2nthrt (problem 3.4.6)

Percentage Accurate: 53.3% → 86.2%

Time: 27.0s

Alternatives: 21

Speedup: 1.9×

• could not determine a ground truth

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 53.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 86.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{-16}:\\ \;\;\;\;\frac{{x}^{\left(\frac{2}{n \cdot 2}\right)}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-69}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right) - \log x}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 0.02:\\ \;\;\;\;\frac{{e}^{\left(\frac{\log x}{n}\right)}}{n \cdot x}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{x}{n}} - {x}^{\mathsf{reciprocal}\left(n\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= (/ 1.0 n) -2e-16)
   (/ (pow x (/ 2.0 (* n 2.0))) (* n x))
   (if (<= (/ 1.0 n) 2e-69)
     (/ (- (log1p x) (log x)) n)
     (if (<= (/ 1.0 n) 0.02)
       (/ (pow E (/ (log x) n)) (* n x))
       (- (exp (/ x n)) (pow x (reciprocal n)))))))

Derivation

1. Split input into 4 regimes

2. if (/.f64 1 n) < -2e-16

   1. Initial program 94.6%

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

      1. reciprocal-define 94.6%

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

      2. reciprocal-define 94.6%

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

   3. Simplified 94.6%

      \[\leadsto \color{blue}{{\left(x + 1\right)}^{\mathsf{reciprocal}\left(n\right)} - {x}^{\mathsf{reciprocal}\left(n\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf 99.9%

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

      1. mul-1-neg 99.9%

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

      2. log-rec 99.9%

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

      3. mul-1-neg 99.9%

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

      4. distribute-neg-frac 99.9%

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

      5. mul-1-neg 99.9%

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

      6. remove-double-neg 99.9%

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

   7. Simplified 99.9%

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

      1. expm1-log1p-u 100.0%

         \[\leadsto \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(e^{\frac{\log x}{n}}\right)\right)}}{n \cdot x} \]

      2. expm1-udef 100.0%

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

      3. log1p-udef 99.9%

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

      4. add-exp-log 99.9%

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

      5. div-inv 99.9%

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

      6. reciprocal-undefine 99.0%

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

      7. pow-to-exp 99.0%

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

   9. Applied egg-rr 99.0%

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

       1. add-exp-log 99.0%

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

       2. log1p-udef 99.0%

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

       3. expm1-udef 99.0%

          \[\leadsto \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({x}^{\mathsf{reciprocal}\left(n\right)}\right)\right)}}{n \cdot x} \]

       4. expm1-log1p-u 99.0%

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

       5. sqr-pow 99.0%

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

       6. pow-prod-up 99.0%

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

       7. reciprocal-undefine 99.0%

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

       8. associate-/l/ 99.0%

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

       9. reciprocal-define 99.0%

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

       10. reciprocal-undefine 99.0%

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

       11. associate-/l/ 99.0%

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

       12. reciprocal-define 99.0%

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

   11. Applied egg-rr 99.0%

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

       1. count-2 99.0%

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

       2. reciprocal-undefine 99.9%

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

       3. associate-*r/ 99.9%

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

       4. metadata-eval 99.9%

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

       5. *-commutative 99.9%

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

   13. Simplified 99.9%

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

   if -2e-16 < (/.f64 1 n) < 1.9999999999999999e-69

   1. Initial program 33.5%

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

      1. reciprocal-define 33.5%

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

      2. reciprocal-define 33.5%

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

   3. Simplified 33.5%

      \[\leadsto \color{blue}{{\left(x + 1\right)}^{\mathsf{reciprocal}\left(n\right)} - {x}^{\mathsf{reciprocal}\left(n\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in n around inf 86.1%

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

      1. +-rgt-identity 86.1%

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

      2. +-rgt-identity 86.1%

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

      3. log1p-def 86.1%

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

   7. Simplified 86.1%

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

   if 1.9999999999999999e-69 < (/.f64 1 n) < 0.0200000000000000004

   1. Initial program 23.2%

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

      1. reciprocal-define 22.7%

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

      2. reciprocal-define 12.8%

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

   3. Simplified 12.8%

      \[\leadsto \color{blue}{{\left(x + 1\right)}^{\mathsf{reciprocal}\left(n\right)} - {x}^{\mathsf{reciprocal}\left(n\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf 67.8%

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

      1. mul-1-neg 67.8%

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

      2. log-rec 67.8%

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

      3. mul-1-neg 67.8%

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

      4. distribute-neg-frac 67.8%

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

      5. mul-1-neg 67.8%

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

      6. remove-double-neg 67.8%

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

   7. Simplified 67.8%

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

      1. *-un-lft-identity 67.8%

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

      2. exp-prod 68.0%

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

   9. Applied egg-rr 68.0%

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

   if 0.0200000000000000004 < (/.f64 1 n)

   1. Initial program 58.6%

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

      1. reciprocal-define 58.6%

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

      2. reciprocal-define 58.6%

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

   3. Simplified 58.6%

      \[\leadsto \color{blue}{{\left(x + 1\right)}^{\mathsf{reciprocal}\left(n\right)} - {x}^{\mathsf{reciprocal}\left(n\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in n around 0 58.6%

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

      1. log1p-def 99.8%

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

   7. Simplified 99.8%

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

   8. Taylor expanded in x around 0 99.8%

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

4. Final simplification 91.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{-16}:\\ \;\;\;\;\frac{{x}^{\left(\frac{2}{n \cdot 2}\right)}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-69}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right) - \log x}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 0.02:\\ \;\;\;\;\frac{{e}^{\left(\frac{\log x}{n}\right)}}{n \cdot x}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{x}{n}} - {x}^{\mathsf{reciprocal}\left(n\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 86.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{-16}:\\ \;\;\;\;\frac{{x}^{\left(\frac{2}{n \cdot 2}\right)}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-69}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right) - \log x}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 0.02:\\ \;\;\;\;\frac{\sqrt[3]{{x}^{\left(\frac{3}{n}\right)}}}{n \cdot x}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{x}{n}} - {x}^{\mathsf{reciprocal}\left(n\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= (/ 1.0 n) -2e-16)
   (/ (pow x (/ 2.0 (* n 2.0))) (* n x))
   (if (<= (/ 1.0 n) 2e-69)
     (/ (- (log1p x) (log x)) n)
     (if (<= (/ 1.0 n) 0.02)
       (/ (cbrt (pow x (/ 3.0 n))) (* n x))
       (- (exp (/ x n)) (pow x (reciprocal n)))))))

Derivation

1. Split input into 4 regimes

2. if (/.f64 1 n) < -2e-16

   1. Initial program 94.6%

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

      1. reciprocal-define 94.6%

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

      2. reciprocal-define 94.6%

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

   3. Simplified 94.6%

      \[\leadsto \color{blue}{{\left(x + 1\right)}^{\mathsf{reciprocal}\left(n\right)} - {x}^{\mathsf{reciprocal}\left(n\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf 99.9%

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

      1. mul-1-neg 99.9%

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

      2. log-rec 99.9%

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

      3. mul-1-neg 99.9%

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

      4. distribute-neg-frac 99.9%

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

      5. mul-1-neg 99.9%

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

      6. remove-double-neg 99.9%

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

   7. Simplified 99.9%

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

      1. expm1-log1p-u 100.0%

         \[\leadsto \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(e^{\frac{\log x}{n}}\right)\right)}}{n \cdot x} \]

      2. expm1-udef 100.0%

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

      3. log1p-udef 99.9%

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

      4. add-exp-log 99.9%

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

      5. div-inv 99.9%

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

      6. reciprocal-undefine 99.0%

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

      7. pow-to-exp 99.0%

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

   9. Applied egg-rr 99.0%

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

       1. add-exp-log 99.0%

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

       2. log1p-udef 99.0%

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

       3. expm1-udef 99.0%

          \[\leadsto \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({x}^{\mathsf{reciprocal}\left(n\right)}\right)\right)}}{n \cdot x} \]

       4. expm1-log1p-u 99.0%

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

       5. sqr-pow 99.0%

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

       6. pow-prod-up 99.0%

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

       7. reciprocal-undefine 99.0%

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

       8. associate-/l/ 99.0%

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

       9. reciprocal-define 99.0%

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

       10. reciprocal-undefine 99.0%

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

       11. associate-/l/ 99.0%

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

       12. reciprocal-define 99.0%

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

   11. Applied egg-rr 99.0%

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

       1. count-2 99.0%

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

       2. reciprocal-undefine 99.9%

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

       3. associate-*r/ 99.9%

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

       4. metadata-eval 99.9%

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

       5. *-commutative 99.9%

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

   13. Simplified 99.9%

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

   if -2e-16 < (/.f64 1 n) < 1.9999999999999999e-69

   1. Initial program 33.5%

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

      1. reciprocal-define 33.5%

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

      2. reciprocal-define 33.5%

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

   3. Simplified 33.5%

      \[\leadsto \color{blue}{{\left(x + 1\right)}^{\mathsf{reciprocal}\left(n\right)} - {x}^{\mathsf{reciprocal}\left(n\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in n around inf 86.1%

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

      1. +-rgt-identity 86.1%

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

      2. +-rgt-identity 86.1%

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

      3. log1p-def 86.1%

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

   7. Simplified 86.1%

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

   if 1.9999999999999999e-69 < (/.f64 1 n) < 0.0200000000000000004

   1. Initial program 23.2%

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

      1. reciprocal-define 22.7%

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

      2. reciprocal-define 12.8%

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

   3. Simplified 12.8%

      \[\leadsto \color{blue}{{\left(x + 1\right)}^{\mathsf{reciprocal}\left(n\right)} - {x}^{\mathsf{reciprocal}\left(n\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf 67.8%

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

      1. mul-1-neg 67.8%

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

      2. log-rec 67.8%

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

      3. mul-1-neg 67.8%

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

      4. distribute-neg-frac 67.8%

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

      5. mul-1-neg 67.8%

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

      6. remove-double-neg 67.8%

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

   7. Simplified 67.8%

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

      1. expm1-log1p-u 67.8%

         \[\leadsto \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(e^{\frac{\log x}{n}}\right)\right)}}{n \cdot x} \]

      2. expm1-udef 67.8%

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

      3. log1p-udef 67.8%

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

      4. add-exp-log 67.8%

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

      5. div-inv 67.8%

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

      6. reciprocal-undefine 60.7%

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

      7. pow-to-exp 60.7%

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

   9. Applied egg-rr 60.7%

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

       1. add-exp-log 60.7%

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

       2. log1p-udef 60.7%

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

       3. expm1-udef 60.7%

          \[\leadsto \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({x}^{\mathsf{reciprocal}\left(n\right)}\right)\right)}}{n \cdot x} \]

       4. expm1-log1p-u 60.7%

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

       5. add-cbrt-cube 60.7%

          \[\leadsto \frac{\color{blue}{\sqrt[3]{\left({x}^{\mathsf{reciprocal}\left(n\right)} \cdot {x}^{\mathsf{reciprocal}\left(n\right)}\right) \cdot {x}^{\mathsf{reciprocal}\left(n\right)}}}}{n \cdot x} \]

       6. pow1/3 60.7%

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

       7. pow3 60.7%

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

       8. pow-pow 60.7%

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

   11. Applied egg-rr 60.7%

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

       1. unpow1/3 60.7%

          \[\leadsto \frac{\color{blue}{\sqrt[3]{{x}^{\left(\mathsf{reciprocal}\left(n\right) \cdot 3\right)}}}}{n \cdot x} \]

       2. *-commutative 60.7%

          \[\leadsto \frac{\sqrt[3]{{x}^{\color{blue}{\left(3 \cdot \mathsf{reciprocal}\left(n\right)\right)}}}}{n \cdot x} \]

       3. reciprocal-define 67.9%

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

       4. associate-*r/ 67.9%

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

       5. metadata-eval 67.9%

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

   13. Simplified 67.9%

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

   if 0.0200000000000000004 < (/.f64 1 n)

   1. Initial program 58.6%

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

      1. reciprocal-define 58.6%

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

      2. reciprocal-define 58.6%

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

   3. Simplified 58.6%

      \[\leadsto \color{blue}{{\left(x + 1\right)}^{\mathsf{reciprocal}\left(n\right)} - {x}^{\mathsf{reciprocal}\left(n\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in n around 0 58.6%

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

      1. log1p-def 99.8%

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

   7. Simplified 99.8%

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

   8. Taylor expanded in x around 0 99.8%

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

4. Final simplification 91.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{-16}:\\ \;\;\;\;\frac{{x}^{\left(\frac{2}{n \cdot 2}\right)}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-69}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right) - \log x}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 0.02:\\ \;\;\;\;\frac{\sqrt[3]{{x}^{\left(\frac{3}{n}\right)}}}{n \cdot x}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{x}{n}} - {x}^{\mathsf{reciprocal}\left(n\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 86.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{{x}^{\left(\frac{2}{n \cdot 2}\right)}}{n \cdot x}\\ \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{-16}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-69}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right) - \log x}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 0.02:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{x}{n}} - {x}^{\mathsf{reciprocal}\left(n\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (/ (pow x (/ 2.0 (* n 2.0))) (* n x))))
   (if (<= (/ 1.0 n) -2e-16)
     t_0
     (if (<= (/ 1.0 n) 2e-69)
       (/ (- (log1p x) (log x)) n)
       (if (<= (/ 1.0 n) 0.02)
         t_0
         (- (exp (/ x n)) (pow x (reciprocal n))))))))

Derivation

1. Split input into 3 regimes

2. if (/.f64 1 n) < -2e-16 or 1.9999999999999999e-69 < (/.f64 1 n) < 0.0200000000000000004

   1. Initial program 82.2%

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

      1. reciprocal-define 82.1%

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

      2. reciprocal-define 80.4%

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

   3. Simplified 80.4%

      \[\leadsto \color{blue}{{\left(x + 1\right)}^{\mathsf{reciprocal}\left(n\right)} - {x}^{\mathsf{reciprocal}\left(n\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf 94.3%

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

      1. mul-1-neg 94.3%

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

      2. log-rec 94.3%

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

      3. mul-1-neg 94.3%

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

      4. distribute-neg-frac 94.3%

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

      5. mul-1-neg 94.3%

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

      6. remove-double-neg 94.3%

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

   7. Simplified 94.3%

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

      1. expm1-log1p-u 94.3%

         \[\leadsto \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(e^{\frac{\log x}{n}}\right)\right)}}{n \cdot x} \]

      2. expm1-udef 94.3%

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

      3. log1p-udef 94.3%

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

      4. add-exp-log 94.3%

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

      5. div-inv 94.3%

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

      6. reciprocal-undefine 92.3%

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

      7. pow-to-exp 92.3%

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

   9. Applied egg-rr 92.3%

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

       1. add-exp-log 92.3%

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

       2. log1p-udef 92.3%

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

       3. expm1-udef 92.3%

          \[\leadsto \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({x}^{\mathsf{reciprocal}\left(n\right)}\right)\right)}}{n \cdot x} \]

       4. expm1-log1p-u 92.3%

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

       5. sqr-pow 92.3%

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

       6. pow-prod-up 92.3%

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

       7. reciprocal-undefine 92.4%

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

       8. associate-/l/ 92.4%

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

       9. reciprocal-define 92.3%

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

       10. reciprocal-undefine 92.4%

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

       11. associate-/l/ 92.4%

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

       12. reciprocal-define 92.3%

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

   11. Applied egg-rr 92.3%

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

       1. count-2 92.3%

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

       2. reciprocal-undefine 94.3%

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

       3. associate-*r/ 94.3%

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

       4. metadata-eval 94.3%

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

       5. *-commutative 94.3%

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

   13. Simplified 94.3%

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

   if -2e-16 < (/.f64 1 n) < 1.9999999999999999e-69

   1. Initial program 33.5%

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

      1. reciprocal-define 33.5%

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

      2. reciprocal-define 33.5%

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

   3. Simplified 33.5%

      \[\leadsto \color{blue}{{\left(x + 1\right)}^{\mathsf{reciprocal}\left(n\right)} - {x}^{\mathsf{reciprocal}\left(n\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in n around inf 86.1%

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

      1. +-rgt-identity 86.1%

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

      2. +-rgt-identity 86.1%

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

      3. log1p-def 86.1%

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

   7. Simplified 86.1%

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

   if 0.0200000000000000004 < (/.f64 1 n)

   1. Initial program 58.6%

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

      1. reciprocal-define 58.6%

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

      2. reciprocal-define 58.6%

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

   3. Simplified 58.6%

      \[\leadsto \color{blue}{{\left(x + 1\right)}^{\mathsf{reciprocal}\left(n\right)} - {x}^{\mathsf{reciprocal}\left(n\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in n around 0 58.6%

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

      1. log1p-def 99.8%

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

   7. Simplified 99.8%

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

   8. Taylor expanded in x around 0 99.8%

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

4. Final simplification 91.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{-16}:\\ \;\;\;\;\frac{{x}^{\left(\frac{2}{n \cdot 2}\right)}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-69}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right) - \log x}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 0.02:\\ \;\;\;\;\frac{{x}^{\left(\frac{2}{n \cdot 2}\right)}}{n \cdot x}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{x}{n}} - {x}^{\mathsf{reciprocal}\left(n\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 82.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{-16}:\\ \;\;\;\;\frac{{x}^{\left(\frac{2}{n \cdot 2}\right)}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-69}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right) - \log x}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-21}:\\ \;\;\;\;{\left(n \cdot \left(x + 0.5\right)\right)}^{-1}\\ \mathbf{elif}\;\frac{1}{n} \leq 4 \cdot 10^{+149}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{reciprocal}\left(\left({x}^{\mathsf{reciprocal}\left(n\right)}\right)\right)}{n \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= (/ 1.0 n) -2e-16)
   (/ (pow x (/ 2.0 (* n 2.0))) (* n x))
   (if (<= (/ 1.0 n) 2e-69)
     (/ (- (log1p x) (log x)) n)
     (if (<= (/ 1.0 n) 5e-21)
       (pow (* n (+ x 0.5)) -1.0)
       (if (<= (/ 1.0 n) 4e+149)
         (- 1.0 (pow x (/ 1.0 n)))
         (/ (reciprocal (pow x (reciprocal n))) (* n x)))))))

Derivation

1. Split input into 5 regimes

2. if (/.f64 1 n) < -2e-16

   1. Initial program 94.6%

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

      1. reciprocal-define 94.6%

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

      2. reciprocal-define 94.6%

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

   3. Simplified 94.6%

      \[\leadsto \color{blue}{{\left(x + 1\right)}^{\mathsf{reciprocal}\left(n\right)} - {x}^{\mathsf{reciprocal}\left(n\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf 99.9%

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

      1. mul-1-neg 99.9%

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

      2. log-rec 99.9%

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

      3. mul-1-neg 99.9%

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

      4. distribute-neg-frac 99.9%

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

      5. mul-1-neg 99.9%

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

      6. remove-double-neg 99.9%

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

   7. Simplified 99.9%

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

      1. expm1-log1p-u 100.0%

         \[\leadsto \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(e^{\frac{\log x}{n}}\right)\right)}}{n \cdot x} \]

      2. expm1-udef 100.0%

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

      3. log1p-udef 99.9%

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

      4. add-exp-log 99.9%

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

      5. div-inv 99.9%

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

      6. reciprocal-undefine 99.0%

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

      7. pow-to-exp 99.0%

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

   9. Applied egg-rr 99.0%

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

       1. add-exp-log 99.0%

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

       2. log1p-udef 99.0%

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

       3. expm1-udef 99.0%

          \[\leadsto \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({x}^{\mathsf{reciprocal}\left(n\right)}\right)\right)}}{n \cdot x} \]

       4. expm1-log1p-u 99.0%

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

       5. sqr-pow 99.0%

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

       6. pow-prod-up 99.0%

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

       7. reciprocal-undefine 99.0%

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

       8. associate-/l/ 99.0%

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

       9. reciprocal-define 99.0%

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

       10. reciprocal-undefine 99.0%

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

       11. associate-/l/ 99.0%

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

       12. reciprocal-define 99.0%

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

   11. Applied egg-rr 99.0%

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

       1. count-2 99.0%

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

       2. reciprocal-undefine 99.9%

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

       3. associate-*r/ 99.9%

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

       4. metadata-eval 99.9%

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

       5. *-commutative 99.9%

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

   13. Simplified 99.9%

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

   if -2e-16 < (/.f64 1 n) < 1.9999999999999999e-69

   1. Initial program 33.5%

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

      1. reciprocal-define 33.5%

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

      2. reciprocal-define 33.5%

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

   3. Simplified 33.5%

      \[\leadsto \color{blue}{{\left(x + 1\right)}^{\mathsf{reciprocal}\left(n\right)} - {x}^{\mathsf{reciprocal}\left(n\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in n around inf 86.1%

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

      1. +-rgt-identity 86.1%

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

      2. +-rgt-identity 86.1%

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

      3. log1p-def 86.1%

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

   7. Simplified 86.1%

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

   if 1.9999999999999999e-69 < (/.f64 1 n) < 4.99999999999999973e-21

   1. Initial program 6.4%

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

      1. reciprocal-define 6.4%

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

      2. reciprocal-define 6.4%

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

   3. Simplified 6.4%

      \[\leadsto \color{blue}{{\left(x + 1\right)}^{\mathsf{reciprocal}\left(n\right)} - {x}^{\mathsf{reciprocal}\left(n\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in n around inf 25.6%

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

      1. +-rgt-identity 25.6%

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

      2. +-rgt-identity 25.6%

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

      3. log1p-def 25.6%

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

   7. Simplified 25.6%

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

      1. clear-num 25.5%

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

      2. inv-pow 25.5%

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

   9. Applied egg-rr 25.5%

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

   10. Taylor expanded in x around inf 82.5%

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

       1. *-commutative 82.5%

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

       2. distribute-lft-out 82.5%

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

   12. Simplified 82.5%

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

   if 4.99999999999999973e-21 < (/.f64 1 n) < 4.0000000000000002e149

   1. Initial program 77.5%

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

      1. reciprocal-define 77.3%

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

      2. reciprocal-define 72.5%

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

   3. Simplified 72.5%

      \[\leadsto \color{blue}{{\left(x + 1\right)}^{\mathsf{reciprocal}\left(n\right)} - {x}^{\mathsf{reciprocal}\left(n\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 69.2%

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

   6. Taylor expanded in n around 0 74.2%

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

   if 4.0000000000000002e149 < (/.f64 1 n)

   1. Initial program 25.5%

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

      1. reciprocal-define 25.5%

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

      2. reciprocal-define 25.5%

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

   3. Simplified 25.5%

      \[\leadsto \color{blue}{{\left(x + 1\right)}^{\mathsf{reciprocal}\left(n\right)} - {x}^{\mathsf{reciprocal}\left(n\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf 0.3%

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

      1. mul-1-neg 0.3%

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

      2. log-rec 0.3%

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

      3. mul-1-neg 0.3%

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

      4. distribute-neg-frac 0.3%

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

      5. mul-1-neg 0.3%

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

      6. remove-double-neg 0.3%

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

   7. Simplified 0.3%

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

      1. add-sqr-sqrt 0.0%

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

      2. sqrt-unprod 78.5%

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

      3. sqr-neg 78.5%

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

      4. distribute-frac-neg 78.5%

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

      5. distribute-frac-neg 78.5%

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

      6. sqrt-unprod 78.5%

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

      7. add-sqr-sqrt 78.5%

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

      8. distribute-frac-neg 78.5%

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

      9. exp-neg 78.5%

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

      10. div-inv 78.5%

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

      11. reciprocal-undefine 78.5%

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

      12. pow-to-exp 78.5%

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

   9. Applied egg-rr 78.5%

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

       1. reciprocal-define 78.5%

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

   11. Simplified 78.5%

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

4. Final simplification 87.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{-16}:\\ \;\;\;\;\frac{{x}^{\left(\frac{2}{n \cdot 2}\right)}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-69}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right) - \log x}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-21}:\\ \;\;\;\;{\left(n \cdot \left(x + 0.5\right)\right)}^{-1}\\ \mathbf{elif}\;\frac{1}{n} \leq 4 \cdot 10^{+149}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{reciprocal}\left(\left({x}^{\mathsf{reciprocal}\left(n\right)}\right)\right)}{n \cdot x}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 52.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\frac{1}{x}}{n}\\ \mathbf{if}\;\frac{1}{n} \leq -5000:\\ \;\;\;\;1 - {x}^{\mathsf{reciprocal}\left(n\right)}\\ \mathbf{elif}\;\frac{1}{n} \leq -2 \cdot 10^{-134}:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq -5 \cdot 10^{-251}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-69}:\\ \;\;\;\;-\frac{\log x}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-21}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{+193}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{reciprocal}\left(\left(n \cdot x\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (/ (/ 1.0 x) n)))
   (if (<= (/ 1.0 n) -5000.0)
     (- 1.0 (pow x (reciprocal n)))
     (if (<= (/ 1.0 n) -2e-134)
       (/ (- x (log x)) n)
       (if (<= (/ 1.0 n) -5e-251)
         t_0
         (if (<= (/ 1.0 n) 2e-69)
           (- (/ (log x) n))
           (if (<= (/ 1.0 n) 5e-21)
             t_0
             (if (<= (/ 1.0 n) 5e+193)
               (- 1.0 (pow x (/ 1.0 n)))
               (reciprocal (* n x))))))))))

Derivation

1. Split input into 6 regimes

2. if (/.f64 1 n) < -5e3

   1. Initial program 100.0%

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

      1. reciprocal-define 100.0%

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

      2. reciprocal-define 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{{\left(x + 1\right)}^{\mathsf{reciprocal}\left(n\right)} - {x}^{\mathsf{reciprocal}\left(n\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 56.6%

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

   if -5e3 < (/.f64 1 n) < -2.00000000000000008e-134

   1. Initial program 14.3%

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

      1. reciprocal-define 14.3%

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

      2. reciprocal-define 14.3%

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

   3. Simplified 14.3%

      \[\leadsto \color{blue}{{\left(x + 1\right)}^{\mathsf{reciprocal}\left(n\right)} - {x}^{\mathsf{reciprocal}\left(n\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in n around inf 73.6%

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

      1. +-rgt-identity 73.6%

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

      2. +-rgt-identity 73.6%

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

      3. log1p-def 73.7%

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

   7. Simplified 73.7%

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

   8. Taylor expanded in x around 0 59.7%

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

      1. neg-mul-1 59.7%

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

      2. sub-neg 59.7%

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

   10. Simplified 59.7%

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

   if -2.00000000000000008e-134 < (/.f64 1 n) < -5.0000000000000003e-251 or 1.9999999999999999e-69 < (/.f64 1 n) < 4.99999999999999973e-21

   1. Initial program 38.7%

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

      1. reciprocal-define 38.7%

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

      2. reciprocal-define 38.7%

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

   3. Simplified 38.7%

      \[\leadsto \color{blue}{{\left(x + 1\right)}^{\mathsf{reciprocal}\left(n\right)} - {x}^{\mathsf{reciprocal}\left(n\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in n around inf 65.8%

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

      1. +-rgt-identity 65.8%

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

      2. +-rgt-identity 65.8%

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

      3. log1p-def 65.9%

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

   7. Simplified 65.9%

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

   8. Taylor expanded in x around inf 73.1%

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

   if -5.0000000000000003e-251 < (/.f64 1 n) < 1.9999999999999999e-69

   1. Initial program 30.3%

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

      1. reciprocal-define 30.3%

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

      2. reciprocal-define 30.3%

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

   3. Simplified 30.3%

      \[\leadsto \color{blue}{{\left(x + 1\right)}^{\mathsf{reciprocal}\left(n\right)} - {x}^{\mathsf{reciprocal}\left(n\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 30.3%

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

   6. Taylor expanded in n around inf 64.7%

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

      1. neg-mul-1 64.7%

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

      2. distribute-neg-frac 64.7%

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

   8. Simplified 64.7%

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

   if 4.99999999999999973e-21 < (/.f64 1 n) < 4.99999999999999972e193

   1. Initial program 76.7%

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

      1. reciprocal-define 76.5%

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

      2. reciprocal-define 72.1%

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

   3. Simplified 72.1%

      \[\leadsto \color{blue}{{\left(x + 1\right)}^{\mathsf{reciprocal}\left(n\right)} - {x}^{\mathsf{reciprocal}\left(n\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 66.2%

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

   6. Taylor expanded in n around 0 70.8%

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

   if 4.99999999999999972e193 < (/.f64 1 n)

   1. Initial program 17.1%

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

      1. reciprocal-define 17.1%

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

      2. reciprocal-define 17.1%

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

   3. Simplified 17.1%

      \[\leadsto \color{blue}{{\left(x + 1\right)}^{\mathsf{reciprocal}\left(n\right)} - {x}^{\mathsf{reciprocal}\left(n\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf 0.1%

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

      1. mul-1-neg 0.1%

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

      2. log-rec 0.1%

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

      3. mul-1-neg 0.1%

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

      4. distribute-neg-frac 0.1%

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

      5. mul-1-neg 0.1%

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

      6. remove-double-neg 0.1%

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

   7. Simplified 0.1%

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

   8. Taylor expanded in n around inf 74.6%

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

      1. associate-/r* 74.6%

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

      2. reciprocal-define 74.6%

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

   10. Simplified 74.6%

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

       1. reciprocal-undefine 74.6%

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

       2. associate-/r* 74.6%

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

       3. reciprocal-define 74.6%

          \[\leadsto \color{blue}{\mathsf{reciprocal}\left(\left(n \cdot x\right)\right)} \]

       4. *-commutative 74.6%

          \[\leadsto \mathsf{reciprocal}\left(\color{blue}{\left(x \cdot n\right)}\right) \]

   12. Applied egg-rr 74.6%

       \[\leadsto \color{blue}{\mathsf{reciprocal}\left(\left(x \cdot n\right)\right)} \]
3. Recombined 6 regimes into one program.

4. Final simplification 65.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -5000:\\ \;\;\;\;1 - {x}^{\mathsf{reciprocal}\left(n\right)}\\ \mathbf{elif}\;\frac{1}{n} \leq -2 \cdot 10^{-134}:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq -5 \cdot 10^{-251}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-69}:\\ \;\;\;\;-\frac{\log x}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-21}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{+193}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{reciprocal}\left(\left(n \cdot x\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 53.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(n \cdot \left(x + 0.5\right)\right)}^{-1}\\ \mathbf{if}\;\frac{1}{n} \leq -5000:\\ \;\;\;\;1 - {x}^{\mathsf{reciprocal}\left(n\right)}\\ \mathbf{elif}\;\frac{1}{n} \leq -2 \cdot 10^{-134}:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq -2 \cdot 10^{-283}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-69}:\\ \;\;\;\;-\frac{\log x}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-21}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{+193}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{reciprocal}\left(\left(n \cdot x\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow (* n (+ x 0.5)) -1.0)))
   (if (<= (/ 1.0 n) -5000.0)
     (- 1.0 (pow x (reciprocal n)))
     (if (<= (/ 1.0 n) -2e-134)
       (/ (- x (log x)) n)
       (if (<= (/ 1.0 n) -2e-283)
         t_0
         (if (<= (/ 1.0 n) 2e-69)
           (- (/ (log x) n))
           (if (<= (/ 1.0 n) 5e-21)
             t_0
             (if (<= (/ 1.0 n) 5e+193)
               (- 1.0 (pow x (/ 1.0 n)))
               (reciprocal (* n x))))))))))

Derivation

1. Split input into 6 regimes

2. if (/.f64 1 n) < -5e3

   1. Initial program 100.0%

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

      1. reciprocal-define 100.0%

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

      2. reciprocal-define 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{{\left(x + 1\right)}^{\mathsf{reciprocal}\left(n\right)} - {x}^{\mathsf{reciprocal}\left(n\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 56.6%

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

   if -5e3 < (/.f64 1 n) < -2.00000000000000008e-134

   1. Initial program 14.3%

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

      1. reciprocal-define 14.3%

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

      2. reciprocal-define 14.3%

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

   3. Simplified 14.3%

      \[\leadsto \color{blue}{{\left(x + 1\right)}^{\mathsf{reciprocal}\left(n\right)} - {x}^{\mathsf{reciprocal}\left(n\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in n around inf 73.6%

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

      1. +-rgt-identity 73.6%

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

      2. +-rgt-identity 73.6%

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

      3. log1p-def 73.7%

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

   7. Simplified 73.7%

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

   8. Taylor expanded in x around 0 59.7%

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

      1. neg-mul-1 59.7%

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

      2. sub-neg 59.7%

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

   10. Simplified 59.7%

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

   if -2.00000000000000008e-134 < (/.f64 1 n) < -1.99999999999999989e-283 or 1.9999999999999999e-69 < (/.f64 1 n) < 4.99999999999999973e-21

   1. Initial program 40.9%

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

      1. reciprocal-define 40.9%

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

      2. reciprocal-define 40.9%

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

   3. Simplified 40.9%

      \[\leadsto \color{blue}{{\left(x + 1\right)}^{\mathsf{reciprocal}\left(n\right)} - {x}^{\mathsf{reciprocal}\left(n\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in n around inf 70.8%

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

      1. +-rgt-identity 70.8%

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

      2. +-rgt-identity 70.8%

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

      3. log1p-def 70.8%

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

   7. Simplified 70.8%

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

      1. clear-num 70.8%

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

      2. inv-pow 70.8%

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

   9. Applied egg-rr 70.8%

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

   10. Taylor expanded in x around inf 72.2%

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

       1. *-commutative 72.2%

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

       2. distribute-lft-out 72.2%

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

   12. Simplified 72.2%

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

   if -1.99999999999999989e-283 < (/.f64 1 n) < 1.9999999999999999e-69

   1. Initial program 27.4%

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

      1. reciprocal-define 27.4%

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

      2. reciprocal-define 27.4%

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

   3. Simplified 27.4%

      \[\leadsto \color{blue}{{\left(x + 1\right)}^{\mathsf{reciprocal}\left(n\right)} - {x}^{\mathsf{reciprocal}\left(n\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 27.4%

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

   6. Taylor expanded in n around inf 66.1%

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

      1. neg-mul-1 66.1%

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

      2. distribute-neg-frac 66.1%

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

   8. Simplified 66.1%

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

   if 4.99999999999999973e-21 < (/.f64 1 n) < 4.99999999999999972e193

   1. Initial program 76.7%

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

      1. reciprocal-define 76.5%

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

      2. reciprocal-define 72.1%

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

   3. Simplified 72.1%

      \[\leadsto \color{blue}{{\left(x + 1\right)}^{\mathsf{reciprocal}\left(n\right)} - {x}^{\mathsf{reciprocal}\left(n\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 66.2%

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

   6. Taylor expanded in n around 0 70.8%

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

   if 4.99999999999999972e193 < (/.f64 1 n)

   1. Initial program 17.1%

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

      1. reciprocal-define 17.1%

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

      2. reciprocal-define 17.1%

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

   3. Simplified 17.1%

      \[\leadsto \color{blue}{{\left(x + 1\right)}^{\mathsf{reciprocal}\left(n\right)} - {x}^{\mathsf{reciprocal}\left(n\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf 0.1%

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

      1. mul-1-neg 0.1%

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

      2. log-rec 0.1%

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

      3. mul-1-neg 0.1%

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

      4. distribute-neg-frac 0.1%

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

      5. mul-1-neg 0.1%

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

      6. remove-double-neg 0.1%

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

   7. Simplified 0.1%

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

   8. Taylor expanded in n around inf 74.6%

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

      1. associate-/r* 74.6%

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

      2. reciprocal-define 74.6%

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

   10. Simplified 74.6%

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

       1. reciprocal-undefine 74.6%

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

       2. associate-/r* 74.6%

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

       3. reciprocal-define 74.6%

          \[\leadsto \color{blue}{\mathsf{reciprocal}\left(\left(n \cdot x\right)\right)} \]

       4. *-commutative 74.6%

          \[\leadsto \mathsf{reciprocal}\left(\color{blue}{\left(x \cdot n\right)}\right) \]

   12. Applied egg-rr 74.6%

       \[\leadsto \color{blue}{\mathsf{reciprocal}\left(\left(x \cdot n\right)\right)} \]
3. Recombined 6 regimes into one program.

4. Final simplification 65.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -5000:\\ \;\;\;\;1 - {x}^{\mathsf{reciprocal}\left(n\right)}\\ \mathbf{elif}\;\frac{1}{n} \leq -2 \cdot 10^{-134}:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq -2 \cdot 10^{-283}:\\ \;\;\;\;{\left(n \cdot \left(x + 0.5\right)\right)}^{-1}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-69}:\\ \;\;\;\;-\frac{\log x}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-21}:\\ \;\;\;\;{\left(n \cdot \left(x + 0.5\right)\right)}^{-1}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{+193}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{reciprocal}\left(\left(n \cdot x\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 67.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(n \cdot \left(x + 0.5\right)\right)}^{-1}\\ \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{-16}:\\ \;\;\;\;\frac{{x}^{\mathsf{reciprocal}\left(n\right)}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq -2 \cdot 10^{-134}:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq -2 \cdot 10^{-283}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-69}:\\ \;\;\;\;-\frac{\log x}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-21}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{+193}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{reciprocal}\left(\left(n \cdot x\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow (* n (+ x 0.5)) -1.0)))
   (if (<= (/ 1.0 n) -2e-16)
     (/ (pow x (reciprocal n)) (* n x))
     (if (<= (/ 1.0 n) -2e-134)
       (/ (- x (log x)) n)
       (if (<= (/ 1.0 n) -2e-283)
         t_0
         (if (<= (/ 1.0 n) 2e-69)
           (- (/ (log x) n))
           (if (<= (/ 1.0 n) 5e-21)
             t_0
             (if (<= (/ 1.0 n) 5e+193)
               (- 1.0 (pow x (/ 1.0 n)))
               (reciprocal (* n x))))))))))

Derivation

1. Split input into 6 regimes

2. if (/.f64 1 n) < -2e-16

   1. Initial program 94.6%

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

      1. reciprocal-define 94.6%

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

      2. reciprocal-define 94.6%

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

   3. Simplified 94.6%

      \[\leadsto \color{blue}{{\left(x + 1\right)}^{\mathsf{reciprocal}\left(n\right)} - {x}^{\mathsf{reciprocal}\left(n\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf 99.9%

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

      1. mul-1-neg 99.9%

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

      2. log-rec 99.9%

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

      3. mul-1-neg 99.9%

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

      4. distribute-neg-frac 99.9%

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

      5. mul-1-neg 99.9%

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

      6. remove-double-neg 99.9%

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

   7. Simplified 99.9%

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

      1. *-un-lft-identity 99.9%

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

      2. times-frac 100.0%

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

      3. remove-double-neg 100.0%

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

      4. distribute-frac-neg 100.0%

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

      5. reciprocal-undefine 96.6%

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

      6. distribute-frac-neg 96.6%

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

      7. remove-double-neg 96.6%

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

      8. div-inv 96.6%

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

      9. reciprocal-undefine 96.7%

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

      10. pow-to-exp 96.7%

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

   9. Applied egg-rr 96.7%

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

       1. reciprocal-define 99.0%

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

       2. times-frac 99.0%

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

       3. *-lft-identity 99.0%

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

       4. *-commutative 99.0%

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

   11. Simplified 99.0%

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

   if -2e-16 < (/.f64 1 n) < -2.00000000000000008e-134

   1. Initial program 16.7%

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

      1. reciprocal-define 16.7%

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

      2. reciprocal-define 16.7%

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

   3. Simplified 16.7%

      \[\leadsto \color{blue}{{\left(x + 1\right)}^{\mathsf{reciprocal}\left(n\right)} - {x}^{\mathsf{reciprocal}\left(n\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in n around inf 89.9%

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

      1. +-rgt-identity 89.9%

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

      2. +-rgt-identity 89.9%

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

      3. log1p-def 90.0%

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

   7. Simplified 90.0%

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

   8. Taylor expanded in x around 0 72.9%

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

      1. neg-mul-1 72.9%

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

      2. sub-neg 72.9%

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

   10. Simplified 72.9%

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

   if -2.00000000000000008e-134 < (/.f64 1 n) < -1.99999999999999989e-283 or 1.9999999999999999e-69 < (/.f64 1 n) < 4.99999999999999973e-21

   1. Initial program 40.9%

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

      1. reciprocal-define 40.9%

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

      2. reciprocal-define 40.9%

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

   3. Simplified 40.9%

      \[\leadsto \color{blue}{{\left(x + 1\right)}^{\mathsf{reciprocal}\left(n\right)} - {x}^{\mathsf{reciprocal}\left(n\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in n around inf 70.8%

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

      1. +-rgt-identity 70.8%

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

      2. +-rgt-identity 70.8%

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

      3. log1p-def 70.8%

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

   7. Simplified 70.8%

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

      1. clear-num 70.8%

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

      2. inv-pow 70.8%

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

   9. Applied egg-rr 70.8%

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

   10. Taylor expanded in x around inf 72.2%

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

       1. *-commutative 72.2%

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

       2. distribute-lft-out 72.2%

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

   12. Simplified 72.2%

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

   if -1.99999999999999989e-283 < (/.f64 1 n) < 1.9999999999999999e-69

   1. Initial program 27.4%

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

      1. reciprocal-define 27.4%

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

      2. reciprocal-define 27.4%

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

   3. Simplified 27.4%

      \[\leadsto \color{blue}{{\left(x + 1\right)}^{\mathsf{reciprocal}\left(n\right)} - {x}^{\mathsf{reciprocal}\left(n\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 27.4%

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

   6. Taylor expanded in n around inf 66.1%

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

      1. neg-mul-1 66.1%

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

      2. distribute-neg-frac 66.1%

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

   8. Simplified 66.1%

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

   if 4.99999999999999973e-21 < (/.f64 1 n) < 4.99999999999999972e193

   1. Initial program 76.7%

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

      1. reciprocal-define 76.5%

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

      2. reciprocal-define 72.1%

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

   3. Simplified 72.1%

      \[\leadsto \color{blue}{{\left(x + 1\right)}^{\mathsf{reciprocal}\left(n\right)} - {x}^{\mathsf{reciprocal}\left(n\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 66.2%

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

   6. Taylor expanded in n around 0 70.8%

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

   if 4.99999999999999972e193 < (/.f64 1 n)

   1. Initial program 17.1%

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

      1. reciprocal-define 17.1%

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

      2. reciprocal-define 17.1%

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

   3. Simplified 17.1%

      \[\leadsto \color{blue}{{\left(x + 1\right)}^{\mathsf{reciprocal}\left(n\right)} - {x}^{\mathsf{reciprocal}\left(n\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf 0.1%

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

      1. mul-1-neg 0.1%

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

      2. log-rec 0.1%

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

      3. mul-1-neg 0.1%

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

      4. distribute-neg-frac 0.1%

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

      5. mul-1-neg 0.1%

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

      6. remove-double-neg 0.1%

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

   7. Simplified 0.1%

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

   8. Taylor expanded in n around inf 74.6%

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

      1. associate-/r* 74.6%

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

      2. reciprocal-define 74.6%

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

   10. Simplified 74.6%

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

       1. reciprocal-undefine 74.6%

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

       2. associate-/r* 74.6%

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

       3. reciprocal-define 74.6%

          \[\leadsto \color{blue}{\mathsf{reciprocal}\left(\left(n \cdot x\right)\right)} \]

       4. *-commutative 74.6%

          \[\leadsto \mathsf{reciprocal}\left(\color{blue}{\left(x \cdot n\right)}\right) \]

   12. Applied egg-rr 74.6%

       \[\leadsto \color{blue}{\mathsf{reciprocal}\left(\left(x \cdot n\right)\right)} \]
3. Recombined 6 regimes into one program.

4. Final simplification 78.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{-16}:\\ \;\;\;\;\frac{{x}^{\mathsf{reciprocal}\left(n\right)}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq -2 \cdot 10^{-134}:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq -2 \cdot 10^{-283}:\\ \;\;\;\;{\left(n \cdot \left(x + 0.5\right)\right)}^{-1}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-69}:\\ \;\;\;\;-\frac{\log x}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-21}:\\ \;\;\;\;{\left(n \cdot \left(x + 0.5\right)\right)}^{-1}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{+193}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{reciprocal}\left(\left(n \cdot x\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 52.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\frac{1}{x}}{n}\\ t_1 := 1 - {x}^{\mathsf{reciprocal}\left(n\right)}\\ t_2 := -\frac{\log x}{n}\\ \mathbf{if}\;\frac{1}{n} \leq -0.0005:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\frac{1}{n} \leq -2 \cdot 10^{-134}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\frac{1}{n} \leq -5 \cdot 10^{-251}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-69}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-21}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{+193}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{reciprocal}\left(\left(n \cdot x\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (/ (/ 1.0 x) n))
        (t_1 (- 1.0 (pow x (reciprocal n))))
        (t_2 (- (/ (log x) n))))
   (if (<= (/ 1.0 n) -0.0005)
     t_1
     (if (<= (/ 1.0 n) -2e-134)
       t_2
       (if (<= (/ 1.0 n) -5e-251)
         t_0
         (if (<= (/ 1.0 n) 2e-69)
           t_2
           (if (<= (/ 1.0 n) 5e-21)
             t_0
             (if (<= (/ 1.0 n) 5e+193) t_1 (reciprocal (* n x))))))))))

Derivation

1. Split input into 4 regimes

2. if (/.f64 1 n) < -5.0000000000000001e-4 or 4.99999999999999973e-21 < (/.f64 1 n) < 4.99999999999999972e193

   1. Initial program 91.3%

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

      1. reciprocal-define 91.2%

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

      2. reciprocal-define 89.8%

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

   3. Simplified 89.8%

      \[\leadsto \color{blue}{{\left(x + 1\right)}^{\mathsf{reciprocal}\left(n\right)} - {x}^{\mathsf{reciprocal}\left(n\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 59.3%

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

   if -5.0000000000000001e-4 < (/.f64 1 n) < -2.00000000000000008e-134 or -5.0000000000000003e-251 < (/.f64 1 n) < 1.9999999999999999e-69

   1. Initial program 27.0%

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

      1. reciprocal-define 27.0%

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

      2. reciprocal-define 27.0%

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

   3. Simplified 27.0%

      \[\leadsto \color{blue}{{\left(x + 1\right)}^{\mathsf{reciprocal}\left(n\right)} - {x}^{\mathsf{reciprocal}\left(n\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 26.9%

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

   6. Taylor expanded in n around inf 64.1%

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

      1. neg-mul-1 64.1%

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

      2. distribute-neg-frac 64.1%

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

   8. Simplified 64.1%

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

   if -2.00000000000000008e-134 < (/.f64 1 n) < -5.0000000000000003e-251 or 1.9999999999999999e-69 < (/.f64 1 n) < 4.99999999999999973e-21

   1. Initial program 38.7%

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

      1. reciprocal-define 38.7%

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

      2. reciprocal-define 38.7%

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

   3. Simplified 38.7%

      \[\leadsto \color{blue}{{\left(x + 1\right)}^{\mathsf{reciprocal}\left(n\right)} - {x}^{\mathsf{reciprocal}\left(n\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in n around inf 65.8%

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

      1. +-rgt-identity 65.8%

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

      2. +-rgt-identity 65.8%

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

      3. log1p-def 65.9%

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

   7. Simplified 65.9%

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

   8. Taylor expanded in x around inf 73.1%

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

   if 4.99999999999999972e193 < (/.f64 1 n)

   1. Initial program 17.1%

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

      1. reciprocal-define 17.1%

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

      2. reciprocal-define 17.1%

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

   3. Simplified 17.1%

      \[\leadsto \color{blue}{{\left(x + 1\right)}^{\mathsf{reciprocal}\left(n\right)} - {x}^{\mathsf{reciprocal}\left(n\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf 0.1%

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

      1. mul-1-neg 0.1%

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

      2. log-rec 0.1%

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

      3. mul-1-neg 0.1%

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

      4. distribute-neg-frac 0.1%

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

      5. mul-1-neg 0.1%

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

      6. remove-double-neg 0.1%

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

   7. Simplified 0.1%

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

   8. Taylor expanded in n around inf 74.6%

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

      1. associate-/r* 74.6%

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

      2. reciprocal-define 74.6%

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

   10. Simplified 74.6%

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

       1. reciprocal-undefine 74.6%

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

       2. associate-/r* 74.6%

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

       3. reciprocal-define 74.6%

          \[\leadsto \color{blue}{\mathsf{reciprocal}\left(\left(n \cdot x\right)\right)} \]

       4. *-commutative 74.6%

          \[\leadsto \mathsf{reciprocal}\left(\color{blue}{\left(x \cdot n\right)}\right) \]

   12. Applied egg-rr 74.6%

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

4. Final simplification 64.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -0.0005:\\ \;\;\;\;1 - {x}^{\mathsf{reciprocal}\left(n\right)}\\ \mathbf{elif}\;\frac{1}{n} \leq -2 \cdot 10^{-134}:\\ \;\;\;\;-\frac{\log x}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq -5 \cdot 10^{-251}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-69}:\\ \;\;\;\;-\frac{\log x}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-21}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{+193}:\\ \;\;\;\;1 - {x}^{\mathsf{reciprocal}\left(n\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{reciprocal}\left(\left(n \cdot x\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 52.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\frac{1}{x}}{n}\\ t_1 := 1 - {x}^{\mathsf{reciprocal}\left(n\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -5000:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\frac{1}{n} \leq -2 \cdot 10^{-134}:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq -5 \cdot 10^{-251}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-69}:\\ \;\;\;\;-\frac{\log x}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-21}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{+193}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{reciprocal}\left(\left(n \cdot x\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (/ (/ 1.0 x) n)) (t_1 (- 1.0 (pow x (reciprocal n)))))
   (if (<= (/ 1.0 n) -5000.0)
     t_1
     (if (<= (/ 1.0 n) -2e-134)
       (/ (- x (log x)) n)
       (if (<= (/ 1.0 n) -5e-251)
         t_0
         (if (<= (/ 1.0 n) 2e-69)
           (- (/ (log x) n))
           (if (<= (/ 1.0 n) 5e-21)
             t_0
             (if (<= (/ 1.0 n) 5e+193) t_1 (reciprocal (* n x))))))))))

Derivation

1. Split input into 5 regimes

2. if (/.f64 1 n) < -5e3 or 4.99999999999999973e-21 < (/.f64 1 n) < 4.99999999999999972e193

   1. Initial program 92.1%

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

      1. reciprocal-define 92.1%

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

      2. reciprocal-define 90.6%

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

   3. Simplified 90.6%

      \[\leadsto \color{blue}{{\left(x + 1\right)}^{\mathsf{reciprocal}\left(n\right)} - {x}^{\mathsf{reciprocal}\left(n\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 59.8%

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

   if -5e3 < (/.f64 1 n) < -2.00000000000000008e-134

   1. Initial program 14.3%

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

      1. reciprocal-define 14.3%

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

      2. reciprocal-define 14.3%

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

   3. Simplified 14.3%

      \[\leadsto \color{blue}{{\left(x + 1\right)}^{\mathsf{reciprocal}\left(n\right)} - {x}^{\mathsf{reciprocal}\left(n\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in n around inf 73.6%

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

      1. +-rgt-identity 73.6%

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

      2. +-rgt-identity 73.6%

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

      3. log1p-def 73.7%

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

   7. Simplified 73.7%

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

   8. Taylor expanded in x around 0 59.7%

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

      1. neg-mul-1 59.7%

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

      2. sub-neg 59.7%

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

   10. Simplified 59.7%

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

   if -2.00000000000000008e-134 < (/.f64 1 n) < -5.0000000000000003e-251 or 1.9999999999999999e-69 < (/.f64 1 n) < 4.99999999999999973e-21

   1. Initial program 38.7%

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

      1. reciprocal-define 38.7%

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

      2. reciprocal-define 38.7%

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

   3. Simplified 38.7%

      \[\leadsto \color{blue}{{\left(x + 1\right)}^{\mathsf{reciprocal}\left(n\right)} - {x}^{\mathsf{reciprocal}\left(n\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in n around inf 65.8%

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

      1. +-rgt-identity 65.8%

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

      2. +-rgt-identity 65.8%

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

      3. log1p-def 65.9%

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

   7. Simplified 65.9%

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

   8. Taylor expanded in x around inf 73.1%

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

   if -5.0000000000000003e-251 < (/.f64 1 n) < 1.9999999999999999e-69

   1. Initial program 30.3%

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

      1. reciprocal-define 30.3%

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

      2. reciprocal-define 30.3%

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

   3. Simplified 30.3%

      \[\leadsto \color{blue}{{\left(x + 1\right)}^{\mathsf{reciprocal}\left(n\right)} - {x}^{\mathsf{reciprocal}\left(n\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 30.3%

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

   6. Taylor expanded in n around inf 64.7%

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

      1. neg-mul-1 64.7%

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

      2. distribute-neg-frac 64.7%

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

   8. Simplified 64.7%

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

   if 4.99999999999999972e193 < (/.f64 1 n)

   1. Initial program 17.1%

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

      1. reciprocal-define 17.1%

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

      2. reciprocal-define 17.1%

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

   3. Simplified 17.1%

      \[\leadsto \color{blue}{{\left(x + 1\right)}^{\mathsf{reciprocal}\left(n\right)} - {x}^{\mathsf{reciprocal}\left(n\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf 0.1%

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

      1. mul-1-neg 0.1%

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

      2. log-rec 0.1%

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

      3. mul-1-neg 0.1%

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

      4. distribute-neg-frac 0.1%

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

      5. mul-1-neg 0.1%

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

      6. remove-double-neg 0.1%

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

   7. Simplified 0.1%

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

   8. Taylor expanded in n around inf 74.6%

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

      1. associate-/r* 74.6%

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

      2. reciprocal-define 74.6%

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

   10. Simplified 74.6%

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

       1. reciprocal-undefine 74.6%

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

       2. associate-/r* 74.6%

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

       3. reciprocal-define 74.6%

          \[\leadsto \color{blue}{\mathsf{reciprocal}\left(\left(n \cdot x\right)\right)} \]

       4. *-commutative 74.6%

          \[\leadsto \mathsf{reciprocal}\left(\color{blue}{\left(x \cdot n\right)}\right) \]

   12. Applied egg-rr 74.6%

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

4. Final simplification 64.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -5000:\\ \;\;\;\;1 - {x}^{\mathsf{reciprocal}\left(n\right)}\\ \mathbf{elif}\;\frac{1}{n} \leq -2 \cdot 10^{-134}:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq -5 \cdot 10^{-251}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-69}:\\ \;\;\;\;-\frac{\log x}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-21}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{+193}:\\ \;\;\;\;1 - {x}^{\mathsf{reciprocal}\left(n\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{reciprocal}\left(\left(n \cdot x\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 82.1% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\mathsf{reciprocal}\left(n\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{-16}:\\ \;\;\;\;\frac{t_0}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-69}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-21}:\\ \;\;\;\;{\left(n \cdot \left(x + 0.5\right)\right)}^{-1}\\ \mathbf{elif}\;\frac{1}{n} \leq 4 \cdot 10^{+149}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{reciprocal}\left(t_0\right)}{n \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (reciprocal n))))
   (if (<= (/ 1.0 n) -2e-16)
     (/ t_0 (* n x))
     (if (<= (/ 1.0 n) 2e-69)
       (/ (log (/ (+ 1.0 x) x)) n)
       (if (<= (/ 1.0 n) 5e-21)
         (pow (* n (+ x 0.5)) -1.0)
         (if (<= (/ 1.0 n) 4e+149)
           (- 1.0 (pow x (/ 1.0 n)))
           (/ (reciprocal t_0) (* n x))))))))

Derivation

1. Split input into 5 regimes

2. if (/.f64 1 n) < -2e-16

   1. Initial program 94.6%

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

      1. reciprocal-define 94.6%

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

      2. reciprocal-define 94.6%

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

   3. Simplified 94.6%

      \[\leadsto \color{blue}{{\left(x + 1\right)}^{\mathsf{reciprocal}\left(n\right)} - {x}^{\mathsf{reciprocal}\left(n\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf 99.9%

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

      1. mul-1-neg 99.9%

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

      2. log-rec 99.9%

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

      3. mul-1-neg 99.9%

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

      4. distribute-neg-frac 99.9%

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

      5. mul-1-neg 99.9%

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

      6. remove-double-neg 99.9%

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

   7. Simplified 99.9%

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

      1. *-un-lft-identity 99.9%

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

      2. times-frac 100.0%

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

      3. remove-double-neg 100.0%

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

      4. distribute-frac-neg 100.0%

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

      5. reciprocal-undefine 96.6%

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

      6. distribute-frac-neg 96.6%

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

      7. remove-double-neg 96.6%

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

      8. div-inv 96.6%

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

      9. reciprocal-undefine 96.7%

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

      10. pow-to-exp 96.7%

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

   9. Applied egg-rr 96.7%

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

       1. reciprocal-define 99.0%

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

       2. times-frac 99.0%

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

       3. *-lft-identity 99.0%

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

       4. *-commutative 99.0%

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

   11. Simplified 99.0%

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

   if -2e-16 < (/.f64 1 n) < 1.9999999999999999e-69

   1. Initial program 33.5%

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

      1. reciprocal-define 33.5%

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

      2. reciprocal-define 33.5%

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

   3. Simplified 33.5%

      \[\leadsto \color{blue}{{\left(x + 1\right)}^{\mathsf{reciprocal}\left(n\right)} - {x}^{\mathsf{reciprocal}\left(n\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in n around inf 86.1%

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

      1. +-rgt-identity 86.1%

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

      2. +-rgt-identity 86.1%

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

      3. log1p-def 86.1%

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

   7. Simplified 86.1%

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

      1. log1p-udef 86.1%

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

      2. diff-log 86.1%

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

      3. +-commutative 86.1%

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

   9. Applied egg-rr 86.1%

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

   if 1.9999999999999999e-69 < (/.f64 1 n) < 4.99999999999999973e-21

   1. Initial program 6.4%

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

      1. reciprocal-define 6.4%

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

      2. reciprocal-define 6.4%

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

   3. Simplified 6.4%

      \[\leadsto \color{blue}{{\left(x + 1\right)}^{\mathsf{reciprocal}\left(n\right)} - {x}^{\mathsf{reciprocal}\left(n\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in n around inf 25.6%

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

      1. +-rgt-identity 25.6%

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

      2. +-rgt-identity 25.6%

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

      3. log1p-def 25.6%

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

   7. Simplified 25.6%

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

      1. clear-num 25.5%

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

      2. inv-pow 25.5%

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

   9. Applied egg-rr 25.5%

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

   10. Taylor expanded in x around inf 82.5%

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

       1. *-commutative 82.5%

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

       2. distribute-lft-out 82.5%

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

   12. Simplified 82.5%

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

   if 4.99999999999999973e-21 < (/.f64 1 n) < 4.0000000000000002e149

   1. Initial program 77.5%

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

      1. reciprocal-define 77.3%

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

      2. reciprocal-define 72.5%

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

   3. Simplified 72.5%

      \[\leadsto \color{blue}{{\left(x + 1\right)}^{\mathsf{reciprocal}\left(n\right)} - {x}^{\mathsf{reciprocal}\left(n\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 69.2%

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

   6. Taylor expanded in n around 0 74.2%

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

   if 4.0000000000000002e149 < (/.f64 1 n)

   1. Initial program 25.5%

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

      1. reciprocal-define 25.5%

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

      2. reciprocal-define 25.5%

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

   3. Simplified 25.5%

      \[\leadsto \color{blue}{{\left(x + 1\right)}^{\mathsf{reciprocal}\left(n\right)} - {x}^{\mathsf{reciprocal}\left(n\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf 0.3%

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

      1. mul-1-neg 0.3%

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

      2. log-rec 0.3%

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

      3. mul-1-neg 0.3%

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

      4. distribute-neg-frac 0.3%

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

      5. mul-1-neg 0.3%

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

      6. remove-double-neg 0.3%

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

   7. Simplified 0.3%

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

      1. add-sqr-sqrt 0.0%

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

      2. sqrt-unprod 78.5%

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

      3. sqr-neg 78.5%

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

      4. distribute-frac-neg 78.5%

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

      5. distribute-frac-neg 78.5%

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

      6. sqrt-unprod 78.5%

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

      7. add-sqr-sqrt 78.5%

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

      8. distribute-frac-neg 78.5%

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

      9. exp-neg 78.5%

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

      10. div-inv 78.5%

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

      11. reciprocal-undefine 78.5%

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

      12. pow-to-exp 78.5%

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

   9. Applied egg-rr 78.5%

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

       1. reciprocal-define 78.5%

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

   11. Simplified 78.5%

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

4. Final simplification 87.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{-16}:\\ \;\;\;\;\frac{{x}^{\mathsf{reciprocal}\left(n\right)}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-69}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-21}:\\ \;\;\;\;{\left(n \cdot \left(x + 0.5\right)\right)}^{-1}\\ \mathbf{elif}\;\frac{1}{n} \leq 4 \cdot 10^{+149}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{reciprocal}\left(\left({x}^{\mathsf{reciprocal}\left(n\right)}\right)\right)}{n \cdot x}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 82.3% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{-16}:\\ \;\;\;\;\frac{{x}^{\left(\frac{2}{n \cdot 2}\right)}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-69}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-21}:\\ \;\;\;\;{\left(n \cdot \left(x + 0.5\right)\right)}^{-1}\\ \mathbf{elif}\;\frac{1}{n} \leq 4 \cdot 10^{+149}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{reciprocal}\left(\left({x}^{\mathsf{reciprocal}\left(n\right)}\right)\right)}{n \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= (/ 1.0 n) -2e-16)
   (/ (pow x (/ 2.0 (* n 2.0))) (* n x))
   (if (<= (/ 1.0 n) 2e-69)
     (/ (log (/ (+ 1.0 x) x)) n)
     (if (<= (/ 1.0 n) 5e-21)
       (pow (* n (+ x 0.5)) -1.0)
       (if (<= (/ 1.0 n) 4e+149)
         (- 1.0 (pow x (/ 1.0 n)))
         (/ (reciprocal (pow x (reciprocal n))) (* n x)))))))

Derivation

1. Split input into 5 regimes

2. if (/.f64 1 n) < -2e-16

   1. Initial program 94.6%

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

      1. reciprocal-define 94.6%

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

      2. reciprocal-define 94.6%

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

   3. Simplified 94.6%

      \[\leadsto \color{blue}{{\left(x + 1\right)}^{\mathsf{reciprocal}\left(n\right)} - {x}^{\mathsf{reciprocal}\left(n\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf 99.9%

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

      1. mul-1-neg 99.9%

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

      2. log-rec 99.9%

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

      3. mul-1-neg 99.9%

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

      4. distribute-neg-frac 99.9%

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

      5. mul-1-neg 99.9%

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

      6. remove-double-neg 99.9%

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

   7. Simplified 99.9%

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

      1. expm1-log1p-u 100.0%

         \[\leadsto \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(e^{\frac{\log x}{n}}\right)\right)}}{n \cdot x} \]

      2. expm1-udef 100.0%

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

      3. log1p-udef 99.9%

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

      4. add-exp-log 99.9%

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

      5. div-inv 99.9%

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

      6. reciprocal-undefine 99.0%

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

      7. pow-to-exp 99.0%

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

   9. Applied egg-rr 99.0%

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

       1. add-exp-log 99.0%

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

       2. log1p-udef 99.0%

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

       3. expm1-udef 99.0%

          \[\leadsto \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({x}^{\mathsf{reciprocal}\left(n\right)}\right)\right)}}{n \cdot x} \]

       4. expm1-log1p-u 99.0%

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

       5. sqr-pow 99.0%

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

       6. pow-prod-up 99.0%

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

       7. reciprocal-undefine 99.0%

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

       8. associate-/l/ 99.0%

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

       9. reciprocal-define 99.0%

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

       10. reciprocal-undefine 99.0%

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

       11. associate-/l/ 99.0%

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

       12. reciprocal-define 99.0%

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

   11. Applied egg-rr 99.0%

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

       1. count-2 99.0%

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

       2. reciprocal-undefine 99.9%

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

       3. associate-*r/ 99.9%

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

       4. metadata-eval 99.9%

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

       5. *-commutative 99.9%

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

   13. Simplified 99.9%

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

   if -2e-16 < (/.f64 1 n) < 1.9999999999999999e-69

   1. Initial program 33.5%

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

      1. reciprocal-define 33.5%

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

      2. reciprocal-define 33.5%

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

   3. Simplified 33.5%

      \[\leadsto \color{blue}{{\left(x + 1\right)}^{\mathsf{reciprocal}\left(n\right)} - {x}^{\mathsf{reciprocal}\left(n\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in n around inf 86.1%

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

      1. +-rgt-identity 86.1%

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

      2. +-rgt-identity 86.1%

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

      3. log1p-def 86.1%

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

   7. Simplified 86.1%

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

      1. log1p-udef 86.1%

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

      2. diff-log 86.1%

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

      3. +-commutative 86.1%

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

   9. Applied egg-rr 86.1%

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

   if 1.9999999999999999e-69 < (/.f64 1 n) < 4.99999999999999973e-21

   1. Initial program 6.4%

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

      1. reciprocal-define 6.4%

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

      2. reciprocal-define 6.4%

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

   3. Simplified 6.4%

      \[\leadsto \color{blue}{{\left(x + 1\right)}^{\mathsf{reciprocal}\left(n\right)} - {x}^{\mathsf{reciprocal}\left(n\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in n around inf 25.6%

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

      1. +-rgt-identity 25.6%

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

      2. +-rgt-identity 25.6%

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

      3. log1p-def 25.6%

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

   7. Simplified 25.6%

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

      1. clear-num 25.5%

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

      2. inv-pow 25.5%

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

   9. Applied egg-rr 25.5%

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

   10. Taylor expanded in x around inf 82.5%

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

       1. *-commutative 82.5%

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

       2. distribute-lft-out 82.5%

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

   12. Simplified 82.5%

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

   if 4.99999999999999973e-21 < (/.f64 1 n) < 4.0000000000000002e149

   1. Initial program 77.5%

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

      1. reciprocal-define 77.3%

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

      2. reciprocal-define 72.5%

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

   3. Simplified 72.5%

      \[\leadsto \color{blue}{{\left(x + 1\right)}^{\mathsf{reciprocal}\left(n\right)} - {x}^{\mathsf{reciprocal}\left(n\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 69.2%

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

   6. Taylor expanded in n around 0 74.2%

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

   if 4.0000000000000002e149 < (/.f64 1 n)

   1. Initial program 25.5%

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

      1. reciprocal-define 25.5%

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

      2. reciprocal-define 25.5%

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

   3. Simplified 25.5%

      \[\leadsto \color{blue}{{\left(x + 1\right)}^{\mathsf{reciprocal}\left(n\right)} - {x}^{\mathsf{reciprocal}\left(n\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf 0.3%

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

      1. mul-1-neg 0.3%

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

      2. log-rec 0.3%

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

      3. mul-1-neg 0.3%

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

      4. distribute-neg-frac 0.3%

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

      5. mul-1-neg 0.3%

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

      6. remove-double-neg 0.3%

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

   7. Simplified 0.3%

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

      1. add-sqr-sqrt 0.0%

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

      2. sqrt-unprod 78.5%

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

      3. sqr-neg 78.5%

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

      4. distribute-frac-neg 78.5%

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

      5. distribute-frac-neg 78.5%

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

      6. sqrt-unprod 78.5%

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

      7. add-sqr-sqrt 78.5%

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

      8. distribute-frac-neg 78.5%

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

      9. exp-neg 78.5%

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

      10. div-inv 78.5%

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

      11. reciprocal-undefine 78.5%

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

      12. pow-to-exp 78.5%

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

   9. Applied egg-rr 78.5%

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

       1. reciprocal-define 78.5%

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

   11. Simplified 78.5%

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

4. Final simplification 87.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{-16}:\\ \;\;\;\;\frac{{x}^{\left(\frac{2}{n \cdot 2}\right)}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-69}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-21}:\\ \;\;\;\;{\left(n \cdot \left(x + 0.5\right)\right)}^{-1}\\ \mathbf{elif}\;\frac{1}{n} \leq 4 \cdot 10^{+149}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{reciprocal}\left(\left({x}^{\mathsf{reciprocal}\left(n\right)}\right)\right)}{n \cdot x}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 80.7% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{-16}:\\ \;\;\;\;\frac{{x}^{\mathsf{reciprocal}\left(n\right)}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-69}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-21}:\\ \;\;\;\;{\left(n \cdot \left(x + 0.5\right)\right)}^{-1}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{+193}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{reciprocal}\left(\left(n \cdot x\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= (/ 1.0 n) -2e-16)
   (/ (pow x (reciprocal n)) (* n x))
   (if (<= (/ 1.0 n) 2e-69)
     (/ (log (/ (+ 1.0 x) x)) n)
     (if (<= (/ 1.0 n) 5e-21)
       (pow (* n (+ x 0.5)) -1.0)
       (if (<= (/ 1.0 n) 5e+193)
         (- 1.0 (pow x (/ 1.0 n)))
         (reciprocal (* n x)))))))

Derivation

1. Split input into 5 regimes

2. if (/.f64 1 n) < -2e-16

   1. Initial program 94.6%

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

      1. reciprocal-define 94.6%

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

      2. reciprocal-define 94.6%

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

   3. Simplified 94.6%

      \[\leadsto \color{blue}{{\left(x + 1\right)}^{\mathsf{reciprocal}\left(n\right)} - {x}^{\mathsf{reciprocal}\left(n\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf 99.9%

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

      1. mul-1-neg 99.9%

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

      2. log-rec 99.9%

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

      3. mul-1-neg 99.9%

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

      4. distribute-neg-frac 99.9%

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

      5. mul-1-neg 99.9%

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

      6. remove-double-neg 99.9%

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

   7. Simplified 99.9%

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

      1. *-un-lft-identity 99.9%

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

      2. times-frac 100.0%

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

      3. remove-double-neg 100.0%

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

      4. distribute-frac-neg 100.0%

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

      5. reciprocal-undefine 96.6%

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

      6. distribute-frac-neg 96.6%

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

      7. remove-double-neg 96.6%

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

      8. div-inv 96.6%

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

      9. reciprocal-undefine 96.7%

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

      10. pow-to-exp 96.7%

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

   9. Applied egg-rr 96.7%

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

       1. reciprocal-define 99.0%

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

       2. times-frac 99.0%

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

       3. *-lft-identity 99.0%

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

       4. *-commutative 99.0%

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

   11. Simplified 99.0%

       \[\leadsto \color{blue}{\frac{{x}^{\mathsf{reciprocal}\left(n\right)}}{x \cdot n}} \]

   if -2e-16 < (/.f64 1 n) < 1.9999999999999999e-69

   1. Initial program 33.5%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   2. Step-by-step derivation

      1. reciprocal-define 33.5%

         \[\leadsto {\left(x + 1\right)}^{\color{blue}{\mathsf{reciprocal}\left(n\right)}} - {x}^{\left(\frac{1}{n}\right)} \]

      2. reciprocal-define 33.5%

         \[\leadsto {\left(x + 1\right)}^{\mathsf{reciprocal}\left(n\right)} - {x}^{\color{blue}{\mathsf{reciprocal}\left(n\right)}} \]

   3. Simplified 33.5%

      \[\leadsto \color{blue}{{\left(x + 1\right)}^{\mathsf{reciprocal}\left(n\right)} - {x}^{\mathsf{reciprocal}\left(n\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in n around inf 86.1%

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
   6. Step-by-step derivation

      1. +-rgt-identity 86.1%

         \[\leadsto \frac{\color{blue}{\left(\log \left(1 + x\right) + 0\right)} - \log x}{n} \]

      2. +-rgt-identity 86.1%

         \[\leadsto \frac{\color{blue}{\log \left(1 + x\right)} - \log x}{n} \]

      3. log1p-def 86.1%

         \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(x\right)} - \log x}{n} \]

   7. Simplified 86.1%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) - \log x}{n}} \]
   8. Step-by-step derivation

      1. log1p-udef 86.1%

         \[\leadsto \frac{\color{blue}{\log \left(1 + x\right)} - \log x}{n} \]

      2. diff-log 86.1%

         \[\leadsto \frac{\color{blue}{\log \left(\frac{1 + x}{x}\right)}}{n} \]

      3. +-commutative 86.1%

         \[\leadsto \frac{\log \left(\frac{\color{blue}{x + 1}}{x}\right)}{n} \]

   9. Applied egg-rr 86.1%

      \[\leadsto \frac{\color{blue}{\log \left(\frac{x + 1}{x}\right)}}{n} \]

   if 1.9999999999999999e-69 < (/.f64 1 n) < 4.99999999999999973e-21

   1. Initial program 6.4%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   2. Step-by-step derivation

      1. reciprocal-define 6.4%

         \[\leadsto {\left(x + 1\right)}^{\color{blue}{\mathsf{reciprocal}\left(n\right)}} - {x}^{\left(\frac{1}{n}\right)} \]

      2. reciprocal-define 6.4%

         \[\leadsto {\left(x + 1\right)}^{\mathsf{reciprocal}\left(n\right)} - {x}^{\color{blue}{\mathsf{reciprocal}\left(n\right)}} \]

   3. Simplified 6.4%

      \[\leadsto \color{blue}{{\left(x + 1\right)}^{\mathsf{reciprocal}\left(n\right)} - {x}^{\mathsf{reciprocal}\left(n\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in n around inf 25.6%

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
   6. Step-by-step derivation

      1. +-rgt-identity 25.6%

         \[\leadsto \frac{\color{blue}{\left(\log \left(1 + x\right) + 0\right)} - \log x}{n} \]

      2. +-rgt-identity 25.6%

         \[\leadsto \frac{\color{blue}{\log \left(1 + x\right)} - \log x}{n} \]

      3. log1p-def 25.6%

         \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(x\right)} - \log x}{n} \]

   7. Simplified 25.6%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) - \log x}{n}} \]
   8. Step-by-step derivation

      1. clear-num 25.5%

         \[\leadsto \color{blue}{\frac{1}{\frac{n}{\mathsf{log1p}\left(x\right) - \log x}}} \]

      2. inv-pow 25.5%

         \[\leadsto \color{blue}{{\left(\frac{n}{\mathsf{log1p}\left(x\right) - \log x}\right)}^{-1}} \]

   9. Applied egg-rr 25.5%

      \[\leadsto \color{blue}{{\left(\frac{n}{\mathsf{log1p}\left(x\right) - \log x}\right)}^{-1}} \]

   10. Taylor expanded in x around inf 82.5%

       \[\leadsto {\color{blue}{\left(0.5 \cdot n + n \cdot x\right)}}^{-1} \]
   11. Step-by-step derivation

       1. *-commutative 82.5%

          \[\leadsto {\left(\color{blue}{n \cdot 0.5} + n \cdot x\right)}^{-1} \]

       2. distribute-lft-out 82.5%

          \[\leadsto {\color{blue}{\left(n \cdot \left(0.5 + x\right)\right)}}^{-1} \]

   12. Simplified 82.5%

       \[\leadsto {\color{blue}{\left(n \cdot \left(0.5 + x\right)\right)}}^{-1} \]

   if 4.99999999999999973e-21 < (/.f64 1 n) < 4.99999999999999972e193

   1. Initial program 76.7%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   2. Step-by-step derivation

      1. reciprocal-define 76.5%

         \[\leadsto {\left(x + 1\right)}^{\color{blue}{\mathsf{reciprocal}\left(n\right)}} - {x}^{\left(\frac{1}{n}\right)} \]

      2. reciprocal-define 72.1%

         \[\leadsto {\left(x + 1\right)}^{\mathsf{reciprocal}\left(n\right)} - {x}^{\color{blue}{\mathsf{reciprocal}\left(n\right)}} \]

   3. Simplified 72.1%

      \[\leadsto \color{blue}{{\left(x + 1\right)}^{\mathsf{reciprocal}\left(n\right)} - {x}^{\mathsf{reciprocal}\left(n\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 66.2%

      \[\leadsto \color{blue}{1} - {x}^{\mathsf{reciprocal}\left(n\right)} \]

   6. Taylor expanded in n around 0 70.8%

      \[\leadsto 1 - {x}^{\color{blue}{\left(\frac{1}{n}\right)}} \]

   if 4.99999999999999972e193 < (/.f64 1 n)

   1. Initial program 17.1%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   2. Step-by-step derivation

      1. reciprocal-define 17.1%

         \[\leadsto {\left(x + 1\right)}^{\color{blue}{\mathsf{reciprocal}\left(n\right)}} - {x}^{\left(\frac{1}{n}\right)} \]

      2. reciprocal-define 17.1%

         \[\leadsto {\left(x + 1\right)}^{\mathsf{reciprocal}\left(n\right)} - {x}^{\color{blue}{\mathsf{reciprocal}\left(n\right)}} \]

   3. Simplified 17.1%

      \[\leadsto \color{blue}{{\left(x + 1\right)}^{\mathsf{reciprocal}\left(n\right)} - {x}^{\mathsf{reciprocal}\left(n\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf 0.1%

      \[\leadsto \color{blue}{\frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}}{n \cdot x}} \]
   6. Step-by-step derivation

      1. mul-1-neg 0.1%

         \[\leadsto \frac{e^{\color{blue}{-\frac{\log \left(\frac{1}{x}\right)}{n}}}}{n \cdot x} \]

      2. log-rec 0.1%

         \[\leadsto \frac{e^{-\frac{\color{blue}{-\log x}}{n}}}{n \cdot x} \]

      3. mul-1-neg 0.1%

         \[\leadsto \frac{e^{-\frac{\color{blue}{-1 \cdot \log x}}{n}}}{n \cdot x} \]

      4. distribute-neg-frac 0.1%

         \[\leadsto \frac{e^{\color{blue}{\frac{--1 \cdot \log x}{n}}}}{n \cdot x} \]

      5. mul-1-neg 0.1%

         \[\leadsto \frac{e^{\frac{-\color{blue}{\left(-\log x\right)}}{n}}}{n \cdot x} \]

      6. remove-double-neg 0.1%

         \[\leadsto \frac{e^{\frac{\color{blue}{\log x}}{n}}}{n \cdot x} \]

   7. Simplified 0.1%

      \[\leadsto \color{blue}{\frac{e^{\frac{\log x}{n}}}{n \cdot x}} \]

   8. Taylor expanded in n around inf 74.6%

      \[\leadsto \color{blue}{\frac{1}{n \cdot x}} \]
   9. Step-by-step derivation

      1. associate-/r* 74.6%

         \[\leadsto \color{blue}{\frac{\frac{1}{n}}{x}} \]

      2. reciprocal-define 74.6%

         \[\leadsto \frac{\color{blue}{\mathsf{reciprocal}\left(n\right)}}{x} \]

   10. Simplified 74.6%

       \[\leadsto \color{blue}{\frac{\mathsf{reciprocal}\left(n\right)}{x}} \]
   11. Step-by-step derivation

       1. reciprocal-undefine 74.6%

          \[\leadsto \frac{\color{blue}{\frac{1}{n}}}{x} \]

       2. associate-/r* 74.6%

          \[\leadsto \color{blue}{\frac{1}{n \cdot x}} \]

       3. reciprocal-define 74.6%

          \[\leadsto \color{blue}{\mathsf{reciprocal}\left(\left(n \cdot x\right)\right)} \]

       4. *-commutative 74.6%

          \[\leadsto \mathsf{reciprocal}\left(\color{blue}{\left(x \cdot n\right)}\right) \]

   12. Applied egg-rr 74.6%

       \[\leadsto \color{blue}{\mathsf{reciprocal}\left(\left(x \cdot n\right)\right)} \]
3. Recombined 5 regimes into one program.

4. Final simplification 86.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{-16}:\\ \;\;\;\;\frac{{x}^{\mathsf{reciprocal}\left(n\right)}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-69}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-21}:\\ \;\;\;\;{\left(n \cdot \left(x + 0.5\right)\right)}^{-1}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{+193}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{reciprocal}\left(\left(n \cdot x\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 43.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\log \mathsf{reciprocal}\left(x\right)}{n}\\ \mathbf{if}\;x \leq 3.5 \cdot 10^{-221}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 1.36 \cdot 10^{-145}:\\ \;\;\;\;\mathsf{reciprocal}\left(\left(n \cdot x\right)\right)\\ \mathbf{elif}\;x \leq 0.55:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (/ (log (reciprocal x)) n)))
   (if (<= x 3.5e-221)
     t_0
     (if (<= x 1.36e-145)
       (reciprocal (* n x))
       (if (<= x 0.55) t_0 (/ (/ 1.0 x) n))))))

Derivation

1. Split input into 3 regimes

2. if x < 3.4999999999999999e-221 or 1.36e-145 < x < 0.55000000000000004

   1. Initial program 44.4%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   2. Step-by-step derivation

      1. reciprocal-define 44.4%

         \[\leadsto {\left(x + 1\right)}^{\color{blue}{\mathsf{reciprocal}\left(n\right)}} - {x}^{\left(\frac{1}{n}\right)} \]

      2. reciprocal-define 43.4%

         \[\leadsto {\left(x + 1\right)}^{\mathsf{reciprocal}\left(n\right)} - {x}^{\color{blue}{\mathsf{reciprocal}\left(n\right)}} \]

   3. Simplified 43.4%

      \[\leadsto \color{blue}{{\left(x + 1\right)}^{\mathsf{reciprocal}\left(n\right)} - {x}^{\mathsf{reciprocal}\left(n\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in n around inf 52.9%

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
   6. Step-by-step derivation

      1. +-rgt-identity 52.9%

         \[\leadsto \frac{\color{blue}{\left(\log \left(1 + x\right) + 0\right)} - \log x}{n} \]

      2. +-rgt-identity 52.9%

         \[\leadsto \frac{\color{blue}{\log \left(1 + x\right)} - \log x}{n} \]

      3. log1p-def 52.9%

         \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(x\right)} - \log x}{n} \]

   7. Simplified 52.9%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) - \log x}{n}} \]
   8. Step-by-step derivation

      1. clear-num 52.9%

         \[\leadsto \color{blue}{\frac{1}{\frac{n}{\mathsf{log1p}\left(x\right) - \log x}}} \]

      2. inv-pow 52.9%

         \[\leadsto \color{blue}{{\left(\frac{n}{\mathsf{log1p}\left(x\right) - \log x}\right)}^{-1}} \]

   9. Applied egg-rr 52.9%

      \[\leadsto \color{blue}{{\left(\frac{n}{\mathsf{log1p}\left(x\right) - \log x}\right)}^{-1}} \]

   10. Taylor expanded in x around 0 50.5%

       \[\leadsto \color{blue}{-1 \cdot \frac{\log x}{n}} \]
   11. Step-by-step derivation

       1. associate-*r/ 50.5%

          \[\leadsto \color{blue}{\frac{-1 \cdot \log x}{n}} \]

       2. neg-mul-1 50.5%

          \[\leadsto \frac{\color{blue}{-\log x}}{n} \]

       3. log-rec 50.5%

          \[\leadsto \frac{\color{blue}{\log \left(\frac{1}{x}\right)}}{n} \]

       4. reciprocal-define 28.8%

          \[\leadsto \frac{\log \color{blue}{\mathsf{reciprocal}\left(x\right)}}{n} \]

   12. Simplified 28.8%

       \[\leadsto \color{blue}{\frac{\log \mathsf{reciprocal}\left(x\right)}{n}} \]

   if 3.4999999999999999e-221 < x < 1.36e-145

   1. Initial program 44.9%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   2. Step-by-step derivation

      1. reciprocal-define 44.9%

         \[\leadsto {\left(x + 1\right)}^{\color{blue}{\mathsf{reciprocal}\left(n\right)}} - {x}^{\left(\frac{1}{n}\right)} \]

      2. reciprocal-define 44.0%

         \[\leadsto {\left(x + 1\right)}^{\mathsf{reciprocal}\left(n\right)} - {x}^{\color{blue}{\mathsf{reciprocal}\left(n\right)}} \]

   3. Simplified 44.0%

      \[\leadsto \color{blue}{{\left(x + 1\right)}^{\mathsf{reciprocal}\left(n\right)} - {x}^{\mathsf{reciprocal}\left(n\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf 27.2%

      \[\leadsto \color{blue}{\frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}}{n \cdot x}} \]
   6. Step-by-step derivation

      1. mul-1-neg 27.2%

         \[\leadsto \frac{e^{\color{blue}{-\frac{\log \left(\frac{1}{x}\right)}{n}}}}{n \cdot x} \]

      2. log-rec 27.2%

         \[\leadsto \frac{e^{-\frac{\color{blue}{-\log x}}{n}}}{n \cdot x} \]

      3. mul-1-neg 27.2%

         \[\leadsto \frac{e^{-\frac{\color{blue}{-1 \cdot \log x}}{n}}}{n \cdot x} \]

      4. distribute-neg-frac 27.2%

         \[\leadsto \frac{e^{\color{blue}{\frac{--1 \cdot \log x}{n}}}}{n \cdot x} \]

      5. mul-1-neg 27.2%

         \[\leadsto \frac{e^{\frac{-\color{blue}{\left(-\log x\right)}}{n}}}{n \cdot x} \]

      6. remove-double-neg 27.2%

         \[\leadsto \frac{e^{\frac{\color{blue}{\log x}}{n}}}{n \cdot x} \]

   7. Simplified 27.2%

      \[\leadsto \color{blue}{\frac{e^{\frac{\log x}{n}}}{n \cdot x}} \]

   8. Taylor expanded in n around inf 38.4%

      \[\leadsto \color{blue}{\frac{1}{n \cdot x}} \]
   9. Step-by-step derivation

      1. associate-/r* 38.4%

         \[\leadsto \color{blue}{\frac{\frac{1}{n}}{x}} \]

      2. reciprocal-define 38.4%

         \[\leadsto \frac{\color{blue}{\mathsf{reciprocal}\left(n\right)}}{x} \]

   10. Simplified 38.4%

       \[\leadsto \color{blue}{\frac{\mathsf{reciprocal}\left(n\right)}{x}} \]
   11. Step-by-step derivation

       1. reciprocal-undefine 38.4%

          \[\leadsto \frac{\color{blue}{\frac{1}{n}}}{x} \]

       2. associate-/r* 38.4%

          \[\leadsto \color{blue}{\frac{1}{n \cdot x}} \]

       3. reciprocal-define 38.4%

          \[\leadsto \color{blue}{\mathsf{reciprocal}\left(\left(n \cdot x\right)\right)} \]

       4. *-commutative 38.4%

          \[\leadsto \mathsf{reciprocal}\left(\color{blue}{\left(x \cdot n\right)}\right) \]

   12. Applied egg-rr 38.4%

       \[\leadsto \color{blue}{\mathsf{reciprocal}\left(\left(x \cdot n\right)\right)} \]

   if 0.55000000000000004 < x

   1. Initial program 69.2%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   2. Step-by-step derivation

      1. reciprocal-define 69.1%

         \[\leadsto {\left(x + 1\right)}^{\color{blue}{\mathsf{reciprocal}\left(n\right)}} - {x}^{\left(\frac{1}{n}\right)} \]

      2. reciprocal-define 69.2%

         \[\leadsto {\left(x + 1\right)}^{\mathsf{reciprocal}\left(n\right)} - {x}^{\color{blue}{\mathsf{reciprocal}\left(n\right)}} \]

   3. Simplified 69.2%

      \[\leadsto \color{blue}{{\left(x + 1\right)}^{\mathsf{reciprocal}\left(n\right)} - {x}^{\mathsf{reciprocal}\left(n\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in n around inf 66.7%

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
   6. Step-by-step derivation

      1. +-rgt-identity 66.7%

         \[\leadsto \frac{\color{blue}{\left(\log \left(1 + x\right) + 0\right)} - \log x}{n} \]

      2. +-rgt-identity 66.7%

         \[\leadsto \frac{\color{blue}{\log \left(1 + x\right)} - \log x}{n} \]

      3. log1p-def 66.7%

         \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(x\right)} - \log x}{n} \]

   7. Simplified 66.7%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) - \log x}{n}} \]

   8. Taylor expanded in x around inf 67.5%

      \[\leadsto \frac{\color{blue}{\frac{1}{x}}}{n} \]
3. Recombined 3 regimes into one program.

4. Final simplification 45.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 3.5 \cdot 10^{-221}:\\ \;\;\;\;\frac{\log \mathsf{reciprocal}\left(x\right)}{n}\\ \mathbf{elif}\;x \leq 1.36 \cdot 10^{-145}:\\ \;\;\;\;\mathsf{reciprocal}\left(\left(n \cdot x\right)\right)\\ \mathbf{elif}\;x \leq 0.55:\\ \;\;\;\;\frac{\log \mathsf{reciprocal}\left(x\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 56.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.55:\\ \;\;\;\;-\frac{\log x}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 0.55) (- (/ (log x) n)) (/ (/ 1.0 x) n)))

C

double code(double x, double n) {
	double tmp;
	if (x <= 0.55) {
		tmp = -(log(x) / n);
	} else {
		tmp = (1.0 / x) / n;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if (x <= 0.55d0) then
        tmp = -(log(x) / n)
    else
        tmp = (1.0d0 / x) / n
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (x <= 0.55) {
		tmp = -(Math.log(x) / n);
	} else {
		tmp = (1.0 / x) / n;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 0.55:
		tmp = -(math.log(x) / n)
	else:
		tmp = (1.0 / x) / n
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 0.55)
		tmp = Float64(-Float64(log(x) / n));
	else
		tmp = Float64(Float64(1.0 / x) / n);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (x <= 0.55)
		tmp = -(log(x) / n);
	else
		tmp = (1.0 / x) / n;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 0.55], (-N[(N[Log[x], $MachinePrecision] / n), $MachinePrecision]), N[(N[(1.0 / x), $MachinePrecision] / n), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.55000000000000004

   1. Initial program 44.6%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   2. Step-by-step derivation

      1. reciprocal-define 44.6%

         \[\leadsto {\left(x + 1\right)}^{\color{blue}{\mathsf{reciprocal}\left(n\right)}} - {x}^{\left(\frac{1}{n}\right)} \]

      2. reciprocal-define 43.6%

         \[\leadsto {\left(x + 1\right)}^{\mathsf{reciprocal}\left(n\right)} - {x}^{\color{blue}{\mathsf{reciprocal}\left(n\right)}} \]

   3. Simplified 43.6%

      \[\leadsto \color{blue}{{\left(x + 1\right)}^{\mathsf{reciprocal}\left(n\right)} - {x}^{\mathsf{reciprocal}\left(n\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 41.7%

      \[\leadsto \color{blue}{1} - {x}^{\mathsf{reciprocal}\left(n\right)} \]

   6. Taylor expanded in n around inf 47.8%

      \[\leadsto \color{blue}{-1 \cdot \frac{\log x}{n}} \]
   7. Step-by-step derivation

      1. neg-mul-1 47.8%

         \[\leadsto \color{blue}{-\frac{\log x}{n}} \]

      2. distribute-neg-frac 47.8%

         \[\leadsto \color{blue}{\frac{-\log x}{n}} \]

   8. Simplified 47.8%

      \[\leadsto \color{blue}{\frac{-\log x}{n}} \]

   if 0.55000000000000004 < x

   1. Initial program 69.2%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   2. Step-by-step derivation

      1. reciprocal-define 69.1%

         \[\leadsto {\left(x + 1\right)}^{\color{blue}{\mathsf{reciprocal}\left(n\right)}} - {x}^{\left(\frac{1}{n}\right)} \]

      2. reciprocal-define 69.2%

         \[\leadsto {\left(x + 1\right)}^{\mathsf{reciprocal}\left(n\right)} - {x}^{\color{blue}{\mathsf{reciprocal}\left(n\right)}} \]

   3. Simplified 69.2%

      \[\leadsto \color{blue}{{\left(x + 1\right)}^{\mathsf{reciprocal}\left(n\right)} - {x}^{\mathsf{reciprocal}\left(n\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in n around inf 66.7%

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
   6. Step-by-step derivation

      1. +-rgt-identity 66.7%

         \[\leadsto \frac{\color{blue}{\left(\log \left(1 + x\right) + 0\right)} - \log x}{n} \]

      2. +-rgt-identity 66.7%

         \[\leadsto \frac{\color{blue}{\log \left(1 + x\right)} - \log x}{n} \]

      3. log1p-def 66.7%

         \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(x\right)} - \log x}{n} \]

   7. Simplified 66.7%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) - \log x}{n}} \]

   8. Taylor expanded in x around inf 67.5%

      \[\leadsto \frac{\color{blue}{\frac{1}{x}}}{n} \]
3. Recombined 2 regimes into one program.

4. Final simplification 55.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.55:\\ \;\;\;\;-\frac{\log x}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 39.9% accurate, 42.2× speedup

Math

\[\begin{array}{l} \\ \frac{1}{n \cdot x} \end{array} \]

FPCore

(FPCore (x n) :precision binary64 (/ 1.0 (* n x)))

C

double code(double x, double n) {
	return 1.0 / (n * x);
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    code = 1.0d0 / (n * x)
end function

Java

public static double code(double x, double n) {
	return 1.0 / (n * x);
}

Python

def code(x, n):
	return 1.0 / (n * x)

Julia

function code(x, n)
	return Float64(1.0 / Float64(n * x))
end

MATLAB

function tmp = code(x, n)
	tmp = 1.0 / (n * x);
end

Wolfram

code[x_, n_] := N[(1.0 / N[(n * x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 54.2%

   \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
2. Step-by-step derivation

   1. reciprocal-define 54.1%

      \[\leadsto {\left(x + 1\right)}^{\color{blue}{\mathsf{reciprocal}\left(n\right)}} - {x}^{\left(\frac{1}{n}\right)} \]

   2. reciprocal-define 53.6%

      \[\leadsto {\left(x + 1\right)}^{\mathsf{reciprocal}\left(n\right)} - {x}^{\color{blue}{\mathsf{reciprocal}\left(n\right)}} \]

3. Simplified 53.6%

   \[\leadsto \color{blue}{{\left(x + 1\right)}^{\mathsf{reciprocal}\left(n\right)} - {x}^{\mathsf{reciprocal}\left(n\right)}} \]
4. Add Preprocessing

5. Taylor expanded in x around inf 54.4%

   \[\leadsto \color{blue}{\frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}}{n \cdot x}} \]
6. Step-by-step derivation

   1. mul-1-neg 54.4%

      \[\leadsto \frac{e^{\color{blue}{-\frac{\log \left(\frac{1}{x}\right)}{n}}}}{n \cdot x} \]

   2. log-rec 54.4%

      \[\leadsto \frac{e^{-\frac{\color{blue}{-\log x}}{n}}}{n \cdot x} \]

   3. mul-1-neg 54.4%

      \[\leadsto \frac{e^{-\frac{\color{blue}{-1 \cdot \log x}}{n}}}{n \cdot x} \]

   4. distribute-neg-frac 54.4%

      \[\leadsto \frac{e^{\color{blue}{\frac{--1 \cdot \log x}{n}}}}{n \cdot x} \]

   5. mul-1-neg 54.4%

      \[\leadsto \frac{e^{\frac{-\color{blue}{\left(-\log x\right)}}{n}}}{n \cdot x} \]

   6. remove-double-neg 54.4%

      \[\leadsto \frac{e^{\frac{\color{blue}{\log x}}{n}}}{n \cdot x} \]

7. Simplified 54.4%

   \[\leadsto \color{blue}{\frac{e^{\frac{\log x}{n}}}{n \cdot x}} \]

8. Taylor expanded in n around inf 38.5%

   \[\leadsto \color{blue}{\frac{1}{n \cdot x}} \]

9. Final simplification 38.5%

   \[\leadsto \frac{1}{n \cdot x} \]
10. Add Preprocessing

Alternative 16: 40.5% accurate, 42.2× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{1}{n}}{x} \end{array} \]

FPCore

(FPCore (x n) :precision binary64 (/ (/ 1.0 n) x))

C

double code(double x, double n) {
	return (1.0 / n) / x;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    code = (1.0d0 / n) / x
end function

Java

public static double code(double x, double n) {
	return (1.0 / n) / x;
}

Python

def code(x, n):
	return (1.0 / n) / x

Julia

function code(x, n)
	return Float64(Float64(1.0 / n) / x)
end

MATLAB

function tmp = code(x, n)
	tmp = (1.0 / n) / x;
end

Wolfram

code[x_, n_] := N[(N[(1.0 / n), $MachinePrecision] / x), $MachinePrecision]

Derivation

1. Initial program 54.2%

   \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
2. Step-by-step derivation

   1. reciprocal-define 54.1%

      \[\leadsto {\left(x + 1\right)}^{\color{blue}{\mathsf{reciprocal}\left(n\right)}} - {x}^{\left(\frac{1}{n}\right)} \]

   2. reciprocal-define 53.6%

      \[\leadsto {\left(x + 1\right)}^{\mathsf{reciprocal}\left(n\right)} - {x}^{\color{blue}{\mathsf{reciprocal}\left(n\right)}} \]

3. Simplified 53.6%

   \[\leadsto \color{blue}{{\left(x + 1\right)}^{\mathsf{reciprocal}\left(n\right)} - {x}^{\mathsf{reciprocal}\left(n\right)}} \]
4. Add Preprocessing

5. Taylor expanded in x around inf 54.4%

   \[\leadsto \color{blue}{\frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}}{n \cdot x}} \]
6. Step-by-step derivation

   1. mul-1-neg 54.4%

      \[\leadsto \frac{e^{\color{blue}{-\frac{\log \left(\frac{1}{x}\right)}{n}}}}{n \cdot x} \]

   2. log-rec 54.4%

      \[\leadsto \frac{e^{-\frac{\color{blue}{-\log x}}{n}}}{n \cdot x} \]

   3. mul-1-neg 54.4%

      \[\leadsto \frac{e^{-\frac{\color{blue}{-1 \cdot \log x}}{n}}}{n \cdot x} \]

   4. distribute-neg-frac 54.4%

      \[\leadsto \frac{e^{\color{blue}{\frac{--1 \cdot \log x}{n}}}}{n \cdot x} \]

   5. mul-1-neg 54.4%

      \[\leadsto \frac{e^{\frac{-\color{blue}{\left(-\log x\right)}}{n}}}{n \cdot x} \]

   6. remove-double-neg 54.4%

      \[\leadsto \frac{e^{\frac{\color{blue}{\log x}}{n}}}{n \cdot x} \]

7. Simplified 54.4%

   \[\leadsto \color{blue}{\frac{e^{\frac{\log x}{n}}}{n \cdot x}} \]

8. Taylor expanded in n around inf 38.5%

   \[\leadsto \color{blue}{\frac{1}{n \cdot x}} \]
9. Step-by-step derivation

   1. associate-/r* 39.2%

      \[\leadsto \color{blue}{\frac{\frac{1}{n}}{x}} \]

   2. reciprocal-define 32.7%

      \[\leadsto \frac{\color{blue}{\mathsf{reciprocal}\left(n\right)}}{x} \]

10. Simplified 32.7%

    \[\leadsto \color{blue}{\frac{\mathsf{reciprocal}\left(n\right)}{x}} \]

11. Taylor expanded in n around 0 39.2%

    \[\leadsto \frac{\color{blue}{\frac{1}{n}}}{x} \]

12. Final simplification 39.2%

    \[\leadsto \frac{\frac{1}{n}}{x} \]
13. Add Preprocessing

Alternative 17: 40.5% accurate, 42.2× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{1}{x}}{n} \end{array} \]

FPCore

(FPCore (x n) :precision binary64 (/ (/ 1.0 x) n))

C

double code(double x, double n) {
	return (1.0 / x) / n;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    code = (1.0d0 / x) / n
end function

Java

public static double code(double x, double n) {
	return (1.0 / x) / n;
}

Python

def code(x, n):
	return (1.0 / x) / n

Julia

function code(x, n)
	return Float64(Float64(1.0 / x) / n)
end

MATLAB

function tmp = code(x, n)
	tmp = (1.0 / x) / n;
end

Wolfram

code[x_, n_] := N[(N[(1.0 / x), $MachinePrecision] / n), $MachinePrecision]

Derivation

1. Initial program 54.2%

   \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
2. Step-by-step derivation

   1. reciprocal-define 54.1%

      \[\leadsto {\left(x + 1\right)}^{\color{blue}{\mathsf{reciprocal}\left(n\right)}} - {x}^{\left(\frac{1}{n}\right)} \]

   2. reciprocal-define 53.6%

      \[\leadsto {\left(x + 1\right)}^{\mathsf{reciprocal}\left(n\right)} - {x}^{\color{blue}{\mathsf{reciprocal}\left(n\right)}} \]

3. Simplified 53.6%

   \[\leadsto \color{blue}{{\left(x + 1\right)}^{\mathsf{reciprocal}\left(n\right)} - {x}^{\mathsf{reciprocal}\left(n\right)}} \]
4. Add Preprocessing

5. Taylor expanded in n around inf 56.2%

   \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
6. Step-by-step derivation

   1. +-rgt-identity 56.2%

      \[\leadsto \frac{\color{blue}{\left(\log \left(1 + x\right) + 0\right)} - \log x}{n} \]

   2. +-rgt-identity 56.2%

      \[\leadsto \frac{\color{blue}{\log \left(1 + x\right)} - \log x}{n} \]

   3. log1p-def 56.2%

      \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(x\right)} - \log x}{n} \]

7. Simplified 56.2%

   \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) - \log x}{n}} \]

8. Taylor expanded in x around inf 39.2%

   \[\leadsto \frac{\color{blue}{\frac{1}{x}}}{n} \]

9. Final simplification 39.2%

   \[\leadsto \frac{\frac{1}{x}}{n} \]
10. Add Preprocessing

Alternative 18: 32.9% accurate, 56.3× speedup

Math

\[\begin{array}{l} \\ \mathsf{reciprocal}\left(\left(n \cdot x\right)\right) \end{array} \]

FPCore

(FPCore (x n) :precision binary64 (reciprocal (* n x)))

Derivation

1. Initial program 54.2%

   \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
2. Step-by-step derivation

   1. reciprocal-define 54.1%

      \[\leadsto {\left(x + 1\right)}^{\color{blue}{\mathsf{reciprocal}\left(n\right)}} - {x}^{\left(\frac{1}{n}\right)} \]

   2. reciprocal-define 53.6%

      \[\leadsto {\left(x + 1\right)}^{\mathsf{reciprocal}\left(n\right)} - {x}^{\color{blue}{\mathsf{reciprocal}\left(n\right)}} \]

3. Simplified 53.6%

   \[\leadsto \color{blue}{{\left(x + 1\right)}^{\mathsf{reciprocal}\left(n\right)} - {x}^{\mathsf{reciprocal}\left(n\right)}} \]
4. Add Preprocessing

5. Taylor expanded in x around inf 54.4%

   \[\leadsto \color{blue}{\frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}}{n \cdot x}} \]
6. Step-by-step derivation

   1. mul-1-neg 54.4%

      \[\leadsto \frac{e^{\color{blue}{-\frac{\log \left(\frac{1}{x}\right)}{n}}}}{n \cdot x} \]

   2. log-rec 54.4%

      \[\leadsto \frac{e^{-\frac{\color{blue}{-\log x}}{n}}}{n \cdot x} \]

   3. mul-1-neg 54.4%

      \[\leadsto \frac{e^{-\frac{\color{blue}{-1 \cdot \log x}}{n}}}{n \cdot x} \]

   4. distribute-neg-frac 54.4%

      \[\leadsto \frac{e^{\color{blue}{\frac{--1 \cdot \log x}{n}}}}{n \cdot x} \]

   5. mul-1-neg 54.4%

      \[\leadsto \frac{e^{\frac{-\color{blue}{\left(-\log x\right)}}{n}}}{n \cdot x} \]

   6. remove-double-neg 54.4%

      \[\leadsto \frac{e^{\frac{\color{blue}{\log x}}{n}}}{n \cdot x} \]

7. Simplified 54.4%

   \[\leadsto \color{blue}{\frac{e^{\frac{\log x}{n}}}{n \cdot x}} \]

8. Taylor expanded in n around inf 38.5%

   \[\leadsto \color{blue}{\frac{1}{n \cdot x}} \]
9. Step-by-step derivation

   1. associate-/r* 39.2%

      \[\leadsto \color{blue}{\frac{\frac{1}{n}}{x}} \]

   2. reciprocal-define 32.7%

      \[\leadsto \frac{\color{blue}{\mathsf{reciprocal}\left(n\right)}}{x} \]

10. Simplified 32.7%

    \[\leadsto \color{blue}{\frac{\mathsf{reciprocal}\left(n\right)}{x}} \]
11. Step-by-step derivation

    1. reciprocal-undefine 39.2%

       \[\leadsto \frac{\color{blue}{\frac{1}{n}}}{x} \]

    2. associate-/r* 38.5%

       \[\leadsto \color{blue}{\frac{1}{n \cdot x}} \]

    3. reciprocal-define 32.3%

       \[\leadsto \color{blue}{\mathsf{reciprocal}\left(\left(n \cdot x\right)\right)} \]

    4. *-commutative 32.3%

       \[\leadsto \mathsf{reciprocal}\left(\color{blue}{\left(x \cdot n\right)}\right) \]

12. Applied egg-rr 32.3%

    \[\leadsto \color{blue}{\mathsf{reciprocal}\left(\left(x \cdot n\right)\right)} \]

13. Final simplification 32.3%

    \[\leadsto \mathsf{reciprocal}\left(\left(n \cdot x\right)\right) \]
14. Add Preprocessing

Alternative 19: 33.1% accurate, 56.3× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{reciprocal}\left(n\right)}{x} \end{array} \]

FPCore

(FPCore (x n) :precision binary64 (/ (reciprocal n) x))

Derivation

1. Initial program 54.2%

   \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
2. Step-by-step derivation

   1. reciprocal-define 54.1%

      \[\leadsto {\left(x + 1\right)}^{\color{blue}{\mathsf{reciprocal}\left(n\right)}} - {x}^{\left(\frac{1}{n}\right)} \]

   2. reciprocal-define 53.6%

      \[\leadsto {\left(x + 1\right)}^{\mathsf{reciprocal}\left(n\right)} - {x}^{\color{blue}{\mathsf{reciprocal}\left(n\right)}} \]

3. Simplified 53.6%

   \[\leadsto \color{blue}{{\left(x + 1\right)}^{\mathsf{reciprocal}\left(n\right)} - {x}^{\mathsf{reciprocal}\left(n\right)}} \]
4. Add Preprocessing

5. Taylor expanded in x around inf 54.4%

   \[\leadsto \color{blue}{\frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}}{n \cdot x}} \]
6. Step-by-step derivation

   1. mul-1-neg 54.4%

      \[\leadsto \frac{e^{\color{blue}{-\frac{\log \left(\frac{1}{x}\right)}{n}}}}{n \cdot x} \]

   2. log-rec 54.4%

      \[\leadsto \frac{e^{-\frac{\color{blue}{-\log x}}{n}}}{n \cdot x} \]

   3. mul-1-neg 54.4%

      \[\leadsto \frac{e^{-\frac{\color{blue}{-1 \cdot \log x}}{n}}}{n \cdot x} \]

   4. distribute-neg-frac 54.4%

      \[\leadsto \frac{e^{\color{blue}{\frac{--1 \cdot \log x}{n}}}}{n \cdot x} \]

   5. mul-1-neg 54.4%

      \[\leadsto \frac{e^{\frac{-\color{blue}{\left(-\log x\right)}}{n}}}{n \cdot x} \]

   6. remove-double-neg 54.4%

      \[\leadsto \frac{e^{\frac{\color{blue}{\log x}}{n}}}{n \cdot x} \]

7. Simplified 54.4%

   \[\leadsto \color{blue}{\frac{e^{\frac{\log x}{n}}}{n \cdot x}} \]

8. Taylor expanded in n around inf 38.5%

   \[\leadsto \color{blue}{\frac{1}{n \cdot x}} \]
9. Step-by-step derivation

   1. associate-/r* 39.2%

      \[\leadsto \color{blue}{\frac{\frac{1}{n}}{x}} \]

   2. reciprocal-define 32.7%

      \[\leadsto \frac{\color{blue}{\mathsf{reciprocal}\left(n\right)}}{x} \]

10. Simplified 32.7%

    \[\leadsto \color{blue}{\frac{\mathsf{reciprocal}\left(n\right)}{x}} \]

11. Final simplification 32.7%

    \[\leadsto \frac{\mathsf{reciprocal}\left(n\right)}{x} \]
12. Add Preprocessing

Alternative 20: 33.1% accurate, 56.3× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{reciprocal}\left(x\right)}{n} \end{array} \]

FPCore

(FPCore (x n) :precision binary64 (/ (reciprocal x) n))

Derivation

1. Initial program 54.2%

   \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
2. Step-by-step derivation

   1. reciprocal-define 54.1%

      \[\leadsto {\left(x + 1\right)}^{\color{blue}{\mathsf{reciprocal}\left(n\right)}} - {x}^{\left(\frac{1}{n}\right)} \]

   2. reciprocal-define 53.6%

      \[\leadsto {\left(x + 1\right)}^{\mathsf{reciprocal}\left(n\right)} - {x}^{\color{blue}{\mathsf{reciprocal}\left(n\right)}} \]

3. Simplified 53.6%

   \[\leadsto \color{blue}{{\left(x + 1\right)}^{\mathsf{reciprocal}\left(n\right)} - {x}^{\mathsf{reciprocal}\left(n\right)}} \]
4. Add Preprocessing

5. Taylor expanded in n around inf 56.2%

   \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
6. Step-by-step derivation

   1. +-rgt-identity 56.2%

      \[\leadsto \frac{\color{blue}{\left(\log \left(1 + x\right) + 0\right)} - \log x}{n} \]

   2. +-rgt-identity 56.2%

      \[\leadsto \frac{\color{blue}{\log \left(1 + x\right)} - \log x}{n} \]

   3. log1p-def 56.2%

      \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(x\right)} - \log x}{n} \]

7. Simplified 56.2%

   \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) - \log x}{n}} \]

8. Taylor expanded in x around inf 39.2%

   \[\leadsto \frac{\color{blue}{\frac{1}{x}}}{n} \]
9. Step-by-step derivation

   1. reciprocal-define 32.7%

      \[\leadsto \frac{\color{blue}{\mathsf{reciprocal}\left(x\right)}}{n} \]

10. Simplified 32.7%

    \[\leadsto \frac{\color{blue}{\mathsf{reciprocal}\left(x\right)}}{n} \]

11. Final simplification 32.7%

    \[\leadsto \frac{\mathsf{reciprocal}\left(x\right)}{n} \]
12. Add Preprocessing

Alternative 21: 4.5% accurate, 70.3× speedup

Math

\[\begin{array}{l} \\ \frac{x}{n} \end{array} \]

FPCore

(FPCore (x n) :precision binary64 (/ x n))

C

double code(double x, double n) {
	return x / n;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    code = x / n
end function

Java

public static double code(double x, double n) {
	return x / n;
}

Python

def code(x, n):
	return x / n

Julia

function code(x, n)
	return Float64(x / n)
end

MATLAB

function tmp = code(x, n)
	tmp = x / n;
end

Wolfram

code[x_, n_] := N[(x / n), $MachinePrecision]

Derivation

1. Initial program 54.2%

   \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
2. Step-by-step derivation

   1. reciprocal-define 54.1%

      \[\leadsto {\left(x + 1\right)}^{\color{blue}{\mathsf{reciprocal}\left(n\right)}} - {x}^{\left(\frac{1}{n}\right)} \]

   2. reciprocal-define 53.6%

      \[\leadsto {\left(x + 1\right)}^{\mathsf{reciprocal}\left(n\right)} - {x}^{\color{blue}{\mathsf{reciprocal}\left(n\right)}} \]

3. Simplified 53.6%

   \[\leadsto \color{blue}{{\left(x + 1\right)}^{\mathsf{reciprocal}\left(n\right)} - {x}^{\mathsf{reciprocal}\left(n\right)}} \]
4. Add Preprocessing

5. Taylor expanded in x around inf 54.4%

   \[\leadsto \color{blue}{\frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}}{n \cdot x}} \]
6. Step-by-step derivation

   1. mul-1-neg 54.4%

      \[\leadsto \frac{e^{\color{blue}{-\frac{\log \left(\frac{1}{x}\right)}{n}}}}{n \cdot x} \]

   2. log-rec 54.4%

      \[\leadsto \frac{e^{-\frac{\color{blue}{-\log x}}{n}}}{n \cdot x} \]

   3. mul-1-neg 54.4%

      \[\leadsto \frac{e^{-\frac{\color{blue}{-1 \cdot \log x}}{n}}}{n \cdot x} \]

   4. distribute-neg-frac 54.4%

      \[\leadsto \frac{e^{\color{blue}{\frac{--1 \cdot \log x}{n}}}}{n \cdot x} \]

   5. mul-1-neg 54.4%

      \[\leadsto \frac{e^{\frac{-\color{blue}{\left(-\log x\right)}}{n}}}{n \cdot x} \]

   6. remove-double-neg 54.4%

      \[\leadsto \frac{e^{\frac{\color{blue}{\log x}}{n}}}{n \cdot x} \]

7. Simplified 54.4%

   \[\leadsto \color{blue}{\frac{e^{\frac{\log x}{n}}}{n \cdot x}} \]

8. Taylor expanded in n around inf 38.5%

   \[\leadsto \color{blue}{\frac{1}{n \cdot x}} \]
9. Step-by-step derivation

   1. associate-/r* 39.2%

      \[\leadsto \color{blue}{\frac{\frac{1}{n}}{x}} \]

   2. reciprocal-define 32.7%

      \[\leadsto \frac{\color{blue}{\mathsf{reciprocal}\left(n\right)}}{x} \]

10. Simplified 32.7%

    \[\leadsto \color{blue}{\frac{\mathsf{reciprocal}\left(n\right)}{x}} \]
11. Step-by-step derivation

    1. reciprocal-undefine 39.2%

       \[\leadsto \frac{\color{blue}{\frac{1}{n}}}{x} \]

    2. associate-/r* 38.5%

       \[\leadsto \color{blue}{\frac{1}{n \cdot x}} \]

    3. reciprocal-define 32.3%

       \[\leadsto \color{blue}{\mathsf{reciprocal}\left(\left(n \cdot x\right)\right)} \]

    4. *-commutative 32.3%

       \[\leadsto \mathsf{reciprocal}\left(\color{blue}{\left(x \cdot n\right)}\right) \]

12. Applied egg-rr 32.3%

    \[\leadsto \color{blue}{\mathsf{reciprocal}\left(\left(x \cdot n\right)\right)} \]
13. Step-by-step derivation

    1. expm1-log1p-u 25.3%

       \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{reciprocal}\left(\left(x \cdot n\right)\right)\right)\right)} \]

    2. expm1-udef 24.5%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\mathsf{reciprocal}\left(\left(x \cdot n\right)\right)\right)} - 1} \]

    3. reciprocal-undefine 24.5%

       \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{1}{x \cdot n}}\right)} - 1 \]

    4. associate-/r* 24.5%

       \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{\frac{1}{x}}{n}}\right)} - 1 \]

    5. add-exp-log 24.5%

       \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{e^{\log \left(\frac{1}{x}\right)}}}{n}\right)} - 1 \]

    6. neg-log 24.5%

       \[\leadsto e^{\mathsf{log1p}\left(\frac{e^{\color{blue}{-\log x}}}{n}\right)} - 1 \]

    7. add-sqr-sqrt 6.1%

       \[\leadsto e^{\mathsf{log1p}\left(\frac{e^{\color{blue}{\sqrt{-\log x} \cdot \sqrt{-\log x}}}}{n}\right)} - 1 \]

    8. sqrt-unprod 13.2%

       \[\leadsto e^{\mathsf{log1p}\left(\frac{e^{\color{blue}{\sqrt{\left(-\log x\right) \cdot \left(-\log x\right)}}}}{n}\right)} - 1 \]

    9. sqr-neg 13.2%

       \[\leadsto e^{\mathsf{log1p}\left(\frac{e^{\sqrt{\color{blue}{\log x \cdot \log x}}}}{n}\right)} - 1 \]

    10. sqrt-unprod 7.1%

        \[\leadsto e^{\mathsf{log1p}\left(\frac{e^{\color{blue}{\sqrt{\log x} \cdot \sqrt{\log x}}}}{n}\right)} - 1 \]

    11. add-sqr-sqrt 9.5%

        \[\leadsto e^{\mathsf{log1p}\left(\frac{e^{\color{blue}{\log x}}}{n}\right)} - 1 \]

    12. add-exp-log 9.5%

        \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{x}}{n}\right)} - 1 \]

14. Applied egg-rr 9.5%

    \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{x}{n}\right)} - 1} \]
15. Step-by-step derivation

    1. expm1-def 3.8%

       \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{x}{n}\right)\right)} \]

    2. expm1-log1p 4.5%

       \[\leadsto \color{blue}{\frac{x}{n}} \]

16. Simplified 4.5%

    \[\leadsto \color{blue}{\frac{x}{n}} \]

17. Final simplification 4.5%

    \[\leadsto \frac{x}{n} \]
18. Add Preprocessing

Reproduce

herbie shell --seed 2024024 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  :precision binary64
  (- (pow (+ x 1.0) (/ 1.0 n)) (pow x (/ 1.0 n))))