2nthrt (problem 3.4.6)

Percentage Accurate: 53.7% → 85.2%

Time: 28.1s

Alternatives: 20

Speedup: 11.7×

• could not determine a ground truth

  1. x = -1.0839270543163638e-218
  2. n = -1.9737286553345202e-289

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: 53.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 85.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ t_1 := \frac{\frac{t\_0}{x}}{n}\\ \mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{-5}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-133}:\\ \;\;\;\;\frac{0.5 \cdot \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}}{n} + \left(\mathsf{log1p}\left(x\right) - \log x\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-7}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;{e}^{\left(\frac{\mathsf{log1p}\left(x\right)}{n}\right)} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))) (t_1 (/ (/ t_0 x) n)))
   (if (<= (/ 1.0 n) -1e-5)
     t_1
     (if (<= (/ 1.0 n) 2e-133)
       (/
        (+
         (* 0.5 (/ (- (pow (log1p x) 2.0) (pow (log x) 2.0)) n))
         (- (log1p x) (log x)))
        n)
       (if (<= (/ 1.0 n) 2e-7) t_1 (- (pow E (/ (log1p x) n)) t_0))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double t_1 = (t_0 / x) / n;
	double tmp;
	if ((1.0 / n) <= -1e-5) {
		tmp = t_1;
	} else if ((1.0 / n) <= 2e-133) {
		tmp = ((0.5 * ((pow(log1p(x), 2.0) - pow(log(x), 2.0)) / n)) + (log1p(x) - log(x))) / n;
	} else if ((1.0 / n) <= 2e-7) {
		tmp = t_1;
	} else {
		tmp = pow(((double) M_E), (log1p(x) / n)) - t_0;
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, (1.0 / n));
	double t_1 = (t_0 / x) / n;
	double tmp;
	if ((1.0 / n) <= -1e-5) {
		tmp = t_1;
	} else if ((1.0 / n) <= 2e-133) {
		tmp = ((0.5 * ((Math.pow(Math.log1p(x), 2.0) - Math.pow(Math.log(x), 2.0)) / n)) + (Math.log1p(x) - Math.log(x))) / n;
	} else if ((1.0 / n) <= 2e-7) {
		tmp = t_1;
	} else {
		tmp = Math.pow(Math.E, (Math.log1p(x) / n)) - t_0;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	t_1 = (t_0 / x) / n
	tmp = 0
	if (1.0 / n) <= -1e-5:
		tmp = t_1
	elif (1.0 / n) <= 2e-133:
		tmp = ((0.5 * ((math.pow(math.log1p(x), 2.0) - math.pow(math.log(x), 2.0)) / n)) + (math.log1p(x) - math.log(x))) / n
	elif (1.0 / n) <= 2e-7:
		tmp = t_1
	else:
		tmp = math.pow(math.e, (math.log1p(x) / n)) - t_0
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	t_1 = Float64(Float64(t_0 / x) / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -1e-5)
		tmp = t_1;
	elseif (Float64(1.0 / n) <= 2e-133)
		tmp = Float64(Float64(Float64(0.5 * Float64(Float64((log1p(x) ^ 2.0) - (log(x) ^ 2.0)) / n)) + Float64(log1p(x) - log(x))) / n);
	elseif (Float64(1.0 / n) <= 2e-7)
		tmp = t_1;
	else
		tmp = Float64((exp(1) ^ Float64(log1p(x) / n)) - t_0);
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(t$95$0 / x), $MachinePrecision] / n), $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -1e-5], t$95$1, If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-133], N[(N[(N[(0.5 * N[(N[(N[Power[N[Log[1 + x], $MachinePrecision], 2.0], $MachinePrecision] - N[Power[N[Log[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision] + N[(N[Log[1 + x], $MachinePrecision] - N[Log[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-7], t$95$1, N[(N[Power[E, N[(N[Log[1 + x], $MachinePrecision] / n), $MachinePrecision]], $MachinePrecision] - t$95$0), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -1.00000000000000008e-5 or 2.0000000000000001e-133 < (/.f64 #s(literal 1 binary64) n) < 1.9999999999999999e-7

   1. Initial program 75.5%

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

   3. Taylor expanded in x around inf

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

      1. associate-/l/ N/A

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

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

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

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

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

      4. log-rec N/A

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

      5. mul-1-neg N/A

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

      6. associate-*r/ N/A

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

      7. associate-*r* N/A

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

      8. metadata-eval N/A

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

      9. *-commutative N/A

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

      10. associate-/l* N/A

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

      11. exp-to-pow N/A

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

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

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

      13. /-lowering-/.f64 92.2

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

   5. Simplified 92.2%

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

   if -1.00000000000000008e-5 < (/.f64 #s(literal 1 binary64) n) < 2.0000000000000001e-133

   1. Initial program 38.7%

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

   3. Taylor expanded in n around inf

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

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

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

   5. Simplified 83.7%

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

   if 1.9999999999999999e-7 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 47.1%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   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. *-lft-identity N/A

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

      2. exp-prod N/A

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

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

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

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

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

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

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

      6. accelerator-lowering-log1p.f64 100.0

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

   6. Applied egg-rr 100.0%

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

4. Final simplification 89.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{-5}:\\ \;\;\;\;\frac{\frac{{x}^{\left(\frac{1}{n}\right)}}{x}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-133}:\\ \;\;\;\;\frac{0.5 \cdot \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}}{n} + \left(\mathsf{log1p}\left(x\right) - \log x\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-7}:\\ \;\;\;\;\frac{\frac{{x}^{\left(\frac{1}{n}\right)}}{x}}{n}\\ \mathbf{else}:\\ \;\;\;\;{e}^{\left(\frac{\mathsf{log1p}\left(x\right)}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 83.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ t_1 := \frac{\frac{t\_0}{x}}{n}\\ \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{-145}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-133}:\\ \;\;\;\;\frac{-1}{n} \cdot \log \left(\frac{x}{1 + x}\right)\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-7}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;{e}^{\left(\frac{\mathsf{log1p}\left(x\right)}{n}\right)} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))) (t_1 (/ (/ t_0 x) n)))
   (if (<= (/ 1.0 n) -5e-145)
     t_1
     (if (<= (/ 1.0 n) 2e-133)
       (* (/ -1.0 n) (log (/ x (+ 1.0 x))))
       (if (<= (/ 1.0 n) 2e-7) t_1 (- (pow E (/ (log1p x) n)) t_0))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double t_1 = (t_0 / x) / n;
	double tmp;
	if ((1.0 / n) <= -5e-145) {
		tmp = t_1;
	} else if ((1.0 / n) <= 2e-133) {
		tmp = (-1.0 / n) * log((x / (1.0 + x)));
	} else if ((1.0 / n) <= 2e-7) {
		tmp = t_1;
	} else {
		tmp = pow(((double) M_E), (log1p(x) / n)) - t_0;
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, (1.0 / n));
	double t_1 = (t_0 / x) / n;
	double tmp;
	if ((1.0 / n) <= -5e-145) {
		tmp = t_1;
	} else if ((1.0 / n) <= 2e-133) {
		tmp = (-1.0 / n) * Math.log((x / (1.0 + x)));
	} else if ((1.0 / n) <= 2e-7) {
		tmp = t_1;
	} else {
		tmp = Math.pow(Math.E, (Math.log1p(x) / n)) - t_0;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	t_1 = (t_0 / x) / n
	tmp = 0
	if (1.0 / n) <= -5e-145:
		tmp = t_1
	elif (1.0 / n) <= 2e-133:
		tmp = (-1.0 / n) * math.log((x / (1.0 + x)))
	elif (1.0 / n) <= 2e-7:
		tmp = t_1
	else:
		tmp = math.pow(math.e, (math.log1p(x) / n)) - t_0
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	t_1 = Float64(Float64(t_0 / x) / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -5e-145)
		tmp = t_1;
	elseif (Float64(1.0 / n) <= 2e-133)
		tmp = Float64(Float64(-1.0 / n) * log(Float64(x / Float64(1.0 + x))));
	elseif (Float64(1.0 / n) <= 2e-7)
		tmp = t_1;
	else
		tmp = Float64((exp(1) ^ Float64(log1p(x) / n)) - t_0);
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(t$95$0 / x), $MachinePrecision] / n), $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -5e-145], t$95$1, If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-133], N[(N[(-1.0 / n), $MachinePrecision] * N[Log[N[(x / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-7], t$95$1, N[(N[Power[E, N[(N[Log[1 + x], $MachinePrecision] / n), $MachinePrecision]], $MachinePrecision] - t$95$0), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -4.9999999999999998e-145 or 2.0000000000000001e-133 < (/.f64 #s(literal 1 binary64) n) < 1.9999999999999999e-7

   1. Initial program 63.3%

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

   3. Taylor expanded in x around inf

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

      1. associate-/l/ N/A

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

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

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

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

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

      4. log-rec N/A

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

      5. mul-1-neg N/A

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

      6. associate-*r/ N/A

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

      7. associate-*r* N/A

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

      8. metadata-eval N/A

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

      9. *-commutative N/A

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

      10. associate-/l* N/A

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

      11. exp-to-pow N/A

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

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

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

      13. /-lowering-/.f64 84.6

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

   5. Simplified 84.6%

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

   if -4.9999999999999998e-145 < (/.f64 #s(literal 1 binary64) n) < 2.0000000000000001e-133

   1. Initial program 44.7%

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

   3. Taylor expanded in n around inf

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

      1. /-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 91.1

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

   5. Simplified 91.1%

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

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

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

      2. diff-log N/A

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

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

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

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

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

      5. +-lowering-+.f64 91.1

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

   7. Applied egg-rr 91.1%

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

      1. frac-2neg N/A

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

      2. log-rec N/A

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

      3. *-lft-identity N/A

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

      4. associate-*l/ N/A

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

      5. distribute-frac-neg2 N/A

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

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

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

      7. neg-mul-1 N/A

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

      8. div-inv N/A

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

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

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

      10. clear-num N/A

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

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

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

      12. /-lowering-/.f64 N/A

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

      13. +-lowering-+.f64 91.2

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

   9. Applied egg-rr 91.2%

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

   if 1.9999999999999999e-7 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 47.1%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   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. *-lft-identity N/A

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

      2. exp-prod N/A

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

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

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

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

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

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

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

      6. accelerator-lowering-log1p.f64 100.0

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

   6. Applied egg-rr 100.0%

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

4. Final simplification 89.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{-145}:\\ \;\;\;\;\frac{\frac{{x}^{\left(\frac{1}{n}\right)}}{x}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-133}:\\ \;\;\;\;\frac{-1}{n} \cdot \log \left(\frac{x}{1 + x}\right)\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-7}:\\ \;\;\;\;\frac{\frac{{x}^{\left(\frac{1}{n}\right)}}{x}}{n}\\ \mathbf{else}:\\ \;\;\;\;{e}^{\left(\frac{\mathsf{log1p}\left(x\right)}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 83.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ t_1 := \frac{\frac{t\_0}{x}}{n}\\ \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{-145}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-133}:\\ \;\;\;\;\frac{-1}{n} \cdot \log \left(\frac{x}{1 + x}\right)\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-7}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{x}{n}} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))) (t_1 (/ (/ t_0 x) n)))
   (if (<= (/ 1.0 n) -5e-145)
     t_1
     (if (<= (/ 1.0 n) 2e-133)
       (* (/ -1.0 n) (log (/ x (+ 1.0 x))))
       (if (<= (/ 1.0 n) 2e-7) 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 / x) / n;
	double tmp;
	if ((1.0 / n) <= -5e-145) {
		tmp = t_1;
	} else if ((1.0 / n) <= 2e-133) {
		tmp = (-1.0 / n) * log((x / (1.0 + x)));
	} else if ((1.0 / n) <= 2e-7) {
		tmp = t_1;
	} else {
		tmp = exp((x / n)) - t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = x ** (1.0d0 / n)
    t_1 = (t_0 / x) / n
    if ((1.0d0 / n) <= (-5d-145)) then
        tmp = t_1
    else if ((1.0d0 / n) <= 2d-133) then
        tmp = ((-1.0d0) / n) * log((x / (1.0d0 + x)))
    else if ((1.0d0 / n) <= 2d-7) then
        tmp = t_1
    else
        tmp = exp((x / n)) - t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, (1.0 / n));
	double t_1 = (t_0 / x) / n;
	double tmp;
	if ((1.0 / n) <= -5e-145) {
		tmp = t_1;
	} else if ((1.0 / n) <= 2e-133) {
		tmp = (-1.0 / n) * Math.log((x / (1.0 + x)));
	} else if ((1.0 / n) <= 2e-7) {
		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 / x) / n
	tmp = 0
	if (1.0 / n) <= -5e-145:
		tmp = t_1
	elif (1.0 / n) <= 2e-133:
		tmp = (-1.0 / n) * math.log((x / (1.0 + x)))
	elif (1.0 / n) <= 2e-7:
		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(Float64(t_0 / x) / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -5e-145)
		tmp = t_1;
	elseif (Float64(1.0 / n) <= 2e-133)
		tmp = Float64(Float64(-1.0 / n) * log(Float64(x / Float64(1.0 + x))));
	elseif (Float64(1.0 / n) <= 2e-7)
		tmp = t_1;
	else
		tmp = Float64(exp(Float64(x / n)) - t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = x ^ (1.0 / n);
	t_1 = (t_0 / x) / n;
	tmp = 0.0;
	if ((1.0 / n) <= -5e-145)
		tmp = t_1;
	elseif ((1.0 / n) <= 2e-133)
		tmp = (-1.0 / n) * log((x / (1.0 + x)));
	elseif ((1.0 / n) <= 2e-7)
		tmp = t_1;
	else
		tmp = exp((x / n)) - t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(t$95$0 / x), $MachinePrecision] / n), $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -5e-145], t$95$1, If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-133], N[(N[(-1.0 / n), $MachinePrecision] * N[Log[N[(x / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-7], 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) < -4.9999999999999998e-145 or 2.0000000000000001e-133 < (/.f64 #s(literal 1 binary64) n) < 1.9999999999999999e-7

   1. Initial program 63.3%

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

   3. Taylor expanded in x around inf

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

      1. associate-/l/ N/A

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

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

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

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

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

      4. log-rec N/A

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

      5. mul-1-neg N/A

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

      6. associate-*r/ N/A

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

      7. associate-*r* N/A

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

      8. metadata-eval N/A

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

      9. *-commutative N/A

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

      10. associate-/l* N/A

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

      11. exp-to-pow N/A

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

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

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

      13. /-lowering-/.f64 84.6

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

   5. Simplified 84.6%

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

   if -4.9999999999999998e-145 < (/.f64 #s(literal 1 binary64) n) < 2.0000000000000001e-133

   1. Initial program 44.7%

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

   3. Taylor expanded in n around inf

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

      1. /-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 91.1

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

   5. Simplified 91.1%

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

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

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

      2. diff-log N/A

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

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

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

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

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

      5. +-lowering-+.f64 91.1

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

   7. Applied egg-rr 91.1%

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

      1. frac-2neg N/A

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

      2. log-rec N/A

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

      3. *-lft-identity N/A

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

      4. associate-*l/ N/A

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

      5. distribute-frac-neg2 N/A

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

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

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

      7. neg-mul-1 N/A

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

      8. div-inv N/A

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

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

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

      10. clear-num N/A

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

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

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

      12. /-lowering-/.f64 N/A

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

      13. +-lowering-+.f64 91.2

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

   9. Applied egg-rr 91.2%

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

   if 1.9999999999999999e-7 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 47.1%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   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: 79.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ t_1 := \frac{\frac{t\_0}{x}}{n}\\ \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{-145}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-133}:\\ \;\;\;\;\frac{-1}{n} \cdot \log \left(\frac{x}{1 + x}\right)\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-7}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\left(1 + x \cdot \left(\frac{1}{n} + \frac{\frac{x \cdot 0.5}{n}}{n}\right)\right) - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))) (t_1 (/ (/ t_0 x) n)))
   (if (<= (/ 1.0 n) -5e-145)
     t_1
     (if (<= (/ 1.0 n) 2e-133)
       (* (/ -1.0 n) (log (/ x (+ 1.0 x))))
       (if (<= (/ 1.0 n) 2e-7)
         t_1
         (- (+ 1.0 (* x (+ (/ 1.0 n) (/ (/ (* x 0.5) n) n)))) t_0))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double t_1 = (t_0 / x) / n;
	double tmp;
	if ((1.0 / n) <= -5e-145) {
		tmp = t_1;
	} else if ((1.0 / n) <= 2e-133) {
		tmp = (-1.0 / n) * log((x / (1.0 + x)));
	} else if ((1.0 / n) <= 2e-7) {
		tmp = t_1;
	} else {
		tmp = (1.0 + (x * ((1.0 / n) + (((x * 0.5) / n) / n)))) - t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = x ** (1.0d0 / n)
    t_1 = (t_0 / x) / n
    if ((1.0d0 / n) <= (-5d-145)) then
        tmp = t_1
    else if ((1.0d0 / n) <= 2d-133) then
        tmp = ((-1.0d0) / n) * log((x / (1.0d0 + x)))
    else if ((1.0d0 / n) <= 2d-7) then
        tmp = t_1
    else
        tmp = (1.0d0 + (x * ((1.0d0 / n) + (((x * 0.5d0) / n) / n)))) - t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, (1.0 / n));
	double t_1 = (t_0 / x) / n;
	double tmp;
	if ((1.0 / n) <= -5e-145) {
		tmp = t_1;
	} else if ((1.0 / n) <= 2e-133) {
		tmp = (-1.0 / n) * Math.log((x / (1.0 + x)));
	} else if ((1.0 / n) <= 2e-7) {
		tmp = t_1;
	} else {
		tmp = (1.0 + (x * ((1.0 / n) + (((x * 0.5) / n) / n)))) - t_0;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	t_1 = (t_0 / x) / n
	tmp = 0
	if (1.0 / n) <= -5e-145:
		tmp = t_1
	elif (1.0 / n) <= 2e-133:
		tmp = (-1.0 / n) * math.log((x / (1.0 + x)))
	elif (1.0 / n) <= 2e-7:
		tmp = t_1
	else:
		tmp = (1.0 + (x * ((1.0 / n) + (((x * 0.5) / n) / n)))) - t_0
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	t_1 = Float64(Float64(t_0 / x) / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -5e-145)
		tmp = t_1;
	elseif (Float64(1.0 / n) <= 2e-133)
		tmp = Float64(Float64(-1.0 / n) * log(Float64(x / Float64(1.0 + x))));
	elseif (Float64(1.0 / n) <= 2e-7)
		tmp = t_1;
	else
		tmp = Float64(Float64(1.0 + Float64(x * Float64(Float64(1.0 / n) + Float64(Float64(Float64(x * 0.5) / n) / n)))) - t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = x ^ (1.0 / n);
	t_1 = (t_0 / x) / n;
	tmp = 0.0;
	if ((1.0 / n) <= -5e-145)
		tmp = t_1;
	elseif ((1.0 / n) <= 2e-133)
		tmp = (-1.0 / n) * log((x / (1.0 + x)));
	elseif ((1.0 / n) <= 2e-7)
		tmp = t_1;
	else
		tmp = (1.0 + (x * ((1.0 / n) + (((x * 0.5) / n) / n)))) - t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(t$95$0 / x), $MachinePrecision] / n), $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -5e-145], t$95$1, If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-133], N[(N[(-1.0 / n), $MachinePrecision] * N[Log[N[(x / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-7], t$95$1, N[(N[(1.0 + N[(x * N[(N[(1.0 / n), $MachinePrecision] + N[(N[(N[(x * 0.5), $MachinePrecision] / n), $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -4.9999999999999998e-145 or 2.0000000000000001e-133 < (/.f64 #s(literal 1 binary64) n) < 1.9999999999999999e-7

   1. Initial program 63.3%

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

   3. Taylor expanded in x around inf

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

      1. associate-/l/ N/A

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

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

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

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

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

      4. log-rec N/A

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

      5. mul-1-neg N/A

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

      6. associate-*r/ N/A

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

      7. associate-*r* N/A

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

      8. metadata-eval N/A

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

      9. *-commutative N/A

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

      10. associate-/l* N/A

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

      11. exp-to-pow N/A

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

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

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

      13. /-lowering-/.f64 84.6

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

   5. Simplified 84.6%

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

   if -4.9999999999999998e-145 < (/.f64 #s(literal 1 binary64) n) < 2.0000000000000001e-133

   1. Initial program 44.7%

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

   3. Taylor expanded in n around inf

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

      1. /-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 91.1

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

   5. Simplified 91.1%

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

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

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

      2. diff-log N/A

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

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

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

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

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

      5. +-lowering-+.f64 91.1

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

   7. Applied egg-rr 91.1%

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

      1. frac-2neg N/A

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

      2. log-rec N/A

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

      3. *-lft-identity N/A

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

      4. associate-*l/ N/A

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

      5. distribute-frac-neg2 N/A

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

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

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

      7. neg-mul-1 N/A

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

      8. div-inv N/A

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

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

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

      10. clear-num N/A

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

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

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

      12. /-lowering-/.f64 N/A

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

      13. +-lowering-+.f64 91.2

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

   9. Applied egg-rr 91.2%

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

   if 1.9999999999999999e-7 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 47.1%

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

   3. Taylor expanded in x around 0

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

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

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

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

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

      3. +-commutative N/A

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

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

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

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

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

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

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

      7. sub-neg N/A

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

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

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

      9. unpow2 N/A

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

      10. associate-/r* N/A

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

      11. associate-*r/ N/A

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

      12. /-lowering-/.f64 N/A

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

      13. associate-*r/ N/A

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

      14. metadata-eval N/A

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

      15. /-lowering-/.f64 N/A

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

      16. associate-*r/ N/A

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

      17. metadata-eval N/A

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

      18. distribute-neg-frac N/A

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

      19. metadata-eval N/A

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

      20. /-lowering-/.f64 68.6

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

   5. Simplified 68.6%

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

   6. Taylor expanded in n around 0

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

      1. associate-*r/ N/A

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

      2. unpow2 N/A

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

      3. associate-/r* N/A

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

      4. associate-*r/ N/A

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

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

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

      6. associate-*r/ N/A

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

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

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

      8. *-commutative N/A

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

      9. *-lowering-*.f64 73.3

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

   8. Simplified 73.3%

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

Alternative 5: 79.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ t_1 := \frac{\frac{t\_0}{x}}{n}\\ \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{-145}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-133}:\\ \;\;\;\;\frac{-1}{n} \cdot \log \left(\frac{x}{1 + x}\right)\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-7}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\frac{1}{n} \leq 8 \cdot 10^{+142}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{0.3333333333333333}{x \cdot \left(x \cdot \left(n \cdot x\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))) (t_1 (/ (/ t_0 x) n)))
   (if (<= (/ 1.0 n) -5e-145)
     t_1
     (if (<= (/ 1.0 n) 2e-133)
       (* (/ -1.0 n) (log (/ x (+ 1.0 x))))
       (if (<= (/ 1.0 n) 2e-7)
         t_1
         (if (<= (/ 1.0 n) 8e+142)
           (- (+ 1.0 (/ x n)) t_0)
           (/ 0.3333333333333333 (* x (* x (* n x))))))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double t_1 = (t_0 / x) / n;
	double tmp;
	if ((1.0 / n) <= -5e-145) {
		tmp = t_1;
	} else if ((1.0 / n) <= 2e-133) {
		tmp = (-1.0 / n) * log((x / (1.0 + x)));
	} else if ((1.0 / n) <= 2e-7) {
		tmp = t_1;
	} else if ((1.0 / n) <= 8e+142) {
		tmp = (1.0 + (x / n)) - t_0;
	} else {
		tmp = 0.3333333333333333 / (x * (x * (n * x)));
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = x ** (1.0d0 / n)
    t_1 = (t_0 / x) / n
    if ((1.0d0 / n) <= (-5d-145)) then
        tmp = t_1
    else if ((1.0d0 / n) <= 2d-133) then
        tmp = ((-1.0d0) / n) * log((x / (1.0d0 + x)))
    else if ((1.0d0 / n) <= 2d-7) then
        tmp = t_1
    else if ((1.0d0 / n) <= 8d+142) then
        tmp = (1.0d0 + (x / n)) - t_0
    else
        tmp = 0.3333333333333333d0 / (x * (x * (n * x)))
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, (1.0 / n));
	double t_1 = (t_0 / x) / n;
	double tmp;
	if ((1.0 / n) <= -5e-145) {
		tmp = t_1;
	} else if ((1.0 / n) <= 2e-133) {
		tmp = (-1.0 / n) * Math.log((x / (1.0 + x)));
	} else if ((1.0 / n) <= 2e-7) {
		tmp = t_1;
	} else if ((1.0 / n) <= 8e+142) {
		tmp = (1.0 + (x / n)) - t_0;
	} else {
		tmp = 0.3333333333333333 / (x * (x * (n * x)));
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	t_1 = (t_0 / x) / n
	tmp = 0
	if (1.0 / n) <= -5e-145:
		tmp = t_1
	elif (1.0 / n) <= 2e-133:
		tmp = (-1.0 / n) * math.log((x / (1.0 + x)))
	elif (1.0 / n) <= 2e-7:
		tmp = t_1
	elif (1.0 / n) <= 8e+142:
		tmp = (1.0 + (x / n)) - t_0
	else:
		tmp = 0.3333333333333333 / (x * (x * (n * x)))
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	t_1 = Float64(Float64(t_0 / x) / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -5e-145)
		tmp = t_1;
	elseif (Float64(1.0 / n) <= 2e-133)
		tmp = Float64(Float64(-1.0 / n) * log(Float64(x / Float64(1.0 + x))));
	elseif (Float64(1.0 / n) <= 2e-7)
		tmp = t_1;
	elseif (Float64(1.0 / n) <= 8e+142)
		tmp = Float64(Float64(1.0 + Float64(x / n)) - t_0);
	else
		tmp = Float64(0.3333333333333333 / Float64(x * Float64(x * Float64(n * x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = x ^ (1.0 / n);
	t_1 = (t_0 / x) / n;
	tmp = 0.0;
	if ((1.0 / n) <= -5e-145)
		tmp = t_1;
	elseif ((1.0 / n) <= 2e-133)
		tmp = (-1.0 / n) * log((x / (1.0 + x)));
	elseif ((1.0 / n) <= 2e-7)
		tmp = t_1;
	elseif ((1.0 / n) <= 8e+142)
		tmp = (1.0 + (x / n)) - t_0;
	else
		tmp = 0.3333333333333333 / (x * (x * (n * x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(t$95$0 / x), $MachinePrecision] / n), $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -5e-145], t$95$1, If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-133], N[(N[(-1.0 / n), $MachinePrecision] * N[Log[N[(x / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-7], t$95$1, If[LessEqual[N[(1.0 / n), $MachinePrecision], 8e+142], N[(N[(1.0 + N[(x / n), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], N[(0.3333333333333333 / N[(x * N[(x * N[(n * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -4.9999999999999998e-145 or 2.0000000000000001e-133 < (/.f64 #s(literal 1 binary64) n) < 1.9999999999999999e-7

   1. Initial program 63.3%

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

   3. Taylor expanded in x around inf

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

      1. associate-/l/ N/A

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

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

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

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

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

      4. log-rec N/A

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

      5. mul-1-neg N/A

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

      6. associate-*r/ N/A

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

      7. associate-*r* N/A

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

      8. metadata-eval N/A

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

      9. *-commutative N/A

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

      10. associate-/l* N/A

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

      11. exp-to-pow N/A

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

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

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

      13. /-lowering-/.f64 84.6

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

   5. Simplified 84.6%

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

   if -4.9999999999999998e-145 < (/.f64 #s(literal 1 binary64) n) < 2.0000000000000001e-133

   1. Initial program 44.7%

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

   3. Taylor expanded in n around inf

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

      1. /-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 91.1

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

   5. Simplified 91.1%

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

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

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

      2. diff-log N/A

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

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

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

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

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

      5. +-lowering-+.f64 91.1

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

   7. Applied egg-rr 91.1%

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

      1. frac-2neg N/A

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

      2. log-rec N/A

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

      3. *-lft-identity N/A

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

      4. associate-*l/ N/A

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

      5. distribute-frac-neg2 N/A

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

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

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

      7. neg-mul-1 N/A

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

      8. div-inv N/A

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

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

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

      10. clear-num N/A

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

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

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

      12. /-lowering-/.f64 N/A

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

      13. +-lowering-+.f64 91.2

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

   9. Applied egg-rr 91.2%

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

   if 1.9999999999999999e-7 < (/.f64 #s(literal 1 binary64) n) < 8.00000000000000041e142

   1. Initial program 80.6%

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

   3. Taylor expanded in 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 74.3

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

   5. Simplified 74.3%

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

   if 8.00000000000000041e142 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 29.1%

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

   3. Taylor expanded in n around inf

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
   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 5.9

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

   5. Simplified 5.9%

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

   6. Taylor expanded in x around inf

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

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

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

   8. Simplified 66.0%

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

   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. *-commutative N/A

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

       3. cube-mult N/A

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

       4. unpow2 N/A

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

       5. associate-*l* N/A

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

       6. *-commutative N/A

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

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

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

       8. *-commutative N/A

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

       9. unpow2 N/A

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

       10. associate-*l* N/A

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

       11. *-commutative N/A

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

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

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

       13. *-commutative N/A

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

       14. *-lowering-*.f64 66.0

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

   11. Simplified 66.0%

       \[\leadsto \color{blue}{\frac{0.3333333333333333}{x \cdot \left(x \cdot \left(x \cdot n\right)\right)}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 84.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{-145}:\\ \;\;\;\;\frac{\frac{{x}^{\left(\frac{1}{n}\right)}}{x}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-133}:\\ \;\;\;\;\frac{-1}{n} \cdot \log \left(\frac{x}{1 + x}\right)\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-7}:\\ \;\;\;\;\frac{\frac{{x}^{\left(\frac{1}{n}\right)}}{x}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 8 \cdot 10^{+142}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.3333333333333333}{x \cdot \left(x \cdot \left(n \cdot x\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 79.0% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ t_1 := \frac{\frac{t\_0}{x}}{n}\\ \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{-145}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-133}:\\ \;\;\;\;\frac{-1}{n} \cdot \log \left(\frac{x}{1 + x}\right)\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-7}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\frac{1}{n} \leq 8 \cdot 10^{+142}:\\ \;\;\;\;1 - t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{0.3333333333333333}{x \cdot \left(x \cdot \left(n \cdot x\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))) (t_1 (/ (/ t_0 x) n)))
   (if (<= (/ 1.0 n) -5e-145)
     t_1
     (if (<= (/ 1.0 n) 2e-133)
       (* (/ -1.0 n) (log (/ x (+ 1.0 x))))
       (if (<= (/ 1.0 n) 2e-7)
         t_1
         (if (<= (/ 1.0 n) 8e+142)
           (- 1.0 t_0)
           (/ 0.3333333333333333 (* x (* x (* n x))))))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double t_1 = (t_0 / x) / n;
	double tmp;
	if ((1.0 / n) <= -5e-145) {
		tmp = t_1;
	} else if ((1.0 / n) <= 2e-133) {
		tmp = (-1.0 / n) * log((x / (1.0 + x)));
	} else if ((1.0 / n) <= 2e-7) {
		tmp = t_1;
	} else if ((1.0 / n) <= 8e+142) {
		tmp = 1.0 - t_0;
	} else {
		tmp = 0.3333333333333333 / (x * (x * (n * x)));
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = x ** (1.0d0 / n)
    t_1 = (t_0 / x) / n
    if ((1.0d0 / n) <= (-5d-145)) then
        tmp = t_1
    else if ((1.0d0 / n) <= 2d-133) then
        tmp = ((-1.0d0) / n) * log((x / (1.0d0 + x)))
    else if ((1.0d0 / n) <= 2d-7) then
        tmp = t_1
    else if ((1.0d0 / n) <= 8d+142) then
        tmp = 1.0d0 - t_0
    else
        tmp = 0.3333333333333333d0 / (x * (x * (n * x)))
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, (1.0 / n));
	double t_1 = (t_0 / x) / n;
	double tmp;
	if ((1.0 / n) <= -5e-145) {
		tmp = t_1;
	} else if ((1.0 / n) <= 2e-133) {
		tmp = (-1.0 / n) * Math.log((x / (1.0 + x)));
	} else if ((1.0 / n) <= 2e-7) {
		tmp = t_1;
	} else if ((1.0 / n) <= 8e+142) {
		tmp = 1.0 - t_0;
	} else {
		tmp = 0.3333333333333333 / (x * (x * (n * x)));
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	t_1 = (t_0 / x) / n
	tmp = 0
	if (1.0 / n) <= -5e-145:
		tmp = t_1
	elif (1.0 / n) <= 2e-133:
		tmp = (-1.0 / n) * math.log((x / (1.0 + x)))
	elif (1.0 / n) <= 2e-7:
		tmp = t_1
	elif (1.0 / n) <= 8e+142:
		tmp = 1.0 - t_0
	else:
		tmp = 0.3333333333333333 / (x * (x * (n * x)))
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	t_1 = Float64(Float64(t_0 / x) / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -5e-145)
		tmp = t_1;
	elseif (Float64(1.0 / n) <= 2e-133)
		tmp = Float64(Float64(-1.0 / n) * log(Float64(x / Float64(1.0 + x))));
	elseif (Float64(1.0 / n) <= 2e-7)
		tmp = t_1;
	elseif (Float64(1.0 / n) <= 8e+142)
		tmp = Float64(1.0 - t_0);
	else
		tmp = Float64(0.3333333333333333 / Float64(x * Float64(x * Float64(n * x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = x ^ (1.0 / n);
	t_1 = (t_0 / x) / n;
	tmp = 0.0;
	if ((1.0 / n) <= -5e-145)
		tmp = t_1;
	elseif ((1.0 / n) <= 2e-133)
		tmp = (-1.0 / n) * log((x / (1.0 + x)));
	elseif ((1.0 / n) <= 2e-7)
		tmp = t_1;
	elseif ((1.0 / n) <= 8e+142)
		tmp = 1.0 - t_0;
	else
		tmp = 0.3333333333333333 / (x * (x * (n * x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(t$95$0 / x), $MachinePrecision] / n), $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -5e-145], t$95$1, If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-133], N[(N[(-1.0 / n), $MachinePrecision] * N[Log[N[(x / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-7], t$95$1, If[LessEqual[N[(1.0 / n), $MachinePrecision], 8e+142], N[(1.0 - t$95$0), $MachinePrecision], N[(0.3333333333333333 / N[(x * N[(x * N[(n * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -4.9999999999999998e-145 or 2.0000000000000001e-133 < (/.f64 #s(literal 1 binary64) n) < 1.9999999999999999e-7

   1. Initial program 63.3%

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

   3. Taylor expanded in x around inf

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

      1. associate-/l/ N/A

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

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

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

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

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

      4. log-rec N/A

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

      5. mul-1-neg N/A

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

      6. associate-*r/ N/A

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

      7. associate-*r* N/A

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

      8. metadata-eval N/A

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

      9. *-commutative N/A

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

      10. associate-/l* N/A

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

      11. exp-to-pow N/A

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

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

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

      13. /-lowering-/.f64 84.6

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

   5. Simplified 84.6%

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

   if -4.9999999999999998e-145 < (/.f64 #s(literal 1 binary64) n) < 2.0000000000000001e-133

   1. Initial program 44.7%

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

   3. Taylor expanded in n around inf

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

      1. /-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 91.1

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

   5. Simplified 91.1%

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

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

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

      2. diff-log N/A

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

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

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

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

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

      5. +-lowering-+.f64 91.1

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

   7. Applied egg-rr 91.1%

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

      1. frac-2neg N/A

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

      2. log-rec N/A

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

      3. *-lft-identity N/A

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

      4. associate-*l/ N/A

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

      5. distribute-frac-neg2 N/A

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

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

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

      7. neg-mul-1 N/A

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

      8. div-inv N/A

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

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

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

      10. clear-num N/A

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

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

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

      12. /-lowering-/.f64 N/A

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

      13. +-lowering-+.f64 91.2

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

   9. Applied egg-rr 91.2%

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

   if 1.9999999999999999e-7 < (/.f64 #s(literal 1 binary64) n) < 8.00000000000000041e142

   1. Initial program 80.6%

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

   3. Taylor expanded in 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 74.2

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

   5. Simplified 74.2%

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

   if 8.00000000000000041e142 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 29.1%

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

   3. Taylor expanded in n around inf

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
   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 5.9

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

   5. Simplified 5.9%

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

   6. Taylor expanded in x around inf

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

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

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

   8. Simplified 66.0%

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

   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. *-commutative N/A

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

       3. cube-mult N/A

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

       4. unpow2 N/A

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

       5. associate-*l* N/A

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

       6. *-commutative N/A

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

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

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

       8. *-commutative N/A

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

       9. unpow2 N/A

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

       10. associate-*l* N/A

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

       11. *-commutative N/A

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

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

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

       13. *-commutative N/A

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

       14. *-lowering-*.f64 66.0

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

   11. Simplified 66.0%

       \[\leadsto \color{blue}{\frac{0.3333333333333333}{x \cdot \left(x \cdot \left(x \cdot n\right)\right)}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 84.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{-145}:\\ \;\;\;\;\frac{\frac{{x}^{\left(\frac{1}{n}\right)}}{x}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-133}:\\ \;\;\;\;\frac{-1}{n} \cdot \log \left(\frac{x}{1 + x}\right)\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-7}:\\ \;\;\;\;\frac{\frac{{x}^{\left(\frac{1}{n}\right)}}{x}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 8 \cdot 10^{+142}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.3333333333333333}{x \cdot \left(x \cdot \left(n \cdot x\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 79.1% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ t_1 := \frac{\frac{t\_0}{x}}{n}\\ \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{-145}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-133}:\\ \;\;\;\;\frac{\log \left(\frac{x}{1 + x}\right)}{0 - n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-7}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\frac{1}{n} \leq 8 \cdot 10^{+142}:\\ \;\;\;\;1 - t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{0.3333333333333333}{x \cdot \left(x \cdot \left(n \cdot x\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))) (t_1 (/ (/ t_0 x) n)))
   (if (<= (/ 1.0 n) -5e-145)
     t_1
     (if (<= (/ 1.0 n) 2e-133)
       (/ (log (/ x (+ 1.0 x))) (- 0.0 n))
       (if (<= (/ 1.0 n) 2e-7)
         t_1
         (if (<= (/ 1.0 n) 8e+142)
           (- 1.0 t_0)
           (/ 0.3333333333333333 (* x (* x (* n x))))))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double t_1 = (t_0 / x) / n;
	double tmp;
	if ((1.0 / n) <= -5e-145) {
		tmp = t_1;
	} else if ((1.0 / n) <= 2e-133) {
		tmp = log((x / (1.0 + x))) / (0.0 - n);
	} else if ((1.0 / n) <= 2e-7) {
		tmp = t_1;
	} else if ((1.0 / n) <= 8e+142) {
		tmp = 1.0 - t_0;
	} else {
		tmp = 0.3333333333333333 / (x * (x * (n * x)));
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = x ** (1.0d0 / n)
    t_1 = (t_0 / x) / n
    if ((1.0d0 / n) <= (-5d-145)) then
        tmp = t_1
    else if ((1.0d0 / n) <= 2d-133) then
        tmp = log((x / (1.0d0 + x))) / (0.0d0 - n)
    else if ((1.0d0 / n) <= 2d-7) then
        tmp = t_1
    else if ((1.0d0 / n) <= 8d+142) then
        tmp = 1.0d0 - t_0
    else
        tmp = 0.3333333333333333d0 / (x * (x * (n * x)))
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, (1.0 / n));
	double t_1 = (t_0 / x) / n;
	double tmp;
	if ((1.0 / n) <= -5e-145) {
		tmp = t_1;
	} else if ((1.0 / n) <= 2e-133) {
		tmp = Math.log((x / (1.0 + x))) / (0.0 - n);
	} else if ((1.0 / n) <= 2e-7) {
		tmp = t_1;
	} else if ((1.0 / n) <= 8e+142) {
		tmp = 1.0 - t_0;
	} else {
		tmp = 0.3333333333333333 / (x * (x * (n * x)));
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	t_1 = (t_0 / x) / n
	tmp = 0
	if (1.0 / n) <= -5e-145:
		tmp = t_1
	elif (1.0 / n) <= 2e-133:
		tmp = math.log((x / (1.0 + x))) / (0.0 - n)
	elif (1.0 / n) <= 2e-7:
		tmp = t_1
	elif (1.0 / n) <= 8e+142:
		tmp = 1.0 - t_0
	else:
		tmp = 0.3333333333333333 / (x * (x * (n * x)))
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	t_1 = Float64(Float64(t_0 / x) / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -5e-145)
		tmp = t_1;
	elseif (Float64(1.0 / n) <= 2e-133)
		tmp = Float64(log(Float64(x / Float64(1.0 + x))) / Float64(0.0 - n));
	elseif (Float64(1.0 / n) <= 2e-7)
		tmp = t_1;
	elseif (Float64(1.0 / n) <= 8e+142)
		tmp = Float64(1.0 - t_0);
	else
		tmp = Float64(0.3333333333333333 / Float64(x * Float64(x * Float64(n * x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = x ^ (1.0 / n);
	t_1 = (t_0 / x) / n;
	tmp = 0.0;
	if ((1.0 / n) <= -5e-145)
		tmp = t_1;
	elseif ((1.0 / n) <= 2e-133)
		tmp = log((x / (1.0 + x))) / (0.0 - n);
	elseif ((1.0 / n) <= 2e-7)
		tmp = t_1;
	elseif ((1.0 / n) <= 8e+142)
		tmp = 1.0 - t_0;
	else
		tmp = 0.3333333333333333 / (x * (x * (n * x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(t$95$0 / x), $MachinePrecision] / n), $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -5e-145], t$95$1, If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-133], N[(N[Log[N[(x / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(0.0 - n), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-7], t$95$1, If[LessEqual[N[(1.0 / n), $MachinePrecision], 8e+142], N[(1.0 - t$95$0), $MachinePrecision], N[(0.3333333333333333 / N[(x * N[(x * N[(n * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -4.9999999999999998e-145 or 2.0000000000000001e-133 < (/.f64 #s(literal 1 binary64) n) < 1.9999999999999999e-7

   1. Initial program 63.3%

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

   3. Taylor expanded in x around inf

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

      1. associate-/l/ N/A

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

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

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

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

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

      4. log-rec N/A

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

      5. mul-1-neg N/A

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

      6. associate-*r/ N/A

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

      7. associate-*r* N/A

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

      8. metadata-eval N/A

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

      9. *-commutative N/A

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

      10. associate-/l* N/A

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

      11. exp-to-pow N/A

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

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

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

      13. /-lowering-/.f64 84.6

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

   5. Simplified 84.6%

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

   if -4.9999999999999998e-145 < (/.f64 #s(literal 1 binary64) n) < 2.0000000000000001e-133

   1. Initial program 44.7%

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

   3. Taylor expanded in n around inf

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

      1. /-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 91.1

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

   5. Simplified 91.1%

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

      1. diff-log N/A

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

      2. clear-num N/A

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

      3. log-rec N/A

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

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

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

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

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

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

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

      7. +-lowering-+.f64 91.2

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

   7. Applied egg-rr 91.2%

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

   if 1.9999999999999999e-7 < (/.f64 #s(literal 1 binary64) n) < 8.00000000000000041e142

   1. Initial program 80.6%

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

   3. Taylor expanded in 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 74.2

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

   5. Simplified 74.2%

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

   if 8.00000000000000041e142 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 29.1%

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

   3. Taylor expanded in n around inf

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
   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 5.9

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

   5. Simplified 5.9%

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

   6. Taylor expanded in x around inf

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

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

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

   8. Simplified 66.0%

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

   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. *-commutative N/A

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

       3. cube-mult N/A

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

       4. unpow2 N/A

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

       5. associate-*l* N/A

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

       6. *-commutative N/A

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

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

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

       8. *-commutative N/A

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

       9. unpow2 N/A

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

       10. associate-*l* N/A

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

       11. *-commutative N/A

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

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

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

       13. *-commutative N/A

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

       14. *-lowering-*.f64 66.0

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

   11. Simplified 66.0%

       \[\leadsto \color{blue}{\frac{0.3333333333333333}{x \cdot \left(x \cdot \left(x \cdot n\right)\right)}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 84.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{-145}:\\ \;\;\;\;\frac{\frac{{x}^{\left(\frac{1}{n}\right)}}{x}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-133}:\\ \;\;\;\;\frac{\log \left(\frac{x}{1 + x}\right)}{0 - n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-7}:\\ \;\;\;\;\frac{\frac{{x}^{\left(\frac{1}{n}\right)}}{x}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 8 \cdot 10^{+142}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.3333333333333333}{x \cdot \left(x \cdot \left(n \cdot x\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 72.5% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{+66}:\\ \;\;\;\;\frac{\frac{0.3333333333333333}{x \cdot \left(x \cdot x\right)}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-133}:\\ \;\;\;\;\frac{\log \left(\frac{x}{1 + x}\right)}{0 - n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-17}:\\ \;\;\;\;\frac{1}{x \cdot \left(n + \frac{n \cdot 0.5}{x}\right)}\\ \mathbf{elif}\;\frac{1}{n} \leq 8 \cdot 10^{+142}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.3333333333333333}{x \cdot \left(x \cdot \left(n \cdot x\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= (/ 1.0 n) -1e+66)
   (/ (/ 0.3333333333333333 (* x (* x x))) n)
   (if (<= (/ 1.0 n) 2e-133)
     (/ (log (/ x (+ 1.0 x))) (- 0.0 n))
     (if (<= (/ 1.0 n) 2e-17)
       (/ 1.0 (* x (+ n (/ (* n 0.5) x))))
       (if (<= (/ 1.0 n) 8e+142)
         (- 1.0 (pow x (/ 1.0 n)))
         (/ 0.3333333333333333 (* x (* x (* n x)))))))))

C

double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -1e+66) {
		tmp = (0.3333333333333333 / (x * (x * x))) / n;
	} else if ((1.0 / n) <= 2e-133) {
		tmp = log((x / (1.0 + x))) / (0.0 - n);
	} else if ((1.0 / n) <= 2e-17) {
		tmp = 1.0 / (x * (n + ((n * 0.5) / x)));
	} else if ((1.0 / n) <= 8e+142) {
		tmp = 1.0 - pow(x, (1.0 / n));
	} else {
		tmp = 0.3333333333333333 / (x * (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) <= (-1d+66)) then
        tmp = (0.3333333333333333d0 / (x * (x * x))) / n
    else if ((1.0d0 / n) <= 2d-133) then
        tmp = log((x / (1.0d0 + x))) / (0.0d0 - n)
    else if ((1.0d0 / n) <= 2d-17) then
        tmp = 1.0d0 / (x * (n + ((n * 0.5d0) / x)))
    else if ((1.0d0 / n) <= 8d+142) then
        tmp = 1.0d0 - (x ** (1.0d0 / n))
    else
        tmp = 0.3333333333333333d0 / (x * (x * (n * x)))
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -1e+66) {
		tmp = (0.3333333333333333 / (x * (x * x))) / n;
	} else if ((1.0 / n) <= 2e-133) {
		tmp = Math.log((x / (1.0 + x))) / (0.0 - n);
	} else if ((1.0 / n) <= 2e-17) {
		tmp = 1.0 / (x * (n + ((n * 0.5) / x)));
	} else if ((1.0 / n) <= 8e+142) {
		tmp = 1.0 - Math.pow(x, (1.0 / n));
	} else {
		tmp = 0.3333333333333333 / (x * (x * (n * x)));
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if (1.0 / n) <= -1e+66:
		tmp = (0.3333333333333333 / (x * (x * x))) / n
	elif (1.0 / n) <= 2e-133:
		tmp = math.log((x / (1.0 + x))) / (0.0 - n)
	elif (1.0 / n) <= 2e-17:
		tmp = 1.0 / (x * (n + ((n * 0.5) / x)))
	elif (1.0 / n) <= 8e+142:
		tmp = 1.0 - math.pow(x, (1.0 / n))
	else:
		tmp = 0.3333333333333333 / (x * (x * (n * x)))
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -1e+66)
		tmp = Float64(Float64(0.3333333333333333 / Float64(x * Float64(x * x))) / n);
	elseif (Float64(1.0 / n) <= 2e-133)
		tmp = Float64(log(Float64(x / Float64(1.0 + x))) / Float64(0.0 - n));
	elseif (Float64(1.0 / n) <= 2e-17)
		tmp = Float64(1.0 / Float64(x * Float64(n + Float64(Float64(n * 0.5) / x))));
	elseif (Float64(1.0 / n) <= 8e+142)
		tmp = Float64(1.0 - (x ^ Float64(1.0 / n)));
	else
		tmp = Float64(0.3333333333333333 / Float64(x * Float64(x * Float64(n * x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if ((1.0 / n) <= -1e+66)
		tmp = (0.3333333333333333 / (x * (x * x))) / n;
	elseif ((1.0 / n) <= 2e-133)
		tmp = log((x / (1.0 + x))) / (0.0 - n);
	elseif ((1.0 / n) <= 2e-17)
		tmp = 1.0 / (x * (n + ((n * 0.5) / x)));
	elseif ((1.0 / n) <= 8e+142)
		tmp = 1.0 - (x ^ (1.0 / n));
	else
		tmp = 0.3333333333333333 / (x * (x * (n * x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[N[(1.0 / n), $MachinePrecision], -1e+66], N[(N[(0.3333333333333333 / N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-133], N[(N[Log[N[(x / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(0.0 - n), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-17], N[(1.0 / N[(x * N[(n + N[(N[(n * 0.5), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 8e+142], N[(1.0 - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(0.3333333333333333 / N[(x * N[(x * N[(n * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -9.99999999999999945e65

   1. Initial program 100.0%

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

   3. Taylor expanded in n around inf

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
   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 50.1

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

   5. Simplified 50.1%

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

   6. Taylor expanded in x around inf

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

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

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

   8. Simplified 48.9%

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

   9. Taylor expanded in x around 0

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

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

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

       2. cube-mult N/A

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

       3. unpow2 N/A

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

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

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

       5. unpow2 N/A

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

       6. *-lowering-*.f64 87.9

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

   11. Simplified 87.9%

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

   if -9.99999999999999945e65 < (/.f64 #s(literal 1 binary64) n) < 2.0000000000000001e-133

   1. Initial program 47.5%

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

   3. Taylor expanded in n around inf

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
   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 78.0

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

   5. Simplified 78.0%

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

      1. diff-log N/A

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

      2. clear-num N/A

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

      3. log-rec N/A

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

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

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

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

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

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

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

      7. +-lowering-+.f64 78.0

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

   7. Applied egg-rr 78.0%

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

   if 2.0000000000000001e-133 < (/.f64 #s(literal 1 binary64) n) < 2.00000000000000014e-17

   1. Initial program 14.7%

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

   3. Taylor expanded in n around inf

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

      1. /-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 43.9

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

   5. Simplified 43.9%

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

      1. clear-num N/A

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

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

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

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

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

      4. diff-log N/A

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

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

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

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

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

      7. +-lowering-+.f64 43.9

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

   7. Applied egg-rr 43.9%

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

   8. Taylor expanded in x around inf

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

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

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

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

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

      3. associate-*r/ N/A

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

      4. metadata-eval N/A

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

      5. distribute-lft-neg-in N/A

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

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

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

      7. *-commutative N/A

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

      8. distribute-rgt-neg-in N/A

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

      9. metadata-eval N/A

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

      10. *-lowering-*.f64 71.1

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

   10. Simplified 71.1%

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

   if 2.00000000000000014e-17 < (/.f64 #s(literal 1 binary64) n) < 8.00000000000000041e142

   1. Initial program 75.2%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   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 68.9

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

   5. Simplified 68.9%

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

   if 8.00000000000000041e142 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 29.1%

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

   3. Taylor expanded in n around inf

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
   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 5.9

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

   5. Simplified 5.9%

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

   6. Taylor expanded in x around inf

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

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

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

   8. Simplified 66.0%

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

   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. *-commutative N/A

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

       3. cube-mult N/A

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

       4. unpow2 N/A

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

       5. associate-*l* N/A

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

       6. *-commutative N/A

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

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

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

       8. *-commutative N/A

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

       9. unpow2 N/A

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

       10. associate-*l* N/A

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

       11. *-commutative N/A

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

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

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

       13. *-commutative N/A

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

       14. *-lowering-*.f64 66.0

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

   11. Simplified 66.0%

       \[\leadsto \color{blue}{\frac{0.3333333333333333}{x \cdot \left(x \cdot \left(x \cdot n\right)\right)}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 77.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{+66}:\\ \;\;\;\;\frac{\frac{0.3333333333333333}{x \cdot \left(x \cdot x\right)}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-133}:\\ \;\;\;\;\frac{\log \left(\frac{x}{1 + x}\right)}{0 - n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-17}:\\ \;\;\;\;\frac{1}{x \cdot \left(n + \frac{n \cdot 0.5}{x}\right)}\\ \mathbf{elif}\;\frac{1}{n} \leq 8 \cdot 10^{+142}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.3333333333333333}{x \cdot \left(x \cdot \left(n \cdot x\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 72.4% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{+66}:\\ \;\;\;\;\frac{\frac{0.3333333333333333}{x \cdot \left(x \cdot x\right)}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-133}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-17}:\\ \;\;\;\;\frac{1}{x \cdot \left(n + \frac{n \cdot 0.5}{x}\right)}\\ \mathbf{elif}\;\frac{1}{n} \leq 8 \cdot 10^{+142}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.3333333333333333}{x \cdot \left(x \cdot \left(n \cdot x\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= (/ 1.0 n) -1e+66)
   (/ (/ 0.3333333333333333 (* x (* x x))) n)
   (if (<= (/ 1.0 n) 2e-133)
     (/ (log (/ (+ 1.0 x) x)) n)
     (if (<= (/ 1.0 n) 2e-17)
       (/ 1.0 (* x (+ n (/ (* n 0.5) x))))
       (if (<= (/ 1.0 n) 8e+142)
         (- 1.0 (pow x (/ 1.0 n)))
         (/ 0.3333333333333333 (* x (* x (* n x)))))))))

C

double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -1e+66) {
		tmp = (0.3333333333333333 / (x * (x * x))) / n;
	} else if ((1.0 / n) <= 2e-133) {
		tmp = log(((1.0 + x) / x)) / n;
	} else if ((1.0 / n) <= 2e-17) {
		tmp = 1.0 / (x * (n + ((n * 0.5) / x)));
	} else if ((1.0 / n) <= 8e+142) {
		tmp = 1.0 - pow(x, (1.0 / n));
	} else {
		tmp = 0.3333333333333333 / (x * (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) <= (-1d+66)) then
        tmp = (0.3333333333333333d0 / (x * (x * x))) / n
    else if ((1.0d0 / n) <= 2d-133) then
        tmp = log(((1.0d0 + x) / x)) / n
    else if ((1.0d0 / n) <= 2d-17) then
        tmp = 1.0d0 / (x * (n + ((n * 0.5d0) / x)))
    else if ((1.0d0 / n) <= 8d+142) then
        tmp = 1.0d0 - (x ** (1.0d0 / n))
    else
        tmp = 0.3333333333333333d0 / (x * (x * (n * x)))
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -1e+66) {
		tmp = (0.3333333333333333 / (x * (x * x))) / n;
	} else if ((1.0 / n) <= 2e-133) {
		tmp = Math.log(((1.0 + x) / x)) / n;
	} else if ((1.0 / n) <= 2e-17) {
		tmp = 1.0 / (x * (n + ((n * 0.5) / x)));
	} else if ((1.0 / n) <= 8e+142) {
		tmp = 1.0 - Math.pow(x, (1.0 / n));
	} else {
		tmp = 0.3333333333333333 / (x * (x * (n * x)));
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if (1.0 / n) <= -1e+66:
		tmp = (0.3333333333333333 / (x * (x * x))) / n
	elif (1.0 / n) <= 2e-133:
		tmp = math.log(((1.0 + x) / x)) / n
	elif (1.0 / n) <= 2e-17:
		tmp = 1.0 / (x * (n + ((n * 0.5) / x)))
	elif (1.0 / n) <= 8e+142:
		tmp = 1.0 - math.pow(x, (1.0 / n))
	else:
		tmp = 0.3333333333333333 / (x * (x * (n * x)))
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -1e+66)
		tmp = Float64(Float64(0.3333333333333333 / Float64(x * Float64(x * x))) / n);
	elseif (Float64(1.0 / n) <= 2e-133)
		tmp = Float64(log(Float64(Float64(1.0 + x) / x)) / n);
	elseif (Float64(1.0 / n) <= 2e-17)
		tmp = Float64(1.0 / Float64(x * Float64(n + Float64(Float64(n * 0.5) / x))));
	elseif (Float64(1.0 / n) <= 8e+142)
		tmp = Float64(1.0 - (x ^ Float64(1.0 / n)));
	else
		tmp = Float64(0.3333333333333333 / Float64(x * Float64(x * Float64(n * x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if ((1.0 / n) <= -1e+66)
		tmp = (0.3333333333333333 / (x * (x * x))) / n;
	elseif ((1.0 / n) <= 2e-133)
		tmp = log(((1.0 + x) / x)) / n;
	elseif ((1.0 / n) <= 2e-17)
		tmp = 1.0 / (x * (n + ((n * 0.5) / x)));
	elseif ((1.0 / n) <= 8e+142)
		tmp = 1.0 - (x ^ (1.0 / n));
	else
		tmp = 0.3333333333333333 / (x * (x * (n * x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[N[(1.0 / n), $MachinePrecision], -1e+66], N[(N[(0.3333333333333333 / N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-133], N[(N[Log[N[(N[(1.0 + x), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-17], N[(1.0 / N[(x * N[(n + N[(N[(n * 0.5), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 8e+142], N[(1.0 - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(0.3333333333333333 / N[(x * N[(x * N[(n * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -9.99999999999999945e65

   1. Initial program 100.0%

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

   3. Taylor expanded in n around inf

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
   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 50.1

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

   5. Simplified 50.1%

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

   6. Taylor expanded in x around inf

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

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

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

   8. Simplified 48.9%

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

   9. Taylor expanded in x around 0

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

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

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

       2. cube-mult N/A

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

       3. unpow2 N/A

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

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

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

       5. unpow2 N/A

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

       6. *-lowering-*.f64 87.9

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

   11. Simplified 87.9%

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

   if -9.99999999999999945e65 < (/.f64 #s(literal 1 binary64) n) < 2.0000000000000001e-133

   1. Initial program 47.5%

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

   3. Taylor expanded in n around inf

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
   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 78.0

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

   5. Simplified 78.0%

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

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

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

      2. diff-log N/A

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

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

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

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

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

      5. +-lowering-+.f64 78.0

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

   7. Applied egg-rr 78.0%

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

   if 2.0000000000000001e-133 < (/.f64 #s(literal 1 binary64) n) < 2.00000000000000014e-17

   1. Initial program 14.7%

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

   3. Taylor expanded in n around inf

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

      1. /-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 43.9

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

   5. Simplified 43.9%

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

      1. clear-num N/A

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

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

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

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

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

      4. diff-log N/A

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

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

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

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

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

      7. +-lowering-+.f64 43.9

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

   7. Applied egg-rr 43.9%

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

   8. Taylor expanded in x around inf

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

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

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

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

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

      3. associate-*r/ N/A

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

      4. metadata-eval N/A

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

      5. distribute-lft-neg-in N/A

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

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

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

      7. *-commutative N/A

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

      8. distribute-rgt-neg-in N/A

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

      9. metadata-eval N/A

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

      10. *-lowering-*.f64 71.1

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

   10. Simplified 71.1%

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

   if 2.00000000000000014e-17 < (/.f64 #s(literal 1 binary64) n) < 8.00000000000000041e142

   1. Initial program 75.2%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   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 68.9

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

   5. Simplified 68.9%

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

   if 8.00000000000000041e142 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 29.1%

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

   3. Taylor expanded in n around inf

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
   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 5.9

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

   5. Simplified 5.9%

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

   6. Taylor expanded in x around inf

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

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

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

   8. Simplified 66.0%

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

   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. *-commutative N/A

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

       3. cube-mult N/A

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

       4. unpow2 N/A

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

       5. associate-*l* N/A

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

       6. *-commutative N/A

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

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

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

       8. *-commutative N/A

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

       9. unpow2 N/A

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

       10. associate-*l* N/A

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

       11. *-commutative N/A

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

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

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

       13. *-commutative N/A

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

       14. *-lowering-*.f64 66.0

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

   11. Simplified 66.0%

       \[\leadsto \color{blue}{\frac{0.3333333333333333}{x \cdot \left(x \cdot \left(x \cdot n\right)\right)}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 77.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{+66}:\\ \;\;\;\;\frac{\frac{0.3333333333333333}{x \cdot \left(x \cdot x\right)}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-133}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-17}:\\ \;\;\;\;\frac{1}{x \cdot \left(n + \frac{n \cdot 0.5}{x}\right)}\\ \mathbf{elif}\;\frac{1}{n} \leq 8 \cdot 10^{+142}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.3333333333333333}{x \cdot \left(x \cdot \left(n \cdot x\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 58.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -0.5 + \frac{0.3333333333333333}{x}\\ t_1 := \frac{t\_0}{x}\\ \mathbf{if}\;x \leq 3.5 \cdot 10^{-88}:\\ \;\;\;\;\frac{1}{\frac{n}{x - \log x}}\\ \mathbf{elif}\;x \leq 7.8 \cdot 10^{+119}:\\ \;\;\;\;\frac{\frac{1}{n}}{\frac{x}{1 - \frac{t\_1}{\frac{x}{t\_0}}}} \cdot \frac{1}{1 - t\_1}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (+ -0.5 (/ 0.3333333333333333 x))) (t_1 (/ t_0 x)))
   (if (<= x 3.5e-88)
     (/ 1.0 (/ n (- x (log x))))
     (if (<= x 7.8e+119)
       (* (/ (/ 1.0 n) (/ x (- 1.0 (/ t_1 (/ x t_0))))) (/ 1.0 (- 1.0 t_1)))
       0.0))))

C

double code(double x, double n) {
	double t_0 = -0.5 + (0.3333333333333333 / x);
	double t_1 = t_0 / x;
	double tmp;
	if (x <= 3.5e-88) {
		tmp = 1.0 / (n / (x - log(x)));
	} else if (x <= 7.8e+119) {
		tmp = ((1.0 / n) / (x / (1.0 - (t_1 / (x / t_0))))) * (1.0 / (1.0 - t_1));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (-0.5d0) + (0.3333333333333333d0 / x)
    t_1 = t_0 / x
    if (x <= 3.5d-88) then
        tmp = 1.0d0 / (n / (x - log(x)))
    else if (x <= 7.8d+119) then
        tmp = ((1.0d0 / n) / (x / (1.0d0 - (t_1 / (x / t_0))))) * (1.0d0 / (1.0d0 - t_1))
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = -0.5 + (0.3333333333333333 / x);
	double t_1 = t_0 / x;
	double tmp;
	if (x <= 3.5e-88) {
		tmp = 1.0 / (n / (x - Math.log(x)));
	} else if (x <= 7.8e+119) {
		tmp = ((1.0 / n) / (x / (1.0 - (t_1 / (x / t_0))))) * (1.0 / (1.0 - t_1));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = -0.5 + (0.3333333333333333 / x)
	t_1 = t_0 / x
	tmp = 0
	if x <= 3.5e-88:
		tmp = 1.0 / (n / (x - math.log(x)))
	elif x <= 7.8e+119:
		tmp = ((1.0 / n) / (x / (1.0 - (t_1 / (x / t_0))))) * (1.0 / (1.0 - t_1))
	else:
		tmp = 0.0
	return tmp

Julia

function code(x, n)
	t_0 = Float64(-0.5 + Float64(0.3333333333333333 / x))
	t_1 = Float64(t_0 / x)
	tmp = 0.0
	if (x <= 3.5e-88)
		tmp = Float64(1.0 / Float64(n / Float64(x - log(x))));
	elseif (x <= 7.8e+119)
		tmp = Float64(Float64(Float64(1.0 / n) / Float64(x / Float64(1.0 - Float64(t_1 / Float64(x / t_0))))) * Float64(1.0 / Float64(1.0 - t_1)));
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = -0.5 + (0.3333333333333333 / x);
	t_1 = t_0 / x;
	tmp = 0.0;
	if (x <= 3.5e-88)
		tmp = 1.0 / (n / (x - log(x)));
	elseif (x <= 7.8e+119)
		tmp = ((1.0 / n) / (x / (1.0 - (t_1 / (x / t_0))))) * (1.0 / (1.0 - t_1));
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(-0.5 + N[(0.3333333333333333 / x), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 / x), $MachinePrecision]}, If[LessEqual[x, 3.5e-88], N[(1.0 / N[(n / N[(x - N[Log[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 7.8e+119], N[(N[(N[(1.0 / n), $MachinePrecision] / N[(x / N[(1.0 - N[(t$95$1 / N[(x / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(1.0 - t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0]]]]

Derivation

1. Split input into 3 regimes

2. if x < 3.5000000000000001e-88

   1. Initial program 42.2%

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

   3. Taylor expanded in n around inf

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
   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 51.5

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

   5. Simplified 51.5%

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

      1. clear-num N/A

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

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

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

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

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

      4. diff-log N/A

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

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

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

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

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

      7. +-lowering-+.f64 51.5

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

   7. Applied egg-rr 51.5%

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

   8. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

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

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

      4. log-lowering-log.f64 51.5

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

   10. Simplified 51.5%

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

   if 3.5000000000000001e-88 < x < 7.7999999999999997e119

   1. Initial program 43.8%

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

   3. Taylor expanded in n around inf

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

      1. /-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 42.8

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

   5. Simplified 42.8%

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

   6. Taylor expanded in x around inf

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

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

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

   8. Simplified 54.6%

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

      1. div-inv N/A

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

      2. clear-num N/A

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

      3. associate-*l/ N/A

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

      4. *-commutative N/A

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

      5. flip-+ N/A

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

      6. associate-/r/ N/A

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

      7. times-frac N/A

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

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

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

   10. Applied egg-rr 60.2%

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

   if 7.7999999999999997e119 < x

   1. Initial program 81.0%

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

   3. Taylor expanded in x around 0

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

      1. remove-double-neg N/A

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

      2. mul-1-neg N/A

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

      3. distribute-neg-frac N/A

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

      4. mul-1-neg N/A

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

      5. log-rec N/A

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

      6. mul-1-neg N/A

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

      7. --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 51.0

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

   5. Simplified 51.0%

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

   6. Taylor expanded in n around inf

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

   8. Simplified 81.0%

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

      1. metadata-eval 81.0

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

   10. Applied egg-rr 81.0%

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

Alternative 11: 58.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -0.5 + \frac{0.3333333333333333}{x}\\ t_1 := \frac{t\_0}{x}\\ \mathbf{if}\;x \leq 3.95 \cdot 10^{-88}:\\ \;\;\;\;\frac{\log x}{0 - n}\\ \mathbf{elif}\;x \leq 1.85 \cdot 10^{+119}:\\ \;\;\;\;\frac{\frac{1}{n}}{\frac{x}{1 - \frac{t\_1}{\frac{x}{t\_0}}}} \cdot \frac{1}{1 - t\_1}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (+ -0.5 (/ 0.3333333333333333 x))) (t_1 (/ t_0 x)))
   (if (<= x 3.95e-88)
     (/ (log x) (- 0.0 n))
     (if (<= x 1.85e+119)
       (* (/ (/ 1.0 n) (/ x (- 1.0 (/ t_1 (/ x t_0))))) (/ 1.0 (- 1.0 t_1)))
       0.0))))

C

double code(double x, double n) {
	double t_0 = -0.5 + (0.3333333333333333 / x);
	double t_1 = t_0 / x;
	double tmp;
	if (x <= 3.95e-88) {
		tmp = log(x) / (0.0 - n);
	} else if (x <= 1.85e+119) {
		tmp = ((1.0 / n) / (x / (1.0 - (t_1 / (x / t_0))))) * (1.0 / (1.0 - t_1));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (-0.5d0) + (0.3333333333333333d0 / x)
    t_1 = t_0 / x
    if (x <= 3.95d-88) then
        tmp = log(x) / (0.0d0 - n)
    else if (x <= 1.85d+119) then
        tmp = ((1.0d0 / n) / (x / (1.0d0 - (t_1 / (x / t_0))))) * (1.0d0 / (1.0d0 - t_1))
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = -0.5 + (0.3333333333333333 / x);
	double t_1 = t_0 / x;
	double tmp;
	if (x <= 3.95e-88) {
		tmp = Math.log(x) / (0.0 - n);
	} else if (x <= 1.85e+119) {
		tmp = ((1.0 / n) / (x / (1.0 - (t_1 / (x / t_0))))) * (1.0 / (1.0 - t_1));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = -0.5 + (0.3333333333333333 / x)
	t_1 = t_0 / x
	tmp = 0
	if x <= 3.95e-88:
		tmp = math.log(x) / (0.0 - n)
	elif x <= 1.85e+119:
		tmp = ((1.0 / n) / (x / (1.0 - (t_1 / (x / t_0))))) * (1.0 / (1.0 - t_1))
	else:
		tmp = 0.0
	return tmp

Julia

function code(x, n)
	t_0 = Float64(-0.5 + Float64(0.3333333333333333 / x))
	t_1 = Float64(t_0 / x)
	tmp = 0.0
	if (x <= 3.95e-88)
		tmp = Float64(log(x) / Float64(0.0 - n));
	elseif (x <= 1.85e+119)
		tmp = Float64(Float64(Float64(1.0 / n) / Float64(x / Float64(1.0 - Float64(t_1 / Float64(x / t_0))))) * Float64(1.0 / Float64(1.0 - t_1)));
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = -0.5 + (0.3333333333333333 / x);
	t_1 = t_0 / x;
	tmp = 0.0;
	if (x <= 3.95e-88)
		tmp = log(x) / (0.0 - n);
	elseif (x <= 1.85e+119)
		tmp = ((1.0 / n) / (x / (1.0 - (t_1 / (x / t_0))))) * (1.0 / (1.0 - t_1));
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(-0.5 + N[(0.3333333333333333 / x), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 / x), $MachinePrecision]}, If[LessEqual[x, 3.95e-88], N[(N[Log[x], $MachinePrecision] / N[(0.0 - n), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.85e+119], N[(N[(N[(1.0 / n), $MachinePrecision] / N[(x / N[(1.0 - N[(t$95$1 / N[(x / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(1.0 - t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0]]]]

Derivation

1. Split input into 3 regimes

2. if x < 3.94999999999999983e-88

   1. Initial program 42.2%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   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 42.2

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

   5. Simplified 42.2%

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

   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. distribute-neg-frac2 N/A

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

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

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

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

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

      5. neg-lowering-neg.f64 51.5

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

   8. Simplified 51.5%

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

   if 3.94999999999999983e-88 < x < 1.85e119

   1. Initial program 43.8%

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

   3. Taylor expanded in n around inf

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

      1. /-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 42.8

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

   5. Simplified 42.8%

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

   6. Taylor expanded in x around inf

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

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

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

   8. Simplified 54.6%

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

      1. div-inv N/A

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

      2. clear-num N/A

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

      3. associate-*l/ N/A

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

      4. *-commutative N/A

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

      5. flip-+ N/A

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

      6. associate-/r/ N/A

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

      7. times-frac N/A

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

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

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

   10. Applied egg-rr 60.2%

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

   if 1.85e119 < x

   1. Initial program 81.0%

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

   3. Taylor expanded in x around 0

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

      1. remove-double-neg N/A

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

      2. mul-1-neg N/A

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

      3. distribute-neg-frac N/A

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

      4. mul-1-neg N/A

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

      5. log-rec N/A

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

      6. mul-1-neg N/A

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

      7. --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 51.0

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

   5. Simplified 51.0%

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

   6. Taylor expanded in n around inf

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

   8. Simplified 81.0%

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

      1. metadata-eval 81.0

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

   10. Applied egg-rr 81.0%

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

4. Final simplification 62.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 3.95 \cdot 10^{-88}:\\ \;\;\;\;\frac{\log x}{0 - n}\\ \mathbf{elif}\;x \leq 1.85 \cdot 10^{+119}:\\ \;\;\;\;\frac{\frac{1}{n}}{\frac{x}{1 - \frac{\frac{-0.5 + \frac{0.3333333333333333}{x}}{x}}{\frac{x}{-0.5 + \frac{0.3333333333333333}{x}}}}} \cdot \frac{1}{1 - \frac{-0.5 + \frac{0.3333333333333333}{x}}{x}}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 56.5% accurate, 10.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -7.6:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \mathbf{elif}\;n \leq 2 \cdot 10^{-88}:\\ \;\;\;\;\frac{\frac{0.3333333333333333}{x \cdot \left(x \cdot x\right)}}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x \cdot \left(n + \frac{n \cdot 0.5}{x}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= n -7.6)
   (/ (/ 1.0 x) n)
   (if (<= n 2e-88)
     (/ (/ 0.3333333333333333 (* x (* x x))) n)
     (/ 1.0 (* x (+ n (/ (* n 0.5) x)))))))

C

double code(double x, double n) {
	double tmp;
	if (n <= -7.6) {
		tmp = (1.0 / x) / n;
	} else if (n <= 2e-88) {
		tmp = (0.3333333333333333 / (x * (x * x))) / n;
	} else {
		tmp = 1.0 / (x * (n + ((n * 0.5) / x)));
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if (n <= (-7.6d0)) then
        tmp = (1.0d0 / x) / n
    else if (n <= 2d-88) then
        tmp = (0.3333333333333333d0 / (x * (x * x))) / n
    else
        tmp = 1.0d0 / (x * (n + ((n * 0.5d0) / x)))
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (n <= -7.6) {
		tmp = (1.0 / x) / n;
	} else if (n <= 2e-88) {
		tmp = (0.3333333333333333 / (x * (x * x))) / n;
	} else {
		tmp = 1.0 / (x * (n + ((n * 0.5) / x)));
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if n <= -7.6:
		tmp = (1.0 / x) / n
	elif n <= 2e-88:
		tmp = (0.3333333333333333 / (x * (x * x))) / n
	else:
		tmp = 1.0 / (x * (n + ((n * 0.5) / x)))
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (n <= -7.6)
		tmp = Float64(Float64(1.0 / x) / n);
	elseif (n <= 2e-88)
		tmp = Float64(Float64(0.3333333333333333 / Float64(x * Float64(x * x))) / n);
	else
		tmp = Float64(1.0 / Float64(x * Float64(n + Float64(Float64(n * 0.5) / x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (n <= -7.6)
		tmp = (1.0 / x) / n;
	elseif (n <= 2e-88)
		tmp = (0.3333333333333333 / (x * (x * x))) / n;
	else
		tmp = 1.0 / (x * (n + ((n * 0.5) / x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[n, -7.6], N[(N[(1.0 / x), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[n, 2e-88], N[(N[(0.3333333333333333 / N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], N[(1.0 / N[(x * N[(n + N[(N[(n * 0.5), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < -7.5999999999999996

   1. Initial program 33.5%

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

   3. Taylor expanded in x around inf

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

      1. associate-/l/ N/A

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

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

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

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

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

      4. log-rec N/A

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

      5. mul-1-neg N/A

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

      6. associate-*r/ N/A

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

      7. associate-*r* N/A

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

      8. metadata-eval N/A

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

      9. *-commutative N/A

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

      10. associate-/l* N/A

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

      11. exp-to-pow N/A

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

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

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

      13. /-lowering-/.f64 60.7

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

   5. Simplified 60.7%

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

   6. Taylor expanded in n around inf

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

   8. Simplified 57.2%

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

   if -7.5999999999999996 < n < 1.99999999999999987e-88

   1. Initial program 78.3%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   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 37.0

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

   5. Simplified 37.0%

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

   6. Taylor expanded in x around inf

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

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

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

   8. Simplified 47.0%

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

   9. Taylor expanded in x around 0

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

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

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

       2. cube-mult N/A

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

       3. unpow2 N/A

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

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

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

       5. unpow2 N/A

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

       6. *-lowering-*.f64 69.2

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

   11. Simplified 69.2%

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

   if 1.99999999999999987e-88 < n

   1. Initial program 40.3%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   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 72.6

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

   5. Simplified 72.6%

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

      1. clear-num N/A

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

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

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

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

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

      4. diff-log N/A

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

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

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

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

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

      7. +-lowering-+.f64 72.7

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

   7. Applied egg-rr 72.7%

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

   8. Taylor expanded in x around inf

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

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

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

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

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

      3. associate-*r/ N/A

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

      4. metadata-eval N/A

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

      5. distribute-lft-neg-in N/A

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

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

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

      7. *-commutative N/A

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

      8. distribute-rgt-neg-in N/A

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

      9. metadata-eval N/A

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

      10. *-lowering-*.f64 51.2

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

   10. Simplified 51.2%

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

Alternative 13: 55.2% accurate, 10.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -100000000000:\\ \;\;\;\;\frac{\frac{0.3333333333333333}{x \cdot \left(x \cdot x\right)}}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \frac{-0.5 + \frac{0.3333333333333333}{x}}{x}}{x}}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= (/ 1.0 n) -100000000000.0)
   (/ (/ 0.3333333333333333 (* x (* x x))) n)
   (/ (/ (+ 1.0 (/ (+ -0.5 (/ 0.3333333333333333 x)) x)) x) n)))

C

double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -100000000000.0) {
		tmp = (0.3333333333333333 / (x * (x * x))) / n;
	} else {
		tmp = ((1.0 + ((-0.5 + (0.3333333333333333 / x)) / x)) / x) / n;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if ((1.0d0 / n) <= (-100000000000.0d0)) then
        tmp = (0.3333333333333333d0 / (x * (x * x))) / n
    else
        tmp = ((1.0d0 + (((-0.5d0) + (0.3333333333333333d0 / x)) / x)) / x) / n
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -100000000000.0) {
		tmp = (0.3333333333333333 / (x * (x * x))) / n;
	} else {
		tmp = ((1.0 + ((-0.5 + (0.3333333333333333 / x)) / x)) / x) / n;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if (1.0 / n) <= -100000000000.0:
		tmp = (0.3333333333333333 / (x * (x * x))) / n
	else:
		tmp = ((1.0 + ((-0.5 + (0.3333333333333333 / x)) / x)) / x) / n
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -100000000000.0)
		tmp = Float64(Float64(0.3333333333333333 / Float64(x * Float64(x * x))) / n);
	else
		tmp = Float64(Float64(Float64(1.0 + Float64(Float64(-0.5 + Float64(0.3333333333333333 / x)) / x)) / x) / n);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if ((1.0 / n) <= -100000000000.0)
		tmp = (0.3333333333333333 / (x * (x * x))) / n;
	else
		tmp = ((1.0 + ((-0.5 + (0.3333333333333333 / x)) / x)) / x) / n;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[N[(1.0 / n), $MachinePrecision], -100000000000.0], N[(N[(0.3333333333333333 / N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], N[(N[(N[(1.0 + N[(N[(-0.5 + N[(0.3333333333333333 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] / n), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -1e11

   1. Initial program 100.0%

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

   3. Taylor expanded in n around inf

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
   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 54.2

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

   5. Simplified 54.2%

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

   6. Taylor expanded in x around inf

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

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

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

   8. Simplified 42.1%

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

   9. Taylor expanded in x around 0

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

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

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

       2. cube-mult N/A

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

       3. unpow2 N/A

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

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

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

       5. unpow2 N/A

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

       6. *-lowering-*.f64 76.3

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

   11. Simplified 76.3%

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

   if -1e11 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 37.4%

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

   3. Taylor expanded in n around inf

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
   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 58.6

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

   5. Simplified 58.6%

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

   6. Taylor expanded in x around inf

      \[\leadsto \frac{\color{blue}{\frac{\left(1 + \frac{\frac{1}{3}}{{x}^{2}}\right) - \frac{1}{2} \cdot \frac{1}{x}}{x}}}{n} \]
   7. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{\left(1 + \frac{\frac{1}{3}}{{x}^{2}}\right) - \frac{1}{2} \cdot \frac{1}{x}}{x}}}{n} \]

   8. Simplified 53.3%

      \[\leadsto \frac{\color{blue}{\frac{1 + \frac{-0.5 + \frac{0.3333333333333333}{x}}{x}}{x}}}{n} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 14: 55.7% accurate, 11.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\frac{1}{x}}{n}\\ \mathbf{if}\;n \leq -4.6:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n \leq 1.08 \cdot 10^{-85}:\\ \;\;\;\;\frac{\frac{0.3333333333333333}{x \cdot \left(x \cdot x\right)}}{n}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (/ (/ 1.0 x) n)))
   (if (<= n -4.6)
     t_0
     (if (<= n 1.08e-85) (/ (/ 0.3333333333333333 (* x (* x x))) n) t_0))))

C

double code(double x, double n) {
	double t_0 = (1.0 / x) / n;
	double tmp;
	if (n <= -4.6) {
		tmp = t_0;
	} else if (n <= 1.08e-85) {
		tmp = (0.3333333333333333 / (x * (x * x))) / n;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (1.0d0 / x) / n
    if (n <= (-4.6d0)) then
        tmp = t_0
    else if (n <= 1.08d-85) then
        tmp = (0.3333333333333333d0 / (x * (x * x))) / n
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = (1.0 / x) / n;
	double tmp;
	if (n <= -4.6) {
		tmp = t_0;
	} else if (n <= 1.08e-85) {
		tmp = (0.3333333333333333 / (x * (x * x))) / n;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = (1.0 / x) / n
	tmp = 0
	if n <= -4.6:
		tmp = t_0
	elif n <= 1.08e-85:
		tmp = (0.3333333333333333 / (x * (x * x))) / n
	else:
		tmp = t_0
	return tmp

Julia

function code(x, n)
	t_0 = Float64(Float64(1.0 / x) / n)
	tmp = 0.0
	if (n <= -4.6)
		tmp = t_0;
	elseif (n <= 1.08e-85)
		tmp = Float64(Float64(0.3333333333333333 / Float64(x * Float64(x * x))) / n);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = (1.0 / x) / n;
	tmp = 0.0;
	if (n <= -4.6)
		tmp = t_0;
	elseif (n <= 1.08e-85)
		tmp = (0.3333333333333333 / (x * (x * x))) / n;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(N[(1.0 / x), $MachinePrecision] / n), $MachinePrecision]}, If[LessEqual[n, -4.6], t$95$0, If[LessEqual[n, 1.08e-85], N[(N[(0.3333333333333333 / N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if n < -4.5999999999999996 or 1.07999999999999997e-85 < n

   1. Initial program 37.1%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}}{n \cdot x}} \]
   4. Step-by-step derivation

      1. associate-/l/ N/A

         \[\leadsto \color{blue}{\frac{\frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}}{x}}{n}} \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}}{x}}{n}} \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}}{x}}}{n} \]

      4. log-rec N/A

         \[\leadsto \frac{\frac{e^{-1 \cdot \frac{\color{blue}{\mathsf{neg}\left(\log x\right)}}{n}}}{x}}{n} \]

      5. mul-1-neg N/A

         \[\leadsto \frac{\frac{e^{-1 \cdot \frac{\color{blue}{-1 \cdot \log x}}{n}}}{x}}{n} \]

      6. associate-*r/ N/A

         \[\leadsto \frac{\frac{e^{\color{blue}{\frac{-1 \cdot \left(-1 \cdot \log x\right)}{n}}}}{x}}{n} \]

      7. associate-*r* N/A

         \[\leadsto \frac{\frac{e^{\frac{\color{blue}{\left(-1 \cdot -1\right) \cdot \log x}}{n}}}{x}}{n} \]

      8. metadata-eval N/A

         \[\leadsto \frac{\frac{e^{\frac{\color{blue}{1} \cdot \log x}{n}}}{x}}{n} \]

      9. *-commutative N/A

         \[\leadsto \frac{\frac{e^{\frac{\color{blue}{\log x \cdot 1}}{n}}}{x}}{n} \]

      10. associate-/l* N/A

          \[\leadsto \frac{\frac{e^{\color{blue}{\log x \cdot \frac{1}{n}}}}{x}}{n} \]

      11. exp-to-pow N/A

          \[\leadsto \frac{\frac{\color{blue}{{x}^{\left(\frac{1}{n}\right)}}}{x}}{n} \]

      12. pow-lowering-pow.f64 N/A

          \[\leadsto \frac{\frac{\color{blue}{{x}^{\left(\frac{1}{n}\right)}}}{x}}{n} \]

      13. /-lowering-/.f64 55.0

          \[\leadsto \frac{\frac{{x}^{\color{blue}{\left(\frac{1}{n}\right)}}}{x}}{n} \]

   5. Simplified 55.0%

      \[\leadsto \color{blue}{\frac{\frac{{x}^{\left(\frac{1}{n}\right)}}{x}}{n}} \]

   6. Taylor expanded in n around inf

      \[\leadsto \frac{\frac{\color{blue}{1}}{x}}{n} \]
   7. Step-by-step derivation

   8. Simplified 53.0%

      \[\leadsto \frac{\frac{\color{blue}{1}}{x}}{n} \]

   if -4.5999999999999996 < n < 1.07999999999999997e-85

   1. Initial program 78.3%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   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 37.0

         \[\leadsto \frac{\mathsf{log1p}\left(x\right) - \color{blue}{\log x}}{n} \]

   5. Simplified 37.0%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) - \log x}{n}} \]

   6. Taylor expanded in x around inf

      \[\leadsto \frac{\color{blue}{\frac{\left(1 + \frac{\frac{1}{3}}{{x}^{2}}\right) - \frac{1}{2} \cdot \frac{1}{x}}{x}}}{n} \]
   7. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{\left(1 + \frac{\frac{1}{3}}{{x}^{2}}\right) - \frac{1}{2} \cdot \frac{1}{x}}{x}}}{n} \]

   8. Simplified 47.0%

      \[\leadsto \frac{\color{blue}{\frac{1 + \frac{-0.5 + \frac{0.3333333333333333}{x}}{x}}{x}}}{n} \]

   9. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{\frac{\frac{1}{3}}{{x}^{3}}}}{n} \]
   10. Step-by-step derivation

       1. /-lowering-/.f64 N/A

          \[\leadsto \frac{\color{blue}{\frac{\frac{1}{3}}{{x}^{3}}}}{n} \]

       2. cube-mult N/A

          \[\leadsto \frac{\frac{\frac{1}{3}}{\color{blue}{x \cdot \left(x \cdot x\right)}}}{n} \]

       3. unpow2 N/A

          \[\leadsto \frac{\frac{\frac{1}{3}}{x \cdot \color{blue}{{x}^{2}}}}{n} \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \frac{\frac{\frac{1}{3}}{\color{blue}{x \cdot {x}^{2}}}}{n} \]

       5. unpow2 N/A

          \[\leadsto \frac{\frac{\frac{1}{3}}{x \cdot \color{blue}{\left(x \cdot x\right)}}}{n} \]

       6. *-lowering-*.f64 69.2

          \[\leadsto \frac{\frac{0.3333333333333333}{x \cdot \color{blue}{\left(x \cdot x\right)}}}{n} \]

   11. Simplified 69.2%

       \[\leadsto \frac{\color{blue}{\frac{0.3333333333333333}{x \cdot \left(x \cdot x\right)}}}{n} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 15: 53.6% accurate, 11.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\frac{1}{x}}{n}\\ \mathbf{if}\;n \leq -5:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n \leq 1.7 \cdot 10^{-86}:\\ \;\;\;\;\frac{0.3333333333333333}{x \cdot \left(x \cdot \left(n \cdot x\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (/ (/ 1.0 x) n)))
   (if (<= n -5.0)
     t_0
     (if (<= n 1.7e-86) (/ 0.3333333333333333 (* x (* x (* n x)))) t_0))))

C

double code(double x, double n) {
	double t_0 = (1.0 / x) / n;
	double tmp;
	if (n <= -5.0) {
		tmp = t_0;
	} else if (n <= 1.7e-86) {
		tmp = 0.3333333333333333 / (x * (x * (n * x)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (1.0d0 / x) / n
    if (n <= (-5.0d0)) then
        tmp = t_0
    else if (n <= 1.7d-86) then
        tmp = 0.3333333333333333d0 / (x * (x * (n * x)))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = (1.0 / x) / n;
	double tmp;
	if (n <= -5.0) {
		tmp = t_0;
	} else if (n <= 1.7e-86) {
		tmp = 0.3333333333333333 / (x * (x * (n * x)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = (1.0 / x) / n
	tmp = 0
	if n <= -5.0:
		tmp = t_0
	elif n <= 1.7e-86:
		tmp = 0.3333333333333333 / (x * (x * (n * x)))
	else:
		tmp = t_0
	return tmp

Julia

function code(x, n)
	t_0 = Float64(Float64(1.0 / x) / n)
	tmp = 0.0
	if (n <= -5.0)
		tmp = t_0;
	elseif (n <= 1.7e-86)
		tmp = Float64(0.3333333333333333 / Float64(x * Float64(x * Float64(n * x))));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = (1.0 / x) / n;
	tmp = 0.0;
	if (n <= -5.0)
		tmp = t_0;
	elseif (n <= 1.7e-86)
		tmp = 0.3333333333333333 / (x * (x * (n * x)));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(N[(1.0 / x), $MachinePrecision] / n), $MachinePrecision]}, If[LessEqual[n, -5.0], t$95$0, If[LessEqual[n, 1.7e-86], N[(0.3333333333333333 / N[(x * N[(x * N[(n * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if n < -5 or 1.7e-86 < n

   1. Initial program 37.1%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}}{n \cdot x}} \]
   4. Step-by-step derivation

      1. associate-/l/ N/A

         \[\leadsto \color{blue}{\frac{\frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}}{x}}{n}} \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}}{x}}{n}} \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}}{x}}}{n} \]

      4. log-rec N/A

         \[\leadsto \frac{\frac{e^{-1 \cdot \frac{\color{blue}{\mathsf{neg}\left(\log x\right)}}{n}}}{x}}{n} \]

      5. mul-1-neg N/A

         \[\leadsto \frac{\frac{e^{-1 \cdot \frac{\color{blue}{-1 \cdot \log x}}{n}}}{x}}{n} \]

      6. associate-*r/ N/A

         \[\leadsto \frac{\frac{e^{\color{blue}{\frac{-1 \cdot \left(-1 \cdot \log x\right)}{n}}}}{x}}{n} \]

      7. associate-*r* N/A

         \[\leadsto \frac{\frac{e^{\frac{\color{blue}{\left(-1 \cdot -1\right) \cdot \log x}}{n}}}{x}}{n} \]

      8. metadata-eval N/A

         \[\leadsto \frac{\frac{e^{\frac{\color{blue}{1} \cdot \log x}{n}}}{x}}{n} \]

      9. *-commutative N/A

         \[\leadsto \frac{\frac{e^{\frac{\color{blue}{\log x \cdot 1}}{n}}}{x}}{n} \]

      10. associate-/l* N/A

          \[\leadsto \frac{\frac{e^{\color{blue}{\log x \cdot \frac{1}{n}}}}{x}}{n} \]

      11. exp-to-pow N/A

          \[\leadsto \frac{\frac{\color{blue}{{x}^{\left(\frac{1}{n}\right)}}}{x}}{n} \]

      12. pow-lowering-pow.f64 N/A

          \[\leadsto \frac{\frac{\color{blue}{{x}^{\left(\frac{1}{n}\right)}}}{x}}{n} \]

      13. /-lowering-/.f64 55.0

          \[\leadsto \frac{\frac{{x}^{\color{blue}{\left(\frac{1}{n}\right)}}}{x}}{n} \]

   5. Simplified 55.0%

      \[\leadsto \color{blue}{\frac{\frac{{x}^{\left(\frac{1}{n}\right)}}{x}}{n}} \]

   6. Taylor expanded in n around inf

      \[\leadsto \frac{\frac{\color{blue}{1}}{x}}{n} \]
   7. Step-by-step derivation

   8. Simplified 53.0%

      \[\leadsto \frac{\frac{\color{blue}{1}}{x}}{n} \]

   if -5 < n < 1.7e-86

   1. Initial program 78.3%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   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 37.0

         \[\leadsto \frac{\mathsf{log1p}\left(x\right) - \color{blue}{\log x}}{n} \]

   5. Simplified 37.0%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) - \log x}{n}} \]

   6. Taylor expanded in x around inf

      \[\leadsto \frac{\color{blue}{\frac{\left(1 + \frac{\frac{1}{3}}{{x}^{2}}\right) - \frac{1}{2} \cdot \frac{1}{x}}{x}}}{n} \]
   7. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{\left(1 + \frac{\frac{1}{3}}{{x}^{2}}\right) - \frac{1}{2} \cdot \frac{1}{x}}{x}}}{n} \]

   8. Simplified 47.0%

      \[\leadsto \frac{\color{blue}{\frac{1 + \frac{-0.5 + \frac{0.3333333333333333}{x}}{x}}{x}}}{n} \]

   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. *-commutative N/A

          \[\leadsto \frac{\frac{1}{3}}{\color{blue}{{x}^{3} \cdot n}} \]

       3. cube-mult N/A

          \[\leadsto \frac{\frac{1}{3}}{\color{blue}{\left(x \cdot \left(x \cdot x\right)\right)} \cdot n} \]

       4. unpow2 N/A

          \[\leadsto \frac{\frac{1}{3}}{\left(x \cdot \color{blue}{{x}^{2}}\right) \cdot n} \]

       5. associate-*l* N/A

          \[\leadsto \frac{\frac{1}{3}}{\color{blue}{x \cdot \left({x}^{2} \cdot n\right)}} \]

       6. *-commutative N/A

          \[\leadsto \frac{\frac{1}{3}}{x \cdot \color{blue}{\left(n \cdot {x}^{2}\right)}} \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \frac{\frac{1}{3}}{\color{blue}{x \cdot \left(n \cdot {x}^{2}\right)}} \]

       8. *-commutative N/A

          \[\leadsto \frac{\frac{1}{3}}{x \cdot \color{blue}{\left({x}^{2} \cdot n\right)}} \]

       9. unpow2 N/A

          \[\leadsto \frac{\frac{1}{3}}{x \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot n\right)} \]

       10. associate-*l* N/A

           \[\leadsto \frac{\frac{1}{3}}{x \cdot \color{blue}{\left(x \cdot \left(x \cdot n\right)\right)}} \]

       11. *-commutative N/A

           \[\leadsto \frac{\frac{1}{3}}{x \cdot \left(x \cdot \color{blue}{\left(n \cdot x\right)}\right)} \]

       12. *-lowering-*.f64 N/A

           \[\leadsto \frac{\frac{1}{3}}{x \cdot \color{blue}{\left(x \cdot \left(n \cdot x\right)\right)}} \]

       13. *-commutative N/A

           \[\leadsto \frac{\frac{1}{3}}{x \cdot \left(x \cdot \color{blue}{\left(x \cdot n\right)}\right)} \]

       14. *-lowering-*.f64 63.2

           \[\leadsto \frac{0.3333333333333333}{x \cdot \left(x \cdot \color{blue}{\left(x \cdot n\right)}\right)} \]

   11. Simplified 63.2%

       \[\leadsto \color{blue}{\frac{0.3333333333333333}{x \cdot \left(x \cdot \left(x \cdot n\right)\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 57.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -5:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \mathbf{elif}\;n \leq 1.7 \cdot 10^{-86}:\\ \;\;\;\;\frac{0.3333333333333333}{x \cdot \left(x \cdot \left(n \cdot x\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 54.8% accurate, 11.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -100000000000:\\ \;\;\;\;\frac{\frac{0.3333333333333333}{x \cdot \left(x \cdot x\right)}}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \frac{0.3333333333333333}{x \cdot x}}{x}}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= (/ 1.0 n) -100000000000.0)
   (/ (/ 0.3333333333333333 (* x (* x x))) n)
   (/ (/ (+ 1.0 (/ 0.3333333333333333 (* x x))) x) n)))

C

double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -100000000000.0) {
		tmp = (0.3333333333333333 / (x * (x * x))) / n;
	} else {
		tmp = ((1.0 + (0.3333333333333333 / (x * x))) / x) / n;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if ((1.0d0 / n) <= (-100000000000.0d0)) then
        tmp = (0.3333333333333333d0 / (x * (x * x))) / n
    else
        tmp = ((1.0d0 + (0.3333333333333333d0 / (x * x))) / x) / n
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -100000000000.0) {
		tmp = (0.3333333333333333 / (x * (x * x))) / n;
	} else {
		tmp = ((1.0 + (0.3333333333333333 / (x * x))) / x) / n;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if (1.0 / n) <= -100000000000.0:
		tmp = (0.3333333333333333 / (x * (x * x))) / n
	else:
		tmp = ((1.0 + (0.3333333333333333 / (x * x))) / x) / n
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -100000000000.0)
		tmp = Float64(Float64(0.3333333333333333 / Float64(x * Float64(x * x))) / n);
	else
		tmp = Float64(Float64(Float64(1.0 + Float64(0.3333333333333333 / Float64(x * x))) / x) / n);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if ((1.0 / n) <= -100000000000.0)
		tmp = (0.3333333333333333 / (x * (x * x))) / n;
	else
		tmp = ((1.0 + (0.3333333333333333 / (x * x))) / x) / n;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[N[(1.0 / n), $MachinePrecision], -100000000000.0], N[(N[(0.3333333333333333 / N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], N[(N[(N[(1.0 + N[(0.3333333333333333 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] / n), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -1e11

   1. Initial program 100.0%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in n around inf

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
   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 54.2

         \[\leadsto \frac{\mathsf{log1p}\left(x\right) - \color{blue}{\log x}}{n} \]

   5. Simplified 54.2%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) - \log x}{n}} \]

   6. Taylor expanded in x around inf

      \[\leadsto \frac{\color{blue}{\frac{\left(1 + \frac{\frac{1}{3}}{{x}^{2}}\right) - \frac{1}{2} \cdot \frac{1}{x}}{x}}}{n} \]
   7. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{\left(1 + \frac{\frac{1}{3}}{{x}^{2}}\right) - \frac{1}{2} \cdot \frac{1}{x}}{x}}}{n} \]

   8. Simplified 42.1%

      \[\leadsto \frac{\color{blue}{\frac{1 + \frac{-0.5 + \frac{0.3333333333333333}{x}}{x}}{x}}}{n} \]

   9. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{\frac{\frac{1}{3}}{{x}^{3}}}}{n} \]
   10. Step-by-step derivation

       1. /-lowering-/.f64 N/A

          \[\leadsto \frac{\color{blue}{\frac{\frac{1}{3}}{{x}^{3}}}}{n} \]

       2. cube-mult N/A

          \[\leadsto \frac{\frac{\frac{1}{3}}{\color{blue}{x \cdot \left(x \cdot x\right)}}}{n} \]

       3. unpow2 N/A

          \[\leadsto \frac{\frac{\frac{1}{3}}{x \cdot \color{blue}{{x}^{2}}}}{n} \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \frac{\frac{\frac{1}{3}}{\color{blue}{x \cdot {x}^{2}}}}{n} \]

       5. unpow2 N/A

          \[\leadsto \frac{\frac{\frac{1}{3}}{x \cdot \color{blue}{\left(x \cdot x\right)}}}{n} \]

       6. *-lowering-*.f64 76.3

          \[\leadsto \frac{\frac{0.3333333333333333}{x \cdot \color{blue}{\left(x \cdot x\right)}}}{n} \]

   11. Simplified 76.3%

       \[\leadsto \frac{\color{blue}{\frac{0.3333333333333333}{x \cdot \left(x \cdot x\right)}}}{n} \]

   if -1e11 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 37.4%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in n around inf

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
   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 58.6

         \[\leadsto \frac{\mathsf{log1p}\left(x\right) - \color{blue}{\log x}}{n} \]

   5. Simplified 58.6%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) - \log x}{n}} \]

   6. Taylor expanded in x around inf

      \[\leadsto \frac{\color{blue}{\frac{\left(1 + \frac{\frac{1}{3}}{{x}^{2}}\right) - \frac{1}{2} \cdot \frac{1}{x}}{x}}}{n} \]
   7. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{\left(1 + \frac{\frac{1}{3}}{{x}^{2}}\right) - \frac{1}{2} \cdot \frac{1}{x}}{x}}}{n} \]

   8. Simplified 53.3%

      \[\leadsto \frac{\color{blue}{\frac{1 + \frac{-0.5 + \frac{0.3333333333333333}{x}}{x}}{x}}}{n} \]

   9. Taylor expanded in x around 0

      \[\leadsto \frac{\frac{1 + \color{blue}{\frac{\frac{1}{3}}{{x}^{2}}}}{x}}{n} \]
   10. Step-by-step derivation

       1. /-lowering-/.f64 N/A

          \[\leadsto \frac{\frac{1 + \color{blue}{\frac{\frac{1}{3}}{{x}^{2}}}}{x}}{n} \]

       2. unpow2 N/A

          \[\leadsto \frac{\frac{1 + \frac{\frac{1}{3}}{\color{blue}{x \cdot x}}}{x}}{n} \]

       3. *-lowering-*.f64 52.9

          \[\leadsto \frac{\frac{1 + \frac{0.3333333333333333}{\color{blue}{x \cdot x}}}{x}}{n} \]

   11. Simplified 52.9%

       \[\leadsto \frac{\frac{1 + \color{blue}{\frac{0.3333333333333333}{x \cdot x}}}{x}}{n} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 17: 46.6% accurate, 14.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\frac{1}{x}}{n}\\ \mathbf{if}\;n \leq -2.8 \cdot 10^{-9}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n \leq -7.2 \cdot 10^{-216}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (/ (/ 1.0 x) n)))
   (if (<= n -2.8e-9) t_0 (if (<= n -7.2e-216) 0.0 t_0))))

C

double code(double x, double n) {
	double t_0 = (1.0 / x) / n;
	double tmp;
	if (n <= -2.8e-9) {
		tmp = t_0;
	} else if (n <= -7.2e-216) {
		tmp = 0.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (1.0d0 / x) / n
    if (n <= (-2.8d-9)) then
        tmp = t_0
    else if (n <= (-7.2d-216)) then
        tmp = 0.0d0
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = (1.0 / x) / n;
	double tmp;
	if (n <= -2.8e-9) {
		tmp = t_0;
	} else if (n <= -7.2e-216) {
		tmp = 0.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = (1.0 / x) / n
	tmp = 0
	if n <= -2.8e-9:
		tmp = t_0
	elif n <= -7.2e-216:
		tmp = 0.0
	else:
		tmp = t_0
	return tmp

Julia

function code(x, n)
	t_0 = Float64(Float64(1.0 / x) / n)
	tmp = 0.0
	if (n <= -2.8e-9)
		tmp = t_0;
	elseif (n <= -7.2e-216)
		tmp = 0.0;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = (1.0 / x) / n;
	tmp = 0.0;
	if (n <= -2.8e-9)
		tmp = t_0;
	elseif (n <= -7.2e-216)
		tmp = 0.0;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(N[(1.0 / x), $MachinePrecision] / n), $MachinePrecision]}, If[LessEqual[n, -2.8e-9], t$95$0, If[LessEqual[n, -7.2e-216], 0.0, t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if n < -2.79999999999999984e-9 or -7.1999999999999998e-216 < n

   1. Initial program 42.9%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}}{n \cdot x}} \]
   4. Step-by-step derivation

      1. associate-/l/ N/A

         \[\leadsto \color{blue}{\frac{\frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}}{x}}{n}} \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}}{x}}{n}} \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}}{x}}}{n} \]

      4. log-rec N/A

         \[\leadsto \frac{\frac{e^{-1 \cdot \frac{\color{blue}{\mathsf{neg}\left(\log x\right)}}{n}}}{x}}{n} \]

      5. mul-1-neg N/A

         \[\leadsto \frac{\frac{e^{-1 \cdot \frac{\color{blue}{-1 \cdot \log x}}{n}}}{x}}{n} \]

      6. associate-*r/ N/A

         \[\leadsto \frac{\frac{e^{\color{blue}{\frac{-1 \cdot \left(-1 \cdot \log x\right)}{n}}}}{x}}{n} \]

      7. associate-*r* N/A

         \[\leadsto \frac{\frac{e^{\frac{\color{blue}{\left(-1 \cdot -1\right) \cdot \log x}}{n}}}{x}}{n} \]

      8. metadata-eval N/A

         \[\leadsto \frac{\frac{e^{\frac{\color{blue}{1} \cdot \log x}{n}}}{x}}{n} \]

      9. *-commutative N/A

         \[\leadsto \frac{\frac{e^{\frac{\color{blue}{\log x \cdot 1}}{n}}}{x}}{n} \]

      10. associate-/l* N/A

          \[\leadsto \frac{\frac{e^{\color{blue}{\log x \cdot \frac{1}{n}}}}{x}}{n} \]

      11. exp-to-pow N/A

          \[\leadsto \frac{\frac{\color{blue}{{x}^{\left(\frac{1}{n}\right)}}}{x}}{n} \]

      12. pow-lowering-pow.f64 N/A

          \[\leadsto \frac{\frac{\color{blue}{{x}^{\left(\frac{1}{n}\right)}}}{x}}{n} \]

      13. /-lowering-/.f64 49.9

          \[\leadsto \frac{\frac{{x}^{\color{blue}{\left(\frac{1}{n}\right)}}}{x}}{n} \]

   5. Simplified 49.9%

      \[\leadsto \color{blue}{\frac{\frac{{x}^{\left(\frac{1}{n}\right)}}{x}}{n}} \]

   6. Taylor expanded in n around inf

      \[\leadsto \frac{\frac{\color{blue}{1}}{x}}{n} \]
   7. Step-by-step derivation

   8. Simplified 50.8%

      \[\leadsto \frac{\frac{\color{blue}{1}}{x}}{n} \]

   if -2.79999999999999984e-9 < n < -7.1999999999999998e-216

   1. Initial program 100.0%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1 - e^{\frac{\log x}{n}}} \]
   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 41.9

          \[\leadsto 1 - {x}^{\color{blue}{\left(\frac{1}{n}\right)}} \]

   5. Simplified 41.9%

      \[\leadsto \color{blue}{1 - {x}^{\left(\frac{1}{n}\right)}} \]

   6. Taylor expanded in n around inf

      \[\leadsto 1 - \color{blue}{1} \]
   7. Step-by-step derivation

   8. Simplified 60.6%

      \[\leadsto 1 - \color{blue}{1} \]
   9. Step-by-step derivation

      1. metadata-eval 60.6

         \[\leadsto \color{blue}{0} \]

   10. Applied egg-rr 60.6%

       \[\leadsto \color{blue}{0} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 18: 46.6% accurate, 14.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\frac{1}{n}}{x}\\ \mathbf{if}\;n \leq -2.8 \cdot 10^{-9}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n \leq -3.2 \cdot 10^{-215}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (/ (/ 1.0 n) x)))
   (if (<= n -2.8e-9) t_0 (if (<= n -3.2e-215) 0.0 t_0))))

C

double code(double x, double n) {
	double t_0 = (1.0 / n) / x;
	double tmp;
	if (n <= -2.8e-9) {
		tmp = t_0;
	} else if (n <= -3.2e-215) {
		tmp = 0.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (1.0d0 / n) / x
    if (n <= (-2.8d-9)) then
        tmp = t_0
    else if (n <= (-3.2d-215)) then
        tmp = 0.0d0
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = (1.0 / n) / x;
	double tmp;
	if (n <= -2.8e-9) {
		tmp = t_0;
	} else if (n <= -3.2e-215) {
		tmp = 0.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = (1.0 / n) / x
	tmp = 0
	if n <= -2.8e-9:
		tmp = t_0
	elif n <= -3.2e-215:
		tmp = 0.0
	else:
		tmp = t_0
	return tmp

Julia

function code(x, n)
	t_0 = Float64(Float64(1.0 / n) / x)
	tmp = 0.0
	if (n <= -2.8e-9)
		tmp = t_0;
	elseif (n <= -3.2e-215)
		tmp = 0.0;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = (1.0 / n) / x;
	tmp = 0.0;
	if (n <= -2.8e-9)
		tmp = t_0;
	elseif (n <= -3.2e-215)
		tmp = 0.0;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(N[(1.0 / n), $MachinePrecision] / x), $MachinePrecision]}, If[LessEqual[n, -2.8e-9], t$95$0, If[LessEqual[n, -3.2e-215], 0.0, t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if n < -2.79999999999999984e-9 or -3.2000000000000001e-215 < n

   1. Initial program 42.9%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}}{n \cdot x}} \]
   4. Step-by-step derivation

      1. associate-/l/ N/A

         \[\leadsto \color{blue}{\frac{\frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}}{x}}{n}} \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}}{x}}{n}} \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}}{x}}}{n} \]

      4. log-rec N/A

         \[\leadsto \frac{\frac{e^{-1 \cdot \frac{\color{blue}{\mathsf{neg}\left(\log x\right)}}{n}}}{x}}{n} \]

      5. mul-1-neg N/A

         \[\leadsto \frac{\frac{e^{-1 \cdot \frac{\color{blue}{-1 \cdot \log x}}{n}}}{x}}{n} \]

      6. associate-*r/ N/A

         \[\leadsto \frac{\frac{e^{\color{blue}{\frac{-1 \cdot \left(-1 \cdot \log x\right)}{n}}}}{x}}{n} \]

      7. associate-*r* N/A

         \[\leadsto \frac{\frac{e^{\frac{\color{blue}{\left(-1 \cdot -1\right) \cdot \log x}}{n}}}{x}}{n} \]

      8. metadata-eval N/A

         \[\leadsto \frac{\frac{e^{\frac{\color{blue}{1} \cdot \log x}{n}}}{x}}{n} \]

      9. *-commutative N/A

         \[\leadsto \frac{\frac{e^{\frac{\color{blue}{\log x \cdot 1}}{n}}}{x}}{n} \]

      10. associate-/l* N/A

          \[\leadsto \frac{\frac{e^{\color{blue}{\log x \cdot \frac{1}{n}}}}{x}}{n} \]

      11. exp-to-pow N/A

          \[\leadsto \frac{\frac{\color{blue}{{x}^{\left(\frac{1}{n}\right)}}}{x}}{n} \]

      12. pow-lowering-pow.f64 N/A

          \[\leadsto \frac{\frac{\color{blue}{{x}^{\left(\frac{1}{n}\right)}}}{x}}{n} \]

      13. /-lowering-/.f64 49.9

          \[\leadsto \frac{\frac{{x}^{\color{blue}{\left(\frac{1}{n}\right)}}}{x}}{n} \]

   5. Simplified 49.9%

      \[\leadsto \color{blue}{\frac{\frac{{x}^{\left(\frac{1}{n}\right)}}{x}}{n}} \]

   6. Taylor expanded in n around inf

      \[\leadsto \color{blue}{\frac{1}{n \cdot x}} \]
   7. Step-by-step derivation

      1. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{1}{n}}{x}} \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{1}{n}}{x}} \]

      3. /-lowering-/.f64 50.7

         \[\leadsto \frac{\color{blue}{\frac{1}{n}}}{x} \]

   8. Simplified 50.7%

      \[\leadsto \color{blue}{\frac{\frac{1}{n}}{x}} \]

   if -2.79999999999999984e-9 < n < -3.2000000000000001e-215

   1. Initial program 100.0%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1 - e^{\frac{\log x}{n}}} \]
   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 41.9

          \[\leadsto 1 - {x}^{\color{blue}{\left(\frac{1}{n}\right)}} \]

   5. Simplified 41.9%

      \[\leadsto \color{blue}{1 - {x}^{\left(\frac{1}{n}\right)}} \]

   6. Taylor expanded in n around inf

      \[\leadsto 1 - \color{blue}{1} \]
   7. Step-by-step derivation

   8. Simplified 60.6%

      \[\leadsto 1 - \color{blue}{1} \]
   9. Step-by-step derivation

      1. metadata-eval 60.6

         \[\leadsto \color{blue}{0} \]

   10. Applied egg-rr 60.6%

       \[\leadsto \color{blue}{0} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 19: 46.0% accurate, 14.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{n \cdot x}\\ \mathbf{if}\;n \leq -2.8 \cdot 10^{-9}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n \leq -3.8 \cdot 10^{-215}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (/ 1.0 (* n x))))
   (if (<= n -2.8e-9) t_0 (if (<= n -3.8e-215) 0.0 t_0))))

C

double code(double x, double n) {
	double t_0 = 1.0 / (n * x);
	double tmp;
	if (n <= -2.8e-9) {
		tmp = t_0;
	} else if (n <= -3.8e-215) {
		tmp = 0.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 1.0d0 / (n * x)
    if (n <= (-2.8d-9)) then
        tmp = t_0
    else if (n <= (-3.8d-215)) then
        tmp = 0.0d0
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = 1.0 / (n * x);
	double tmp;
	if (n <= -2.8e-9) {
		tmp = t_0;
	} else if (n <= -3.8e-215) {
		tmp = 0.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = 1.0 / (n * x)
	tmp = 0
	if n <= -2.8e-9:
		tmp = t_0
	elif n <= -3.8e-215:
		tmp = 0.0
	else:
		tmp = t_0
	return tmp

Julia

function code(x, n)
	t_0 = Float64(1.0 / Float64(n * x))
	tmp = 0.0
	if (n <= -2.8e-9)
		tmp = t_0;
	elseif (n <= -3.8e-215)
		tmp = 0.0;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = 1.0 / (n * x);
	tmp = 0.0;
	if (n <= -2.8e-9)
		tmp = t_0;
	elseif (n <= -3.8e-215)
		tmp = 0.0;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(1.0 / N[(n * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[n, -2.8e-9], t$95$0, If[LessEqual[n, -3.8e-215], 0.0, t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if n < -2.79999999999999984e-9 or -3.79999999999999977e-215 < n

   1. Initial program 42.9%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in n around inf

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
   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 56.4

         \[\leadsto \frac{\mathsf{log1p}\left(x\right) - \color{blue}{\log x}}{n} \]

   5. Simplified 56.4%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) - \log x}{n}} \]

   6. Taylor expanded in x around inf

      \[\leadsto \frac{\color{blue}{\frac{\left(1 + \frac{\frac{1}{3}}{{x}^{2}}\right) - \frac{1}{2} \cdot \frac{1}{x}}{x}}}{n} \]
   7. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{\left(1 + \frac{\frac{1}{3}}{{x}^{2}}\right) - \frac{1}{2} \cdot \frac{1}{x}}{x}}}{n} \]

   8. Simplified 54.5%

      \[\leadsto \frac{\color{blue}{\frac{1 + \frac{-0.5 + \frac{0.3333333333333333}{x}}{x}}{x}}}{n} \]

   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 49.3

          \[\leadsto \frac{1}{\color{blue}{x \cdot n}} \]

   11. Simplified 49.3%

       \[\leadsto \color{blue}{\frac{1}{x \cdot n}} \]

   if -2.79999999999999984e-9 < n < -3.79999999999999977e-215

   1. Initial program 100.0%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1 - e^{\frac{\log x}{n}}} \]
   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 41.9

          \[\leadsto 1 - {x}^{\color{blue}{\left(\frac{1}{n}\right)}} \]

   5. Simplified 41.9%

      \[\leadsto \color{blue}{1 - {x}^{\left(\frac{1}{n}\right)}} \]

   6. Taylor expanded in n around inf

      \[\leadsto 1 - \color{blue}{1} \]
   7. Step-by-step derivation

   8. Simplified 60.6%

      \[\leadsto 1 - \color{blue}{1} \]
   9. Step-by-step derivation

      1. metadata-eval 60.6

         \[\leadsto \color{blue}{0} \]

   10. Applied egg-rr 60.6%

       \[\leadsto \color{blue}{0} \]
3. Recombined 2 regimes into one program.

4. Final simplification 51.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -2.8 \cdot 10^{-9}:\\ \;\;\;\;\frac{1}{n \cdot x}\\ \mathbf{elif}\;n \leq -3.8 \cdot 10^{-215}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{n \cdot x}\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 31.3% accurate, 211.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (x n) :precision binary64 0.0)

C

double code(double x, double n) {
	return 0.0;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    code = 0.0d0
end function

Java

public static double code(double x, double n) {
	return 0.0;
}

Python

def code(x, n):
	return 0.0

Julia

function code(x, n)
	return 0.0
end

MATLAB

function tmp = code(x, n)
	tmp = 0.0;
end

Wolfram

code[x_, n_] := 0.0

Derivation

1. Initial program 54.0%

   \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
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 39.9

       \[\leadsto 1 - {x}^{\color{blue}{\left(\frac{1}{n}\right)}} \]

5. Simplified 39.9%

   \[\leadsto \color{blue}{1 - {x}^{\left(\frac{1}{n}\right)}} \]

6. Taylor expanded in n around inf

   \[\leadsto 1 - \color{blue}{1} \]
7. Step-by-step derivation

8. Simplified 32.8%

   \[\leadsto 1 - \color{blue}{1} \]
9. Step-by-step derivation

   1. metadata-eval 32.8

      \[\leadsto \color{blue}{0} \]

10. Applied egg-rr 32.8%

    \[\leadsto \color{blue}{0} \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024191 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  :precision binary64
  (- (pow (+ x 1.0) (/ 1.0 n)) (pow x (/ 1.0 n))))