2nthrt (problem 3.4.6)

Percentage Accurate: 52.9% → 98.1%

Time: 41.5s

Alternatives: 12

Speedup: 1.8×

• could not determine a ground truth

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 52.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ t_1 := \sqrt{t\_0}\\ \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{-11}:\\ \;\;\;\;t\_0 \cdot \frac{1}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-7}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(\frac{1}{x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t\_1, -t\_1, e^{\frac{\mathsf{log1p}\left(x\right)}{n}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))) (t_1 (sqrt t_0)))
   (if (<= (/ 1.0 n) -5e-11)
     (* t_0 (/ 1.0 (* n x)))
     (if (<= (/ 1.0 n) 2e-7)
       (/ (log1p (/ 1.0 x)) n)
       (fma t_1 (- t_1) (exp (/ (log1p x) n)))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double t_1 = sqrt(t_0);
	double tmp;
	if ((1.0 / n) <= -5e-11) {
		tmp = t_0 * (1.0 / (n * x));
	} else if ((1.0 / n) <= 2e-7) {
		tmp = log1p((1.0 / x)) / n;
	} else {
		tmp = fma(t_1, -t_1, exp((log1p(x) / n)));
	}
	return tmp;
}

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	t_1 = sqrt(t_0)
	tmp = 0.0
	if (Float64(1.0 / n) <= -5e-11)
		tmp = Float64(t_0 * Float64(1.0 / Float64(n * x)));
	elseif (Float64(1.0 / n) <= 2e-7)
		tmp = Float64(log1p(Float64(1.0 / x)) / n);
	else
		tmp = fma(t_1, Float64(-t_1), exp(Float64(log1p(x) / n)));
	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[Sqrt[t$95$0], $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -5e-11], N[(t$95$0 * N[(1.0 / N[(n * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-7], N[(N[Log[1 + N[(1.0 / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], N[(t$95$1 * (-t$95$1) + N[Exp[N[(N[Log[1 + x], $MachinePrecision] / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 1 n) < -5.00000000000000018e-11

   1. Initial program 97.2%

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

   3. Taylor expanded in x around inf 100.0%

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

      1. mul-1-neg 100.0%

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

      2. log-rec 100.0%

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

      3. mul-1-neg 100.0%

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

      4. distribute-neg-frac 100.0%

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

      5. mul-1-neg 100.0%

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

      6. remove-double-neg 100.0%

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

      7. *-commutative 100.0%

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

   5. Simplified 100.0%

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

      1. div-inv 100.0%

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

      2. div-inv 100.0%

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

      3. inv-pow 100.0%

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

      4. metadata-eval 100.0%

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

      5. pow-prod-up 0.0%

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

      6. exp-to-pow 0.0%

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

      7. pow-prod-up 100.0%

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

      8. metadata-eval 100.0%

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

      9. inv-pow 100.0%

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

   7. Applied egg-rr 100.0%

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

   if -5.00000000000000018e-11 < (/.f64 1 n) < 1.9999999999999999e-7

   1. Initial program 23.3%

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

   3. Taylor expanded in n around inf 76.3%

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

      1. diff-log 76.5%

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

   5. Applied egg-rr 76.5%

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

      1. log1p-expm1-u 76.5%

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

      2. expm1-undefine 76.5%

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

      3. add-exp-log 76.5%

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

   7. Applied egg-rr 76.5%

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

      1. *-lft-identity 76.5%

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

      2. associate-*l/ 75.1%

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

      3. distribute-rgt-in 75.1%

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

      4. *-lft-identity 75.1%

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

      5. rgt-mult-inverse 76.5%

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

      6. associate--l+ 98.7%

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

      7. metadata-eval 98.7%

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

   9. Simplified 98.7%

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

   10. Taylor expanded in n around 0 76.5%

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

       1. log1p-define 98.7%

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

   12. Simplified 98.7%

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

   if 1.9999999999999999e-7 < (/.f64 1 n)

   1. Initial program 42.8%

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

      1. sub-neg 42.8%

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

      2. +-commutative 42.8%

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

      3. sqr-pow 42.8%

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

      4. distribute-rgt-neg-in 42.8%

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

      5. fma-define 42.9%

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

      6. sqrt-pow1 42.9%

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

      7. sqrt-pow1 42.9%

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

      8. pow-to-exp 42.9%

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

      9. un-div-inv 42.9%

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

      10. +-commutative 42.9%

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

      11. log1p-define 94.6%

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

   4. Applied egg-rr 94.6%

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

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{-11}:\\ \;\;\;\;{x}^{\left(\frac{1}{n}\right)} \cdot \frac{1}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-7}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(\frac{1}{x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\sqrt{{x}^{\left(\frac{1}{n}\right)}}, -\sqrt{{x}^{\left(\frac{1}{n}\right)}}, e^{\frac{\mathsf{log1p}\left(x\right)}{n}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 98.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{-11}:\\ \;\;\;\;t\_0 \cdot \frac{1}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-7}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(\frac{1}{x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{x}{n}} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -5e-11)
     (* t_0 (/ 1.0 (* n x)))
     (if (<= (/ 1.0 n) 2e-7) (/ (log1p (/ 1.0 x)) n) (- (exp (/ x n)) t_0)))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -5e-11) {
		tmp = t_0 * (1.0 / (n * x));
	} else if ((1.0 / n) <= 2e-7) {
		tmp = log1p((1.0 / x)) / n;
	} else {
		tmp = exp((x / n)) - t_0;
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -5e-11) {
		tmp = t_0 * (1.0 / (n * x));
	} else if ((1.0 / n) <= 2e-7) {
		tmp = Math.log1p((1.0 / x)) / n;
	} else {
		tmp = Math.exp((x / n)) - t_0;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	tmp = 0
	if (1.0 / n) <= -5e-11:
		tmp = t_0 * (1.0 / (n * x))
	elif (1.0 / n) <= 2e-7:
		tmp = math.log1p((1.0 / x)) / n
	else:
		tmp = math.exp((x / n)) - t_0
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -5e-11)
		tmp = Float64(t_0 * Float64(1.0 / Float64(n * x)));
	elseif (Float64(1.0 / n) <= 2e-7)
		tmp = Float64(log1p(Float64(1.0 / x)) / n);
	else
		tmp = Float64(exp(Float64(x / n)) - t_0);
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -5e-11], N[(t$95$0 * N[(1.0 / N[(n * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-7], N[(N[Log[1 + N[(1.0 / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], N[(N[Exp[N[(x / n), $MachinePrecision]], $MachinePrecision] - t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 1 n) < -5.00000000000000018e-11

   1. Initial program 97.2%

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

   3. Taylor expanded in x around inf 100.0%

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

      1. mul-1-neg 100.0%

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

      2. log-rec 100.0%

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

      3. mul-1-neg 100.0%

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

      4. distribute-neg-frac 100.0%

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

      5. mul-1-neg 100.0%

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

      6. remove-double-neg 100.0%

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

      7. *-commutative 100.0%

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

   5. Simplified 100.0%

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

      1. div-inv 100.0%

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

      2. div-inv 100.0%

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

      3. inv-pow 100.0%

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

      4. metadata-eval 100.0%

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

      5. pow-prod-up 0.0%

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

      6. exp-to-pow 0.0%

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

      7. pow-prod-up 100.0%

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

      8. metadata-eval 100.0%

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

      9. inv-pow 100.0%

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

   7. Applied egg-rr 100.0%

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

   if -5.00000000000000018e-11 < (/.f64 1 n) < 1.9999999999999999e-7

   1. Initial program 23.3%

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

   3. Taylor expanded in n around inf 76.3%

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

      1. diff-log 76.5%

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

   5. Applied egg-rr 76.5%

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

      1. log1p-expm1-u 76.5%

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

      2. expm1-undefine 76.5%

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

      3. add-exp-log 76.5%

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

   7. Applied egg-rr 76.5%

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

      1. *-lft-identity 76.5%

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

      2. associate-*l/ 75.1%

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

      3. distribute-rgt-in 75.1%

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

      4. *-lft-identity 75.1%

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

      5. rgt-mult-inverse 76.5%

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

      6. associate--l+ 98.7%

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

      7. metadata-eval 98.7%

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

   9. Simplified 98.7%

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

   10. Taylor expanded in n around 0 76.5%

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

       1. log1p-define 98.7%

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

   12. Simplified 98.7%

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

   if 1.9999999999999999e-7 < (/.f64 1 n)

   1. Initial program 42.8%

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

   3. Taylor expanded in n around 0 42.8%

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

      1. log1p-define 94.5%

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

   5. Simplified 94.5%

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

   6. Taylor expanded in x around 0 94.5%

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

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{-11}:\\ \;\;\;\;{x}^{\left(\frac{1}{n}\right)} \cdot \frac{1}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-7}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(\frac{1}{x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{x}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 93.1% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{-11}:\\ \;\;\;\;t\_0 \cdot \frac{1}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-7}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(\frac{1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{+117}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - t\_0\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{{x}^{-3}}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -5e-11)
     (* t_0 (/ 1.0 (* n x)))
     (if (<= (/ 1.0 n) 2e-7)
       (/ (log1p (/ 1.0 x)) n)
       (if (<= (/ 1.0 n) 5e+117)
         (- (+ 1.0 (/ x n)) t_0)
         (* 0.3333333333333333 (/ (pow x -3.0) n)))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -5e-11) {
		tmp = t_0 * (1.0 / (n * x));
	} else if ((1.0 / n) <= 2e-7) {
		tmp = log1p((1.0 / x)) / n;
	} else if ((1.0 / n) <= 5e+117) {
		tmp = (1.0 + (x / n)) - t_0;
	} else {
		tmp = 0.3333333333333333 * (pow(x, -3.0) / n);
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -5e-11) {
		tmp = t_0 * (1.0 / (n * x));
	} else if ((1.0 / n) <= 2e-7) {
		tmp = Math.log1p((1.0 / x)) / n;
	} else if ((1.0 / n) <= 5e+117) {
		tmp = (1.0 + (x / n)) - t_0;
	} else {
		tmp = 0.3333333333333333 * (Math.pow(x, -3.0) / n);
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	tmp = 0
	if (1.0 / n) <= -5e-11:
		tmp = t_0 * (1.0 / (n * x))
	elif (1.0 / n) <= 2e-7:
		tmp = math.log1p((1.0 / x)) / n
	elif (1.0 / n) <= 5e+117:
		tmp = (1.0 + (x / n)) - t_0
	else:
		tmp = 0.3333333333333333 * (math.pow(x, -3.0) / n)
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -5e-11)
		tmp = Float64(t_0 * Float64(1.0 / Float64(n * x)));
	elseif (Float64(1.0 / n) <= 2e-7)
		tmp = Float64(log1p(Float64(1.0 / x)) / n);
	elseif (Float64(1.0 / n) <= 5e+117)
		tmp = Float64(Float64(1.0 + Float64(x / n)) - t_0);
	else
		tmp = Float64(0.3333333333333333 * Float64((x ^ -3.0) / n));
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -5e-11], N[(t$95$0 * N[(1.0 / N[(n * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-7], N[(N[Log[1 + N[(1.0 / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e+117], N[(N[(1.0 + N[(x / n), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], N[(0.3333333333333333 * N[(N[Power[x, -3.0], $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 1 n) < -5.00000000000000018e-11

   1. Initial program 97.2%

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

   3. Taylor expanded in x around inf 100.0%

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

      1. mul-1-neg 100.0%

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

      2. log-rec 100.0%

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

      3. mul-1-neg 100.0%

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

      4. distribute-neg-frac 100.0%

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

      5. mul-1-neg 100.0%

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

      6. remove-double-neg 100.0%

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

      7. *-commutative 100.0%

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

   5. Simplified 100.0%

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

      1. div-inv 100.0%

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

      2. div-inv 100.0%

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

      3. inv-pow 100.0%

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

      4. metadata-eval 100.0%

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

      5. pow-prod-up 0.0%

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

      6. exp-to-pow 0.0%

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

      7. pow-prod-up 100.0%

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

      8. metadata-eval 100.0%

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

      9. inv-pow 100.0%

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

   7. Applied egg-rr 100.0%

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

   if -5.00000000000000018e-11 < (/.f64 1 n) < 1.9999999999999999e-7

   1. Initial program 23.3%

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

   3. Taylor expanded in n around inf 76.3%

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

      1. diff-log 76.5%

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

   5. Applied egg-rr 76.5%

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

      1. log1p-expm1-u 76.5%

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

      2. expm1-undefine 76.5%

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

      3. add-exp-log 76.5%

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

   7. Applied egg-rr 76.5%

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

      1. *-lft-identity 76.5%

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

      2. associate-*l/ 75.1%

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

      3. distribute-rgt-in 75.1%

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

      4. *-lft-identity 75.1%

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

      5. rgt-mult-inverse 76.5%

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

      6. associate--l+ 98.7%

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

      7. metadata-eval 98.7%

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

   9. Simplified 98.7%

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

   10. Taylor expanded in n around 0 76.5%

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

       1. log1p-define 98.7%

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

   12. Simplified 98.7%

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

   if 1.9999999999999999e-7 < (/.f64 1 n) < 4.99999999999999983e117

   1. Initial program 72.9%

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

   3. Taylor expanded in x around 0 73.6%

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

   if 4.99999999999999983e117 < (/.f64 1 n)

   1. Initial program 17.1%

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

   3. Taylor expanded in n around inf 10.8%

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

   4. Taylor expanded in x around inf 43.3%

      \[\leadsto \frac{\color{blue}{\left(0.3333333333333333 \cdot \frac{1}{{x}^{3}} + \frac{1}{x}\right) - 0.5 \cdot \frac{1}{{x}^{2}}}}{n} \]
   5. Step-by-step derivation

      1. sub-neg 43.3%

         \[\leadsto \frac{\color{blue}{\left(0.3333333333333333 \cdot \frac{1}{{x}^{3}} + \frac{1}{x}\right) + \left(-0.5 \cdot \frac{1}{{x}^{2}}\right)}}{n} \]

      2. +-commutative 43.3%

         \[\leadsto \frac{\color{blue}{\left(\frac{1}{x} + 0.3333333333333333 \cdot \frac{1}{{x}^{3}}\right)} + \left(-0.5 \cdot \frac{1}{{x}^{2}}\right)}{n} \]

      3. associate-+l+ 43.3%

         \[\leadsto \frac{\color{blue}{\frac{1}{x} + \left(0.3333333333333333 \cdot \frac{1}{{x}^{3}} + \left(-0.5 \cdot \frac{1}{{x}^{2}}\right)\right)}}{n} \]

      4. associate-*r/ 43.3%

         \[\leadsto \frac{\frac{1}{x} + \left(\color{blue}{\frac{0.3333333333333333 \cdot 1}{{x}^{3}}} + \left(-0.5 \cdot \frac{1}{{x}^{2}}\right)\right)}{n} \]

      5. metadata-eval 43.3%

         \[\leadsto \frac{\frac{1}{x} + \left(\frac{\color{blue}{0.3333333333333333}}{{x}^{3}} + \left(-0.5 \cdot \frac{1}{{x}^{2}}\right)\right)}{n} \]

      6. associate-*r/ 43.3%

         \[\leadsto \frac{\frac{1}{x} + \left(\frac{0.3333333333333333}{{x}^{3}} + \left(-\color{blue}{\frac{0.5 \cdot 1}{{x}^{2}}}\right)\right)}{n} \]

      7. metadata-eval 43.3%

         \[\leadsto \frac{\frac{1}{x} + \left(\frac{0.3333333333333333}{{x}^{3}} + \left(-\frac{\color{blue}{0.5}}{{x}^{2}}\right)\right)}{n} \]

      8. distribute-neg-frac 43.3%

         \[\leadsto \frac{\frac{1}{x} + \left(\frac{0.3333333333333333}{{x}^{3}} + \color{blue}{\frac{-0.5}{{x}^{2}}}\right)}{n} \]

      9. metadata-eval 43.3%

         \[\leadsto \frac{\frac{1}{x} + \left(\frac{0.3333333333333333}{{x}^{3}} + \frac{\color{blue}{-0.5}}{{x}^{2}}\right)}{n} \]

   6. Simplified 43.3%

      \[\leadsto \frac{\color{blue}{\frac{1}{x} + \left(\frac{0.3333333333333333}{{x}^{3}} + \frac{-0.5}{{x}^{2}}\right)}}{n} \]

   7. Taylor expanded in x around 0 81.9%

      \[\leadsto \color{blue}{\frac{0.3333333333333333}{n \cdot {x}^{3}}} \]
   8. Step-by-step derivation

      1. associate-/r* 81.9%

         \[\leadsto \color{blue}{\frac{\frac{0.3333333333333333}{n}}{{x}^{3}}} \]

   9. Simplified 81.9%

      \[\leadsto \color{blue}{\frac{\frac{0.3333333333333333}{n}}{{x}^{3}}} \]
   10. Step-by-step derivation

       1. div-inv 81.9%

          \[\leadsto \color{blue}{\frac{0.3333333333333333}{n} \cdot \frac{1}{{x}^{3}}} \]

       2. div-inv 81.9%

          \[\leadsto \color{blue}{\left(0.3333333333333333 \cdot \frac{1}{n}\right)} \cdot \frac{1}{{x}^{3}} \]

       3. inv-pow 81.9%

          \[\leadsto \left(0.3333333333333333 \cdot \color{blue}{{n}^{-1}}\right) \cdot \frac{1}{{x}^{3}} \]

       4. metadata-eval 81.9%

          \[\leadsto \left(0.3333333333333333 \cdot {n}^{\color{blue}{\left(-0.5 + -0.5\right)}}\right) \cdot \frac{1}{{x}^{3}} \]

       5. pow-prod-up 81.9%

          \[\leadsto \left(0.3333333333333333 \cdot \color{blue}{\left({n}^{-0.5} \cdot {n}^{-0.5}\right)}\right) \cdot \frac{1}{{x}^{3}} \]

       6. metadata-eval 81.9%

          \[\leadsto \left(0.3333333333333333 \cdot \left({n}^{-0.5} \cdot {n}^{-0.5}\right)\right) \cdot \frac{\color{blue}{{1}^{3}}}{{x}^{3}} \]

       7. cube-div 81.9%

          \[\leadsto \left(0.3333333333333333 \cdot \left({n}^{-0.5} \cdot {n}^{-0.5}\right)\right) \cdot \color{blue}{{\left(\frac{1}{x}\right)}^{3}} \]

       8. pow3 81.9%

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

       9. associate-*l* 81.9%

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

       10. pow-prod-up 81.9%

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

       11. metadata-eval 81.9%

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

       12. inv-pow 81.9%

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

       13. pow3 81.9%

           \[\leadsto 0.3333333333333333 \cdot \left(\frac{1}{n} \cdot \color{blue}{{\left(\frac{1}{x}\right)}^{3}}\right) \]

       14. inv-pow 81.9%

           \[\leadsto 0.3333333333333333 \cdot \left(\frac{1}{n} \cdot {\color{blue}{\left({x}^{-1}\right)}}^{3}\right) \]

       15. pow-pow 81.9%

           \[\leadsto 0.3333333333333333 \cdot \left(\frac{1}{n} \cdot \color{blue}{{x}^{\left(-1 \cdot 3\right)}}\right) \]

       16. metadata-eval 81.9%

           \[\leadsto 0.3333333333333333 \cdot \left(\frac{1}{n} \cdot {x}^{\color{blue}{-3}}\right) \]

   11. Applied egg-rr 81.9%

       \[\leadsto \color{blue}{0.3333333333333333 \cdot \left(\frac{1}{n} \cdot {x}^{-3}\right)} \]
   12. Step-by-step derivation

       1. *-commutative 81.9%

          \[\leadsto 0.3333333333333333 \cdot \color{blue}{\left({x}^{-3} \cdot \frac{1}{n}\right)} \]

       2. associate-*r/ 81.9%

          \[\leadsto 0.3333333333333333 \cdot \color{blue}{\frac{{x}^{-3} \cdot 1}{n}} \]

       3. *-commutative 81.9%

          \[\leadsto 0.3333333333333333 \cdot \frac{\color{blue}{1 \cdot {x}^{-3}}}{n} \]

       4. *-lft-identity 81.9%

          \[\leadsto 0.3333333333333333 \cdot \frac{\color{blue}{{x}^{-3}}}{n} \]

   13. Simplified 81.9%

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

4. Final simplification 95.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{-11}:\\ \;\;\;\;{x}^{\left(\frac{1}{n}\right)} \cdot \frac{1}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-7}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(\frac{1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{+117}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{{x}^{-3}}{n}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 84.5% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.3333333333333333 \cdot \frac{{x}^{-3}}{n}\\ \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{+22}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-7}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(\frac{1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{+117}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (* 0.3333333333333333 (/ (pow x -3.0) n))))
   (if (<= (/ 1.0 n) -2e+22)
     t_0
     (if (<= (/ 1.0 n) 2e-7)
       (/ (log1p (/ 1.0 x)) n)
       (if (<= (/ 1.0 n) 5e+117) (- 1.0 (pow x (/ 1.0 n))) t_0)))))

C

double code(double x, double n) {
	double t_0 = 0.3333333333333333 * (pow(x, -3.0) / n);
	double tmp;
	if ((1.0 / n) <= -2e+22) {
		tmp = t_0;
	} else if ((1.0 / n) <= 2e-7) {
		tmp = log1p((1.0 / x)) / n;
	} else if ((1.0 / n) <= 5e+117) {
		tmp = 1.0 - pow(x, (1.0 / n));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double t_0 = 0.3333333333333333 * (Math.pow(x, -3.0) / n);
	double tmp;
	if ((1.0 / n) <= -2e+22) {
		tmp = t_0;
	} else if ((1.0 / n) <= 2e-7) {
		tmp = Math.log1p((1.0 / x)) / n;
	} else if ((1.0 / n) <= 5e+117) {
		tmp = 1.0 - Math.pow(x, (1.0 / n));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = 0.3333333333333333 * (math.pow(x, -3.0) / n)
	tmp = 0
	if (1.0 / n) <= -2e+22:
		tmp = t_0
	elif (1.0 / n) <= 2e-7:
		tmp = math.log1p((1.0 / x)) / n
	elif (1.0 / n) <= 5e+117:
		tmp = 1.0 - math.pow(x, (1.0 / n))
	else:
		tmp = t_0
	return tmp

Julia

function code(x, n)
	t_0 = Float64(0.3333333333333333 * Float64((x ^ -3.0) / n))
	tmp = 0.0
	if (Float64(1.0 / n) <= -2e+22)
		tmp = t_0;
	elseif (Float64(1.0 / n) <= 2e-7)
		tmp = Float64(log1p(Float64(1.0 / x)) / n);
	elseif (Float64(1.0 / n) <= 5e+117)
		tmp = Float64(1.0 - (x ^ Float64(1.0 / n)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(0.3333333333333333 * N[(N[Power[x, -3.0], $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -2e+22], t$95$0, If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-7], N[(N[Log[1 + N[(1.0 / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e+117], N[(1.0 - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 1 n) < -2e22 or 4.99999999999999983e117 < (/.f64 1 n)

   1. Initial program 80.0%

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

   3. Taylor expanded in n around inf 43.7%

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

   4. Taylor expanded in x around inf 25.0%

      \[\leadsto \frac{\color{blue}{\left(0.3333333333333333 \cdot \frac{1}{{x}^{3}} + \frac{1}{x}\right) - 0.5 \cdot \frac{1}{{x}^{2}}}}{n} \]
   5. Step-by-step derivation

      1. sub-neg 25.0%

         \[\leadsto \frac{\color{blue}{\left(0.3333333333333333 \cdot \frac{1}{{x}^{3}} + \frac{1}{x}\right) + \left(-0.5 \cdot \frac{1}{{x}^{2}}\right)}}{n} \]

      2. +-commutative 25.0%

         \[\leadsto \frac{\color{blue}{\left(\frac{1}{x} + 0.3333333333333333 \cdot \frac{1}{{x}^{3}}\right)} + \left(-0.5 \cdot \frac{1}{{x}^{2}}\right)}{n} \]

      3. associate-+l+ 25.0%

         \[\leadsto \frac{\color{blue}{\frac{1}{x} + \left(0.3333333333333333 \cdot \frac{1}{{x}^{3}} + \left(-0.5 \cdot \frac{1}{{x}^{2}}\right)\right)}}{n} \]

      4. associate-*r/ 25.0%

         \[\leadsto \frac{\frac{1}{x} + \left(\color{blue}{\frac{0.3333333333333333 \cdot 1}{{x}^{3}}} + \left(-0.5 \cdot \frac{1}{{x}^{2}}\right)\right)}{n} \]

      5. metadata-eval 25.0%

         \[\leadsto \frac{\frac{1}{x} + \left(\frac{\color{blue}{0.3333333333333333}}{{x}^{3}} + \left(-0.5 \cdot \frac{1}{{x}^{2}}\right)\right)}{n} \]

      6. associate-*r/ 25.0%

         \[\leadsto \frac{\frac{1}{x} + \left(\frac{0.3333333333333333}{{x}^{3}} + \left(-\color{blue}{\frac{0.5 \cdot 1}{{x}^{2}}}\right)\right)}{n} \]

      7. metadata-eval 25.0%

         \[\leadsto \frac{\frac{1}{x} + \left(\frac{0.3333333333333333}{{x}^{3}} + \left(-\frac{\color{blue}{0.5}}{{x}^{2}}\right)\right)}{n} \]

      8. distribute-neg-frac 25.0%

         \[\leadsto \frac{\frac{1}{x} + \left(\frac{0.3333333333333333}{{x}^{3}} + \color{blue}{\frac{-0.5}{{x}^{2}}}\right)}{n} \]

      9. metadata-eval 25.0%

         \[\leadsto \frac{\frac{1}{x} + \left(\frac{0.3333333333333333}{{x}^{3}} + \frac{\color{blue}{-0.5}}{{x}^{2}}\right)}{n} \]

   6. Simplified 25.0%

      \[\leadsto \frac{\color{blue}{\frac{1}{x} + \left(\frac{0.3333333333333333}{{x}^{3}} + \frac{-0.5}{{x}^{2}}\right)}}{n} \]

   7. Taylor expanded in x around 0 71.4%

      \[\leadsto \color{blue}{\frac{0.3333333333333333}{n \cdot {x}^{3}}} \]
   8. Step-by-step derivation

      1. associate-/r* 71.4%

         \[\leadsto \color{blue}{\frac{\frac{0.3333333333333333}{n}}{{x}^{3}}} \]

   9. Simplified 71.4%

      \[\leadsto \color{blue}{\frac{\frac{0.3333333333333333}{n}}{{x}^{3}}} \]
   10. Step-by-step derivation

       1. div-inv 71.4%

          \[\leadsto \color{blue}{\frac{0.3333333333333333}{n} \cdot \frac{1}{{x}^{3}}} \]

       2. div-inv 71.4%

          \[\leadsto \color{blue}{\left(0.3333333333333333 \cdot \frac{1}{n}\right)} \cdot \frac{1}{{x}^{3}} \]

       3. inv-pow 71.4%

          \[\leadsto \left(0.3333333333333333 \cdot \color{blue}{{n}^{-1}}\right) \cdot \frac{1}{{x}^{3}} \]

       4. metadata-eval 71.4%

          \[\leadsto \left(0.3333333333333333 \cdot {n}^{\color{blue}{\left(-0.5 + -0.5\right)}}\right) \cdot \frac{1}{{x}^{3}} \]

       5. pow-prod-up 19.8%

          \[\leadsto \left(0.3333333333333333 \cdot \color{blue}{\left({n}^{-0.5} \cdot {n}^{-0.5}\right)}\right) \cdot \frac{1}{{x}^{3}} \]

       6. metadata-eval 19.8%

          \[\leadsto \left(0.3333333333333333 \cdot \left({n}^{-0.5} \cdot {n}^{-0.5}\right)\right) \cdot \frac{\color{blue}{{1}^{3}}}{{x}^{3}} \]

       7. cube-div 19.8%

          \[\leadsto \left(0.3333333333333333 \cdot \left({n}^{-0.5} \cdot {n}^{-0.5}\right)\right) \cdot \color{blue}{{\left(\frac{1}{x}\right)}^{3}} \]

       8. pow3 19.8%

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

       9. associate-*l* 19.8%

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

       10. pow-prod-up 71.4%

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

       11. metadata-eval 71.4%

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

       12. inv-pow 71.4%

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

       13. pow3 71.4%

           \[\leadsto 0.3333333333333333 \cdot \left(\frac{1}{n} \cdot \color{blue}{{\left(\frac{1}{x}\right)}^{3}}\right) \]

       14. inv-pow 71.4%

           \[\leadsto 0.3333333333333333 \cdot \left(\frac{1}{n} \cdot {\color{blue}{\left({x}^{-1}\right)}}^{3}\right) \]

       15. pow-pow 71.4%

           \[\leadsto 0.3333333333333333 \cdot \left(\frac{1}{n} \cdot \color{blue}{{x}^{\left(-1 \cdot 3\right)}}\right) \]

       16. metadata-eval 71.4%

           \[\leadsto 0.3333333333333333 \cdot \left(\frac{1}{n} \cdot {x}^{\color{blue}{-3}}\right) \]

   11. Applied egg-rr 71.4%

       \[\leadsto \color{blue}{0.3333333333333333 \cdot \left(\frac{1}{n} \cdot {x}^{-3}\right)} \]
   12. Step-by-step derivation

       1. *-commutative 71.4%

          \[\leadsto 0.3333333333333333 \cdot \color{blue}{\left({x}^{-3} \cdot \frac{1}{n}\right)} \]

       2. associate-*r/ 71.4%

          \[\leadsto 0.3333333333333333 \cdot \color{blue}{\frac{{x}^{-3} \cdot 1}{n}} \]

       3. *-commutative 71.4%

          \[\leadsto 0.3333333333333333 \cdot \frac{\color{blue}{1 \cdot {x}^{-3}}}{n} \]

       4. *-lft-identity 71.4%

          \[\leadsto 0.3333333333333333 \cdot \frac{\color{blue}{{x}^{-3}}}{n} \]

   13. Simplified 71.4%

       \[\leadsto \color{blue}{0.3333333333333333 \cdot \frac{{x}^{-3}}{n}} \]

   if -2e22 < (/.f64 1 n) < 1.9999999999999999e-7

   1. Initial program 23.1%

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

   3. Taylor expanded in n around inf 75.3%

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

      1. diff-log 75.5%

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

   5. Applied egg-rr 75.5%

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

      1. log1p-expm1-u 75.5%

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

      2. expm1-undefine 75.5%

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

      3. add-exp-log 75.5%

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

   7. Applied egg-rr 75.5%

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

      1. *-lft-identity 75.5%

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

      2. associate-*l/ 74.2%

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

      3. distribute-rgt-in 74.2%

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

      4. *-lft-identity 74.2%

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

      5. rgt-mult-inverse 75.5%

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

      6. associate--l+ 97.9%

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

      7. metadata-eval 97.9%

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

   9. Simplified 97.9%

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

   10. Taylor expanded in n around 0 75.5%

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

       1. log1p-define 97.9%

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

   12. Simplified 97.9%

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

   if 1.9999999999999999e-7 < (/.f64 1 n) < 4.99999999999999983e117

   1. Initial program 72.9%

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

   3. Taylor expanded in x around 0 72.9%

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

4. Final simplification 87.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{+22}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{{x}^{-3}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-7}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(\frac{1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{+117}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{{x}^{-3}}{n}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 93.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{-11}:\\ \;\;\;\;\frac{t\_0}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-7}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(\frac{1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{+117}:\\ \;\;\;\;1 - t\_0\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{{x}^{-3}}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -5e-11)
     (/ t_0 (* n x))
     (if (<= (/ 1.0 n) 2e-7)
       (/ (log1p (/ 1.0 x)) n)
       (if (<= (/ 1.0 n) 5e+117)
         (- 1.0 t_0)
         (* 0.3333333333333333 (/ (pow x -3.0) n)))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -5e-11) {
		tmp = t_0 / (n * x);
	} else if ((1.0 / n) <= 2e-7) {
		tmp = log1p((1.0 / x)) / n;
	} else if ((1.0 / n) <= 5e+117) {
		tmp = 1.0 - t_0;
	} else {
		tmp = 0.3333333333333333 * (pow(x, -3.0) / n);
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -5e-11) {
		tmp = t_0 / (n * x);
	} else if ((1.0 / n) <= 2e-7) {
		tmp = Math.log1p((1.0 / x)) / n;
	} else if ((1.0 / n) <= 5e+117) {
		tmp = 1.0 - t_0;
	} else {
		tmp = 0.3333333333333333 * (Math.pow(x, -3.0) / n);
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	tmp = 0
	if (1.0 / n) <= -5e-11:
		tmp = t_0 / (n * x)
	elif (1.0 / n) <= 2e-7:
		tmp = math.log1p((1.0 / x)) / n
	elif (1.0 / n) <= 5e+117:
		tmp = 1.0 - t_0
	else:
		tmp = 0.3333333333333333 * (math.pow(x, -3.0) / n)
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -5e-11)
		tmp = Float64(t_0 / Float64(n * x));
	elseif (Float64(1.0 / n) <= 2e-7)
		tmp = Float64(log1p(Float64(1.0 / x)) / n);
	elseif (Float64(1.0 / n) <= 5e+117)
		tmp = Float64(1.0 - t_0);
	else
		tmp = Float64(0.3333333333333333 * Float64((x ^ -3.0) / n));
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -5e-11], N[(t$95$0 / N[(n * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-7], N[(N[Log[1 + N[(1.0 / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e+117], N[(1.0 - t$95$0), $MachinePrecision], N[(0.3333333333333333 * N[(N[Power[x, -3.0], $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 1 n) < -5.00000000000000018e-11

   1. Initial program 97.2%

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

   3. Taylor expanded in x around inf 100.0%

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

      1. mul-1-neg 100.0%

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

      2. log-rec 100.0%

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

      3. mul-1-neg 100.0%

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

      4. distribute-neg-frac 100.0%

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

      5. mul-1-neg 100.0%

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

      6. remove-double-neg 100.0%

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

      7. *-commutative 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in x around 0 100.0%

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

      1. *-rgt-identity 100.0%

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

      2. associate-*r/ 100.0%

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

      3. exp-to-pow 100.0%

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

      4. *-commutative 100.0%

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

   8. Simplified 100.0%

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

   if -5.00000000000000018e-11 < (/.f64 1 n) < 1.9999999999999999e-7

   1. Initial program 23.3%

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

   3. Taylor expanded in n around inf 76.3%

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

      1. diff-log 76.5%

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

   5. Applied egg-rr 76.5%

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

      1. log1p-expm1-u 76.5%

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

      2. expm1-undefine 76.5%

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

      3. add-exp-log 76.5%

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

   7. Applied egg-rr 76.5%

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

      1. *-lft-identity 76.5%

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

      2. associate-*l/ 75.1%

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

      3. distribute-rgt-in 75.1%

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

      4. *-lft-identity 75.1%

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

      5. rgt-mult-inverse 76.5%

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

      6. associate--l+ 98.7%

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

      7. metadata-eval 98.7%

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

   9. Simplified 98.7%

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

   10. Taylor expanded in n around 0 76.5%

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

       1. log1p-define 98.7%

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

   12. Simplified 98.7%

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

   if 1.9999999999999999e-7 < (/.f64 1 n) < 4.99999999999999983e117

   1. Initial program 72.9%

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

   3. Taylor expanded in x around 0 72.9%

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

   if 4.99999999999999983e117 < (/.f64 1 n)

   1. Initial program 17.1%

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

   3. Taylor expanded in n around inf 10.8%

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

   4. Taylor expanded in x around inf 43.3%

      \[\leadsto \frac{\color{blue}{\left(0.3333333333333333 \cdot \frac{1}{{x}^{3}} + \frac{1}{x}\right) - 0.5 \cdot \frac{1}{{x}^{2}}}}{n} \]
   5. Step-by-step derivation

      1. sub-neg 43.3%

         \[\leadsto \frac{\color{blue}{\left(0.3333333333333333 \cdot \frac{1}{{x}^{3}} + \frac{1}{x}\right) + \left(-0.5 \cdot \frac{1}{{x}^{2}}\right)}}{n} \]

      2. +-commutative 43.3%

         \[\leadsto \frac{\color{blue}{\left(\frac{1}{x} + 0.3333333333333333 \cdot \frac{1}{{x}^{3}}\right)} + \left(-0.5 \cdot \frac{1}{{x}^{2}}\right)}{n} \]

      3. associate-+l+ 43.3%

         \[\leadsto \frac{\color{blue}{\frac{1}{x} + \left(0.3333333333333333 \cdot \frac{1}{{x}^{3}} + \left(-0.5 \cdot \frac{1}{{x}^{2}}\right)\right)}}{n} \]

      4. associate-*r/ 43.3%

         \[\leadsto \frac{\frac{1}{x} + \left(\color{blue}{\frac{0.3333333333333333 \cdot 1}{{x}^{3}}} + \left(-0.5 \cdot \frac{1}{{x}^{2}}\right)\right)}{n} \]

      5. metadata-eval 43.3%

         \[\leadsto \frac{\frac{1}{x} + \left(\frac{\color{blue}{0.3333333333333333}}{{x}^{3}} + \left(-0.5 \cdot \frac{1}{{x}^{2}}\right)\right)}{n} \]

      6. associate-*r/ 43.3%

         \[\leadsto \frac{\frac{1}{x} + \left(\frac{0.3333333333333333}{{x}^{3}} + \left(-\color{blue}{\frac{0.5 \cdot 1}{{x}^{2}}}\right)\right)}{n} \]

      7. metadata-eval 43.3%

         \[\leadsto \frac{\frac{1}{x} + \left(\frac{0.3333333333333333}{{x}^{3}} + \left(-\frac{\color{blue}{0.5}}{{x}^{2}}\right)\right)}{n} \]

      8. distribute-neg-frac 43.3%

         \[\leadsto \frac{\frac{1}{x} + \left(\frac{0.3333333333333333}{{x}^{3}} + \color{blue}{\frac{-0.5}{{x}^{2}}}\right)}{n} \]

      9. metadata-eval 43.3%

         \[\leadsto \frac{\frac{1}{x} + \left(\frac{0.3333333333333333}{{x}^{3}} + \frac{\color{blue}{-0.5}}{{x}^{2}}\right)}{n} \]

   6. Simplified 43.3%

      \[\leadsto \frac{\color{blue}{\frac{1}{x} + \left(\frac{0.3333333333333333}{{x}^{3}} + \frac{-0.5}{{x}^{2}}\right)}}{n} \]

   7. Taylor expanded in x around 0 81.9%

      \[\leadsto \color{blue}{\frac{0.3333333333333333}{n \cdot {x}^{3}}} \]
   8. Step-by-step derivation

      1. associate-/r* 81.9%

         \[\leadsto \color{blue}{\frac{\frac{0.3333333333333333}{n}}{{x}^{3}}} \]

   9. Simplified 81.9%

      \[\leadsto \color{blue}{\frac{\frac{0.3333333333333333}{n}}{{x}^{3}}} \]
   10. Step-by-step derivation

       1. div-inv 81.9%

          \[\leadsto \color{blue}{\frac{0.3333333333333333}{n} \cdot \frac{1}{{x}^{3}}} \]

       2. div-inv 81.9%

          \[\leadsto \color{blue}{\left(0.3333333333333333 \cdot \frac{1}{n}\right)} \cdot \frac{1}{{x}^{3}} \]

       3. inv-pow 81.9%

          \[\leadsto \left(0.3333333333333333 \cdot \color{blue}{{n}^{-1}}\right) \cdot \frac{1}{{x}^{3}} \]

       4. metadata-eval 81.9%

          \[\leadsto \left(0.3333333333333333 \cdot {n}^{\color{blue}{\left(-0.5 + -0.5\right)}}\right) \cdot \frac{1}{{x}^{3}} \]

       5. pow-prod-up 81.9%

          \[\leadsto \left(0.3333333333333333 \cdot \color{blue}{\left({n}^{-0.5} \cdot {n}^{-0.5}\right)}\right) \cdot \frac{1}{{x}^{3}} \]

       6. metadata-eval 81.9%

          \[\leadsto \left(0.3333333333333333 \cdot \left({n}^{-0.5} \cdot {n}^{-0.5}\right)\right) \cdot \frac{\color{blue}{{1}^{3}}}{{x}^{3}} \]

       7. cube-div 81.9%

          \[\leadsto \left(0.3333333333333333 \cdot \left({n}^{-0.5} \cdot {n}^{-0.5}\right)\right) \cdot \color{blue}{{\left(\frac{1}{x}\right)}^{3}} \]

       8. pow3 81.9%

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

       9. associate-*l* 81.9%

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

       10. pow-prod-up 81.9%

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

       11. metadata-eval 81.9%

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

       12. inv-pow 81.9%

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

       13. pow3 81.9%

           \[\leadsto 0.3333333333333333 \cdot \left(\frac{1}{n} \cdot \color{blue}{{\left(\frac{1}{x}\right)}^{3}}\right) \]

       14. inv-pow 81.9%

           \[\leadsto 0.3333333333333333 \cdot \left(\frac{1}{n} \cdot {\color{blue}{\left({x}^{-1}\right)}}^{3}\right) \]

       15. pow-pow 81.9%

           \[\leadsto 0.3333333333333333 \cdot \left(\frac{1}{n} \cdot \color{blue}{{x}^{\left(-1 \cdot 3\right)}}\right) \]

       16. metadata-eval 81.9%

           \[\leadsto 0.3333333333333333 \cdot \left(\frac{1}{n} \cdot {x}^{\color{blue}{-3}}\right) \]

   11. Applied egg-rr 81.9%

       \[\leadsto \color{blue}{0.3333333333333333 \cdot \left(\frac{1}{n} \cdot {x}^{-3}\right)} \]
   12. Step-by-step derivation

       1. *-commutative 81.9%

          \[\leadsto 0.3333333333333333 \cdot \color{blue}{\left({x}^{-3} \cdot \frac{1}{n}\right)} \]

       2. associate-*r/ 81.9%

          \[\leadsto 0.3333333333333333 \cdot \color{blue}{\frac{{x}^{-3} \cdot 1}{n}} \]

       3. *-commutative 81.9%

          \[\leadsto 0.3333333333333333 \cdot \frac{\color{blue}{1 \cdot {x}^{-3}}}{n} \]

       4. *-lft-identity 81.9%

          \[\leadsto 0.3333333333333333 \cdot \frac{\color{blue}{{x}^{-3}}}{n} \]

   13. Simplified 81.9%

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

4. Final simplification 95.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{-11}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-7}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(\frac{1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{+117}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{{x}^{-3}}{n}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 93.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{-11}:\\ \;\;\;\;t\_0 \cdot \frac{1}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-7}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(\frac{1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{+117}:\\ \;\;\;\;1 - t\_0\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{{x}^{-3}}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -5e-11)
     (* t_0 (/ 1.0 (* n x)))
     (if (<= (/ 1.0 n) 2e-7)
       (/ (log1p (/ 1.0 x)) n)
       (if (<= (/ 1.0 n) 5e+117)
         (- 1.0 t_0)
         (* 0.3333333333333333 (/ (pow x -3.0) n)))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -5e-11) {
		tmp = t_0 * (1.0 / (n * x));
	} else if ((1.0 / n) <= 2e-7) {
		tmp = log1p((1.0 / x)) / n;
	} else if ((1.0 / n) <= 5e+117) {
		tmp = 1.0 - t_0;
	} else {
		tmp = 0.3333333333333333 * (pow(x, -3.0) / n);
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -5e-11) {
		tmp = t_0 * (1.0 / (n * x));
	} else if ((1.0 / n) <= 2e-7) {
		tmp = Math.log1p((1.0 / x)) / n;
	} else if ((1.0 / n) <= 5e+117) {
		tmp = 1.0 - t_0;
	} else {
		tmp = 0.3333333333333333 * (Math.pow(x, -3.0) / n);
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	tmp = 0
	if (1.0 / n) <= -5e-11:
		tmp = t_0 * (1.0 / (n * x))
	elif (1.0 / n) <= 2e-7:
		tmp = math.log1p((1.0 / x)) / n
	elif (1.0 / n) <= 5e+117:
		tmp = 1.0 - t_0
	else:
		tmp = 0.3333333333333333 * (math.pow(x, -3.0) / n)
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -5e-11)
		tmp = Float64(t_0 * Float64(1.0 / Float64(n * x)));
	elseif (Float64(1.0 / n) <= 2e-7)
		tmp = Float64(log1p(Float64(1.0 / x)) / n);
	elseif (Float64(1.0 / n) <= 5e+117)
		tmp = Float64(1.0 - t_0);
	else
		tmp = Float64(0.3333333333333333 * Float64((x ^ -3.0) / n));
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -5e-11], N[(t$95$0 * N[(1.0 / N[(n * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-7], N[(N[Log[1 + N[(1.0 / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e+117], N[(1.0 - t$95$0), $MachinePrecision], N[(0.3333333333333333 * N[(N[Power[x, -3.0], $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 1 n) < -5.00000000000000018e-11

   1. Initial program 97.2%

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

   3. Taylor expanded in x around inf 100.0%

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

      1. mul-1-neg 100.0%

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

      2. log-rec 100.0%

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

      3. mul-1-neg 100.0%

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

      4. distribute-neg-frac 100.0%

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

      5. mul-1-neg 100.0%

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

      6. remove-double-neg 100.0%

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

      7. *-commutative 100.0%

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

   5. Simplified 100.0%

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

      1. div-inv 100.0%

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

      2. div-inv 100.0%

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

      3. inv-pow 100.0%

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

      4. metadata-eval 100.0%

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

      5. pow-prod-up 0.0%

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

      6. exp-to-pow 0.0%

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

      7. pow-prod-up 100.0%

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

      8. metadata-eval 100.0%

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

      9. inv-pow 100.0%

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

   7. Applied egg-rr 100.0%

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

   if -5.00000000000000018e-11 < (/.f64 1 n) < 1.9999999999999999e-7

   1. Initial program 23.3%

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

   3. Taylor expanded in n around inf 76.3%

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

      1. diff-log 76.5%

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

   5. Applied egg-rr 76.5%

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

      1. log1p-expm1-u 76.5%

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

      2. expm1-undefine 76.5%

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

      3. add-exp-log 76.5%

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

   7. Applied egg-rr 76.5%

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

      1. *-lft-identity 76.5%

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

      2. associate-*l/ 75.1%

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

      3. distribute-rgt-in 75.1%

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

      4. *-lft-identity 75.1%

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

      5. rgt-mult-inverse 76.5%

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

      6. associate--l+ 98.7%

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

      7. metadata-eval 98.7%

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

   9. Simplified 98.7%

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

   10. Taylor expanded in n around 0 76.5%

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

       1. log1p-define 98.7%

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

   12. Simplified 98.7%

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

   if 1.9999999999999999e-7 < (/.f64 1 n) < 4.99999999999999983e117

   1. Initial program 72.9%

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

   3. Taylor expanded in x around 0 72.9%

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

   if 4.99999999999999983e117 < (/.f64 1 n)

   1. Initial program 17.1%

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

   3. Taylor expanded in n around inf 10.8%

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

   4. Taylor expanded in x around inf 43.3%

      \[\leadsto \frac{\color{blue}{\left(0.3333333333333333 \cdot \frac{1}{{x}^{3}} + \frac{1}{x}\right) - 0.5 \cdot \frac{1}{{x}^{2}}}}{n} \]
   5. Step-by-step derivation

      1. sub-neg 43.3%

         \[\leadsto \frac{\color{blue}{\left(0.3333333333333333 \cdot \frac{1}{{x}^{3}} + \frac{1}{x}\right) + \left(-0.5 \cdot \frac{1}{{x}^{2}}\right)}}{n} \]

      2. +-commutative 43.3%

         \[\leadsto \frac{\color{blue}{\left(\frac{1}{x} + 0.3333333333333333 \cdot \frac{1}{{x}^{3}}\right)} + \left(-0.5 \cdot \frac{1}{{x}^{2}}\right)}{n} \]

      3. associate-+l+ 43.3%

         \[\leadsto \frac{\color{blue}{\frac{1}{x} + \left(0.3333333333333333 \cdot \frac{1}{{x}^{3}} + \left(-0.5 \cdot \frac{1}{{x}^{2}}\right)\right)}}{n} \]

      4. associate-*r/ 43.3%

         \[\leadsto \frac{\frac{1}{x} + \left(\color{blue}{\frac{0.3333333333333333 \cdot 1}{{x}^{3}}} + \left(-0.5 \cdot \frac{1}{{x}^{2}}\right)\right)}{n} \]

      5. metadata-eval 43.3%

         \[\leadsto \frac{\frac{1}{x} + \left(\frac{\color{blue}{0.3333333333333333}}{{x}^{3}} + \left(-0.5 \cdot \frac{1}{{x}^{2}}\right)\right)}{n} \]

      6. associate-*r/ 43.3%

         \[\leadsto \frac{\frac{1}{x} + \left(\frac{0.3333333333333333}{{x}^{3}} + \left(-\color{blue}{\frac{0.5 \cdot 1}{{x}^{2}}}\right)\right)}{n} \]

      7. metadata-eval 43.3%

         \[\leadsto \frac{\frac{1}{x} + \left(\frac{0.3333333333333333}{{x}^{3}} + \left(-\frac{\color{blue}{0.5}}{{x}^{2}}\right)\right)}{n} \]

      8. distribute-neg-frac 43.3%

         \[\leadsto \frac{\frac{1}{x} + \left(\frac{0.3333333333333333}{{x}^{3}} + \color{blue}{\frac{-0.5}{{x}^{2}}}\right)}{n} \]

      9. metadata-eval 43.3%

         \[\leadsto \frac{\frac{1}{x} + \left(\frac{0.3333333333333333}{{x}^{3}} + \frac{\color{blue}{-0.5}}{{x}^{2}}\right)}{n} \]

   6. Simplified 43.3%

      \[\leadsto \frac{\color{blue}{\frac{1}{x} + \left(\frac{0.3333333333333333}{{x}^{3}} + \frac{-0.5}{{x}^{2}}\right)}}{n} \]

   7. Taylor expanded in x around 0 81.9%

      \[\leadsto \color{blue}{\frac{0.3333333333333333}{n \cdot {x}^{3}}} \]
   8. Step-by-step derivation

      1. associate-/r* 81.9%

         \[\leadsto \color{blue}{\frac{\frac{0.3333333333333333}{n}}{{x}^{3}}} \]

   9. Simplified 81.9%

      \[\leadsto \color{blue}{\frac{\frac{0.3333333333333333}{n}}{{x}^{3}}} \]
   10. Step-by-step derivation

       1. div-inv 81.9%

          \[\leadsto \color{blue}{\frac{0.3333333333333333}{n} \cdot \frac{1}{{x}^{3}}} \]

       2. div-inv 81.9%

          \[\leadsto \color{blue}{\left(0.3333333333333333 \cdot \frac{1}{n}\right)} \cdot \frac{1}{{x}^{3}} \]

       3. inv-pow 81.9%

          \[\leadsto \left(0.3333333333333333 \cdot \color{blue}{{n}^{-1}}\right) \cdot \frac{1}{{x}^{3}} \]

       4. metadata-eval 81.9%

          \[\leadsto \left(0.3333333333333333 \cdot {n}^{\color{blue}{\left(-0.5 + -0.5\right)}}\right) \cdot \frac{1}{{x}^{3}} \]

       5. pow-prod-up 81.9%

          \[\leadsto \left(0.3333333333333333 \cdot \color{blue}{\left({n}^{-0.5} \cdot {n}^{-0.5}\right)}\right) \cdot \frac{1}{{x}^{3}} \]

       6. metadata-eval 81.9%

          \[\leadsto \left(0.3333333333333333 \cdot \left({n}^{-0.5} \cdot {n}^{-0.5}\right)\right) \cdot \frac{\color{blue}{{1}^{3}}}{{x}^{3}} \]

       7. cube-div 81.9%

          \[\leadsto \left(0.3333333333333333 \cdot \left({n}^{-0.5} \cdot {n}^{-0.5}\right)\right) \cdot \color{blue}{{\left(\frac{1}{x}\right)}^{3}} \]

       8. pow3 81.9%

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

       9. associate-*l* 81.9%

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

       10. pow-prod-up 81.9%

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

       11. metadata-eval 81.9%

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

       12. inv-pow 81.9%

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

       13. pow3 81.9%

           \[\leadsto 0.3333333333333333 \cdot \left(\frac{1}{n} \cdot \color{blue}{{\left(\frac{1}{x}\right)}^{3}}\right) \]

       14. inv-pow 81.9%

           \[\leadsto 0.3333333333333333 \cdot \left(\frac{1}{n} \cdot {\color{blue}{\left({x}^{-1}\right)}}^{3}\right) \]

       15. pow-pow 81.9%

           \[\leadsto 0.3333333333333333 \cdot \left(\frac{1}{n} \cdot \color{blue}{{x}^{\left(-1 \cdot 3\right)}}\right) \]

       16. metadata-eval 81.9%

           \[\leadsto 0.3333333333333333 \cdot \left(\frac{1}{n} \cdot {x}^{\color{blue}{-3}}\right) \]

   11. Applied egg-rr 81.9%

       \[\leadsto \color{blue}{0.3333333333333333 \cdot \left(\frac{1}{n} \cdot {x}^{-3}\right)} \]
   12. Step-by-step derivation

       1. *-commutative 81.9%

          \[\leadsto 0.3333333333333333 \cdot \color{blue}{\left({x}^{-3} \cdot \frac{1}{n}\right)} \]

       2. associate-*r/ 81.9%

          \[\leadsto 0.3333333333333333 \cdot \color{blue}{\frac{{x}^{-3} \cdot 1}{n}} \]

       3. *-commutative 81.9%

          \[\leadsto 0.3333333333333333 \cdot \frac{\color{blue}{1 \cdot {x}^{-3}}}{n} \]

       4. *-lft-identity 81.9%

          \[\leadsto 0.3333333333333333 \cdot \frac{\color{blue}{{x}^{-3}}}{n} \]

   13. Simplified 81.9%

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

4. Final simplification 95.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{-11}:\\ \;\;\;\;{x}^{\left(\frac{1}{n}\right)} \cdot \frac{1}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-7}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(\frac{1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{+117}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{{x}^{-3}}{n}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 73.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{log1p}\left(\frac{1}{x}\right)}{n}\\ \mathbf{if}\;n \leq -6.4:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n \leq -3.5 \cdot 10^{-296}:\\ \;\;\;\;0\\ \mathbf{elif}\;n \leq 1.6 \cdot 10^{-122}:\\ \;\;\;\;\frac{1}{n \cdot x}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (/ (log1p (/ 1.0 x)) n)))
   (if (<= n -6.4)
     t_0
     (if (<= n -3.5e-296) 0.0 (if (<= n 1.6e-122) (/ 1.0 (* n x)) t_0)))))

C

double code(double x, double n) {
	double t_0 = log1p((1.0 / x)) / n;
	double tmp;
	if (n <= -6.4) {
		tmp = t_0;
	} else if (n <= -3.5e-296) {
		tmp = 0.0;
	} else if (n <= 1.6e-122) {
		tmp = 1.0 / (n * x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double t_0 = Math.log1p((1.0 / x)) / n;
	double tmp;
	if (n <= -6.4) {
		tmp = t_0;
	} else if (n <= -3.5e-296) {
		tmp = 0.0;
	} else if (n <= 1.6e-122) {
		tmp = 1.0 / (n * x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.log1p((1.0 / x)) / n
	tmp = 0
	if n <= -6.4:
		tmp = t_0
	elif n <= -3.5e-296:
		tmp = 0.0
	elif n <= 1.6e-122:
		tmp = 1.0 / (n * x)
	else:
		tmp = t_0
	return tmp

Julia

function code(x, n)
	t_0 = Float64(log1p(Float64(1.0 / x)) / n)
	tmp = 0.0
	if (n <= -6.4)
		tmp = t_0;
	elseif (n <= -3.5e-296)
		tmp = 0.0;
	elseif (n <= 1.6e-122)
		tmp = Float64(1.0 / Float64(n * x));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(N[Log[1 + N[(1.0 / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision]}, If[LessEqual[n, -6.4], t$95$0, If[LessEqual[n, -3.5e-296], 0.0, If[LessEqual[n, 1.6e-122], N[(1.0 / N[(n * x), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if n < -6.4000000000000004 or 1.6000000000000001e-122 < n

   1. Initial program 28.4%

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

   3. Taylor expanded in n around inf 68.2%

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

      1. diff-log 68.4%

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

   5. Applied egg-rr 68.4%

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

      1. log1p-expm1-u 68.4%

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

      2. expm1-undefine 68.4%

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

      3. add-exp-log 68.4%

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

   7. Applied egg-rr 68.4%

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

      1. *-lft-identity 68.4%

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

      2. associate-*l/ 67.2%

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

      3. distribute-rgt-in 67.2%

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

      4. *-lft-identity 67.2%

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

      5. rgt-mult-inverse 68.4%

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

      6. associate--l+ 88.4%

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

      7. metadata-eval 88.4%

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

   9. Simplified 88.4%

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

   10. Taylor expanded in n around 0 68.4%

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

       1. log1p-define 88.4%

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

   12. Simplified 88.4%

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

   if -6.4000000000000004 < n < -3.4999999999999999e-296

   1. Initial program 100.0%

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

      1. sub-neg 100.0%

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

      2. +-commutative 100.0%

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

      3. sqr-pow 100.0%

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

      4. distribute-rgt-neg-in 100.0%

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

      5. fma-define 100.0%

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

      6. sqrt-pow1 100.0%

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

      7. sqrt-pow1 100.0%

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

      8. pow-to-exp 100.0%

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

      9. un-div-inv 100.0%

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

      10. +-commutative 100.0%

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

      11. log1p-define 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in x around inf 57.1%

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

      1. distribute-rgt1-in 57.1%

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

      2. metadata-eval 57.1%

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

      3. mul0-lft 57.8%

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

   7. Simplified 57.8%

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

   if -3.4999999999999999e-296 < n < 1.6000000000000001e-122

   1. Initial program 27.4%

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

   3. Taylor expanded in n around inf 10.9%

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

   4. Taylor expanded in x around inf 64.8%

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

      1. *-commutative 64.8%

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

   6. Simplified 64.8%

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

4. Final simplification 78.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -6.4:\\ \;\;\;\;\frac{\mathsf{log1p}\left(\frac{1}{x}\right)}{n}\\ \mathbf{elif}\;n \leq -3.5 \cdot 10^{-296}:\\ \;\;\;\;0\\ \mathbf{elif}\;n \leq 1.6 \cdot 10^{-122}:\\ \;\;\;\;\frac{1}{n \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(\frac{1}{x}\right)}{n}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 58.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\log x}{-n}\\ \mathbf{if}\;x \leq 1.45 \cdot 10^{-266}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 4.3 \cdot 10^{-245}:\\ \;\;\;\;\frac{1}{n \cdot x}\\ \mathbf{elif}\;x \leq 0.16:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 9.2 \cdot 10^{+183}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (/ (log x) (- n))))
   (if (<= x 1.45e-266)
     t_0
     (if (<= x 4.3e-245)
       (/ 1.0 (* n x))
       (if (<= x 0.16) t_0 (if (<= x 9.2e+183) (/ (/ 1.0 x) n) 0.0))))))

C

double code(double x, double n) {
	double t_0 = log(x) / -n;
	double tmp;
	if (x <= 1.45e-266) {
		tmp = t_0;
	} else if (x <= 4.3e-245) {
		tmp = 1.0 / (n * x);
	} else if (x <= 0.16) {
		tmp = t_0;
	} else if (x <= 9.2e+183) {
		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) :: t_0
    real(8) :: tmp
    t_0 = log(x) / -n
    if (x <= 1.45d-266) then
        tmp = t_0
    else if (x <= 4.3d-245) then
        tmp = 1.0d0 / (n * x)
    else if (x <= 0.16d0) then
        tmp = t_0
    else if (x <= 9.2d+183) 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 t_0 = Math.log(x) / -n;
	double tmp;
	if (x <= 1.45e-266) {
		tmp = t_0;
	} else if (x <= 4.3e-245) {
		tmp = 1.0 / (n * x);
	} else if (x <= 0.16) {
		tmp = t_0;
	} else if (x <= 9.2e+183) {
		tmp = (1.0 / x) / n;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.log(x) / -n
	tmp = 0
	if x <= 1.45e-266:
		tmp = t_0
	elif x <= 4.3e-245:
		tmp = 1.0 / (n * x)
	elif x <= 0.16:
		tmp = t_0
	elif x <= 9.2e+183:
		tmp = (1.0 / x) / n
	else:
		tmp = 0.0
	return tmp

Julia

function code(x, n)
	t_0 = Float64(log(x) / Float64(-n))
	tmp = 0.0
	if (x <= 1.45e-266)
		tmp = t_0;
	elseif (x <= 4.3e-245)
		tmp = Float64(1.0 / Float64(n * x));
	elseif (x <= 0.16)
		tmp = t_0;
	elseif (x <= 9.2e+183)
		tmp = Float64(Float64(1.0 / x) / n);
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = log(x) / -n;
	tmp = 0.0;
	if (x <= 1.45e-266)
		tmp = t_0;
	elseif (x <= 4.3e-245)
		tmp = 1.0 / (n * x);
	elseif (x <= 0.16)
		tmp = t_0;
	elseif (x <= 9.2e+183)
		tmp = (1.0 / x) / n;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(N[Log[x], $MachinePrecision] / (-n)), $MachinePrecision]}, If[LessEqual[x, 1.45e-266], t$95$0, If[LessEqual[x, 4.3e-245], N[(1.0 / N[(n * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.16], t$95$0, If[LessEqual[x, 9.2e+183], N[(N[(1.0 / x), $MachinePrecision] / n), $MachinePrecision], 0.0]]]]]

Derivation

1. Split input into 4 regimes

2. if x < 1.44999999999999998e-266 or 4.30000000000000003e-245 < x < 0.160000000000000003

   1. Initial program 32.8%

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

   3. Taylor expanded in n around inf 60.1%

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

   4. Taylor expanded in x around 0 59.1%

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

      1. neg-mul-1 59.1%

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

   6. Simplified 59.1%

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

   if 1.44999999999999998e-266 < x < 4.30000000000000003e-245

   1. Initial program 80.6%

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

   3. Taylor expanded in n around inf 6.7%

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

   4. Taylor expanded in x around inf 100.0%

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

      1. *-commutative 100.0%

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

   6. Simplified 100.0%

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

   if 0.160000000000000003 < x < 9.1999999999999992e183

   1. Initial program 49.3%

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

   3. Taylor expanded in n around inf 50.2%

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

   4. Taylor expanded in x around inf 64.4%

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

   if 9.1999999999999992e183 < x

   1. Initial program 83.2%

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

      1. sub-neg 83.2%

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

      2. +-commutative 83.2%

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

      3. sqr-pow 83.2%

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

      4. distribute-rgt-neg-in 83.2%

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

      5. fma-define 82.9%

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

      6. sqrt-pow1 82.9%

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

      7. sqrt-pow1 82.9%

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

      8. pow-to-exp 82.9%

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

      9. un-div-inv 82.9%

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

      10. +-commutative 82.9%

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

      11. log1p-define 82.9%

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

   4. Applied egg-rr 82.9%

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

   5. Taylor expanded in x around inf 83.2%

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

      1. distribute-rgt1-in 83.2%

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

      2. metadata-eval 83.2%

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

      3. mul0-lft 83.2%

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

   7. Simplified 83.2%

      \[\leadsto \color{blue}{0} \]
3. Recombined 4 regimes into one program.

4. Final simplification 65.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.45 \cdot 10^{-266}:\\ \;\;\;\;\frac{\log x}{-n}\\ \mathbf{elif}\;x \leq 4.3 \cdot 10^{-245}:\\ \;\;\;\;\frac{1}{n \cdot x}\\ \mathbf{elif}\;x \leq 0.16:\\ \;\;\;\;\frac{\log x}{-n}\\ \mathbf{elif}\;x \leq 9.2 \cdot 10^{+183}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 82.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -3.1 \lor \neg \left(n \leq 2.6 \cdot 10^{-118}\right):\\ \;\;\;\;\frac{\mathsf{log1p}\left(\frac{1}{x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{{x}^{-3}}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (or (<= n -3.1) (not (<= n 2.6e-118)))
   (/ (log1p (/ 1.0 x)) n)
   (* 0.3333333333333333 (/ (pow x -3.0) n))))

C

double code(double x, double n) {
	double tmp;
	if ((n <= -3.1) || !(n <= 2.6e-118)) {
		tmp = log1p((1.0 / x)) / n;
	} else {
		tmp = 0.3333333333333333 * (pow(x, -3.0) / n);
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double tmp;
	if ((n <= -3.1) || !(n <= 2.6e-118)) {
		tmp = Math.log1p((1.0 / x)) / n;
	} else {
		tmp = 0.3333333333333333 * (Math.pow(x, -3.0) / n);
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if (n <= -3.1) or not (n <= 2.6e-118):
		tmp = math.log1p((1.0 / x)) / n
	else:
		tmp = 0.3333333333333333 * (math.pow(x, -3.0) / n)
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if ((n <= -3.1) || !(n <= 2.6e-118))
		tmp = Float64(log1p(Float64(1.0 / x)) / n);
	else
		tmp = Float64(0.3333333333333333 * Float64((x ^ -3.0) / n));
	end
	return tmp
end

Wolfram

code[x_, n_] := If[Or[LessEqual[n, -3.1], N[Not[LessEqual[n, 2.6e-118]], $MachinePrecision]], N[(N[Log[1 + N[(1.0 / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], N[(0.3333333333333333 * N[(N[Power[x, -3.0], $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < -3.10000000000000009 or 2.6e-118 < n

   1. Initial program 28.4%

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

   3. Taylor expanded in n around inf 68.2%

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

      1. diff-log 68.4%

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

   5. Applied egg-rr 68.4%

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

      1. log1p-expm1-u 68.4%

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

      2. expm1-undefine 68.4%

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

      3. add-exp-log 68.4%

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

   7. Applied egg-rr 68.4%

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

      1. *-lft-identity 68.4%

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

      2. associate-*l/ 67.2%

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

      3. distribute-rgt-in 67.2%

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

      4. *-lft-identity 67.2%

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

      5. rgt-mult-inverse 68.4%

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

      6. associate--l+ 88.4%

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

      7. metadata-eval 88.4%

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

   9. Simplified 88.4%

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

   10. Taylor expanded in n around 0 68.4%

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

       1. log1p-define 88.4%

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

   12. Simplified 88.4%

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

   if -3.10000000000000009 < n < 2.6e-118

   1. Initial program 80.0%

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

   3. Taylor expanded in n around inf 43.7%

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

   4. Taylor expanded in x around inf 25.0%

      \[\leadsto \frac{\color{blue}{\left(0.3333333333333333 \cdot \frac{1}{{x}^{3}} + \frac{1}{x}\right) - 0.5 \cdot \frac{1}{{x}^{2}}}}{n} \]
   5. Step-by-step derivation

      1. sub-neg 25.0%

         \[\leadsto \frac{\color{blue}{\left(0.3333333333333333 \cdot \frac{1}{{x}^{3}} + \frac{1}{x}\right) + \left(-0.5 \cdot \frac{1}{{x}^{2}}\right)}}{n} \]

      2. +-commutative 25.0%

         \[\leadsto \frac{\color{blue}{\left(\frac{1}{x} + 0.3333333333333333 \cdot \frac{1}{{x}^{3}}\right)} + \left(-0.5 \cdot \frac{1}{{x}^{2}}\right)}{n} \]

      3. associate-+l+ 25.0%

         \[\leadsto \frac{\color{blue}{\frac{1}{x} + \left(0.3333333333333333 \cdot \frac{1}{{x}^{3}} + \left(-0.5 \cdot \frac{1}{{x}^{2}}\right)\right)}}{n} \]

      4. associate-*r/ 25.0%

         \[\leadsto \frac{\frac{1}{x} + \left(\color{blue}{\frac{0.3333333333333333 \cdot 1}{{x}^{3}}} + \left(-0.5 \cdot \frac{1}{{x}^{2}}\right)\right)}{n} \]

      5. metadata-eval 25.0%

         \[\leadsto \frac{\frac{1}{x} + \left(\frac{\color{blue}{0.3333333333333333}}{{x}^{3}} + \left(-0.5 \cdot \frac{1}{{x}^{2}}\right)\right)}{n} \]

      6. associate-*r/ 25.0%

         \[\leadsto \frac{\frac{1}{x} + \left(\frac{0.3333333333333333}{{x}^{3}} + \left(-\color{blue}{\frac{0.5 \cdot 1}{{x}^{2}}}\right)\right)}{n} \]

      7. metadata-eval 25.0%

         \[\leadsto \frac{\frac{1}{x} + \left(\frac{0.3333333333333333}{{x}^{3}} + \left(-\frac{\color{blue}{0.5}}{{x}^{2}}\right)\right)}{n} \]

      8. distribute-neg-frac 25.0%

         \[\leadsto \frac{\frac{1}{x} + \left(\frac{0.3333333333333333}{{x}^{3}} + \color{blue}{\frac{-0.5}{{x}^{2}}}\right)}{n} \]

      9. metadata-eval 25.0%

         \[\leadsto \frac{\frac{1}{x} + \left(\frac{0.3333333333333333}{{x}^{3}} + \frac{\color{blue}{-0.5}}{{x}^{2}}\right)}{n} \]

   6. Simplified 25.0%

      \[\leadsto \frac{\color{blue}{\frac{1}{x} + \left(\frac{0.3333333333333333}{{x}^{3}} + \frac{-0.5}{{x}^{2}}\right)}}{n} \]

   7. Taylor expanded in x around 0 71.4%

      \[\leadsto \color{blue}{\frac{0.3333333333333333}{n \cdot {x}^{3}}} \]
   8. Step-by-step derivation

      1. associate-/r* 71.4%

         \[\leadsto \color{blue}{\frac{\frac{0.3333333333333333}{n}}{{x}^{3}}} \]

   9. Simplified 71.4%

      \[\leadsto \color{blue}{\frac{\frac{0.3333333333333333}{n}}{{x}^{3}}} \]
   10. Step-by-step derivation

       1. div-inv 71.4%

          \[\leadsto \color{blue}{\frac{0.3333333333333333}{n} \cdot \frac{1}{{x}^{3}}} \]

       2. div-inv 71.4%

          \[\leadsto \color{blue}{\left(0.3333333333333333 \cdot \frac{1}{n}\right)} \cdot \frac{1}{{x}^{3}} \]

       3. inv-pow 71.4%

          \[\leadsto \left(0.3333333333333333 \cdot \color{blue}{{n}^{-1}}\right) \cdot \frac{1}{{x}^{3}} \]

       4. metadata-eval 71.4%

          \[\leadsto \left(0.3333333333333333 \cdot {n}^{\color{blue}{\left(-0.5 + -0.5\right)}}\right) \cdot \frac{1}{{x}^{3}} \]

       5. pow-prod-up 19.8%

          \[\leadsto \left(0.3333333333333333 \cdot \color{blue}{\left({n}^{-0.5} \cdot {n}^{-0.5}\right)}\right) \cdot \frac{1}{{x}^{3}} \]

       6. metadata-eval 19.8%

          \[\leadsto \left(0.3333333333333333 \cdot \left({n}^{-0.5} \cdot {n}^{-0.5}\right)\right) \cdot \frac{\color{blue}{{1}^{3}}}{{x}^{3}} \]

       7. cube-div 19.8%

          \[\leadsto \left(0.3333333333333333 \cdot \left({n}^{-0.5} \cdot {n}^{-0.5}\right)\right) \cdot \color{blue}{{\left(\frac{1}{x}\right)}^{3}} \]

       8. pow3 19.8%

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

       9. associate-*l* 19.8%

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

       10. pow-prod-up 71.4%

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

       11. metadata-eval 71.4%

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

       12. inv-pow 71.4%

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

       13. pow3 71.4%

           \[\leadsto 0.3333333333333333 \cdot \left(\frac{1}{n} \cdot \color{blue}{{\left(\frac{1}{x}\right)}^{3}}\right) \]

       14. inv-pow 71.4%

           \[\leadsto 0.3333333333333333 \cdot \left(\frac{1}{n} \cdot {\color{blue}{\left({x}^{-1}\right)}}^{3}\right) \]

       15. pow-pow 71.4%

           \[\leadsto 0.3333333333333333 \cdot \left(\frac{1}{n} \cdot \color{blue}{{x}^{\left(-1 \cdot 3\right)}}\right) \]

       16. metadata-eval 71.4%

           \[\leadsto 0.3333333333333333 \cdot \left(\frac{1}{n} \cdot {x}^{\color{blue}{-3}}\right) \]

   11. Applied egg-rr 71.4%

       \[\leadsto \color{blue}{0.3333333333333333 \cdot \left(\frac{1}{n} \cdot {x}^{-3}\right)} \]
   12. Step-by-step derivation

       1. *-commutative 71.4%

          \[\leadsto 0.3333333333333333 \cdot \color{blue}{\left({x}^{-3} \cdot \frac{1}{n}\right)} \]

       2. associate-*r/ 71.4%

          \[\leadsto 0.3333333333333333 \cdot \color{blue}{\frac{{x}^{-3} \cdot 1}{n}} \]

       3. *-commutative 71.4%

          \[\leadsto 0.3333333333333333 \cdot \frac{\color{blue}{1 \cdot {x}^{-3}}}{n} \]

       4. *-lft-identity 71.4%

          \[\leadsto 0.3333333333333333 \cdot \frac{\color{blue}{{x}^{-3}}}{n} \]

   13. Simplified 71.4%

       \[\leadsto \color{blue}{0.3333333333333333 \cdot \frac{{x}^{-3}}{n}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 82.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -3.1 \lor \neg \left(n \leq 2.6 \cdot 10^{-118}\right):\\ \;\;\;\;\frac{\mathsf{log1p}\left(\frac{1}{x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{{x}^{-3}}{n}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 45.9% accurate, 14.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -12.5 \lor \neg \left(n \leq -3.2 \cdot 10^{-295}\right):\\ \;\;\;\;\frac{1}{n \cdot x}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (or (<= n -12.5) (not (<= n -3.2e-295))) (/ 1.0 (* n x)) 0.0))

C

double code(double x, double n) {
	double tmp;
	if ((n <= -12.5) || !(n <= -3.2e-295)) {
		tmp = 1.0 / (n * x);
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if ((n <= (-12.5d0)) .or. (.not. (n <= (-3.2d-295)))) then
        tmp = 1.0d0 / (n * x)
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if ((n <= -12.5) || !(n <= -3.2e-295)) {
		tmp = 1.0 / (n * x);
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if (n <= -12.5) or not (n <= -3.2e-295):
		tmp = 1.0 / (n * x)
	else:
		tmp = 0.0
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if ((n <= -12.5) || !(n <= -3.2e-295))
		tmp = Float64(1.0 / Float64(n * x));
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if ((n <= -12.5) || ~((n <= -3.2e-295)))
		tmp = 1.0 / (n * x);
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[Or[LessEqual[n, -12.5], N[Not[LessEqual[n, -3.2e-295]], $MachinePrecision]], N[(1.0 / N[(n * x), $MachinePrecision]), $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if n < -12.5 or -3.2e-295 < n

   1. Initial program 28.3%

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

   3. Taylor expanded in n around inf 61.1%

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

   4. Taylor expanded in x around inf 44.2%

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

      1. *-commutative 44.2%

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

   6. Simplified 44.2%

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

   if -12.5 < n < -3.2e-295

   1. Initial program 100.0%

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

      1. sub-neg 100.0%

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

      2. +-commutative 100.0%

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

      3. sqr-pow 100.0%

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

      4. distribute-rgt-neg-in 100.0%

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

      5. fma-define 100.0%

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

      6. sqrt-pow1 100.0%

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

      7. sqrt-pow1 100.0%

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

      8. pow-to-exp 100.0%

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

      9. un-div-inv 100.0%

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

      10. +-commutative 100.0%

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

      11. log1p-define 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in x around inf 57.1%

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

      1. distribute-rgt1-in 57.1%

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

      2. metadata-eval 57.1%

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

      3. mul0-lft 57.8%

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

   7. Simplified 57.8%

      \[\leadsto \color{blue}{0} \]
3. Recombined 2 regimes into one program.

4. Final simplification 47.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -12.5 \lor \neg \left(n \leq -3.2 \cdot 10^{-295}\right):\\ \;\;\;\;\frac{1}{n \cdot x}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 46.4% accurate, 14.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -1.9 \lor \neg \left(n \leq -3.3 \cdot 10^{-295}\right):\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (or (<= n -1.9) (not (<= n -3.3e-295))) (/ (/ 1.0 x) n) 0.0))

C

double code(double x, double n) {
	double tmp;
	if ((n <= -1.9) || !(n <= -3.3e-295)) {
		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 ((n <= (-1.9d0)) .or. (.not. (n <= (-3.3d-295)))) 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 ((n <= -1.9) || !(n <= -3.3e-295)) {
		tmp = (1.0 / x) / n;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if (n <= -1.9) or not (n <= -3.3e-295):
		tmp = (1.0 / x) / n
	else:
		tmp = 0.0
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if ((n <= -1.9) || !(n <= -3.3e-295))
		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 ((n <= -1.9) || ~((n <= -3.3e-295)))
		tmp = (1.0 / x) / n;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[Or[LessEqual[n, -1.9], N[Not[LessEqual[n, -3.3e-295]], $MachinePrecision]], N[(N[(1.0 / x), $MachinePrecision] / n), $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if n < -1.8999999999999999 or -3.2999999999999998e-295 < n

   1. Initial program 28.3%

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

   3. Taylor expanded in n around inf 61.1%

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

   4. Taylor expanded in x around inf 44.3%

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

   if -1.8999999999999999 < n < -3.2999999999999998e-295

   1. Initial program 100.0%

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

      1. sub-neg 100.0%

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

      2. +-commutative 100.0%

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

      3. sqr-pow 100.0%

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

      4. distribute-rgt-neg-in 100.0%

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

      5. fma-define 100.0%

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

      6. sqrt-pow1 100.0%

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

      7. sqrt-pow1 100.0%

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

      8. pow-to-exp 100.0%

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

      9. un-div-inv 100.0%

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

      10. +-commutative 100.0%

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

      11. log1p-define 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in x around inf 57.1%

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

      1. distribute-rgt1-in 57.1%

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

      2. metadata-eval 57.1%

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

      3. mul0-lft 57.8%

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

   7. Simplified 57.8%

      \[\leadsto \color{blue}{0} \]
3. Recombined 2 regimes into one program.

4. Final simplification 47.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -1.9 \lor \neg \left(n \leq -3.3 \cdot 10^{-295}\right):\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 30.1% accurate, 211.0× speedup

Math

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

FPCore

(FPCore (x n) :precision binary64 0.0)

C

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

Fortran

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

Java

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

Python

def code(x, n):
	return 0.0

Julia

function code(x, n)
	return 0.0
end

MATLAB

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

Wolfram

code[x_, n_] := 0.0

Derivation

1. Initial program 45.9%

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

   1. sub-neg 45.9%

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

   2. +-commutative 45.9%

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

   3. sqr-pow 45.9%

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

   4. distribute-rgt-neg-in 45.9%

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

   5. fma-define 45.8%

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

   6. sqrt-pow1 45.8%

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

   7. sqrt-pow1 45.8%

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

   8. pow-to-exp 45.8%

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

   9. un-div-inv 45.8%

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

   10. +-commutative 45.8%

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

   11. log1p-define 53.7%

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

4. Applied egg-rr 53.7%

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

5. Taylor expanded in x around inf 27.7%

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

   1. distribute-rgt1-in 27.7%

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

   2. metadata-eval 27.7%

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

   3. mul0-lft 27.9%

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

7. Simplified 27.9%

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

8. Final simplification 27.9%

   \[\leadsto 0 \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024048 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  :precision binary64
  (- (pow (+ x 1.0) (/ 1.0 n)) (pow x (/ 1.0 n))))