2nthrt (problem 3.4.6)

Percentage Accurate: 53.4% → 86.5%

Time: 25.9s

Alternatives: 20

Speedup: 5.2×

• Could not converge on a ground truth

  1. x = -1.4089662910261981e+31
  2. n = -6.731318857791573e-302

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: 53.4% 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: 86.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\frac{e^{\frac{\log x}{n}}}{n}}{x}\\ \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{-11}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-50}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right) - \log x}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 1:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;{\left(e^{-1}\right)}^{\left(\frac{\mathsf{log1p}\left(x\right)}{-n}\right)} - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (/ (/ (exp (/ (log x) n)) n) x)))
   (if (<= (/ 1.0 n) -2e-11)
     t_0
     (if (<= (/ 1.0 n) 2e-50)
       (/ (- (log1p x) (log x)) n)
       (if (<= (/ 1.0 n) 1.0)
         t_0
         (- (pow (exp -1.0) (/ (log1p x) (- n))) (pow x (/ 1.0 n))))))))

C

double code(double x, double n) {
	double t_0 = (exp((log(x) / n)) / n) / x;
	double tmp;
	if ((1.0 / n) <= -2e-11) {
		tmp = t_0;
	} else if ((1.0 / n) <= 2e-50) {
		tmp = (log1p(x) - log(x)) / n;
	} else if ((1.0 / n) <= 1.0) {
		tmp = t_0;
	} else {
		tmp = pow(exp(-1.0), (log1p(x) / -n)) - pow(x, (1.0 / n));
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double t_0 = (Math.exp((Math.log(x) / n)) / n) / x;
	double tmp;
	if ((1.0 / n) <= -2e-11) {
		tmp = t_0;
	} else if ((1.0 / n) <= 2e-50) {
		tmp = (Math.log1p(x) - Math.log(x)) / n;
	} else if ((1.0 / n) <= 1.0) {
		tmp = t_0;
	} else {
		tmp = Math.pow(Math.exp(-1.0), (Math.log1p(x) / -n)) - Math.pow(x, (1.0 / n));
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = (math.exp((math.log(x) / n)) / n) / x
	tmp = 0
	if (1.0 / n) <= -2e-11:
		tmp = t_0
	elif (1.0 / n) <= 2e-50:
		tmp = (math.log1p(x) - math.log(x)) / n
	elif (1.0 / n) <= 1.0:
		tmp = t_0
	else:
		tmp = math.pow(math.exp(-1.0), (math.log1p(x) / -n)) - math.pow(x, (1.0 / n))
	return tmp

Julia

function code(x, n)
	t_0 = Float64(Float64(exp(Float64(log(x) / n)) / n) / x)
	tmp = 0.0
	if (Float64(1.0 / n) <= -2e-11)
		tmp = t_0;
	elseif (Float64(1.0 / n) <= 2e-50)
		tmp = Float64(Float64(log1p(x) - log(x)) / n);
	elseif (Float64(1.0 / n) <= 1.0)
		tmp = t_0;
	else
		tmp = Float64((exp(-1.0) ^ Float64(log1p(x) / Float64(-n))) - (x ^ Float64(1.0 / n)));
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(N[(N[Exp[N[(N[Log[x], $MachinePrecision] / n), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision] / x), $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -2e-11], t$95$0, If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-50], N[(N[(N[Log[1 + x], $MachinePrecision] - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 1.0], t$95$0, N[(N[Power[N[Exp[-1.0], $MachinePrecision], 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) < -1.99999999999999988e-11 or 2.00000000000000002e-50 < (/.f64 #s(literal 1 binary64) n) < 1

   1. Initial program 88.0%

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

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{\frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}}{n} + \left(\frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}} \cdot \left(\frac{1}{2} \cdot \frac{1}{{n}^{2}} - \frac{1}{2} \cdot \frac{1}{n}\right)}{x} + \frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}} \cdot \left(\left(\frac{1}{6} \cdot \frac{1}{{n}^{3}} + \frac{1}{3} \cdot \frac{1}{n}\right) - \frac{1}{2} \cdot \frac{1}{{n}^{2}}\right)}{{x}^{2}}\right)}{x}} \]

   5. Simplified 51.5%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\frac{{x}^{\left(\frac{1}{n}\right)}}{x}, \left(\frac{0.5}{n \cdot n} + \frac{-0.5}{n}\right) + \frac{\frac{0.16666666666666666}{n \cdot \left(n \cdot n\right)} + \left(\frac{0.3333333333333333}{n} + \frac{-0.5}{n \cdot n}\right)}{x}, \frac{{x}^{\left(\frac{1}{n}\right)}}{n}\right)}{x}} \]

   6. Taylor expanded in x around inf

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

      1. mul-1-neg N/A

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

      2. log-rec N/A

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

      3. distribute-frac-neg N/A

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

      4. remove-double-neg N/A

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

      5. /-lowering-/.f64 N/A

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

      6. exp-lowering-exp.f64 N/A

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

      7. /-lowering-/.f64 N/A

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

      8. log-lowering-log.f64 96.2

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

   8. Simplified 96.2%

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

   if -1.99999999999999988e-11 < (/.f64 #s(literal 1 binary64) n) < 2.00000000000000002e-50

   1. Initial program 31.3%

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

   3. Taylor expanded in n around inf

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

      1. /-lowering-/.f64 N/A

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

      2. --lowering--.f64 N/A

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

      3. accelerator-lowering-log1p.f64 N/A

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

      4. log-lowering-log.f64 82.6

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

   5. Simplified 82.6%

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

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

   1. Initial program 61.7%

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

      1. pow-to-exp N/A

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

      2. exp-lowering-exp.f64 N/A

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

      3. un-div-inv N/A

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

      4. /-lowering-/.f64 N/A

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

      5. +-commutative N/A

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

      6. accelerator-lowering-log1p.f64 99.8

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

   4. Applied egg-rr 99.8%

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

      1. frac-2neg N/A

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

      2. distribute-frac-neg2 N/A

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

      3. exp-neg N/A

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

      4. /-lowering-/.f64 N/A

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

      5. exp-lowering-exp.f64 N/A

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

      6. /-lowering-/.f64 N/A

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

      7. neg-lowering-neg.f64 N/A

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

      8. accelerator-lowering-log1p.f64 99.7

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

   6. Applied egg-rr 99.7%

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

      1. rec-exp N/A

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

      2. neg-mul-1 N/A

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

      3. exp-prod N/A

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

      4. pow-lowering-pow.f64 N/A

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

      5. exp-lowering-exp.f64 N/A

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

      6. remove-double-neg N/A

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

      7. neg-lowering-neg.f64 N/A

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

      8. distribute-frac-neg N/A

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

      9. remove-double-neg N/A

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

      10. /-lowering-/.f64 N/A

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

      11. accelerator-lowering-log1p.f64 99.8

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

   8. Applied egg-rr 99.8%

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

4. Final simplification 90.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{-11}:\\ \;\;\;\;\frac{\frac{e^{\frac{\log x}{n}}}{n}}{x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-50}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right) - \log x}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 1:\\ \;\;\;\;\frac{\frac{e^{\frac{\log x}{n}}}{n}}{x}\\ \mathbf{else}:\\ \;\;\;\;{\left(e^{-1}\right)}^{\left(\frac{\mathsf{log1p}\left(x\right)}{-n}\right)} - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 86.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\frac{e^{\frac{\log x}{n}}}{n}}{x}\\ \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{-11}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-50}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right) - \log x}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 1:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;{e}^{\left(\frac{x}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (/ (/ (exp (/ (log x) n)) n) x)))
   (if (<= (/ 1.0 n) -2e-11)
     t_0
     (if (<= (/ 1.0 n) 2e-50)
       (/ (- (log1p x) (log x)) n)
       (if (<= (/ 1.0 n) 1.0) t_0 (- (pow E (/ x n)) (pow x (/ 1.0 n))))))))

C

double code(double x, double n) {
	double t_0 = (exp((log(x) / n)) / n) / x;
	double tmp;
	if ((1.0 / n) <= -2e-11) {
		tmp = t_0;
	} else if ((1.0 / n) <= 2e-50) {
		tmp = (log1p(x) - log(x)) / n;
	} else if ((1.0 / n) <= 1.0) {
		tmp = t_0;
	} else {
		tmp = pow(((double) M_E), (x / n)) - pow(x, (1.0 / n));
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double t_0 = (Math.exp((Math.log(x) / n)) / n) / x;
	double tmp;
	if ((1.0 / n) <= -2e-11) {
		tmp = t_0;
	} else if ((1.0 / n) <= 2e-50) {
		tmp = (Math.log1p(x) - Math.log(x)) / n;
	} else if ((1.0 / n) <= 1.0) {
		tmp = t_0;
	} else {
		tmp = Math.pow(Math.E, (x / n)) - Math.pow(x, (1.0 / n));
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = (math.exp((math.log(x) / n)) / n) / x
	tmp = 0
	if (1.0 / n) <= -2e-11:
		tmp = t_0
	elif (1.0 / n) <= 2e-50:
		tmp = (math.log1p(x) - math.log(x)) / n
	elif (1.0 / n) <= 1.0:
		tmp = t_0
	else:
		tmp = math.pow(math.e, (x / n)) - math.pow(x, (1.0 / n))
	return tmp

Julia

function code(x, n)
	t_0 = Float64(Float64(exp(Float64(log(x) / n)) / n) / x)
	tmp = 0.0
	if (Float64(1.0 / n) <= -2e-11)
		tmp = t_0;
	elseif (Float64(1.0 / n) <= 2e-50)
		tmp = Float64(Float64(log1p(x) - log(x)) / n);
	elseif (Float64(1.0 / n) <= 1.0)
		tmp = t_0;
	else
		tmp = Float64((exp(1) ^ Float64(x / n)) - (x ^ Float64(1.0 / n)));
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(N[(N[Exp[N[(N[Log[x], $MachinePrecision] / n), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision] / x), $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -2e-11], t$95$0, If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-50], N[(N[(N[Log[1 + x], $MachinePrecision] - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 1.0], t$95$0, N[(N[Power[E, 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) < -1.99999999999999988e-11 or 2.00000000000000002e-50 < (/.f64 #s(literal 1 binary64) n) < 1

   1. Initial program 88.0%

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

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{\frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}}{n} + \left(\frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}} \cdot \left(\frac{1}{2} \cdot \frac{1}{{n}^{2}} - \frac{1}{2} \cdot \frac{1}{n}\right)}{x} + \frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}} \cdot \left(\left(\frac{1}{6} \cdot \frac{1}{{n}^{3}} + \frac{1}{3} \cdot \frac{1}{n}\right) - \frac{1}{2} \cdot \frac{1}{{n}^{2}}\right)}{{x}^{2}}\right)}{x}} \]

   5. Simplified 51.5%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\frac{{x}^{\left(\frac{1}{n}\right)}}{x}, \left(\frac{0.5}{n \cdot n} + \frac{-0.5}{n}\right) + \frac{\frac{0.16666666666666666}{n \cdot \left(n \cdot n\right)} + \left(\frac{0.3333333333333333}{n} + \frac{-0.5}{n \cdot n}\right)}{x}, \frac{{x}^{\left(\frac{1}{n}\right)}}{n}\right)}{x}} \]

   6. Taylor expanded in x around inf

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

      1. mul-1-neg N/A

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

      2. log-rec N/A

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

      3. distribute-frac-neg N/A

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

      4. remove-double-neg N/A

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

      5. /-lowering-/.f64 N/A

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

      6. exp-lowering-exp.f64 N/A

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

      7. /-lowering-/.f64 N/A

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

      8. log-lowering-log.f64 96.2

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

   8. Simplified 96.2%

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

   if -1.99999999999999988e-11 < (/.f64 #s(literal 1 binary64) n) < 2.00000000000000002e-50

   1. Initial program 31.3%

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

   3. Taylor expanded in n around inf

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

      1. /-lowering-/.f64 N/A

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

      2. --lowering--.f64 N/A

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

      3. accelerator-lowering-log1p.f64 N/A

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

      4. log-lowering-log.f64 82.6

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

   5. Simplified 82.6%

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

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

   1. Initial program 61.7%

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

      1. pow-to-exp N/A

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

      2. exp-lowering-exp.f64 N/A

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

      3. un-div-inv N/A

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

      4. /-lowering-/.f64 N/A

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

      5. +-commutative N/A

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

      6. accelerator-lowering-log1p.f64 99.8

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

   4. Applied egg-rr 99.8%

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

      1. clear-num N/A

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

      2. div-inv N/A

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

      3. clear-num N/A

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

      4. exp-prod N/A

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

      5. pow-lowering-pow.f64 N/A

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

      6. exp-lowering-exp.f64 N/A

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

      7. /-lowering-/.f64 N/A

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

      8. accelerator-lowering-log1p.f64 99.8

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

   6. Applied egg-rr 99.8%

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

   7. Taylor expanded in x around 0

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

      1. /-lowering-/.f64 99.8

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

   9. Simplified 99.8%

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

4. Final simplification 90.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{-11}:\\ \;\;\;\;\frac{\frac{e^{\frac{\log x}{n}}}{n}}{x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-50}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right) - \log x}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 1:\\ \;\;\;\;\frac{\frac{e^{\frac{\log x}{n}}}{n}}{x}\\ \mathbf{else}:\\ \;\;\;\;{e}^{\left(\frac{x}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 86.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\frac{e^{\frac{\log x}{n}}}{n}}{x}\\ \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{-11}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-50}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right) - \log x}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 1:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{x}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (/ (/ (exp (/ (log x) n)) n) x)))
   (if (<= (/ 1.0 n) -2e-11)
     t_0
     (if (<= (/ 1.0 n) 2e-50)
       (/ (- (log1p x) (log x)) n)
       (if (<= (/ 1.0 n) 1.0) t_0 (- (exp (/ x n)) (pow x (/ 1.0 n))))))))

C

double code(double x, double n) {
	double t_0 = (exp((log(x) / n)) / n) / x;
	double tmp;
	if ((1.0 / n) <= -2e-11) {
		tmp = t_0;
	} else if ((1.0 / n) <= 2e-50) {
		tmp = (log1p(x) - log(x)) / n;
	} else if ((1.0 / n) <= 1.0) {
		tmp = t_0;
	} else {
		tmp = exp((x / n)) - pow(x, (1.0 / n));
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double t_0 = (Math.exp((Math.log(x) / n)) / n) / x;
	double tmp;
	if ((1.0 / n) <= -2e-11) {
		tmp = t_0;
	} else if ((1.0 / n) <= 2e-50) {
		tmp = (Math.log1p(x) - Math.log(x)) / n;
	} else if ((1.0 / n) <= 1.0) {
		tmp = t_0;
	} else {
		tmp = Math.exp((x / n)) - Math.pow(x, (1.0 / n));
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = (math.exp((math.log(x) / n)) / n) / x
	tmp = 0
	if (1.0 / n) <= -2e-11:
		tmp = t_0
	elif (1.0 / n) <= 2e-50:
		tmp = (math.log1p(x) - math.log(x)) / n
	elif (1.0 / n) <= 1.0:
		tmp = t_0
	else:
		tmp = math.exp((x / n)) - math.pow(x, (1.0 / n))
	return tmp

Julia

function code(x, n)
	t_0 = Float64(Float64(exp(Float64(log(x) / n)) / n) / x)
	tmp = 0.0
	if (Float64(1.0 / n) <= -2e-11)
		tmp = t_0;
	elseif (Float64(1.0 / n) <= 2e-50)
		tmp = Float64(Float64(log1p(x) - log(x)) / n);
	elseif (Float64(1.0 / n) <= 1.0)
		tmp = t_0;
	else
		tmp = Float64(exp(Float64(x / n)) - (x ^ Float64(1.0 / n)));
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(N[(N[Exp[N[(N[Log[x], $MachinePrecision] / n), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision] / x), $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -2e-11], t$95$0, If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-50], N[(N[(N[Log[1 + x], $MachinePrecision] - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 1.0], t$95$0, 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) < -1.99999999999999988e-11 or 2.00000000000000002e-50 < (/.f64 #s(literal 1 binary64) n) < 1

   1. Initial program 88.0%

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

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{\frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}}{n} + \left(\frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}} \cdot \left(\frac{1}{2} \cdot \frac{1}{{n}^{2}} - \frac{1}{2} \cdot \frac{1}{n}\right)}{x} + \frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}} \cdot \left(\left(\frac{1}{6} \cdot \frac{1}{{n}^{3}} + \frac{1}{3} \cdot \frac{1}{n}\right) - \frac{1}{2} \cdot \frac{1}{{n}^{2}}\right)}{{x}^{2}}\right)}{x}} \]

   5. Simplified 51.5%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\frac{{x}^{\left(\frac{1}{n}\right)}}{x}, \left(\frac{0.5}{n \cdot n} + \frac{-0.5}{n}\right) + \frac{\frac{0.16666666666666666}{n \cdot \left(n \cdot n\right)} + \left(\frac{0.3333333333333333}{n} + \frac{-0.5}{n \cdot n}\right)}{x}, \frac{{x}^{\left(\frac{1}{n}\right)}}{n}\right)}{x}} \]

   6. Taylor expanded in x around inf

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

      1. mul-1-neg N/A

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

      2. log-rec N/A

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

      3. distribute-frac-neg N/A

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

      4. remove-double-neg N/A

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

      5. /-lowering-/.f64 N/A

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

      6. exp-lowering-exp.f64 N/A

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

      7. /-lowering-/.f64 N/A

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

      8. log-lowering-log.f64 96.2

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

   8. Simplified 96.2%

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

   if -1.99999999999999988e-11 < (/.f64 #s(literal 1 binary64) n) < 2.00000000000000002e-50

   1. Initial program 31.3%

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

   3. Taylor expanded in n around inf

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

      1. /-lowering-/.f64 N/A

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

      2. --lowering--.f64 N/A

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

      3. accelerator-lowering-log1p.f64 N/A

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

      4. log-lowering-log.f64 82.6

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

   5. Simplified 82.6%

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

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

   1. Initial program 61.7%

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

      1. pow-to-exp N/A

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

      2. exp-lowering-exp.f64 N/A

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

      3. un-div-inv N/A

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

      4. /-lowering-/.f64 N/A

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

      5. +-commutative N/A

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

      6. accelerator-lowering-log1p.f64 99.8

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

   4. Applied egg-rr 99.8%

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

   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. /-lowering-/.f64 99.8

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

   7. Simplified 99.8%

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

Alternative 4: 86.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ t_1 := \frac{t\_0}{n \cdot x}\\ \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{-11}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-50}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right) - \log x}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 1:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{x}{n}} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))) (t_1 (/ t_0 (* n x))))
   (if (<= (/ 1.0 n) -2e-11)
     t_1
     (if (<= (/ 1.0 n) 2e-50)
       (/ (- (log1p x) (log x)) n)
       (if (<= (/ 1.0 n) 1.0) t_1 (- (exp (/ x n)) t_0))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double t_1 = t_0 / (n * x);
	double tmp;
	if ((1.0 / n) <= -2e-11) {
		tmp = t_1;
	} else if ((1.0 / n) <= 2e-50) {
		tmp = (log1p(x) - log(x)) / n;
	} else if ((1.0 / n) <= 1.0) {
		tmp = t_1;
	} else {
		tmp = exp((x / n)) - t_0;
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, (1.0 / n));
	double t_1 = t_0 / (n * x);
	double tmp;
	if ((1.0 / n) <= -2e-11) {
		tmp = t_1;
	} else if ((1.0 / n) <= 2e-50) {
		tmp = (Math.log1p(x) - Math.log(x)) / n;
	} else if ((1.0 / n) <= 1.0) {
		tmp = t_1;
	} else {
		tmp = Math.exp((x / n)) - t_0;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	t_1 = t_0 / (n * x)
	tmp = 0
	if (1.0 / n) <= -2e-11:
		tmp = t_1
	elif (1.0 / n) <= 2e-50:
		tmp = (math.log1p(x) - math.log(x)) / n
	elif (1.0 / n) <= 1.0:
		tmp = t_1
	else:
		tmp = math.exp((x / n)) - t_0
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	t_1 = Float64(t_0 / Float64(n * x))
	tmp = 0.0
	if (Float64(1.0 / n) <= -2e-11)
		tmp = t_1;
	elseif (Float64(1.0 / n) <= 2e-50)
		tmp = Float64(Float64(log1p(x) - log(x)) / n);
	elseif (Float64(1.0 / n) <= 1.0)
		tmp = t_1;
	else
		tmp = Float64(exp(Float64(x / n)) - t_0);
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 / N[(n * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -2e-11], t$95$1, If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-50], N[(N[(N[Log[1 + x], $MachinePrecision] - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 1.0], t$95$1, N[(N[Exp[N[(x / n), $MachinePrecision]], $MachinePrecision] - t$95$0), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -1.99999999999999988e-11 or 2.00000000000000002e-50 < (/.f64 #s(literal 1 binary64) n) < 1

   1. Initial program 88.0%

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

   3. Taylor expanded in x around inf

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

      1. /-lowering-/.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. pow-lowering-pow.f64 N/A

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

      11. /-lowering-/.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. *-lowering-*.f64 96.2

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

   5. Simplified 96.2%

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

   if -1.99999999999999988e-11 < (/.f64 #s(literal 1 binary64) n) < 2.00000000000000002e-50

   1. Initial program 31.3%

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

   3. Taylor expanded in n around inf

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

      1. /-lowering-/.f64 N/A

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

      2. --lowering--.f64 N/A

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

      3. accelerator-lowering-log1p.f64 N/A

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

      4. log-lowering-log.f64 82.6

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

   5. Simplified 82.6%

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

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

   1. Initial program 61.7%

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

      1. pow-to-exp N/A

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

      2. exp-lowering-exp.f64 N/A

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

      3. un-div-inv N/A

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

      4. /-lowering-/.f64 N/A

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

      5. +-commutative N/A

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

      6. accelerator-lowering-log1p.f64 99.8

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

   4. Applied egg-rr 99.8%

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

   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. /-lowering-/.f64 99.8

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

   7. Simplified 99.8%

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

4. Final simplification 90.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{-11}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-50}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right) - \log x}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 1:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{n \cdot x}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{x}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 84.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ t_1 := \frac{t\_0}{n \cdot x}\\ \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{-11}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-50}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right) - \log x}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 1:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \frac{\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.3333333333333333, -0.5\right) + \frac{\mathsf{fma}\left(x, -0.5, 0.5\right)}{n}, \mathsf{fma}\left(0.16666666666666666, \frac{x \cdot \frac{x}{n}}{n}, 1\right)\right)}{n}, 1 - t\_0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))) (t_1 (/ t_0 (* n x))))
   (if (<= (/ 1.0 n) -2e-11)
     t_1
     (if (<= (/ 1.0 n) 2e-50)
       (/ (- (log1p x) (log x)) n)
       (if (<= (/ 1.0 n) 1.0)
         t_1
         (fma
          x
          (/
           (fma
            x
            (+ (fma x 0.3333333333333333 -0.5) (/ (fma x -0.5 0.5) n))
            (fma 0.16666666666666666 (/ (* x (/ x n)) n) 1.0))
           n)
          (- 1.0 t_0)))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double t_1 = t_0 / (n * x);
	double tmp;
	if ((1.0 / n) <= -2e-11) {
		tmp = t_1;
	} else if ((1.0 / n) <= 2e-50) {
		tmp = (log1p(x) - log(x)) / n;
	} else if ((1.0 / n) <= 1.0) {
		tmp = t_1;
	} else {
		tmp = fma(x, (fma(x, (fma(x, 0.3333333333333333, -0.5) + (fma(x, -0.5, 0.5) / n)), fma(0.16666666666666666, ((x * (x / n)) / n), 1.0)) / n), (1.0 - t_0));
	}
	return tmp;
}

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	t_1 = Float64(t_0 / Float64(n * x))
	tmp = 0.0
	if (Float64(1.0 / n) <= -2e-11)
		tmp = t_1;
	elseif (Float64(1.0 / n) <= 2e-50)
		tmp = Float64(Float64(log1p(x) - log(x)) / n);
	elseif (Float64(1.0 / n) <= 1.0)
		tmp = t_1;
	else
		tmp = fma(x, Float64(fma(x, Float64(fma(x, 0.3333333333333333, -0.5) + Float64(fma(x, -0.5, 0.5) / n)), fma(0.16666666666666666, Float64(Float64(x * Float64(x / n)) / n), 1.0)) / n), Float64(1.0 - t_0));
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 / N[(n * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -2e-11], t$95$1, If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-50], N[(N[(N[Log[1 + x], $MachinePrecision] - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 1.0], t$95$1, N[(x * N[(N[(x * N[(N[(x * 0.3333333333333333 + -0.5), $MachinePrecision] + N[(N[(x * -0.5 + 0.5), $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision] + N[(0.16666666666666666 * N[(N[(x * N[(x / n), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision] + N[(1.0 - t$95$0), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -1.99999999999999988e-11 or 2.00000000000000002e-50 < (/.f64 #s(literal 1 binary64) n) < 1

   1. Initial program 88.0%

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

   3. Taylor expanded in x around inf

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

      1. /-lowering-/.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. pow-lowering-pow.f64 N/A

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

      11. /-lowering-/.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. *-lowering-*.f64 96.2

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

   5. Simplified 96.2%

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

   if -1.99999999999999988e-11 < (/.f64 #s(literal 1 binary64) n) < 2.00000000000000002e-50

   1. Initial program 31.3%

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

   3. Taylor expanded in n around inf

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

      1. /-lowering-/.f64 N/A

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

      2. --lowering--.f64 N/A

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

      3. accelerator-lowering-log1p.f64 N/A

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

      4. log-lowering-log.f64 82.6

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

   5. Simplified 82.6%

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

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

   1. Initial program 61.7%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(1 + x \cdot \left(x \cdot \left(\left(\frac{1}{2} \cdot \frac{1}{{n}^{2}} + x \cdot \left(\left(\frac{1}{6} \cdot \frac{1}{{n}^{3}} + \frac{1}{3} \cdot \frac{1}{n}\right) - \frac{1}{2} \cdot \frac{1}{{n}^{2}}\right)\right) - \frac{1}{2} \cdot \frac{1}{n}\right) + \frac{1}{n}\right)\right) - e^{\frac{\log x}{n}}} \]

   5. Simplified 48.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{0.16666666666666666}{n \cdot \left(n \cdot n\right)} + \left(\frac{0.3333333333333333}{n} + \frac{-0.5}{n \cdot n}\right), \frac{0.5}{n \cdot n} + \frac{-0.5}{n}\right), \frac{1}{n}\right), 1 - {x}^{\left(\frac{1}{n}\right)}\right)} \]

   6. Taylor expanded in n around inf

      \[\leadsto \mathsf{fma}\left(x, \color{blue}{\frac{1 + \left(\frac{1}{6} \cdot \frac{{x}^{2}}{{n}^{2}} + \left(x \cdot \left(\frac{1}{3} \cdot x - \frac{1}{2}\right) + \frac{x \cdot \left(\frac{1}{2} + \frac{-1}{2} \cdot x\right)}{n}\right)\right)}{n}}, 1 - {x}^{\left(\frac{1}{n}\right)}\right) \]
   7. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{\frac{1 + \left(\frac{1}{6} \cdot \frac{{x}^{2}}{{n}^{2}} + \left(x \cdot \left(\frac{1}{3} \cdot x - \frac{1}{2}\right) + \frac{x \cdot \left(\frac{1}{2} + \frac{-1}{2} \cdot x\right)}{n}\right)\right)}{n}}, 1 - {x}^{\left(\frac{1}{n}\right)}\right) \]

   8. Simplified 74.0%

      \[\leadsto \mathsf{fma}\left(x, \color{blue}{\frac{\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.3333333333333333, -0.5\right) + \frac{\mathsf{fma}\left(x, -0.5, 0.5\right)}{n}, \mathsf{fma}\left(0.16666666666666666, \frac{x \cdot x}{n \cdot n}, 1\right)\right)}{n}}, 1 - {x}^{\left(\frac{1}{n}\right)}\right) \]
   9. Step-by-step derivation

      1. times-frac N/A

         \[\leadsto \mathsf{fma}\left(x, \frac{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{1}{3}, \frac{-1}{2}\right) + \frac{\mathsf{fma}\left(x, \frac{-1}{2}, \frac{1}{2}\right)}{n}, \mathsf{fma}\left(\frac{1}{6}, \color{blue}{\frac{x}{n} \cdot \frac{x}{n}}, 1\right)\right)}{n}, 1 - {x}^{\left(\frac{1}{n}\right)}\right) \]

      2. associate-*r/ N/A

         \[\leadsto \mathsf{fma}\left(x, \frac{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{1}{3}, \frac{-1}{2}\right) + \frac{\mathsf{fma}\left(x, \frac{-1}{2}, \frac{1}{2}\right)}{n}, \mathsf{fma}\left(\frac{1}{6}, \color{blue}{\frac{\frac{x}{n} \cdot x}{n}}, 1\right)\right)}{n}, 1 - {x}^{\left(\frac{1}{n}\right)}\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(x, \frac{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{1}{3}, \frac{-1}{2}\right) + \frac{\mathsf{fma}\left(x, \frac{-1}{2}, \frac{1}{2}\right)}{n}, \mathsf{fma}\left(\frac{1}{6}, \color{blue}{\frac{\frac{x}{n} \cdot x}{n}}, 1\right)\right)}{n}, 1 - {x}^{\left(\frac{1}{n}\right)}\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(x, \frac{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{1}{3}, \frac{-1}{2}\right) + \frac{\mathsf{fma}\left(x, \frac{-1}{2}, \frac{1}{2}\right)}{n}, \mathsf{fma}\left(\frac{1}{6}, \frac{\color{blue}{\frac{x}{n} \cdot x}}{n}, 1\right)\right)}{n}, 1 - {x}^{\left(\frac{1}{n}\right)}\right) \]

      5. /-lowering-/.f64 87.2

         \[\leadsto \mathsf{fma}\left(x, \frac{\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.3333333333333333, -0.5\right) + \frac{\mathsf{fma}\left(x, -0.5, 0.5\right)}{n}, \mathsf{fma}\left(0.16666666666666666, \frac{\color{blue}{\frac{x}{n}} \cdot x}{n}, 1\right)\right)}{n}, 1 - {x}^{\left(\frac{1}{n}\right)}\right) \]

   10. Applied egg-rr 87.2%

       \[\leadsto \mathsf{fma}\left(x, \frac{\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.3333333333333333, -0.5\right) + \frac{\mathsf{fma}\left(x, -0.5, 0.5\right)}{n}, \mathsf{fma}\left(0.16666666666666666, \color{blue}{\frac{\frac{x}{n} \cdot x}{n}}, 1\right)\right)}{n}, 1 - {x}^{\left(\frac{1}{n}\right)}\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 88.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{-11}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-50}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right) - \log x}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 1:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{n \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \frac{\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.3333333333333333, -0.5\right) + \frac{\mathsf{fma}\left(x, -0.5, 0.5\right)}{n}, \mathsf{fma}\left(0.16666666666666666, \frac{x \cdot \frac{x}{n}}{n}, 1\right)\right)}{n}, 1 - {x}^{\left(\frac{1}{n}\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 68.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ t_1 := \frac{t\_0}{n \cdot x}\\ \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{-11}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-190}:\\ \;\;\;\;-\frac{\log x}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 1:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \frac{\mathsf{fma}\left(x, -0.5 + \frac{0.5}{n}, 1\right)}{n}, 1 - t\_0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))) (t_1 (/ t_0 (* n x))))
   (if (<= (/ 1.0 n) -2e-11)
     t_1
     (if (<= (/ 1.0 n) 5e-190)
       (- (/ (log x) n))
       (if (<= (/ 1.0 n) 1.0)
         t_1
         (fma x (/ (fma x (+ -0.5 (/ 0.5 n)) 1.0) n) (- 1.0 t_0)))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double t_1 = t_0 / (n * x);
	double tmp;
	if ((1.0 / n) <= -2e-11) {
		tmp = t_1;
	} else if ((1.0 / n) <= 5e-190) {
		tmp = -(log(x) / n);
	} else if ((1.0 / n) <= 1.0) {
		tmp = t_1;
	} else {
		tmp = fma(x, (fma(x, (-0.5 + (0.5 / n)), 1.0) / n), (1.0 - t_0));
	}
	return tmp;
}

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	t_1 = Float64(t_0 / Float64(n * x))
	tmp = 0.0
	if (Float64(1.0 / n) <= -2e-11)
		tmp = t_1;
	elseif (Float64(1.0 / n) <= 5e-190)
		tmp = Float64(-Float64(log(x) / n));
	elseif (Float64(1.0 / n) <= 1.0)
		tmp = t_1;
	else
		tmp = fma(x, Float64(fma(x, Float64(-0.5 + Float64(0.5 / n)), 1.0) / n), Float64(1.0 - t_0));
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 / N[(n * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -2e-11], t$95$1, If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e-190], (-N[(N[Log[x], $MachinePrecision] / n), $MachinePrecision]), If[LessEqual[N[(1.0 / n), $MachinePrecision], 1.0], t$95$1, N[(x * N[(N[(x * N[(-0.5 + N[(0.5 / n), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / n), $MachinePrecision] + N[(1.0 - t$95$0), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -1.99999999999999988e-11 or 5.00000000000000034e-190 < (/.f64 #s(literal 1 binary64) n) < 1

   1. Initial program 76.8%

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

   3. Taylor expanded in x around inf

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

      1. /-lowering-/.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. pow-lowering-pow.f64 N/A

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

      11. /-lowering-/.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. *-lowering-*.f64 87.9

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

   5. Simplified 87.9%

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

   if -1.99999999999999988e-11 < (/.f64 #s(literal 1 binary64) n) < 5.00000000000000034e-190

   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 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. --lowering--.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. pow-lowering-pow.f64 N/A

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

      17. /-lowering-/.f64 32.0

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

   5. Simplified 32.0%

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

   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. neg-lowering-neg.f64 N/A

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

      3. /-lowering-/.f64 N/A

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

      4. log-lowering-log.f64 56.8

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

   8. Simplified 56.8%

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

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

   1. Initial program 61.7%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(1 + x \cdot \left(x \cdot \left(\left(\frac{1}{2} \cdot \frac{1}{{n}^{2}} + x \cdot \left(\left(\frac{1}{6} \cdot \frac{1}{{n}^{3}} + \frac{1}{3} \cdot \frac{1}{n}\right) - \frac{1}{2} \cdot \frac{1}{{n}^{2}}\right)\right) - \frac{1}{2} \cdot \frac{1}{n}\right) + \frac{1}{n}\right)\right) - e^{\frac{\log x}{n}}} \]

   5. Simplified 48.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{0.16666666666666666}{n \cdot \left(n \cdot n\right)} + \left(\frac{0.3333333333333333}{n} + \frac{-0.5}{n \cdot n}\right), \frac{0.5}{n \cdot n} + \frac{-0.5}{n}\right), \frac{1}{n}\right), 1 - {x}^{\left(\frac{1}{n}\right)}\right)} \]

   6. Taylor expanded in n around inf

      \[\leadsto \mathsf{fma}\left(x, \color{blue}{\frac{1 + \left(\frac{1}{6} \cdot \frac{{x}^{2}}{{n}^{2}} + \left(x \cdot \left(\frac{1}{3} \cdot x - \frac{1}{2}\right) + \frac{x \cdot \left(\frac{1}{2} + \frac{-1}{2} \cdot x\right)}{n}\right)\right)}{n}}, 1 - {x}^{\left(\frac{1}{n}\right)}\right) \]
   7. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{\frac{1 + \left(\frac{1}{6} \cdot \frac{{x}^{2}}{{n}^{2}} + \left(x \cdot \left(\frac{1}{3} \cdot x - \frac{1}{2}\right) + \frac{x \cdot \left(\frac{1}{2} + \frac{-1}{2} \cdot x\right)}{n}\right)\right)}{n}}, 1 - {x}^{\left(\frac{1}{n}\right)}\right) \]

   8. Simplified 74.0%

      \[\leadsto \mathsf{fma}\left(x, \color{blue}{\frac{\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.3333333333333333, -0.5\right) + \frac{\mathsf{fma}\left(x, -0.5, 0.5\right)}{n}, \mathsf{fma}\left(0.16666666666666666, \frac{x \cdot x}{n \cdot n}, 1\right)\right)}{n}}, 1 - {x}^{\left(\frac{1}{n}\right)}\right) \]

   9. Taylor expanded in x around 0

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

       1. +-commutative N/A

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

       2. accelerator-lowering-fma.f64 N/A

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

       3. sub-neg N/A

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

       4. metadata-eval N/A

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

       5. +-lowering-+.f64 N/A

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

       6. associate-*r/ N/A

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

       7. metadata-eval N/A

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

       8. /-lowering-/.f64 75.1

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

   11. Simplified 75.1%

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

4. Final simplification 75.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{-11}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-190}:\\ \;\;\;\;-\frac{\log x}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 1:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{n \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \frac{\mathsf{fma}\left(x, -0.5 + \frac{0.5}{n}, 1\right)}{n}, 1 - {x}^{\left(\frac{1}{n}\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 70.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;x \leq 8 \cdot 10^{-5}:\\ \;\;\;\;\mathsf{fma}\left(x, \frac{\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.3333333333333333, -0.5\right) + \frac{\mathsf{fma}\left(x, -0.5, 0.5\right)}{n}, \mathsf{fma}\left(0.16666666666666666, \frac{x \cdot \frac{x}{n}}{n}, 1\right)\right)}{n}, 1 - t\_0\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0}{n \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= x 8e-5)
     (fma
      x
      (/
       (fma
        x
        (+ (fma x 0.3333333333333333 -0.5) (/ (fma x -0.5 0.5) n))
        (fma 0.16666666666666666 (/ (* x (/ x n)) n) 1.0))
       n)
      (- 1.0 t_0))
     (/ t_0 (* n x)))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double tmp;
	if (x <= 8e-5) {
		tmp = fma(x, (fma(x, (fma(x, 0.3333333333333333, -0.5) + (fma(x, -0.5, 0.5) / n)), fma(0.16666666666666666, ((x * (x / n)) / n), 1.0)) / n), (1.0 - t_0));
	} else {
		tmp = t_0 / (n * x);
	}
	return tmp;
}

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (x <= 8e-5)
		tmp = fma(x, Float64(fma(x, Float64(fma(x, 0.3333333333333333, -0.5) + Float64(fma(x, -0.5, 0.5) / n)), fma(0.16666666666666666, Float64(Float64(x * Float64(x / n)) / n), 1.0)) / n), Float64(1.0 - t_0));
	else
		tmp = Float64(t_0 / Float64(n * x));
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, 8e-5], N[(x * N[(N[(x * N[(N[(x * 0.3333333333333333 + -0.5), $MachinePrecision] + N[(N[(x * -0.5 + 0.5), $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision] + N[(0.16666666666666666 * N[(N[(x * N[(x / n), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision] + N[(1.0 - t$95$0), $MachinePrecision]), $MachinePrecision], N[(t$95$0 / N[(n * x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 8.00000000000000065e-5

   1. Initial program 50.7%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(1 + x \cdot \left(x \cdot \left(\left(\frac{1}{2} \cdot \frac{1}{{n}^{2}} + x \cdot \left(\left(\frac{1}{6} \cdot \frac{1}{{n}^{3}} + \frac{1}{3} \cdot \frac{1}{n}\right) - \frac{1}{2} \cdot \frac{1}{{n}^{2}}\right)\right) - \frac{1}{2} \cdot \frac{1}{n}\right) + \frac{1}{n}\right)\right) - e^{\frac{\log x}{n}}} \]

   5. Simplified 33.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{0.16666666666666666}{n \cdot \left(n \cdot n\right)} + \left(\frac{0.3333333333333333}{n} + \frac{-0.5}{n \cdot n}\right), \frac{0.5}{n \cdot n} + \frac{-0.5}{n}\right), \frac{1}{n}\right), 1 - {x}^{\left(\frac{1}{n}\right)}\right)} \]

   6. Taylor expanded in n around inf

      \[\leadsto \mathsf{fma}\left(x, \color{blue}{\frac{1 + \left(\frac{1}{6} \cdot \frac{{x}^{2}}{{n}^{2}} + \left(x \cdot \left(\frac{1}{3} \cdot x - \frac{1}{2}\right) + \frac{x \cdot \left(\frac{1}{2} + \frac{-1}{2} \cdot x\right)}{n}\right)\right)}{n}}, 1 - {x}^{\left(\frac{1}{n}\right)}\right) \]
   7. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{\frac{1 + \left(\frac{1}{6} \cdot \frac{{x}^{2}}{{n}^{2}} + \left(x \cdot \left(\frac{1}{3} \cdot x - \frac{1}{2}\right) + \frac{x \cdot \left(\frac{1}{2} + \frac{-1}{2} \cdot x\right)}{n}\right)\right)}{n}}, 1 - {x}^{\left(\frac{1}{n}\right)}\right) \]

   8. Simplified 49.6%

      \[\leadsto \mathsf{fma}\left(x, \color{blue}{\frac{\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.3333333333333333, -0.5\right) + \frac{\mathsf{fma}\left(x, -0.5, 0.5\right)}{n}, \mathsf{fma}\left(0.16666666666666666, \frac{x \cdot x}{n \cdot n}, 1\right)\right)}{n}}, 1 - {x}^{\left(\frac{1}{n}\right)}\right) \]
   9. Step-by-step derivation

      1. times-frac N/A

         \[\leadsto \mathsf{fma}\left(x, \frac{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{1}{3}, \frac{-1}{2}\right) + \frac{\mathsf{fma}\left(x, \frac{-1}{2}, \frac{1}{2}\right)}{n}, \mathsf{fma}\left(\frac{1}{6}, \color{blue}{\frac{x}{n} \cdot \frac{x}{n}}, 1\right)\right)}{n}, 1 - {x}^{\left(\frac{1}{n}\right)}\right) \]

      2. associate-*r/ N/A

         \[\leadsto \mathsf{fma}\left(x, \frac{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{1}{3}, \frac{-1}{2}\right) + \frac{\mathsf{fma}\left(x, \frac{-1}{2}, \frac{1}{2}\right)}{n}, \mathsf{fma}\left(\frac{1}{6}, \color{blue}{\frac{\frac{x}{n} \cdot x}{n}}, 1\right)\right)}{n}, 1 - {x}^{\left(\frac{1}{n}\right)}\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(x, \frac{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{1}{3}, \frac{-1}{2}\right) + \frac{\mathsf{fma}\left(x, \frac{-1}{2}, \frac{1}{2}\right)}{n}, \mathsf{fma}\left(\frac{1}{6}, \color{blue}{\frac{\frac{x}{n} \cdot x}{n}}, 1\right)\right)}{n}, 1 - {x}^{\left(\frac{1}{n}\right)}\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(x, \frac{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{1}{3}, \frac{-1}{2}\right) + \frac{\mathsf{fma}\left(x, \frac{-1}{2}, \frac{1}{2}\right)}{n}, \mathsf{fma}\left(\frac{1}{6}, \frac{\color{blue}{\frac{x}{n} \cdot x}}{n}, 1\right)\right)}{n}, 1 - {x}^{\left(\frac{1}{n}\right)}\right) \]

      5. /-lowering-/.f64 57.8

         \[\leadsto \mathsf{fma}\left(x, \frac{\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.3333333333333333, -0.5\right) + \frac{\mathsf{fma}\left(x, -0.5, 0.5\right)}{n}, \mathsf{fma}\left(0.16666666666666666, \frac{\color{blue}{\frac{x}{n}} \cdot x}{n}, 1\right)\right)}{n}, 1 - {x}^{\left(\frac{1}{n}\right)}\right) \]

   10. Applied egg-rr 57.8%

       \[\leadsto \mathsf{fma}\left(x, \frac{\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.3333333333333333, -0.5\right) + \frac{\mathsf{fma}\left(x, -0.5, 0.5\right)}{n}, \mathsf{fma}\left(0.16666666666666666, \color{blue}{\frac{\frac{x}{n} \cdot x}{n}}, 1\right)\right)}{n}, 1 - {x}^{\left(\frac{1}{n}\right)}\right) \]

   if 8.00000000000000065e-5 < x

   1. Initial program 70.4%

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

   3. Taylor expanded in x around 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. /-lowering-/.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. pow-lowering-pow.f64 N/A

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

      11. /-lowering-/.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. *-lowering-*.f64 98.0

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

   5. Simplified 98.0%

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

4. Final simplification 74.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 8 \cdot 10^{-5}:\\ \;\;\;\;\mathsf{fma}\left(x, \frac{\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.3333333333333333, -0.5\right) + \frac{\mathsf{fma}\left(x, -0.5, 0.5\right)}{n}, \mathsf{fma}\left(0.16666666666666666, \frac{x \cdot \frac{x}{n}}{n}, 1\right)\right)}{n}, 1 - {x}^{\left(\frac{1}{n}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{n \cdot x}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 68.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ t_1 := \frac{t\_0}{n \cdot x}\\ \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{-11}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-190}:\\ \;\;\;\;-\frac{\log x}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 1:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{+114}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - t\_0\\ \mathbf{else}:\\ \;\;\;\;-1 + e^{\frac{x}{n}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))) (t_1 (/ t_0 (* n x))))
   (if (<= (/ 1.0 n) -2e-11)
     t_1
     (if (<= (/ 1.0 n) 5e-190)
       (- (/ (log x) n))
       (if (<= (/ 1.0 n) 1.0)
         t_1
         (if (<= (/ 1.0 n) 5e+114)
           (- (+ 1.0 (/ x n)) t_0)
           (+ -1.0 (exp (/ x n)))))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double t_1 = t_0 / (n * x);
	double tmp;
	if ((1.0 / n) <= -2e-11) {
		tmp = t_1;
	} else if ((1.0 / n) <= 5e-190) {
		tmp = -(log(x) / n);
	} else if ((1.0 / n) <= 1.0) {
		tmp = t_1;
	} else if ((1.0 / n) <= 5e+114) {
		tmp = (1.0 + (x / n)) - t_0;
	} else {
		tmp = -1.0 + exp((x / n));
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = x ** (1.0d0 / n)
    t_1 = t_0 / (n * x)
    if ((1.0d0 / n) <= (-2d-11)) then
        tmp = t_1
    else if ((1.0d0 / n) <= 5d-190) then
        tmp = -(log(x) / n)
    else if ((1.0d0 / n) <= 1.0d0) then
        tmp = t_1
    else if ((1.0d0 / n) <= 5d+114) then
        tmp = (1.0d0 + (x / n)) - t_0
    else
        tmp = (-1.0d0) + exp((x / n))
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, (1.0 / n));
	double t_1 = t_0 / (n * x);
	double tmp;
	if ((1.0 / n) <= -2e-11) {
		tmp = t_1;
	} else if ((1.0 / n) <= 5e-190) {
		tmp = -(Math.log(x) / n);
	} else if ((1.0 / n) <= 1.0) {
		tmp = t_1;
	} else if ((1.0 / n) <= 5e+114) {
		tmp = (1.0 + (x / n)) - t_0;
	} else {
		tmp = -1.0 + Math.exp((x / n));
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	t_1 = t_0 / (n * x)
	tmp = 0
	if (1.0 / n) <= -2e-11:
		tmp = t_1
	elif (1.0 / n) <= 5e-190:
		tmp = -(math.log(x) / n)
	elif (1.0 / n) <= 1.0:
		tmp = t_1
	elif (1.0 / n) <= 5e+114:
		tmp = (1.0 + (x / n)) - t_0
	else:
		tmp = -1.0 + math.exp((x / n))
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	t_1 = Float64(t_0 / Float64(n * x))
	tmp = 0.0
	if (Float64(1.0 / n) <= -2e-11)
		tmp = t_1;
	elseif (Float64(1.0 / n) <= 5e-190)
		tmp = Float64(-Float64(log(x) / n));
	elseif (Float64(1.0 / n) <= 1.0)
		tmp = t_1;
	elseif (Float64(1.0 / n) <= 5e+114)
		tmp = Float64(Float64(1.0 + Float64(x / n)) - t_0);
	else
		tmp = Float64(-1.0 + exp(Float64(x / n)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = x ^ (1.0 / n);
	t_1 = t_0 / (n * x);
	tmp = 0.0;
	if ((1.0 / n) <= -2e-11)
		tmp = t_1;
	elseif ((1.0 / n) <= 5e-190)
		tmp = -(log(x) / n);
	elseif ((1.0 / n) <= 1.0)
		tmp = t_1;
	elseif ((1.0 / n) <= 5e+114)
		tmp = (1.0 + (x / n)) - t_0;
	else
		tmp = -1.0 + exp((x / n));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 / N[(n * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -2e-11], t$95$1, If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e-190], (-N[(N[Log[x], $MachinePrecision] / n), $MachinePrecision]), If[LessEqual[N[(1.0 / n), $MachinePrecision], 1.0], t$95$1, If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e+114], N[(N[(1.0 + N[(x / n), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], N[(-1.0 + N[Exp[N[(x / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -1.99999999999999988e-11 or 5.00000000000000034e-190 < (/.f64 #s(literal 1 binary64) n) < 1

   1. Initial program 76.8%

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

   3. Taylor expanded in x around inf

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

      1. /-lowering-/.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. pow-lowering-pow.f64 N/A

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

      11. /-lowering-/.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. *-lowering-*.f64 87.9

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

   5. Simplified 87.9%

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

   if -1.99999999999999988e-11 < (/.f64 #s(literal 1 binary64) n) < 5.00000000000000034e-190

   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 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. --lowering--.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. pow-lowering-pow.f64 N/A

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

      17. /-lowering-/.f64 32.0

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

   5. Simplified 32.0%

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

   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. neg-lowering-neg.f64 N/A

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

      3. /-lowering-/.f64 N/A

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

      4. log-lowering-log.f64 56.8

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

   8. Simplified 56.8%

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

   if 1 < (/.f64 #s(literal 1 binary64) n) < 5.0000000000000001e114

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(1 + \frac{x}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   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. +-lowering-+.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. /-lowering-/.f64 90.3

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

   5. Simplified 90.3%

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

   if 5.0000000000000001e114 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 23.6%

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

      1. pow-to-exp N/A

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

      2. exp-lowering-exp.f64 N/A

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

      3. un-div-inv N/A

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

      4. /-lowering-/.f64 N/A

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

      5. +-commutative N/A

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

      6. accelerator-lowering-log1p.f64 99.5

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

   4. Applied egg-rr 99.5%

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

   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. /-lowering-/.f64 99.5

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

   7. Simplified 99.5%

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

   8. Taylor expanded in n around inf

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

   10. Simplified 73.1%

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

4. Final simplification 76.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{-11}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-190}:\\ \;\;\;\;-\frac{\log x}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 1:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{+114}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;-1 + e^{\frac{x}{n}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 69.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;x \leq 6.5 \cdot 10^{-167}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - t\_0\\ \mathbf{elif}\;x \leq 4.5 \cdot 10^{-134}:\\ \;\;\;\;\frac{\log x}{\left(n \cdot n\right) \cdot \frac{-1}{n}}\\ \mathbf{elif}\;x \leq 7.5 \cdot 10^{-5}:\\ \;\;\;\;\mathsf{fma}\left(x, \frac{\mathsf{fma}\left(x, -0.5 + \frac{0.5}{n}, \mathsf{fma}\left(0.16666666666666666, \frac{x \cdot x}{n \cdot n}, 1\right)\right)}{n}, 1 - t\_0\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0}{n \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= x 6.5e-167)
     (- (+ 1.0 (/ x n)) t_0)
     (if (<= x 4.5e-134)
       (/ (log x) (* (* n n) (/ -1.0 n)))
       (if (<= x 7.5e-5)
         (fma
          x
          (/
           (fma
            x
            (+ -0.5 (/ 0.5 n))
            (fma 0.16666666666666666 (/ (* x x) (* n n)) 1.0))
           n)
          (- 1.0 t_0))
         (/ t_0 (* n x)))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double tmp;
	if (x <= 6.5e-167) {
		tmp = (1.0 + (x / n)) - t_0;
	} else if (x <= 4.5e-134) {
		tmp = log(x) / ((n * n) * (-1.0 / n));
	} else if (x <= 7.5e-5) {
		tmp = fma(x, (fma(x, (-0.5 + (0.5 / n)), fma(0.16666666666666666, ((x * x) / (n * n)), 1.0)) / n), (1.0 - t_0));
	} else {
		tmp = t_0 / (n * x);
	}
	return tmp;
}

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (x <= 6.5e-167)
		tmp = Float64(Float64(1.0 + Float64(x / n)) - t_0);
	elseif (x <= 4.5e-134)
		tmp = Float64(log(x) / Float64(Float64(n * n) * Float64(-1.0 / n)));
	elseif (x <= 7.5e-5)
		tmp = fma(x, Float64(fma(x, Float64(-0.5 + Float64(0.5 / n)), fma(0.16666666666666666, Float64(Float64(x * x) / Float64(n * n)), 1.0)) / n), Float64(1.0 - t_0));
	else
		tmp = Float64(t_0 / Float64(n * x));
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, 6.5e-167], N[(N[(1.0 + N[(x / n), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[x, 4.5e-134], N[(N[Log[x], $MachinePrecision] / N[(N[(n * n), $MachinePrecision] * N[(-1.0 / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 7.5e-5], N[(x * N[(N[(x * N[(-0.5 + N[(0.5 / n), $MachinePrecision]), $MachinePrecision] + N[(0.16666666666666666 * N[(N[(x * x), $MachinePrecision] / N[(n * n), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision] + N[(1.0 - t$95$0), $MachinePrecision]), $MachinePrecision], N[(t$95$0 / N[(n * x), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if x < 6.49999999999999973e-167

   1. Initial program 58.2%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(1 + \frac{x}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   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. +-lowering-+.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. /-lowering-/.f64 58.4

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

   5. Simplified 58.4%

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

   if 6.49999999999999973e-167 < x < 4.5000000000000005e-134

   1. Initial program 30.4%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   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. --lowering--.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. pow-lowering-pow.f64 N/A

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

      17. /-lowering-/.f64 30.4

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

   5. Simplified 30.4%

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

      1. flip-- N/A

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

      2. /-lowering-/.f64 N/A

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

      3. metadata-eval N/A

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

      4. --lowering--.f64 N/A

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

      5. pow-sqr N/A

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

      6. pow-lowering-pow.f64 N/A

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

      7. *-lowering-*.f64 N/A

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

      8. /-lowering-/.f64 N/A

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

      9. +-lowering-+.f64 N/A

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

      10. pow-lowering-pow.f64 N/A

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

      11. /-lowering-/.f64 9.4

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

   7. Applied egg-rr 9.4%

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

   8. Taylor expanded in n around inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

      4. /-lowering-/.f64 N/A

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

      5. log-lowering-log.f64 N/A

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

      6. mul-1-neg N/A

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

      7. neg-lowering-neg.f64 46.2

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

   10. Simplified 46.2%

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

       1. neg-sub0 N/A

          \[\leadsto \frac{\log x}{\color{blue}{0 - n}} \]

       2. flip-- N/A

          \[\leadsto \frac{\log x}{\color{blue}{\frac{0 \cdot 0 - n \cdot n}{0 + n}}} \]

       3. div-inv N/A

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

       4. +-lft-identity N/A

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

       5. *-lowering-*.f64 N/A

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

       6. metadata-eval N/A

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

       7. sub0-neg N/A

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

       8. neg-lowering-neg.f64 N/A

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

       9. *-lowering-*.f64 N/A

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

       10. /-lowering-/.f64 69.9

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

   12. Applied egg-rr 69.9%

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

   if 4.5000000000000005e-134 < x < 7.49999999999999934e-5

   1. Initial program 48.7%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(1 + x \cdot \left(x \cdot \left(\left(\frac{1}{2} \cdot \frac{1}{{n}^{2}} + x \cdot \left(\left(\frac{1}{6} \cdot \frac{1}{{n}^{3}} + \frac{1}{3} \cdot \frac{1}{n}\right) - \frac{1}{2} \cdot \frac{1}{{n}^{2}}\right)\right) - \frac{1}{2} \cdot \frac{1}{n}\right) + \frac{1}{n}\right)\right) - e^{\frac{\log x}{n}}} \]

   5. Simplified 33.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{0.16666666666666666}{n \cdot \left(n \cdot n\right)} + \left(\frac{0.3333333333333333}{n} + \frac{-0.5}{n \cdot n}\right), \frac{0.5}{n \cdot n} + \frac{-0.5}{n}\right), \frac{1}{n}\right), 1 - {x}^{\left(\frac{1}{n}\right)}\right)} \]

   6. Taylor expanded in n around inf

      \[\leadsto \mathsf{fma}\left(x, \color{blue}{\frac{1 + \left(\frac{1}{6} \cdot \frac{{x}^{2}}{{n}^{2}} + \left(x \cdot \left(\frac{1}{3} \cdot x - \frac{1}{2}\right) + \frac{x \cdot \left(\frac{1}{2} + \frac{-1}{2} \cdot x\right)}{n}\right)\right)}{n}}, 1 - {x}^{\left(\frac{1}{n}\right)}\right) \]
   7. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{\frac{1 + \left(\frac{1}{6} \cdot \frac{{x}^{2}}{{n}^{2}} + \left(x \cdot \left(\frac{1}{3} \cdot x - \frac{1}{2}\right) + \frac{x \cdot \left(\frac{1}{2} + \frac{-1}{2} \cdot x\right)}{n}\right)\right)}{n}}, 1 - {x}^{\left(\frac{1}{n}\right)}\right) \]

   8. Simplified 58.6%

      \[\leadsto \mathsf{fma}\left(x, \color{blue}{\frac{\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.3333333333333333, -0.5\right) + \frac{\mathsf{fma}\left(x, -0.5, 0.5\right)}{n}, \mathsf{fma}\left(0.16666666666666666, \frac{x \cdot x}{n \cdot n}, 1\right)\right)}{n}}, 1 - {x}^{\left(\frac{1}{n}\right)}\right) \]

   9. Taylor expanded in x around 0

      \[\leadsto \mathsf{fma}\left(x, \frac{\mathsf{fma}\left(x, \color{blue}{\frac{1}{2} \cdot \frac{1}{n} - \frac{1}{2}}, \mathsf{fma}\left(\frac{1}{6}, \frac{x \cdot x}{n \cdot n}, 1\right)\right)}{n}, 1 - {x}^{\left(\frac{1}{n}\right)}\right) \]
   10. Step-by-step derivation

       1. sub-neg N/A

          \[\leadsto \mathsf{fma}\left(x, \frac{\mathsf{fma}\left(x, \color{blue}{\frac{1}{2} \cdot \frac{1}{n} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}, \mathsf{fma}\left(\frac{1}{6}, \frac{x \cdot x}{n \cdot n}, 1\right)\right)}{n}, 1 - {x}^{\left(\frac{1}{n}\right)}\right) \]

       2. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(x, \frac{\mathsf{fma}\left(x, \frac{1}{2} \cdot \frac{1}{n} + \color{blue}{\frac{-1}{2}}, \mathsf{fma}\left(\frac{1}{6}, \frac{x \cdot x}{n \cdot n}, 1\right)\right)}{n}, 1 - {x}^{\left(\frac{1}{n}\right)}\right) \]

       3. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{fma}\left(x, \frac{\mathsf{fma}\left(x, \color{blue}{\frac{1}{2} \cdot \frac{1}{n} + \frac{-1}{2}}, \mathsf{fma}\left(\frac{1}{6}, \frac{x \cdot x}{n \cdot n}, 1\right)\right)}{n}, 1 - {x}^{\left(\frac{1}{n}\right)}\right) \]

       4. associate-*r/ N/A

          \[\leadsto \mathsf{fma}\left(x, \frac{\mathsf{fma}\left(x, \color{blue}{\frac{\frac{1}{2} \cdot 1}{n}} + \frac{-1}{2}, \mathsf{fma}\left(\frac{1}{6}, \frac{x \cdot x}{n \cdot n}, 1\right)\right)}{n}, 1 - {x}^{\left(\frac{1}{n}\right)}\right) \]

       5. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(x, \frac{\mathsf{fma}\left(x, \frac{\color{blue}{\frac{1}{2}}}{n} + \frac{-1}{2}, \mathsf{fma}\left(\frac{1}{6}, \frac{x \cdot x}{n \cdot n}, 1\right)\right)}{n}, 1 - {x}^{\left(\frac{1}{n}\right)}\right) \]

       6. /-lowering-/.f64 58.6

          \[\leadsto \mathsf{fma}\left(x, \frac{\mathsf{fma}\left(x, \color{blue}{\frac{0.5}{n}} + -0.5, \mathsf{fma}\left(0.16666666666666666, \frac{x \cdot x}{n \cdot n}, 1\right)\right)}{n}, 1 - {x}^{\left(\frac{1}{n}\right)}\right) \]

   11. Simplified 58.6%

       \[\leadsto \mathsf{fma}\left(x, \frac{\mathsf{fma}\left(x, \color{blue}{\frac{0.5}{n} + -0.5}, \mathsf{fma}\left(0.16666666666666666, \frac{x \cdot x}{n \cdot n}, 1\right)\right)}{n}, 1 - {x}^{\left(\frac{1}{n}\right)}\right) \]

   if 7.49999999999999934e-5 < x

   1. Initial program 70.4%

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

   3. Taylor expanded in x around 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. /-lowering-/.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. pow-lowering-pow.f64 N/A

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

      11. /-lowering-/.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. *-lowering-*.f64 98.0

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

   5. Simplified 98.0%

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

4. Final simplification 75.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 6.5 \cdot 10^{-167}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;x \leq 4.5 \cdot 10^{-134}:\\ \;\;\;\;\frac{\log x}{\left(n \cdot n\right) \cdot \frac{-1}{n}}\\ \mathbf{elif}\;x \leq 7.5 \cdot 10^{-5}:\\ \;\;\;\;\mathsf{fma}\left(x, \frac{\mathsf{fma}\left(x, -0.5 + \frac{0.5}{n}, \mathsf{fma}\left(0.16666666666666666, \frac{x \cdot x}{n \cdot n}, 1\right)\right)}{n}, 1 - {x}^{\left(\frac{1}{n}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{n \cdot x}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 68.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ t_1 := \frac{t\_0}{n \cdot x}\\ \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{-11}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-190}:\\ \;\;\;\;-\frac{\log x}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 1:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{+114}:\\ \;\;\;\;1 - t\_0\\ \mathbf{else}:\\ \;\;\;\;-1 + e^{\frac{x}{n}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))) (t_1 (/ t_0 (* n x))))
   (if (<= (/ 1.0 n) -2e-11)
     t_1
     (if (<= (/ 1.0 n) 5e-190)
       (- (/ (log x) n))
       (if (<= (/ 1.0 n) 1.0)
         t_1
         (if (<= (/ 1.0 n) 5e+114) (- 1.0 t_0) (+ -1.0 (exp (/ x n)))))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double t_1 = t_0 / (n * x);
	double tmp;
	if ((1.0 / n) <= -2e-11) {
		tmp = t_1;
	} else if ((1.0 / n) <= 5e-190) {
		tmp = -(log(x) / n);
	} else if ((1.0 / n) <= 1.0) {
		tmp = t_1;
	} else if ((1.0 / n) <= 5e+114) {
		tmp = 1.0 - t_0;
	} else {
		tmp = -1.0 + exp((x / n));
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = x ** (1.0d0 / n)
    t_1 = t_0 / (n * x)
    if ((1.0d0 / n) <= (-2d-11)) then
        tmp = t_1
    else if ((1.0d0 / n) <= 5d-190) then
        tmp = -(log(x) / n)
    else if ((1.0d0 / n) <= 1.0d0) then
        tmp = t_1
    else if ((1.0d0 / n) <= 5d+114) then
        tmp = 1.0d0 - t_0
    else
        tmp = (-1.0d0) + exp((x / n))
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, (1.0 / n));
	double t_1 = t_0 / (n * x);
	double tmp;
	if ((1.0 / n) <= -2e-11) {
		tmp = t_1;
	} else if ((1.0 / n) <= 5e-190) {
		tmp = -(Math.log(x) / n);
	} else if ((1.0 / n) <= 1.0) {
		tmp = t_1;
	} else if ((1.0 / n) <= 5e+114) {
		tmp = 1.0 - t_0;
	} else {
		tmp = -1.0 + Math.exp((x / n));
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	t_1 = t_0 / (n * x)
	tmp = 0
	if (1.0 / n) <= -2e-11:
		tmp = t_1
	elif (1.0 / n) <= 5e-190:
		tmp = -(math.log(x) / n)
	elif (1.0 / n) <= 1.0:
		tmp = t_1
	elif (1.0 / n) <= 5e+114:
		tmp = 1.0 - t_0
	else:
		tmp = -1.0 + math.exp((x / n))
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	t_1 = Float64(t_0 / Float64(n * x))
	tmp = 0.0
	if (Float64(1.0 / n) <= -2e-11)
		tmp = t_1;
	elseif (Float64(1.0 / n) <= 5e-190)
		tmp = Float64(-Float64(log(x) / n));
	elseif (Float64(1.0 / n) <= 1.0)
		tmp = t_1;
	elseif (Float64(1.0 / n) <= 5e+114)
		tmp = Float64(1.0 - t_0);
	else
		tmp = Float64(-1.0 + exp(Float64(x / n)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = x ^ (1.0 / n);
	t_1 = t_0 / (n * x);
	tmp = 0.0;
	if ((1.0 / n) <= -2e-11)
		tmp = t_1;
	elseif ((1.0 / n) <= 5e-190)
		tmp = -(log(x) / n);
	elseif ((1.0 / n) <= 1.0)
		tmp = t_1;
	elseif ((1.0 / n) <= 5e+114)
		tmp = 1.0 - t_0;
	else
		tmp = -1.0 + exp((x / n));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 / N[(n * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -2e-11], t$95$1, If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e-190], (-N[(N[Log[x], $MachinePrecision] / n), $MachinePrecision]), If[LessEqual[N[(1.0 / n), $MachinePrecision], 1.0], t$95$1, If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e+114], N[(1.0 - t$95$0), $MachinePrecision], N[(-1.0 + N[Exp[N[(x / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -1.99999999999999988e-11 or 5.00000000000000034e-190 < (/.f64 #s(literal 1 binary64) n) < 1

   1. Initial program 76.8%

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

   3. Taylor expanded in x around inf

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

      1. /-lowering-/.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. pow-lowering-pow.f64 N/A

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

      11. /-lowering-/.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. *-lowering-*.f64 87.9

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

   5. Simplified 87.9%

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

   if -1.99999999999999988e-11 < (/.f64 #s(literal 1 binary64) n) < 5.00000000000000034e-190

   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 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. --lowering--.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. pow-lowering-pow.f64 N/A

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

      17. /-lowering-/.f64 32.0

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

   5. Simplified 32.0%

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

   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. neg-lowering-neg.f64 N/A

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

      3. /-lowering-/.f64 N/A

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

      4. log-lowering-log.f64 56.8

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

   8. Simplified 56.8%

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

   if 1 < (/.f64 #s(literal 1 binary64) n) < 5.0000000000000001e114

   1. Initial program 99.7%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   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. --lowering--.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. pow-lowering-pow.f64 N/A

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

      17. /-lowering-/.f64 90.0

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

   5. Simplified 90.0%

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

   if 5.0000000000000001e114 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 23.6%

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

      1. pow-to-exp N/A

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

      2. exp-lowering-exp.f64 N/A

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

      3. un-div-inv N/A

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

      4. /-lowering-/.f64 N/A

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

      5. +-commutative N/A

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

      6. accelerator-lowering-log1p.f64 99.5

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

   4. Applied egg-rr 99.5%

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

   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. /-lowering-/.f64 99.5

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

   7. Simplified 99.5%

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

   8. Taylor expanded in n around inf

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

   10. Simplified 73.1%

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

4. Final simplification 76.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{-11}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-190}:\\ \;\;\;\;-\frac{\log x}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 1:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{+114}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;-1 + e^{\frac{x}{n}}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 60.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -1:\\ \;\;\;\;\frac{0.3333333333333333}{n \cdot \left(x \cdot \left(x \cdot x\right)\right)}\\ \mathbf{elif}\;\frac{1}{n} \leq -2 \cdot 10^{-294}:\\ \;\;\;\;-\frac{\log x}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 1:\\ \;\;\;\;\frac{\frac{1}{n}}{x}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{+114}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;-1 + e^{\frac{x}{n}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= (/ 1.0 n) -1.0)
   (/ 0.3333333333333333 (* n (* x (* x x))))
   (if (<= (/ 1.0 n) -2e-294)
     (- (/ (log x) n))
     (if (<= (/ 1.0 n) 1.0)
       (/ (/ 1.0 n) x)
       (if (<= (/ 1.0 n) 5e+114)
         (- 1.0 (pow x (/ 1.0 n)))
         (+ -1.0 (exp (/ x n))))))))

C

double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -1.0) {
		tmp = 0.3333333333333333 / (n * (x * (x * x)));
	} else if ((1.0 / n) <= -2e-294) {
		tmp = -(log(x) / n);
	} else if ((1.0 / n) <= 1.0) {
		tmp = (1.0 / n) / x;
	} else if ((1.0 / n) <= 5e+114) {
		tmp = 1.0 - pow(x, (1.0 / n));
	} else {
		tmp = -1.0 + exp((x / n));
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if ((1.0d0 / n) <= (-1.0d0)) then
        tmp = 0.3333333333333333d0 / (n * (x * (x * x)))
    else if ((1.0d0 / n) <= (-2d-294)) then
        tmp = -(log(x) / n)
    else if ((1.0d0 / n) <= 1.0d0) then
        tmp = (1.0d0 / n) / x
    else if ((1.0d0 / n) <= 5d+114) then
        tmp = 1.0d0 - (x ** (1.0d0 / n))
    else
        tmp = (-1.0d0) + exp((x / n))
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -1.0) {
		tmp = 0.3333333333333333 / (n * (x * (x * x)));
	} else if ((1.0 / n) <= -2e-294) {
		tmp = -(Math.log(x) / n);
	} else if ((1.0 / n) <= 1.0) {
		tmp = (1.0 / n) / x;
	} else if ((1.0 / n) <= 5e+114) {
		tmp = 1.0 - Math.pow(x, (1.0 / n));
	} else {
		tmp = -1.0 + Math.exp((x / n));
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if (1.0 / n) <= -1.0:
		tmp = 0.3333333333333333 / (n * (x * (x * x)))
	elif (1.0 / n) <= -2e-294:
		tmp = -(math.log(x) / n)
	elif (1.0 / n) <= 1.0:
		tmp = (1.0 / n) / x
	elif (1.0 / n) <= 5e+114:
		tmp = 1.0 - math.pow(x, (1.0 / n))
	else:
		tmp = -1.0 + math.exp((x / n))
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -1.0)
		tmp = Float64(0.3333333333333333 / Float64(n * Float64(x * Float64(x * x))));
	elseif (Float64(1.0 / n) <= -2e-294)
		tmp = Float64(-Float64(log(x) / n));
	elseif (Float64(1.0 / n) <= 1.0)
		tmp = Float64(Float64(1.0 / n) / x);
	elseif (Float64(1.0 / n) <= 5e+114)
		tmp = Float64(1.0 - (x ^ Float64(1.0 / n)));
	else
		tmp = Float64(-1.0 + exp(Float64(x / n)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if ((1.0 / n) <= -1.0)
		tmp = 0.3333333333333333 / (n * (x * (x * x)));
	elseif ((1.0 / n) <= -2e-294)
		tmp = -(log(x) / n);
	elseif ((1.0 / n) <= 1.0)
		tmp = (1.0 / n) / x;
	elseif ((1.0 / n) <= 5e+114)
		tmp = 1.0 - (x ^ (1.0 / n));
	else
		tmp = -1.0 + exp((x / n));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[N[(1.0 / n), $MachinePrecision], -1.0], N[(0.3333333333333333 / N[(n * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], -2e-294], (-N[(N[Log[x], $MachinePrecision] / n), $MachinePrecision]), If[LessEqual[N[(1.0 / n), $MachinePrecision], 1.0], N[(N[(1.0 / n), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e+114], N[(1.0 - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(-1.0 + N[Exp[N[(x / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{\frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}}{n} + \left(\frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}} \cdot \left(\frac{1}{2} \cdot \frac{1}{{n}^{2}} - \frac{1}{2} \cdot \frac{1}{n}\right)}{x} + \frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}} \cdot \left(\left(\frac{1}{6} \cdot \frac{1}{{n}^{3}} + \frac{1}{3} \cdot \frac{1}{n}\right) - \frac{1}{2} \cdot \frac{1}{{n}^{2}}\right)}{{x}^{2}}\right)}{x}} \]

   5. Simplified 47.7%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\frac{{x}^{\left(\frac{1}{n}\right)}}{x}, \left(\frac{0.5}{n \cdot n} + \frac{-0.5}{n}\right) + \frac{\frac{0.16666666666666666}{n \cdot \left(n \cdot n\right)} + \left(\frac{0.3333333333333333}{n} + \frac{-0.5}{n \cdot n}\right)}{x}, \frac{{x}^{\left(\frac{1}{n}\right)}}{n}\right)}{x}} \]

   6. Taylor expanded in n around inf

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

      1. /-lowering-/.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. associate-+l+ N/A

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

      5. +-lowering-+.f64 N/A

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

      6. associate-*r/ N/A

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

      7. metadata-eval N/A

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

      8. /-lowering-/.f64 N/A

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

      9. unpow2 N/A

         \[\leadsto \frac{\frac{\frac{\frac{1}{3}}{\color{blue}{x \cdot x}} + \left(1 + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{x}\right)\right)\right)}{n}}{x} \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \frac{\frac{\frac{\frac{1}{3}}{\color{blue}{x \cdot x}} + \left(1 + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{x}\right)\right)\right)}{n}}{x} \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \frac{\frac{\frac{\frac{1}{3}}{x \cdot x} + \color{blue}{\left(1 + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{x}\right)\right)\right)}}{n}}{x} \]

      12. associate-*r/ N/A

          \[\leadsto \frac{\frac{\frac{\frac{1}{3}}{x \cdot x} + \left(1 + \left(\mathsf{neg}\left(\color{blue}{\frac{\frac{1}{2} \cdot 1}{x}}\right)\right)\right)}{n}}{x} \]

      13. metadata-eval N/A

          \[\leadsto \frac{\frac{\frac{\frac{1}{3}}{x \cdot x} + \left(1 + \left(\mathsf{neg}\left(\frac{\color{blue}{\frac{1}{2}}}{x}\right)\right)\right)}{n}}{x} \]

      14. distribute-neg-frac N/A

          \[\leadsto \frac{\frac{\frac{\frac{1}{3}}{x \cdot x} + \left(1 + \color{blue}{\frac{\mathsf{neg}\left(\frac{1}{2}\right)}{x}}\right)}{n}}{x} \]

      15. metadata-eval N/A

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

      16. /-lowering-/.f64 48.6

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

   8. Simplified 48.6%

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

   9. Taylor expanded in x around 0

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

       1. /-lowering-/.f64 N/A

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

       2. *-lowering-*.f64 N/A

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

       3. cube-mult N/A

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

       4. unpow2 N/A

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

       5. *-lowering-*.f64 N/A

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

       6. unpow2 N/A

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

       7. *-lowering-*.f64 82.6

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

   11. Simplified 82.6%

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

   if -1 < (/.f64 #s(literal 1 binary64) n) < -2.00000000000000003e-294

   1. Initial program 24.2%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1 - e^{\frac{\log x}{n}}} \]
   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. --lowering--.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. pow-lowering-pow.f64 N/A

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

      17. /-lowering-/.f64 24.1

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

   5. Simplified 24.1%

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

   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. neg-lowering-neg.f64 N/A

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

      3. /-lowering-/.f64 N/A

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

      4. log-lowering-log.f64 56.0

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

   8. Simplified 56.0%

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

   if -2.00000000000000003e-294 < (/.f64 #s(literal 1 binary64) n) < 1

   1. Initial program 35.2%

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

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{\frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}}{n} + \left(\frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}} \cdot \left(\frac{1}{2} \cdot \frac{1}{{n}^{2}} - \frac{1}{2} \cdot \frac{1}{n}\right)}{x} + \frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}} \cdot \left(\left(\frac{1}{6} \cdot \frac{1}{{n}^{3}} + \frac{1}{3} \cdot \frac{1}{n}\right) - \frac{1}{2} \cdot \frac{1}{{n}^{2}}\right)}{{x}^{2}}\right)}{x}} \]

   5. Simplified 57.5%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\frac{{x}^{\left(\frac{1}{n}\right)}}{x}, \left(\frac{0.5}{n \cdot n} + \frac{-0.5}{n}\right) + \frac{\frac{0.16666666666666666}{n \cdot \left(n \cdot n\right)} + \left(\frac{0.3333333333333333}{n} + \frac{-0.5}{n \cdot n}\right)}{x}, \frac{{x}^{\left(\frac{1}{n}\right)}}{n}\right)}{x}} \]

   6. Taylor expanded in n around inf

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

      1. /-lowering-/.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. associate-+l+ N/A

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

      5. +-lowering-+.f64 N/A

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

      6. associate-*r/ N/A

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

      7. metadata-eval N/A

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

      8. /-lowering-/.f64 N/A

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

      9. unpow2 N/A

         \[\leadsto \frac{\frac{\frac{\frac{1}{3}}{\color{blue}{x \cdot x}} + \left(1 + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{x}\right)\right)\right)}{n}}{x} \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \frac{\frac{\frac{\frac{1}{3}}{\color{blue}{x \cdot x}} + \left(1 + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{x}\right)\right)\right)}{n}}{x} \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \frac{\frac{\frac{\frac{1}{3}}{x \cdot x} + \color{blue}{\left(1 + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{x}\right)\right)\right)}}{n}}{x} \]

      12. associate-*r/ N/A

          \[\leadsto \frac{\frac{\frac{\frac{1}{3}}{x \cdot x} + \left(1 + \left(\mathsf{neg}\left(\color{blue}{\frac{\frac{1}{2} \cdot 1}{x}}\right)\right)\right)}{n}}{x} \]

      13. metadata-eval N/A

          \[\leadsto \frac{\frac{\frac{\frac{1}{3}}{x \cdot x} + \left(1 + \left(\mathsf{neg}\left(\frac{\color{blue}{\frac{1}{2}}}{x}\right)\right)\right)}{n}}{x} \]

      14. distribute-neg-frac N/A

          \[\leadsto \frac{\frac{\frac{\frac{1}{3}}{x \cdot x} + \left(1 + \color{blue}{\frac{\mathsf{neg}\left(\frac{1}{2}\right)}{x}}\right)}{n}}{x} \]

      15. metadata-eval N/A

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

      16. /-lowering-/.f64 53.3

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

   8. Simplified 53.3%

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

   9. Taylor expanded in x around inf

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

       1. /-lowering-/.f64 53.5

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

   11. Simplified 53.5%

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

   if 1 < (/.f64 #s(literal 1 binary64) n) < 5.0000000000000001e114

   1. Initial program 99.7%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   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. --lowering--.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. pow-lowering-pow.f64 N/A

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

      17. /-lowering-/.f64 90.0

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

   5. Simplified 90.0%

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

   if 5.0000000000000001e114 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 23.6%

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

      1. pow-to-exp N/A

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

      2. exp-lowering-exp.f64 N/A

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

      3. un-div-inv N/A

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

      4. /-lowering-/.f64 N/A

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

      5. +-commutative N/A

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

      6. accelerator-lowering-log1p.f64 99.5

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

   4. Applied egg-rr 99.5%

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

   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. /-lowering-/.f64 99.5

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

   7. Simplified 99.5%

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

   8. Taylor expanded in n around inf

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

   10. Simplified 73.1%

       \[\leadsto e^{\frac{x}{n}} - \color{blue}{1} \]
3. Recombined 5 regimes into one program.

4. Final simplification 68.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -1:\\ \;\;\;\;\frac{0.3333333333333333}{n \cdot \left(x \cdot \left(x \cdot x\right)\right)}\\ \mathbf{elif}\;\frac{1}{n} \leq -2 \cdot 10^{-294}:\\ \;\;\;\;-\frac{\log x}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 1:\\ \;\;\;\;\frac{\frac{1}{n}}{x}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{+114}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;-1 + e^{\frac{x}{n}}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 56.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -1:\\ \;\;\;\;\frac{0.3333333333333333}{n \cdot \left(x \cdot \left(x \cdot x\right)\right)}\\ \mathbf{elif}\;\frac{1}{n} \leq -2 \cdot 10^{-294}:\\ \;\;\;\;-\frac{\log x}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 500000:\\ \;\;\;\;\frac{\frac{1}{n}}{x}\\ \mathbf{else}:\\ \;\;\;\;-1 + e^{\frac{x}{n}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= (/ 1.0 n) -1.0)
   (/ 0.3333333333333333 (* n (* x (* x x))))
   (if (<= (/ 1.0 n) -2e-294)
     (- (/ (log x) n))
     (if (<= (/ 1.0 n) 500000.0) (/ (/ 1.0 n) x) (+ -1.0 (exp (/ x n)))))))

C

double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -1.0) {
		tmp = 0.3333333333333333 / (n * (x * (x * x)));
	} else if ((1.0 / n) <= -2e-294) {
		tmp = -(log(x) / n);
	} else if ((1.0 / n) <= 500000.0) {
		tmp = (1.0 / n) / x;
	} else {
		tmp = -1.0 + exp((x / n));
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if ((1.0d0 / n) <= (-1.0d0)) then
        tmp = 0.3333333333333333d0 / (n * (x * (x * x)))
    else if ((1.0d0 / n) <= (-2d-294)) then
        tmp = -(log(x) / n)
    else if ((1.0d0 / n) <= 500000.0d0) then
        tmp = (1.0d0 / n) / x
    else
        tmp = (-1.0d0) + exp((x / n))
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -1.0) {
		tmp = 0.3333333333333333 / (n * (x * (x * x)));
	} else if ((1.0 / n) <= -2e-294) {
		tmp = -(Math.log(x) / n);
	} else if ((1.0 / n) <= 500000.0) {
		tmp = (1.0 / n) / x;
	} else {
		tmp = -1.0 + Math.exp((x / n));
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if (1.0 / n) <= -1.0:
		tmp = 0.3333333333333333 / (n * (x * (x * x)))
	elif (1.0 / n) <= -2e-294:
		tmp = -(math.log(x) / n)
	elif (1.0 / n) <= 500000.0:
		tmp = (1.0 / n) / x
	else:
		tmp = -1.0 + math.exp((x / n))
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -1.0)
		tmp = Float64(0.3333333333333333 / Float64(n * Float64(x * Float64(x * x))));
	elseif (Float64(1.0 / n) <= -2e-294)
		tmp = Float64(-Float64(log(x) / n));
	elseif (Float64(1.0 / n) <= 500000.0)
		tmp = Float64(Float64(1.0 / n) / x);
	else
		tmp = Float64(-1.0 + exp(Float64(x / n)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if ((1.0 / n) <= -1.0)
		tmp = 0.3333333333333333 / (n * (x * (x * x)));
	elseif ((1.0 / n) <= -2e-294)
		tmp = -(log(x) / n);
	elseif ((1.0 / n) <= 500000.0)
		tmp = (1.0 / n) / x;
	else
		tmp = -1.0 + exp((x / n));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[N[(1.0 / n), $MachinePrecision], -1.0], N[(0.3333333333333333 / N[(n * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], -2e-294], (-N[(N[Log[x], $MachinePrecision] / n), $MachinePrecision]), If[LessEqual[N[(1.0 / n), $MachinePrecision], 500000.0], N[(N[(1.0 / n), $MachinePrecision] / x), $MachinePrecision], N[(-1.0 + N[Exp[N[(x / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{\frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}}{n} + \left(\frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}} \cdot \left(\frac{1}{2} \cdot \frac{1}{{n}^{2}} - \frac{1}{2} \cdot \frac{1}{n}\right)}{x} + \frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}} \cdot \left(\left(\frac{1}{6} \cdot \frac{1}{{n}^{3}} + \frac{1}{3} \cdot \frac{1}{n}\right) - \frac{1}{2} \cdot \frac{1}{{n}^{2}}\right)}{{x}^{2}}\right)}{x}} \]

   5. Simplified 47.7%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\frac{{x}^{\left(\frac{1}{n}\right)}}{x}, \left(\frac{0.5}{n \cdot n} + \frac{-0.5}{n}\right) + \frac{\frac{0.16666666666666666}{n \cdot \left(n \cdot n\right)} + \left(\frac{0.3333333333333333}{n} + \frac{-0.5}{n \cdot n}\right)}{x}, \frac{{x}^{\left(\frac{1}{n}\right)}}{n}\right)}{x}} \]

   6. Taylor expanded in n around inf

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

      1. /-lowering-/.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. associate-+l+ N/A

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

      5. +-lowering-+.f64 N/A

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

      6. associate-*r/ N/A

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

      7. metadata-eval N/A

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

      8. /-lowering-/.f64 N/A

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

      9. unpow2 N/A

         \[\leadsto \frac{\frac{\frac{\frac{1}{3}}{\color{blue}{x \cdot x}} + \left(1 + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{x}\right)\right)\right)}{n}}{x} \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \frac{\frac{\frac{\frac{1}{3}}{\color{blue}{x \cdot x}} + \left(1 + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{x}\right)\right)\right)}{n}}{x} \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \frac{\frac{\frac{\frac{1}{3}}{x \cdot x} + \color{blue}{\left(1 + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{x}\right)\right)\right)}}{n}}{x} \]

      12. associate-*r/ N/A

          \[\leadsto \frac{\frac{\frac{\frac{1}{3}}{x \cdot x} + \left(1 + \left(\mathsf{neg}\left(\color{blue}{\frac{\frac{1}{2} \cdot 1}{x}}\right)\right)\right)}{n}}{x} \]

      13. metadata-eval N/A

          \[\leadsto \frac{\frac{\frac{\frac{1}{3}}{x \cdot x} + \left(1 + \left(\mathsf{neg}\left(\frac{\color{blue}{\frac{1}{2}}}{x}\right)\right)\right)}{n}}{x} \]

      14. distribute-neg-frac N/A

          \[\leadsto \frac{\frac{\frac{\frac{1}{3}}{x \cdot x} + \left(1 + \color{blue}{\frac{\mathsf{neg}\left(\frac{1}{2}\right)}{x}}\right)}{n}}{x} \]

      15. metadata-eval N/A

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

      16. /-lowering-/.f64 48.6

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

   8. Simplified 48.6%

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

   9. Taylor expanded in x around 0

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

       1. /-lowering-/.f64 N/A

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

       2. *-lowering-*.f64 N/A

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

       3. cube-mult N/A

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

       4. unpow2 N/A

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

       5. *-lowering-*.f64 N/A

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

       6. unpow2 N/A

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

       7. *-lowering-*.f64 82.6

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

   11. Simplified 82.6%

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

   if -1 < (/.f64 #s(literal 1 binary64) n) < -2.00000000000000003e-294

   1. Initial program 24.2%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1 - e^{\frac{\log x}{n}}} \]
   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. --lowering--.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. pow-lowering-pow.f64 N/A

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

      17. /-lowering-/.f64 24.1

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

   5. Simplified 24.1%

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

   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. neg-lowering-neg.f64 N/A

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

      3. /-lowering-/.f64 N/A

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

      4. log-lowering-log.f64 56.0

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

   8. Simplified 56.0%

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

   if -2.00000000000000003e-294 < (/.f64 #s(literal 1 binary64) n) < 5e5

   1. Initial program 38.1%

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

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{\frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}}{n} + \left(\frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}} \cdot \left(\frac{1}{2} \cdot \frac{1}{{n}^{2}} - \frac{1}{2} \cdot \frac{1}{n}\right)}{x} + \frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}} \cdot \left(\left(\frac{1}{6} \cdot \frac{1}{{n}^{3}} + \frac{1}{3} \cdot \frac{1}{n}\right) - \frac{1}{2} \cdot \frac{1}{{n}^{2}}\right)}{{x}^{2}}\right)}{x}} \]

   5. Simplified 55.0%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\frac{{x}^{\left(\frac{1}{n}\right)}}{x}, \left(\frac{0.5}{n \cdot n} + \frac{-0.5}{n}\right) + \frac{\frac{0.16666666666666666}{n \cdot \left(n \cdot n\right)} + \left(\frac{0.3333333333333333}{n} + \frac{-0.5}{n \cdot n}\right)}{x}, \frac{{x}^{\left(\frac{1}{n}\right)}}{n}\right)}{x}} \]

   6. Taylor expanded in n around inf

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

      1. /-lowering-/.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. associate-+l+ N/A

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

      5. +-lowering-+.f64 N/A

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

      6. associate-*r/ N/A

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

      7. metadata-eval N/A

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

      8. /-lowering-/.f64 N/A

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

      9. unpow2 N/A

         \[\leadsto \frac{\frac{\frac{\frac{1}{3}}{\color{blue}{x \cdot x}} + \left(1 + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{x}\right)\right)\right)}{n}}{x} \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \frac{\frac{\frac{\frac{1}{3}}{\color{blue}{x \cdot x}} + \left(1 + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{x}\right)\right)\right)}{n}}{x} \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \frac{\frac{\frac{\frac{1}{3}}{x \cdot x} + \color{blue}{\left(1 + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{x}\right)\right)\right)}}{n}}{x} \]

      12. associate-*r/ N/A

          \[\leadsto \frac{\frac{\frac{\frac{1}{3}}{x \cdot x} + \left(1 + \left(\mathsf{neg}\left(\color{blue}{\frac{\frac{1}{2} \cdot 1}{x}}\right)\right)\right)}{n}}{x} \]

      13. metadata-eval N/A

          \[\leadsto \frac{\frac{\frac{\frac{1}{3}}{x \cdot x} + \left(1 + \left(\mathsf{neg}\left(\frac{\color{blue}{\frac{1}{2}}}{x}\right)\right)\right)}{n}}{x} \]

      14. distribute-neg-frac N/A

          \[\leadsto \frac{\frac{\frac{\frac{1}{3}}{x \cdot x} + \left(1 + \color{blue}{\frac{\mathsf{neg}\left(\frac{1}{2}\right)}{x}}\right)}{n}}{x} \]

      15. metadata-eval N/A

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

      16. /-lowering-/.f64 51.1

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

   8. Simplified 51.1%

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

   9. Taylor expanded in x around inf

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

       1. /-lowering-/.f64 51.3

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

   11. Simplified 51.3%

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

   if 5e5 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 58.5%

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

      1. pow-to-exp N/A

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

      2. exp-lowering-exp.f64 N/A

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

      3. un-div-inv N/A

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

      4. /-lowering-/.f64 N/A

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

      5. +-commutative N/A

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

      6. accelerator-lowering-log1p.f64 99.7

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

   4. Applied egg-rr 99.7%

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

   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. /-lowering-/.f64 99.7

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

   7. Simplified 99.7%

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

   8. Taylor expanded in n around inf

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

   10. Simplified 46.4%

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

4. Final simplification 62.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -1:\\ \;\;\;\;\frac{0.3333333333333333}{n \cdot \left(x \cdot \left(x \cdot x\right)\right)}\\ \mathbf{elif}\;\frac{1}{n} \leq -2 \cdot 10^{-294}:\\ \;\;\;\;-\frac{\log x}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 500000:\\ \;\;\;\;\frac{\frac{1}{n}}{x}\\ \mathbf{else}:\\ \;\;\;\;-1 + e^{\frac{x}{n}}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 54.9% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -1:\\ \;\;\;\;\frac{0.3333333333333333}{n \cdot \left(x \cdot \left(x \cdot x\right)\right)}\\ \mathbf{elif}\;\frac{1}{n} \leq -2 \cdot 10^{-294}:\\ \;\;\;\;-\frac{\log x}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{0.3333333333333333}{x \cdot x} + \left(1 + \frac{-0.5}{x}\right)}{n}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= (/ 1.0 n) -1.0)
   (/ 0.3333333333333333 (* n (* x (* x x))))
   (if (<= (/ 1.0 n) -2e-294)
     (- (/ (log x) n))
     (/ (/ (+ (/ 0.3333333333333333 (* x x)) (+ 1.0 (/ -0.5 x))) n) x))))

C

double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -1.0) {
		tmp = 0.3333333333333333 / (n * (x * (x * x)));
	} else if ((1.0 / n) <= -2e-294) {
		tmp = -(log(x) / n);
	} else {
		tmp = (((0.3333333333333333 / (x * x)) + (1.0 + (-0.5 / x))) / n) / x;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if ((1.0d0 / n) <= (-1.0d0)) then
        tmp = 0.3333333333333333d0 / (n * (x * (x * x)))
    else if ((1.0d0 / n) <= (-2d-294)) then
        tmp = -(log(x) / n)
    else
        tmp = (((0.3333333333333333d0 / (x * x)) + (1.0d0 + ((-0.5d0) / x))) / n) / x
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -1.0) {
		tmp = 0.3333333333333333 / (n * (x * (x * x)));
	} else if ((1.0 / n) <= -2e-294) {
		tmp = -(Math.log(x) / n);
	} else {
		tmp = (((0.3333333333333333 / (x * x)) + (1.0 + (-0.5 / x))) / n) / x;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if (1.0 / n) <= -1.0:
		tmp = 0.3333333333333333 / (n * (x * (x * x)))
	elif (1.0 / n) <= -2e-294:
		tmp = -(math.log(x) / n)
	else:
		tmp = (((0.3333333333333333 / (x * x)) + (1.0 + (-0.5 / x))) / n) / x
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -1.0)
		tmp = Float64(0.3333333333333333 / Float64(n * Float64(x * Float64(x * x))));
	elseif (Float64(1.0 / n) <= -2e-294)
		tmp = Float64(-Float64(log(x) / n));
	else
		tmp = Float64(Float64(Float64(Float64(0.3333333333333333 / Float64(x * x)) + Float64(1.0 + Float64(-0.5 / x))) / n) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if ((1.0 / n) <= -1.0)
		tmp = 0.3333333333333333 / (n * (x * (x * x)));
	elseif ((1.0 / n) <= -2e-294)
		tmp = -(log(x) / n);
	else
		tmp = (((0.3333333333333333 / (x * x)) + (1.0 + (-0.5 / x))) / n) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[N[(1.0 / n), $MachinePrecision], -1.0], N[(0.3333333333333333 / N[(n * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], -2e-294], (-N[(N[Log[x], $MachinePrecision] / n), $MachinePrecision]), N[(N[(N[(N[(0.3333333333333333 / N[(x * x), $MachinePrecision]), $MachinePrecision] + N[(1.0 + N[(-0.5 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision] / x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{\frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}}{n} + \left(\frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}} \cdot \left(\frac{1}{2} \cdot \frac{1}{{n}^{2}} - \frac{1}{2} \cdot \frac{1}{n}\right)}{x} + \frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}} \cdot \left(\left(\frac{1}{6} \cdot \frac{1}{{n}^{3}} + \frac{1}{3} \cdot \frac{1}{n}\right) - \frac{1}{2} \cdot \frac{1}{{n}^{2}}\right)}{{x}^{2}}\right)}{x}} \]

   5. Simplified 47.7%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\frac{{x}^{\left(\frac{1}{n}\right)}}{x}, \left(\frac{0.5}{n \cdot n} + \frac{-0.5}{n}\right) + \frac{\frac{0.16666666666666666}{n \cdot \left(n \cdot n\right)} + \left(\frac{0.3333333333333333}{n} + \frac{-0.5}{n \cdot n}\right)}{x}, \frac{{x}^{\left(\frac{1}{n}\right)}}{n}\right)}{x}} \]

   6. Taylor expanded in n around inf

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

      1. /-lowering-/.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. associate-+l+ N/A

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

      5. +-lowering-+.f64 N/A

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

      6. associate-*r/ N/A

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

      7. metadata-eval N/A

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

      8. /-lowering-/.f64 N/A

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

      9. unpow2 N/A

         \[\leadsto \frac{\frac{\frac{\frac{1}{3}}{\color{blue}{x \cdot x}} + \left(1 + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{x}\right)\right)\right)}{n}}{x} \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \frac{\frac{\frac{\frac{1}{3}}{\color{blue}{x \cdot x}} + \left(1 + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{x}\right)\right)\right)}{n}}{x} \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \frac{\frac{\frac{\frac{1}{3}}{x \cdot x} + \color{blue}{\left(1 + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{x}\right)\right)\right)}}{n}}{x} \]

      12. associate-*r/ N/A

          \[\leadsto \frac{\frac{\frac{\frac{1}{3}}{x \cdot x} + \left(1 + \left(\mathsf{neg}\left(\color{blue}{\frac{\frac{1}{2} \cdot 1}{x}}\right)\right)\right)}{n}}{x} \]

      13. metadata-eval N/A

          \[\leadsto \frac{\frac{\frac{\frac{1}{3}}{x \cdot x} + \left(1 + \left(\mathsf{neg}\left(\frac{\color{blue}{\frac{1}{2}}}{x}\right)\right)\right)}{n}}{x} \]

      14. distribute-neg-frac N/A

          \[\leadsto \frac{\frac{\frac{\frac{1}{3}}{x \cdot x} + \left(1 + \color{blue}{\frac{\mathsf{neg}\left(\frac{1}{2}\right)}{x}}\right)}{n}}{x} \]

      15. metadata-eval N/A

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

      16. /-lowering-/.f64 48.6

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

   8. Simplified 48.6%

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

   9. Taylor expanded in x around 0

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

       1. /-lowering-/.f64 N/A

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

       2. *-lowering-*.f64 N/A

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

       3. cube-mult N/A

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

       4. unpow2 N/A

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

       5. *-lowering-*.f64 N/A

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

       6. unpow2 N/A

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

       7. *-lowering-*.f64 82.6

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

   11. Simplified 82.6%

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

   if -1 < (/.f64 #s(literal 1 binary64) n) < -2.00000000000000003e-294

   1. Initial program 24.2%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1 - e^{\frac{\log x}{n}}} \]
   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. --lowering--.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. pow-lowering-pow.f64 N/A

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

      17. /-lowering-/.f64 24.1

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

   5. Simplified 24.1%

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

   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. neg-lowering-neg.f64 N/A

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

      3. /-lowering-/.f64 N/A

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

      4. log-lowering-log.f64 56.0

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

   8. Simplified 56.0%

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

   if -2.00000000000000003e-294 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 45.4%

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

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{\frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}}{n} + \left(\frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}} \cdot \left(\frac{1}{2} \cdot \frac{1}{{n}^{2}} - \frac{1}{2} \cdot \frac{1}{n}\right)}{x} + \frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}} \cdot \left(\left(\frac{1}{6} \cdot \frac{1}{{n}^{3}} + \frac{1}{3} \cdot \frac{1}{n}\right) - \frac{1}{2} \cdot \frac{1}{{n}^{2}}\right)}{{x}^{2}}\right)}{x}} \]

   5. Simplified 35.7%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\frac{{x}^{\left(\frac{1}{n}\right)}}{x}, \left(\frac{0.5}{n \cdot n} + \frac{-0.5}{n}\right) + \frac{\frac{0.16666666666666666}{n \cdot \left(n \cdot n\right)} + \left(\frac{0.3333333333333333}{n} + \frac{-0.5}{n \cdot n}\right)}{x}, \frac{{x}^{\left(\frac{1}{n}\right)}}{n}\right)}{x}} \]

   6. Taylor expanded in n around inf

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

      1. /-lowering-/.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. associate-+l+ N/A

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

      5. +-lowering-+.f64 N/A

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

      6. associate-*r/ N/A

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

      7. metadata-eval N/A

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

      8. /-lowering-/.f64 N/A

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

      9. unpow2 N/A

         \[\leadsto \frac{\frac{\frac{\frac{1}{3}}{\color{blue}{x \cdot x}} + \left(1 + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{x}\right)\right)\right)}{n}}{x} \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \frac{\frac{\frac{\frac{1}{3}}{\color{blue}{x \cdot x}} + \left(1 + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{x}\right)\right)\right)}{n}}{x} \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \frac{\frac{\frac{\frac{1}{3}}{x \cdot x} + \color{blue}{\left(1 + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{x}\right)\right)\right)}}{n}}{x} \]

      12. associate-*r/ N/A

          \[\leadsto \frac{\frac{\frac{\frac{1}{3}}{x \cdot x} + \left(1 + \left(\mathsf{neg}\left(\color{blue}{\frac{\frac{1}{2} \cdot 1}{x}}\right)\right)\right)}{n}}{x} \]

      13. metadata-eval N/A

          \[\leadsto \frac{\frac{\frac{\frac{1}{3}}{x \cdot x} + \left(1 + \left(\mathsf{neg}\left(\frac{\color{blue}{\frac{1}{2}}}{x}\right)\right)\right)}{n}}{x} \]

      14. distribute-neg-frac N/A

          \[\leadsto \frac{\frac{\frac{\frac{1}{3}}{x \cdot x} + \left(1 + \color{blue}{\frac{\mathsf{neg}\left(\frac{1}{2}\right)}{x}}\right)}{n}}{x} \]

      15. metadata-eval N/A

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

      16. /-lowering-/.f64 41.7

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

   8. Simplified 41.7%

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

Alternative 14: 54.9% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -0.0001:\\ \;\;\;\;\frac{0.3333333333333333}{n \cdot \left(x \cdot \left(x \cdot x\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{0.3333333333333333}{x \cdot x} + \left(1 + \frac{-0.5}{x}\right)}{n}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= (/ 1.0 n) -0.0001)
   (/ 0.3333333333333333 (* n (* x (* x x))))
   (/ (/ (+ (/ 0.3333333333333333 (* x x)) (+ 1.0 (/ -0.5 x))) n) x)))

C

double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -0.0001) {
		tmp = 0.3333333333333333 / (n * (x * (x * x)));
	} else {
		tmp = (((0.3333333333333333 / (x * x)) + (1.0 + (-0.5 / x))) / n) / x;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if ((1.0d0 / n) <= (-0.0001d0)) then
        tmp = 0.3333333333333333d0 / (n * (x * (x * x)))
    else
        tmp = (((0.3333333333333333d0 / (x * x)) + (1.0d0 + ((-0.5d0) / x))) / n) / x
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -0.0001) {
		tmp = 0.3333333333333333 / (n * (x * (x * x)));
	} else {
		tmp = (((0.3333333333333333 / (x * x)) + (1.0 + (-0.5 / x))) / n) / x;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if (1.0 / n) <= -0.0001:
		tmp = 0.3333333333333333 / (n * (x * (x * x)))
	else:
		tmp = (((0.3333333333333333 / (x * x)) + (1.0 + (-0.5 / x))) / n) / x
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -0.0001)
		tmp = Float64(0.3333333333333333 / Float64(n * Float64(x * Float64(x * x))));
	else
		tmp = Float64(Float64(Float64(Float64(0.3333333333333333 / Float64(x * x)) + Float64(1.0 + Float64(-0.5 / x))) / n) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if ((1.0 / n) <= -0.0001)
		tmp = 0.3333333333333333 / (n * (x * (x * x)));
	else
		tmp = (((0.3333333333333333 / (x * x)) + (1.0 + (-0.5 / x))) / n) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[N[(1.0 / n), $MachinePrecision], -0.0001], N[(0.3333333333333333 / N[(n * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(0.3333333333333333 / N[(x * x), $MachinePrecision]), $MachinePrecision] + N[(1.0 + N[(-0.5 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -1.00000000000000005e-4

   1. Initial program 99.9%

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

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{\frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}}{n} + \left(\frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}} \cdot \left(\frac{1}{2} \cdot \frac{1}{{n}^{2}} - \frac{1}{2} \cdot \frac{1}{n}\right)}{x} + \frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}} \cdot \left(\left(\frac{1}{6} \cdot \frac{1}{{n}^{3}} + \frac{1}{3} \cdot \frac{1}{n}\right) - \frac{1}{2} \cdot \frac{1}{{n}^{2}}\right)}{{x}^{2}}\right)}{x}} \]

   5. Simplified 46.7%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\frac{{x}^{\left(\frac{1}{n}\right)}}{x}, \left(\frac{0.5}{n \cdot n} + \frac{-0.5}{n}\right) + \frac{\frac{0.16666666666666666}{n \cdot \left(n \cdot n\right)} + \left(\frac{0.3333333333333333}{n} + \frac{-0.5}{n \cdot n}\right)}{x}, \frac{{x}^{\left(\frac{1}{n}\right)}}{n}\right)}{x}} \]

   6. Taylor expanded in n around inf

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

      1. /-lowering-/.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. associate-+l+ N/A

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

      5. +-lowering-+.f64 N/A

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

      6. associate-*r/ N/A

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

      7. metadata-eval N/A

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

      8. /-lowering-/.f64 N/A

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

      9. unpow2 N/A

         \[\leadsto \frac{\frac{\frac{\frac{1}{3}}{\color{blue}{x \cdot x}} + \left(1 + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{x}\right)\right)\right)}{n}}{x} \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \frac{\frac{\frac{\frac{1}{3}}{\color{blue}{x \cdot x}} + \left(1 + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{x}\right)\right)\right)}{n}}{x} \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \frac{\frac{\frac{\frac{1}{3}}{x \cdot x} + \color{blue}{\left(1 + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{x}\right)\right)\right)}}{n}}{x} \]

      12. associate-*r/ N/A

          \[\leadsto \frac{\frac{\frac{\frac{1}{3}}{x \cdot x} + \left(1 + \left(\mathsf{neg}\left(\color{blue}{\frac{\frac{1}{2} \cdot 1}{x}}\right)\right)\right)}{n}}{x} \]

      13. metadata-eval N/A

          \[\leadsto \frac{\frac{\frac{\frac{1}{3}}{x \cdot x} + \left(1 + \left(\mathsf{neg}\left(\frac{\color{blue}{\frac{1}{2}}}{x}\right)\right)\right)}{n}}{x} \]

      14. distribute-neg-frac N/A

          \[\leadsto \frac{\frac{\frac{\frac{1}{3}}{x \cdot x} + \left(1 + \color{blue}{\frac{\mathsf{neg}\left(\frac{1}{2}\right)}{x}}\right)}{n}}{x} \]

      15. metadata-eval N/A

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

      16. /-lowering-/.f64 47.6

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

   8. Simplified 47.6%

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

   9. Taylor expanded in x around 0

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

       1. /-lowering-/.f64 N/A

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

       2. *-lowering-*.f64 N/A

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

       3. cube-mult N/A

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

       4. unpow2 N/A

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

       5. *-lowering-*.f64 N/A

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

       6. unpow2 N/A

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

       7. *-lowering-*.f64 80.8

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

   11. Simplified 80.8%

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

   if -1.00000000000000005e-4 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 36.6%

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

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{\frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}}{n} + \left(\frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}} \cdot \left(\frac{1}{2} \cdot \frac{1}{{n}^{2}} - \frac{1}{2} \cdot \frac{1}{n}\right)}{x} + \frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}} \cdot \left(\left(\frac{1}{6} \cdot \frac{1}{{n}^{3}} + \frac{1}{3} \cdot \frac{1}{n}\right) - \frac{1}{2} \cdot \frac{1}{{n}^{2}}\right)}{{x}^{2}}\right)}{x}} \]

   5. Simplified 39.2%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\frac{{x}^{\left(\frac{1}{n}\right)}}{x}, \left(\frac{0.5}{n \cdot n} + \frac{-0.5}{n}\right) + \frac{\frac{0.16666666666666666}{n \cdot \left(n \cdot n\right)} + \left(\frac{0.3333333333333333}{n} + \frac{-0.5}{n \cdot n}\right)}{x}, \frac{{x}^{\left(\frac{1}{n}\right)}}{n}\right)}{x}} \]

   6. Taylor expanded in n around inf

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

      1. /-lowering-/.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. associate-+l+ N/A

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

      5. +-lowering-+.f64 N/A

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

      6. associate-*r/ N/A

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

      7. metadata-eval N/A

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

      8. /-lowering-/.f64 N/A

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

      9. unpow2 N/A

         \[\leadsto \frac{\frac{\frac{\frac{1}{3}}{\color{blue}{x \cdot x}} + \left(1 + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{x}\right)\right)\right)}{n}}{x} \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \frac{\frac{\frac{\frac{1}{3}}{\color{blue}{x \cdot x}} + \left(1 + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{x}\right)\right)\right)}{n}}{x} \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \frac{\frac{\frac{\frac{1}{3}}{x \cdot x} + \color{blue}{\left(1 + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{x}\right)\right)\right)}}{n}}{x} \]

      12. associate-*r/ N/A

          \[\leadsto \frac{\frac{\frac{\frac{1}{3}}{x \cdot x} + \left(1 + \left(\mathsf{neg}\left(\color{blue}{\frac{\frac{1}{2} \cdot 1}{x}}\right)\right)\right)}{n}}{x} \]

      13. metadata-eval N/A

          \[\leadsto \frac{\frac{\frac{\frac{1}{3}}{x \cdot x} + \left(1 + \left(\mathsf{neg}\left(\frac{\color{blue}{\frac{1}{2}}}{x}\right)\right)\right)}{n}}{x} \]

      14. distribute-neg-frac N/A

          \[\leadsto \frac{\frac{\frac{\frac{1}{3}}{x \cdot x} + \left(1 + \color{blue}{\frac{\mathsf{neg}\left(\frac{1}{2}\right)}{x}}\right)}{n}}{x} \]

      15. metadata-eval N/A

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

      16. /-lowering-/.f64 42.5

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

   8. Simplified 42.5%

      \[\leadsto \frac{\color{blue}{\frac{\frac{0.3333333333333333}{x \cdot x} + \left(1 + \frac{-0.5}{x}\right)}{n}}}{x} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 15: 54.2% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -0.0001:\\ \;\;\;\;\frac{0.3333333333333333}{n \cdot \left(x \cdot \left(x \cdot x\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{0.3333333333333333}{x \cdot x} + \left(1 + \frac{-0.5}{x}\right)\right) \cdot \frac{1}{n \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= (/ 1.0 n) -0.0001)
   (/ 0.3333333333333333 (* n (* x (* x x))))
   (* (+ (/ 0.3333333333333333 (* x x)) (+ 1.0 (/ -0.5 x))) (/ 1.0 (* n x)))))

C

double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -0.0001) {
		tmp = 0.3333333333333333 / (n * (x * (x * x)));
	} else {
		tmp = ((0.3333333333333333 / (x * x)) + (1.0 + (-0.5 / x))) * (1.0 / (n * x));
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if ((1.0d0 / n) <= (-0.0001d0)) then
        tmp = 0.3333333333333333d0 / (n * (x * (x * x)))
    else
        tmp = ((0.3333333333333333d0 / (x * x)) + (1.0d0 + ((-0.5d0) / x))) * (1.0d0 / (n * x))
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -0.0001) {
		tmp = 0.3333333333333333 / (n * (x * (x * x)));
	} else {
		tmp = ((0.3333333333333333 / (x * x)) + (1.0 + (-0.5 / x))) * (1.0 / (n * x));
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if (1.0 / n) <= -0.0001:
		tmp = 0.3333333333333333 / (n * (x * (x * x)))
	else:
		tmp = ((0.3333333333333333 / (x * x)) + (1.0 + (-0.5 / x))) * (1.0 / (n * x))
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -0.0001)
		tmp = Float64(0.3333333333333333 / Float64(n * Float64(x * Float64(x * x))));
	else
		tmp = Float64(Float64(Float64(0.3333333333333333 / Float64(x * x)) + Float64(1.0 + Float64(-0.5 / x))) * Float64(1.0 / Float64(n * x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if ((1.0 / n) <= -0.0001)
		tmp = 0.3333333333333333 / (n * (x * (x * x)));
	else
		tmp = ((0.3333333333333333 / (x * x)) + (1.0 + (-0.5 / x))) * (1.0 / (n * x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[N[(1.0 / n), $MachinePrecision], -0.0001], N[(0.3333333333333333 / N[(n * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(0.3333333333333333 / N[(x * x), $MachinePrecision]), $MachinePrecision] + N[(1.0 + N[(-0.5 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(n * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -1.00000000000000005e-4

   1. Initial program 99.9%

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

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{\frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}}{n} + \left(\frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}} \cdot \left(\frac{1}{2} \cdot \frac{1}{{n}^{2}} - \frac{1}{2} \cdot \frac{1}{n}\right)}{x} + \frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}} \cdot \left(\left(\frac{1}{6} \cdot \frac{1}{{n}^{3}} + \frac{1}{3} \cdot \frac{1}{n}\right) - \frac{1}{2} \cdot \frac{1}{{n}^{2}}\right)}{{x}^{2}}\right)}{x}} \]

   5. Simplified 46.7%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\frac{{x}^{\left(\frac{1}{n}\right)}}{x}, \left(\frac{0.5}{n \cdot n} + \frac{-0.5}{n}\right) + \frac{\frac{0.16666666666666666}{n \cdot \left(n \cdot n\right)} + \left(\frac{0.3333333333333333}{n} + \frac{-0.5}{n \cdot n}\right)}{x}, \frac{{x}^{\left(\frac{1}{n}\right)}}{n}\right)}{x}} \]

   6. Taylor expanded in n around inf

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

      1. /-lowering-/.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. associate-+l+ N/A

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

      5. +-lowering-+.f64 N/A

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

      6. associate-*r/ N/A

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

      7. metadata-eval N/A

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

      8. /-lowering-/.f64 N/A

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

      9. unpow2 N/A

         \[\leadsto \frac{\frac{\frac{\frac{1}{3}}{\color{blue}{x \cdot x}} + \left(1 + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{x}\right)\right)\right)}{n}}{x} \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \frac{\frac{\frac{\frac{1}{3}}{\color{blue}{x \cdot x}} + \left(1 + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{x}\right)\right)\right)}{n}}{x} \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \frac{\frac{\frac{\frac{1}{3}}{x \cdot x} + \color{blue}{\left(1 + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{x}\right)\right)\right)}}{n}}{x} \]

      12. associate-*r/ N/A

          \[\leadsto \frac{\frac{\frac{\frac{1}{3}}{x \cdot x} + \left(1 + \left(\mathsf{neg}\left(\color{blue}{\frac{\frac{1}{2} \cdot 1}{x}}\right)\right)\right)}{n}}{x} \]

      13. metadata-eval N/A

          \[\leadsto \frac{\frac{\frac{\frac{1}{3}}{x \cdot x} + \left(1 + \left(\mathsf{neg}\left(\frac{\color{blue}{\frac{1}{2}}}{x}\right)\right)\right)}{n}}{x} \]

      14. distribute-neg-frac N/A

          \[\leadsto \frac{\frac{\frac{\frac{1}{3}}{x \cdot x} + \left(1 + \color{blue}{\frac{\mathsf{neg}\left(\frac{1}{2}\right)}{x}}\right)}{n}}{x} \]

      15. metadata-eval N/A

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

      16. /-lowering-/.f64 47.6

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

   8. Simplified 47.6%

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

   9. Taylor expanded in x around 0

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

       1. /-lowering-/.f64 N/A

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

       2. *-lowering-*.f64 N/A

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

       3. cube-mult N/A

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

       4. unpow2 N/A

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

       5. *-lowering-*.f64 N/A

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

       6. unpow2 N/A

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

       7. *-lowering-*.f64 80.8

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

   11. Simplified 80.8%

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

   if -1.00000000000000005e-4 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 36.6%

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

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{\frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}}{n} + \left(\frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}} \cdot \left(\frac{1}{2} \cdot \frac{1}{{n}^{2}} - \frac{1}{2} \cdot \frac{1}{n}\right)}{x} + \frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}} \cdot \left(\left(\frac{1}{6} \cdot \frac{1}{{n}^{3}} + \frac{1}{3} \cdot \frac{1}{n}\right) - \frac{1}{2} \cdot \frac{1}{{n}^{2}}\right)}{{x}^{2}}\right)}{x}} \]

   5. Simplified 39.2%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\frac{{x}^{\left(\frac{1}{n}\right)}}{x}, \left(\frac{0.5}{n \cdot n} + \frac{-0.5}{n}\right) + \frac{\frac{0.16666666666666666}{n \cdot \left(n \cdot n\right)} + \left(\frac{0.3333333333333333}{n} + \frac{-0.5}{n \cdot n}\right)}{x}, \frac{{x}^{\left(\frac{1}{n}\right)}}{n}\right)}{x}} \]

   6. Taylor expanded in n around inf

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

      1. /-lowering-/.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. associate-+l+ N/A

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

      5. +-lowering-+.f64 N/A

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

      6. associate-*r/ N/A

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

      7. metadata-eval N/A

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

      8. /-lowering-/.f64 N/A

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

      9. unpow2 N/A

         \[\leadsto \frac{\frac{\frac{\frac{1}{3}}{\color{blue}{x \cdot x}} + \left(1 + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{x}\right)\right)\right)}{n}}{x} \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \frac{\frac{\frac{\frac{1}{3}}{\color{blue}{x \cdot x}} + \left(1 + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{x}\right)\right)\right)}{n}}{x} \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \frac{\frac{\frac{\frac{1}{3}}{x \cdot x} + \color{blue}{\left(1 + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{x}\right)\right)\right)}}{n}}{x} \]

      12. associate-*r/ N/A

          \[\leadsto \frac{\frac{\frac{\frac{1}{3}}{x \cdot x} + \left(1 + \left(\mathsf{neg}\left(\color{blue}{\frac{\frac{1}{2} \cdot 1}{x}}\right)\right)\right)}{n}}{x} \]

      13. metadata-eval N/A

          \[\leadsto \frac{\frac{\frac{\frac{1}{3}}{x \cdot x} + \left(1 + \left(\mathsf{neg}\left(\frac{\color{blue}{\frac{1}{2}}}{x}\right)\right)\right)}{n}}{x} \]

      14. distribute-neg-frac N/A

          \[\leadsto \frac{\frac{\frac{\frac{1}{3}}{x \cdot x} + \left(1 + \color{blue}{\frac{\mathsf{neg}\left(\frac{1}{2}\right)}{x}}\right)}{n}}{x} \]

      15. metadata-eval N/A

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

      16. /-lowering-/.f64 42.5

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

   8. Simplified 42.5%

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

      1. clear-num N/A

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

      2. associate-/r/ N/A

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

      3. *-rgt-identity N/A

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

      4. times-frac N/A

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

      5. remove-double-neg N/A

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

      6. metadata-eval N/A

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

      7. frac-2neg N/A

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

      8. clear-num N/A

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

      9. metadata-eval N/A

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

      10. frac-2neg N/A

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

      11. flip-+ N/A

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

      12. clear-num N/A

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

      13. clear-num N/A

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

   10. Applied egg-rr 42.1%

       \[\leadsto \color{blue}{\frac{1}{x \cdot n} \cdot \left(\frac{0.3333333333333333}{x \cdot x} + \left(1 + \frac{-0.5}{x}\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 55.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -0.0001:\\ \;\;\;\;\frac{0.3333333333333333}{n \cdot \left(x \cdot \left(x \cdot x\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{0.3333333333333333}{x \cdot x} + \left(1 + \frac{-0.5}{x}\right)\right) \cdot \frac{1}{n \cdot x}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 54.2% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -0.0001:\\ \;\;\;\;\frac{0.3333333333333333}{n \cdot \left(x \cdot \left(x \cdot x\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.3333333333333333}{x \cdot x} + \left(1 + \frac{-0.5}{x}\right)}{n \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= (/ 1.0 n) -0.0001)
   (/ 0.3333333333333333 (* n (* x (* x x))))
   (/ (+ (/ 0.3333333333333333 (* x x)) (+ 1.0 (/ -0.5 x))) (* n x))))

C

double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -0.0001) {
		tmp = 0.3333333333333333 / (n * (x * (x * x)));
	} else {
		tmp = ((0.3333333333333333 / (x * x)) + (1.0 + (-0.5 / x))) / (n * x);
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if ((1.0d0 / n) <= (-0.0001d0)) then
        tmp = 0.3333333333333333d0 / (n * (x * (x * x)))
    else
        tmp = ((0.3333333333333333d0 / (x * x)) + (1.0d0 + ((-0.5d0) / x))) / (n * x)
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -0.0001) {
		tmp = 0.3333333333333333 / (n * (x * (x * x)));
	} else {
		tmp = ((0.3333333333333333 / (x * x)) + (1.0 + (-0.5 / x))) / (n * x);
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if (1.0 / n) <= -0.0001:
		tmp = 0.3333333333333333 / (n * (x * (x * x)))
	else:
		tmp = ((0.3333333333333333 / (x * x)) + (1.0 + (-0.5 / x))) / (n * x)
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -0.0001)
		tmp = Float64(0.3333333333333333 / Float64(n * Float64(x * Float64(x * x))));
	else
		tmp = Float64(Float64(Float64(0.3333333333333333 / Float64(x * x)) + Float64(1.0 + Float64(-0.5 / x))) / Float64(n * x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if ((1.0 / n) <= -0.0001)
		tmp = 0.3333333333333333 / (n * (x * (x * x)));
	else
		tmp = ((0.3333333333333333 / (x * x)) + (1.0 + (-0.5 / x))) / (n * x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[N[(1.0 / n), $MachinePrecision], -0.0001], N[(0.3333333333333333 / N[(n * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(0.3333333333333333 / N[(x * x), $MachinePrecision]), $MachinePrecision] + N[(1.0 + N[(-0.5 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(n * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -1.00000000000000005e-4

   1. Initial program 99.9%

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

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{\frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}}{n} + \left(\frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}} \cdot \left(\frac{1}{2} \cdot \frac{1}{{n}^{2}} - \frac{1}{2} \cdot \frac{1}{n}\right)}{x} + \frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}} \cdot \left(\left(\frac{1}{6} \cdot \frac{1}{{n}^{3}} + \frac{1}{3} \cdot \frac{1}{n}\right) - \frac{1}{2} \cdot \frac{1}{{n}^{2}}\right)}{{x}^{2}}\right)}{x}} \]

   5. Simplified 46.7%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\frac{{x}^{\left(\frac{1}{n}\right)}}{x}, \left(\frac{0.5}{n \cdot n} + \frac{-0.5}{n}\right) + \frac{\frac{0.16666666666666666}{n \cdot \left(n \cdot n\right)} + \left(\frac{0.3333333333333333}{n} + \frac{-0.5}{n \cdot n}\right)}{x}, \frac{{x}^{\left(\frac{1}{n}\right)}}{n}\right)}{x}} \]

   6. Taylor expanded in n around inf

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

      1. /-lowering-/.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. associate-+l+ N/A

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

      5. +-lowering-+.f64 N/A

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

      6. associate-*r/ N/A

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

      7. metadata-eval N/A

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

      8. /-lowering-/.f64 N/A

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

      9. unpow2 N/A

         \[\leadsto \frac{\frac{\frac{\frac{1}{3}}{\color{blue}{x \cdot x}} + \left(1 + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{x}\right)\right)\right)}{n}}{x} \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \frac{\frac{\frac{\frac{1}{3}}{\color{blue}{x \cdot x}} + \left(1 + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{x}\right)\right)\right)}{n}}{x} \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \frac{\frac{\frac{\frac{1}{3}}{x \cdot x} + \color{blue}{\left(1 + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{x}\right)\right)\right)}}{n}}{x} \]

      12. associate-*r/ N/A

          \[\leadsto \frac{\frac{\frac{\frac{1}{3}}{x \cdot x} + \left(1 + \left(\mathsf{neg}\left(\color{blue}{\frac{\frac{1}{2} \cdot 1}{x}}\right)\right)\right)}{n}}{x} \]

      13. metadata-eval N/A

          \[\leadsto \frac{\frac{\frac{\frac{1}{3}}{x \cdot x} + \left(1 + \left(\mathsf{neg}\left(\frac{\color{blue}{\frac{1}{2}}}{x}\right)\right)\right)}{n}}{x} \]

      14. distribute-neg-frac N/A

          \[\leadsto \frac{\frac{\frac{\frac{1}{3}}{x \cdot x} + \left(1 + \color{blue}{\frac{\mathsf{neg}\left(\frac{1}{2}\right)}{x}}\right)}{n}}{x} \]

      15. metadata-eval N/A

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

      16. /-lowering-/.f64 47.6

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

   8. Simplified 47.6%

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

   9. Taylor expanded in x around 0

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

       1. /-lowering-/.f64 N/A

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

       2. *-lowering-*.f64 N/A

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

       3. cube-mult N/A

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

       4. unpow2 N/A

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

       5. *-lowering-*.f64 N/A

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

       6. unpow2 N/A

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

       7. *-lowering-*.f64 80.8

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

   11. Simplified 80.8%

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

   if -1.00000000000000005e-4 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 36.6%

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

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{\frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}}{n} + \left(\frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}} \cdot \left(\frac{1}{2} \cdot \frac{1}{{n}^{2}} - \frac{1}{2} \cdot \frac{1}{n}\right)}{x} + \frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}} \cdot \left(\left(\frac{1}{6} \cdot \frac{1}{{n}^{3}} + \frac{1}{3} \cdot \frac{1}{n}\right) - \frac{1}{2} \cdot \frac{1}{{n}^{2}}\right)}{{x}^{2}}\right)}{x}} \]

   5. Simplified 39.2%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\frac{{x}^{\left(\frac{1}{n}\right)}}{x}, \left(\frac{0.5}{n \cdot n} + \frac{-0.5}{n}\right) + \frac{\frac{0.16666666666666666}{n \cdot \left(n \cdot n\right)} + \left(\frac{0.3333333333333333}{n} + \frac{-0.5}{n \cdot n}\right)}{x}, \frac{{x}^{\left(\frac{1}{n}\right)}}{n}\right)}{x}} \]

   6. Taylor expanded in n around inf

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

      1. /-lowering-/.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. associate-+l+ N/A

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

      5. +-lowering-+.f64 N/A

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

      6. associate-*r/ N/A

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

      7. metadata-eval N/A

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

      8. /-lowering-/.f64 N/A

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

      9. unpow2 N/A

         \[\leadsto \frac{\frac{\frac{1}{3}}{\color{blue}{x \cdot x}} + \left(1 + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{x}\right)\right)\right)}{n \cdot x} \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \frac{\frac{\frac{1}{3}}{\color{blue}{x \cdot x}} + \left(1 + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{x}\right)\right)\right)}{n \cdot x} \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \frac{\frac{\frac{1}{3}}{x \cdot x} + \color{blue}{\left(1 + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{x}\right)\right)\right)}}{n \cdot x} \]

      12. associate-*r/ N/A

          \[\leadsto \frac{\frac{\frac{1}{3}}{x \cdot x} + \left(1 + \left(\mathsf{neg}\left(\color{blue}{\frac{\frac{1}{2} \cdot 1}{x}}\right)\right)\right)}{n \cdot x} \]

      13. metadata-eval N/A

          \[\leadsto \frac{\frac{\frac{1}{3}}{x \cdot x} + \left(1 + \left(\mathsf{neg}\left(\frac{\color{blue}{\frac{1}{2}}}{x}\right)\right)\right)}{n \cdot x} \]

      14. distribute-neg-frac N/A

          \[\leadsto \frac{\frac{\frac{1}{3}}{x \cdot x} + \left(1 + \color{blue}{\frac{\mathsf{neg}\left(\frac{1}{2}\right)}{x}}\right)}{n \cdot x} \]

      15. metadata-eval N/A

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

      16. /-lowering-/.f64 N/A

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

      17. *-commutative N/A

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

      18. *-lowering-*.f64 42.1

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

   8. Simplified 42.1%

      \[\leadsto \color{blue}{\frac{\frac{0.3333333333333333}{x \cdot x} + \left(1 + \frac{-0.5}{x}\right)}{x \cdot n}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 55.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -0.0001:\\ \;\;\;\;\frac{0.3333333333333333}{n \cdot \left(x \cdot \left(x \cdot x\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.3333333333333333}{x \cdot x} + \left(1 + \frac{-0.5}{x}\right)}{n \cdot x}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 53.6% accurate, 5.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -1:\\ \;\;\;\;\frac{0.3333333333333333}{n \cdot \left(x \cdot \left(x \cdot x\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{n}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= (/ 1.0 n) -1.0)
   (/ 0.3333333333333333 (* n (* x (* x x))))
   (/ (/ 1.0 n) x)))

C

double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -1.0) {
		tmp = 0.3333333333333333 / (n * (x * (x * x)));
	} else {
		tmp = (1.0 / n) / x;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if ((1.0d0 / n) <= (-1.0d0)) then
        tmp = 0.3333333333333333d0 / (n * (x * (x * x)))
    else
        tmp = (1.0d0 / n) / x
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -1.0) {
		tmp = 0.3333333333333333 / (n * (x * (x * x)));
	} else {
		tmp = (1.0 / n) / x;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if (1.0 / n) <= -1.0:
		tmp = 0.3333333333333333 / (n * (x * (x * x)))
	else:
		tmp = (1.0 / n) / x
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -1.0)
		tmp = Float64(0.3333333333333333 / Float64(n * Float64(x * Float64(x * x))));
	else
		tmp = Float64(Float64(1.0 / n) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if ((1.0 / n) <= -1.0)
		tmp = 0.3333333333333333 / (n * (x * (x * x)));
	else
		tmp = (1.0 / n) / x;
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{\frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}}{n} + \left(\frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}} \cdot \left(\frac{1}{2} \cdot \frac{1}{{n}^{2}} - \frac{1}{2} \cdot \frac{1}{n}\right)}{x} + \frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}} \cdot \left(\left(\frac{1}{6} \cdot \frac{1}{{n}^{3}} + \frac{1}{3} \cdot \frac{1}{n}\right) - \frac{1}{2} \cdot \frac{1}{{n}^{2}}\right)}{{x}^{2}}\right)}{x}} \]

   5. Simplified 47.7%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\frac{{x}^{\left(\frac{1}{n}\right)}}{x}, \left(\frac{0.5}{n \cdot n} + \frac{-0.5}{n}\right) + \frac{\frac{0.16666666666666666}{n \cdot \left(n \cdot n\right)} + \left(\frac{0.3333333333333333}{n} + \frac{-0.5}{n \cdot n}\right)}{x}, \frac{{x}^{\left(\frac{1}{n}\right)}}{n}\right)}{x}} \]

   6. Taylor expanded in n around inf

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

      1. /-lowering-/.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. associate-+l+ N/A

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

      5. +-lowering-+.f64 N/A

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

      6. associate-*r/ N/A

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

      7. metadata-eval N/A

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

      8. /-lowering-/.f64 N/A

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

      9. unpow2 N/A

         \[\leadsto \frac{\frac{\frac{\frac{1}{3}}{\color{blue}{x \cdot x}} + \left(1 + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{x}\right)\right)\right)}{n}}{x} \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \frac{\frac{\frac{\frac{1}{3}}{\color{blue}{x \cdot x}} + \left(1 + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{x}\right)\right)\right)}{n}}{x} \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \frac{\frac{\frac{\frac{1}{3}}{x \cdot x} + \color{blue}{\left(1 + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{x}\right)\right)\right)}}{n}}{x} \]

      12. associate-*r/ N/A

          \[\leadsto \frac{\frac{\frac{\frac{1}{3}}{x \cdot x} + \left(1 + \left(\mathsf{neg}\left(\color{blue}{\frac{\frac{1}{2} \cdot 1}{x}}\right)\right)\right)}{n}}{x} \]

      13. metadata-eval N/A

          \[\leadsto \frac{\frac{\frac{\frac{1}{3}}{x \cdot x} + \left(1 + \left(\mathsf{neg}\left(\frac{\color{blue}{\frac{1}{2}}}{x}\right)\right)\right)}{n}}{x} \]

      14. distribute-neg-frac N/A

          \[\leadsto \frac{\frac{\frac{\frac{1}{3}}{x \cdot x} + \left(1 + \color{blue}{\frac{\mathsf{neg}\left(\frac{1}{2}\right)}{x}}\right)}{n}}{x} \]

      15. metadata-eval N/A

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

      16. /-lowering-/.f64 48.6

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

   8. Simplified 48.6%

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

   9. Taylor expanded in x around 0

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

       1. /-lowering-/.f64 N/A

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

       2. *-lowering-*.f64 N/A

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

       3. cube-mult N/A

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

       4. unpow2 N/A

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

       5. *-lowering-*.f64 N/A

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

       6. unpow2 N/A

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

       7. *-lowering-*.f64 82.6

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

   11. Simplified 82.6%

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

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

   1. Initial program 37.3%

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

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{\frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}}{n} + \left(\frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}} \cdot \left(\frac{1}{2} \cdot \frac{1}{{n}^{2}} - \frac{1}{2} \cdot \frac{1}{n}\right)}{x} + \frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}} \cdot \left(\left(\frac{1}{6} \cdot \frac{1}{{n}^{3}} + \frac{1}{3} \cdot \frac{1}{n}\right) - \frac{1}{2} \cdot \frac{1}{{n}^{2}}\right)}{{x}^{2}}\right)}{x}} \]

   5. Simplified 38.8%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\frac{{x}^{\left(\frac{1}{n}\right)}}{x}, \left(\frac{0.5}{n \cdot n} + \frac{-0.5}{n}\right) + \frac{\frac{0.16666666666666666}{n \cdot \left(n \cdot n\right)} + \left(\frac{0.3333333333333333}{n} + \frac{-0.5}{n \cdot n}\right)}{x}, \frac{{x}^{\left(\frac{1}{n}\right)}}{n}\right)}{x}} \]

   6. Taylor expanded in n around inf

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

      1. /-lowering-/.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. associate-+l+ N/A

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

      5. +-lowering-+.f64 N/A

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

      6. associate-*r/ N/A

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

      7. metadata-eval N/A

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

      8. /-lowering-/.f64 N/A

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

      9. unpow2 N/A

         \[\leadsto \frac{\frac{\frac{\frac{1}{3}}{\color{blue}{x \cdot x}} + \left(1 + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{x}\right)\right)\right)}{n}}{x} \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \frac{\frac{\frac{\frac{1}{3}}{\color{blue}{x \cdot x}} + \left(1 + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{x}\right)\right)\right)}{n}}{x} \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \frac{\frac{\frac{\frac{1}{3}}{x \cdot x} + \color{blue}{\left(1 + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{x}\right)\right)\right)}}{n}}{x} \]

      12. associate-*r/ N/A

          \[\leadsto \frac{\frac{\frac{\frac{1}{3}}{x \cdot x} + \left(1 + \left(\mathsf{neg}\left(\color{blue}{\frac{\frac{1}{2} \cdot 1}{x}}\right)\right)\right)}{n}}{x} \]

      13. metadata-eval N/A

          \[\leadsto \frac{\frac{\frac{\frac{1}{3}}{x \cdot x} + \left(1 + \left(\mathsf{neg}\left(\frac{\color{blue}{\frac{1}{2}}}{x}\right)\right)\right)}{n}}{x} \]

      14. distribute-neg-frac N/A

          \[\leadsto \frac{\frac{\frac{\frac{1}{3}}{x \cdot x} + \left(1 + \color{blue}{\frac{\mathsf{neg}\left(\frac{1}{2}\right)}{x}}\right)}{n}}{x} \]

      15. metadata-eval N/A

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

      16. /-lowering-/.f64 42.0

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

   8. Simplified 42.0%

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

   9. Taylor expanded in x around inf

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

       1. /-lowering-/.f64 41.5

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

   11. Simplified 41.5%

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

Alternative 18: 43.1% accurate, 8.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 5 \cdot 10^{+65}:\\ \;\;\;\;\frac{\frac{1}{n}}{x}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n) :precision binary64 (if (<= x 5e+65) (/ (/ 1.0 n) x) 0.0))

C

double code(double x, double n) {
	double tmp;
	if (x <= 5e+65) {
		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 <= 5d+65) 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 <= 5e+65) {
		tmp = (1.0 / n) / x;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 5e+65:
		tmp = (1.0 / n) / x
	else:
		tmp = 0.0
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 5e+65)
		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 <= 5e+65)
		tmp = (1.0 / n) / x;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 5e+65], N[(N[(1.0 / n), $MachinePrecision] / x), $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if x < 4.99999999999999973e65

   1. Initial program 48.4%

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

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{\frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}}{n} + \left(\frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}} \cdot \left(\frac{1}{2} \cdot \frac{1}{{n}^{2}} - \frac{1}{2} \cdot \frac{1}{n}\right)}{x} + \frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}} \cdot \left(\left(\frac{1}{6} \cdot \frac{1}{{n}^{3}} + \frac{1}{3} \cdot \frac{1}{n}\right) - \frac{1}{2} \cdot \frac{1}{{n}^{2}}\right)}{{x}^{2}}\right)}{x}} \]

   5. Simplified 25.6%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\frac{{x}^{\left(\frac{1}{n}\right)}}{x}, \left(\frac{0.5}{n \cdot n} + \frac{-0.5}{n}\right) + \frac{\frac{0.16666666666666666}{n \cdot \left(n \cdot n\right)} + \left(\frac{0.3333333333333333}{n} + \frac{-0.5}{n \cdot n}\right)}{x}, \frac{{x}^{\left(\frac{1}{n}\right)}}{n}\right)}{x}} \]

   6. Taylor expanded in n around inf

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

      1. /-lowering-/.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. associate-+l+ N/A

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

      5. +-lowering-+.f64 N/A

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

      6. associate-*r/ N/A

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

      7. metadata-eval N/A

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

      8. /-lowering-/.f64 N/A

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

      9. unpow2 N/A

         \[\leadsto \frac{\frac{\frac{\frac{1}{3}}{\color{blue}{x \cdot x}} + \left(1 + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{x}\right)\right)\right)}{n}}{x} \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \frac{\frac{\frac{\frac{1}{3}}{\color{blue}{x \cdot x}} + \left(1 + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{x}\right)\right)\right)}{n}}{x} \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \frac{\frac{\frac{\frac{1}{3}}{x \cdot x} + \color{blue}{\left(1 + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{x}\right)\right)\right)}}{n}}{x} \]

      12. associate-*r/ N/A

          \[\leadsto \frac{\frac{\frac{\frac{1}{3}}{x \cdot x} + \left(1 + \left(\mathsf{neg}\left(\color{blue}{\frac{\frac{1}{2} \cdot 1}{x}}\right)\right)\right)}{n}}{x} \]

      13. metadata-eval N/A

          \[\leadsto \frac{\frac{\frac{\frac{1}{3}}{x \cdot x} + \left(1 + \left(\mathsf{neg}\left(\frac{\color{blue}{\frac{1}{2}}}{x}\right)\right)\right)}{n}}{x} \]

      14. distribute-neg-frac N/A

          \[\leadsto \frac{\frac{\frac{\frac{1}{3}}{x \cdot x} + \left(1 + \color{blue}{\frac{\mathsf{neg}\left(\frac{1}{2}\right)}{x}}\right)}{n}}{x} \]

      15. metadata-eval N/A

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

      16. /-lowering-/.f64 38.9

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

   8. Simplified 38.9%

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

   9. Taylor expanded in x around inf

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

       1. /-lowering-/.f64 29.0

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

   11. Simplified 29.0%

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

   if 4.99999999999999973e65 < x

   1. Initial program 79.2%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1 - e^{\frac{\log x}{n}}} \]
   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. --lowering--.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. pow-lowering-pow.f64 N/A

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

      17. /-lowering-/.f64 36.6

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

   5. Simplified 36.6%

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

   6. Taylor expanded in n around inf

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

   8. Simplified 79.2%

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

      1. metadata-eval 79.2

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

   10. Applied egg-rr 79.2%

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

Alternative 19: 43.1% accurate, 10.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2 \cdot 10^{+69}:\\ \;\;\;\;\frac{1}{n \cdot x}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n) :precision binary64 (if (<= x 2e+69) (/ 1.0 (* n x)) 0.0))

C

double code(double x, double n) {
	double tmp;
	if (x <= 2e+69) {
		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 <= 2d+69) 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 <= 2e+69) {
		tmp = 1.0 / (n * x);
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 2e+69:
		tmp = 1.0 / (n * x)
	else:
		tmp = 0.0
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 2e+69)
		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 <= 2e+69)
		tmp = 1.0 / (n * x);
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 2e+69], N[(1.0 / N[(n * x), $MachinePrecision]), $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if x < 2.0000000000000001e69

   1. Initial program 48.4%

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

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{\frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}}{n} + \left(\frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}} \cdot \left(\frac{1}{2} \cdot \frac{1}{{n}^{2}} - \frac{1}{2} \cdot \frac{1}{n}\right)}{x} + \frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}} \cdot \left(\left(\frac{1}{6} \cdot \frac{1}{{n}^{3}} + \frac{1}{3} \cdot \frac{1}{n}\right) - \frac{1}{2} \cdot \frac{1}{{n}^{2}}\right)}{{x}^{2}}\right)}{x}} \]

   5. Simplified 25.6%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\frac{{x}^{\left(\frac{1}{n}\right)}}{x}, \left(\frac{0.5}{n \cdot n} + \frac{-0.5}{n}\right) + \frac{\frac{0.16666666666666666}{n \cdot \left(n \cdot n\right)} + \left(\frac{0.3333333333333333}{n} + \frac{-0.5}{n \cdot n}\right)}{x}, \frac{{x}^{\left(\frac{1}{n}\right)}}{n}\right)}{x}} \]

   6. Taylor expanded in n around inf

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

      1. /-lowering-/.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. associate-+l+ N/A

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

      5. +-lowering-+.f64 N/A

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

      6. associate-*r/ N/A

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

      7. metadata-eval N/A

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

      8. /-lowering-/.f64 N/A

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

      9. unpow2 N/A

         \[\leadsto \frac{\frac{\frac{\frac{1}{3}}{\color{blue}{x \cdot x}} + \left(1 + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{x}\right)\right)\right)}{n}}{x} \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \frac{\frac{\frac{\frac{1}{3}}{\color{blue}{x \cdot x}} + \left(1 + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{x}\right)\right)\right)}{n}}{x} \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \frac{\frac{\frac{\frac{1}{3}}{x \cdot x} + \color{blue}{\left(1 + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{x}\right)\right)\right)}}{n}}{x} \]

      12. associate-*r/ N/A

          \[\leadsto \frac{\frac{\frac{\frac{1}{3}}{x \cdot x} + \left(1 + \left(\mathsf{neg}\left(\color{blue}{\frac{\frac{1}{2} \cdot 1}{x}}\right)\right)\right)}{n}}{x} \]

      13. metadata-eval N/A

          \[\leadsto \frac{\frac{\frac{\frac{1}{3}}{x \cdot x} + \left(1 + \left(\mathsf{neg}\left(\frac{\color{blue}{\frac{1}{2}}}{x}\right)\right)\right)}{n}}{x} \]

      14. distribute-neg-frac N/A

          \[\leadsto \frac{\frac{\frac{\frac{1}{3}}{x \cdot x} + \left(1 + \color{blue}{\frac{\mathsf{neg}\left(\frac{1}{2}\right)}{x}}\right)}{n}}{x} \]

      15. metadata-eval N/A

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

      16. /-lowering-/.f64 38.9

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

   8. Simplified 38.9%

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

   9. Taylor expanded in x around inf

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

       1. /-lowering-/.f64 N/A

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

       2. *-commutative N/A

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

       3. *-lowering-*.f64 28.9

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

   11. Simplified 28.9%

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

   if 2.0000000000000001e69 < x

   1. Initial program 79.2%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1 - e^{\frac{\log x}{n}}} \]
   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. --lowering--.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. pow-lowering-pow.f64 N/A

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

      17. /-lowering-/.f64 36.6

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

   5. Simplified 36.6%

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

   6. Taylor expanded in n around inf

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

   8. Simplified 79.2%

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

      1. metadata-eval 79.2

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

   10. Applied egg-rr 79.2%

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

4. Final simplification 46.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2 \cdot 10^{+69}:\\ \;\;\;\;\frac{1}{n \cdot x}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 31.2% 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 58.9%

   \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
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. --lowering--.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. pow-lowering-pow.f64 N/A

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

   17. /-lowering-/.f64 42.5

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

5. Simplified 42.5%

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

6. Taylor expanded in n around inf

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

8. Simplified 30.8%

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

   1. metadata-eval 30.8

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

10. Applied egg-rr 30.8%

    \[\leadsto \color{blue}{0} \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024205 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  :precision binary64
  (- (pow (+ x 1.0) (/ 1.0 n)) (pow x (/ 1.0 n))))