2nthrt (problem 3.4.6)

Percentage Accurate: 53.5% → 86.0%

Time: 26.2s

Alternatives: 20

Speedup: 1.7×

• Could not converge on a ground truth

  1. x = -2.534528426288909e+157
  2. n = -1.2385946526059064e-99

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 53.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 86.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -4 \cdot 10^{-14}:\\ \;\;\;\;\frac{1}{\left(n \cdot x\right) \cdot {x}^{\left(\frac{-1}{n}\right)}}\\ \mathbf{elif}\;\frac{1}{n} \leq 4 \cdot 10^{-20}:\\ \;\;\;\;\frac{\log \left(\frac{x}{1 + x}\right)}{-n}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= (/ 1.0 n) -4e-14)
   (/ 1.0 (* (* n x) (pow x (/ -1.0 n))))
   (if (<= (/ 1.0 n) 4e-20)
     (/ (log (/ x (+ 1.0 x))) (- n))
     (- (exp (/ (log1p x) n)) (pow x (/ 1.0 n))))))

C

double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -4e-14) {
		tmp = 1.0 / ((n * x) * pow(x, (-1.0 / n)));
	} else if ((1.0 / n) <= 4e-20) {
		tmp = log((x / (1.0 + x))) / -n;
	} else {
		tmp = exp((log1p(x) / n)) - pow(x, (1.0 / n));
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -4e-14) {
		tmp = 1.0 / ((n * x) * Math.pow(x, (-1.0 / n)));
	} else if ((1.0 / n) <= 4e-20) {
		tmp = Math.log((x / (1.0 + x))) / -n;
	} else {
		tmp = Math.exp((Math.log1p(x) / n)) - Math.pow(x, (1.0 / n));
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if (1.0 / n) <= -4e-14:
		tmp = 1.0 / ((n * x) * math.pow(x, (-1.0 / n)))
	elif (1.0 / n) <= 4e-20:
		tmp = math.log((x / (1.0 + x))) / -n
	else:
		tmp = math.exp((math.log1p(x) / n)) - math.pow(x, (1.0 / n))
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -4e-14)
		tmp = Float64(1.0 / Float64(Float64(n * x) * (x ^ Float64(-1.0 / n))));
	elseif (Float64(1.0 / n) <= 4e-20)
		tmp = Float64(log(Float64(x / Float64(1.0 + x))) / Float64(-n));
	else
		tmp = Float64(exp(Float64(log1p(x) / n)) - (x ^ Float64(1.0 / n)));
	end
	return tmp
end

Wolfram

code[x_, n_] := If[LessEqual[N[(1.0 / n), $MachinePrecision], -4e-14], N[(1.0 / N[(N[(n * x), $MachinePrecision] * N[Power[x, N[(-1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 4e-20], N[(N[Log[N[(x / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-n)), $MachinePrecision], N[(N[Exp[N[(N[Log[1 + x], $MachinePrecision] / n), $MachinePrecision]], $MachinePrecision] - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -4e-14

   1. Initial program 95.5%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. log-rec N/A

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

      3. mul-1-neg N/A

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

      4. associate-*r/ N/A

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

      5. associate-*r* N/A

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

      6. metadata-eval N/A

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

      9. exp-to-pow N/A

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

      10. lower-pow.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 99.9

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

   5. Simplified 99.9%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. clear-num N/A

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

      5. lower-/.f64 N/A

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

      6. div-inv N/A

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

      7. lower-*.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. pow-flip N/A

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

      10. lower-pow.f64 N/A

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

      11. lower-neg.f64 99.9

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

   7. Applied egg-rr 99.9%

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

   if -4e-14 < (/.f64 #s(literal 1 binary64) n) < 3.99999999999999978e-20

   1. Initial program 32.0%

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

   3. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-log1p.f64 N/A

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

      4. lower-log.f64 78.1

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

   5. Simplified 78.1%

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

      1. diff-log N/A

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

      2. clear-num N/A

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

      3. log-rec N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-log.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 78.3

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

   7. Applied egg-rr 78.3%

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

   if 3.99999999999999978e-20 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 58.0%

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

      1. lift-+.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. pow-to-exp N/A

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

      4. lower-exp.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. un-div-inv N/A

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

      7. lower-/.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-log1p.f64 94.7

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

   4. Applied egg-rr 94.7%

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

4. Final simplification 88.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -4 \cdot 10^{-14}:\\ \;\;\;\;\frac{1}{\left(n \cdot x\right) \cdot {x}^{\left(\frac{-1}{n}\right)}}\\ \mathbf{elif}\;\frac{1}{n} \leq 4 \cdot 10^{-20}:\\ \;\;\;\;\frac{\log \left(\frac{x}{1 + x}\right)}{-n}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 78.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ t_1 := {\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - t\_0\\ t_2 := 1 - t\_0\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;\frac{\log \left(\frac{x}{1 + x}\right)}{-n}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n)))
        (t_1 (- (pow (+ 1.0 x) (/ 1.0 n)) t_0))
        (t_2 (- 1.0 t_0)))
   (if (<= t_1 (- INFINITY))
     t_2
     (if (<= t_1 0.0) (/ (log (/ x (+ 1.0 x))) (- n)) t_2))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double t_1 = pow((1.0 + x), (1.0 / n)) - t_0;
	double t_2 = 1.0 - t_0;
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = t_2;
	} else if (t_1 <= 0.0) {
		tmp = log((x / (1.0 + x))) / -n;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, (1.0 / n));
	double t_1 = Math.pow((1.0 + x), (1.0 / n)) - t_0;
	double t_2 = 1.0 - t_0;
	double tmp;
	if (t_1 <= -Double.POSITIVE_INFINITY) {
		tmp = t_2;
	} else if (t_1 <= 0.0) {
		tmp = Math.log((x / (1.0 + x))) / -n;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	t_1 = math.pow((1.0 + x), (1.0 / n)) - t_0
	t_2 = 1.0 - t_0
	tmp = 0
	if t_1 <= -math.inf:
		tmp = t_2
	elif t_1 <= 0.0:
		tmp = math.log((x / (1.0 + x))) / -n
	else:
		tmp = t_2
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	t_1 = Float64((Float64(1.0 + x) ^ Float64(1.0 / n)) - t_0)
	t_2 = Float64(1.0 - t_0)
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = t_2;
	elseif (t_1 <= 0.0)
		tmp = Float64(log(Float64(x / Float64(1.0 + x))) / Float64(-n));
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = x ^ (1.0 / n);
	t_1 = ((1.0 + x) ^ (1.0 / n)) - t_0;
	t_2 = 1.0 - t_0;
	tmp = 0.0;
	if (t_1 <= -Inf)
		tmp = t_2;
	elseif (t_1 <= 0.0)
		tmp = log((x / (1.0 + x))) / -n;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[Power[N[(1.0 + x), $MachinePrecision], N[(1.0 / n), $MachinePrecision]], $MachinePrecision] - t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(1.0 - t$95$0), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], t$95$2, If[LessEqual[t$95$1, 0.0], N[(N[Log[N[(x / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-n)), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (pow.f64 (+.f64 x #s(literal 1 binary64)) (/.f64 #s(literal 1 binary64) n)) (pow.f64 x (/.f64 #s(literal 1 binary64) n))) < -inf.0 or 0.0 < (-.f64 (pow.f64 (+.f64 x #s(literal 1 binary64)) (/.f64 #s(literal 1 binary64) n)) (pow.f64 x (/.f64 #s(literal 1 binary64) n)))

   1. Initial program 81.0%

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

   3. Taylor expanded in x around 0

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

      1. remove-double-neg N/A

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

      2. mul-1-neg N/A

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

      3. distribute-neg-frac N/A

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

      4. mul-1-neg N/A

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

      5. log-rec N/A

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

      6. mul-1-neg N/A

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

      7. lower--.f64 N/A

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

      8. log-rec N/A

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

      9. mul-1-neg N/A

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

      10. associate-*r/ N/A

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

      11. associate-*r* N/A

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

      12. metadata-eval N/A

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

      13. *-commutative N/A

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

      14. associate-/l* N/A

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

      15. exp-to-pow N/A

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

      16. lower-pow.f64 N/A

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

      17. lower-/.f64 79.4

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

   5. Simplified 79.4%

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

   if -inf.0 < (-.f64 (pow.f64 (+.f64 x #s(literal 1 binary64)) (/.f64 #s(literal 1 binary64) n)) (pow.f64 x (/.f64 #s(literal 1 binary64) n))) < 0.0

   1. Initial program 45.7%

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

   3. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-log1p.f64 N/A

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

      4. lower-log.f64 78.6

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

   5. Simplified 78.6%

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

      1. diff-log N/A

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

      2. clear-num N/A

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

      3. log-rec N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-log.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 78.3

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

   7. Applied egg-rr 78.3%

      \[\leadsto \frac{\color{blue}{-\log \left(\frac{x}{x + 1}\right)}}{n} \]
3. Recombined 2 regimes into one program.

4. Final simplification 78.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \leq -\infty:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;{\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \leq 0:\\ \;\;\;\;\frac{\log \left(\frac{x}{1 + x}\right)}{-n}\\ \mathbf{else}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 78.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ t_1 := {\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - t\_0\\ t_2 := 1 - t\_0\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n)))
        (t_1 (- (pow (+ 1.0 x) (/ 1.0 n)) t_0))
        (t_2 (- 1.0 t_0)))
   (if (<= t_1 (- INFINITY))
     t_2
     (if (<= t_1 0.0) (/ (log (/ (+ 1.0 x) x)) n) t_2))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double t_1 = pow((1.0 + x), (1.0 / n)) - t_0;
	double t_2 = 1.0 - t_0;
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = t_2;
	} else if (t_1 <= 0.0) {
		tmp = log(((1.0 + x) / x)) / n;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, (1.0 / n));
	double t_1 = Math.pow((1.0 + x), (1.0 / n)) - t_0;
	double t_2 = 1.0 - t_0;
	double tmp;
	if (t_1 <= -Double.POSITIVE_INFINITY) {
		tmp = t_2;
	} else if (t_1 <= 0.0) {
		tmp = Math.log(((1.0 + x) / x)) / n;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	t_1 = math.pow((1.0 + x), (1.0 / n)) - t_0
	t_2 = 1.0 - t_0
	tmp = 0
	if t_1 <= -math.inf:
		tmp = t_2
	elif t_1 <= 0.0:
		tmp = math.log(((1.0 + x) / x)) / n
	else:
		tmp = t_2
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	t_1 = Float64((Float64(1.0 + x) ^ Float64(1.0 / n)) - t_0)
	t_2 = Float64(1.0 - t_0)
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = t_2;
	elseif (t_1 <= 0.0)
		tmp = Float64(log(Float64(Float64(1.0 + x) / x)) / n);
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = x ^ (1.0 / n);
	t_1 = ((1.0 + x) ^ (1.0 / n)) - t_0;
	t_2 = 1.0 - t_0;
	tmp = 0.0;
	if (t_1 <= -Inf)
		tmp = t_2;
	elseif (t_1 <= 0.0)
		tmp = log(((1.0 + x) / x)) / n;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[Power[N[(1.0 + x), $MachinePrecision], N[(1.0 / n), $MachinePrecision]], $MachinePrecision] - t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(1.0 - t$95$0), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], t$95$2, If[LessEqual[t$95$1, 0.0], N[(N[Log[N[(N[(1.0 + x), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (pow.f64 (+.f64 x #s(literal 1 binary64)) (/.f64 #s(literal 1 binary64) n)) (pow.f64 x (/.f64 #s(literal 1 binary64) n))) < -inf.0 or 0.0 < (-.f64 (pow.f64 (+.f64 x #s(literal 1 binary64)) (/.f64 #s(literal 1 binary64) n)) (pow.f64 x (/.f64 #s(literal 1 binary64) n)))

   1. Initial program 81.0%

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

   3. Taylor expanded in x around 0

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

      1. remove-double-neg N/A

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

      2. mul-1-neg N/A

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

      3. distribute-neg-frac N/A

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

      4. mul-1-neg N/A

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

      5. log-rec N/A

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

      6. mul-1-neg N/A

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

      7. lower--.f64 N/A

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

      8. log-rec N/A

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

      9. mul-1-neg N/A

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

      10. associate-*r/ N/A

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

      11. associate-*r* N/A

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

      12. metadata-eval N/A

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

      13. *-commutative N/A

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

      14. associate-/l* N/A

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

      15. exp-to-pow N/A

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

      16. lower-pow.f64 N/A

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

      17. lower-/.f64 79.4

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

   5. Simplified 79.4%

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

   if -inf.0 < (-.f64 (pow.f64 (+.f64 x #s(literal 1 binary64)) (/.f64 #s(literal 1 binary64) n)) (pow.f64 x (/.f64 #s(literal 1 binary64) n))) < 0.0

   1. Initial program 45.7%

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

   3. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-log1p.f64 N/A

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

      4. lower-log.f64 78.6

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

   5. Simplified 78.6%

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

      1. lift-log1p.f64 N/A

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

      2. lift-log.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-/.f64 78.6

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

      5. lift--.f64 N/A

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

      6. lift-log1p.f64 N/A

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

      7. lift-log.f64 N/A

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

      8. diff-log N/A

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

      9. lower-log.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 78.3

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

   7. Applied egg-rr 78.3%

      \[\leadsto \color{blue}{\frac{\log \left(\frac{x + 1}{x}\right)}{n}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 78.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \leq -\infty:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;{\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \leq 0:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 86.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -4 \cdot 10^{-14}:\\ \;\;\;\;\frac{1}{\left(n \cdot x\right) \cdot {x}^{\left(\frac{-1}{n}\right)}}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-16}:\\ \;\;\;\;\frac{\log \left(\frac{x}{1 + x}\right)}{-n}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{x}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= (/ 1.0 n) -4e-14)
   (/ 1.0 (* (* n x) (pow x (/ -1.0 n))))
   (if (<= (/ 1.0 n) 1e-16)
     (/ (log (/ x (+ 1.0 x))) (- n))
     (- (exp (/ x n)) (pow x (/ 1.0 n))))))

C

double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -4e-14) {
		tmp = 1.0 / ((n * x) * pow(x, (-1.0 / n)));
	} else if ((1.0 / n) <= 1e-16) {
		tmp = log((x / (1.0 + x))) / -n;
	} else {
		tmp = exp((x / n)) - pow(x, (1.0 / n));
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if ((1.0d0 / n) <= (-4d-14)) then
        tmp = 1.0d0 / ((n * x) * (x ** ((-1.0d0) / n)))
    else if ((1.0d0 / n) <= 1d-16) then
        tmp = log((x / (1.0d0 + x))) / -n
    else
        tmp = exp((x / n)) - (x ** (1.0d0 / n))
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -4e-14) {
		tmp = 1.0 / ((n * x) * Math.pow(x, (-1.0 / n)));
	} else if ((1.0 / n) <= 1e-16) {
		tmp = Math.log((x / (1.0 + x))) / -n;
	} else {
		tmp = Math.exp((x / n)) - Math.pow(x, (1.0 / n));
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if (1.0 / n) <= -4e-14:
		tmp = 1.0 / ((n * x) * math.pow(x, (-1.0 / n)))
	elif (1.0 / n) <= 1e-16:
		tmp = math.log((x / (1.0 + x))) / -n
	else:
		tmp = math.exp((x / n)) - math.pow(x, (1.0 / n))
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -4e-14)
		tmp = Float64(1.0 / Float64(Float64(n * x) * (x ^ Float64(-1.0 / n))));
	elseif (Float64(1.0 / n) <= 1e-16)
		tmp = Float64(log(Float64(x / Float64(1.0 + x))) / Float64(-n));
	else
		tmp = Float64(exp(Float64(x / n)) - (x ^ Float64(1.0 / n)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if ((1.0 / n) <= -4e-14)
		tmp = 1.0 / ((n * x) * (x ^ (-1.0 / n)));
	elseif ((1.0 / n) <= 1e-16)
		tmp = log((x / (1.0 + x))) / -n;
	else
		tmp = exp((x / n)) - (x ^ (1.0 / n));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[N[(1.0 / n), $MachinePrecision], -4e-14], N[(1.0 / N[(N[(n * x), $MachinePrecision] * N[Power[x, N[(-1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 1e-16], N[(N[Log[N[(x / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-n)), $MachinePrecision], N[(N[Exp[N[(x / n), $MachinePrecision]], $MachinePrecision] - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -4e-14

   1. Initial program 95.5%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. log-rec N/A

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

      3. mul-1-neg N/A

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

      4. associate-*r/ N/A

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

      5. associate-*r* N/A

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

      6. metadata-eval N/A

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

      9. exp-to-pow N/A

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

      10. lower-pow.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 99.9

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

   5. Simplified 99.9%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. clear-num N/A

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

      5. lower-/.f64 N/A

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

      6. div-inv N/A

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

      7. lower-*.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. pow-flip N/A

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

      10. lower-pow.f64 N/A

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

      11. lower-neg.f64 99.9

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

   7. Applied egg-rr 99.9%

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

   if -4e-14 < (/.f64 #s(literal 1 binary64) n) < 9.9999999999999998e-17

   1. Initial program 31.6%

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

   3. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-log1p.f64 N/A

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

      4. lower-log.f64 77.0

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

   5. Simplified 77.0%

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

      1. diff-log N/A

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

      2. clear-num N/A

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

      3. log-rec N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-log.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 77.2

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

   7. Applied egg-rr 77.2%

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

   if 9.9999999999999998e-17 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 60.5%

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

      1. lift-+.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. pow-to-exp N/A

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

      4. lower-exp.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. un-div-inv N/A

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

      7. lower-/.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-log1p.f64 99.0

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

   4. Applied egg-rr 99.0%

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

   5. Taylor expanded in x around 0

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

      1. lower-/.f64 98.9

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

   7. Simplified 98.9%

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

4. Final simplification 88.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -4 \cdot 10^{-14}:\\ \;\;\;\;\frac{1}{\left(n \cdot x\right) \cdot {x}^{\left(\frac{-1}{n}\right)}}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-16}:\\ \;\;\;\;\frac{\log \left(\frac{x}{1 + x}\right)}{-n}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{x}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 82.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -4 \cdot 10^{-14}:\\ \;\;\;\;\frac{1}{\left(n \cdot x\right) \cdot {x}^{\left(\frac{-1}{n}\right)}}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-16}:\\ \;\;\;\;\frac{\log \left(\frac{x}{1 + x}\right)}{-n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{+223}:\\ \;\;\;\;\left(1 - \frac{\mathsf{fma}\left(-0.5, \frac{x \cdot x}{n}, -x\right)}{n}\right) - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{x}{n}} + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= (/ 1.0 n) -4e-14)
   (/ 1.0 (* (* n x) (pow x (/ -1.0 n))))
   (if (<= (/ 1.0 n) 1e-16)
     (/ (log (/ x (+ 1.0 x))) (- n))
     (if (<= (/ 1.0 n) 2e+223)
       (- (- 1.0 (/ (fma -0.5 (/ (* x x) n) (- x)) n)) (pow x (/ 1.0 n)))
       (+ (exp (/ x n)) -1.0)))))

C

double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -4e-14) {
		tmp = 1.0 / ((n * x) * pow(x, (-1.0 / n)));
	} else if ((1.0 / n) <= 1e-16) {
		tmp = log((x / (1.0 + x))) / -n;
	} else if ((1.0 / n) <= 2e+223) {
		tmp = (1.0 - (fma(-0.5, ((x * x) / n), -x) / n)) - pow(x, (1.0 / n));
	} else {
		tmp = exp((x / n)) + -1.0;
	}
	return tmp;
}

Julia

function code(x, n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -4e-14)
		tmp = Float64(1.0 / Float64(Float64(n * x) * (x ^ Float64(-1.0 / n))));
	elseif (Float64(1.0 / n) <= 1e-16)
		tmp = Float64(log(Float64(x / Float64(1.0 + x))) / Float64(-n));
	elseif (Float64(1.0 / n) <= 2e+223)
		tmp = Float64(Float64(1.0 - Float64(fma(-0.5, Float64(Float64(x * x) / n), Float64(-x)) / n)) - (x ^ Float64(1.0 / n)));
	else
		tmp = Float64(exp(Float64(x / n)) + -1.0);
	end
	return tmp
end

Wolfram

code[x_, n_] := If[LessEqual[N[(1.0 / n), $MachinePrecision], -4e-14], N[(1.0 / N[(N[(n * x), $MachinePrecision] * N[Power[x, N[(-1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 1e-16], N[(N[Log[N[(x / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-n)), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e+223], N[(N[(1.0 - N[(N[(-0.5 * N[(N[(x * x), $MachinePrecision] / n), $MachinePrecision] + (-x)), $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision] - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Exp[N[(x / n), $MachinePrecision]], $MachinePrecision] + -1.0), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -4e-14

   1. Initial program 95.5%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. log-rec N/A

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

      3. mul-1-neg N/A

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

      4. associate-*r/ N/A

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

      5. associate-*r* N/A

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

      6. metadata-eval N/A

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

      9. exp-to-pow N/A

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

      10. lower-pow.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 99.9

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

   5. Simplified 99.9%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. clear-num N/A

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

      5. lower-/.f64 N/A

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

      6. div-inv N/A

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

      7. lower-*.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. pow-flip N/A

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

      10. lower-pow.f64 N/A

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

      11. lower-neg.f64 99.9

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

   7. Applied egg-rr 99.9%

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

   if -4e-14 < (/.f64 #s(literal 1 binary64) n) < 9.9999999999999998e-17

   1. Initial program 31.6%

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

   3. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-log1p.f64 N/A

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

      4. lower-log.f64 77.0

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

   5. Simplified 77.0%

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

      1. diff-log N/A

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

      2. clear-num N/A

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

      3. log-rec N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-log.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 77.2

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

   7. Applied egg-rr 77.2%

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

   if 9.9999999999999998e-17 < (/.f64 #s(literal 1 binary64) n) < 2.00000000000000009e223

   1. Initial program 75.4%

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

      1. lift-+.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. pow-to-exp N/A

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

      4. lower-exp.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. un-div-inv N/A

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

      7. lower-/.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-log1p.f64 98.7

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

   4. Applied egg-rr 98.7%

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

   5. Taylor expanded in x around 0

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

      1. lower-/.f64 98.6

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

   7. Simplified 98.6%

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

   8. Taylor expanded in n around -inf

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. mul-1-neg N/A

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

      11. lower-neg.f64 76.6

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

   10. Simplified 76.6%

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

   if 2.00000000000000009e223 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 14.3%

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

      1. lift-+.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. pow-to-exp N/A

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

      4. lower-exp.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. un-div-inv N/A

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

      7. lower-/.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-log1p.f64 100.0

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in x around 0

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

      1. lower-/.f64 100.0

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

   7. Simplified 100.0%

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

   8. Taylor expanded in n around inf

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

   10. Simplified 82.3%

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

4. Final simplification 84.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -4 \cdot 10^{-14}:\\ \;\;\;\;\frac{1}{\left(n \cdot x\right) \cdot {x}^{\left(\frac{-1}{n}\right)}}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-16}:\\ \;\;\;\;\frac{\log \left(\frac{x}{1 + x}\right)}{-n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{+223}:\\ \;\;\;\;\left(1 - \frac{\mathsf{fma}\left(-0.5, \frac{x \cdot x}{n}, -x\right)}{n}\right) - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{x}{n}} + -1\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 82.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -4 \cdot 10^{-14}:\\ \;\;\;\;\frac{1}{\left(n \cdot x\right) \cdot {x}^{\left(\frac{-1}{n}\right)}}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-16}:\\ \;\;\;\;\frac{\log \left(\frac{x}{1 + x}\right)}{-n}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \frac{1 + \mathsf{fma}\left(x, -0.5, \frac{x}{n} \cdot 0.5\right)}{n}, 1\right) - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= (/ 1.0 n) -4e-14)
   (/ 1.0 (* (* n x) (pow x (/ -1.0 n))))
   (if (<= (/ 1.0 n) 1e-16)
     (/ (log (/ x (+ 1.0 x))) (- n))
     (-
      (fma x (/ (+ 1.0 (fma x -0.5 (* (/ x n) 0.5))) n) 1.0)
      (pow x (/ 1.0 n))))))

C

double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -4e-14) {
		tmp = 1.0 / ((n * x) * pow(x, (-1.0 / n)));
	} else if ((1.0 / n) <= 1e-16) {
		tmp = log((x / (1.0 + x))) / -n;
	} else {
		tmp = fma(x, ((1.0 + fma(x, -0.5, ((x / n) * 0.5))) / n), 1.0) - pow(x, (1.0 / n));
	}
	return tmp;
}

Julia

function code(x, n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -4e-14)
		tmp = Float64(1.0 / Float64(Float64(n * x) * (x ^ Float64(-1.0 / n))));
	elseif (Float64(1.0 / n) <= 1e-16)
		tmp = Float64(log(Float64(x / Float64(1.0 + x))) / Float64(-n));
	else
		tmp = Float64(fma(x, Float64(Float64(1.0 + fma(x, -0.5, Float64(Float64(x / n) * 0.5))) / n), 1.0) - (x ^ Float64(1.0 / n)));
	end
	return tmp
end

Wolfram

code[x_, n_] := If[LessEqual[N[(1.0 / n), $MachinePrecision], -4e-14], N[(1.0 / N[(N[(n * x), $MachinePrecision] * N[Power[x, N[(-1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 1e-16], N[(N[Log[N[(x / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-n)), $MachinePrecision], N[(N[(x * N[(N[(1.0 + N[(x * -0.5 + N[(N[(x / n), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision] + 1.0), $MachinePrecision] - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -4e-14

   1. Initial program 95.5%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. log-rec N/A

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

      3. mul-1-neg N/A

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

      4. associate-*r/ N/A

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

      5. associate-*r* N/A

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

      6. metadata-eval N/A

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

      9. exp-to-pow N/A

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

      10. lower-pow.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 99.9

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

   5. Simplified 99.9%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. clear-num N/A

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

      5. lower-/.f64 N/A

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

      6. div-inv N/A

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

      7. lower-*.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. pow-flip N/A

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

      10. lower-pow.f64 N/A

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

      11. lower-neg.f64 99.9

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

   7. Applied egg-rr 99.9%

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

   if -4e-14 < (/.f64 #s(literal 1 binary64) n) < 9.9999999999999998e-17

   1. Initial program 31.6%

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

   3. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-log1p.f64 N/A

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

      4. lower-log.f64 77.0

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

   5. Simplified 77.0%

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

      1. diff-log N/A

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

      2. clear-num N/A

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

      3. log-rec N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-log.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 77.2

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

   7. Applied egg-rr 77.2%

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

   if 9.9999999999999998e-17 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 60.5%

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

   3. Taylor expanded in x around 0

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

      1. remove-double-neg N/A

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

      2. mul-1-neg N/A

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

      3. distribute-neg-frac N/A

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

      4. mul-1-neg N/A

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

      5. log-rec N/A

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

      6. mul-1-neg N/A

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

      7. lower--.f64 N/A

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

   5. Simplified 70.5%

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

   6. Taylor expanded in n around inf

      \[\leadsto \mathsf{fma}\left(x, \color{blue}{\frac{1 + \left(\frac{-1}{2} \cdot x + \frac{1}{2} \cdot \frac{x}{n}\right)}{n}}, 1\right) - {x}^{\left(\frac{1}{n}\right)} \]
   7. Step-by-step derivation

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 78.1

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

   8. Simplified 78.1%

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

4. Final simplification 84.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -4 \cdot 10^{-14}:\\ \;\;\;\;\frac{1}{\left(n \cdot x\right) \cdot {x}^{\left(\frac{-1}{n}\right)}}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-16}:\\ \;\;\;\;\frac{\log \left(\frac{x}{1 + x}\right)}{-n}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \frac{1 + \mathsf{fma}\left(x, -0.5, \frac{x}{n} \cdot 0.5\right)}{n}, 1\right) - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 82.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -4 \cdot 10^{-14}:\\ \;\;\;\;\frac{1}{\left(n \cdot x\right) \cdot {x}^{\left(\frac{-1}{n}\right)}}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-16}:\\ \;\;\;\;\frac{\log \left(\frac{x}{1 + x}\right)}{-n}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{0.5}{n \cdot n}, \frac{1}{n}\right), 1\right) - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= (/ 1.0 n) -4e-14)
   (/ 1.0 (* (* n x) (pow x (/ -1.0 n))))
   (if (<= (/ 1.0 n) 1e-16)
     (/ (log (/ x (+ 1.0 x))) (- n))
     (- (fma x (fma x (/ 0.5 (* n n)) (/ 1.0 n)) 1.0) (pow x (/ 1.0 n))))))

C

double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -4e-14) {
		tmp = 1.0 / ((n * x) * pow(x, (-1.0 / n)));
	} else if ((1.0 / n) <= 1e-16) {
		tmp = log((x / (1.0 + x))) / -n;
	} else {
		tmp = fma(x, fma(x, (0.5 / (n * n)), (1.0 / n)), 1.0) - pow(x, (1.0 / n));
	}
	return tmp;
}

Julia

function code(x, n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -4e-14)
		tmp = Float64(1.0 / Float64(Float64(n * x) * (x ^ Float64(-1.0 / n))));
	elseif (Float64(1.0 / n) <= 1e-16)
		tmp = Float64(log(Float64(x / Float64(1.0 + x))) / Float64(-n));
	else
		tmp = Float64(fma(x, fma(x, Float64(0.5 / Float64(n * n)), Float64(1.0 / n)), 1.0) - (x ^ Float64(1.0 / n)));
	end
	return tmp
end

Wolfram

code[x_, n_] := If[LessEqual[N[(1.0 / n), $MachinePrecision], -4e-14], N[(1.0 / N[(N[(n * x), $MachinePrecision] * N[Power[x, N[(-1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 1e-16], N[(N[Log[N[(x / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-n)), $MachinePrecision], N[(N[(x * N[(x * N[(0.5 / N[(n * n), $MachinePrecision]), $MachinePrecision] + N[(1.0 / n), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -4e-14

   1. Initial program 95.5%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. log-rec N/A

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

      3. mul-1-neg N/A

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

      4. associate-*r/ N/A

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

      5. associate-*r* N/A

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

      6. metadata-eval N/A

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

      9. exp-to-pow N/A

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

      10. lower-pow.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 99.9

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

   5. Simplified 99.9%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. clear-num N/A

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

      5. lower-/.f64 N/A

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

      6. div-inv N/A

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

      7. lower-*.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. pow-flip N/A

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

      10. lower-pow.f64 N/A

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

      11. lower-neg.f64 99.9

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

   7. Applied egg-rr 99.9%

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

   if -4e-14 < (/.f64 #s(literal 1 binary64) n) < 9.9999999999999998e-17

   1. Initial program 31.6%

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

   3. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-log1p.f64 N/A

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

      4. lower-log.f64 77.0

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

   5. Simplified 77.0%

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

      1. diff-log N/A

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

      2. clear-num N/A

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

      3. log-rec N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-log.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 77.2

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

   7. Applied egg-rr 77.2%

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

   if 9.9999999999999998e-17 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 60.5%

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

   3. Taylor expanded in x around 0

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

      1. remove-double-neg N/A

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

      2. mul-1-neg N/A

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

      3. distribute-neg-frac N/A

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

      4. mul-1-neg N/A

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

      5. log-rec N/A

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

      6. mul-1-neg N/A

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

      7. lower--.f64 N/A

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

   5. Simplified 70.5%

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

   6. Taylor expanded in n around 0

      \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \color{blue}{\frac{\frac{1}{2}}{{n}^{2}}}, \frac{1}{n}\right), 1\right) - {x}^{\left(\frac{1}{n}\right)} \]
   7. Step-by-step derivation

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 70.5

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

   8. Simplified 70.5%

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

4. Final simplification 83.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -4 \cdot 10^{-14}:\\ \;\;\;\;\frac{1}{\left(n \cdot x\right) \cdot {x}^{\left(\frac{-1}{n}\right)}}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-16}:\\ \;\;\;\;\frac{\log \left(\frac{x}{1 + x}\right)}{-n}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{0.5}{n \cdot n}, \frac{1}{n}\right), 1\right) - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 82.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -4 \cdot 10^{-14}:\\ \;\;\;\;\frac{1}{\left(n \cdot x\right) \cdot {x}^{\left(\frac{-1}{n}\right)}}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-16}:\\ \;\;\;\;\frac{\log \left(\frac{x}{1 + x}\right)}{-n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{+202}:\\ \;\;\;\;\frac{n + x}{n} - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{x}{n}} + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= (/ 1.0 n) -4e-14)
   (/ 1.0 (* (* n x) (pow x (/ -1.0 n))))
   (if (<= (/ 1.0 n) 1e-16)
     (/ (log (/ x (+ 1.0 x))) (- n))
     (if (<= (/ 1.0 n) 5e+202)
       (- (/ (+ n x) n) (pow x (/ 1.0 n)))
       (+ (exp (/ x n)) -1.0)))))

C

double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -4e-14) {
		tmp = 1.0 / ((n * x) * pow(x, (-1.0 / n)));
	} else if ((1.0 / n) <= 1e-16) {
		tmp = log((x / (1.0 + x))) / -n;
	} else if ((1.0 / n) <= 5e+202) {
		tmp = ((n + x) / n) - pow(x, (1.0 / n));
	} else {
		tmp = exp((x / n)) + -1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if ((1.0d0 / n) <= (-4d-14)) then
        tmp = 1.0d0 / ((n * x) * (x ** ((-1.0d0) / n)))
    else if ((1.0d0 / n) <= 1d-16) then
        tmp = log((x / (1.0d0 + x))) / -n
    else if ((1.0d0 / n) <= 5d+202) then
        tmp = ((n + x) / n) - (x ** (1.0d0 / n))
    else
        tmp = exp((x / n)) + (-1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -4e-14) {
		tmp = 1.0 / ((n * x) * Math.pow(x, (-1.0 / n)));
	} else if ((1.0 / n) <= 1e-16) {
		tmp = Math.log((x / (1.0 + x))) / -n;
	} else if ((1.0 / n) <= 5e+202) {
		tmp = ((n + x) / n) - Math.pow(x, (1.0 / n));
	} else {
		tmp = Math.exp((x / n)) + -1.0;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if (1.0 / n) <= -4e-14:
		tmp = 1.0 / ((n * x) * math.pow(x, (-1.0 / n)))
	elif (1.0 / n) <= 1e-16:
		tmp = math.log((x / (1.0 + x))) / -n
	elif (1.0 / n) <= 5e+202:
		tmp = ((n + x) / n) - math.pow(x, (1.0 / n))
	else:
		tmp = math.exp((x / n)) + -1.0
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -4e-14)
		tmp = Float64(1.0 / Float64(Float64(n * x) * (x ^ Float64(-1.0 / n))));
	elseif (Float64(1.0 / n) <= 1e-16)
		tmp = Float64(log(Float64(x / Float64(1.0 + x))) / Float64(-n));
	elseif (Float64(1.0 / n) <= 5e+202)
		tmp = Float64(Float64(Float64(n + x) / n) - (x ^ Float64(1.0 / n)));
	else
		tmp = Float64(exp(Float64(x / n)) + -1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if ((1.0 / n) <= -4e-14)
		tmp = 1.0 / ((n * x) * (x ^ (-1.0 / n)));
	elseif ((1.0 / n) <= 1e-16)
		tmp = log((x / (1.0 + x))) / -n;
	elseif ((1.0 / n) <= 5e+202)
		tmp = ((n + x) / n) - (x ^ (1.0 / n));
	else
		tmp = exp((x / n)) + -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[N[(1.0 / n), $MachinePrecision], -4e-14], N[(1.0 / N[(N[(n * x), $MachinePrecision] * N[Power[x, N[(-1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 1e-16], N[(N[Log[N[(x / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-n)), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e+202], N[(N[(N[(n + x), $MachinePrecision] / n), $MachinePrecision] - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Exp[N[(x / n), $MachinePrecision]], $MachinePrecision] + -1.0), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -4e-14

   1. Initial program 95.5%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. log-rec N/A

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

      3. mul-1-neg N/A

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

      4. associate-*r/ N/A

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

      5. associate-*r* N/A

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

      6. metadata-eval N/A

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

      9. exp-to-pow N/A

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

      10. lower-pow.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 99.9

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

   5. Simplified 99.9%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. clear-num N/A

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

      5. lower-/.f64 N/A

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

      6. div-inv N/A

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

      7. lower-*.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. pow-flip N/A

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

      10. lower-pow.f64 N/A

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

      11. lower-neg.f64 99.9

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

   7. Applied egg-rr 99.9%

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

   if -4e-14 < (/.f64 #s(literal 1 binary64) n) < 9.9999999999999998e-17

   1. Initial program 31.6%

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

   3. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-log1p.f64 N/A

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

      4. lower-log.f64 77.0

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

   5. Simplified 77.0%

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

      1. diff-log N/A

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

      2. clear-num N/A

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

      3. log-rec N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-log.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 77.2

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

   7. Applied egg-rr 77.2%

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

   if 9.9999999999999998e-17 < (/.f64 #s(literal 1 binary64) n) < 4.9999999999999999e202

   1. Initial program 77.6%

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

   3. Taylor expanded in x around 0

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

      1. remove-double-neg N/A

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

      2. mul-1-neg N/A

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

      3. distribute-neg-frac N/A

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

      4. mul-1-neg N/A

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

      5. log-rec N/A

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

      6. mul-1-neg N/A

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

      7. lower--.f64 N/A

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

   5. Simplified 68.3%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 74.0

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

   8. Simplified 74.0%

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

   9. Taylor expanded in n around 0

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

       1. lower-/.f64 N/A

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

       2. +-commutative N/A

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

       3. lower-+.f64 74.0

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

   11. Simplified 74.0%

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

   if 4.9999999999999999e202 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 19.3%

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

      1. lift-+.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. pow-to-exp N/A

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

      4. lower-exp.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. un-div-inv N/A

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

      7. lower-/.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-log1p.f64 100.0

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in x around 0

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

      1. lower-/.f64 100.0

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

   7. Simplified 100.0%

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

   8. Taylor expanded in n around inf

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

   10. Simplified 77.7%

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

4. Final simplification 84.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -4 \cdot 10^{-14}:\\ \;\;\;\;\frac{1}{\left(n \cdot x\right) \cdot {x}^{\left(\frac{-1}{n}\right)}}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-16}:\\ \;\;\;\;\frac{\log \left(\frac{x}{1 + x}\right)}{-n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{+202}:\\ \;\;\;\;\frac{n + x}{n} - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{x}{n}} + -1\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 82.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -4 \cdot 10^{-14}:\\ \;\;\;\;\frac{\frac{t\_0}{n}}{x}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-16}:\\ \;\;\;\;\frac{\log \left(\frac{x}{1 + x}\right)}{-n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{+202}:\\ \;\;\;\;\frac{n + x}{n} - t\_0\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{x}{n}} + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -4e-14)
     (/ (/ t_0 n) x)
     (if (<= (/ 1.0 n) 1e-16)
       (/ (log (/ x (+ 1.0 x))) (- n))
       (if (<= (/ 1.0 n) 5e+202)
         (- (/ (+ n x) n) t_0)
         (+ (exp (/ x n)) -1.0))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -4e-14) {
		tmp = (t_0 / n) / x;
	} else if ((1.0 / n) <= 1e-16) {
		tmp = log((x / (1.0 + x))) / -n;
	} else if ((1.0 / n) <= 5e+202) {
		tmp = ((n + x) / n) - t_0;
	} else {
		tmp = exp((x / n)) + -1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x ** (1.0d0 / n)
    if ((1.0d0 / n) <= (-4d-14)) then
        tmp = (t_0 / n) / x
    else if ((1.0d0 / n) <= 1d-16) then
        tmp = log((x / (1.0d0 + x))) / -n
    else if ((1.0d0 / n) <= 5d+202) then
        tmp = ((n + x) / n) - t_0
    else
        tmp = exp((x / n)) + (-1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -4e-14) {
		tmp = (t_0 / n) / x;
	} else if ((1.0 / n) <= 1e-16) {
		tmp = Math.log((x / (1.0 + x))) / -n;
	} else if ((1.0 / n) <= 5e+202) {
		tmp = ((n + x) / n) - t_0;
	} else {
		tmp = Math.exp((x / n)) + -1.0;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	tmp = 0
	if (1.0 / n) <= -4e-14:
		tmp = (t_0 / n) / x
	elif (1.0 / n) <= 1e-16:
		tmp = math.log((x / (1.0 + x))) / -n
	elif (1.0 / n) <= 5e+202:
		tmp = ((n + x) / n) - t_0
	else:
		tmp = math.exp((x / n)) + -1.0
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -4e-14)
		tmp = Float64(Float64(t_0 / n) / x);
	elseif (Float64(1.0 / n) <= 1e-16)
		tmp = Float64(log(Float64(x / Float64(1.0 + x))) / Float64(-n));
	elseif (Float64(1.0 / n) <= 5e+202)
		tmp = Float64(Float64(Float64(n + x) / n) - t_0);
	else
		tmp = Float64(exp(Float64(x / n)) + -1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = x ^ (1.0 / n);
	tmp = 0.0;
	if ((1.0 / n) <= -4e-14)
		tmp = (t_0 / n) / x;
	elseif ((1.0 / n) <= 1e-16)
		tmp = log((x / (1.0 + x))) / -n;
	elseif ((1.0 / n) <= 5e+202)
		tmp = ((n + x) / n) - t_0;
	else
		tmp = exp((x / n)) + -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -4e-14], N[(N[(t$95$0 / n), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 1e-16], N[(N[Log[N[(x / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-n)), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e+202], N[(N[(N[(n + x), $MachinePrecision] / n), $MachinePrecision] - t$95$0), $MachinePrecision], N[(N[Exp[N[(x / n), $MachinePrecision]], $MachinePrecision] + -1.0), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -4e-14

   1. Initial program 95.5%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. log-rec N/A

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

      3. mul-1-neg N/A

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

      4. associate-*r/ N/A

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

      5. associate-*r* N/A

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

      6. metadata-eval N/A

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

      9. exp-to-pow N/A

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

      10. lower-pow.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 99.9

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

   5. Simplified 99.9%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 99.9

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

   7. Applied egg-rr 99.9%

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

   if -4e-14 < (/.f64 #s(literal 1 binary64) n) < 9.9999999999999998e-17

   1. Initial program 31.6%

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

   3. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-log1p.f64 N/A

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

      4. lower-log.f64 77.0

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

   5. Simplified 77.0%

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

      1. diff-log N/A

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

      2. clear-num N/A

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

      3. log-rec N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-log.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 77.2

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

   7. Applied egg-rr 77.2%

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

   if 9.9999999999999998e-17 < (/.f64 #s(literal 1 binary64) n) < 4.9999999999999999e202

   1. Initial program 77.6%

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

   3. Taylor expanded in x around 0

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

      1. remove-double-neg N/A

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

      2. mul-1-neg N/A

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

      3. distribute-neg-frac N/A

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

      4. mul-1-neg N/A

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

      5. log-rec N/A

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

      6. mul-1-neg N/A

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

      7. lower--.f64 N/A

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

   5. Simplified 68.3%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 74.0

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

   8. Simplified 74.0%

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

   9. Taylor expanded in n around 0

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

       1. lower-/.f64 N/A

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

       2. +-commutative N/A

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

       3. lower-+.f64 74.0

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

   11. Simplified 74.0%

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

   if 4.9999999999999999e202 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 19.3%

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

      1. lift-+.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. pow-to-exp N/A

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

      4. lower-exp.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. un-div-inv N/A

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

      7. lower-/.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-log1p.f64 100.0

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in x around 0

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

      1. lower-/.f64 100.0

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

   7. Simplified 100.0%

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

   8. Taylor expanded in n around inf

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

   10. Simplified 77.7%

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

4. Final simplification 84.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -4 \cdot 10^{-14}:\\ \;\;\;\;\frac{\frac{{x}^{\left(\frac{1}{n}\right)}}{n}}{x}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-16}:\\ \;\;\;\;\frac{\log \left(\frac{x}{1 + x}\right)}{-n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{+202}:\\ \;\;\;\;\frac{n + x}{n} - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{x}{n}} + -1\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 82.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -4 \cdot 10^{-14}:\\ \;\;\;\;\frac{t\_0}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-16}:\\ \;\;\;\;\frac{\log \left(\frac{x}{1 + x}\right)}{-n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{+202}:\\ \;\;\;\;\frac{n + x}{n} - t\_0\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{x}{n}} + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -4e-14)
     (/ t_0 (* n x))
     (if (<= (/ 1.0 n) 1e-16)
       (/ (log (/ x (+ 1.0 x))) (- n))
       (if (<= (/ 1.0 n) 5e+202)
         (- (/ (+ n x) n) t_0)
         (+ (exp (/ x n)) -1.0))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -4e-14) {
		tmp = t_0 / (n * x);
	} else if ((1.0 / n) <= 1e-16) {
		tmp = log((x / (1.0 + x))) / -n;
	} else if ((1.0 / n) <= 5e+202) {
		tmp = ((n + x) / n) - t_0;
	} else {
		tmp = exp((x / n)) + -1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x ** (1.0d0 / n)
    if ((1.0d0 / n) <= (-4d-14)) then
        tmp = t_0 / (n * x)
    else if ((1.0d0 / n) <= 1d-16) then
        tmp = log((x / (1.0d0 + x))) / -n
    else if ((1.0d0 / n) <= 5d+202) then
        tmp = ((n + x) / n) - t_0
    else
        tmp = exp((x / n)) + (-1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -4e-14) {
		tmp = t_0 / (n * x);
	} else if ((1.0 / n) <= 1e-16) {
		tmp = Math.log((x / (1.0 + x))) / -n;
	} else if ((1.0 / n) <= 5e+202) {
		tmp = ((n + x) / n) - t_0;
	} else {
		tmp = Math.exp((x / n)) + -1.0;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	tmp = 0
	if (1.0 / n) <= -4e-14:
		tmp = t_0 / (n * x)
	elif (1.0 / n) <= 1e-16:
		tmp = math.log((x / (1.0 + x))) / -n
	elif (1.0 / n) <= 5e+202:
		tmp = ((n + x) / n) - t_0
	else:
		tmp = math.exp((x / n)) + -1.0
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -4e-14)
		tmp = Float64(t_0 / Float64(n * x));
	elseif (Float64(1.0 / n) <= 1e-16)
		tmp = Float64(log(Float64(x / Float64(1.0 + x))) / Float64(-n));
	elseif (Float64(1.0 / n) <= 5e+202)
		tmp = Float64(Float64(Float64(n + x) / n) - t_0);
	else
		tmp = Float64(exp(Float64(x / n)) + -1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = x ^ (1.0 / n);
	tmp = 0.0;
	if ((1.0 / n) <= -4e-14)
		tmp = t_0 / (n * x);
	elseif ((1.0 / n) <= 1e-16)
		tmp = log((x / (1.0 + x))) / -n;
	elseif ((1.0 / n) <= 5e+202)
		tmp = ((n + x) / n) - t_0;
	else
		tmp = exp((x / n)) + -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -4e-14], N[(t$95$0 / N[(n * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 1e-16], N[(N[Log[N[(x / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-n)), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e+202], N[(N[(N[(n + x), $MachinePrecision] / n), $MachinePrecision] - t$95$0), $MachinePrecision], N[(N[Exp[N[(x / n), $MachinePrecision]], $MachinePrecision] + -1.0), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -4e-14

   1. Initial program 95.5%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. log-rec N/A

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

      3. mul-1-neg N/A

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

      4. associate-*r/ N/A

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

      5. associate-*r* N/A

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

      6. metadata-eval N/A

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

      9. exp-to-pow N/A

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

      10. lower-pow.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 99.9

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

   5. Simplified 99.9%

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

   if -4e-14 < (/.f64 #s(literal 1 binary64) n) < 9.9999999999999998e-17

   1. Initial program 31.6%

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

   3. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-log1p.f64 N/A

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

      4. lower-log.f64 77.0

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

   5. Simplified 77.0%

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

      1. diff-log N/A

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

      2. clear-num N/A

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

      3. log-rec N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-log.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 77.2

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

   7. Applied egg-rr 77.2%

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

   if 9.9999999999999998e-17 < (/.f64 #s(literal 1 binary64) n) < 4.9999999999999999e202

   1. Initial program 77.6%

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

   3. Taylor expanded in x around 0

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

      1. remove-double-neg N/A

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

      2. mul-1-neg N/A

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

      3. distribute-neg-frac N/A

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

      4. mul-1-neg N/A

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

      5. log-rec N/A

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

      6. mul-1-neg N/A

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

      7. lower--.f64 N/A

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

   5. Simplified 68.3%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 74.0

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

   8. Simplified 74.0%

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

   9. Taylor expanded in n around 0

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

       1. lower-/.f64 N/A

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

       2. +-commutative N/A

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

       3. lower-+.f64 74.0

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

   11. Simplified 74.0%

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

   if 4.9999999999999999e202 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 19.3%

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

      1. lift-+.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. pow-to-exp N/A

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

      4. lower-exp.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. un-div-inv N/A

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

      7. lower-/.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-log1p.f64 100.0

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in x around 0

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

      1. lower-/.f64 100.0

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

   7. Simplified 100.0%

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

   8. Taylor expanded in n around inf

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

   10. Simplified 77.7%

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

4. Final simplification 84.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -4 \cdot 10^{-14}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-16}:\\ \;\;\;\;\frac{\log \left(\frac{x}{1 + x}\right)}{-n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{+202}:\\ \;\;\;\;\frac{n + x}{n} - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{x}{n}} + -1\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 82.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -4 \cdot 10^{-14}:\\ \;\;\;\;\frac{t\_0}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-16}:\\ \;\;\;\;\frac{\log \left(\frac{x}{1 + x}\right)}{-n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{+202}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - t\_0\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{x}{n}} + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -4e-14)
     (/ t_0 (* n x))
     (if (<= (/ 1.0 n) 1e-16)
       (/ (log (/ x (+ 1.0 x))) (- n))
       (if (<= (/ 1.0 n) 5e+202)
         (- (+ 1.0 (/ x n)) t_0)
         (+ (exp (/ x n)) -1.0))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -4e-14) {
		tmp = t_0 / (n * x);
	} else if ((1.0 / n) <= 1e-16) {
		tmp = log((x / (1.0 + x))) / -n;
	} else if ((1.0 / n) <= 5e+202) {
		tmp = (1.0 + (x / n)) - t_0;
	} else {
		tmp = exp((x / n)) + -1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x ** (1.0d0 / n)
    if ((1.0d0 / n) <= (-4d-14)) then
        tmp = t_0 / (n * x)
    else if ((1.0d0 / n) <= 1d-16) then
        tmp = log((x / (1.0d0 + x))) / -n
    else if ((1.0d0 / n) <= 5d+202) then
        tmp = (1.0d0 + (x / n)) - t_0
    else
        tmp = exp((x / n)) + (-1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -4e-14) {
		tmp = t_0 / (n * x);
	} else if ((1.0 / n) <= 1e-16) {
		tmp = Math.log((x / (1.0 + x))) / -n;
	} else if ((1.0 / n) <= 5e+202) {
		tmp = (1.0 + (x / n)) - t_0;
	} else {
		tmp = Math.exp((x / n)) + -1.0;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	tmp = 0
	if (1.0 / n) <= -4e-14:
		tmp = t_0 / (n * x)
	elif (1.0 / n) <= 1e-16:
		tmp = math.log((x / (1.0 + x))) / -n
	elif (1.0 / n) <= 5e+202:
		tmp = (1.0 + (x / n)) - t_0
	else:
		tmp = math.exp((x / n)) + -1.0
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -4e-14)
		tmp = Float64(t_0 / Float64(n * x));
	elseif (Float64(1.0 / n) <= 1e-16)
		tmp = Float64(log(Float64(x / Float64(1.0 + x))) / Float64(-n));
	elseif (Float64(1.0 / n) <= 5e+202)
		tmp = Float64(Float64(1.0 + Float64(x / n)) - t_0);
	else
		tmp = Float64(exp(Float64(x / n)) + -1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = x ^ (1.0 / n);
	tmp = 0.0;
	if ((1.0 / n) <= -4e-14)
		tmp = t_0 / (n * x);
	elseif ((1.0 / n) <= 1e-16)
		tmp = log((x / (1.0 + x))) / -n;
	elseif ((1.0 / n) <= 5e+202)
		tmp = (1.0 + (x / n)) - t_0;
	else
		tmp = exp((x / n)) + -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -4e-14], N[(t$95$0 / N[(n * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 1e-16], N[(N[Log[N[(x / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-n)), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e+202], N[(N[(1.0 + N[(x / n), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], N[(N[Exp[N[(x / n), $MachinePrecision]], $MachinePrecision] + -1.0), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -4e-14

   1. Initial program 95.5%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. log-rec N/A

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

      3. mul-1-neg N/A

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

      4. associate-*r/ N/A

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

      5. associate-*r* N/A

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

      6. metadata-eval N/A

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

      9. exp-to-pow N/A

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

      10. lower-pow.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 99.9

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

   5. Simplified 99.9%

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

   if -4e-14 < (/.f64 #s(literal 1 binary64) n) < 9.9999999999999998e-17

   1. Initial program 31.6%

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

   3. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-log1p.f64 N/A

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

      4. lower-log.f64 77.0

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

   5. Simplified 77.0%

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

      1. diff-log N/A

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

      2. clear-num N/A

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

      3. log-rec N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-log.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 77.2

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

   7. Applied egg-rr 77.2%

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

   if 9.9999999999999998e-17 < (/.f64 #s(literal 1 binary64) n) < 4.9999999999999999e202

   1. Initial program 77.6%

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

   3. Taylor expanded in x around 0

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

      1. *-rgt-identity N/A

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

      2. associate-*r/ N/A

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

      3. lower-+.f64 N/A

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

      4. associate-*r/ N/A

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

      5. *-rgt-identity N/A

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

      6. lower-/.f64 74.0

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

   5. Simplified 74.0%

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

   if 4.9999999999999999e202 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 19.3%

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

      1. lift-+.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. pow-to-exp N/A

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

      4. lower-exp.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. un-div-inv N/A

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

      7. lower-/.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-log1p.f64 100.0

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in x around 0

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

      1. lower-/.f64 100.0

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

   7. Simplified 100.0%

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

   8. Taylor expanded in n around inf

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

   10. Simplified 77.7%

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

4. Final simplification 84.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -4 \cdot 10^{-14}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-16}:\\ \;\;\;\;\frac{\log \left(\frac{x}{1 + x}\right)}{-n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{+202}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{x}{n}} + -1\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 82.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -4 \cdot 10^{-14}:\\ \;\;\;\;\frac{t\_0}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-16}:\\ \;\;\;\;\frac{\log \left(\frac{x}{1 + x}\right)}{-n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{+165}:\\ \;\;\;\;1 - t\_0\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{x}{n}} + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -4e-14)
     (/ t_0 (* n x))
     (if (<= (/ 1.0 n) 1e-16)
       (/ (log (/ x (+ 1.0 x))) (- n))
       (if (<= (/ 1.0 n) 2e+165) (- 1.0 t_0) (+ (exp (/ x n)) -1.0))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -4e-14) {
		tmp = t_0 / (n * x);
	} else if ((1.0 / n) <= 1e-16) {
		tmp = log((x / (1.0 + x))) / -n;
	} else if ((1.0 / n) <= 2e+165) {
		tmp = 1.0 - t_0;
	} else {
		tmp = exp((x / n)) + -1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x ** (1.0d0 / n)
    if ((1.0d0 / n) <= (-4d-14)) then
        tmp = t_0 / (n * x)
    else if ((1.0d0 / n) <= 1d-16) then
        tmp = log((x / (1.0d0 + x))) / -n
    else if ((1.0d0 / n) <= 2d+165) then
        tmp = 1.0d0 - t_0
    else
        tmp = exp((x / n)) + (-1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -4e-14) {
		tmp = t_0 / (n * x);
	} else if ((1.0 / n) <= 1e-16) {
		tmp = Math.log((x / (1.0 + x))) / -n;
	} else if ((1.0 / n) <= 2e+165) {
		tmp = 1.0 - t_0;
	} else {
		tmp = Math.exp((x / n)) + -1.0;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	tmp = 0
	if (1.0 / n) <= -4e-14:
		tmp = t_0 / (n * x)
	elif (1.0 / n) <= 1e-16:
		tmp = math.log((x / (1.0 + x))) / -n
	elif (1.0 / n) <= 2e+165:
		tmp = 1.0 - t_0
	else:
		tmp = math.exp((x / n)) + -1.0
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -4e-14)
		tmp = Float64(t_0 / Float64(n * x));
	elseif (Float64(1.0 / n) <= 1e-16)
		tmp = Float64(log(Float64(x / Float64(1.0 + x))) / Float64(-n));
	elseif (Float64(1.0 / n) <= 2e+165)
		tmp = Float64(1.0 - t_0);
	else
		tmp = Float64(exp(Float64(x / n)) + -1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = x ^ (1.0 / n);
	tmp = 0.0;
	if ((1.0 / n) <= -4e-14)
		tmp = t_0 / (n * x);
	elseif ((1.0 / n) <= 1e-16)
		tmp = log((x / (1.0 + x))) / -n;
	elseif ((1.0 / n) <= 2e+165)
		tmp = 1.0 - t_0;
	else
		tmp = exp((x / n)) + -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -4e-14], N[(t$95$0 / N[(n * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 1e-16], N[(N[Log[N[(x / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-n)), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e+165], N[(1.0 - t$95$0), $MachinePrecision], N[(N[Exp[N[(x / n), $MachinePrecision]], $MachinePrecision] + -1.0), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -4e-14

   1. Initial program 95.5%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. log-rec N/A

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

      3. mul-1-neg N/A

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

      4. associate-*r/ N/A

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

      5. associate-*r* N/A

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

      6. metadata-eval N/A

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

      9. exp-to-pow N/A

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

      10. lower-pow.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 99.9

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

   5. Simplified 99.9%

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

   if -4e-14 < (/.f64 #s(literal 1 binary64) n) < 9.9999999999999998e-17

   1. Initial program 31.6%

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

   3. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-log1p.f64 N/A

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

      4. lower-log.f64 77.0

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

   5. Simplified 77.0%

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

      1. diff-log N/A

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

      2. clear-num N/A

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

      3. log-rec N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-log.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 77.2

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

   7. Applied egg-rr 77.2%

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

   if 9.9999999999999998e-17 < (/.f64 #s(literal 1 binary64) n) < 1.9999999999999998e165

   1. Initial program 81.7%

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

   3. Taylor expanded in x around 0

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

      1. remove-double-neg N/A

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

      2. mul-1-neg N/A

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

      3. distribute-neg-frac N/A

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

      4. mul-1-neg N/A

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

      5. log-rec N/A

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

      6. mul-1-neg N/A

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

      7. lower--.f64 N/A

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

      8. log-rec N/A

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

      9. mul-1-neg N/A

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

      10. associate-*r/ N/A

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

      11. associate-*r* N/A

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

      12. metadata-eval N/A

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

      13. *-commutative N/A

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

      14. associate-/l* N/A

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

      15. exp-to-pow N/A

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

      16. lower-pow.f64 N/A

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

      17. lower-/.f64 76.6

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

   5. Simplified 76.6%

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

   if 1.9999999999999998e165 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 27.4%

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

      1. lift-+.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. pow-to-exp N/A

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

      4. lower-exp.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. un-div-inv N/A

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

      7. lower-/.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-log1p.f64 100.0

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in x around 0

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

      1. lower-/.f64 100.0

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

   7. Simplified 100.0%

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

   8. Taylor expanded in n around inf

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

   10. Simplified 71.2%

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

4. Final simplification 84.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -4 \cdot 10^{-14}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-16}:\\ \;\;\;\;\frac{\log \left(\frac{x}{1 + x}\right)}{-n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{+165}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{x}{n}} + -1\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 57.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 5 \cdot 10^{-235}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;x \leq 3.3 \cdot 10^{-174}:\\ \;\;\;\;-\frac{\log x}{n}\\ \mathbf{elif}\;x \leq 8 \cdot 10^{-61}:\\ \;\;\;\;\frac{1}{n \cdot x} + \frac{\frac{0.3333333333333333}{n \cdot x} + \frac{-0.5}{n}}{x \cdot x}\\ \mathbf{elif}\;x \leq 0.88:\\ \;\;\;\;\frac{1}{\frac{n}{x - \log x}}\\ \mathbf{elif}\;x \leq 6.4 \cdot 10^{+81}:\\ \;\;\;\;\frac{\frac{1 + \left(\frac{-0.5 + \frac{0.3333333333333333}{x}}{x} - \frac{0.25}{x \cdot \left(x \cdot x\right)}\right)}{n}}{x}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 5e-235)
   (- 1.0 (pow x (/ 1.0 n)))
   (if (<= x 3.3e-174)
     (- (/ (log x) n))
     (if (<= x 8e-61)
       (+
        (/ 1.0 (* n x))
        (/ (+ (/ 0.3333333333333333 (* n x)) (/ -0.5 n)) (* x x)))
       (if (<= x 0.88)
         (/ 1.0 (/ n (- x (log x))))
         (if (<= x 6.4e+81)
           (/
            (/
             (+
              1.0
              (-
               (/ (+ -0.5 (/ 0.3333333333333333 x)) x)
               (/ 0.25 (* x (* x x)))))
             n)
            x)
           0.0))))))

C

double code(double x, double n) {
	double tmp;
	if (x <= 5e-235) {
		tmp = 1.0 - pow(x, (1.0 / n));
	} else if (x <= 3.3e-174) {
		tmp = -(log(x) / n);
	} else if (x <= 8e-61) {
		tmp = (1.0 / (n * x)) + (((0.3333333333333333 / (n * x)) + (-0.5 / n)) / (x * x));
	} else if (x <= 0.88) {
		tmp = 1.0 / (n / (x - log(x)));
	} else if (x <= 6.4e+81) {
		tmp = ((1.0 + (((-0.5 + (0.3333333333333333 / x)) / x) - (0.25 / (x * (x * x))))) / 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 (x <= 5d-235) then
        tmp = 1.0d0 - (x ** (1.0d0 / n))
    else if (x <= 3.3d-174) then
        tmp = -(log(x) / n)
    else if (x <= 8d-61) then
        tmp = (1.0d0 / (n * x)) + (((0.3333333333333333d0 / (n * x)) + ((-0.5d0) / n)) / (x * x))
    else if (x <= 0.88d0) then
        tmp = 1.0d0 / (n / (x - log(x)))
    else if (x <= 6.4d+81) then
        tmp = ((1.0d0 + ((((-0.5d0) + (0.3333333333333333d0 / x)) / x) - (0.25d0 / (x * (x * x))))) / n) / x
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (x <= 5e-235) {
		tmp = 1.0 - Math.pow(x, (1.0 / n));
	} else if (x <= 3.3e-174) {
		tmp = -(Math.log(x) / n);
	} else if (x <= 8e-61) {
		tmp = (1.0 / (n * x)) + (((0.3333333333333333 / (n * x)) + (-0.5 / n)) / (x * x));
	} else if (x <= 0.88) {
		tmp = 1.0 / (n / (x - Math.log(x)));
	} else if (x <= 6.4e+81) {
		tmp = ((1.0 + (((-0.5 + (0.3333333333333333 / x)) / x) - (0.25 / (x * (x * x))))) / n) / x;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 5e-235:
		tmp = 1.0 - math.pow(x, (1.0 / n))
	elif x <= 3.3e-174:
		tmp = -(math.log(x) / n)
	elif x <= 8e-61:
		tmp = (1.0 / (n * x)) + (((0.3333333333333333 / (n * x)) + (-0.5 / n)) / (x * x))
	elif x <= 0.88:
		tmp = 1.0 / (n / (x - math.log(x)))
	elif x <= 6.4e+81:
		tmp = ((1.0 + (((-0.5 + (0.3333333333333333 / x)) / x) - (0.25 / (x * (x * x))))) / n) / x
	else:
		tmp = 0.0
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 5e-235)
		tmp = Float64(1.0 - (x ^ Float64(1.0 / n)));
	elseif (x <= 3.3e-174)
		tmp = Float64(-Float64(log(x) / n));
	elseif (x <= 8e-61)
		tmp = Float64(Float64(1.0 / Float64(n * x)) + Float64(Float64(Float64(0.3333333333333333 / Float64(n * x)) + Float64(-0.5 / n)) / Float64(x * x)));
	elseif (x <= 0.88)
		tmp = Float64(1.0 / Float64(n / Float64(x - log(x))));
	elseif (x <= 6.4e+81)
		tmp = Float64(Float64(Float64(1.0 + Float64(Float64(Float64(-0.5 + Float64(0.3333333333333333 / x)) / x) - Float64(0.25 / Float64(x * Float64(x * x))))) / n) / x);
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (x <= 5e-235)
		tmp = 1.0 - (x ^ (1.0 / n));
	elseif (x <= 3.3e-174)
		tmp = -(log(x) / n);
	elseif (x <= 8e-61)
		tmp = (1.0 / (n * x)) + (((0.3333333333333333 / (n * x)) + (-0.5 / n)) / (x * x));
	elseif (x <= 0.88)
		tmp = 1.0 / (n / (x - log(x)));
	elseif (x <= 6.4e+81)
		tmp = ((1.0 + (((-0.5 + (0.3333333333333333 / x)) / x) - (0.25 / (x * (x * x))))) / n) / x;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 5e-235], N[(1.0 - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 3.3e-174], (-N[(N[Log[x], $MachinePrecision] / n), $MachinePrecision]), If[LessEqual[x, 8e-61], N[(N[(1.0 / N[(n * x), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(0.3333333333333333 / N[(n * x), $MachinePrecision]), $MachinePrecision] + N[(-0.5 / n), $MachinePrecision]), $MachinePrecision] / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.88], N[(1.0 / N[(n / N[(x - N[Log[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 6.4e+81], N[(N[(N[(1.0 + N[(N[(N[(-0.5 + N[(0.3333333333333333 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] - N[(0.25 / N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision] / x), $MachinePrecision], 0.0]]]]]

Derivation

1. Split input into 6 regimes

2. if x < 4.9999999999999998e-235

   1. Initial program 67.6%

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

   3. Taylor expanded in x around 0

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

      1. remove-double-neg N/A

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

      2. mul-1-neg N/A

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

      3. distribute-neg-frac N/A

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

      4. mul-1-neg N/A

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

      5. log-rec N/A

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

      6. mul-1-neg N/A

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

      7. lower--.f64 N/A

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

      8. log-rec N/A

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

      9. mul-1-neg N/A

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

      10. associate-*r/ N/A

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

      11. associate-*r* N/A

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

      12. metadata-eval N/A

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

      13. *-commutative N/A

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

      14. associate-/l* N/A

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

      15. exp-to-pow N/A

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

      16. lower-pow.f64 N/A

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

      17. lower-/.f64 67.6

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

   5. Simplified 67.6%

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

   if 4.9999999999999998e-235 < x < 3.3000000000000001e-174

   1. Initial program 25.4%

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

   3. Taylor expanded in x around 0

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

      1. remove-double-neg N/A

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

      2. mul-1-neg N/A

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

      3. distribute-neg-frac N/A

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

      4. mul-1-neg N/A

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

      5. log-rec N/A

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

      6. mul-1-neg N/A

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

      7. lower--.f64 N/A

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

      8. log-rec N/A

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

      9. mul-1-neg N/A

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

      10. associate-*r/ N/A

          \[\leadsto 1 - e^{\color{blue}{\frac{-1 \cdot \left(-1 \cdot \log x\right)}{n}}} \]

      11. associate-*r* N/A

          \[\leadsto 1 - e^{\frac{\color{blue}{\left(-1 \cdot -1\right) \cdot \log x}}{n}} \]

      12. metadata-eval N/A

          \[\leadsto 1 - e^{\frac{\color{blue}{1} \cdot \log x}{n}} \]

      13. *-commutative N/A

          \[\leadsto 1 - e^{\frac{\color{blue}{\log x \cdot 1}}{n}} \]

      14. associate-/l* N/A

          \[\leadsto 1 - e^{\color{blue}{\log x \cdot \frac{1}{n}}} \]

      15. exp-to-pow N/A

          \[\leadsto 1 - \color{blue}{{x}^{\left(\frac{1}{n}\right)}} \]

      16. lower-pow.f64 N/A

          \[\leadsto 1 - \color{blue}{{x}^{\left(\frac{1}{n}\right)}} \]

      17. lower-/.f64 25.4

          \[\leadsto 1 - {x}^{\color{blue}{\left(\frac{1}{n}\right)}} \]

   5. Simplified 25.4%

      \[\leadsto \color{blue}{1 - {x}^{\left(\frac{1}{n}\right)}} \]

   6. Taylor expanded in n around inf

      \[\leadsto \color{blue}{-1 \cdot \frac{\log x}{n}} \]
   7. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{\log x}{n}\right)} \]

      2. lower-neg.f64 N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{\log x}{n}\right)} \]

      3. lower-/.f64 N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\frac{\log x}{n}}\right) \]

      4. lower-log.f64 80.1

         \[\leadsto -\frac{\color{blue}{\log x}}{n} \]

   8. Simplified 80.1%

      \[\leadsto \color{blue}{-\frac{\log x}{n}} \]

   if 3.3000000000000001e-174 < x < 8.0000000000000003e-61

   1. Initial program 54.0%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in n around inf

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]

      2. lower--.f64 N/A

         \[\leadsto \frac{\color{blue}{\log \left(1 + x\right) - \log x}}{n} \]

      3. lower-log1p.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(x\right)} - \log x}{n} \]

      4. lower-log.f64 36.5

         \[\leadsto \frac{\mathsf{log1p}\left(x\right) - \color{blue}{\log x}}{n} \]

   5. Simplified 36.5%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) - \log x}{n}} \]

   6. Taylor expanded in x around -inf

      \[\leadsto \color{blue}{-1 \cdot \frac{-1 \cdot \frac{\frac{1}{3} \cdot \frac{1}{n \cdot x} - \frac{1}{2} \cdot \frac{1}{n}}{x} - \frac{1}{n}}{x}} \]
   7. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{-1 \cdot \frac{\frac{1}{3} \cdot \frac{1}{n \cdot x} - \frac{1}{2} \cdot \frac{1}{n}}{x} - \frac{1}{n}}{x}\right)} \]

      2. lower-neg.f64 N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{-1 \cdot \frac{\frac{1}{3} \cdot \frac{1}{n \cdot x} - \frac{1}{2} \cdot \frac{1}{n}}{x} - \frac{1}{n}}{x}\right)} \]

      3. div-sub N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\left(\frac{-1 \cdot \frac{\frac{1}{3} \cdot \frac{1}{n \cdot x} - \frac{1}{2} \cdot \frac{1}{n}}{x}}{x} - \frac{\frac{1}{n}}{x}\right)}\right) \]

      4. associate-/r* N/A

         \[\leadsto \mathsf{neg}\left(\left(\frac{-1 \cdot \frac{\frac{1}{3} \cdot \frac{1}{n \cdot x} - \frac{1}{2} \cdot \frac{1}{n}}{x}}{x} - \color{blue}{\frac{1}{n \cdot x}}\right)\right) \]

      5. lower--.f64 N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\left(\frac{-1 \cdot \frac{\frac{1}{3} \cdot \frac{1}{n \cdot x} - \frac{1}{2} \cdot \frac{1}{n}}{x}}{x} - \frac{1}{n \cdot x}\right)}\right) \]

   8. Simplified 57.4%

      \[\leadsto \color{blue}{-\left(\frac{-\left(\frac{0.3333333333333333}{x \cdot n} + \frac{-0.5}{n}\right)}{x \cdot x} - \frac{1}{x \cdot n}\right)} \]

   if 8.0000000000000003e-61 < x < 0.880000000000000004

   1. Initial program 38.8%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(1 + x \cdot \left(x \cdot \left(\frac{1}{2} \cdot \frac{1}{{n}^{2}} - \frac{1}{2} \cdot \frac{1}{n}\right) + \frac{1}{n}\right)\right) - e^{\frac{\log x}{n}}} \]
   4. Step-by-step derivation

      1. remove-double-neg N/A

         \[\leadsto \left(1 + x \cdot \left(x \cdot \left(\frac{1}{2} \cdot \frac{1}{{n}^{2}} - \frac{1}{2} \cdot \frac{1}{n}\right) + \frac{1}{n}\right)\right) - e^{\frac{\color{blue}{\mathsf{neg}\left(\left(\mathsf{neg}\left(\log x\right)\right)\right)}}{n}} \]

      2. mul-1-neg N/A

         \[\leadsto \left(1 + x \cdot \left(x \cdot \left(\frac{1}{2} \cdot \frac{1}{{n}^{2}} - \frac{1}{2} \cdot \frac{1}{n}\right) + \frac{1}{n}\right)\right) - e^{\frac{\mathsf{neg}\left(\color{blue}{-1 \cdot \log x}\right)}{n}} \]

      3. distribute-neg-frac N/A

         \[\leadsto \left(1 + x \cdot \left(x \cdot \left(\frac{1}{2} \cdot \frac{1}{{n}^{2}} - \frac{1}{2} \cdot \frac{1}{n}\right) + \frac{1}{n}\right)\right) - e^{\color{blue}{\mathsf{neg}\left(\frac{-1 \cdot \log x}{n}\right)}} \]

      4. mul-1-neg N/A

         \[\leadsto \left(1 + x \cdot \left(x \cdot \left(\frac{1}{2} \cdot \frac{1}{{n}^{2}} - \frac{1}{2} \cdot \frac{1}{n}\right) + \frac{1}{n}\right)\right) - e^{\mathsf{neg}\left(\frac{\color{blue}{\mathsf{neg}\left(\log x\right)}}{n}\right)} \]

      5. log-rec N/A

         \[\leadsto \left(1 + x \cdot \left(x \cdot \left(\frac{1}{2} \cdot \frac{1}{{n}^{2}} - \frac{1}{2} \cdot \frac{1}{n}\right) + \frac{1}{n}\right)\right) - e^{\mathsf{neg}\left(\frac{\color{blue}{\log \left(\frac{1}{x}\right)}}{n}\right)} \]

      6. mul-1-neg N/A

         \[\leadsto \left(1 + x \cdot \left(x \cdot \left(\frac{1}{2} \cdot \frac{1}{{n}^{2}} - \frac{1}{2} \cdot \frac{1}{n}\right) + \frac{1}{n}\right)\right) - e^{\color{blue}{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}} \]

      7. lower--.f64 N/A

         \[\leadsto \color{blue}{\left(1 + x \cdot \left(x \cdot \left(\frac{1}{2} \cdot \frac{1}{{n}^{2}} - \frac{1}{2} \cdot \frac{1}{n}\right) + \frac{1}{n}\right)\right) - e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}} \]

   5. Simplified 25.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{0.5}{n \cdot n} + \frac{-0.5}{n}, \frac{1}{n}\right), 1\right) - {x}^{\left(\frac{1}{n}\right)}} \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(1 + \frac{x}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{x}{n} + 1\right)} - {x}^{\left(\frac{1}{n}\right)} \]

      2. lower-+.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{x}{n} + 1\right)} - {x}^{\left(\frac{1}{n}\right)} \]

      3. lower-/.f64 35.3

         \[\leadsto \left(\color{blue}{\frac{x}{n}} + 1\right) - {x}^{\left(\frac{1}{n}\right)} \]

   8. Simplified 35.3%

      \[\leadsto \color{blue}{\left(\frac{x}{n} + 1\right)} - {x}^{\left(\frac{1}{n}\right)} \]

   9. Taylor expanded in n around inf

      \[\leadsto \color{blue}{\frac{x - \log x}{n}} \]
   10. Step-by-step derivation

       1. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{x - \log x}{n}} \]

       2. lower--.f64 N/A

          \[\leadsto \frac{\color{blue}{x - \log x}}{n} \]

       3. lower-log.f64 48.1

          \[\leadsto \frac{x - \color{blue}{\log x}}{n} \]

   11. Simplified 48.1%

       \[\leadsto \color{blue}{\frac{x - \log x}{n}} \]
   12. Step-by-step derivation

       1. lift-log.f64 N/A

          \[\leadsto \frac{x - \color{blue}{\log x}}{n} \]

       2. lift--.f64 N/A

          \[\leadsto \frac{\color{blue}{x - \log x}}{n} \]

       3. clear-num N/A

          \[\leadsto \color{blue}{\frac{1}{\frac{n}{x - \log x}}} \]

       4. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{1}{\frac{n}{x - \log x}}} \]

       5. lower-/.f64 48.1

          \[\leadsto \frac{1}{\color{blue}{\frac{n}{x - \log x}}} \]

   13. Applied egg-rr 48.1%

       \[\leadsto \color{blue}{\frac{1}{\frac{n}{x - \log x}}} \]

   if 0.880000000000000004 < x < 6.4e81

   1. Initial program 39.0%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{\frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}}{n} + \left(\frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}} \cdot \left(\frac{1}{2} \cdot \frac{1}{{n}^{2}} - \frac{1}{2} \cdot \frac{1}{n}\right)}{x} + \left(\frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}} \cdot \left(\left(\frac{1}{24} \cdot \frac{1}{{n}^{4}} + \frac{11}{24} \cdot \frac{1}{{n}^{2}}\right) - \left(\frac{1}{4} \cdot \frac{1}{n} + \frac{1}{4} \cdot \frac{1}{{n}^{3}}\right)\right)}{{x}^{3}} + \frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}} \cdot \left(\left(\frac{1}{6} \cdot \frac{1}{{n}^{3}} + \frac{1}{3} \cdot \frac{1}{n}\right) - \frac{1}{2} \cdot \frac{1}{{n}^{2}}\right)}{{x}^{2}}\right)\right)}{x}} \]

   5. Simplified 79.8%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left({x}^{\left(\frac{1}{n}\right)}, \left(\frac{0.5}{x \cdot \left(n \cdot n\right)} - \frac{0.5}{x \cdot n}\right) + \left(\frac{\frac{0.16666666666666666}{n \cdot \left(n \cdot n\right)} + \left(\frac{0.3333333333333333}{n} + \frac{-0.5}{n \cdot n}\right)}{x \cdot x} + \frac{\frac{0.041666666666666664}{{n}^{4}} + \left(\left(\frac{0.4583333333333333}{n \cdot n} + \frac{-0.25}{n}\right) + \frac{-0.25}{n \cdot \left(n \cdot n\right)}\right)}{x \cdot \left(x \cdot x\right)}\right), \frac{{x}^{\left(\frac{1}{n}\right)}}{n}\right)}{x}} \]

   6. Taylor expanded in n around inf

      \[\leadsto \frac{\color{blue}{\frac{\left(1 + \frac{1}{3} \cdot \frac{1}{{x}^{2}}\right) - \left(\frac{1}{4} \cdot \frac{1}{{x}^{3}} + \frac{1}{2} \cdot \frac{1}{x}\right)}{n}}}{x} \]
   7. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{\left(1 + \frac{1}{3} \cdot \frac{1}{{x}^{2}}\right) - \left(\frac{1}{4} \cdot \frac{1}{{x}^{3}} + \frac{1}{2} \cdot \frac{1}{x}\right)}{n}}}{x} \]

   8. Simplified 70.0%

      \[\leadsto \frac{\color{blue}{\frac{1 + \left(\frac{-0.5 + \frac{0.3333333333333333}{x}}{x} - \frac{0.25}{x \cdot \left(x \cdot x\right)}\right)}{n}}}{x} \]

   if 6.4e81 < x

   1. Initial program 77.6%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1 - e^{\frac{\log x}{n}}} \]
   4. Step-by-step derivation

      1. remove-double-neg N/A

         \[\leadsto 1 - e^{\frac{\color{blue}{\mathsf{neg}\left(\left(\mathsf{neg}\left(\log x\right)\right)\right)}}{n}} \]

      2. mul-1-neg N/A

         \[\leadsto 1 - e^{\frac{\mathsf{neg}\left(\color{blue}{-1 \cdot \log x}\right)}{n}} \]

      3. distribute-neg-frac N/A

         \[\leadsto 1 - e^{\color{blue}{\mathsf{neg}\left(\frac{-1 \cdot \log x}{n}\right)}} \]

      4. mul-1-neg N/A

         \[\leadsto 1 - e^{\mathsf{neg}\left(\frac{\color{blue}{\mathsf{neg}\left(\log x\right)}}{n}\right)} \]

      5. log-rec N/A

         \[\leadsto 1 - e^{\mathsf{neg}\left(\frac{\color{blue}{\log \left(\frac{1}{x}\right)}}{n}\right)} \]

      6. mul-1-neg N/A

         \[\leadsto 1 - e^{\color{blue}{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}} \]

      7. lower--.f64 N/A

         \[\leadsto \color{blue}{1 - e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}} \]

      8. log-rec N/A

         \[\leadsto 1 - e^{-1 \cdot \frac{\color{blue}{\mathsf{neg}\left(\log x\right)}}{n}} \]

      9. mul-1-neg N/A

         \[\leadsto 1 - e^{-1 \cdot \frac{\color{blue}{-1 \cdot \log x}}{n}} \]

      10. associate-*r/ N/A

          \[\leadsto 1 - e^{\color{blue}{\frac{-1 \cdot \left(-1 \cdot \log x\right)}{n}}} \]

      11. associate-*r* N/A

          \[\leadsto 1 - e^{\frac{\color{blue}{\left(-1 \cdot -1\right) \cdot \log x}}{n}} \]

      12. metadata-eval N/A

          \[\leadsto 1 - e^{\frac{\color{blue}{1} \cdot \log x}{n}} \]

      13. *-commutative N/A

          \[\leadsto 1 - e^{\frac{\color{blue}{\log x \cdot 1}}{n}} \]

      14. associate-/l* N/A

          \[\leadsto 1 - e^{\color{blue}{\log x \cdot \frac{1}{n}}} \]

      15. exp-to-pow N/A

          \[\leadsto 1 - \color{blue}{{x}^{\left(\frac{1}{n}\right)}} \]

      16. lower-pow.f64 N/A

          \[\leadsto 1 - \color{blue}{{x}^{\left(\frac{1}{n}\right)}} \]

      17. lower-/.f64 42.8

          \[\leadsto 1 - {x}^{\color{blue}{\left(\frac{1}{n}\right)}} \]

   5. Simplified 42.8%

      \[\leadsto \color{blue}{1 - {x}^{\left(\frac{1}{n}\right)}} \]

   6. Taylor expanded in n around inf

      \[\leadsto 1 - \color{blue}{1} \]
   7. Step-by-step derivation

   8. Simplified 77.6%

      \[\leadsto 1 - \color{blue}{1} \]
   9. Step-by-step derivation

      1. metadata-eval 77.6

         \[\leadsto \color{blue}{0} \]

   10. Applied egg-rr 77.6%

       \[\leadsto \color{blue}{0} \]
3. Recombined 6 regimes into one program.

4. Final simplification 68.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 5 \cdot 10^{-235}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;x \leq 3.3 \cdot 10^{-174}:\\ \;\;\;\;-\frac{\log x}{n}\\ \mathbf{elif}\;x \leq 8 \cdot 10^{-61}:\\ \;\;\;\;\frac{1}{n \cdot x} + \frac{\frac{0.3333333333333333}{n \cdot x} + \frac{-0.5}{n}}{x \cdot x}\\ \mathbf{elif}\;x \leq 0.88:\\ \;\;\;\;\frac{1}{\frac{n}{x - \log x}}\\ \mathbf{elif}\;x \leq 6.4 \cdot 10^{+81}:\\ \;\;\;\;\frac{\frac{1 + \left(\frac{-0.5 + \frac{0.3333333333333333}{x}}{x} - \frac{0.25}{x \cdot \left(x \cdot x\right)}\right)}{n}}{x}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 57.5% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 4.8 \cdot 10^{-235}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;x \leq 5.2 \cdot 10^{-174}:\\ \;\;\;\;-\frac{\log x}{n}\\ \mathbf{elif}\;x \leq 7.5 \cdot 10^{-61}:\\ \;\;\;\;\frac{1}{n \cdot x} + \frac{\frac{0.3333333333333333}{n \cdot x} + \frac{-0.5}{n}}{x \cdot x}\\ \mathbf{elif}\;x \leq 0.88:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{elif}\;x \leq 3.7 \cdot 10^{+82}:\\ \;\;\;\;\frac{\frac{1 + \left(\frac{-0.5 + \frac{0.3333333333333333}{x}}{x} - \frac{0.25}{x \cdot \left(x \cdot x\right)}\right)}{n}}{x}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 4.8e-235)
   (- 1.0 (pow x (/ 1.0 n)))
   (if (<= x 5.2e-174)
     (- (/ (log x) n))
     (if (<= x 7.5e-61)
       (+
        (/ 1.0 (* n x))
        (/ (+ (/ 0.3333333333333333 (* n x)) (/ -0.5 n)) (* x x)))
       (if (<= x 0.88)
         (/ (- x (log x)) n)
         (if (<= x 3.7e+82)
           (/
            (/
             (+
              1.0
              (-
               (/ (+ -0.5 (/ 0.3333333333333333 x)) x)
               (/ 0.25 (* x (* x x)))))
             n)
            x)
           0.0))))))

C

double code(double x, double n) {
	double tmp;
	if (x <= 4.8e-235) {
		tmp = 1.0 - pow(x, (1.0 / n));
	} else if (x <= 5.2e-174) {
		tmp = -(log(x) / n);
	} else if (x <= 7.5e-61) {
		tmp = (1.0 / (n * x)) + (((0.3333333333333333 / (n * x)) + (-0.5 / n)) / (x * x));
	} else if (x <= 0.88) {
		tmp = (x - log(x)) / n;
	} else if (x <= 3.7e+82) {
		tmp = ((1.0 + (((-0.5 + (0.3333333333333333 / x)) / x) - (0.25 / (x * (x * x))))) / 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 (x <= 4.8d-235) then
        tmp = 1.0d0 - (x ** (1.0d0 / n))
    else if (x <= 5.2d-174) then
        tmp = -(log(x) / n)
    else if (x <= 7.5d-61) then
        tmp = (1.0d0 / (n * x)) + (((0.3333333333333333d0 / (n * x)) + ((-0.5d0) / n)) / (x * x))
    else if (x <= 0.88d0) then
        tmp = (x - log(x)) / n
    else if (x <= 3.7d+82) then
        tmp = ((1.0d0 + ((((-0.5d0) + (0.3333333333333333d0 / x)) / x) - (0.25d0 / (x * (x * x))))) / n) / x
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (x <= 4.8e-235) {
		tmp = 1.0 - Math.pow(x, (1.0 / n));
	} else if (x <= 5.2e-174) {
		tmp = -(Math.log(x) / n);
	} else if (x <= 7.5e-61) {
		tmp = (1.0 / (n * x)) + (((0.3333333333333333 / (n * x)) + (-0.5 / n)) / (x * x));
	} else if (x <= 0.88) {
		tmp = (x - Math.log(x)) / n;
	} else if (x <= 3.7e+82) {
		tmp = ((1.0 + (((-0.5 + (0.3333333333333333 / x)) / x) - (0.25 / (x * (x * x))))) / n) / x;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 4.8e-235:
		tmp = 1.0 - math.pow(x, (1.0 / n))
	elif x <= 5.2e-174:
		tmp = -(math.log(x) / n)
	elif x <= 7.5e-61:
		tmp = (1.0 / (n * x)) + (((0.3333333333333333 / (n * x)) + (-0.5 / n)) / (x * x))
	elif x <= 0.88:
		tmp = (x - math.log(x)) / n
	elif x <= 3.7e+82:
		tmp = ((1.0 + (((-0.5 + (0.3333333333333333 / x)) / x) - (0.25 / (x * (x * x))))) / n) / x
	else:
		tmp = 0.0
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 4.8e-235)
		tmp = Float64(1.0 - (x ^ Float64(1.0 / n)));
	elseif (x <= 5.2e-174)
		tmp = Float64(-Float64(log(x) / n));
	elseif (x <= 7.5e-61)
		tmp = Float64(Float64(1.0 / Float64(n * x)) + Float64(Float64(Float64(0.3333333333333333 / Float64(n * x)) + Float64(-0.5 / n)) / Float64(x * x)));
	elseif (x <= 0.88)
		tmp = Float64(Float64(x - log(x)) / n);
	elseif (x <= 3.7e+82)
		tmp = Float64(Float64(Float64(1.0 + Float64(Float64(Float64(-0.5 + Float64(0.3333333333333333 / x)) / x) - Float64(0.25 / Float64(x * Float64(x * x))))) / n) / x);
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (x <= 4.8e-235)
		tmp = 1.0 - (x ^ (1.0 / n));
	elseif (x <= 5.2e-174)
		tmp = -(log(x) / n);
	elseif (x <= 7.5e-61)
		tmp = (1.0 / (n * x)) + (((0.3333333333333333 / (n * x)) + (-0.5 / n)) / (x * x));
	elseif (x <= 0.88)
		tmp = (x - log(x)) / n;
	elseif (x <= 3.7e+82)
		tmp = ((1.0 + (((-0.5 + (0.3333333333333333 / x)) / x) - (0.25 / (x * (x * x))))) / n) / x;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 4.8e-235], N[(1.0 - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 5.2e-174], (-N[(N[Log[x], $MachinePrecision] / n), $MachinePrecision]), If[LessEqual[x, 7.5e-61], N[(N[(1.0 / N[(n * x), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(0.3333333333333333 / N[(n * x), $MachinePrecision]), $MachinePrecision] + N[(-0.5 / n), $MachinePrecision]), $MachinePrecision] / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.88], N[(N[(x - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[x, 3.7e+82], N[(N[(N[(1.0 + N[(N[(N[(-0.5 + N[(0.3333333333333333 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] - N[(0.25 / N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision] / x), $MachinePrecision], 0.0]]]]]

Derivation

1. Split input into 6 regimes

2. if x < 4.80000000000000022e-235

   1. Initial program 67.6%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1 - e^{\frac{\log x}{n}}} \]
   4. Step-by-step derivation

      1. remove-double-neg N/A

         \[\leadsto 1 - e^{\frac{\color{blue}{\mathsf{neg}\left(\left(\mathsf{neg}\left(\log x\right)\right)\right)}}{n}} \]

      2. mul-1-neg N/A

         \[\leadsto 1 - e^{\frac{\mathsf{neg}\left(\color{blue}{-1 \cdot \log x}\right)}{n}} \]

      3. distribute-neg-frac N/A

         \[\leadsto 1 - e^{\color{blue}{\mathsf{neg}\left(\frac{-1 \cdot \log x}{n}\right)}} \]

      4. mul-1-neg N/A

         \[\leadsto 1 - e^{\mathsf{neg}\left(\frac{\color{blue}{\mathsf{neg}\left(\log x\right)}}{n}\right)} \]

      5. log-rec N/A

         \[\leadsto 1 - e^{\mathsf{neg}\left(\frac{\color{blue}{\log \left(\frac{1}{x}\right)}}{n}\right)} \]

      6. mul-1-neg N/A

         \[\leadsto 1 - e^{\color{blue}{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}} \]

      7. lower--.f64 N/A

         \[\leadsto \color{blue}{1 - e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}} \]

      8. log-rec N/A

         \[\leadsto 1 - e^{-1 \cdot \frac{\color{blue}{\mathsf{neg}\left(\log x\right)}}{n}} \]

      9. mul-1-neg N/A

         \[\leadsto 1 - e^{-1 \cdot \frac{\color{blue}{-1 \cdot \log x}}{n}} \]

      10. associate-*r/ N/A

          \[\leadsto 1 - e^{\color{blue}{\frac{-1 \cdot \left(-1 \cdot \log x\right)}{n}}} \]

      11. associate-*r* N/A

          \[\leadsto 1 - e^{\frac{\color{blue}{\left(-1 \cdot -1\right) \cdot \log x}}{n}} \]

      12. metadata-eval N/A

          \[\leadsto 1 - e^{\frac{\color{blue}{1} \cdot \log x}{n}} \]

      13. *-commutative N/A

          \[\leadsto 1 - e^{\frac{\color{blue}{\log x \cdot 1}}{n}} \]

      14. associate-/l* N/A

          \[\leadsto 1 - e^{\color{blue}{\log x \cdot \frac{1}{n}}} \]

      15. exp-to-pow N/A

          \[\leadsto 1 - \color{blue}{{x}^{\left(\frac{1}{n}\right)}} \]

      16. lower-pow.f64 N/A

          \[\leadsto 1 - \color{blue}{{x}^{\left(\frac{1}{n}\right)}} \]

      17. lower-/.f64 67.6

          \[\leadsto 1 - {x}^{\color{blue}{\left(\frac{1}{n}\right)}} \]

   5. Simplified 67.6%

      \[\leadsto \color{blue}{1 - {x}^{\left(\frac{1}{n}\right)}} \]

   if 4.80000000000000022e-235 < x < 5.2000000000000004e-174

   1. Initial program 25.4%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1 - e^{\frac{\log x}{n}}} \]
   4. Step-by-step derivation

      1. remove-double-neg N/A

         \[\leadsto 1 - e^{\frac{\color{blue}{\mathsf{neg}\left(\left(\mathsf{neg}\left(\log x\right)\right)\right)}}{n}} \]

      2. mul-1-neg N/A

         \[\leadsto 1 - e^{\frac{\mathsf{neg}\left(\color{blue}{-1 \cdot \log x}\right)}{n}} \]

      3. distribute-neg-frac N/A

         \[\leadsto 1 - e^{\color{blue}{\mathsf{neg}\left(\frac{-1 \cdot \log x}{n}\right)}} \]

      4. mul-1-neg N/A

         \[\leadsto 1 - e^{\mathsf{neg}\left(\frac{\color{blue}{\mathsf{neg}\left(\log x\right)}}{n}\right)} \]

      5. log-rec N/A

         \[\leadsto 1 - e^{\mathsf{neg}\left(\frac{\color{blue}{\log \left(\frac{1}{x}\right)}}{n}\right)} \]

      6. mul-1-neg N/A

         \[\leadsto 1 - e^{\color{blue}{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}} \]

      7. lower--.f64 N/A

         \[\leadsto \color{blue}{1 - e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}} \]

      8. log-rec N/A

         \[\leadsto 1 - e^{-1 \cdot \frac{\color{blue}{\mathsf{neg}\left(\log x\right)}}{n}} \]

      9. mul-1-neg N/A

         \[\leadsto 1 - e^{-1 \cdot \frac{\color{blue}{-1 \cdot \log x}}{n}} \]

      10. associate-*r/ N/A

          \[\leadsto 1 - e^{\color{blue}{\frac{-1 \cdot \left(-1 \cdot \log x\right)}{n}}} \]

      11. associate-*r* N/A

          \[\leadsto 1 - e^{\frac{\color{blue}{\left(-1 \cdot -1\right) \cdot \log x}}{n}} \]

      12. metadata-eval N/A

          \[\leadsto 1 - e^{\frac{\color{blue}{1} \cdot \log x}{n}} \]

      13. *-commutative N/A

          \[\leadsto 1 - e^{\frac{\color{blue}{\log x \cdot 1}}{n}} \]

      14. associate-/l* N/A

          \[\leadsto 1 - e^{\color{blue}{\log x \cdot \frac{1}{n}}} \]

      15. exp-to-pow N/A

          \[\leadsto 1 - \color{blue}{{x}^{\left(\frac{1}{n}\right)}} \]

      16. lower-pow.f64 N/A

          \[\leadsto 1 - \color{blue}{{x}^{\left(\frac{1}{n}\right)}} \]

      17. lower-/.f64 25.4

          \[\leadsto 1 - {x}^{\color{blue}{\left(\frac{1}{n}\right)}} \]

   5. Simplified 25.4%

      \[\leadsto \color{blue}{1 - {x}^{\left(\frac{1}{n}\right)}} \]

   6. Taylor expanded in n around inf

      \[\leadsto \color{blue}{-1 \cdot \frac{\log x}{n}} \]
   7. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{\log x}{n}\right)} \]

      2. lower-neg.f64 N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{\log x}{n}\right)} \]

      3. lower-/.f64 N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\frac{\log x}{n}}\right) \]

      4. lower-log.f64 80.1

         \[\leadsto -\frac{\color{blue}{\log x}}{n} \]

   8. Simplified 80.1%

      \[\leadsto \color{blue}{-\frac{\log x}{n}} \]

   if 5.2000000000000004e-174 < x < 7.50000000000000047e-61

   1. Initial program 54.0%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in n around inf

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]

      2. lower--.f64 N/A

         \[\leadsto \frac{\color{blue}{\log \left(1 + x\right) - \log x}}{n} \]

      3. lower-log1p.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(x\right)} - \log x}{n} \]

      4. lower-log.f64 36.5

         \[\leadsto \frac{\mathsf{log1p}\left(x\right) - \color{blue}{\log x}}{n} \]

   5. Simplified 36.5%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) - \log x}{n}} \]

   6. Taylor expanded in x around -inf

      \[\leadsto \color{blue}{-1 \cdot \frac{-1 \cdot \frac{\frac{1}{3} \cdot \frac{1}{n \cdot x} - \frac{1}{2} \cdot \frac{1}{n}}{x} - \frac{1}{n}}{x}} \]
   7. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{-1 \cdot \frac{\frac{1}{3} \cdot \frac{1}{n \cdot x} - \frac{1}{2} \cdot \frac{1}{n}}{x} - \frac{1}{n}}{x}\right)} \]

      2. lower-neg.f64 N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{-1 \cdot \frac{\frac{1}{3} \cdot \frac{1}{n \cdot x} - \frac{1}{2} \cdot \frac{1}{n}}{x} - \frac{1}{n}}{x}\right)} \]

      3. div-sub N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\left(\frac{-1 \cdot \frac{\frac{1}{3} \cdot \frac{1}{n \cdot x} - \frac{1}{2} \cdot \frac{1}{n}}{x}}{x} - \frac{\frac{1}{n}}{x}\right)}\right) \]

      4. associate-/r* N/A

         \[\leadsto \mathsf{neg}\left(\left(\frac{-1 \cdot \frac{\frac{1}{3} \cdot \frac{1}{n \cdot x} - \frac{1}{2} \cdot \frac{1}{n}}{x}}{x} - \color{blue}{\frac{1}{n \cdot x}}\right)\right) \]

      5. lower--.f64 N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\left(\frac{-1 \cdot \frac{\frac{1}{3} \cdot \frac{1}{n \cdot x} - \frac{1}{2} \cdot \frac{1}{n}}{x}}{x} - \frac{1}{n \cdot x}\right)}\right) \]

   8. Simplified 57.4%

      \[\leadsto \color{blue}{-\left(\frac{-\left(\frac{0.3333333333333333}{x \cdot n} + \frac{-0.5}{n}\right)}{x \cdot x} - \frac{1}{x \cdot n}\right)} \]

   if 7.50000000000000047e-61 < x < 0.880000000000000004

   1. Initial program 38.8%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(1 + x \cdot \left(x \cdot \left(\frac{1}{2} \cdot \frac{1}{{n}^{2}} - \frac{1}{2} \cdot \frac{1}{n}\right) + \frac{1}{n}\right)\right) - e^{\frac{\log x}{n}}} \]
   4. Step-by-step derivation

      1. remove-double-neg N/A

         \[\leadsto \left(1 + x \cdot \left(x \cdot \left(\frac{1}{2} \cdot \frac{1}{{n}^{2}} - \frac{1}{2} \cdot \frac{1}{n}\right) + \frac{1}{n}\right)\right) - e^{\frac{\color{blue}{\mathsf{neg}\left(\left(\mathsf{neg}\left(\log x\right)\right)\right)}}{n}} \]

      2. mul-1-neg N/A

         \[\leadsto \left(1 + x \cdot \left(x \cdot \left(\frac{1}{2} \cdot \frac{1}{{n}^{2}} - \frac{1}{2} \cdot \frac{1}{n}\right) + \frac{1}{n}\right)\right) - e^{\frac{\mathsf{neg}\left(\color{blue}{-1 \cdot \log x}\right)}{n}} \]

      3. distribute-neg-frac N/A

         \[\leadsto \left(1 + x \cdot \left(x \cdot \left(\frac{1}{2} \cdot \frac{1}{{n}^{2}} - \frac{1}{2} \cdot \frac{1}{n}\right) + \frac{1}{n}\right)\right) - e^{\color{blue}{\mathsf{neg}\left(\frac{-1 \cdot \log x}{n}\right)}} \]

      4. mul-1-neg N/A

         \[\leadsto \left(1 + x \cdot \left(x \cdot \left(\frac{1}{2} \cdot \frac{1}{{n}^{2}} - \frac{1}{2} \cdot \frac{1}{n}\right) + \frac{1}{n}\right)\right) - e^{\mathsf{neg}\left(\frac{\color{blue}{\mathsf{neg}\left(\log x\right)}}{n}\right)} \]

      5. log-rec N/A

         \[\leadsto \left(1 + x \cdot \left(x \cdot \left(\frac{1}{2} \cdot \frac{1}{{n}^{2}} - \frac{1}{2} \cdot \frac{1}{n}\right) + \frac{1}{n}\right)\right) - e^{\mathsf{neg}\left(\frac{\color{blue}{\log \left(\frac{1}{x}\right)}}{n}\right)} \]

      6. mul-1-neg N/A

         \[\leadsto \left(1 + x \cdot \left(x \cdot \left(\frac{1}{2} \cdot \frac{1}{{n}^{2}} - \frac{1}{2} \cdot \frac{1}{n}\right) + \frac{1}{n}\right)\right) - e^{\color{blue}{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}} \]

      7. lower--.f64 N/A

         \[\leadsto \color{blue}{\left(1 + x \cdot \left(x \cdot \left(\frac{1}{2} \cdot \frac{1}{{n}^{2}} - \frac{1}{2} \cdot \frac{1}{n}\right) + \frac{1}{n}\right)\right) - e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}} \]

   5. Simplified 25.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{0.5}{n \cdot n} + \frac{-0.5}{n}, \frac{1}{n}\right), 1\right) - {x}^{\left(\frac{1}{n}\right)}} \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(1 + \frac{x}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{x}{n} + 1\right)} - {x}^{\left(\frac{1}{n}\right)} \]

      2. lower-+.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{x}{n} + 1\right)} - {x}^{\left(\frac{1}{n}\right)} \]

      3. lower-/.f64 35.3

         \[\leadsto \left(\color{blue}{\frac{x}{n}} + 1\right) - {x}^{\left(\frac{1}{n}\right)} \]

   8. Simplified 35.3%

      \[\leadsto \color{blue}{\left(\frac{x}{n} + 1\right)} - {x}^{\left(\frac{1}{n}\right)} \]

   9. Taylor expanded in n around inf

      \[\leadsto \color{blue}{\frac{x - \log x}{n}} \]
   10. Step-by-step derivation

       1. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{x - \log x}{n}} \]

       2. lower--.f64 N/A

          \[\leadsto \frac{\color{blue}{x - \log x}}{n} \]

       3. lower-log.f64 48.1

          \[\leadsto \frac{x - \color{blue}{\log x}}{n} \]

   11. Simplified 48.1%

       \[\leadsto \color{blue}{\frac{x - \log x}{n}} \]

   if 0.880000000000000004 < x < 3.7000000000000002e82

   1. Initial program 39.0%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{\frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}}{n} + \left(\frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}} \cdot \left(\frac{1}{2} \cdot \frac{1}{{n}^{2}} - \frac{1}{2} \cdot \frac{1}{n}\right)}{x} + \left(\frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}} \cdot \left(\left(\frac{1}{24} \cdot \frac{1}{{n}^{4}} + \frac{11}{24} \cdot \frac{1}{{n}^{2}}\right) - \left(\frac{1}{4} \cdot \frac{1}{n} + \frac{1}{4} \cdot \frac{1}{{n}^{3}}\right)\right)}{{x}^{3}} + \frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}} \cdot \left(\left(\frac{1}{6} \cdot \frac{1}{{n}^{3}} + \frac{1}{3} \cdot \frac{1}{n}\right) - \frac{1}{2} \cdot \frac{1}{{n}^{2}}\right)}{{x}^{2}}\right)\right)}{x}} \]

   5. Simplified 79.8%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left({x}^{\left(\frac{1}{n}\right)}, \left(\frac{0.5}{x \cdot \left(n \cdot n\right)} - \frac{0.5}{x \cdot n}\right) + \left(\frac{\frac{0.16666666666666666}{n \cdot \left(n \cdot n\right)} + \left(\frac{0.3333333333333333}{n} + \frac{-0.5}{n \cdot n}\right)}{x \cdot x} + \frac{\frac{0.041666666666666664}{{n}^{4}} + \left(\left(\frac{0.4583333333333333}{n \cdot n} + \frac{-0.25}{n}\right) + \frac{-0.25}{n \cdot \left(n \cdot n\right)}\right)}{x \cdot \left(x \cdot x\right)}\right), \frac{{x}^{\left(\frac{1}{n}\right)}}{n}\right)}{x}} \]

   6. Taylor expanded in n around inf

      \[\leadsto \frac{\color{blue}{\frac{\left(1 + \frac{1}{3} \cdot \frac{1}{{x}^{2}}\right) - \left(\frac{1}{4} \cdot \frac{1}{{x}^{3}} + \frac{1}{2} \cdot \frac{1}{x}\right)}{n}}}{x} \]
   7. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{\left(1 + \frac{1}{3} \cdot \frac{1}{{x}^{2}}\right) - \left(\frac{1}{4} \cdot \frac{1}{{x}^{3}} + \frac{1}{2} \cdot \frac{1}{x}\right)}{n}}}{x} \]

   8. Simplified 70.0%

      \[\leadsto \frac{\color{blue}{\frac{1 + \left(\frac{-0.5 + \frac{0.3333333333333333}{x}}{x} - \frac{0.25}{x \cdot \left(x \cdot x\right)}\right)}{n}}}{x} \]

   if 3.7000000000000002e82 < x

   1. Initial program 77.6%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1 - e^{\frac{\log x}{n}}} \]
   4. Step-by-step derivation

      1. remove-double-neg N/A

         \[\leadsto 1 - e^{\frac{\color{blue}{\mathsf{neg}\left(\left(\mathsf{neg}\left(\log x\right)\right)\right)}}{n}} \]

      2. mul-1-neg N/A

         \[\leadsto 1 - e^{\frac{\mathsf{neg}\left(\color{blue}{-1 \cdot \log x}\right)}{n}} \]

      3. distribute-neg-frac N/A

         \[\leadsto 1 - e^{\color{blue}{\mathsf{neg}\left(\frac{-1 \cdot \log x}{n}\right)}} \]

      4. mul-1-neg N/A

         \[\leadsto 1 - e^{\mathsf{neg}\left(\frac{\color{blue}{\mathsf{neg}\left(\log x\right)}}{n}\right)} \]

      5. log-rec N/A

         \[\leadsto 1 - e^{\mathsf{neg}\left(\frac{\color{blue}{\log \left(\frac{1}{x}\right)}}{n}\right)} \]

      6. mul-1-neg N/A

         \[\leadsto 1 - e^{\color{blue}{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}} \]

      7. lower--.f64 N/A

         \[\leadsto \color{blue}{1 - e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}} \]

      8. log-rec N/A

         \[\leadsto 1 - e^{-1 \cdot \frac{\color{blue}{\mathsf{neg}\left(\log x\right)}}{n}} \]

      9. mul-1-neg N/A

         \[\leadsto 1 - e^{-1 \cdot \frac{\color{blue}{-1 \cdot \log x}}{n}} \]

      10. associate-*r/ N/A

          \[\leadsto 1 - e^{\color{blue}{\frac{-1 \cdot \left(-1 \cdot \log x\right)}{n}}} \]

      11. associate-*r* N/A

          \[\leadsto 1 - e^{\frac{\color{blue}{\left(-1 \cdot -1\right) \cdot \log x}}{n}} \]

      12. metadata-eval N/A

          \[\leadsto 1 - e^{\frac{\color{blue}{1} \cdot \log x}{n}} \]

      13. *-commutative N/A

          \[\leadsto 1 - e^{\frac{\color{blue}{\log x \cdot 1}}{n}} \]

      14. associate-/l* N/A

          \[\leadsto 1 - e^{\color{blue}{\log x \cdot \frac{1}{n}}} \]

      15. exp-to-pow N/A

          \[\leadsto 1 - \color{blue}{{x}^{\left(\frac{1}{n}\right)}} \]

      16. lower-pow.f64 N/A

          \[\leadsto 1 - \color{blue}{{x}^{\left(\frac{1}{n}\right)}} \]

      17. lower-/.f64 42.8

          \[\leadsto 1 - {x}^{\color{blue}{\left(\frac{1}{n}\right)}} \]

   5. Simplified 42.8%

      \[\leadsto \color{blue}{1 - {x}^{\left(\frac{1}{n}\right)}} \]

   6. Taylor expanded in n around inf

      \[\leadsto 1 - \color{blue}{1} \]
   7. Step-by-step derivation

   8. Simplified 77.6%

      \[\leadsto 1 - \color{blue}{1} \]
   9. Step-by-step derivation

      1. metadata-eval 77.6

         \[\leadsto \color{blue}{0} \]

   10. Applied egg-rr 77.6%

       \[\leadsto \color{blue}{0} \]
3. Recombined 6 regimes into one program.

4. Final simplification 68.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 4.8 \cdot 10^{-235}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;x \leq 5.2 \cdot 10^{-174}:\\ \;\;\;\;-\frac{\log x}{n}\\ \mathbf{elif}\;x \leq 7.5 \cdot 10^{-61}:\\ \;\;\;\;\frac{1}{n \cdot x} + \frac{\frac{0.3333333333333333}{n \cdot x} + \frac{-0.5}{n}}{x \cdot x}\\ \mathbf{elif}\;x \leq 0.88:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{elif}\;x \leq 3.7 \cdot 10^{+82}:\\ \;\;\;\;\frac{\frac{1 + \left(\frac{-0.5 + \frac{0.3333333333333333}{x}}{x} - \frac{0.25}{x \cdot \left(x \cdot x\right)}\right)}{n}}{x}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 55.4% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{n \cdot x}\\ \mathbf{if}\;x \leq 8.5 \cdot 10^{-265}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 5.2 \cdot 10^{-174}:\\ \;\;\;\;-\frac{\log x}{n}\\ \mathbf{elif}\;x \leq 7.5 \cdot 10^{-61}:\\ \;\;\;\;t\_0 + \frac{\frac{0.3333333333333333}{n \cdot x} + \frac{-0.5}{n}}{x \cdot x}\\ \mathbf{elif}\;x \leq 0.88:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{elif}\;x \leq 3.4 \cdot 10^{+82}:\\ \;\;\;\;\frac{\frac{1 + \left(\frac{-0.5 + \frac{0.3333333333333333}{x}}{x} - \frac{0.25}{x \cdot \left(x \cdot x\right)}\right)}{n}}{x}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (/ 1.0 (* n x))))
   (if (<= x 8.5e-265)
     t_0
     (if (<= x 5.2e-174)
       (- (/ (log x) n))
       (if (<= x 7.5e-61)
         (+ t_0 (/ (+ (/ 0.3333333333333333 (* n x)) (/ -0.5 n)) (* x x)))
         (if (<= x 0.88)
           (/ (- x (log x)) n)
           (if (<= x 3.4e+82)
             (/
              (/
               (+
                1.0
                (-
                 (/ (+ -0.5 (/ 0.3333333333333333 x)) x)
                 (/ 0.25 (* x (* x x)))))
               n)
              x)
             0.0)))))))

C

double code(double x, double n) {
	double t_0 = 1.0 / (n * x);
	double tmp;
	if (x <= 8.5e-265) {
		tmp = t_0;
	} else if (x <= 5.2e-174) {
		tmp = -(log(x) / n);
	} else if (x <= 7.5e-61) {
		tmp = t_0 + (((0.3333333333333333 / (n * x)) + (-0.5 / n)) / (x * x));
	} else if (x <= 0.88) {
		tmp = (x - log(x)) / n;
	} else if (x <= 3.4e+82) {
		tmp = ((1.0 + (((-0.5 + (0.3333333333333333 / x)) / x) - (0.25 / (x * (x * x))))) / 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) :: t_0
    real(8) :: tmp
    t_0 = 1.0d0 / (n * x)
    if (x <= 8.5d-265) then
        tmp = t_0
    else if (x <= 5.2d-174) then
        tmp = -(log(x) / n)
    else if (x <= 7.5d-61) then
        tmp = t_0 + (((0.3333333333333333d0 / (n * x)) + ((-0.5d0) / n)) / (x * x))
    else if (x <= 0.88d0) then
        tmp = (x - log(x)) / n
    else if (x <= 3.4d+82) then
        tmp = ((1.0d0 + ((((-0.5d0) + (0.3333333333333333d0 / x)) / x) - (0.25d0 / (x * (x * x))))) / n) / x
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = 1.0 / (n * x);
	double tmp;
	if (x <= 8.5e-265) {
		tmp = t_0;
	} else if (x <= 5.2e-174) {
		tmp = -(Math.log(x) / n);
	} else if (x <= 7.5e-61) {
		tmp = t_0 + (((0.3333333333333333 / (n * x)) + (-0.5 / n)) / (x * x));
	} else if (x <= 0.88) {
		tmp = (x - Math.log(x)) / n;
	} else if (x <= 3.4e+82) {
		tmp = ((1.0 + (((-0.5 + (0.3333333333333333 / x)) / x) - (0.25 / (x * (x * x))))) / n) / x;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = 1.0 / (n * x)
	tmp = 0
	if x <= 8.5e-265:
		tmp = t_0
	elif x <= 5.2e-174:
		tmp = -(math.log(x) / n)
	elif x <= 7.5e-61:
		tmp = t_0 + (((0.3333333333333333 / (n * x)) + (-0.5 / n)) / (x * x))
	elif x <= 0.88:
		tmp = (x - math.log(x)) / n
	elif x <= 3.4e+82:
		tmp = ((1.0 + (((-0.5 + (0.3333333333333333 / x)) / x) - (0.25 / (x * (x * x))))) / n) / x
	else:
		tmp = 0.0
	return tmp

Julia

function code(x, n)
	t_0 = Float64(1.0 / Float64(n * x))
	tmp = 0.0
	if (x <= 8.5e-265)
		tmp = t_0;
	elseif (x <= 5.2e-174)
		tmp = Float64(-Float64(log(x) / n));
	elseif (x <= 7.5e-61)
		tmp = Float64(t_0 + Float64(Float64(Float64(0.3333333333333333 / Float64(n * x)) + Float64(-0.5 / n)) / Float64(x * x)));
	elseif (x <= 0.88)
		tmp = Float64(Float64(x - log(x)) / n);
	elseif (x <= 3.4e+82)
		tmp = Float64(Float64(Float64(1.0 + Float64(Float64(Float64(-0.5 + Float64(0.3333333333333333 / x)) / x) - Float64(0.25 / Float64(x * Float64(x * x))))) / n) / x);
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = 1.0 / (n * x);
	tmp = 0.0;
	if (x <= 8.5e-265)
		tmp = t_0;
	elseif (x <= 5.2e-174)
		tmp = -(log(x) / n);
	elseif (x <= 7.5e-61)
		tmp = t_0 + (((0.3333333333333333 / (n * x)) + (-0.5 / n)) / (x * x));
	elseif (x <= 0.88)
		tmp = (x - log(x)) / n;
	elseif (x <= 3.4e+82)
		tmp = ((1.0 + (((-0.5 + (0.3333333333333333 / x)) / x) - (0.25 / (x * (x * x))))) / n) / x;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(1.0 / N[(n * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 8.5e-265], t$95$0, If[LessEqual[x, 5.2e-174], (-N[(N[Log[x], $MachinePrecision] / n), $MachinePrecision]), If[LessEqual[x, 7.5e-61], N[(t$95$0 + N[(N[(N[(0.3333333333333333 / N[(n * x), $MachinePrecision]), $MachinePrecision] + N[(-0.5 / n), $MachinePrecision]), $MachinePrecision] / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.88], N[(N[(x - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[x, 3.4e+82], N[(N[(N[(1.0 + N[(N[(N[(-0.5 + N[(0.3333333333333333 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] - N[(0.25 / N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision] / x), $MachinePrecision], 0.0]]]]]]

Derivation

1. Split input into 6 regimes

2. if x < 8.4999999999999997e-265

   1. Initial program 74.7%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}}{n \cdot x}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}}{n \cdot x}} \]

      2. log-rec N/A

         \[\leadsto \frac{e^{-1 \cdot \frac{\color{blue}{\mathsf{neg}\left(\log x\right)}}{n}}}{n \cdot x} \]

      3. mul-1-neg N/A

         \[\leadsto \frac{e^{-1 \cdot \frac{\color{blue}{-1 \cdot \log x}}{n}}}{n \cdot x} \]

      4. associate-*r/ N/A

         \[\leadsto \frac{e^{\color{blue}{\frac{-1 \cdot \left(-1 \cdot \log x\right)}{n}}}}{n \cdot x} \]

      5. associate-*r* N/A

         \[\leadsto \frac{e^{\frac{\color{blue}{\left(-1 \cdot -1\right) \cdot \log x}}{n}}}{n \cdot x} \]

      6. metadata-eval N/A

         \[\leadsto \frac{e^{\frac{\color{blue}{1} \cdot \log x}{n}}}{n \cdot x} \]

      7. *-commutative N/A

         \[\leadsto \frac{e^{\frac{\color{blue}{\log x \cdot 1}}{n}}}{n \cdot x} \]

      8. associate-/l* N/A

         \[\leadsto \frac{e^{\color{blue}{\log x \cdot \frac{1}{n}}}}{n \cdot x} \]

      9. exp-to-pow N/A

         \[\leadsto \frac{\color{blue}{{x}^{\left(\frac{1}{n}\right)}}}{n \cdot x} \]

      10. lower-pow.f64 N/A

          \[\leadsto \frac{\color{blue}{{x}^{\left(\frac{1}{n}\right)}}}{n \cdot x} \]

      11. lower-/.f64 N/A

          \[\leadsto \frac{{x}^{\color{blue}{\left(\frac{1}{n}\right)}}}{n \cdot x} \]

      12. *-commutative N/A

          \[\leadsto \frac{{x}^{\left(\frac{1}{n}\right)}}{\color{blue}{x \cdot n}} \]

      13. lower-*.f64 39.2

          \[\leadsto \frac{{x}^{\left(\frac{1}{n}\right)}}{\color{blue}{x \cdot n}} \]

   5. Simplified 39.2%

      \[\leadsto \color{blue}{\frac{{x}^{\left(\frac{1}{n}\right)}}{x \cdot n}} \]

   6. Taylor expanded in n around inf

      \[\leadsto \color{blue}{\frac{1}{n \cdot x}} \]
   7. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{n \cdot x}} \]

      2. *-commutative N/A

         \[\leadsto \frac{1}{\color{blue}{x \cdot n}} \]

      3. lower-*.f64 47.9

         \[\leadsto \frac{1}{\color{blue}{x \cdot n}} \]

   8. Simplified 47.9%

      \[\leadsto \color{blue}{\frac{1}{x \cdot n}} \]

   if 8.4999999999999997e-265 < x < 5.2000000000000004e-174

   1. Initial program 36.9%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1 - e^{\frac{\log x}{n}}} \]
   4. Step-by-step derivation

      1. remove-double-neg N/A

         \[\leadsto 1 - e^{\frac{\color{blue}{\mathsf{neg}\left(\left(\mathsf{neg}\left(\log x\right)\right)\right)}}{n}} \]

      2. mul-1-neg N/A

         \[\leadsto 1 - e^{\frac{\mathsf{neg}\left(\color{blue}{-1 \cdot \log x}\right)}{n}} \]

      3. distribute-neg-frac N/A

         \[\leadsto 1 - e^{\color{blue}{\mathsf{neg}\left(\frac{-1 \cdot \log x}{n}\right)}} \]

      4. mul-1-neg N/A

         \[\leadsto 1 - e^{\mathsf{neg}\left(\frac{\color{blue}{\mathsf{neg}\left(\log x\right)}}{n}\right)} \]

      5. log-rec N/A

         \[\leadsto 1 - e^{\mathsf{neg}\left(\frac{\color{blue}{\log \left(\frac{1}{x}\right)}}{n}\right)} \]

      6. mul-1-neg N/A

         \[\leadsto 1 - e^{\color{blue}{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}} \]

      7. lower--.f64 N/A

         \[\leadsto \color{blue}{1 - e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}} \]

      8. log-rec N/A

         \[\leadsto 1 - e^{-1 \cdot \frac{\color{blue}{\mathsf{neg}\left(\log x\right)}}{n}} \]

      9. mul-1-neg N/A

         \[\leadsto 1 - e^{-1 \cdot \frac{\color{blue}{-1 \cdot \log x}}{n}} \]

      10. associate-*r/ N/A

          \[\leadsto 1 - e^{\color{blue}{\frac{-1 \cdot \left(-1 \cdot \log x\right)}{n}}} \]

      11. associate-*r* N/A

          \[\leadsto 1 - e^{\frac{\color{blue}{\left(-1 \cdot -1\right) \cdot \log x}}{n}} \]

      12. metadata-eval N/A

          \[\leadsto 1 - e^{\frac{\color{blue}{1} \cdot \log x}{n}} \]

      13. *-commutative N/A

          \[\leadsto 1 - e^{\frac{\color{blue}{\log x \cdot 1}}{n}} \]

      14. associate-/l* N/A

          \[\leadsto 1 - e^{\color{blue}{\log x \cdot \frac{1}{n}}} \]

      15. exp-to-pow N/A

          \[\leadsto 1 - \color{blue}{{x}^{\left(\frac{1}{n}\right)}} \]

      16. lower-pow.f64 N/A

          \[\leadsto 1 - \color{blue}{{x}^{\left(\frac{1}{n}\right)}} \]

      17. lower-/.f64 36.9

          \[\leadsto 1 - {x}^{\color{blue}{\left(\frac{1}{n}\right)}} \]

   5. Simplified 36.9%

      \[\leadsto \color{blue}{1 - {x}^{\left(\frac{1}{n}\right)}} \]

   6. Taylor expanded in n around inf

      \[\leadsto \color{blue}{-1 \cdot \frac{\log x}{n}} \]
   7. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{\log x}{n}\right)} \]

      2. lower-neg.f64 N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{\log x}{n}\right)} \]

      3. lower-/.f64 N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\frac{\log x}{n}}\right) \]

      4. lower-log.f64 68.4

         \[\leadsto -\frac{\color{blue}{\log x}}{n} \]

   8. Simplified 68.4%

      \[\leadsto \color{blue}{-\frac{\log x}{n}} \]

   if 5.2000000000000004e-174 < x < 7.50000000000000047e-61

   1. Initial program 54.0%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in n around inf

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]

      2. lower--.f64 N/A

         \[\leadsto \frac{\color{blue}{\log \left(1 + x\right) - \log x}}{n} \]

      3. lower-log1p.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(x\right)} - \log x}{n} \]

      4. lower-log.f64 36.5

         \[\leadsto \frac{\mathsf{log1p}\left(x\right) - \color{blue}{\log x}}{n} \]

   5. Simplified 36.5%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) - \log x}{n}} \]

   6. Taylor expanded in x around -inf

      \[\leadsto \color{blue}{-1 \cdot \frac{-1 \cdot \frac{\frac{1}{3} \cdot \frac{1}{n \cdot x} - \frac{1}{2} \cdot \frac{1}{n}}{x} - \frac{1}{n}}{x}} \]
   7. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{-1 \cdot \frac{\frac{1}{3} \cdot \frac{1}{n \cdot x} - \frac{1}{2} \cdot \frac{1}{n}}{x} - \frac{1}{n}}{x}\right)} \]

      2. lower-neg.f64 N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{-1 \cdot \frac{\frac{1}{3} \cdot \frac{1}{n \cdot x} - \frac{1}{2} \cdot \frac{1}{n}}{x} - \frac{1}{n}}{x}\right)} \]

      3. div-sub N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\left(\frac{-1 \cdot \frac{\frac{1}{3} \cdot \frac{1}{n \cdot x} - \frac{1}{2} \cdot \frac{1}{n}}{x}}{x} - \frac{\frac{1}{n}}{x}\right)}\right) \]

      4. associate-/r* N/A

         \[\leadsto \mathsf{neg}\left(\left(\frac{-1 \cdot \frac{\frac{1}{3} \cdot \frac{1}{n \cdot x} - \frac{1}{2} \cdot \frac{1}{n}}{x}}{x} - \color{blue}{\frac{1}{n \cdot x}}\right)\right) \]

      5. lower--.f64 N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\left(\frac{-1 \cdot \frac{\frac{1}{3} \cdot \frac{1}{n \cdot x} - \frac{1}{2} \cdot \frac{1}{n}}{x}}{x} - \frac{1}{n \cdot x}\right)}\right) \]

   8. Simplified 57.4%

      \[\leadsto \color{blue}{-\left(\frac{-\left(\frac{0.3333333333333333}{x \cdot n} + \frac{-0.5}{n}\right)}{x \cdot x} - \frac{1}{x \cdot n}\right)} \]

   if 7.50000000000000047e-61 < x < 0.880000000000000004

   1. Initial program 38.8%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(1 + x \cdot \left(x \cdot \left(\frac{1}{2} \cdot \frac{1}{{n}^{2}} - \frac{1}{2} \cdot \frac{1}{n}\right) + \frac{1}{n}\right)\right) - e^{\frac{\log x}{n}}} \]
   4. Step-by-step derivation

      1. remove-double-neg N/A

         \[\leadsto \left(1 + x \cdot \left(x \cdot \left(\frac{1}{2} \cdot \frac{1}{{n}^{2}} - \frac{1}{2} \cdot \frac{1}{n}\right) + \frac{1}{n}\right)\right) - e^{\frac{\color{blue}{\mathsf{neg}\left(\left(\mathsf{neg}\left(\log x\right)\right)\right)}}{n}} \]

      2. mul-1-neg N/A

         \[\leadsto \left(1 + x \cdot \left(x \cdot \left(\frac{1}{2} \cdot \frac{1}{{n}^{2}} - \frac{1}{2} \cdot \frac{1}{n}\right) + \frac{1}{n}\right)\right) - e^{\frac{\mathsf{neg}\left(\color{blue}{-1 \cdot \log x}\right)}{n}} \]

      3. distribute-neg-frac N/A

         \[\leadsto \left(1 + x \cdot \left(x \cdot \left(\frac{1}{2} \cdot \frac{1}{{n}^{2}} - \frac{1}{2} \cdot \frac{1}{n}\right) + \frac{1}{n}\right)\right) - e^{\color{blue}{\mathsf{neg}\left(\frac{-1 \cdot \log x}{n}\right)}} \]

      4. mul-1-neg N/A

         \[\leadsto \left(1 + x \cdot \left(x \cdot \left(\frac{1}{2} \cdot \frac{1}{{n}^{2}} - \frac{1}{2} \cdot \frac{1}{n}\right) + \frac{1}{n}\right)\right) - e^{\mathsf{neg}\left(\frac{\color{blue}{\mathsf{neg}\left(\log x\right)}}{n}\right)} \]

      5. log-rec N/A

         \[\leadsto \left(1 + x \cdot \left(x \cdot \left(\frac{1}{2} \cdot \frac{1}{{n}^{2}} - \frac{1}{2} \cdot \frac{1}{n}\right) + \frac{1}{n}\right)\right) - e^{\mathsf{neg}\left(\frac{\color{blue}{\log \left(\frac{1}{x}\right)}}{n}\right)} \]

      6. mul-1-neg N/A

         \[\leadsto \left(1 + x \cdot \left(x \cdot \left(\frac{1}{2} \cdot \frac{1}{{n}^{2}} - \frac{1}{2} \cdot \frac{1}{n}\right) + \frac{1}{n}\right)\right) - e^{\color{blue}{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}} \]

      7. lower--.f64 N/A

         \[\leadsto \color{blue}{\left(1 + x \cdot \left(x \cdot \left(\frac{1}{2} \cdot \frac{1}{{n}^{2}} - \frac{1}{2} \cdot \frac{1}{n}\right) + \frac{1}{n}\right)\right) - e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}} \]

   5. Simplified 25.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{0.5}{n \cdot n} + \frac{-0.5}{n}, \frac{1}{n}\right), 1\right) - {x}^{\left(\frac{1}{n}\right)}} \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(1 + \frac{x}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{x}{n} + 1\right)} - {x}^{\left(\frac{1}{n}\right)} \]

      2. lower-+.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{x}{n} + 1\right)} - {x}^{\left(\frac{1}{n}\right)} \]

      3. lower-/.f64 35.3

         \[\leadsto \left(\color{blue}{\frac{x}{n}} + 1\right) - {x}^{\left(\frac{1}{n}\right)} \]

   8. Simplified 35.3%

      \[\leadsto \color{blue}{\left(\frac{x}{n} + 1\right)} - {x}^{\left(\frac{1}{n}\right)} \]

   9. Taylor expanded in n around inf

      \[\leadsto \color{blue}{\frac{x - \log x}{n}} \]
   10. Step-by-step derivation

       1. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{x - \log x}{n}} \]

       2. lower--.f64 N/A

          \[\leadsto \frac{\color{blue}{x - \log x}}{n} \]

       3. lower-log.f64 48.1

          \[\leadsto \frac{x - \color{blue}{\log x}}{n} \]

   11. Simplified 48.1%

       \[\leadsto \color{blue}{\frac{x - \log x}{n}} \]

   if 0.880000000000000004 < x < 3.39999999999999994e82

   1. Initial program 39.0%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{\frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}}{n} + \left(\frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}} \cdot \left(\frac{1}{2} \cdot \frac{1}{{n}^{2}} - \frac{1}{2} \cdot \frac{1}{n}\right)}{x} + \left(\frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}} \cdot \left(\left(\frac{1}{24} \cdot \frac{1}{{n}^{4}} + \frac{11}{24} \cdot \frac{1}{{n}^{2}}\right) - \left(\frac{1}{4} \cdot \frac{1}{n} + \frac{1}{4} \cdot \frac{1}{{n}^{3}}\right)\right)}{{x}^{3}} + \frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}} \cdot \left(\left(\frac{1}{6} \cdot \frac{1}{{n}^{3}} + \frac{1}{3} \cdot \frac{1}{n}\right) - \frac{1}{2} \cdot \frac{1}{{n}^{2}}\right)}{{x}^{2}}\right)\right)}{x}} \]

   5. Simplified 79.8%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left({x}^{\left(\frac{1}{n}\right)}, \left(\frac{0.5}{x \cdot \left(n \cdot n\right)} - \frac{0.5}{x \cdot n}\right) + \left(\frac{\frac{0.16666666666666666}{n \cdot \left(n \cdot n\right)} + \left(\frac{0.3333333333333333}{n} + \frac{-0.5}{n \cdot n}\right)}{x \cdot x} + \frac{\frac{0.041666666666666664}{{n}^{4}} + \left(\left(\frac{0.4583333333333333}{n \cdot n} + \frac{-0.25}{n}\right) + \frac{-0.25}{n \cdot \left(n \cdot n\right)}\right)}{x \cdot \left(x \cdot x\right)}\right), \frac{{x}^{\left(\frac{1}{n}\right)}}{n}\right)}{x}} \]

   6. Taylor expanded in n around inf

      \[\leadsto \frac{\color{blue}{\frac{\left(1 + \frac{1}{3} \cdot \frac{1}{{x}^{2}}\right) - \left(\frac{1}{4} \cdot \frac{1}{{x}^{3}} + \frac{1}{2} \cdot \frac{1}{x}\right)}{n}}}{x} \]
   7. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{\left(1 + \frac{1}{3} \cdot \frac{1}{{x}^{2}}\right) - \left(\frac{1}{4} \cdot \frac{1}{{x}^{3}} + \frac{1}{2} \cdot \frac{1}{x}\right)}{n}}}{x} \]

   8. Simplified 70.0%

      \[\leadsto \frac{\color{blue}{\frac{1 + \left(\frac{-0.5 + \frac{0.3333333333333333}{x}}{x} - \frac{0.25}{x \cdot \left(x \cdot x\right)}\right)}{n}}}{x} \]

   if 3.39999999999999994e82 < x

   1. Initial program 77.6%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1 - e^{\frac{\log x}{n}}} \]
   4. Step-by-step derivation

      1. remove-double-neg N/A

         \[\leadsto 1 - e^{\frac{\color{blue}{\mathsf{neg}\left(\left(\mathsf{neg}\left(\log x\right)\right)\right)}}{n}} \]

      2. mul-1-neg N/A

         \[\leadsto 1 - e^{\frac{\mathsf{neg}\left(\color{blue}{-1 \cdot \log x}\right)}{n}} \]

      3. distribute-neg-frac N/A

         \[\leadsto 1 - e^{\color{blue}{\mathsf{neg}\left(\frac{-1 \cdot \log x}{n}\right)}} \]

      4. mul-1-neg N/A

         \[\leadsto 1 - e^{\mathsf{neg}\left(\frac{\color{blue}{\mathsf{neg}\left(\log x\right)}}{n}\right)} \]

      5. log-rec N/A

         \[\leadsto 1 - e^{\mathsf{neg}\left(\frac{\color{blue}{\log \left(\frac{1}{x}\right)}}{n}\right)} \]

      6. mul-1-neg N/A

         \[\leadsto 1 - e^{\color{blue}{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}} \]

      7. lower--.f64 N/A

         \[\leadsto \color{blue}{1 - e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}} \]

      8. log-rec N/A

         \[\leadsto 1 - e^{-1 \cdot \frac{\color{blue}{\mathsf{neg}\left(\log x\right)}}{n}} \]

      9. mul-1-neg N/A

         \[\leadsto 1 - e^{-1 \cdot \frac{\color{blue}{-1 \cdot \log x}}{n}} \]

      10. associate-*r/ N/A

          \[\leadsto 1 - e^{\color{blue}{\frac{-1 \cdot \left(-1 \cdot \log x\right)}{n}}} \]

      11. associate-*r* N/A

          \[\leadsto 1 - e^{\frac{\color{blue}{\left(-1 \cdot -1\right) \cdot \log x}}{n}} \]

      12. metadata-eval N/A

          \[\leadsto 1 - e^{\frac{\color{blue}{1} \cdot \log x}{n}} \]

      13. *-commutative N/A

          \[\leadsto 1 - e^{\frac{\color{blue}{\log x \cdot 1}}{n}} \]

      14. associate-/l* N/A

          \[\leadsto 1 - e^{\color{blue}{\log x \cdot \frac{1}{n}}} \]

      15. exp-to-pow N/A

          \[\leadsto 1 - \color{blue}{{x}^{\left(\frac{1}{n}\right)}} \]

      16. lower-pow.f64 N/A

          \[\leadsto 1 - \color{blue}{{x}^{\left(\frac{1}{n}\right)}} \]

      17. lower-/.f64 42.8

          \[\leadsto 1 - {x}^{\color{blue}{\left(\frac{1}{n}\right)}} \]

   5. Simplified 42.8%

      \[\leadsto \color{blue}{1 - {x}^{\left(\frac{1}{n}\right)}} \]

   6. Taylor expanded in n around inf

      \[\leadsto 1 - \color{blue}{1} \]
   7. Step-by-step derivation

   8. Simplified 77.6%

      \[\leadsto 1 - \color{blue}{1} \]
   9. Step-by-step derivation

      1. metadata-eval 77.6

         \[\leadsto \color{blue}{0} \]

   10. Applied egg-rr 77.6%

       \[\leadsto \color{blue}{0} \]
3. Recombined 6 regimes into one program.

4. Final simplification 65.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 8.5 \cdot 10^{-265}:\\ \;\;\;\;\frac{1}{n \cdot x}\\ \mathbf{elif}\;x \leq 5.2 \cdot 10^{-174}:\\ \;\;\;\;-\frac{\log x}{n}\\ \mathbf{elif}\;x \leq 7.5 \cdot 10^{-61}:\\ \;\;\;\;\frac{1}{n \cdot x} + \frac{\frac{0.3333333333333333}{n \cdot x} + \frac{-0.5}{n}}{x \cdot x}\\ \mathbf{elif}\;x \leq 0.88:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{elif}\;x \leq 3.4 \cdot 10^{+82}:\\ \;\;\;\;\frac{\frac{1 + \left(\frac{-0.5 + \frac{0.3333333333333333}{x}}{x} - \frac{0.25}{x \cdot \left(x \cdot x\right)}\right)}{n}}{x}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 52.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 9.2 \cdot 10^{-265}:\\ \;\;\;\;\frac{1}{n \cdot x}\\ \mathbf{elif}\;x \leq 8.5 \cdot 10^{-175}:\\ \;\;\;\;-\frac{\log x}{n}\\ \mathbf{elif}\;x \leq 2.3 \cdot 10^{+81}:\\ \;\;\;\;\frac{\frac{1 + \frac{-0.5 + \frac{0.3333333333333333}{x}}{x}}{x}}{n}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 9.2e-265)
   (/ 1.0 (* n x))
   (if (<= x 8.5e-175)
     (- (/ (log x) n))
     (if (<= x 2.3e+81)
       (/ (/ (+ 1.0 (/ (+ -0.5 (/ 0.3333333333333333 x)) x)) x) n)
       0.0))))

C

double code(double x, double n) {
	double tmp;
	if (x <= 9.2e-265) {
		tmp = 1.0 / (n * x);
	} else if (x <= 8.5e-175) {
		tmp = -(log(x) / n);
	} else if (x <= 2.3e+81) {
		tmp = ((1.0 + ((-0.5 + (0.3333333333333333 / x)) / x)) / x) / n;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if (x <= 9.2d-265) then
        tmp = 1.0d0 / (n * x)
    else if (x <= 8.5d-175) then
        tmp = -(log(x) / n)
    else if (x <= 2.3d+81) then
        tmp = ((1.0d0 + (((-0.5d0) + (0.3333333333333333d0 / x)) / x)) / x) / n
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (x <= 9.2e-265) {
		tmp = 1.0 / (n * x);
	} else if (x <= 8.5e-175) {
		tmp = -(Math.log(x) / n);
	} else if (x <= 2.3e+81) {
		tmp = ((1.0 + ((-0.5 + (0.3333333333333333 / x)) / x)) / x) / n;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 9.2e-265:
		tmp = 1.0 / (n * x)
	elif x <= 8.5e-175:
		tmp = -(math.log(x) / n)
	elif x <= 2.3e+81:
		tmp = ((1.0 + ((-0.5 + (0.3333333333333333 / x)) / x)) / x) / n
	else:
		tmp = 0.0
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 9.2e-265)
		tmp = Float64(1.0 / Float64(n * x));
	elseif (x <= 8.5e-175)
		tmp = Float64(-Float64(log(x) / n));
	elseif (x <= 2.3e+81)
		tmp = Float64(Float64(Float64(1.0 + Float64(Float64(-0.5 + Float64(0.3333333333333333 / x)) / x)) / x) / n);
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (x <= 9.2e-265)
		tmp = 1.0 / (n * x);
	elseif (x <= 8.5e-175)
		tmp = -(log(x) / n);
	elseif (x <= 2.3e+81)
		tmp = ((1.0 + ((-0.5 + (0.3333333333333333 / x)) / x)) / x) / n;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 9.2e-265], N[(1.0 / N[(n * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 8.5e-175], (-N[(N[Log[x], $MachinePrecision] / n), $MachinePrecision]), If[LessEqual[x, 2.3e+81], N[(N[(N[(1.0 + N[(N[(-0.5 + N[(0.3333333333333333 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] / n), $MachinePrecision], 0.0]]]

Derivation

1. Split input into 4 regimes

2. if x < 9.1999999999999996e-265

   1. Initial program 74.7%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}}{n \cdot x}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}}{n \cdot x}} \]

      2. log-rec N/A

         \[\leadsto \frac{e^{-1 \cdot \frac{\color{blue}{\mathsf{neg}\left(\log x\right)}}{n}}}{n \cdot x} \]

      3. mul-1-neg N/A

         \[\leadsto \frac{e^{-1 \cdot \frac{\color{blue}{-1 \cdot \log x}}{n}}}{n \cdot x} \]

      4. associate-*r/ N/A

         \[\leadsto \frac{e^{\color{blue}{\frac{-1 \cdot \left(-1 \cdot \log x\right)}{n}}}}{n \cdot x} \]

      5. associate-*r* N/A

         \[\leadsto \frac{e^{\frac{\color{blue}{\left(-1 \cdot -1\right) \cdot \log x}}{n}}}{n \cdot x} \]

      6. metadata-eval N/A

         \[\leadsto \frac{e^{\frac{\color{blue}{1} \cdot \log x}{n}}}{n \cdot x} \]

      7. *-commutative N/A

         \[\leadsto \frac{e^{\frac{\color{blue}{\log x \cdot 1}}{n}}}{n \cdot x} \]

      8. associate-/l* N/A

         \[\leadsto \frac{e^{\color{blue}{\log x \cdot \frac{1}{n}}}}{n \cdot x} \]

      9. exp-to-pow N/A

         \[\leadsto \frac{\color{blue}{{x}^{\left(\frac{1}{n}\right)}}}{n \cdot x} \]

      10. lower-pow.f64 N/A

          \[\leadsto \frac{\color{blue}{{x}^{\left(\frac{1}{n}\right)}}}{n \cdot x} \]

      11. lower-/.f64 N/A

          \[\leadsto \frac{{x}^{\color{blue}{\left(\frac{1}{n}\right)}}}{n \cdot x} \]

      12. *-commutative N/A

          \[\leadsto \frac{{x}^{\left(\frac{1}{n}\right)}}{\color{blue}{x \cdot n}} \]

      13. lower-*.f64 39.2

          \[\leadsto \frac{{x}^{\left(\frac{1}{n}\right)}}{\color{blue}{x \cdot n}} \]

   5. Simplified 39.2%

      \[\leadsto \color{blue}{\frac{{x}^{\left(\frac{1}{n}\right)}}{x \cdot n}} \]

   6. Taylor expanded in n around inf

      \[\leadsto \color{blue}{\frac{1}{n \cdot x}} \]
   7. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{n \cdot x}} \]

      2. *-commutative N/A

         \[\leadsto \frac{1}{\color{blue}{x \cdot n}} \]

      3. lower-*.f64 47.9

         \[\leadsto \frac{1}{\color{blue}{x \cdot n}} \]

   8. Simplified 47.9%

      \[\leadsto \color{blue}{\frac{1}{x \cdot n}} \]

   if 9.1999999999999996e-265 < x < 8.5000000000000005e-175

   1. Initial program 36.9%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1 - e^{\frac{\log x}{n}}} \]
   4. Step-by-step derivation

      1. remove-double-neg N/A

         \[\leadsto 1 - e^{\frac{\color{blue}{\mathsf{neg}\left(\left(\mathsf{neg}\left(\log x\right)\right)\right)}}{n}} \]

      2. mul-1-neg N/A

         \[\leadsto 1 - e^{\frac{\mathsf{neg}\left(\color{blue}{-1 \cdot \log x}\right)}{n}} \]

      3. distribute-neg-frac N/A

         \[\leadsto 1 - e^{\color{blue}{\mathsf{neg}\left(\frac{-1 \cdot \log x}{n}\right)}} \]

      4. mul-1-neg N/A

         \[\leadsto 1 - e^{\mathsf{neg}\left(\frac{\color{blue}{\mathsf{neg}\left(\log x\right)}}{n}\right)} \]

      5. log-rec N/A

         \[\leadsto 1 - e^{\mathsf{neg}\left(\frac{\color{blue}{\log \left(\frac{1}{x}\right)}}{n}\right)} \]

      6. mul-1-neg N/A

         \[\leadsto 1 - e^{\color{blue}{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}} \]

      7. lower--.f64 N/A

         \[\leadsto \color{blue}{1 - e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}} \]

      8. log-rec N/A

         \[\leadsto 1 - e^{-1 \cdot \frac{\color{blue}{\mathsf{neg}\left(\log x\right)}}{n}} \]

      9. mul-1-neg N/A

         \[\leadsto 1 - e^{-1 \cdot \frac{\color{blue}{-1 \cdot \log x}}{n}} \]

      10. associate-*r/ N/A

          \[\leadsto 1 - e^{\color{blue}{\frac{-1 \cdot \left(-1 \cdot \log x\right)}{n}}} \]

      11. associate-*r* N/A

          \[\leadsto 1 - e^{\frac{\color{blue}{\left(-1 \cdot -1\right) \cdot \log x}}{n}} \]

      12. metadata-eval N/A

          \[\leadsto 1 - e^{\frac{\color{blue}{1} \cdot \log x}{n}} \]

      13. *-commutative N/A

          \[\leadsto 1 - e^{\frac{\color{blue}{\log x \cdot 1}}{n}} \]

      14. associate-/l* N/A

          \[\leadsto 1 - e^{\color{blue}{\log x \cdot \frac{1}{n}}} \]

      15. exp-to-pow N/A

          \[\leadsto 1 - \color{blue}{{x}^{\left(\frac{1}{n}\right)}} \]

      16. lower-pow.f64 N/A

          \[\leadsto 1 - \color{blue}{{x}^{\left(\frac{1}{n}\right)}} \]

      17. lower-/.f64 36.9

          \[\leadsto 1 - {x}^{\color{blue}{\left(\frac{1}{n}\right)}} \]

   5. Simplified 36.9%

      \[\leadsto \color{blue}{1 - {x}^{\left(\frac{1}{n}\right)}} \]

   6. Taylor expanded in n around inf

      \[\leadsto \color{blue}{-1 \cdot \frac{\log x}{n}} \]
   7. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{\log x}{n}\right)} \]

      2. lower-neg.f64 N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{\log x}{n}\right)} \]

      3. lower-/.f64 N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\frac{\log x}{n}}\right) \]

      4. lower-log.f64 68.4

         \[\leadsto -\frac{\color{blue}{\log x}}{n} \]

   8. Simplified 68.4%

      \[\leadsto \color{blue}{-\frac{\log x}{n}} \]

   if 8.5000000000000005e-175 < x < 2.2999999999999999e81

   1. Initial program 45.8%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in n around inf

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]

      2. lower--.f64 N/A

         \[\leadsto \frac{\color{blue}{\log \left(1 + x\right) - \log x}}{n} \]

      3. lower-log1p.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(x\right)} - \log x}{n} \]

      4. lower-log.f64 41.2

         \[\leadsto \frac{\mathsf{log1p}\left(x\right) - \color{blue}{\log x}}{n} \]

   5. Simplified 41.2%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) - \log x}{n}} \]

   6. Taylor expanded in x around inf

      \[\leadsto \frac{\color{blue}{\frac{\left(1 + \frac{\frac{1}{3}}{{x}^{2}}\right) - \frac{1}{2} \cdot \frac{1}{x}}{x}}}{n} \]
   7. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{\left(1 + \frac{\frac{1}{3}}{{x}^{2}}\right) - \frac{1}{2} \cdot \frac{1}{x}}{x}}}{n} \]

   8. Simplified 50.6%

      \[\leadsto \frac{\color{blue}{\frac{1 + \frac{-0.5 + \frac{0.3333333333333333}{x}}{x}}{x}}}{n} \]

   if 2.2999999999999999e81 < x

   1. Initial program 77.6%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1 - e^{\frac{\log x}{n}}} \]
   4. Step-by-step derivation

      1. remove-double-neg N/A

         \[\leadsto 1 - e^{\frac{\color{blue}{\mathsf{neg}\left(\left(\mathsf{neg}\left(\log x\right)\right)\right)}}{n}} \]

      2. mul-1-neg N/A

         \[\leadsto 1 - e^{\frac{\mathsf{neg}\left(\color{blue}{-1 \cdot \log x}\right)}{n}} \]

      3. distribute-neg-frac N/A

         \[\leadsto 1 - e^{\color{blue}{\mathsf{neg}\left(\frac{-1 \cdot \log x}{n}\right)}} \]

      4. mul-1-neg N/A

         \[\leadsto 1 - e^{\mathsf{neg}\left(\frac{\color{blue}{\mathsf{neg}\left(\log x\right)}}{n}\right)} \]

      5. log-rec N/A

         \[\leadsto 1 - e^{\mathsf{neg}\left(\frac{\color{blue}{\log \left(\frac{1}{x}\right)}}{n}\right)} \]

      6. mul-1-neg N/A

         \[\leadsto 1 - e^{\color{blue}{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}} \]

      7. lower--.f64 N/A

         \[\leadsto \color{blue}{1 - e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}} \]

      8. log-rec N/A

         \[\leadsto 1 - e^{-1 \cdot \frac{\color{blue}{\mathsf{neg}\left(\log x\right)}}{n}} \]

      9. mul-1-neg N/A

         \[\leadsto 1 - e^{-1 \cdot \frac{\color{blue}{-1 \cdot \log x}}{n}} \]

      10. associate-*r/ N/A

          \[\leadsto 1 - e^{\color{blue}{\frac{-1 \cdot \left(-1 \cdot \log x\right)}{n}}} \]

      11. associate-*r* N/A

          \[\leadsto 1 - e^{\frac{\color{blue}{\left(-1 \cdot -1\right) \cdot \log x}}{n}} \]

      12. metadata-eval N/A

          \[\leadsto 1 - e^{\frac{\color{blue}{1} \cdot \log x}{n}} \]

      13. *-commutative N/A

          \[\leadsto 1 - e^{\frac{\color{blue}{\log x \cdot 1}}{n}} \]

      14. associate-/l* N/A

          \[\leadsto 1 - e^{\color{blue}{\log x \cdot \frac{1}{n}}} \]

      15. exp-to-pow N/A

          \[\leadsto 1 - \color{blue}{{x}^{\left(\frac{1}{n}\right)}} \]

      16. lower-pow.f64 N/A

          \[\leadsto 1 - \color{blue}{{x}^{\left(\frac{1}{n}\right)}} \]

      17. lower-/.f64 42.8

          \[\leadsto 1 - {x}^{\color{blue}{\left(\frac{1}{n}\right)}} \]

   5. Simplified 42.8%

      \[\leadsto \color{blue}{1 - {x}^{\left(\frac{1}{n}\right)}} \]

   6. Taylor expanded in n around inf

      \[\leadsto 1 - \color{blue}{1} \]
   7. Step-by-step derivation

   8. Simplified 77.6%

      \[\leadsto 1 - \color{blue}{1} \]
   9. Step-by-step derivation

      1. metadata-eval 77.6

         \[\leadsto \color{blue}{0} \]

   10. Applied egg-rr 77.6%

       \[\leadsto \color{blue}{0} \]
3. Recombined 4 regimes into one program.

4. Final simplification 61.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 9.2 \cdot 10^{-265}:\\ \;\;\;\;\frac{1}{n \cdot x}\\ \mathbf{elif}\;x \leq 8.5 \cdot 10^{-175}:\\ \;\;\;\;-\frac{\log x}{n}\\ \mathbf{elif}\;x \leq 2.3 \cdot 10^{+81}:\\ \;\;\;\;\frac{\frac{1 + \frac{-0.5 + \frac{0.3333333333333333}{x}}{x}}{x}}{n}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 48.9% accurate, 4.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2.5 \cdot 10^{+82}:\\ \;\;\;\;\frac{\frac{1 + \frac{-0.5 + \frac{0.3333333333333333}{x}}{x}}{x}}{n}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 2.5e+82)
   (/ (/ (+ 1.0 (/ (+ -0.5 (/ 0.3333333333333333 x)) x)) x) n)
   0.0))

C

double code(double x, double n) {
	double tmp;
	if (x <= 2.5e+82) {
		tmp = ((1.0 + ((-0.5 + (0.3333333333333333 / x)) / x)) / x) / n;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if (x <= 2.5d+82) then
        tmp = ((1.0d0 + (((-0.5d0) + (0.3333333333333333d0 / x)) / x)) / x) / n
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (x <= 2.5e+82) {
		tmp = ((1.0 + ((-0.5 + (0.3333333333333333 / x)) / x)) / x) / n;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 2.5e+82:
		tmp = ((1.0 + ((-0.5 + (0.3333333333333333 / x)) / x)) / x) / n
	else:
		tmp = 0.0
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 2.5e+82)
		tmp = Float64(Float64(Float64(1.0 + Float64(Float64(-0.5 + Float64(0.3333333333333333 / x)) / x)) / x) / n);
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (x <= 2.5e+82)
		tmp = ((1.0 + ((-0.5 + (0.3333333333333333 / x)) / x)) / x) / n;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 2.5e+82], N[(N[(N[(1.0 + N[(N[(-0.5 + N[(0.3333333333333333 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] / n), $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if x < 2.50000000000000008e82

   1. Initial program 48.1%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in n around inf

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]

      2. lower--.f64 N/A

         \[\leadsto \frac{\color{blue}{\log \left(1 + x\right) - \log x}}{n} \]

      3. lower-log1p.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(x\right)} - \log x}{n} \]

      4. lower-log.f64 44.2

         \[\leadsto \frac{\mathsf{log1p}\left(x\right) - \color{blue}{\log x}}{n} \]

   5. Simplified 44.2%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) - \log x}{n}} \]

   6. Taylor expanded in x around inf

      \[\leadsto \frac{\color{blue}{\frac{\left(1 + \frac{\frac{1}{3}}{{x}^{2}}\right) - \frac{1}{2} \cdot \frac{1}{x}}{x}}}{n} \]
   7. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{\left(1 + \frac{\frac{1}{3}}{{x}^{2}}\right) - \frac{1}{2} \cdot \frac{1}{x}}{x}}}{n} \]

   8. Simplified 42.8%

      \[\leadsto \frac{\color{blue}{\frac{1 + \frac{-0.5 + \frac{0.3333333333333333}{x}}{x}}{x}}}{n} \]

   if 2.50000000000000008e82 < x

   1. Initial program 77.6%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1 - e^{\frac{\log x}{n}}} \]
   4. Step-by-step derivation

      1. remove-double-neg N/A

         \[\leadsto 1 - e^{\frac{\color{blue}{\mathsf{neg}\left(\left(\mathsf{neg}\left(\log x\right)\right)\right)}}{n}} \]

      2. mul-1-neg N/A

         \[\leadsto 1 - e^{\frac{\mathsf{neg}\left(\color{blue}{-1 \cdot \log x}\right)}{n}} \]

      3. distribute-neg-frac N/A

         \[\leadsto 1 - e^{\color{blue}{\mathsf{neg}\left(\frac{-1 \cdot \log x}{n}\right)}} \]

      4. mul-1-neg N/A

         \[\leadsto 1 - e^{\mathsf{neg}\left(\frac{\color{blue}{\mathsf{neg}\left(\log x\right)}}{n}\right)} \]

      5. log-rec N/A

         \[\leadsto 1 - e^{\mathsf{neg}\left(\frac{\color{blue}{\log \left(\frac{1}{x}\right)}}{n}\right)} \]

      6. mul-1-neg N/A

         \[\leadsto 1 - e^{\color{blue}{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}} \]

      7. lower--.f64 N/A

         \[\leadsto \color{blue}{1 - e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}} \]

      8. log-rec N/A

         \[\leadsto 1 - e^{-1 \cdot \frac{\color{blue}{\mathsf{neg}\left(\log x\right)}}{n}} \]

      9. mul-1-neg N/A

         \[\leadsto 1 - e^{-1 \cdot \frac{\color{blue}{-1 \cdot \log x}}{n}} \]

      10. associate-*r/ N/A

          \[\leadsto 1 - e^{\color{blue}{\frac{-1 \cdot \left(-1 \cdot \log x\right)}{n}}} \]

      11. associate-*r* N/A

          \[\leadsto 1 - e^{\frac{\color{blue}{\left(-1 \cdot -1\right) \cdot \log x}}{n}} \]

      12. metadata-eval N/A

          \[\leadsto 1 - e^{\frac{\color{blue}{1} \cdot \log x}{n}} \]

      13. *-commutative N/A

          \[\leadsto 1 - e^{\frac{\color{blue}{\log x \cdot 1}}{n}} \]

      14. associate-/l* N/A

          \[\leadsto 1 - e^{\color{blue}{\log x \cdot \frac{1}{n}}} \]

      15. exp-to-pow N/A

          \[\leadsto 1 - \color{blue}{{x}^{\left(\frac{1}{n}\right)}} \]

      16. lower-pow.f64 N/A

          \[\leadsto 1 - \color{blue}{{x}^{\left(\frac{1}{n}\right)}} \]

      17. lower-/.f64 42.8

          \[\leadsto 1 - {x}^{\color{blue}{\left(\frac{1}{n}\right)}} \]

   5. Simplified 42.8%

      \[\leadsto \color{blue}{1 - {x}^{\left(\frac{1}{n}\right)}} \]

   6. Taylor expanded in n around inf

      \[\leadsto 1 - \color{blue}{1} \]
   7. Step-by-step derivation

   8. Simplified 77.6%

      \[\leadsto 1 - \color{blue}{1} \]
   9. Step-by-step derivation

      1. metadata-eval 77.6

         \[\leadsto \color{blue}{0} \]

   10. Applied egg-rr 77.6%

       \[\leadsto \color{blue}{0} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 18: 43.6% accurate, 8.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 5.8 \cdot 10^{+81}:\\ \;\;\;\;\frac{\frac{1}{n}}{x}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n) :precision binary64 (if (<= x 5.8e+81) (/ (/ 1.0 n) x) 0.0))

C

double code(double x, double n) {
	double tmp;
	if (x <= 5.8e+81) {
		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 (x <= 5.8d+81) 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 (x <= 5.8e+81) {
		tmp = (1.0 / n) / x;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 5.8e+81:
		tmp = (1.0 / n) / x
	else:
		tmp = 0.0
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 5.8e+81)
		tmp = Float64(Float64(1.0 / n) / x);
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (x <= 5.8e+81)
		tmp = (1.0 / n) / x;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 5.8e+81], N[(N[(1.0 / n), $MachinePrecision] / x), $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if x < 5.7999999999999999e81

   1. Initial program 48.1%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}}{n \cdot x}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}}{n \cdot x}} \]

      2. log-rec N/A

         \[\leadsto \frac{e^{-1 \cdot \frac{\color{blue}{\mathsf{neg}\left(\log x\right)}}{n}}}{n \cdot x} \]

      3. mul-1-neg N/A

         \[\leadsto \frac{e^{-1 \cdot \frac{\color{blue}{-1 \cdot \log x}}{n}}}{n \cdot x} \]

      4. associate-*r/ N/A

         \[\leadsto \frac{e^{\color{blue}{\frac{-1 \cdot \left(-1 \cdot \log x\right)}{n}}}}{n \cdot x} \]

      5. associate-*r* N/A

         \[\leadsto \frac{e^{\frac{\color{blue}{\left(-1 \cdot -1\right) \cdot \log x}}{n}}}{n \cdot x} \]

      6. metadata-eval N/A

         \[\leadsto \frac{e^{\frac{\color{blue}{1} \cdot \log x}{n}}}{n \cdot x} \]

      7. *-commutative N/A

         \[\leadsto \frac{e^{\frac{\color{blue}{\log x \cdot 1}}{n}}}{n \cdot x} \]

      8. associate-/l* N/A

         \[\leadsto \frac{e^{\color{blue}{\log x \cdot \frac{1}{n}}}}{n \cdot x} \]

      9. exp-to-pow N/A

         \[\leadsto \frac{\color{blue}{{x}^{\left(\frac{1}{n}\right)}}}{n \cdot x} \]

      10. lower-pow.f64 N/A

          \[\leadsto \frac{\color{blue}{{x}^{\left(\frac{1}{n}\right)}}}{n \cdot x} \]

      11. lower-/.f64 N/A

          \[\leadsto \frac{{x}^{\color{blue}{\left(\frac{1}{n}\right)}}}{n \cdot x} \]

      12. *-commutative N/A

          \[\leadsto \frac{{x}^{\left(\frac{1}{n}\right)}}{\color{blue}{x \cdot n}} \]

      13. lower-*.f64 42.9

          \[\leadsto \frac{{x}^{\left(\frac{1}{n}\right)}}{\color{blue}{x \cdot n}} \]

   5. Simplified 42.9%

      \[\leadsto \color{blue}{\frac{{x}^{\left(\frac{1}{n}\right)}}{x \cdot n}} \]
   6. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{{x}^{\color{blue}{\left(\frac{1}{n}\right)}}}{x \cdot n} \]

      2. lift-pow.f64 N/A

         \[\leadsto \frac{\color{blue}{{x}^{\left(\frac{1}{n}\right)}}}{x \cdot n} \]

      3. *-commutative N/A

         \[\leadsto \frac{{x}^{\left(\frac{1}{n}\right)}}{\color{blue}{n \cdot x}} \]

      4. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{{x}^{\left(\frac{1}{n}\right)}}{n}}{x}} \]

      5. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{{x}^{\left(\frac{1}{n}\right)}}{n}}{x}} \]

      6. lower-/.f64 43.8

         \[\leadsto \frac{\color{blue}{\frac{{x}^{\left(\frac{1}{n}\right)}}{n}}}{x} \]

   7. Applied egg-rr 43.8%

      \[\leadsto \color{blue}{\frac{\frac{{x}^{\left(\frac{1}{n}\right)}}{n}}{x}} \]

   8. Taylor expanded in n around inf

      \[\leadsto \frac{\color{blue}{\frac{1}{n}}}{x} \]
   9. Step-by-step derivation

      1. lower-/.f64 33.8

         \[\leadsto \frac{\color{blue}{\frac{1}{n}}}{x} \]

   10. Simplified 33.8%

       \[\leadsto \frac{\color{blue}{\frac{1}{n}}}{x} \]

   if 5.7999999999999999e81 < x

   1. Initial program 77.6%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1 - e^{\frac{\log x}{n}}} \]
   4. Step-by-step derivation

      1. remove-double-neg N/A

         \[\leadsto 1 - e^{\frac{\color{blue}{\mathsf{neg}\left(\left(\mathsf{neg}\left(\log x\right)\right)\right)}}{n}} \]

      2. mul-1-neg N/A

         \[\leadsto 1 - e^{\frac{\mathsf{neg}\left(\color{blue}{-1 \cdot \log x}\right)}{n}} \]

      3. distribute-neg-frac N/A

         \[\leadsto 1 - e^{\color{blue}{\mathsf{neg}\left(\frac{-1 \cdot \log x}{n}\right)}} \]

      4. mul-1-neg N/A

         \[\leadsto 1 - e^{\mathsf{neg}\left(\frac{\color{blue}{\mathsf{neg}\left(\log x\right)}}{n}\right)} \]

      5. log-rec N/A

         \[\leadsto 1 - e^{\mathsf{neg}\left(\frac{\color{blue}{\log \left(\frac{1}{x}\right)}}{n}\right)} \]

      6. mul-1-neg N/A

         \[\leadsto 1 - e^{\color{blue}{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}} \]

      7. lower--.f64 N/A

         \[\leadsto \color{blue}{1 - e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}} \]

      8. log-rec N/A

         \[\leadsto 1 - e^{-1 \cdot \frac{\color{blue}{\mathsf{neg}\left(\log x\right)}}{n}} \]

      9. mul-1-neg N/A

         \[\leadsto 1 - e^{-1 \cdot \frac{\color{blue}{-1 \cdot \log x}}{n}} \]

      10. associate-*r/ N/A

          \[\leadsto 1 - e^{\color{blue}{\frac{-1 \cdot \left(-1 \cdot \log x\right)}{n}}} \]

      11. associate-*r* N/A

          \[\leadsto 1 - e^{\frac{\color{blue}{\left(-1 \cdot -1\right) \cdot \log x}}{n}} \]

      12. metadata-eval N/A

          \[\leadsto 1 - e^{\frac{\color{blue}{1} \cdot \log x}{n}} \]

      13. *-commutative N/A

          \[\leadsto 1 - e^{\frac{\color{blue}{\log x \cdot 1}}{n}} \]

      14. associate-/l* N/A

          \[\leadsto 1 - e^{\color{blue}{\log x \cdot \frac{1}{n}}} \]

      15. exp-to-pow N/A

          \[\leadsto 1 - \color{blue}{{x}^{\left(\frac{1}{n}\right)}} \]

      16. lower-pow.f64 N/A

          \[\leadsto 1 - \color{blue}{{x}^{\left(\frac{1}{n}\right)}} \]

      17. lower-/.f64 42.8

          \[\leadsto 1 - {x}^{\color{blue}{\left(\frac{1}{n}\right)}} \]

   5. Simplified 42.8%

      \[\leadsto \color{blue}{1 - {x}^{\left(\frac{1}{n}\right)}} \]

   6. Taylor expanded in n around inf

      \[\leadsto 1 - \color{blue}{1} \]
   7. Step-by-step derivation

   8. Simplified 77.6%

      \[\leadsto 1 - \color{blue}{1} \]
   9. Step-by-step derivation

      1. metadata-eval 77.6

         \[\leadsto \color{blue}{0} \]

   10. Applied egg-rr 77.6%

       \[\leadsto \color{blue}{0} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 19: 43.5% accurate, 10.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 7.5 \cdot 10^{+82}:\\ \;\;\;\;\frac{1}{n \cdot x}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n) :precision binary64 (if (<= x 7.5e+82) (/ 1.0 (* n x)) 0.0))

C

double code(double x, double n) {
	double tmp;
	if (x <= 7.5e+82) {
		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 (x <= 7.5d+82) 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 (x <= 7.5e+82) {
		tmp = 1.0 / (n * x);
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 7.5e+82:
		tmp = 1.0 / (n * x)
	else:
		tmp = 0.0
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 7.5e+82)
		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 (x <= 7.5e+82)
		tmp = 1.0 / (n * x);
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 7.5e+82], N[(1.0 / N[(n * x), $MachinePrecision]), $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if x < 7.4999999999999999e82

   1. Initial program 48.1%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}}{n \cdot x}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}}{n \cdot x}} \]

      2. log-rec N/A

         \[\leadsto \frac{e^{-1 \cdot \frac{\color{blue}{\mathsf{neg}\left(\log x\right)}}{n}}}{n \cdot x} \]

      3. mul-1-neg N/A

         \[\leadsto \frac{e^{-1 \cdot \frac{\color{blue}{-1 \cdot \log x}}{n}}}{n \cdot x} \]

      4. associate-*r/ N/A

         \[\leadsto \frac{e^{\color{blue}{\frac{-1 \cdot \left(-1 \cdot \log x\right)}{n}}}}{n \cdot x} \]

      5. associate-*r* N/A

         \[\leadsto \frac{e^{\frac{\color{blue}{\left(-1 \cdot -1\right) \cdot \log x}}{n}}}{n \cdot x} \]

      6. metadata-eval N/A

         \[\leadsto \frac{e^{\frac{\color{blue}{1} \cdot \log x}{n}}}{n \cdot x} \]

      7. *-commutative N/A

         \[\leadsto \frac{e^{\frac{\color{blue}{\log x \cdot 1}}{n}}}{n \cdot x} \]

      8. associate-/l* N/A

         \[\leadsto \frac{e^{\color{blue}{\log x \cdot \frac{1}{n}}}}{n \cdot x} \]

      9. exp-to-pow N/A

         \[\leadsto \frac{\color{blue}{{x}^{\left(\frac{1}{n}\right)}}}{n \cdot x} \]

      10. lower-pow.f64 N/A

          \[\leadsto \frac{\color{blue}{{x}^{\left(\frac{1}{n}\right)}}}{n \cdot x} \]

      11. lower-/.f64 N/A

          \[\leadsto \frac{{x}^{\color{blue}{\left(\frac{1}{n}\right)}}}{n \cdot x} \]

      12. *-commutative N/A

          \[\leadsto \frac{{x}^{\left(\frac{1}{n}\right)}}{\color{blue}{x \cdot n}} \]

      13. lower-*.f64 42.9

          \[\leadsto \frac{{x}^{\left(\frac{1}{n}\right)}}{\color{blue}{x \cdot n}} \]

   5. Simplified 42.9%

      \[\leadsto \color{blue}{\frac{{x}^{\left(\frac{1}{n}\right)}}{x \cdot n}} \]

   6. Taylor expanded in n around inf

      \[\leadsto \color{blue}{\frac{1}{n \cdot x}} \]
   7. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{n \cdot x}} \]

      2. *-commutative N/A

         \[\leadsto \frac{1}{\color{blue}{x \cdot n}} \]

      3. lower-*.f64 33.2

         \[\leadsto \frac{1}{\color{blue}{x \cdot n}} \]

   8. Simplified 33.2%

      \[\leadsto \color{blue}{\frac{1}{x \cdot n}} \]

   if 7.4999999999999999e82 < x

   1. Initial program 77.6%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1 - e^{\frac{\log x}{n}}} \]
   4. Step-by-step derivation

      1. remove-double-neg N/A

         \[\leadsto 1 - e^{\frac{\color{blue}{\mathsf{neg}\left(\left(\mathsf{neg}\left(\log x\right)\right)\right)}}{n}} \]

      2. mul-1-neg N/A

         \[\leadsto 1 - e^{\frac{\mathsf{neg}\left(\color{blue}{-1 \cdot \log x}\right)}{n}} \]

      3. distribute-neg-frac N/A

         \[\leadsto 1 - e^{\color{blue}{\mathsf{neg}\left(\frac{-1 \cdot \log x}{n}\right)}} \]

      4. mul-1-neg N/A

         \[\leadsto 1 - e^{\mathsf{neg}\left(\frac{\color{blue}{\mathsf{neg}\left(\log x\right)}}{n}\right)} \]

      5. log-rec N/A

         \[\leadsto 1 - e^{\mathsf{neg}\left(\frac{\color{blue}{\log \left(\frac{1}{x}\right)}}{n}\right)} \]

      6. mul-1-neg N/A

         \[\leadsto 1 - e^{\color{blue}{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}} \]

      7. lower--.f64 N/A

         \[\leadsto \color{blue}{1 - e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}} \]

      8. log-rec N/A

         \[\leadsto 1 - e^{-1 \cdot \frac{\color{blue}{\mathsf{neg}\left(\log x\right)}}{n}} \]

      9. mul-1-neg N/A

         \[\leadsto 1 - e^{-1 \cdot \frac{\color{blue}{-1 \cdot \log x}}{n}} \]

      10. associate-*r/ N/A

          \[\leadsto 1 - e^{\color{blue}{\frac{-1 \cdot \left(-1 \cdot \log x\right)}{n}}} \]

      11. associate-*r* N/A

          \[\leadsto 1 - e^{\frac{\color{blue}{\left(-1 \cdot -1\right) \cdot \log x}}{n}} \]

      12. metadata-eval N/A

          \[\leadsto 1 - e^{\frac{\color{blue}{1} \cdot \log x}{n}} \]

      13. *-commutative N/A

          \[\leadsto 1 - e^{\frac{\color{blue}{\log x \cdot 1}}{n}} \]

      14. associate-/l* N/A

          \[\leadsto 1 - e^{\color{blue}{\log x \cdot \frac{1}{n}}} \]

      15. exp-to-pow N/A

          \[\leadsto 1 - \color{blue}{{x}^{\left(\frac{1}{n}\right)}} \]

      16. lower-pow.f64 N/A

          \[\leadsto 1 - \color{blue}{{x}^{\left(\frac{1}{n}\right)}} \]

      17. lower-/.f64 42.8

          \[\leadsto 1 - {x}^{\color{blue}{\left(\frac{1}{n}\right)}} \]

   5. Simplified 42.8%

      \[\leadsto \color{blue}{1 - {x}^{\left(\frac{1}{n}\right)}} \]

   6. Taylor expanded in n around inf

      \[\leadsto 1 - \color{blue}{1} \]
   7. Step-by-step derivation

   8. Simplified 77.6%

      \[\leadsto 1 - \color{blue}{1} \]
   9. Step-by-step derivation

      1. metadata-eval 77.6

         \[\leadsto \color{blue}{0} \]

   10. Applied egg-rr 77.6%

       \[\leadsto \color{blue}{0} \]
3. Recombined 2 regimes into one program.

4. Final simplification 47.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 7.5 \cdot 10^{+82}:\\ \;\;\;\;\frac{1}{n \cdot x}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 31.1% accurate, 231.0× 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 57.4%

   \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{1 - e^{\frac{\log x}{n}}} \]
4. Step-by-step derivation

   1. remove-double-neg N/A

      \[\leadsto 1 - e^{\frac{\color{blue}{\mathsf{neg}\left(\left(\mathsf{neg}\left(\log x\right)\right)\right)}}{n}} \]

   2. mul-1-neg N/A

      \[\leadsto 1 - e^{\frac{\mathsf{neg}\left(\color{blue}{-1 \cdot \log x}\right)}{n}} \]

   3. distribute-neg-frac N/A

      \[\leadsto 1 - e^{\color{blue}{\mathsf{neg}\left(\frac{-1 \cdot \log x}{n}\right)}} \]

   4. mul-1-neg N/A

      \[\leadsto 1 - e^{\mathsf{neg}\left(\frac{\color{blue}{\mathsf{neg}\left(\log x\right)}}{n}\right)} \]

   5. log-rec N/A

      \[\leadsto 1 - e^{\mathsf{neg}\left(\frac{\color{blue}{\log \left(\frac{1}{x}\right)}}{n}\right)} \]

   6. mul-1-neg N/A

      \[\leadsto 1 - e^{\color{blue}{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}} \]

   7. lower--.f64 N/A

      \[\leadsto \color{blue}{1 - e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}} \]

   8. log-rec N/A

      \[\leadsto 1 - e^{-1 \cdot \frac{\color{blue}{\mathsf{neg}\left(\log x\right)}}{n}} \]

   9. mul-1-neg N/A

      \[\leadsto 1 - e^{-1 \cdot \frac{\color{blue}{-1 \cdot \log x}}{n}} \]

   10. associate-*r/ N/A

       \[\leadsto 1 - e^{\color{blue}{\frac{-1 \cdot \left(-1 \cdot \log x\right)}{n}}} \]

   11. associate-*r* N/A

       \[\leadsto 1 - e^{\frac{\color{blue}{\left(-1 \cdot -1\right) \cdot \log x}}{n}} \]

   12. metadata-eval N/A

       \[\leadsto 1 - e^{\frac{\color{blue}{1} \cdot \log x}{n}} \]

   13. *-commutative N/A

       \[\leadsto 1 - e^{\frac{\color{blue}{\log x \cdot 1}}{n}} \]

   14. associate-/l* N/A

       \[\leadsto 1 - e^{\color{blue}{\log x \cdot \frac{1}{n}}} \]

   15. exp-to-pow N/A

       \[\leadsto 1 - \color{blue}{{x}^{\left(\frac{1}{n}\right)}} \]

   16. lower-pow.f64 N/A

       \[\leadsto 1 - \color{blue}{{x}^{\left(\frac{1}{n}\right)}} \]

   17. lower-/.f64 42.9

       \[\leadsto 1 - {x}^{\color{blue}{\left(\frac{1}{n}\right)}} \]

5. Simplified 42.9%

   \[\leadsto \color{blue}{1 - {x}^{\left(\frac{1}{n}\right)}} \]

6. Taylor expanded in n around inf

   \[\leadsto 1 - \color{blue}{1} \]
7. Step-by-step derivation

8. Simplified 31.2%

   \[\leadsto 1 - \color{blue}{1} \]
9. Step-by-step derivation

   1. metadata-eval 31.2

      \[\leadsto \color{blue}{0} \]

10. Applied egg-rr 31.2%

    \[\leadsto \color{blue}{0} \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024207 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  :precision binary64
  (- (pow (+ x 1.0) (/ 1.0 n)) (pow x (/ 1.0 n))))