0.5 \cdot \frac{1}{n}}{x} - \frac{1}{n}}{x}} \]

   8. Simplified 0.1%

      \[\leadsto \color{blue}{-\frac{\left(-\frac{\left(-\frac{\frac{0.25}{x \cdot n} - \frac{0.3333333333333333}{n}}{x}\right) - \frac{0.5}{n}}{x}\right) - \frac{1}{n}}{x}} \]
   9. Step-by-step derivation

      1. add-sqr-sqrt 0.1%

         \[\leadsto -\frac{\left(-\frac{\left(-\color{blue}{\sqrt{\frac{\frac{0.25}{x \cdot n} - \frac{0.3333333333333333}{n}}{x}} \cdot \sqrt{\frac{\frac{0.25}{x \cdot n} - \frac{0.3333333333333333}{n}}{x}}}\right) - \frac{0.5}{n}}{x}\right) - \frac{1}{n}}{x} \]

      2. sqrt-unprod 0.1%

         \[\leadsto -\frac{\left(-\frac{\left(-\color{blue}{\sqrt{\frac{\frac{0.25}{x \cdot n} - \frac{0.3333333333333333}{n}}{x} \cdot \frac{\frac{0.25}{x \cdot n} - \frac{0.3333333333333333}{n}}{x}}}\right) - \frac{0.5}{n}}{x}\right) - \frac{1}{n}}{x} \]

      3. sqr-neg 0.1%

         \[\leadsto -\frac{\left(-\frac{\left(-\sqrt{\color{blue}{\left(-\frac{\frac{0.25}{x \cdot n} - \frac{0.3333333333333333}{n}}{x}\right) \cdot \left(-\frac{\frac{0.25}{x \cdot n} - \frac{0.3333333333333333}{n}}{x}\right)}}\right) - \frac{0.5}{n}}{x}\right) - \frac{1}{n}}{x} \]

      4. sqrt-unprod 0.0%

         \[\leadsto -\frac{\left(-\frac{\left(-\color{blue}{\sqrt{-\frac{\frac{0.25}{x \cdot n} - \frac{0.3333333333333333}{n}}{x}} \cdot \sqrt{-\frac{\frac{0.25}{x \cdot n} - \frac{0.3333333333333333}{n}}{x}}}\right) - \frac{0.5}{n}}{x}\right) - \frac{1}{n}}{x} \]

      5. add-sqr-sqrt 76.4%

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

      6. neg-sub0 76.4%

         \[\leadsto -\frac{\left(-\frac{\left(-\color{blue}{\left(0 - \frac{\frac{0.25}{x \cdot n} - \frac{0.3333333333333333}{n}}{x}\right)}\right) - \frac{0.5}{n}}{x}\right) - \frac{1}{n}}{x} \]

      7. associate-/r* 76.4%

         \[\leadsto -\frac{\left(-\frac{\left(-\left(0 - \frac{\color{blue}{\frac{\frac{0.25}{x}}{n}} - \frac{0.3333333333333333}{n}}{x}\right)\right) - \frac{0.5}{n}}{x}\right) - \frac{1}{n}}{x} \]

      8. sub-div 76.4%

         \[\leadsto -\frac{\left(-\frac{\left(-\left(0 - \frac{\color{blue}{\frac{\frac{0.25}{x} - 0.3333333333333333}{n}}}{x}\right)\right) - \frac{0.5}{n}}{x}\right) - \frac{1}{n}}{x} \]

   10. Applied egg-rr 76.4%

       \[\leadsto -\frac{\left(-\frac{\left(-\color{blue}{\left(0 - \frac{\frac{\frac{0.25}{x} - 0.3333333333333333}{n}}{x}\right)}\right) - \frac{0.5}{n}}{x}\right) - \frac{1}{n}}{x} \]
   11. Step-by-step derivation

       1. neg-sub0 76.4%

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

       2. distribute-neg-frac 76.4%

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

       3. distribute-neg-frac 76.4%

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

       4. mul-1-neg 76.4%

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

       5. sub-neg 76.4%

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

       6. metadata-eval 76.4%

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

       7. associate-*r/ 76.4%

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

       8. metadata-eval 76.4%

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

       9. distribute-lft-in 76.4%

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

       10. neg-mul-1 76.4%

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

       11. associate-*r/ 76.4%

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

       12. metadata-eval 76.4%

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

       13. distribute-neg-frac 76.4%

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

       14. metadata-eval 76.4%

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

       15. metadata-eval 76.4%

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

   12. Simplified 76.4%

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

   13. Taylor expanded in x around 0 76.4%

       \[\leadsto -\frac{\left(-\frac{\left(-\frac{\color{blue}{\frac{-0.25}{n \cdot x}}}{x}\right) - \frac{0.5}{n}}{x}\right) - \frac{1}{n}}{x} \]
   14. Step-by-step derivation

       1. *-commutative 76.4%

          \[\leadsto -\frac{\left(-\frac{\left(-\frac{\frac{-0.25}{\color{blue}{x \cdot n}}}{x}\right) - \frac{0.5}{n}}{x}\right) - \frac{1}{n}}{x} \]

   15. Simplified 76.4%

       \[\leadsto -\frac{\left(-\frac{\left(-\frac{\color{blue}{\frac{-0.25}{x \cdot n}}}{x}\right) - \frac{0.5}{n}}{x}\right) - \frac{1}{n}}{x} \]
3. Recombined 4 regimes into one program.

4. Final simplification 85.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -4 \cdot 10^{-68}:\\ \;\;\;\;\frac{\frac{{x}^{\left(\frac{1}{n}\right)}}{n}}{x}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-13}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{+197}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{n} - \frac{\frac{\frac{-0.25}{n \cdot x}}{x} + \frac{0.5}{n}}{x}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 56.6% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 3.9 \cdot 10^{-184}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;x \leq 2.6 \cdot 10^{-89}:\\ \;\;\;\;\frac{\log x}{-n}\\ \mathbf{elif}\;x \leq 2.4 \cdot 10^{-84}:\\ \;\;\;\;\frac{\frac{-1 + \frac{0.5 + \frac{-0.3333333333333333}{x}}{x}}{-x}}{n}\\ \mathbf{elif}\;x \leq 0.9:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{n} + \frac{\frac{\frac{0.3333333333333333 - \frac{0.25}{x}}{x}}{n} + \frac{-0.5}{n}}{x}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 3.9e-184)
   (- 1.0 (pow x (/ 1.0 n)))
   (if (<= x 2.6e-89)
     (/ (log x) (- n))
     (if (<= x 2.4e-84)
       (/ (/ (+ -1.0 (/ (+ 0.5 (/ -0.3333333333333333 x)) x)) (- x)) n)
       (if (<= x 0.9)
         (/ (- x (log x)) n)
         (/
          (+
           (/ 1.0 n)
           (/ (+ (/ (/ (- 0.3333333333333333 (/ 0.25 x)) x) n) (/ -0.5 n)) x))
          x))))))

C

double code(double x, double n) {
	double tmp;
	if (x <= 3.9e-184) {
		tmp = 1.0 - pow(x, (1.0 / n));
	} else if (x <= 2.6e-89) {
		tmp = log(x) / -n;
	} else if (x <= 2.4e-84) {
		tmp = ((-1.0 + ((0.5 + (-0.3333333333333333 / x)) / x)) / -x) / n;
	} else if (x <= 0.9) {
		tmp = (x - log(x)) / n;
	} else {
		tmp = ((1.0 / n) + (((((0.3333333333333333 - (0.25 / x)) / x) / n) + (-0.5 / n)) / x)) / x;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if (x <= 3.9d-184) then
        tmp = 1.0d0 - (x ** (1.0d0 / n))
    else if (x <= 2.6d-89) then
        tmp = log(x) / -n
    else if (x <= 2.4d-84) then
        tmp = (((-1.0d0) + ((0.5d0 + ((-0.3333333333333333d0) / x)) / x)) / -x) / n
    else if (x <= 0.9d0) then
        tmp = (x - log(x)) / n
    else
        tmp = ((1.0d0 / n) + (((((0.3333333333333333d0 - (0.25d0 / x)) / x) / n) + ((-0.5d0) / n)) / x)) / x
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (x <= 3.9e-184) {
		tmp = 1.0 - Math.pow(x, (1.0 / n));
	} else if (x <= 2.6e-89) {
		tmp = Math.log(x) / -n;
	} else if (x <= 2.4e-84) {
		tmp = ((-1.0 + ((0.5 + (-0.3333333333333333 / x)) / x)) / -x) / n;
	} else if (x <= 0.9) {
		tmp = (x - Math.log(x)) / n;
	} else {
		tmp = ((1.0 / n) + (((((0.3333333333333333 - (0.25 / x)) / x) / n) + (-0.5 / n)) / x)) / x;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 3.9e-184:
		tmp = 1.0 - math.pow(x, (1.0 / n))
	elif x <= 2.6e-89:
		tmp = math.log(x) / -n
	elif x <= 2.4e-84:
		tmp = ((-1.0 + ((0.5 + (-0.3333333333333333 / x)) / x)) / -x) / n
	elif x <= 0.9:
		tmp = (x - math.log(x)) / n
	else:
		tmp = ((1.0 / n) + (((((0.3333333333333333 - (0.25 / x)) / x) / n) + (-0.5 / n)) / x)) / x
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 3.9e-184)
		tmp = Float64(1.0 - (x ^ Float64(1.0 / n)));
	elseif (x <= 2.6e-89)
		tmp = Float64(log(x) / Float64(-n));
	elseif (x <= 2.4e-84)
		tmp = Float64(Float64(Float64(-1.0 + Float64(Float64(0.5 + Float64(-0.3333333333333333 / x)) / x)) / Float64(-x)) / n);
	elseif (x <= 0.9)
		tmp = Float64(Float64(x - log(x)) / n);
	else
		tmp = Float64(Float64(Float64(1.0 / n) + Float64(Float64(Float64(Float64(Float64(0.3333333333333333 - Float64(0.25 / x)) / x) / n) + Float64(-0.5 / n)) / x)) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (x <= 3.9e-184)
		tmp = 1.0 - (x ^ (1.0 / n));
	elseif (x <= 2.6e-89)
		tmp = log(x) / -n;
	elseif (x <= 2.4e-84)
		tmp = ((-1.0 + ((0.5 + (-0.3333333333333333 / x)) / x)) / -x) / n;
	elseif (x <= 0.9)
		tmp = (x - log(x)) / n;
	else
		tmp = ((1.0 / n) + (((((0.3333333333333333 - (0.25 / x)) / x) / n) + (-0.5 / n)) / x)) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 3.9e-184], N[(1.0 - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 2.6e-89], N[(N[Log[x], $MachinePrecision] / (-n)), $MachinePrecision], If[LessEqual[x, 2.4e-84], N[(N[(N[(-1.0 + N[(N[(0.5 + N[(-0.3333333333333333 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / (-x)), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[x, 0.9], N[(N[(x - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], N[(N[(N[(1.0 / n), $MachinePrecision] + N[(N[(N[(N[(N[(0.3333333333333333 - N[(0.25 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] / n), $MachinePrecision] + N[(-0.5 / n), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if x < 3.89999999999999994e-184

   1. Initial program 68.3%

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

   3. Taylor expanded in x around 0 68.3%

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

      1. *-rgt-identity 68.3%

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

      2. associate-/l* 68.3%

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

      3. exp-to-pow 68.3%

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

   5. Simplified 68.3%

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

   if 3.89999999999999994e-184 < x < 2.5999999999999999e-89

   1. Initial program 40.5%

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

   3. Taylor expanded in n around inf 63.7%

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

      1. log1p-define 63.7%

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

   5. Simplified 63.7%

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

   6. Taylor expanded in x around 0 63.7%

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

      1. neg-mul-1 63.7%

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

   8. Simplified 63.7%

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

   if 2.5999999999999999e-89 < x < 2.40000000000000017e-84

   1. Initial program 80.6%

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

   3. Taylor expanded in n around inf 4.4%

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

      1. log1p-define 4.4%

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

   5. Simplified 4.4%

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

      1. expm1-log1p-u 4.4%

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

   7. Applied egg-rr 4.4%

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

   8. Taylor expanded in x around -inf 100.0%

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

      1. mul-1-neg 100.0%

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

      2. distribute-neg-frac2 100.0%

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

      3. sub-neg 100.0%

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

      4. associate-*r/ 100.0%

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

      5. sub-neg 100.0%

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

      6. metadata-eval 100.0%

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

      7. distribute-lft-in 100.0%

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

      8. neg-mul-1 100.0%

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

      9. associate-*r/ 100.0%

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

      10. metadata-eval 100.0%

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

      11. distribute-neg-frac 100.0%

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

      12. metadata-eval 100.0%

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

      13. metadata-eval 100.0%

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

      14. metadata-eval 100.0%

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

   10. Simplified 100.0%

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

   if 2.40000000000000017e-84 < x < 0.900000000000000022

   1. Initial program 31.9%

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

   3. Taylor expanded in n around inf 59.2%

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

      1. log1p-define 59.2%

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

   5. Simplified 59.2%

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

   6. Taylor expanded in x around 0 55.1%

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

   if 0.900000000000000022 < x

   1. Initial program 69.1%

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

   3. Taylor expanded in n around inf 70.1%

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

      1. log1p-define 70.1%

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

   5. Simplified 70.1%

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

      1. add-exp-log 70.1%

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

   7. Applied egg-rr 70.1%

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

   8. Taylor expanded in x around -inf 67.0%

      \[\leadsto \color{blue}{-1 \cdot \frac{-1 \cdot \frac{-1 \cdot \frac{0.25 \cdot \frac{1}{n \cdot x} - 0.3333333333333333 \cdot \frac{1}{n}}{x} - 0.5 \cdot \frac{1}{n}}{x} - \frac{1}{n}}{x}} \]

   10. Simplified 67.0%

       \[\leadsto \color{blue}{\frac{\frac{\frac{\frac{0.3333333333333333 - \frac{0.25}{x}}{x}}{n} + \frac{-0.5}{n}}{x} + \frac{1}{n}}{x}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 65.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 3.9 \cdot 10^{-184}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;x \leq 2.6 \cdot 10^{-89}:\\ \;\;\;\;\frac{\log x}{-n}\\ \mathbf{elif}\;x \leq 2.4 \cdot 10^{-84}:\\ \;\;\;\;\frac{\frac{-1 + \frac{0.5 + \frac{-0.3333333333333333}{x}}{x}}{-x}}{n}\\ \mathbf{elif}\;x \leq 0.9:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{n} + \frac{\frac{\frac{0.3333333333333333 - \frac{0.25}{x}}{x}}{n} + \frac{-0.5}{n}}{x}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 57.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.12 \cdot 10^{-89}:\\ \;\;\;\;\frac{\log x}{-n}\\ \mathbf{elif}\;x \leq 2.4 \cdot 10^{-84}:\\ \;\;\;\;\frac{\frac{-1 + \frac{0.5 + \frac{-0.3333333333333333}{x}}{x}}{-x}}{n}\\ \mathbf{elif}\;x \leq 0.9:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{n} + \frac{\frac{\frac{0.3333333333333333 - \frac{0.25}{x}}{x}}{n} + \frac{-0.5}{n}}{x}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 1.12e-89)
   (/ (log x) (- n))
   (if (<= x 2.4e-84)
     (/ (/ (+ -1.0 (/ (+ 0.5 (/ -0.3333333333333333 x)) x)) (- x)) n)
     (if (<= x 0.9)
       (/ (- x (log x)) n)
       (/
        (+
         (/ 1.0 n)
         (/ (+ (/ (/ (- 0.3333333333333333 (/ 0.25 x)) x) n) (/ -0.5 n)) x))
        x)))))

C

double code(double x, double n) {
	double tmp;
	if (x <= 1.12e-89) {
		tmp = log(x) / -n;
	} else if (x <= 2.4e-84) {
		tmp = ((-1.0 + ((0.5 + (-0.3333333333333333 / x)) / x)) / -x) / n;
	} else if (x <= 0.9) {
		tmp = (x - log(x)) / n;
	} else {
		tmp = ((1.0 / n) + (((((0.3333333333333333 - (0.25 / x)) / x) / n) + (-0.5 / n)) / x)) / x;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if (x <= 1.12d-89) then
        tmp = log(x) / -n
    else if (x <= 2.4d-84) then
        tmp = (((-1.0d0) + ((0.5d0 + ((-0.3333333333333333d0) / x)) / x)) / -x) / n
    else if (x <= 0.9d0) then
        tmp = (x - log(x)) / n
    else
        tmp = ((1.0d0 / n) + (((((0.3333333333333333d0 - (0.25d0 / x)) / x) / n) + ((-0.5d0) / n)) / x)) / x
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (x <= 1.12e-89) {
		tmp = Math.log(x) / -n;
	} else if (x <= 2.4e-84) {
		tmp = ((-1.0 + ((0.5 + (-0.3333333333333333 / x)) / x)) / -x) / n;
	} else if (x <= 0.9) {
		tmp = (x - Math.log(x)) / n;
	} else {
		tmp = ((1.0 / n) + (((((0.3333333333333333 - (0.25 / x)) / x) / n) + (-0.5 / n)) / x)) / x;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 1.12e-89:
		tmp = math.log(x) / -n
	elif x <= 2.4e-84:
		tmp = ((-1.0 + ((0.5 + (-0.3333333333333333 / x)) / x)) / -x) / n
	elif x <= 0.9:
		tmp = (x - math.log(x)) / n
	else:
		tmp = ((1.0 / n) + (((((0.3333333333333333 - (0.25 / x)) / x) / n) + (-0.5 / n)) / x)) / x
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 1.12e-89)
		tmp = Float64(log(x) / Float64(-n));
	elseif (x <= 2.4e-84)
		tmp = Float64(Float64(Float64(-1.0 + Float64(Float64(0.5 + Float64(-0.3333333333333333 / x)) / x)) / Float64(-x)) / n);
	elseif (x <= 0.9)
		tmp = Float64(Float64(x - log(x)) / n);
	else
		tmp = Float64(Float64(Float64(1.0 / n) + Float64(Float64(Float64(Float64(Float64(0.3333333333333333 - Float64(0.25 / x)) / x) / n) + Float64(-0.5 / n)) / x)) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (x <= 1.12e-89)
		tmp = log(x) / -n;
	elseif (x <= 2.4e-84)
		tmp = ((-1.0 + ((0.5 + (-0.3333333333333333 / x)) / x)) / -x) / n;
	elseif (x <= 0.9)
		tmp = (x - log(x)) / n;
	else
		tmp = ((1.0 / n) + (((((0.3333333333333333 - (0.25 / x)) / x) / n) + (-0.5 / n)) / x)) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 1.12e-89], N[(N[Log[x], $MachinePrecision] / (-n)), $MachinePrecision], If[LessEqual[x, 2.4e-84], N[(N[(N[(-1.0 + N[(N[(0.5 + N[(-0.3333333333333333 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / (-x)), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[x, 0.9], N[(N[(x - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], N[(N[(N[(1.0 / n), $MachinePrecision] + N[(N[(N[(N[(N[(0.3333333333333333 - N[(0.25 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] / n), $MachinePrecision] + N[(-0.5 / n), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < 1.12e-89

   1. Initial program 55.4%

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

   3. Taylor expanded in n around inf 48.3%

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

      1. log1p-define 48.3%

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

   5. Simplified 48.3%

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

   6. Taylor expanded in x around 0 48.3%

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

      1. neg-mul-1 48.3%

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

   8. Simplified 48.3%

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

   if 1.12e-89 < x < 2.40000000000000017e-84

   1. Initial program 80.6%

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

   3. Taylor expanded in n around inf 4.4%

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

      1. log1p-define 4.4%

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

   5. Simplified 4.4%

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

      1. expm1-log1p-u 4.4%

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

   7. Applied egg-rr 4.4%

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

   8. Taylor expanded in x around -inf 100.0%

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

      1. mul-1-neg 100.0%

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

      2. distribute-neg-frac2 100.0%

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

      3. sub-neg 100.0%

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

      4. associate-*r/ 100.0%

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

      5. sub-neg 100.0%

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

      6. metadata-eval 100.0%

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

      7. distribute-lft-in 100.0%

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

      8. neg-mul-1 100.0%

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

      9. associate-*r/ 100.0%

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

      10. metadata-eval 100.0%

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

      11. distribute-neg-frac 100.0%

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

      12. metadata-eval 100.0%

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

      13. metadata-eval 100.0%

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

      14. metadata-eval 100.0%

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

   10. Simplified 100.0%

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

   if 2.40000000000000017e-84 < x < 0.900000000000000022

   1. Initial program 31.9%

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

   3. Taylor expanded in n around inf 59.2%

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

      1. log1p-define 59.2%

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

   5. Simplified 59.2%

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

   6. Taylor expanded in x around 0 55.1%

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

   if 0.900000000000000022 < x

   1. Initial program 69.1%

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

   3. Taylor expanded in n around inf 70.1%

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

      1. log1p-define 70.1%

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

   5. Simplified 70.1%

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

      1. add-exp-log 70.1%

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

   7. Applied egg-rr 70.1%

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

   8. Taylor expanded in x around -inf 67.0%

      \[\leadsto \color{blue}{-1 \cdot \frac{-1 \cdot \frac{-1 \cdot \frac{0.25 \cdot \frac{1}{n \cdot x} - 0.3333333333333333 \cdot \frac{1}{n}}{x} - 0.5 \cdot \frac{1}{n}}{x} - \frac{1}{n}}{x}} \]

   10. Simplified 67.0%

       \[\leadsto \color{blue}{\frac{\frac{\frac{\frac{0.3333333333333333 - \frac{0.25}{x}}{x}}{n} + \frac{-0.5}{n}}{x} + \frac{1}{n}}{x}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 58.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.12 \cdot 10^{-89}:\\ \;\;\;\;\frac{\log x}{-n}\\ \mathbf{elif}\;x \leq 2.4 \cdot 10^{-84}:\\ \;\;\;\;\frac{\frac{-1 + \frac{0.5 + \frac{-0.3333333333333333}{x}}{x}}{-x}}{n}\\ \mathbf{elif}\;x \leq 0.9:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{n} + \frac{\frac{\frac{0.3333333333333333 - \frac{0.25}{x}}{x}}{n} + \frac{-0.5}{n}}{x}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 57.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\log x}{-n}\\ \mathbf{if}\;x \leq 1.7 \cdot 10^{-89}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 2.4 \cdot 10^{-84}:\\ \;\;\;\;\frac{\frac{-1 + \frac{0.5 + \frac{-0.3333333333333333}{x}}{x}}{-x}}{n}\\ \mathbf{elif}\;x \leq 0.72:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{n} + \frac{\frac{\frac{0.3333333333333333 - \frac{0.25}{x}}{x}}{n} + \frac{-0.5}{n}}{x}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (/ (log x) (- n))))
   (if (<= x 1.7e-89)
     t_0
     (if (<= x 2.4e-84)
       (/ (/ (+ -1.0 (/ (+ 0.5 (/ -0.3333333333333333 x)) x)) (- x)) n)
       (if (<= x 0.72)
         t_0
         (/
          (+
           (/ 1.0 n)
           (/ (+ (/ (/ (- 0.3333333333333333 (/ 0.25 x)) x) n) (/ -0.5 n)) x))
          x))))))

C

double code(double x, double n) {
	double t_0 = log(x) / -n;
	double tmp;
	if (x <= 1.7e-89) {
		tmp = t_0;
	} else if (x <= 2.4e-84) {
		tmp = ((-1.0 + ((0.5 + (-0.3333333333333333 / x)) / x)) / -x) / n;
	} else if (x <= 0.72) {
		tmp = t_0;
	} else {
		tmp = ((1.0 / n) + (((((0.3333333333333333 - (0.25 / x)) / x) / n) + (-0.5 / n)) / x)) / x;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: tmp
    t_0 = log(x) / -n
    if (x <= 1.7d-89) then
        tmp = t_0
    else if (x <= 2.4d-84) then
        tmp = (((-1.0d0) + ((0.5d0 + ((-0.3333333333333333d0) / x)) / x)) / -x) / n
    else if (x <= 0.72d0) then
        tmp = t_0
    else
        tmp = ((1.0d0 / n) + (((((0.3333333333333333d0 - (0.25d0 / x)) / x) / n) + ((-0.5d0) / n)) / x)) / x
    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 <= 1.7e-89) {
		tmp = t_0;
	} else if (x <= 2.4e-84) {
		tmp = ((-1.0 + ((0.5 + (-0.3333333333333333 / x)) / x)) / -x) / n;
	} else if (x <= 0.72) {
		tmp = t_0;
	} else {
		tmp = ((1.0 / n) + (((((0.3333333333333333 - (0.25 / x)) / x) / n) + (-0.5 / n)) / x)) / x;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.log(x) / -n
	tmp = 0
	if x <= 1.7e-89:
		tmp = t_0
	elif x <= 2.4e-84:
		tmp = ((-1.0 + ((0.5 + (-0.3333333333333333 / x)) / x)) / -x) / n
	elif x <= 0.72:
		tmp = t_0
	else:
		tmp = ((1.0 / n) + (((((0.3333333333333333 - (0.25 / x)) / x) / n) + (-0.5 / n)) / x)) / x
	return tmp

Julia

function code(x, n)
	t_0 = Float64(log(x) / Float64(-n))
	tmp = 0.0
	if (x <= 1.7e-89)
		tmp = t_0;
	elseif (x <= 2.4e-84)
		tmp = Float64(Float64(Float64(-1.0 + Float64(Float64(0.5 + Float64(-0.3333333333333333 / x)) / x)) / Float64(-x)) / n);
	elseif (x <= 0.72)
		tmp = t_0;
	else
		tmp = Float64(Float64(Float64(1.0 / n) + Float64(Float64(Float64(Float64(Float64(0.3333333333333333 - Float64(0.25 / x)) / x) / n) + Float64(-0.5 / n)) / x)) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = log(x) / -n;
	tmp = 0.0;
	if (x <= 1.7e-89)
		tmp = t_0;
	elseif (x <= 2.4e-84)
		tmp = ((-1.0 + ((0.5 + (-0.3333333333333333 / x)) / x)) / -x) / n;
	elseif (x <= 0.72)
		tmp = t_0;
	else
		tmp = ((1.0 / n) + (((((0.3333333333333333 - (0.25 / x)) / x) / n) + (-0.5 / n)) / x)) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(N[Log[x], $MachinePrecision] / (-n)), $MachinePrecision]}, If[LessEqual[x, 1.7e-89], t$95$0, If[LessEqual[x, 2.4e-84], N[(N[(N[(-1.0 + N[(N[(0.5 + N[(-0.3333333333333333 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / (-x)), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[x, 0.72], t$95$0, N[(N[(N[(1.0 / n), $MachinePrecision] + N[(N[(N[(N[(N[(0.3333333333333333 - N[(0.25 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] / n), $MachinePrecision] + N[(-0.5 / n), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if x < 1.7e-89 or 2.40000000000000017e-84 < x < 0.71999999999999997

   1. Initial program 48.2%

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

   3. Taylor expanded in n around inf 51.6%

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

      1. log1p-define 51.6%

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

   5. Simplified 51.6%

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

   6. Taylor expanded in x around 0 50.0%

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

      1. neg-mul-1 50.0%

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

   8. Simplified 50.0%

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

   if 1.7e-89 < x < 2.40000000000000017e-84

   1. Initial program 80.6%

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

   3. Taylor expanded in n around inf 4.4%

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

      1. log1p-define 4.4%

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

   5. Simplified 4.4%

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

      1. expm1-log1p-u 4.4%

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

   7. Applied egg-rr 4.4%

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

   8. Taylor expanded in x around -inf 100.0%

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

      1. mul-1-neg 100.0%

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

      2. distribute-neg-frac2 100.0%

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

      3. sub-neg 100.0%

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

      4. associate-*r/ 100.0%

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

      5. sub-neg 100.0%

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

      6. metadata-eval 100.0%

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

      7. distribute-lft-in 100.0%

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

      8. neg-mul-1 100.0%

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

      9. associate-*r/ 100.0%

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

      10. metadata-eval 100.0%

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

      11. distribute-neg-frac 100.0%

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

      12. metadata-eval 100.0%

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

      13. metadata-eval 100.0%

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

      14. metadata-eval 100.0%

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

   10. Simplified 100.0%

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

   if 0.71999999999999997 < x

   1. Initial program 69.1%

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

   3. Taylor expanded in n around inf 70.1%

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

      1. log1p-define 70.1%

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

   5. Simplified 70.1%

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

      1. add-exp-log 70.1%

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

   7. Applied egg-rr 70.1%

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

   8. Taylor expanded in x around -inf 67.0%

      \[\leadsto \color{blue}{-1 \cdot \frac{-1 \cdot \frac{-1 \cdot \frac{0.25 \cdot \frac{1}{n \cdot x} - 0.3333333333333333 \cdot \frac{1}{n}}{x} - 0.5 \cdot \frac{1}{n}}{x} - \frac{1}{n}}{x}} \]

   10. Simplified 67.0%

       \[\leadsto \color{blue}{\frac{\frac{\frac{\frac{0.3333333333333333 - \frac{0.25}{x}}{x}}{n} + \frac{-0.5}{n}}{x} + \frac{1}{n}}{x}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 58.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.7 \cdot 10^{-89}:\\ \;\;\;\;\frac{\log x}{-n}\\ \mathbf{elif}\;x \leq 2.4 \cdot 10^{-84}:\\ \;\;\;\;\frac{\frac{-1 + \frac{0.5 + \frac{-0.3333333333333333}{x}}{x}}{-x}}{n}\\ \mathbf{elif}\;x \leq 0.72:\\ \;\;\;\;\frac{\log x}{-n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{n} + \frac{\frac{\frac{0.3333333333333333 - \frac{0.25}{x}}{x}}{n} + \frac{-0.5}{n}}{x}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 47.0% accurate, 11.1× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{1}{n} + \frac{-0.5 + \frac{-0.3333333333333333 + \frac{0.25}{x}}{x}}{n \cdot x}}{x} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (/
  (+ (/ 1.0 n) (/ (+ -0.5 (/ (+ -0.3333333333333333 (/ 0.25 x)) x)) (* n x)))
  x))

C

double code(double x, double n) {
	return ((1.0 / n) + ((-0.5 + ((-0.3333333333333333 + (0.25 / x)) / x)) / (n * x))) / x;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    code = ((1.0d0 / n) + (((-0.5d0) + (((-0.3333333333333333d0) + (0.25d0 / x)) / x)) / (n * x))) / x
end function

Java

public static double code(double x, double n) {
	return ((1.0 / n) + ((-0.5 + ((-0.3333333333333333 + (0.25 / x)) / x)) / (n * x))) / x;
}

Python

def code(x, n):
	return ((1.0 / n) + ((-0.5 + ((-0.3333333333333333 + (0.25 / x)) / x)) / (n * x))) / x

Julia

function code(x, n)
	return Float64(Float64(Float64(1.0 / n) + Float64(Float64(-0.5 + Float64(Float64(-0.3333333333333333 + Float64(0.25 / x)) / x)) / Float64(n * x))) / x)
end

MATLAB

function tmp = code(x, n)
	tmp = ((1.0 / n) + ((-0.5 + ((-0.3333333333333333 + (0.25 / x)) / x)) / (n * x))) / x;
end

Wolfram

code[x_, n_] := N[(N[(N[(1.0 / n), $MachinePrecision] + N[(N[(-0.5 + N[(N[(-0.3333333333333333 + N[(0.25 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / N[(n * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]

Derivation

1. Initial program 58.1%

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

3. Taylor expanded in n around inf 58.9%

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

   1. log1p-define 58.9%

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

5. Simplified 58.9%

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

6. Taylor expanded in x around -inf 30.3%

   \[\leadsto \color{blue}{-1 \cdot \frac{-1 \cdot \frac{-1 \cdot \frac{0.25 \cdot \frac{1}{n \cdot x} - 0.3333333333333333 \cdot \frac{1}{n}}{x} - 0.5 \cdot \frac{1}{n}}{x} - \frac{1}{n}}{x}} \]
7. Step-by-step derivation

   1. mul-1-neg 30.3%

      \[\leadsto \color{blue}{-\frac{-1 \cdot \frac{-1 \cdot \frac{0.25 \cdot \frac{1}{n \cdot x} - 0.3333333333333333 \cdot \frac{1}{n}}{x} - 0.5 \cdot \frac{1}{n}}{x} - \frac{1}{n}}{x}} \]

8. Simplified 30.3%

   \[\leadsto \color{blue}{-\frac{\left(-\frac{\left(-\frac{\frac{0.25}{x \cdot n} - \frac{0.3333333333333333}{n}}{x}\right) - \frac{0.5}{n}}{x}\right) - \frac{1}{n}}{x}} \]
9. Step-by-step derivation

   1. add-sqr-sqrt 22.9%

      \[\leadsto -\frac{\left(-\frac{\left(-\color{blue}{\sqrt{\frac{\frac{0.25}{x \cdot n} - \frac{0.3333333333333333}{n}}{x}} \cdot \sqrt{\frac{\frac{0.25}{x \cdot n} - \frac{0.3333333333333333}{n}}{x}}}\right) - \frac{0.5}{n}}{x}\right) - \frac{1}{n}}{x} \]

   2. sqrt-unprod 45.6%

      \[\leadsto -\frac{\left(-\frac{\left(-\color{blue}{\sqrt{\frac{\frac{0.25}{x \cdot n} - \frac{0.3333333333333333}{n}}{x} \cdot \frac{\frac{0.25}{x \cdot n} - \frac{0.3333333333333333}{n}}{x}}}\right) - \frac{0.5}{n}}{x}\right) - \frac{1}{n}}{x} \]

   3. sqr-neg 45.6%

      \[\leadsto -\frac{\left(-\frac{\left(-\sqrt{\color{blue}{\left(-\frac{\frac{0.25}{x \cdot n} - \frac{0.3333333333333333}{n}}{x}\right) \cdot \left(-\frac{\frac{0.25}{x \cdot n} - \frac{0.3333333333333333}{n}}{x}\right)}}\right) - \frac{0.5}{n}}{x}\right) - \frac{1}{n}}{x} \]

   4. sqrt-unprod 36.4%

      \[\leadsto -\frac{\left(-\frac{\left(-\color{blue}{\sqrt{-\frac{\frac{0.25}{x \cdot n} - \frac{0.3333333333333333}{n}}{x}} \cdot \sqrt{-\frac{\frac{0.25}{x \cdot n} - \frac{0.3333333333333333}{n}}{x}}}\right) - \frac{0.5}{n}}{x}\right) - \frac{1}{n}}{x} \]

   5. add-sqr-sqrt 48.9%

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

   6. neg-sub0 48.9%

      \[\leadsto -\frac{\left(-\frac{\left(-\color{blue}{\left(0 - \frac{\frac{0.25}{x \cdot n} - \frac{0.3333333333333333}{n}}{x}\right)}\right) - \frac{0.5}{n}}{x}\right) - \frac{1}{n}}{x} \]

   7. associate-/r* 48.9%

      \[\leadsto -\frac{\left(-\frac{\left(-\left(0 - \frac{\color{blue}{\frac{\frac{0.25}{x}}{n}} - \frac{0.3333333333333333}{n}}{x}\right)\right) - \frac{0.5}{n}}{x}\right) - \frac{1}{n}}{x} \]

   8. sub-div 48.9%

      \[\leadsto -\frac{\left(-\frac{\left(-\left(0 - \frac{\color{blue}{\frac{\frac{0.25}{x} - 0.3333333333333333}{n}}}{x}\right)\right) - \frac{0.5}{n}}{x}\right) - \frac{1}{n}}{x} \]

10. Applied egg-rr 48.9%

    \[\leadsto -\frac{\left(-\frac{\left(-\color{blue}{\left(0 - \frac{\frac{\frac{0.25}{x} - 0.3333333333333333}{n}}{x}\right)}\right) - \frac{0.5}{n}}{x}\right) - \frac{1}{n}}{x} \]
11. Step-by-step derivation

    1. neg-sub0 48.9%

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

    2. distribute-neg-frac 48.9%

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

    3. distribute-neg-frac 48.9%

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

    4. mul-1-neg 48.9%

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

    5. sub-neg 48.9%

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

    6. metadata-eval 48.9%

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

    7. associate-*r/ 48.9%

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

    8. metadata-eval 48.9%

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

    9. distribute-lft-in 48.9%

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

    10. neg-mul-1 48.9%

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

    11. associate-*r/ 48.9%

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

    12. metadata-eval 48.9%

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

    13. distribute-neg-frac 48.9%

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

    14. metadata-eval 48.9%

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

    15. metadata-eval 48.9%

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

12. Simplified 48.9%

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

13. Taylor expanded in n around -inf 48.9%

    \[\leadsto -\frac{\left(-\color{blue}{\frac{-1 \cdot \frac{0.3333333333333333 - 0.25 \cdot \frac{1}{x}}{x} - 0.5}{n \cdot x}}\right) - \frac{1}{n}}{x} \]
14. Step-by-step derivation

15. Simplified 48.9%

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

16. Final simplification 48.9%

    \[\leadsto \frac{\frac{1}{n} + \frac{-0.5 + \frac{-0.3333333333333333 + \frac{0.25}{x}}{x}}{n \cdot x}}{x} \]
17. Add Preprocessing

Alternative 12: 47.0% accurate, 11.1× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{1}{n} - \frac{\frac{\frac{-0.25}{n \cdot x}}{x} + \frac{0.5}{n}}{x}}{x} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (/ (- (/ 1.0 n) (/ (+ (/ (/ -0.25 (* n x)) x) (/ 0.5 n)) x)) x))

C

double code(double x, double n) {
	return ((1.0 / n) - ((((-0.25 / (n * x)) / x) + (0.5 / n)) / x)) / x;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    code = ((1.0d0 / n) - (((((-0.25d0) / (n * x)) / x) + (0.5d0 / n)) / x)) / x
end function

Java

public static double code(double x, double n) {
	return ((1.0 / n) - ((((-0.25 / (n * x)) / x) + (0.5 / n)) / x)) / x;
}

Python

def code(x, n):
	return ((1.0 / n) - ((((-0.25 / (n * x)) / x) + (0.5 / n)) / x)) / x

Julia

function code(x, n)
	return Float64(Float64(Float64(1.0 / n) - Float64(Float64(Float64(Float64(-0.25 / Float64(n * x)) / x) + Float64(0.5 / n)) / x)) / x)
end

MATLAB

function tmp = code(x, n)
	tmp = ((1.0 / n) - ((((-0.25 / (n * x)) / x) + (0.5 / n)) / x)) / x;
end

Wolfram

code[x_, n_] := N[(N[(N[(1.0 / n), $MachinePrecision] - N[(N[(N[(N[(-0.25 / N[(n * x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] + N[(0.5 / n), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]

Derivation

1. Initial program 58.1%

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

3. Taylor expanded in n around inf 58.9%

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

   1. log1p-define 58.9%

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

5. Simplified 58.9%

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

6. Taylor expanded in x around -inf 30.3%

   \[\leadsto \color{blue}{-1 \cdot \frac{-1 \cdot \frac{-1 \cdot \frac{0.25 \cdot \frac{1}{n \cdot x} - 0.3333333333333333 \cdot \frac{1}{n}}{x} - 0.5 \cdot \frac{1}{n}}{x} - \frac{1}{n}}{x}} \]
7. Step-by-step derivation

   1. mul-1-neg 30.3%

      \[\leadsto \color{blue}{-\frac{-1 \cdot \frac{-1 \cdot \frac{0.25 \cdot \frac{1}{n \cdot x} - 0.3333333333333333 \cdot \frac{1}{n}}{x} - 0.5 \cdot \frac{1}{n}}{x} - \frac{1}{n}}{x}} \]

8. Simplified 30.3%

   \[\leadsto \color{blue}{-\frac{\left(-\frac{\left(-\frac{\frac{0.25}{x \cdot n} - \frac{0.3333333333333333}{n}}{x}\right) - \frac{0.5}{n}}{x}\right) - \frac{1}{n}}{x}} \]
9. Step-by-step derivation

   1. add-sqr-sqrt 22.9%

      \[\leadsto -\frac{\left(-\frac{\left(-\color{blue}{\sqrt{\frac{\frac{0.25}{x \cdot n} - \frac{0.3333333333333333}{n}}{x}} \cdot \sqrt{\frac{\frac{0.25}{x \cdot n} - \frac{0.3333333333333333}{n}}{x}}}\right) - \frac{0.5}{n}}{x}\right) - \frac{1}{n}}{x} \]

   2. sqrt-unprod 45.6%

      \[\leadsto -\frac{\left(-\frac{\left(-\color{blue}{\sqrt{\frac{\frac{0.25}{x \cdot n} - \frac{0.3333333333333333}{n}}{x} \cdot \frac{\frac{0.25}{x \cdot n} - \frac{0.3333333333333333}{n}}{x}}}\right) - \frac{0.5}{n}}{x}\right) - \frac{1}{n}}{x} \]

   3. sqr-neg 45.6%

      \[\leadsto -\frac{\left(-\frac{\left(-\sqrt{\color{blue}{\left(-\frac{\frac{0.25}{x \cdot n} - \frac{0.3333333333333333}{n}}{x}\right) \cdot \left(-\frac{\frac{0.25}{x \cdot n} - \frac{0.3333333333333333}{n}}{x}\right)}}\right) - \frac{0.5}{n}}{x}\right) - \frac{1}{n}}{x} \]

   4. sqrt-unprod 36.4%

      \[\leadsto -\frac{\left(-\frac{\left(-\color{blue}{\sqrt{-\frac{\frac{0.25}{x \cdot n} - \frac{0.3333333333333333}{n}}{x}} \cdot \sqrt{-\frac{\frac{0.25}{x \cdot n} - \frac{0.3333333333333333}{n}}{x}}}\right) - \frac{0.5}{n}}{x}\right) - \frac{1}{n}}{x} \]

   5. add-sqr-sqrt 48.9%

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

   6. neg-sub0 48.9%

      \[\leadsto -\frac{\left(-\frac{\left(-\color{blue}{\left(0 - \frac{\frac{0.25}{x \cdot n} - \frac{0.3333333333333333}{n}}{x}\right)}\right) - \frac{0.5}{n}}{x}\right) - \frac{1}{n}}{x} \]

   7. associate-/r* 48.9%

      \[\leadsto -\frac{\left(-\frac{\left(-\left(0 - \frac{\color{blue}{\frac{\frac{0.25}{x}}{n}} - \frac{0.3333333333333333}{n}}{x}\right)\right) - \frac{0.5}{n}}{x}\right) - \frac{1}{n}}{x} \]

   8. sub-div 48.9%

      \[\leadsto -\frac{\left(-\frac{\left(-\left(0 - \frac{\color{blue}{\frac{\frac{0.25}{x} - 0.3333333333333333}{n}}}{x}\right)\right) - \frac{0.5}{n}}{x}\right) - \frac{1}{n}}{x} \]

10. Applied egg-rr 48.9%

    \[\leadsto -\frac{\left(-\frac{\left(-\color{blue}{\left(0 - \frac{\frac{\frac{0.25}{x} - 0.3333333333333333}{n}}{x}\right)}\right) - \frac{0.5}{n}}{x}\right) - \frac{1}{n}}{x} \]
11. Step-by-step derivation

    1. neg-sub0 48.9%

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

    2. distribute-neg-frac 48.9%

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

    3. distribute-neg-frac 48.9%

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

    4. mul-1-neg 48.9%

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

    5. sub-neg 48.9%

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

    6. metadata-eval 48.9%

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

    7. associate-*r/ 48.9%

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

    8. metadata-eval 48.9%

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

    9. distribute-lft-in 48.9%

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

    10. neg-mul-1 48.9%

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

    11. associate-*r/ 48.9%

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

    12. metadata-eval 48.9%

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

    13. distribute-neg-frac 48.9%

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

    14. metadata-eval 48.9%

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

    15. metadata-eval 48.9%

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

12. Simplified 48.9%

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

13. Taylor expanded in x around 0 48.9%

    \[\leadsto -\frac{\left(-\frac{\left(-\frac{\color{blue}{\frac{-0.25}{n \cdot x}}}{x}\right) - \frac{0.5}{n}}{x}\right) - \frac{1}{n}}{x} \]
14. Step-by-step derivation

    1. *-commutative 48.9%

       \[\leadsto -\frac{\left(-\frac{\left(-\frac{\frac{-0.25}{\color{blue}{x \cdot n}}}{x}\right) - \frac{0.5}{n}}{x}\right) - \frac{1}{n}}{x} \]

15. Simplified 48.9%

    \[\leadsto -\frac{\left(-\frac{\left(-\frac{\color{blue}{\frac{-0.25}{x \cdot n}}}{x}\right) - \frac{0.5}{n}}{x}\right) - \frac{1}{n}}{x} \]

16. Final simplification 48.9%

    \[\leadsto \frac{\frac{1}{n} - \frac{\frac{\frac{-0.25}{n \cdot x}}{x} + \frac{0.5}{n}}{x}}{x} \]
17. Add Preprocessing

Alternative 13: 46.2% accurate, 12.4× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{1}{n} + \frac{\frac{0.3333333333333333}{n \cdot x} - \frac{0.5}{n}}{x}}{x} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (/ (+ (/ 1.0 n) (/ (- (/ 0.3333333333333333 (* n x)) (/ 0.5 n)) x)) x))

C

double code(double x, double n) {
	return ((1.0 / n) + (((0.3333333333333333 / (n * x)) - (0.5 / n)) / x)) / x;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    code = ((1.0d0 / n) + (((0.3333333333333333d0 / (n * x)) - (0.5d0 / n)) / x)) / x
end function

Java

public static double code(double x, double n) {
	return ((1.0 / n) + (((0.3333333333333333 / (n * x)) - (0.5 / n)) / x)) / x;
}

Python

def code(x, n):
	return ((1.0 / n) + (((0.3333333333333333 / (n * x)) - (0.5 / n)) / x)) / x

Julia

function code(x, n)
	return Float64(Float64(Float64(1.0 / n) + Float64(Float64(Float64(0.3333333333333333 / Float64(n * x)) - Float64(0.5 / n)) / x)) / x)
end

MATLAB

function tmp = code(x, n)
	tmp = ((1.0 / n) + (((0.3333333333333333 / (n * x)) - (0.5 / n)) / x)) / x;
end

Wolfram

code[x_, n_] := N[(N[(N[(1.0 / n), $MachinePrecision] + N[(N[(N[(0.3333333333333333 / N[(n * x), $MachinePrecision]), $MachinePrecision] - N[(0.5 / n), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]

Derivation

1. Initial program 58.1%

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

3. Taylor expanded in n around inf 58.9%

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

   1. log1p-define 58.9%

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

5. Simplified 58.9%

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

6. Taylor expanded in x around -inf 48.8%

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

   1. mul-1-neg 48.8%

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

   2. mul-1-neg 48.8%

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

   3. associate-*r/ 48.8%

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

   4. metadata-eval 48.8%

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

   5. *-commutative 48.8%

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

   6. associate-*r/ 48.8%

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

   7. metadata-eval 48.8%

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

8. Simplified 48.8%

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

9. Final simplification 48.8%

   \[\leadsto \frac{\frac{1}{n} + \frac{\frac{0.3333333333333333}{n \cdot x} - \frac{0.5}{n}}{x}}{x} \]
10. Add Preprocessing

Alternative 14: 46.2% accurate, 15.1× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{-1 + \frac{0.5 + \frac{-0.3333333333333333}{x}}{x}}{-x}}{n} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (/ (/ (+ -1.0 (/ (+ 0.5 (/ -0.3333333333333333 x)) x)) (- x)) n))

C

double code(double x, double n) {
	return ((-1.0 + ((0.5 + (-0.3333333333333333 / x)) / x)) / -x) / n;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    code = (((-1.0d0) + ((0.5d0 + ((-0.3333333333333333d0) / x)) / x)) / -x) / n
end function

Java

public static double code(double x, double n) {
	return ((-1.0 + ((0.5 + (-0.3333333333333333 / x)) / x)) / -x) / n;
}

Python

def code(x, n):
	return ((-1.0 + ((0.5 + (-0.3333333333333333 / x)) / x)) / -x) / n

Julia

function code(x, n)
	return Float64(Float64(Float64(-1.0 + Float64(Float64(0.5 + Float64(-0.3333333333333333 / x)) / x)) / Float64(-x)) / n)
end

MATLAB

function tmp = code(x, n)
	tmp = ((-1.0 + ((0.5 + (-0.3333333333333333 / x)) / x)) / -x) / n;
end

Wolfram

code[x_, n_] := N[(N[(N[(-1.0 + N[(N[(0.5 + N[(-0.3333333333333333 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / (-x)), $MachinePrecision] / n), $MachinePrecision]

Derivation

1. Initial program 58.1%

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

3. Taylor expanded in n around inf 58.9%

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

   1. log1p-define 58.9%

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

5. Simplified 58.9%

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

   1. expm1-log1p-u 58.6%

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

7. Applied egg-rr 58.6%

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

8. Taylor expanded in x around -inf 48.8%

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

   1. mul-1-neg 48.8%

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

   2. distribute-neg-frac2 48.8%

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

   3. sub-neg 48.8%

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

   4. associate-*r/ 48.8%

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

   5. sub-neg 48.8%

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

   6. metadata-eval 48.8%

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

   7. distribute-lft-in 48.8%

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

   8. neg-mul-1 48.8%

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

   9. associate-*r/ 48.8%

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

   10. metadata-eval 48.8%

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

   11. distribute-neg-frac 48.8%

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

   12. metadata-eval 48.8%

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

   13. metadata-eval 48.8%

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

   14. metadata-eval 48.8%

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

10. Simplified 48.8%

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

11. Final simplification 48.8%

    \[\leadsto \frac{\frac{-1 + \frac{0.5 + \frac{-0.3333333333333333}{x}}{x}}{-x}}{n} \]
12. Add Preprocessing

Alternative 15: 40.2% 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 58.1%

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

3. Taylor expanded in n around inf 58.9%

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

   1. log1p-define 58.9%

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

5. Simplified 58.9%

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

6. Taylor expanded in x around inf 40.1%

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

   1. *-commutative 40.1%

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

8. Simplified 40.1%

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

9. Final simplification 40.1%

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

Alternative 16: 40.7% 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 58.1%

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

3. Taylor expanded in n around inf 58.9%

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

   1. log1p-define 58.9%

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

5. Simplified 58.9%

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

6. Taylor expanded in x around inf 40.1%

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

   1. *-commutative 40.1%

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

8. Simplified 40.1%

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

9. Taylor expanded in x around 0 40.1%

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

    1. associate-/r* 40.7%

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

11. Simplified 40.7%

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

12. Final simplification 40.7%

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

Reproduce

herbie shell --seed 2024079 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  :precision binary64
  (- (pow (+ x 1.0) (/ 1.0 n)) (pow x (/ 1.0 n))))