2nthrt (problem 3.4.6)

Percentage Accurate: 53.5% → 85.8%

Time: 39.8s

Alternatives: 15

Speedup: 1.9×

• could not determine a ground truth

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 53.5% 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: 85.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -4 \cdot 10^{-72}:\\ \;\;\;\;\frac{t\_0}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-18}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right) + \left(0.5 \cdot \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}}{n} - \log x\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 1000000:\\ \;\;\;\;\frac{\mathsf{fma}\left(t\_0, \frac{\mathsf{fma}\left(0.5, {n}^{-2}, \frac{-0.5}{n}\right)}{x}, \frac{t\_0}{n}\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -4e-72)
     (/ t_0 (* n x))
     (if (<= (/ 1.0 n) 2e-18)
       (/
        (+
         (log1p x)
         (- (* 0.5 (/ (- (pow (log1p x) 2.0) (pow (log x) 2.0)) n)) (log x)))
        n)
       (if (<= (/ 1.0 n) 1000000.0)
         (/ (fma t_0 (/ (fma 0.5 (pow n -2.0) (/ -0.5 n)) x) (/ t_0 n)) x)
         (- (exp (/ (log1p x) n)) t_0))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -4e-72) {
		tmp = t_0 / (n * x);
	} else if ((1.0 / n) <= 2e-18) {
		tmp = (log1p(x) + ((0.5 * ((pow(log1p(x), 2.0) - pow(log(x), 2.0)) / n)) - log(x))) / n;
	} else if ((1.0 / n) <= 1000000.0) {
		tmp = fma(t_0, (fma(0.5, pow(n, -2.0), (-0.5 / n)) / x), (t_0 / n)) / x;
	} else {
		tmp = exp((log1p(x) / n)) - t_0;
	}
	return tmp;
}

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -4e-72)
		tmp = Float64(t_0 / Float64(n * x));
	elseif (Float64(1.0 / n) <= 2e-18)
		tmp = Float64(Float64(log1p(x) + Float64(Float64(0.5 * Float64(Float64((log1p(x) ^ 2.0) - (log(x) ^ 2.0)) / n)) - log(x))) / n);
	elseif (Float64(1.0 / n) <= 1000000.0)
		tmp = Float64(fma(t_0, Float64(fma(0.5, (n ^ -2.0), Float64(-0.5 / n)) / x), Float64(t_0 / n)) / x);
	else
		tmp = Float64(exp(Float64(log1p(x) / n)) - t_0);
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -4e-72], N[(t$95$0 / N[(n * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-18], N[(N[(N[Log[1 + x], $MachinePrecision] + N[(N[(0.5 * N[(N[(N[Power[N[Log[1 + x], $MachinePrecision], 2.0], $MachinePrecision] - N[Power[N[Log[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision] - N[Log[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 1000000.0], N[(N[(t$95$0 * N[(N[(0.5 * N[Power[n, -2.0], $MachinePrecision] + N[(-0.5 / n), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] + N[(t$95$0 / n), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], N[(N[Exp[N[(N[Log[1 + x], $MachinePrecision] / n), $MachinePrecision]], $MachinePrecision] - t$95$0), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -3.9999999999999999e-72

   1. Initial program 83.8%

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

   3. Taylor expanded in x around inf 93.2%

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

      1. log-rec 93.2%

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

      2. mul-1-neg 93.2%

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

      3. neg-mul-1 93.2%

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

      4. mul-1-neg 93.2%

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

      5. distribute-frac-neg 93.2%

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

      6. remove-double-neg 93.2%

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

      7. *-rgt-identity 93.2%

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

      8. associate-/l* 93.2%

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

      9. exp-to-pow 93.2%

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

      10. *-commutative 93.2%

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

   5. Simplified 93.2%

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

   if -3.9999999999999999e-72 < (/.f64 #s(literal 1 binary64) n) < 2.0000000000000001e-18

   1. Initial program 31.5%

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

   3. Taylor expanded in n around inf 83.0%

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

      1. associate--l+ 83.0%

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

      2. log1p-define 83.0%

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

      3. +-commutative 83.0%

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

      4. associate--r+ 83.0%

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

      5. distribute-lft-out-- 83.0%

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

      6. div-sub 83.0%

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

      7. log1p-define 83.0%

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

   5. Simplified 83.0%

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

   if 2.0000000000000001e-18 < (/.f64 #s(literal 1 binary64) n) < 1e6

   1. Initial program 2.1%

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

   3. Taylor expanded in x around inf 100.0%

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

   5. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left({x}^{\left(\frac{1}{n}\right)}, \frac{\mathsf{fma}\left(0.5, {n}^{-2}, \frac{-0.5}{n}\right)}{x}, \frac{{x}^{\left(\frac{1}{n}\right)}}{n}\right)}{x}} \]

   if 1e6 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 63.2%

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

   3. Taylor expanded in n around 0 63.2%

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

      1. log1p-define 100.0%

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

      2. *-rgt-identity 100.0%

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

      3. associate-*l/ 100.0%

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

      4. associate-/l* 100.0%

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

      5. exp-to-pow 100.0%

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

   5. Simplified 100.0%

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

4. Final simplification 89.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -4 \cdot 10^{-72}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-18}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right) + \left(0.5 \cdot \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}}{n} - \log x\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 1000000:\\ \;\;\;\;\frac{\mathsf{fma}\left({x}^{\left(\frac{1}{n}\right)}, \frac{\mathsf{fma}\left(0.5, {n}^{-2}, \frac{-0.5}{n}\right)}{x}, \frac{{x}^{\left(\frac{1}{n}\right)}}{n}\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 85.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -4 \cdot 10^{-72}:\\ \;\;\;\;\frac{t\_0}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-18}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right) - \log x}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 1000000:\\ \;\;\;\;\frac{\mathsf{fma}\left(t\_0, \frac{\mathsf{fma}\left(0.5, {n}^{-2}, \frac{-0.5}{n}\right)}{x}, \frac{t\_0}{n}\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -4e-72)
     (/ t_0 (* n x))
     (if (<= (/ 1.0 n) 2e-18)
       (/ (- (log1p x) (log x)) n)
       (if (<= (/ 1.0 n) 1000000.0)
         (/ (fma t_0 (/ (fma 0.5 (pow n -2.0) (/ -0.5 n)) x) (/ t_0 n)) x)
         (- (exp (/ (log1p x) n)) t_0))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -4e-72) {
		tmp = t_0 / (n * x);
	} else if ((1.0 / n) <= 2e-18) {
		tmp = (log1p(x) - log(x)) / n;
	} else if ((1.0 / n) <= 1000000.0) {
		tmp = fma(t_0, (fma(0.5, pow(n, -2.0), (-0.5 / n)) / x), (t_0 / n)) / x;
	} else {
		tmp = exp((log1p(x) / n)) - t_0;
	}
	return tmp;
}

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -4e-72)
		tmp = Float64(t_0 / Float64(n * x));
	elseif (Float64(1.0 / n) <= 2e-18)
		tmp = Float64(Float64(log1p(x) - log(x)) / n);
	elseif (Float64(1.0 / n) <= 1000000.0)
		tmp = Float64(fma(t_0, Float64(fma(0.5, (n ^ -2.0), Float64(-0.5 / n)) / x), Float64(t_0 / n)) / x);
	else
		tmp = Float64(exp(Float64(log1p(x) / n)) - t_0);
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -4e-72], N[(t$95$0 / N[(n * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-18], N[(N[(N[Log[1 + x], $MachinePrecision] - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 1000000.0], N[(N[(t$95$0 * N[(N[(0.5 * N[Power[n, -2.0], $MachinePrecision] + N[(-0.5 / n), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] + N[(t$95$0 / n), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], N[(N[Exp[N[(N[Log[1 + x], $MachinePrecision] / n), $MachinePrecision]], $MachinePrecision] - t$95$0), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -3.9999999999999999e-72

   1. Initial program 83.8%

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

   3. Taylor expanded in x around inf 93.2%

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

      1. log-rec 93.2%

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

      2. mul-1-neg 93.2%

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

      3. neg-mul-1 93.2%

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

      4. mul-1-neg 93.2%

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

      5. distribute-frac-neg 93.2%

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

      6. remove-double-neg 93.2%

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

      7. *-rgt-identity 93.2%

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

      8. associate-/l* 93.2%

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

      9. exp-to-pow 93.2%

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

      10. *-commutative 93.2%

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

   5. Simplified 93.2%

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

   if -3.9999999999999999e-72 < (/.f64 #s(literal 1 binary64) n) < 2.0000000000000001e-18

   1. Initial program 31.5%

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

   3. Taylor expanded in n around inf 83.0%

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

      1. log1p-define 83.0%

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

   5. Simplified 83.0%

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

   if 2.0000000000000001e-18 < (/.f64 #s(literal 1 binary64) n) < 1e6

   1. Initial program 2.1%

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

   3. Taylor expanded in x around inf 100.0%

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

   5. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left({x}^{\left(\frac{1}{n}\right)}, \frac{\mathsf{fma}\left(0.5, {n}^{-2}, \frac{-0.5}{n}\right)}{x}, \frac{{x}^{\left(\frac{1}{n}\right)}}{n}\right)}{x}} \]

   if 1e6 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 63.2%

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

   3. Taylor expanded in n around 0 63.2%

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

      1. log1p-define 100.0%

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

      2. *-rgt-identity 100.0%

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

      3. associate-*l/ 100.0%

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

      4. associate-/l* 100.0%

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

      5. exp-to-pow 100.0%

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

   5. Simplified 100.0%

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

4. Final simplification 89.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -4 \cdot 10^{-72}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-18}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right) - \log x}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 1000000:\\ \;\;\;\;\frac{\mathsf{fma}\left({x}^{\left(\frac{1}{n}\right)}, \frac{\mathsf{fma}\left(0.5, {n}^{-2}, \frac{-0.5}{n}\right)}{x}, \frac{{x}^{\left(\frac{1}{n}\right)}}{n}\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 85.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ t_1 := \frac{t\_0}{n \cdot x}\\ \mathbf{if}\;\frac{1}{n} \leq -4 \cdot 10^{-72}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-18}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right) - \log x}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 1000000:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))) (t_1 (/ t_0 (* n x))))
   (if (<= (/ 1.0 n) -4e-72)
     t_1
     (if (<= (/ 1.0 n) 2e-18)
       (/ (- (log1p x) (log x)) n)
       (if (<= (/ 1.0 n) 1000000.0) t_1 (- (exp (/ (log1p x) n)) t_0))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double t_1 = t_0 / (n * x);
	double tmp;
	if ((1.0 / n) <= -4e-72) {
		tmp = t_1;
	} else if ((1.0 / n) <= 2e-18) {
		tmp = (log1p(x) - log(x)) / n;
	} else if ((1.0 / n) <= 1000000.0) {
		tmp = t_1;
	} else {
		tmp = exp((log1p(x) / n)) - t_0;
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, (1.0 / n));
	double t_1 = t_0 / (n * x);
	double tmp;
	if ((1.0 / n) <= -4e-72) {
		tmp = t_1;
	} else if ((1.0 / n) <= 2e-18) {
		tmp = (Math.log1p(x) - Math.log(x)) / n;
	} else if ((1.0 / n) <= 1000000.0) {
		tmp = t_1;
	} else {
		tmp = Math.exp((Math.log1p(x) / n)) - t_0;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	t_1 = t_0 / (n * x)
	tmp = 0
	if (1.0 / n) <= -4e-72:
		tmp = t_1
	elif (1.0 / n) <= 2e-18:
		tmp = (math.log1p(x) - math.log(x)) / n
	elif (1.0 / n) <= 1000000.0:
		tmp = t_1
	else:
		tmp = math.exp((math.log1p(x) / n)) - t_0
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	t_1 = Float64(t_0 / Float64(n * x))
	tmp = 0.0
	if (Float64(1.0 / n) <= -4e-72)
		tmp = t_1;
	elseif (Float64(1.0 / n) <= 2e-18)
		tmp = Float64(Float64(log1p(x) - log(x)) / n);
	elseif (Float64(1.0 / n) <= 1000000.0)
		tmp = t_1;
	else
		tmp = Float64(exp(Float64(log1p(x) / n)) - t_0);
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 / N[(n * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -4e-72], t$95$1, If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-18], N[(N[(N[Log[1 + x], $MachinePrecision] - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 1000000.0], t$95$1, N[(N[Exp[N[(N[Log[1 + x], $MachinePrecision] / n), $MachinePrecision]], $MachinePrecision] - t$95$0), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -3.9999999999999999e-72 or 2.0000000000000001e-18 < (/.f64 #s(literal 1 binary64) n) < 1e6

   1. Initial program 78.7%

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

   3. Taylor expanded in x around inf 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. log-rec 93.6%

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

      2. mul-1-neg 93.6%

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

      3. neg-mul-1 93.6%

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

      4. mul-1-neg 93.6%

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

      5. distribute-frac-neg 93.6%

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

      6. remove-double-neg 93.6%

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

      7. *-rgt-identity 93.6%

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

      8. associate-/l* 93.6%

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

      9. exp-to-pow 93.6%

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

      10. *-commutative 93.6%

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

   5. Simplified 93.6%

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

   if -3.9999999999999999e-72 < (/.f64 #s(literal 1 binary64) n) < 2.0000000000000001e-18

   1. Initial program 31.5%

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

   3. Taylor expanded in n around inf 83.0%

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

      1. log1p-define 83.0%

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

   5. Simplified 83.0%

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

   if 1e6 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 63.2%

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

   3. Taylor expanded in n around 0 63.2%

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

      1. log1p-define 100.0%

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

      2. *-rgt-identity 100.0%

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

      3. associate-*l/ 100.0%

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

      4. associate-/l* 100.0%

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

      5. exp-to-pow 100.0%

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

   5. Simplified 100.0%

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

4. Final simplification 89.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -4 \cdot 10^{-72}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-18}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right) - \log x}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 1000000:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{n \cdot x}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 82.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ t_1 := \frac{t\_0}{n \cdot x}\\ \mathbf{if}\;\frac{1}{n} \leq -4 \cdot 10^{-72}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-18}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right) - \log x}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 1000000:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\left(1 + x \cdot \left(\frac{1}{n} + x \cdot \frac{-0.5 + \frac{0.5}{n}}{n}\right)\right) - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))) (t_1 (/ t_0 (* n x))))
   (if (<= (/ 1.0 n) -4e-72)
     t_1
     (if (<= (/ 1.0 n) 2e-18)
       (/ (- (log1p x) (log x)) n)
       (if (<= (/ 1.0 n) 1000000.0)
         t_1
         (-
          (+ 1.0 (* x (+ (/ 1.0 n) (* x (/ (+ -0.5 (/ 0.5 n)) n)))))
          t_0))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double t_1 = t_0 / (n * x);
	double tmp;
	if ((1.0 / n) <= -4e-72) {
		tmp = t_1;
	} else if ((1.0 / n) <= 2e-18) {
		tmp = (log1p(x) - log(x)) / n;
	} else if ((1.0 / n) <= 1000000.0) {
		tmp = t_1;
	} else {
		tmp = (1.0 + (x * ((1.0 / n) + (x * ((-0.5 + (0.5 / n)) / n))))) - t_0;
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, (1.0 / n));
	double t_1 = t_0 / (n * x);
	double tmp;
	if ((1.0 / n) <= -4e-72) {
		tmp = t_1;
	} else if ((1.0 / n) <= 2e-18) {
		tmp = (Math.log1p(x) - Math.log(x)) / n;
	} else if ((1.0 / n) <= 1000000.0) {
		tmp = t_1;
	} else {
		tmp = (1.0 + (x * ((1.0 / n) + (x * ((-0.5 + (0.5 / n)) / n))))) - t_0;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	t_1 = t_0 / (n * x)
	tmp = 0
	if (1.0 / n) <= -4e-72:
		tmp = t_1
	elif (1.0 / n) <= 2e-18:
		tmp = (math.log1p(x) - math.log(x)) / n
	elif (1.0 / n) <= 1000000.0:
		tmp = t_1
	else:
		tmp = (1.0 + (x * ((1.0 / n) + (x * ((-0.5 + (0.5 / n)) / n))))) - t_0
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	t_1 = Float64(t_0 / Float64(n * x))
	tmp = 0.0
	if (Float64(1.0 / n) <= -4e-72)
		tmp = t_1;
	elseif (Float64(1.0 / n) <= 2e-18)
		tmp = Float64(Float64(log1p(x) - log(x)) / n);
	elseif (Float64(1.0 / n) <= 1000000.0)
		tmp = t_1;
	else
		tmp = Float64(Float64(1.0 + Float64(x * Float64(Float64(1.0 / n) + Float64(x * Float64(Float64(-0.5 + Float64(0.5 / n)) / n))))) - t_0);
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 / N[(n * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -4e-72], t$95$1, If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-18], N[(N[(N[Log[1 + x], $MachinePrecision] - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 1000000.0], t$95$1, N[(N[(1.0 + N[(x * N[(N[(1.0 / n), $MachinePrecision] + N[(x * N[(N[(-0.5 + N[(0.5 / n), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -3.9999999999999999e-72 or 2.0000000000000001e-18 < (/.f64 #s(literal 1 binary64) n) < 1e6

   1. Initial program 78.7%

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

   3. Taylor expanded in x around inf 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. log-rec 93.6%

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

      2. mul-1-neg 93.6%

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

      3. neg-mul-1 93.6%

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

      4. mul-1-neg 93.6%

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

      5. distribute-frac-neg 93.6%

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

      6. remove-double-neg 93.6%

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

      7. *-rgt-identity 93.6%

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

      8. associate-/l* 93.6%

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

      9. exp-to-pow 93.6%

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

      10. *-commutative 93.6%

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

   5. Simplified 93.6%

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

   if -3.9999999999999999e-72 < (/.f64 #s(literal 1 binary64) n) < 2.0000000000000001e-18

   1. Initial program 31.5%

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

   3. Taylor expanded in n around inf 83.0%

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

      1. log1p-define 83.0%

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

   5. Simplified 83.0%

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

   if 1e6 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 63.2%

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

   3. Taylor expanded in x around 0 81.6%

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

   4. Taylor expanded in n around inf 81.6%

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

      1. sub-neg 81.6%

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

      2. associate-*r/ 81.6%

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

      3. metadata-eval 81.6%

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

      4. metadata-eval 81.6%

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

   6. Simplified 81.6%

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

4. Final simplification 86.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -4 \cdot 10^{-72}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-18}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right) - \log x}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 1000000:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{n \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + x \cdot \left(\frac{1}{n} + x \cdot \frac{-0.5 + \frac{0.5}{n}}{n}\right)\right) - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 70.3% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;x \leq 5.4 \cdot 10^{-170}:\\ \;\;\;\;\left(1 + x \cdot \frac{1 + \left(x \cdot -0.5 + 0.5 \cdot \frac{x}{n}\right)}{n}\right) - t\_0\\ \mathbf{elif}\;x \leq 3.1 \cdot 10^{-5}:\\ \;\;\;\;\frac{x \cdot \left(1 + x \cdot -0.5\right) + \log \left(\frac{1}{x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0}{n \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= x 5.4e-170)
     (- (+ 1.0 (* x (/ (+ 1.0 (+ (* x -0.5) (* 0.5 (/ x n)))) n))) t_0)
     (if (<= x 3.1e-5)
       (/ (+ (* x (+ 1.0 (* x -0.5))) (log (/ 1.0 x))) n)
       (/ t_0 (* n x))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double tmp;
	if (x <= 5.4e-170) {
		tmp = (1.0 + (x * ((1.0 + ((x * -0.5) + (0.5 * (x / n)))) / n))) - t_0;
	} else if (x <= 3.1e-5) {
		tmp = ((x * (1.0 + (x * -0.5))) + log((1.0 / x))) / n;
	} else {
		tmp = t_0 / (n * x);
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x ** (1.0d0 / n)
    if (x <= 5.4d-170) then
        tmp = (1.0d0 + (x * ((1.0d0 + ((x * (-0.5d0)) + (0.5d0 * (x / n)))) / n))) - t_0
    else if (x <= 3.1d-5) then
        tmp = ((x * (1.0d0 + (x * (-0.5d0)))) + log((1.0d0 / x))) / n
    else
        tmp = t_0 / (n * x)
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, (1.0 / n));
	double tmp;
	if (x <= 5.4e-170) {
		tmp = (1.0 + (x * ((1.0 + ((x * -0.5) + (0.5 * (x / n)))) / n))) - t_0;
	} else if (x <= 3.1e-5) {
		tmp = ((x * (1.0 + (x * -0.5))) + Math.log((1.0 / x))) / n;
	} else {
		tmp = t_0 / (n * x);
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	tmp = 0
	if x <= 5.4e-170:
		tmp = (1.0 + (x * ((1.0 + ((x * -0.5) + (0.5 * (x / n)))) / n))) - t_0
	elif x <= 3.1e-5:
		tmp = ((x * (1.0 + (x * -0.5))) + math.log((1.0 / x))) / n
	else:
		tmp = t_0 / (n * x)
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (x <= 5.4e-170)
		tmp = Float64(Float64(1.0 + Float64(x * Float64(Float64(1.0 + Float64(Float64(x * -0.5) + Float64(0.5 * Float64(x / n)))) / n))) - t_0);
	elseif (x <= 3.1e-5)
		tmp = Float64(Float64(Float64(x * Float64(1.0 + Float64(x * -0.5))) + log(Float64(1.0 / x))) / n);
	else
		tmp = Float64(t_0 / Float64(n * x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = x ^ (1.0 / n);
	tmp = 0.0;
	if (x <= 5.4e-170)
		tmp = (1.0 + (x * ((1.0 + ((x * -0.5) + (0.5 * (x / n)))) / n))) - t_0;
	elseif (x <= 3.1e-5)
		tmp = ((x * (1.0 + (x * -0.5))) + log((1.0 / x))) / n;
	else
		tmp = t_0 / (n * x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, 5.4e-170], N[(N[(1.0 + N[(x * N[(N[(1.0 + N[(N[(x * -0.5), $MachinePrecision] + N[(0.5 * N[(x / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[x, 3.1e-5], N[(N[(N[(x * N[(1.0 + N[(x * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[Log[N[(1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], N[(t$95$0 / N[(n * x), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < 5.3999999999999997e-170

   1. Initial program 53.7%

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

   3. Taylor expanded in x around 0 42.2%

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

   4. Taylor expanded in n around inf 54.3%

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

   if 5.3999999999999997e-170 < x < 3.10000000000000014e-5

   1. Initial program 37.2%

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

   3. Taylor expanded in x around 0 33.6%

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

   4. Taylor expanded in n around inf 53.7%

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

   5. Taylor expanded in x around inf 53.8%

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

   if 3.10000000000000014e-5 < x

   1. Initial program 65.8%

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

   3. Taylor expanded in x around inf 96.1%

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

      1. log-rec 96.1%

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

      2. mul-1-neg 96.1%

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

      3. neg-mul-1 96.1%

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

      4. mul-1-neg 96.1%

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

      5. distribute-frac-neg 96.1%

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

      6. remove-double-neg 96.1%

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

      7. *-rgt-identity 96.1%

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

      8. associate-/l* 96.1%

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

      9. exp-to-pow 96.1%

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

      10. *-commutative 96.1%

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

   5. Simplified 96.1%

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

4. Final simplification 73.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 5.4 \cdot 10^{-170}:\\ \;\;\;\;\left(1 + x \cdot \frac{1 + \left(x \cdot -0.5 + 0.5 \cdot \frac{x}{n}\right)}{n}\right) - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;x \leq 3.1 \cdot 10^{-5}:\\ \;\;\;\;\frac{x \cdot \left(1 + x \cdot -0.5\right) + \log \left(\frac{1}{x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{n \cdot x}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 70.3% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;x \leq 4.2 \cdot 10^{-169}:\\ \;\;\;\;\frac{n + x}{n} - t\_0\\ \mathbf{elif}\;x \leq 4.3 \cdot 10^{-6}:\\ \;\;\;\;\frac{x \cdot \left(1 + x \cdot -0.5\right) + \log \left(\frac{1}{x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0}{n \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= x 4.2e-169)
     (- (/ (+ n x) n) t_0)
     (if (<= x 4.3e-6)
       (/ (+ (* x (+ 1.0 (* x -0.5))) (log (/ 1.0 x))) n)
       (/ t_0 (* n x))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double tmp;
	if (x <= 4.2e-169) {
		tmp = ((n + x) / n) - t_0;
	} else if (x <= 4.3e-6) {
		tmp = ((x * (1.0 + (x * -0.5))) + log((1.0 / x))) / n;
	} else {
		tmp = t_0 / (n * x);
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x ** (1.0d0 / n)
    if (x <= 4.2d-169) then
        tmp = ((n + x) / n) - t_0
    else if (x <= 4.3d-6) then
        tmp = ((x * (1.0d0 + (x * (-0.5d0)))) + log((1.0d0 / x))) / n
    else
        tmp = t_0 / (n * x)
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, (1.0 / n));
	double tmp;
	if (x <= 4.2e-169) {
		tmp = ((n + x) / n) - t_0;
	} else if (x <= 4.3e-6) {
		tmp = ((x * (1.0 + (x * -0.5))) + Math.log((1.0 / x))) / n;
	} else {
		tmp = t_0 / (n * x);
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	tmp = 0
	if x <= 4.2e-169:
		tmp = ((n + x) / n) - t_0
	elif x <= 4.3e-6:
		tmp = ((x * (1.0 + (x * -0.5))) + math.log((1.0 / x))) / n
	else:
		tmp = t_0 / (n * x)
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (x <= 4.2e-169)
		tmp = Float64(Float64(Float64(n + x) / n) - t_0);
	elseif (x <= 4.3e-6)
		tmp = Float64(Float64(Float64(x * Float64(1.0 + Float64(x * -0.5))) + log(Float64(1.0 / x))) / n);
	else
		tmp = Float64(t_0 / Float64(n * x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = x ^ (1.0 / n);
	tmp = 0.0;
	if (x <= 4.2e-169)
		tmp = ((n + x) / n) - t_0;
	elseif (x <= 4.3e-6)
		tmp = ((x * (1.0 + (x * -0.5))) + log((1.0 / x))) / n;
	else
		tmp = t_0 / (n * x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, 4.2e-169], N[(N[(N[(n + x), $MachinePrecision] / n), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[x, 4.3e-6], N[(N[(N[(x * N[(1.0 + N[(x * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[Log[N[(1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], N[(t$95$0 / N[(n * x), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < 4.2000000000000001e-169

   1. Initial program 53.7%

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

   3. Taylor expanded in x around 0 54.3%

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

   4. Taylor expanded in n around 0 54.3%

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

      1. +-commutative 54.3%

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

   6. Simplified 54.3%

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

   if 4.2000000000000001e-169 < x < 4.30000000000000033e-6

   1. Initial program 37.2%

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

   3. Taylor expanded in x around 0 33.6%

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

   4. Taylor expanded in n around inf 53.7%

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

   5. Taylor expanded in x around inf 53.8%

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

   if 4.30000000000000033e-6 < x

   1. Initial program 65.8%

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

   3. Taylor expanded in x around inf 96.1%

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

      1. log-rec 96.1%

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

      2. mul-1-neg 96.1%

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

      3. neg-mul-1 96.1%

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

      4. mul-1-neg 96.1%

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

      5. distribute-frac-neg 96.1%

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

      6. remove-double-neg 96.1%

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

      7. *-rgt-identity 96.1%

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

      8. associate-/l* 96.1%

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

      9. exp-to-pow 96.1%

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

      10. *-commutative 96.1%

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

   5. Simplified 96.1%

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

4. Final simplification 73.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 4.2 \cdot 10^{-169}:\\ \;\;\;\;\frac{n + x}{n} - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;x \leq 4.3 \cdot 10^{-6}:\\ \;\;\;\;\frac{x \cdot \left(1 + x \cdot -0.5\right) + \log \left(\frac{1}{x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{n \cdot x}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 70.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;x \leq 6.5 \cdot 10^{-170}:\\ \;\;\;\;\frac{n + x}{n} - t\_0\\ \mathbf{elif}\;x \leq 1.45 \cdot 10^{-5}:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0}{n \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= x 6.5e-170)
     (- (/ (+ n x) n) t_0)
     (if (<= x 1.45e-5) (/ (- x (log x)) n) (/ t_0 (* n x))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double tmp;
	if (x <= 6.5e-170) {
		tmp = ((n + x) / n) - t_0;
	} else if (x <= 1.45e-5) {
		tmp = (x - log(x)) / n;
	} else {
		tmp = t_0 / (n * x);
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x ** (1.0d0 / n)
    if (x <= 6.5d-170) then
        tmp = ((n + x) / n) - t_0
    else if (x <= 1.45d-5) then
        tmp = (x - log(x)) / n
    else
        tmp = t_0 / (n * x)
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, (1.0 / n));
	double tmp;
	if (x <= 6.5e-170) {
		tmp = ((n + x) / n) - t_0;
	} else if (x <= 1.45e-5) {
		tmp = (x - Math.log(x)) / n;
	} else {
		tmp = t_0 / (n * x);
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	tmp = 0
	if x <= 6.5e-170:
		tmp = ((n + x) / n) - t_0
	elif x <= 1.45e-5:
		tmp = (x - math.log(x)) / n
	else:
		tmp = t_0 / (n * x)
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (x <= 6.5e-170)
		tmp = Float64(Float64(Float64(n + x) / n) - t_0);
	elseif (x <= 1.45e-5)
		tmp = Float64(Float64(x - log(x)) / n);
	else
		tmp = Float64(t_0 / Float64(n * x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = x ^ (1.0 / n);
	tmp = 0.0;
	if (x <= 6.5e-170)
		tmp = ((n + x) / n) - t_0;
	elseif (x <= 1.45e-5)
		tmp = (x - log(x)) / n;
	else
		tmp = t_0 / (n * x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, 6.5e-170], N[(N[(N[(n + x), $MachinePrecision] / n), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[x, 1.45e-5], N[(N[(x - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], N[(t$95$0 / N[(n * x), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < 6.50000000000000035e-170

   1. Initial program 53.7%

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

   3. Taylor expanded in x around 0 54.3%

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

   4. Taylor expanded in n around 0 54.3%

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

      1. +-commutative 54.3%

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

   6. Simplified 54.3%

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

   if 6.50000000000000035e-170 < x < 1.45e-5

   1. Initial program 37.2%

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

   3. Taylor expanded in x around 0 36.7%

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

   4. Taylor expanded in n around inf 53.7%

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

   if 1.45e-5 < x

   1. Initial program 65.8%

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

   3. Taylor expanded in x around inf 96.1%

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

      1. log-rec 96.1%

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

      2. mul-1-neg 96.1%

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

      3. neg-mul-1 96.1%

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

      4. mul-1-neg 96.1%

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

      5. distribute-frac-neg 96.1%

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

      6. remove-double-neg 96.1%

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

      7. *-rgt-identity 96.1%

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

      8. associate-/l* 96.1%

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

      9. exp-to-pow 96.1%

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

      10. *-commutative 96.1%

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

   5. Simplified 96.1%

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

4. Final simplification 73.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 6.5 \cdot 10^{-170}:\\ \;\;\;\;\frac{n + x}{n} - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;x \leq 1.45 \cdot 10^{-5}:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{n \cdot x}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 70.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;x \leq 6.4 \cdot 10^{-170}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - t\_0\\ \mathbf{elif}\;x \leq 4.8 \cdot 10^{-6}:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0}{n \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= x 6.4e-170)
     (- (+ 1.0 (/ x n)) t_0)
     (if (<= x 4.8e-6) (/ (- x (log x)) n) (/ t_0 (* n x))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double tmp;
	if (x <= 6.4e-170) {
		tmp = (1.0 + (x / n)) - t_0;
	} else if (x <= 4.8e-6) {
		tmp = (x - log(x)) / n;
	} else {
		tmp = t_0 / (n * x);
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x ** (1.0d0 / n)
    if (x <= 6.4d-170) then
        tmp = (1.0d0 + (x / n)) - t_0
    else if (x <= 4.8d-6) then
        tmp = (x - log(x)) / n
    else
        tmp = t_0 / (n * x)
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, (1.0 / n));
	double tmp;
	if (x <= 6.4e-170) {
		tmp = (1.0 + (x / n)) - t_0;
	} else if (x <= 4.8e-6) {
		tmp = (x - Math.log(x)) / n;
	} else {
		tmp = t_0 / (n * x);
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	tmp = 0
	if x <= 6.4e-170:
		tmp = (1.0 + (x / n)) - t_0
	elif x <= 4.8e-6:
		tmp = (x - math.log(x)) / n
	else:
		tmp = t_0 / (n * x)
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (x <= 6.4e-170)
		tmp = Float64(Float64(1.0 + Float64(x / n)) - t_0);
	elseif (x <= 4.8e-6)
		tmp = Float64(Float64(x - log(x)) / n);
	else
		tmp = Float64(t_0 / Float64(n * x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = x ^ (1.0 / n);
	tmp = 0.0;
	if (x <= 6.4e-170)
		tmp = (1.0 + (x / n)) - t_0;
	elseif (x <= 4.8e-6)
		tmp = (x - log(x)) / n;
	else
		tmp = t_0 / (n * x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, 6.4e-170], N[(N[(1.0 + N[(x / n), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[x, 4.8e-6], N[(N[(x - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], N[(t$95$0 / N[(n * x), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < 6.3999999999999999e-170

   1. Initial program 53.7%

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

   3. Taylor expanded in x around 0 54.3%

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

   if 6.3999999999999999e-170 < x < 4.7999999999999998e-6

   1. Initial program 37.2%

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

   3. Taylor expanded in x around 0 36.7%

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

   4. Taylor expanded in n around inf 53.7%

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

   if 4.7999999999999998e-6 < x

   1. Initial program 65.8%

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

   3. Taylor expanded in x around inf 96.1%

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

      1. log-rec 96.1%

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

      2. mul-1-neg 96.1%

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

      3. neg-mul-1 96.1%

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

      4. mul-1-neg 96.1%

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

      5. distribute-frac-neg 96.1%

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

      6. remove-double-neg 96.1%

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

      7. *-rgt-identity 96.1%

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

      8. associate-/l* 96.1%

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

      9. exp-to-pow 96.1%

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

      10. *-commutative 96.1%

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

   5. Simplified 96.1%

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

4. Final simplification 73.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 6.4 \cdot 10^{-170}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;x \leq 4.8 \cdot 10^{-6}:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{n \cdot x}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 70.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;x \leq 1.1 \cdot 10^{-169}:\\ \;\;\;\;1 - t\_0\\ \mathbf{elif}\;x \leq 2.15 \cdot 10^{-5}:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0}{n \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= x 1.1e-169)
     (- 1.0 t_0)
     (if (<= x 2.15e-5) (/ (- x (log x)) n) (/ t_0 (* n x))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double tmp;
	if (x <= 1.1e-169) {
		tmp = 1.0 - t_0;
	} else if (x <= 2.15e-5) {
		tmp = (x - log(x)) / n;
	} else {
		tmp = t_0 / (n * x);
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x ** (1.0d0 / n)
    if (x <= 1.1d-169) then
        tmp = 1.0d0 - t_0
    else if (x <= 2.15d-5) then
        tmp = (x - log(x)) / n
    else
        tmp = t_0 / (n * x)
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, (1.0 / n));
	double tmp;
	if (x <= 1.1e-169) {
		tmp = 1.0 - t_0;
	} else if (x <= 2.15e-5) {
		tmp = (x - Math.log(x)) / n;
	} else {
		tmp = t_0 / (n * x);
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	tmp = 0
	if x <= 1.1e-169:
		tmp = 1.0 - t_0
	elif x <= 2.15e-5:
		tmp = (x - math.log(x)) / n
	else:
		tmp = t_0 / (n * x)
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (x <= 1.1e-169)
		tmp = Float64(1.0 - t_0);
	elseif (x <= 2.15e-5)
		tmp = Float64(Float64(x - log(x)) / n);
	else
		tmp = Float64(t_0 / Float64(n * x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = x ^ (1.0 / n);
	tmp = 0.0;
	if (x <= 1.1e-169)
		tmp = 1.0 - t_0;
	elseif (x <= 2.15e-5)
		tmp = (x - log(x)) / n;
	else
		tmp = t_0 / (n * x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, 1.1e-169], N[(1.0 - t$95$0), $MachinePrecision], If[LessEqual[x, 2.15e-5], N[(N[(x - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], N[(t$95$0 / N[(n * x), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < 1.10000000000000004e-169

   1. Initial program 53.7%

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

   3. Taylor expanded in x around 0 53.7%

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

      1. *-rgt-identity 53.7%

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

      2. associate-*l/ 53.7%

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

      3. associate-/l* 53.7%

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

      4. exp-to-pow 53.7%

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

   5. Simplified 53.7%

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

   if 1.10000000000000004e-169 < x < 2.1500000000000001e-5

   1. Initial program 37.2%

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

   3. Taylor expanded in x around 0 36.7%

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

   4. Taylor expanded in n around inf 53.7%

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

   if 2.1500000000000001e-5 < x

   1. Initial program 65.8%

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

   3. Taylor expanded in x around inf 96.1%

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

      1. log-rec 96.1%

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

      2. mul-1-neg 96.1%

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

      3. neg-mul-1 96.1%

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

      4. mul-1-neg 96.1%

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

      5. distribute-frac-neg 96.1%

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

      6. remove-double-neg 96.1%

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

      7. *-rgt-identity 96.1%

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

      8. associate-/l* 96.1%

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

      9. exp-to-pow 96.1%

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

      10. *-commutative 96.1%

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

   5. Simplified 96.1%

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

4. Final simplification 72.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.1 \cdot 10^{-169}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;x \leq 2.15 \cdot 10^{-5}:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{n \cdot x}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 56.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 5.8 \cdot 10^{-170}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;x \leq 7.2 \cdot 10^{-7}:\\ \;\;\;\;\frac{\log x}{-n}\\ \mathbf{elif}\;x \leq 1.4 \cdot 10^{+24}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 5.8e-170)
   (- 1.0 (pow x (/ 1.0 n)))
   (if (<= x 7.2e-7)
     (/ (log x) (- n))
     (if (<= x 1.4e+24) (/ (/ 1.0 x) n) 0.0))))

C

double code(double x, double n) {
	double tmp;
	if (x <= 5.8e-170) {
		tmp = 1.0 - pow(x, (1.0 / n));
	} else if (x <= 7.2e-7) {
		tmp = log(x) / -n;
	} else if (x <= 1.4e+24) {
		tmp = (1.0 / x) / n;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if (x <= 5.8d-170) then
        tmp = 1.0d0 - (x ** (1.0d0 / n))
    else if (x <= 7.2d-7) then
        tmp = log(x) / -n
    else if (x <= 1.4d+24) then
        tmp = (1.0d0 / x) / n
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (x <= 5.8e-170) {
		tmp = 1.0 - Math.pow(x, (1.0 / n));
	} else if (x <= 7.2e-7) {
		tmp = Math.log(x) / -n;
	} else if (x <= 1.4e+24) {
		tmp = (1.0 / x) / n;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 5.8e-170:
		tmp = 1.0 - math.pow(x, (1.0 / n))
	elif x <= 7.2e-7:
		tmp = math.log(x) / -n
	elif x <= 1.4e+24:
		tmp = (1.0 / x) / n
	else:
		tmp = 0.0
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 5.8e-170)
		tmp = Float64(1.0 - (x ^ Float64(1.0 / n)));
	elseif (x <= 7.2e-7)
		tmp = Float64(log(x) / Float64(-n));
	elseif (x <= 1.4e+24)
		tmp = Float64(Float64(1.0 / x) / n);
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (x <= 5.8e-170)
		tmp = 1.0 - (x ^ (1.0 / n));
	elseif (x <= 7.2e-7)
		tmp = log(x) / -n;
	elseif (x <= 1.4e+24)
		tmp = (1.0 / x) / n;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 5.8e-170], N[(1.0 - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 7.2e-7], N[(N[Log[x], $MachinePrecision] / (-n)), $MachinePrecision], If[LessEqual[x, 1.4e+24], N[(N[(1.0 / x), $MachinePrecision] / n), $MachinePrecision], 0.0]]]

Derivation

1. Split input into 4 regimes

2. if x < 5.8000000000000001e-170

   1. Initial program 53.7%

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

   3. Taylor expanded in x around 0 53.7%

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

      1. *-rgt-identity 53.7%

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

      2. associate-*l/ 53.7%

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

      3. associate-/l* 53.7%

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

      4. exp-to-pow 53.7%

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

   5. Simplified 53.7%

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

   if 5.8000000000000001e-170 < x < 7.19999999999999989e-7

   1. Initial program 36.4%

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

   3. Taylor expanded in x around 0 36.4%

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

      1. *-rgt-identity 36.4%

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

      2. associate-*l/ 36.4%

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

      3. associate-/l* 36.4%

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

      4. exp-to-pow 36.4%

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

   5. Simplified 36.4%

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

   6. Taylor expanded in n around inf 54.2%

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

      1. associate-*r/ 54.2%

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

      2. neg-mul-1 54.2%

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

   8. Simplified 54.2%

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

   if 7.19999999999999989e-7 < x < 1.4000000000000001e24

   1. Initial program 31.5%

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

   3. Taylor expanded in n around inf 57.3%

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

      1. associate--l+ 57.3%

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

      2. log1p-define 57.3%

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

      3. +-commutative 57.3%

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

      4. associate--r+ 57.3%

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

      5. distribute-lft-out-- 57.3%

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

      6. div-sub 57.3%

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

      7. log1p-define 57.3%

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

   5. Simplified 57.3%

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

   6. Taylor expanded in x around inf 61.0%

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

      1. +-commutative 61.0%

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

      2. mul-1-neg 61.0%

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

      3. log-rec 61.0%

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

      4. neg-mul-1 61.0%

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

      5. associate-*r/ 61.0%

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

      6. mul-1-neg 61.0%

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

      7. remove-double-neg 61.0%

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

   8. Simplified 61.0%

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

   9. Taylor expanded in n around inf 52.9%

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

   if 1.4000000000000001e24 < x

   1. Initial program 69.7%

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

   3. Taylor expanded in x around 0 33.9%

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

      1. *-rgt-identity 33.9%

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

      2. associate-*l/ 33.9%

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

      3. associate-/l* 33.9%

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

      4. exp-to-pow 33.9%

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

   5. Simplified 33.9%

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

      1. add-log-exp 33.9%

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

   7. Applied egg-rr 33.9%

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

   8. Taylor expanded in n around inf 69.8%

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

4. Final simplification 60.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 5.8 \cdot 10^{-170}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;x \leq 7.2 \cdot 10^{-7}:\\ \;\;\;\;\frac{\log x}{-n}\\ \mathbf{elif}\;x \leq 1.4 \cdot 10^{+24}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 57.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 7.2 \cdot 10^{-7}:\\ \;\;\;\;\frac{\log x}{-n}\\ \mathbf{elif}\;x \leq 2.2 \cdot 10^{+24}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 7.2e-7) (/ (log x) (- n)) (if (<= x 2.2e+24) (/ (/ 1.0 x) n) 0.0)))

C

double code(double x, double n) {
	double tmp;
	if (x <= 7.2e-7) {
		tmp = log(x) / -n;
	} else if (x <= 2.2e+24) {
		tmp = (1.0 / x) / n;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if (x <= 7.2d-7) then
        tmp = log(x) / -n
    else if (x <= 2.2d+24) then
        tmp = (1.0d0 / x) / n
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (x <= 7.2e-7) {
		tmp = Math.log(x) / -n;
	} else if (x <= 2.2e+24) {
		tmp = (1.0 / x) / n;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 7.2e-7:
		tmp = math.log(x) / -n
	elif x <= 2.2e+24:
		tmp = (1.0 / x) / n
	else:
		tmp = 0.0
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 7.2e-7)
		tmp = Float64(log(x) / Float64(-n));
	elseif (x <= 2.2e+24)
		tmp = Float64(Float64(1.0 / x) / n);
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (x <= 7.2e-7)
		tmp = log(x) / -n;
	elseif (x <= 2.2e+24)
		tmp = (1.0 / x) / n;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 7.2e-7], N[(N[Log[x], $MachinePrecision] / (-n)), $MachinePrecision], If[LessEqual[x, 2.2e+24], N[(N[(1.0 / x), $MachinePrecision] / n), $MachinePrecision], 0.0]]

Derivation

1. Split input into 3 regimes

2. if x < 7.19999999999999989e-7

   1. Initial program 44.5%

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

   3. Taylor expanded in x around 0 44.5%

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

      1. *-rgt-identity 44.5%

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

      2. associate-*l/ 44.5%

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

      3. associate-/l* 44.5%

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

      4. exp-to-pow 44.5%

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

   5. Simplified 44.5%

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

   6. Taylor expanded in n around inf 50.8%

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

      1. associate-*r/ 50.8%

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

      2. neg-mul-1 50.8%

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

   8. Simplified 50.8%

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

   if 7.19999999999999989e-7 < x < 2.20000000000000002e24

   1. Initial program 31.5%

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

   3. Taylor expanded in n around inf 57.3%

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

      1. associate--l+ 57.3%

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

      2. log1p-define 57.3%

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

      3. +-commutative 57.3%

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

      4. associate--r+ 57.3%

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

      5. distribute-lft-out-- 57.3%

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

      6. div-sub 57.3%

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

      7. log1p-define 57.3%

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

   5. Simplified 57.3%

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

   6. Taylor expanded in x around inf 61.0%

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

      1. +-commutative 61.0%

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

      2. mul-1-neg 61.0%

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

      3. log-rec 61.0%

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

      4. neg-mul-1 61.0%

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

      5. associate-*r/ 61.0%

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

      6. mul-1-neg 61.0%

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

      7. remove-double-neg 61.0%

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

   8. Simplified 61.0%

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

   9. Taylor expanded in n around inf 52.9%

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

   if 2.20000000000000002e24 < x

   1. Initial program 69.7%

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

   3. Taylor expanded in x around 0 33.9%

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

      1. *-rgt-identity 33.9%

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

      2. associate-*l/ 33.9%

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

      3. associate-/l* 33.9%

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

      4. exp-to-pow 33.9%

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

   5. Simplified 33.9%

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

      1. add-log-exp 33.9%

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

   7. Applied egg-rr 33.9%

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

   8. Taylor expanded in n around inf 69.8%

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

4. Final simplification 58.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 7.2 \cdot 10^{-7}:\\ \;\;\;\;\frac{\log x}{-n}\\ \mathbf{elif}\;x \leq 2.2 \cdot 10^{+24}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 45.4% accurate, 17.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{+39}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= (/ 1.0 n) -5e+39) 0.0 (/ (/ 1.0 x) n)))

C

double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -5e+39) {
		tmp = 0.0;
	} else {
		tmp = (1.0 / x) / n;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if ((1.0d0 / n) <= (-5d+39)) then
        tmp = 0.0d0
    else
        tmp = (1.0d0 / x) / n
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -5e+39) {
		tmp = 0.0;
	} else {
		tmp = (1.0 / x) / n;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if (1.0 / n) <= -5e+39:
		tmp = 0.0
	else:
		tmp = (1.0 / x) / n
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -5e+39)
		tmp = 0.0;
	else
		tmp = Float64(Float64(1.0 / x) / n);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if ((1.0 / n) <= -5e+39)
		tmp = 0.0;
	else
		tmp = (1.0 / x) / n;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[N[(1.0 / n), $MachinePrecision], -5e+39], 0.0, N[(N[(1.0 / x), $MachinePrecision] / n), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -5.00000000000000015e39

   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 x around 0 45.2%

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

      1. *-rgt-identity 45.2%

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

      2. associate-*l/ 45.2%

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

      3. associate-/l* 45.2%

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

      4. exp-to-pow 45.2%

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

   5. Simplified 45.2%

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

      1. add-log-exp 45.2%

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

   7. Applied egg-rr 45.2%

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

   8. Taylor expanded in n around inf 57.2%

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

   if -5.00000000000000015e39 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 37.6%

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

   3. Taylor expanded in n around inf 56.4%

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

      1. associate--l+ 56.4%

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

      2. log1p-define 56.4%

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

      3. +-commutative 56.4%

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

      4. associate--r+ 56.4%

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

      5. distribute-lft-out-- 56.4%

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

      6. div-sub 56.4%

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

      7. log1p-define 56.4%

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

   5. Simplified 56.4%

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

   6. Taylor expanded in x around inf 38.2%

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

      1. +-commutative 38.2%

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

      2. mul-1-neg 38.2%

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

      3. log-rec 38.2%

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

      4. neg-mul-1 38.2%

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

      5. associate-*r/ 38.2%

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

      6. mul-1-neg 38.2%

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

      7. remove-double-neg 38.2%

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

   8. Simplified 38.2%

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

   9. Taylor expanded in n around inf 44.2%

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

4. Final simplification 47.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{+39}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 41.0% accurate, 42.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 54.4%

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

3. Taylor expanded in n around inf 64.8%

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

   1. associate--l+ 49.9%

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

   2. log1p-define 49.9%

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

   3. +-commutative 49.9%

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

   4. associate--r+ 64.8%

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

   5. distribute-lft-out-- 64.8%

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

   6. div-sub 64.8%

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

   7. log1p-define 64.8%

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

5. Simplified 64.8%

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

6. Taylor expanded in x around inf 38.5%

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

   1. +-commutative 38.5%

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

   2. mul-1-neg 38.5%

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

   3. log-rec 38.5%

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

   4. neg-mul-1 38.5%

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

   5. associate-*r/ 38.5%

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

   6. mul-1-neg 38.5%

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

   7. remove-double-neg 38.5%

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

8. Simplified 38.5%

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

9. Taylor expanded in n around inf 40.1%

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

Alternative 14: 40.5% 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 54.4%

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

3. Taylor expanded in n around inf 64.8%

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

   1. associate--l+ 49.9%

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

   2. log1p-define 49.9%

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

   3. +-commutative 49.9%

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

   4. associate--r+ 64.8%

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

   5. distribute-lft-out-- 64.8%

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

   6. div-sub 64.8%

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

   7. log1p-define 64.8%

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

5. Simplified 64.8%

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

6. Taylor expanded in x around inf 38.5%

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

   1. +-commutative 38.5%

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

   2. mul-1-neg 38.5%

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

   3. log-rec 38.5%

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

   4. neg-mul-1 38.5%

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

   5. associate-*r/ 38.5%

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

   6. mul-1-neg 38.5%

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

   7. remove-double-neg 38.5%

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

8. Simplified 38.5%

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

9. Taylor expanded in n around inf 39.6%

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

    1. *-commutative 39.6%

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

11. Simplified 39.6%

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

12. Final simplification 39.6%

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

Alternative 15: 4.5% accurate, 70.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 54.4%

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

3. Taylor expanded in x around 0 32.0%

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

4. Taylor expanded in x around inf 4.9%

   \[\leadsto \color{blue}{\frac{x}{n}} \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024160 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  :precision binary64
  (- (pow (+ x 1.0) (/ 1.0 n)) (pow x (/ 1.0 n))))