2nthrt (problem 3.4.6)

Percentage Accurate: 53.3% → 94.5%

Time: 19.1s

Alternatives: 17

Speedup: 2.5×

• could not determine a ground truth

  1. x = -6.287722190784108e-268
  2. n = -1.567809629880825e+175

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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, n)
use fmin_fmax_functions
    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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, n)
use fmin_fmax_functions
    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: 94.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -0.05:\\ \;\;\;\;\frac{e^{\frac{\log x}{n}}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2:\\ \;\;\;\;\frac{\mathsf{log1p}\left(\frac{1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{+134}:\\ \;\;\;\;\left(\frac{x}{n} - -1\right) - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\left(-\mathsf{expm1}\left(\frac{\log \left(\frac{x - -1}{x}\right)}{n}\right)\right) \cdot -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= (/ 1.0 n) -0.05)
   (/ (exp (/ (log x) n)) (* n x))
   (if (<= (/ 1.0 n) 2.0)
     (/ (log1p (/ 1.0 x)) n)
     (if (<= (/ 1.0 n) 1e+134)
       (- (- (/ x n) -1.0) (pow x (/ 1.0 n)))
       (* 1.0 (* (- (expm1 (/ (log (/ (- x -1.0) x)) n))) -1.0))))))

C

double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -0.05) {
		tmp = exp((log(x) / n)) / (n * x);
	} else if ((1.0 / n) <= 2.0) {
		tmp = log1p((1.0 / x)) / n;
	} else if ((1.0 / n) <= 1e+134) {
		tmp = ((x / n) - -1.0) - pow(x, (1.0 / n));
	} else {
		tmp = 1.0 * (-expm1((log(((x - -1.0) / x)) / n)) * -1.0);
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -0.05) {
		tmp = Math.exp((Math.log(x) / n)) / (n * x);
	} else if ((1.0 / n) <= 2.0) {
		tmp = Math.log1p((1.0 / x)) / n;
	} else if ((1.0 / n) <= 1e+134) {
		tmp = ((x / n) - -1.0) - Math.pow(x, (1.0 / n));
	} else {
		tmp = 1.0 * (-Math.expm1((Math.log(((x - -1.0) / x)) / n)) * -1.0);
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if (1.0 / n) <= -0.05:
		tmp = math.exp((math.log(x) / n)) / (n * x)
	elif (1.0 / n) <= 2.0:
		tmp = math.log1p((1.0 / x)) / n
	elif (1.0 / n) <= 1e+134:
		tmp = ((x / n) - -1.0) - math.pow(x, (1.0 / n))
	else:
		tmp = 1.0 * (-math.expm1((math.log(((x - -1.0) / x)) / n)) * -1.0)
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -0.05)
		tmp = Float64(exp(Float64(log(x) / n)) / Float64(n * x));
	elseif (Float64(1.0 / n) <= 2.0)
		tmp = Float64(log1p(Float64(1.0 / x)) / n);
	elseif (Float64(1.0 / n) <= 1e+134)
		tmp = Float64(Float64(Float64(x / n) - -1.0) - (x ^ Float64(1.0 / n)));
	else
		tmp = Float64(1.0 * Float64(Float64(-expm1(Float64(log(Float64(Float64(x - -1.0) / x)) / n))) * -1.0));
	end
	return tmp
end

Wolfram

code[x_, n_] := If[LessEqual[N[(1.0 / n), $MachinePrecision], -0.05], N[(N[Exp[N[(N[Log[x], $MachinePrecision] / n), $MachinePrecision]], $MachinePrecision] / N[(n * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2.0], N[(N[Log[1 + N[(1.0 / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 1e+134], N[(N[(N[(x / n), $MachinePrecision] - -1.0), $MachinePrecision] - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(1.0 * N[((-N[(Exp[N[(N[Log[N[(N[(x - -1.0), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision]] - 1), $MachinePrecision]) * -1.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -0.050000000000000003

   1. Initial program 53.3%

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

   2. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower-exp.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-log.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 57.8

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

   4. Applied rewrites 57.8%

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

      1. lift-*.f64 N/A

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

      2. mul-1-neg N/A

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

      3. lift-/.f64 N/A

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

      4. distribute-neg-frac2 N/A

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

      5. lift-log.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. log-rec N/A

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

      8. frac-2neg N/A

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

      9. lower-/.f64 N/A

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

      10. lower-log.f64 57.8

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

   6. Applied rewrites 57.8%

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

   if -0.050000000000000003 < (/.f64 #s(literal 1 binary64) n) < 2

   1. Initial program 53.3%

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

   2. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-log.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-log.f64 58.9

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

   4. Applied rewrites 58.9%

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

      1. lift--.f64 N/A

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

      2. lift-log.f64 N/A

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

      3. lift-log.f64 N/A

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

      4. diff-log N/A

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. *-lft-identity N/A

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

      8. add-to-fraction N/A

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

      9. lower-log1p.f64 N/A

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

      10. lower-/.f64 58.3

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

   6. Applied rewrites 58.3%

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

   if 2 < (/.f64 #s(literal 1 binary64) n) < 9.99999999999999921e133

   1. Initial program 53.3%

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

   2. Taylor expanded in x around 0

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

      1. lower-+.f64 N/A

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

      2. lower-/.f64 30.9

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

   4. Applied rewrites 30.9%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. add-flip N/A

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

      4. metadata-eval N/A

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

      5. lower--.f64 30.9

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

   6. Applied rewrites 30.9%

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

   if 9.99999999999999921e133 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 53.3%

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

      1. lift--.f64 N/A

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

      2. sub-to-mult N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. add-flip N/A

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

      7. metadata-eval N/A

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

      8. lower--.f64 N/A

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

      9. sub-negate-rev N/A

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

      10. lower-neg.f64 N/A

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

      11. lift-pow.f64 N/A

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

      12. pow-to-exp N/A

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

      13. lift-pow.f64 N/A

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

      14. pow-to-exp N/A

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

      15. div-exp N/A

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

      16. lower-expm1.f64 N/A

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

      17. lower--.f64 N/A

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

   3. Applied rewrites 78.6%

      \[\leadsto \color{blue}{{\left(x - -1\right)}^{\left(\frac{1}{n}\right)} \cdot \left(-\mathsf{expm1}\left(\frac{\log x}{n} - \frac{\log \left(x - -1\right)}{n}\right)\right)} \]

   4. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1} \cdot \left(-\mathsf{expm1}\left(\frac{\log x}{n} - \frac{\log \left(x - -1\right)}{n}\right)\right) \]
   5. Step-by-step derivation

   6. Applied rewrites 78.6%

      \[\leadsto \color{blue}{1} \cdot \left(-\mathsf{expm1}\left(\frac{\log x}{n} - \frac{\log \left(x - -1\right)}{n}\right)\right) \]
   7. Step-by-step derivation

      1. lift-neg.f64 N/A

         \[\leadsto 1 \cdot \color{blue}{\left(\mathsf{neg}\left(\mathsf{expm1}\left(\frac{\log x}{n} - \frac{\log \left(x - -1\right)}{n}\right)\right)\right)} \]

      2. lift-expm1.f64 N/A

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

      3. sub-to-mult N/A

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

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

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

      5. lower-*.f64 N/A

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

   8. Applied rewrites 71.6%

      \[\leadsto 1 \cdot \color{blue}{\left(\left(-\mathsf{expm1}\left(\frac{\log \left(\frac{x - -1}{x}\right)}{n}\right)\right) \cdot \left(-e^{\frac{\log \left(\frac{x}{x - -1}\right)}{n}}\right)\right)} \]

   9. Taylor expanded in x around inf

      \[\leadsto 1 \cdot \left(\left(-\mathsf{expm1}\left(\frac{\log \left(\frac{x - -1}{x}\right)}{n}\right)\right) \cdot \color{blue}{-1}\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 64.4%

       \[\leadsto 1 \cdot \left(\left(-\mathsf{expm1}\left(\frac{\log \left(\frac{x - -1}{x}\right)}{n}\right)\right) \cdot \color{blue}{-1}\right) \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 2: 94.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -0.05:\\ \;\;\;\;\frac{e^{\frac{\log x}{n}}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2:\\ \;\;\;\;\frac{\mathsf{log1p}\left(\frac{1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{+134}:\\ \;\;\;\;\left(\frac{x}{n} - -1\right) - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{1 + -1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}{n \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= (/ 1.0 n) -0.05)
   (/ (exp (/ (log x) n)) (* n x))
   (if (<= (/ 1.0 n) 2.0)
     (/ (log1p (/ 1.0 x)) n)
     (if (<= (/ 1.0 n) 1e+134)
       (- (- (/ x n) -1.0) (pow x (/ 1.0 n)))
       (* -1.0 (/ (+ 1.0 (* -1.0 (/ (log (/ 1.0 x)) n))) (* n x)))))))

C

double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -0.05) {
		tmp = exp((log(x) / n)) / (n * x);
	} else if ((1.0 / n) <= 2.0) {
		tmp = log1p((1.0 / x)) / n;
	} else if ((1.0 / n) <= 1e+134) {
		tmp = ((x / n) - -1.0) - pow(x, (1.0 / n));
	} else {
		tmp = -1.0 * ((1.0 + (-1.0 * (log((1.0 / x)) / n))) / (n * x));
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -0.05) {
		tmp = Math.exp((Math.log(x) / n)) / (n * x);
	} else if ((1.0 / n) <= 2.0) {
		tmp = Math.log1p((1.0 / x)) / n;
	} else if ((1.0 / n) <= 1e+134) {
		tmp = ((x / n) - -1.0) - Math.pow(x, (1.0 / n));
	} else {
		tmp = -1.0 * ((1.0 + (-1.0 * (Math.log((1.0 / x)) / n))) / (n * x));
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if (1.0 / n) <= -0.05:
		tmp = math.exp((math.log(x) / n)) / (n * x)
	elif (1.0 / n) <= 2.0:
		tmp = math.log1p((1.0 / x)) / n
	elif (1.0 / n) <= 1e+134:
		tmp = ((x / n) - -1.0) - math.pow(x, (1.0 / n))
	else:
		tmp = -1.0 * ((1.0 + (-1.0 * (math.log((1.0 / x)) / n))) / (n * x))
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -0.05)
		tmp = Float64(exp(Float64(log(x) / n)) / Float64(n * x));
	elseif (Float64(1.0 / n) <= 2.0)
		tmp = Float64(log1p(Float64(1.0 / x)) / n);
	elseif (Float64(1.0 / n) <= 1e+134)
		tmp = Float64(Float64(Float64(x / n) - -1.0) - (x ^ Float64(1.0 / n)));
	else
		tmp = Float64(-1.0 * Float64(Float64(1.0 + Float64(-1.0 * Float64(log(Float64(1.0 / x)) / n))) / Float64(n * x)));
	end
	return tmp
end

Wolfram

code[x_, n_] := If[LessEqual[N[(1.0 / n), $MachinePrecision], -0.05], N[(N[Exp[N[(N[Log[x], $MachinePrecision] / n), $MachinePrecision]], $MachinePrecision] / N[(n * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2.0], N[(N[Log[1 + N[(1.0 / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 1e+134], N[(N[(N[(x / n), $MachinePrecision] - -1.0), $MachinePrecision] - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(-1.0 * N[(N[(1.0 + N[(-1.0 * N[(N[Log[N[(1.0 / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(n * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -0.050000000000000003

   1. Initial program 53.3%

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

   2. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower-exp.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-log.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 57.8

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

   4. Applied rewrites 57.8%

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

      1. lift-*.f64 N/A

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

      2. mul-1-neg N/A

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

      3. lift-/.f64 N/A

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

      4. distribute-neg-frac2 N/A

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

      5. lift-log.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. log-rec N/A

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

      8. frac-2neg N/A

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

      9. lower-/.f64 N/A

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

      10. lower-log.f64 57.8

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

   6. Applied rewrites 57.8%

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

   if -0.050000000000000003 < (/.f64 #s(literal 1 binary64) n) < 2

   1. Initial program 53.3%

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

   2. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-log.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-log.f64 58.9

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

   4. Applied rewrites 58.9%

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

      1. lift--.f64 N/A

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

      2. lift-log.f64 N/A

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

      3. lift-log.f64 N/A

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

      4. diff-log N/A

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. *-lft-identity N/A

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

      8. add-to-fraction N/A

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

      9. lower-log1p.f64 N/A

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

      10. lower-/.f64 58.3

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

   6. Applied rewrites 58.3%

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

   if 2 < (/.f64 #s(literal 1 binary64) n) < 9.99999999999999921e133

   1. Initial program 53.3%

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

   2. Taylor expanded in x around 0

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

      1. lower-+.f64 N/A

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

      2. lower-/.f64 30.9

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

   4. Applied rewrites 30.9%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. add-flip N/A

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

      4. metadata-eval N/A

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

      5. lower--.f64 30.9

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

   6. Applied rewrites 30.9%

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

   if 9.99999999999999921e133 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 53.3%

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

   2. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

   4. Applied rewrites 64.5%

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

   5. Taylor expanded in x around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-log.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 21.5

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

   7. Applied rewrites 21.5%

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

Alternative 3: 94.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\log x}{n}\\ \mathbf{if}\;\frac{1}{n} \leq -0.05:\\ \;\;\;\;\frac{e^{t\_0}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 4 \cdot 10^{-5}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(\frac{1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{+134}:\\ \;\;\;\;1 \cdot \left(-\mathsf{expm1}\left(t\_0 - \frac{x}{n}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{1 + -1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}{n \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (/ (log x) n)))
   (if (<= (/ 1.0 n) -0.05)
     (/ (exp t_0) (* n x))
     (if (<= (/ 1.0 n) 4e-5)
       (/ (log1p (/ 1.0 x)) n)
       (if (<= (/ 1.0 n) 1e+134)
         (* 1.0 (- (expm1 (- t_0 (/ x n)))))
         (* -1.0 (/ (+ 1.0 (* -1.0 (/ (log (/ 1.0 x)) n))) (* n x))))))))

C

double code(double x, double n) {
	double t_0 = log(x) / n;
	double tmp;
	if ((1.0 / n) <= -0.05) {
		tmp = exp(t_0) / (n * x);
	} else if ((1.0 / n) <= 4e-5) {
		tmp = log1p((1.0 / x)) / n;
	} else if ((1.0 / n) <= 1e+134) {
		tmp = 1.0 * -expm1((t_0 - (x / n)));
	} else {
		tmp = -1.0 * ((1.0 + (-1.0 * (log((1.0 / x)) / n))) / (n * x));
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double t_0 = Math.log(x) / n;
	double tmp;
	if ((1.0 / n) <= -0.05) {
		tmp = Math.exp(t_0) / (n * x);
	} else if ((1.0 / n) <= 4e-5) {
		tmp = Math.log1p((1.0 / x)) / n;
	} else if ((1.0 / n) <= 1e+134) {
		tmp = 1.0 * -Math.expm1((t_0 - (x / n)));
	} else {
		tmp = -1.0 * ((1.0 + (-1.0 * (Math.log((1.0 / x)) / n))) / (n * x));
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.log(x) / n
	tmp = 0
	if (1.0 / n) <= -0.05:
		tmp = math.exp(t_0) / (n * x)
	elif (1.0 / n) <= 4e-5:
		tmp = math.log1p((1.0 / x)) / n
	elif (1.0 / n) <= 1e+134:
		tmp = 1.0 * -math.expm1((t_0 - (x / n)))
	else:
		tmp = -1.0 * ((1.0 + (-1.0 * (math.log((1.0 / x)) / n))) / (n * x))
	return tmp

Julia

function code(x, n)
	t_0 = Float64(log(x) / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -0.05)
		tmp = Float64(exp(t_0) / Float64(n * x));
	elseif (Float64(1.0 / n) <= 4e-5)
		tmp = Float64(log1p(Float64(1.0 / x)) / n);
	elseif (Float64(1.0 / n) <= 1e+134)
		tmp = Float64(1.0 * Float64(-expm1(Float64(t_0 - Float64(x / n)))));
	else
		tmp = Float64(-1.0 * Float64(Float64(1.0 + Float64(-1.0 * Float64(log(Float64(1.0 / x)) / n))) / Float64(n * x)));
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(N[Log[x], $MachinePrecision] / n), $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -0.05], N[(N[Exp[t$95$0], $MachinePrecision] / N[(n * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 4e-5], N[(N[Log[1 + N[(1.0 / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 1e+134], N[(1.0 * (-N[(Exp[N[(t$95$0 - N[(x / n), $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision])), $MachinePrecision], N[(-1.0 * N[(N[(1.0 + N[(-1.0 * N[(N[Log[N[(1.0 / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(n * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -0.050000000000000003

   1. Initial program 53.3%

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

   2. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower-exp.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-log.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 57.8

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

   4. Applied rewrites 57.8%

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

      1. lift-*.f64 N/A

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

      2. mul-1-neg N/A

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

      3. lift-/.f64 N/A

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

      4. distribute-neg-frac2 N/A

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

      5. lift-log.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. log-rec N/A

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

      8. frac-2neg N/A

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

      9. lower-/.f64 N/A

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

      10. lower-log.f64 57.8

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

   6. Applied rewrites 57.8%

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

   if -0.050000000000000003 < (/.f64 #s(literal 1 binary64) n) < 4.00000000000000033e-5

   1. Initial program 53.3%

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

   2. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-log.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-log.f64 58.9

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

   4. Applied rewrites 58.9%

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

      1. lift--.f64 N/A

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

      2. lift-log.f64 N/A

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

      3. lift-log.f64 N/A

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

      4. diff-log N/A

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. *-lft-identity N/A

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

      8. add-to-fraction N/A

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

      9. lower-log1p.f64 N/A

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

      10. lower-/.f64 58.3

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

   6. Applied rewrites 58.3%

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

   if 4.00000000000000033e-5 < (/.f64 #s(literal 1 binary64) n) < 9.99999999999999921e133

   1. Initial program 53.3%

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

      1. lift--.f64 N/A

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

      2. sub-to-mult N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. add-flip N/A

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

      7. metadata-eval N/A

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

      8. lower--.f64 N/A

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

      9. sub-negate-rev N/A

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

      10. lower-neg.f64 N/A

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

      11. lift-pow.f64 N/A

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

      12. pow-to-exp N/A

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

      13. lift-pow.f64 N/A

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

      14. pow-to-exp N/A

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

      15. div-exp N/A

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

      16. lower-expm1.f64 N/A

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

      17. lower--.f64 N/A

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

   3. Applied rewrites 78.6%

      \[\leadsto \color{blue}{{\left(x - -1\right)}^{\left(\frac{1}{n}\right)} \cdot \left(-\mathsf{expm1}\left(\frac{\log x}{n} - \frac{\log \left(x - -1\right)}{n}\right)\right)} \]

   4. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1} \cdot \left(-\mathsf{expm1}\left(\frac{\log x}{n} - \frac{\log \left(x - -1\right)}{n}\right)\right) \]
   5. Step-by-step derivation

   6. Applied rewrites 78.6%

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

   7. Taylor expanded in x around 0

      \[\leadsto 1 \cdot \left(-\mathsf{expm1}\left(\frac{\log x}{n} - \color{blue}{\frac{x}{n}}\right)\right) \]
   8. Step-by-step derivation

      1. lower-/.f64 50.7

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

   9. Applied rewrites 50.7%

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

   if 9.99999999999999921e133 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 53.3%

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

   2. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

   4. Applied rewrites 64.5%

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

   5. Taylor expanded in x around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-log.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 21.5

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

   7. Applied rewrites 21.5%

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

Alternative 4: 94.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\log x}{n}\\ \mathbf{if}\;\frac{1}{n} \leq -0.05:\\ \;\;\;\;\frac{e^{t\_0}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 4 \cdot 10^{-5}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(\frac{1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{+158}:\\ \;\;\;\;1 \cdot \left(-\mathsf{expm1}\left(t\_0 - \frac{x}{n}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{n}{x}}{n \cdot n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (/ (log x) n)))
   (if (<= (/ 1.0 n) -0.05)
     (/ (exp t_0) (* n x))
     (if (<= (/ 1.0 n) 4e-5)
       (/ (log1p (/ 1.0 x)) n)
       (if (<= (/ 1.0 n) 2e+158)
         (* 1.0 (- (expm1 (- t_0 (/ x n)))))
         (/ (/ n x) (* n n)))))))

C

double code(double x, double n) {
	double t_0 = log(x) / n;
	double tmp;
	if ((1.0 / n) <= -0.05) {
		tmp = exp(t_0) / (n * x);
	} else if ((1.0 / n) <= 4e-5) {
		tmp = log1p((1.0 / x)) / n;
	} else if ((1.0 / n) <= 2e+158) {
		tmp = 1.0 * -expm1((t_0 - (x / n)));
	} else {
		tmp = (n / x) / (n * n);
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double t_0 = Math.log(x) / n;
	double tmp;
	if ((1.0 / n) <= -0.05) {
		tmp = Math.exp(t_0) / (n * x);
	} else if ((1.0 / n) <= 4e-5) {
		tmp = Math.log1p((1.0 / x)) / n;
	} else if ((1.0 / n) <= 2e+158) {
		tmp = 1.0 * -Math.expm1((t_0 - (x / n)));
	} else {
		tmp = (n / x) / (n * n);
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.log(x) / n
	tmp = 0
	if (1.0 / n) <= -0.05:
		tmp = math.exp(t_0) / (n * x)
	elif (1.0 / n) <= 4e-5:
		tmp = math.log1p((1.0 / x)) / n
	elif (1.0 / n) <= 2e+158:
		tmp = 1.0 * -math.expm1((t_0 - (x / n)))
	else:
		tmp = (n / x) / (n * n)
	return tmp

Julia

function code(x, n)
	t_0 = Float64(log(x) / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -0.05)
		tmp = Float64(exp(t_0) / Float64(n * x));
	elseif (Float64(1.0 / n) <= 4e-5)
		tmp = Float64(log1p(Float64(1.0 / x)) / n);
	elseif (Float64(1.0 / n) <= 2e+158)
		tmp = Float64(1.0 * Float64(-expm1(Float64(t_0 - Float64(x / n)))));
	else
		tmp = Float64(Float64(n / x) / Float64(n * n));
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(N[Log[x], $MachinePrecision] / n), $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -0.05], N[(N[Exp[t$95$0], $MachinePrecision] / N[(n * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 4e-5], N[(N[Log[1 + N[(1.0 / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e+158], N[(1.0 * (-N[(Exp[N[(t$95$0 - N[(x / n), $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision])), $MachinePrecision], N[(N[(n / x), $MachinePrecision] / N[(n * n), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -0.050000000000000003

   1. Initial program 53.3%

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

   2. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower-exp.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-log.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 57.8

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

   4. Applied rewrites 57.8%

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

      1. lift-*.f64 N/A

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

      2. mul-1-neg N/A

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

      3. lift-/.f64 N/A

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

      4. distribute-neg-frac2 N/A

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

      5. lift-log.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. log-rec N/A

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

      8. frac-2neg N/A

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

      9. lower-/.f64 N/A

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

      10. lower-log.f64 57.8

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

   6. Applied rewrites 57.8%

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

   if -0.050000000000000003 < (/.f64 #s(literal 1 binary64) n) < 4.00000000000000033e-5

   1. Initial program 53.3%

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

   2. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-log.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-log.f64 58.9

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

   4. Applied rewrites 58.9%

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

      1. lift--.f64 N/A

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

      2. lift-log.f64 N/A

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

      3. lift-log.f64 N/A

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

      4. diff-log N/A

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. *-lft-identity N/A

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

      8. add-to-fraction N/A

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

      9. lower-log1p.f64 N/A

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

      10. lower-/.f64 58.3

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

   6. Applied rewrites 58.3%

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

   if 4.00000000000000033e-5 < (/.f64 #s(literal 1 binary64) n) < 1.99999999999999991e158

   1. Initial program 53.3%

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

      1. lift--.f64 N/A

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

      2. sub-to-mult N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. add-flip N/A

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

      7. metadata-eval N/A

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

      8. lower--.f64 N/A

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

      9. sub-negate-rev N/A

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

      10. lower-neg.f64 N/A

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

      11. lift-pow.f64 N/A

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

      12. pow-to-exp N/A

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

      13. lift-pow.f64 N/A

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

      14. pow-to-exp N/A

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

      15. div-exp N/A

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

      16. lower-expm1.f64 N/A

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

      17. lower--.f64 N/A

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

   3. Applied rewrites 78.6%

      \[\leadsto \color{blue}{{\left(x - -1\right)}^{\left(\frac{1}{n}\right)} \cdot \left(-\mathsf{expm1}\left(\frac{\log x}{n} - \frac{\log \left(x - -1\right)}{n}\right)\right)} \]

   4. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1} \cdot \left(-\mathsf{expm1}\left(\frac{\log x}{n} - \frac{\log \left(x - -1\right)}{n}\right)\right) \]
   5. Step-by-step derivation

   6. Applied rewrites 78.6%

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

   7. Taylor expanded in x around 0

      \[\leadsto 1 \cdot \left(-\mathsf{expm1}\left(\frac{\log x}{n} - \color{blue}{\frac{x}{n}}\right)\right) \]
   8. Step-by-step derivation

      1. lower-/.f64 50.7

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

   9. Applied rewrites 50.7%

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

   if 1.99999999999999991e158 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 53.3%

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

   2. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-log.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-log.f64 58.9

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

   4. Applied rewrites 58.9%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. div-sub N/A

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

      4. frac-sub N/A

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

      5. *-commutative N/A

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. add-flip N/A

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

      9. metadata-eval N/A

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

      10. lift--.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-/.f64 N/A

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

   6. Applied rewrites 48.1%

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

   7. Taylor expanded in x around inf

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

      1. lower-/.f64 40.5

         \[\leadsto \frac{\frac{n}{x}}{n \cdot n} \]

   9. Applied rewrites 40.5%

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

Alternative 5: 94.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\log x}{n}\\ \mathbf{if}\;\frac{1}{n} \leq -0.05:\\ \;\;\;\;\frac{e^{t\_0}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2:\\ \;\;\;\;\frac{\mathsf{log1p}\left(\frac{1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{+158}:\\ \;\;\;\;1 \cdot \left(-\mathsf{expm1}\left(t\_0\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{n}{x}}{n \cdot n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (/ (log x) n)))
   (if (<= (/ 1.0 n) -0.05)
     (/ (exp t_0) (* n x))
     (if (<= (/ 1.0 n) 2.0)
       (/ (log1p (/ 1.0 x)) n)
       (if (<= (/ 1.0 n) 2e+158)
         (* 1.0 (- (expm1 t_0)))
         (/ (/ n x) (* n n)))))))

C

double code(double x, double n) {
	double t_0 = log(x) / n;
	double tmp;
	if ((1.0 / n) <= -0.05) {
		tmp = exp(t_0) / (n * x);
	} else if ((1.0 / n) <= 2.0) {
		tmp = log1p((1.0 / x)) / n;
	} else if ((1.0 / n) <= 2e+158) {
		tmp = 1.0 * -expm1(t_0);
	} else {
		tmp = (n / x) / (n * n);
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double t_0 = Math.log(x) / n;
	double tmp;
	if ((1.0 / n) <= -0.05) {
		tmp = Math.exp(t_0) / (n * x);
	} else if ((1.0 / n) <= 2.0) {
		tmp = Math.log1p((1.0 / x)) / n;
	} else if ((1.0 / n) <= 2e+158) {
		tmp = 1.0 * -Math.expm1(t_0);
	} else {
		tmp = (n / x) / (n * n);
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.log(x) / n
	tmp = 0
	if (1.0 / n) <= -0.05:
		tmp = math.exp(t_0) / (n * x)
	elif (1.0 / n) <= 2.0:
		tmp = math.log1p((1.0 / x)) / n
	elif (1.0 / n) <= 2e+158:
		tmp = 1.0 * -math.expm1(t_0)
	else:
		tmp = (n / x) / (n * n)
	return tmp

Julia

function code(x, n)
	t_0 = Float64(log(x) / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -0.05)
		tmp = Float64(exp(t_0) / Float64(n * x));
	elseif (Float64(1.0 / n) <= 2.0)
		tmp = Float64(log1p(Float64(1.0 / x)) / n);
	elseif (Float64(1.0 / n) <= 2e+158)
		tmp = Float64(1.0 * Float64(-expm1(t_0)));
	else
		tmp = Float64(Float64(n / x) / Float64(n * n));
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(N[Log[x], $MachinePrecision] / n), $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -0.05], N[(N[Exp[t$95$0], $MachinePrecision] / N[(n * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2.0], N[(N[Log[1 + N[(1.0 / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e+158], N[(1.0 * (-N[(Exp[t$95$0] - 1), $MachinePrecision])), $MachinePrecision], N[(N[(n / x), $MachinePrecision] / N[(n * n), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -0.050000000000000003

   1. Initial program 53.3%

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

   2. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower-exp.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-log.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 57.8

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

   4. Applied rewrites 57.8%

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

      1. lift-*.f64 N/A

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

      2. mul-1-neg N/A

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

      3. lift-/.f64 N/A

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

      4. distribute-neg-frac2 N/A

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

      5. lift-log.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. log-rec N/A

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

      8. frac-2neg N/A

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

      9. lower-/.f64 N/A

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

      10. lower-log.f64 57.8

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

   6. Applied rewrites 57.8%

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

   if -0.050000000000000003 < (/.f64 #s(literal 1 binary64) n) < 2

   1. Initial program 53.3%

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

   2. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-log.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-log.f64 58.9

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

   4. Applied rewrites 58.9%

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

      1. lift--.f64 N/A

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

      2. lift-log.f64 N/A

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

      3. lift-log.f64 N/A

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

      4. diff-log N/A

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. *-lft-identity N/A

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

      8. add-to-fraction N/A

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

      9. lower-log1p.f64 N/A

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

      10. lower-/.f64 58.3

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

   6. Applied rewrites 58.3%

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

   if 2 < (/.f64 #s(literal 1 binary64) n) < 1.99999999999999991e158

   1. Initial program 53.3%

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

      1. lift--.f64 N/A

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

      2. sub-to-mult N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. add-flip N/A

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

      7. metadata-eval N/A

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

      8. lower--.f64 N/A

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

      9. sub-negate-rev N/A

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

      10. lower-neg.f64 N/A

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

      11. lift-pow.f64 N/A

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

      12. pow-to-exp N/A

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

      13. lift-pow.f64 N/A

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

      14. pow-to-exp N/A

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

      15. div-exp N/A

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

      16. lower-expm1.f64 N/A

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

      17. lower--.f64 N/A

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

   3. Applied rewrites 78.6%

      \[\leadsto \color{blue}{{\left(x - -1\right)}^{\left(\frac{1}{n}\right)} \cdot \left(-\mathsf{expm1}\left(\frac{\log x}{n} - \frac{\log \left(x - -1\right)}{n}\right)\right)} \]

   4. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1} \cdot \left(-\mathsf{expm1}\left(\frac{\log x}{n} - \frac{\log \left(x - -1\right)}{n}\right)\right) \]
   5. Step-by-step derivation

   6. Applied rewrites 78.6%

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

   7. Taylor expanded in x around 0

      \[\leadsto 1 \cdot \left(-\mathsf{expm1}\left(\color{blue}{\frac{\log x}{n}}\right)\right) \]
   8. Step-by-step derivation

      1. lower-/.f64 N/A

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

      2. lower-log.f64 50.9

         \[\leadsto 1 \cdot \left(-\mathsf{expm1}\left(\frac{\log x}{n}\right)\right) \]

   9. Applied rewrites 50.9%

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

   if 1.99999999999999991e158 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 53.3%

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

   2. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-log.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-log.f64 58.9

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

   4. Applied rewrites 58.9%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. div-sub N/A

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

      4. frac-sub N/A

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

      5. *-commutative N/A

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. add-flip N/A

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

      9. metadata-eval N/A

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

      10. lift--.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-/.f64 N/A

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

   6. Applied rewrites 48.1%

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

   7. Taylor expanded in x around inf

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

      1. lower-/.f64 40.5

         \[\leadsto \frac{\frac{n}{x}}{n \cdot n} \]

   9. Applied rewrites 40.5%

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

Alternative 6: 94.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -0.05:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2:\\ \;\;\;\;\frac{\mathsf{log1p}\left(\frac{1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{+158}:\\ \;\;\;\;1 \cdot \left(-\mathsf{expm1}\left(\frac{\log x}{n}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{n}{x}}{n \cdot n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= (/ 1.0 n) -0.05)
   (/ (pow x (/ 1.0 n)) (* n x))
   (if (<= (/ 1.0 n) 2.0)
     (/ (log1p (/ 1.0 x)) n)
     (if (<= (/ 1.0 n) 2e+158)
       (* 1.0 (- (expm1 (/ (log x) n))))
       (/ (/ n x) (* n n))))))

C

double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -0.05) {
		tmp = pow(x, (1.0 / n)) / (n * x);
	} else if ((1.0 / n) <= 2.0) {
		tmp = log1p((1.0 / x)) / n;
	} else if ((1.0 / n) <= 2e+158) {
		tmp = 1.0 * -expm1((log(x) / n));
	} else {
		tmp = (n / x) / (n * n);
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -0.05) {
		tmp = Math.pow(x, (1.0 / n)) / (n * x);
	} else if ((1.0 / n) <= 2.0) {
		tmp = Math.log1p((1.0 / x)) / n;
	} else if ((1.0 / n) <= 2e+158) {
		tmp = 1.0 * -Math.expm1((Math.log(x) / n));
	} else {
		tmp = (n / x) / (n * n);
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if (1.0 / n) <= -0.05:
		tmp = math.pow(x, (1.0 / n)) / (n * x)
	elif (1.0 / n) <= 2.0:
		tmp = math.log1p((1.0 / x)) / n
	elif (1.0 / n) <= 2e+158:
		tmp = 1.0 * -math.expm1((math.log(x) / n))
	else:
		tmp = (n / x) / (n * n)
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -0.05)
		tmp = Float64((x ^ Float64(1.0 / n)) / Float64(n * x));
	elseif (Float64(1.0 / n) <= 2.0)
		tmp = Float64(log1p(Float64(1.0 / x)) / n);
	elseif (Float64(1.0 / n) <= 2e+158)
		tmp = Float64(1.0 * Float64(-expm1(Float64(log(x) / n))));
	else
		tmp = Float64(Float64(n / x) / Float64(n * n));
	end
	return tmp
end

Wolfram

code[x_, n_] := If[LessEqual[N[(1.0 / n), $MachinePrecision], -0.05], N[(N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision] / N[(n * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2.0], N[(N[Log[1 + N[(1.0 / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e+158], N[(1.0 * (-N[(Exp[N[(N[Log[x], $MachinePrecision] / n), $MachinePrecision]] - 1), $MachinePrecision])), $MachinePrecision], N[(N[(n / x), $MachinePrecision] / N[(n * n), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -0.050000000000000003

   1. Initial program 53.3%

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

   2. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower-exp.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-log.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 57.8

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

   4. Applied rewrites 57.8%

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

      1. lift-exp.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lift-/.f64 N/A

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

      5. distribute-neg-frac2 N/A

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

      6. lift-log.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. log-rec N/A

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

      9. frac-2neg N/A

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

      10. mult-flip N/A

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

      11. pow-to-exp N/A

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

      12. lower-pow.f64 N/A

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

      13. lower-/.f64 57.8

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

   6. Applied rewrites 57.8%

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

   if -0.050000000000000003 < (/.f64 #s(literal 1 binary64) n) < 2

   1. Initial program 53.3%

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

   2. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-log.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-log.f64 58.9

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

   4. Applied rewrites 58.9%

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

      1. lift--.f64 N/A

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

      2. lift-log.f64 N/A

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

      3. lift-log.f64 N/A

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

      4. diff-log N/A

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. *-lft-identity N/A

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

      8. add-to-fraction N/A

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

      9. lower-log1p.f64 N/A

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

      10. lower-/.f64 58.3

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

   6. Applied rewrites 58.3%

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

   if 2 < (/.f64 #s(literal 1 binary64) n) < 1.99999999999999991e158

   1. Initial program 53.3%

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

      1. lift--.f64 N/A

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

      2. sub-to-mult N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. add-flip N/A

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

      7. metadata-eval N/A

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

      8. lower--.f64 N/A

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

      9. sub-negate-rev N/A

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

      10. lower-neg.f64 N/A

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

      11. lift-pow.f64 N/A

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

      12. pow-to-exp N/A

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

      13. lift-pow.f64 N/A

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

      14. pow-to-exp N/A

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

      15. div-exp N/A

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

      16. lower-expm1.f64 N/A

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

      17. lower--.f64 N/A

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

   3. Applied rewrites 78.6%

      \[\leadsto \color{blue}{{\left(x - -1\right)}^{\left(\frac{1}{n}\right)} \cdot \left(-\mathsf{expm1}\left(\frac{\log x}{n} - \frac{\log \left(x - -1\right)}{n}\right)\right)} \]

   4. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1} \cdot \left(-\mathsf{expm1}\left(\frac{\log x}{n} - \frac{\log \left(x - -1\right)}{n}\right)\right) \]
   5. Step-by-step derivation

   6. Applied rewrites 78.6%

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

   7. Taylor expanded in x around 0

      \[\leadsto 1 \cdot \left(-\mathsf{expm1}\left(\color{blue}{\frac{\log x}{n}}\right)\right) \]
   8. Step-by-step derivation

      1. lower-/.f64 N/A

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

      2. lower-log.f64 50.9

         \[\leadsto 1 \cdot \left(-\mathsf{expm1}\left(\frac{\log x}{n}\right)\right) \]

   9. Applied rewrites 50.9%

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

   if 1.99999999999999991e158 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 53.3%

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

   2. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-log.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-log.f64 58.9

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

   4. Applied rewrites 58.9%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. div-sub N/A

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

      4. frac-sub N/A

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

      5. *-commutative N/A

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. add-flip N/A

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

      9. metadata-eval N/A

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

      10. lift--.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-/.f64 N/A

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

   6. Applied rewrites 48.1%

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

   7. Taylor expanded in x around inf

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

      1. lower-/.f64 40.5

         \[\leadsto \frac{\frac{n}{x}}{n \cdot n} \]

   9. Applied rewrites 40.5%

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

Alternative 7: 94.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -0.05:\\ \;\;\;\;\frac{e^{\frac{\log x}{n}}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2:\\ \;\;\;\;\frac{\mathsf{log1p}\left(\frac{1}{x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + x \cdot \mathsf{fma}\left(x, 0.5 \cdot \frac{1}{{n}^{2}} - 0.5 \cdot \frac{1}{n}, \frac{1}{n}\right)\right) - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= (/ 1.0 n) -0.05)
   (/ (exp (/ (log x) n)) (* n x))
   (if (<= (/ 1.0 n) 2.0)
     (/ (log1p (/ 1.0 x)) n)
     (-
      (+
       1.0
       (*
        x
        (fma x (- (* 0.5 (/ 1.0 (pow n 2.0))) (* 0.5 (/ 1.0 n))) (/ 1.0 n))))
      (pow x (/ 1.0 n))))))

C

double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -0.05) {
		tmp = exp((log(x) / n)) / (n * x);
	} else if ((1.0 / n) <= 2.0) {
		tmp = log1p((1.0 / x)) / n;
	} else {
		tmp = (1.0 + (x * fma(x, ((0.5 * (1.0 / pow(n, 2.0))) - (0.5 * (1.0 / n))), (1.0 / n)))) - pow(x, (1.0 / n));
	}
	return tmp;
}

Julia

function code(x, n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -0.05)
		tmp = Float64(exp(Float64(log(x) / n)) / Float64(n * x));
	elseif (Float64(1.0 / n) <= 2.0)
		tmp = Float64(log1p(Float64(1.0 / x)) / n);
	else
		tmp = Float64(Float64(1.0 + Float64(x * fma(x, Float64(Float64(0.5 * Float64(1.0 / (n ^ 2.0))) - Float64(0.5 * Float64(1.0 / n))), Float64(1.0 / n)))) - (x ^ Float64(1.0 / n)));
	end
	return tmp
end

Wolfram

code[x_, n_] := If[LessEqual[N[(1.0 / n), $MachinePrecision], -0.05], N[(N[Exp[N[(N[Log[x], $MachinePrecision] / n), $MachinePrecision]], $MachinePrecision] / N[(n * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2.0], N[(N[Log[1 + N[(1.0 / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], N[(N[(1.0 + N[(x * N[(x * N[(N[(0.5 * N[(1.0 / N[Power[n, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(0.5 * N[(1.0 / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -0.050000000000000003

   1. Initial program 53.3%

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

   2. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower-exp.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-log.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 57.8

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

   4. Applied rewrites 57.8%

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

      1. lift-*.f64 N/A

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

      2. mul-1-neg N/A

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

      3. lift-/.f64 N/A

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

      4. distribute-neg-frac2 N/A

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

      5. lift-log.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. log-rec N/A

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

      8. frac-2neg N/A

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

      9. lower-/.f64 N/A

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

      10. lower-log.f64 57.8

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

   6. Applied rewrites 57.8%

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

   if -0.050000000000000003 < (/.f64 #s(literal 1 binary64) n) < 2

   1. Initial program 53.3%

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

   2. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-log.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-log.f64 58.9

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

   4. Applied rewrites 58.9%

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

      1. lift--.f64 N/A

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

      2. lift-log.f64 N/A

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

      3. lift-log.f64 N/A

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

      4. diff-log N/A

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. *-lft-identity N/A

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

      8. add-to-fraction N/A

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

      9. lower-log1p.f64 N/A

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

      10. lower-/.f64 58.3

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

   6. Applied rewrites 58.3%

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

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

   1. Initial program 53.3%

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

   2. Taylor expanded in x around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 22.7

          \[\leadsto \left(1 + x \cdot \mathsf{fma}\left(x, 0.5 \cdot \frac{1}{{n}^{2}} - 0.5 \cdot \frac{1}{n}, \frac{1}{n}\right)\right) - {x}^{\left(\frac{1}{n}\right)} \]

   4. Applied rewrites 22.7%

      \[\leadsto \color{blue}{\left(1 + x \cdot \mathsf{fma}\left(x, 0.5 \cdot \frac{1}{{n}^{2}} - 0.5 \cdot \frac{1}{n}, \frac{1}{n}\right)\right)} - {x}^{\left(\frac{1}{n}\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 8: 86.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 \cdot \left(-\mathsf{expm1}\left(\frac{\log x}{n}\right)\right)\\ \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{+150}:\\ \;\;\;\;\frac{1}{n \cdot \frac{n}{\frac{n}{x}}}\\ \mathbf{elif}\;\frac{1}{n} \leq -1 \cdot 10^{+79}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\frac{1}{n} \leq -20000:\\ \;\;\;\;\frac{\log \left(\frac{x - -1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2:\\ \;\;\;\;\frac{\mathsf{log1p}\left(\frac{1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{+158}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{n}{x}}{n \cdot n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (* 1.0 (- (expm1 (/ (log x) n))))))
   (if (<= (/ 1.0 n) -2e+150)
     (/ 1.0 (* n (/ n (/ n x))))
     (if (<= (/ 1.0 n) -1e+79)
       t_0
       (if (<= (/ 1.0 n) -20000.0)
         (/ (log (/ (- x -1.0) x)) n)
         (if (<= (/ 1.0 n) 2.0)
           (/ (log1p (/ 1.0 x)) n)
           (if (<= (/ 1.0 n) 2e+158) t_0 (/ (/ n x) (* n n)))))))))

C

double code(double x, double n) {
	double t_0 = 1.0 * -expm1((log(x) / n));
	double tmp;
	if ((1.0 / n) <= -2e+150) {
		tmp = 1.0 / (n * (n / (n / x)));
	} else if ((1.0 / n) <= -1e+79) {
		tmp = t_0;
	} else if ((1.0 / n) <= -20000.0) {
		tmp = log(((x - -1.0) / x)) / n;
	} else if ((1.0 / n) <= 2.0) {
		tmp = log1p((1.0 / x)) / n;
	} else if ((1.0 / n) <= 2e+158) {
		tmp = t_0;
	} else {
		tmp = (n / x) / (n * n);
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double t_0 = 1.0 * -Math.expm1((Math.log(x) / n));
	double tmp;
	if ((1.0 / n) <= -2e+150) {
		tmp = 1.0 / (n * (n / (n / x)));
	} else if ((1.0 / n) <= -1e+79) {
		tmp = t_0;
	} else if ((1.0 / n) <= -20000.0) {
		tmp = Math.log(((x - -1.0) / x)) / n;
	} else if ((1.0 / n) <= 2.0) {
		tmp = Math.log1p((1.0 / x)) / n;
	} else if ((1.0 / n) <= 2e+158) {
		tmp = t_0;
	} else {
		tmp = (n / x) / (n * n);
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = 1.0 * -math.expm1((math.log(x) / n))
	tmp = 0
	if (1.0 / n) <= -2e+150:
		tmp = 1.0 / (n * (n / (n / x)))
	elif (1.0 / n) <= -1e+79:
		tmp = t_0
	elif (1.0 / n) <= -20000.0:
		tmp = math.log(((x - -1.0) / x)) / n
	elif (1.0 / n) <= 2.0:
		tmp = math.log1p((1.0 / x)) / n
	elif (1.0 / n) <= 2e+158:
		tmp = t_0
	else:
		tmp = (n / x) / (n * n)
	return tmp

Julia

function code(x, n)
	t_0 = Float64(1.0 * Float64(-expm1(Float64(log(x) / n))))
	tmp = 0.0
	if (Float64(1.0 / n) <= -2e+150)
		tmp = Float64(1.0 / Float64(n * Float64(n / Float64(n / x))));
	elseif (Float64(1.0 / n) <= -1e+79)
		tmp = t_0;
	elseif (Float64(1.0 / n) <= -20000.0)
		tmp = Float64(log(Float64(Float64(x - -1.0) / x)) / n);
	elseif (Float64(1.0 / n) <= 2.0)
		tmp = Float64(log1p(Float64(1.0 / x)) / n);
	elseif (Float64(1.0 / n) <= 2e+158)
		tmp = t_0;
	else
		tmp = Float64(Float64(n / x) / Float64(n * n));
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(1.0 * (-N[(Exp[N[(N[Log[x], $MachinePrecision] / n), $MachinePrecision]] - 1), $MachinePrecision])), $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -2e+150], N[(1.0 / N[(n * N[(n / N[(n / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], -1e+79], t$95$0, If[LessEqual[N[(1.0 / n), $MachinePrecision], -20000.0], N[(N[Log[N[(N[(x - -1.0), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2.0], N[(N[Log[1 + N[(1.0 / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e+158], t$95$0, N[(N[(n / x), $MachinePrecision] / N[(n * n), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 5 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -1.99999999999999996e150

   1. Initial program 53.3%

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

   2. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-log.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-log.f64 58.9

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

   4. Applied rewrites 58.9%

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

      1. lift-/.f64 N/A

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

      2. div-flip N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 58.9

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

      5. lift--.f64 N/A

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

      6. lift-log.f64 N/A

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

      7. lift-log.f64 N/A

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

      8. diff-log N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. *-lft-identity N/A

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

      12. add-to-fraction N/A

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

      13. lower-log.f64 N/A

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

      14. add-to-fraction N/A

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

      15. *-lft-identity N/A

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

      16. lift-+.f64 N/A

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

      17. lower-/.f64 58.9

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

      18. lift-+.f64 N/A

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

      19. add-flip N/A

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

      20. metadata-eval N/A

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

      21. lift--.f64 58.9

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

   6. Applied rewrites 58.9%

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

      1. lift-/.f64 N/A

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

      2. div-flip N/A

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

      3. lift-log.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. log-div N/A

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

      6. lift-log.f64 N/A

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

      7. lift-log.f64 N/A

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

      8. sub-div N/A

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

      9. frac-sub N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. lift--.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. div-flip N/A

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

      16. lift-*.f64 N/A

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

      17. associate-/l* N/A

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

      18. lower-*.f64 N/A

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

   8. Applied rewrites 58.6%

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

   9. Taylor expanded in x around inf

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

       1. lower-/.f64 46.2

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

   11. Applied rewrites 46.2%

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

   if -1.99999999999999996e150 < (/.f64 #s(literal 1 binary64) n) < -9.99999999999999967e78 or 2 < (/.f64 #s(literal 1 binary64) n) < 1.99999999999999991e158

   1. Initial program 53.3%

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

      1. lift--.f64 N/A

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

      2. sub-to-mult N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. add-flip N/A

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

      7. metadata-eval N/A

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

      8. lower--.f64 N/A

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

      9. sub-negate-rev N/A

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

      10. lower-neg.f64 N/A

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

      11. lift-pow.f64 N/A

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

      12. pow-to-exp N/A

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

      13. lift-pow.f64 N/A

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

      14. pow-to-exp N/A

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

      15. div-exp N/A

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

      16. lower-expm1.f64 N/A

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

      17. lower--.f64 N/A

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

   3. Applied rewrites 78.6%

      \[\leadsto \color{blue}{{\left(x - -1\right)}^{\left(\frac{1}{n}\right)} \cdot \left(-\mathsf{expm1}\left(\frac{\log x}{n} - \frac{\log \left(x - -1\right)}{n}\right)\right)} \]

   4. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1} \cdot \left(-\mathsf{expm1}\left(\frac{\log x}{n} - \frac{\log \left(x - -1\right)}{n}\right)\right) \]
   5. Step-by-step derivation

   6. Applied rewrites 78.6%

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

   7. Taylor expanded in x around 0

      \[\leadsto 1 \cdot \left(-\mathsf{expm1}\left(\color{blue}{\frac{\log x}{n}}\right)\right) \]
   8. Step-by-step derivation

      1. lower-/.f64 N/A

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

      2. lower-log.f64 50.9

         \[\leadsto 1 \cdot \left(-\mathsf{expm1}\left(\frac{\log x}{n}\right)\right) \]

   9. Applied rewrites 50.9%

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

   if -9.99999999999999967e78 < (/.f64 #s(literal 1 binary64) n) < -2e4

   1. Initial program 53.3%

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

   2. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-log.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-log.f64 58.9

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

   4. Applied rewrites 58.9%

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

      1. lift--.f64 N/A

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

      2. lift-log.f64 N/A

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

      3. lift-log.f64 N/A

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

      4. diff-log N/A

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. *-lft-identity N/A

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

      8. add-to-fraction N/A

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

      9. lower-log.f64 N/A

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

      10. add-to-fraction N/A

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

      11. *-lft-identity N/A

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

      12. lift-+.f64 N/A

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

      13. lower-/.f64 59.0

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

      14. lift-+.f64 N/A

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

      15. add-flip N/A

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

      16. metadata-eval N/A

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

      17. lift--.f64 59.0

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

   6. Applied rewrites 59.0%

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

   if -2e4 < (/.f64 #s(literal 1 binary64) n) < 2

   1. Initial program 53.3%

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

   2. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-log.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-log.f64 58.9

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

   4. Applied rewrites 58.9%

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

      1. lift--.f64 N/A

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

      2. lift-log.f64 N/A

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

      3. lift-log.f64 N/A

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

      4. diff-log N/A

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. *-lft-identity N/A

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

      8. add-to-fraction N/A

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

      9. lower-log1p.f64 N/A

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

      10. lower-/.f64 58.3

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

   6. Applied rewrites 58.3%

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

   if 1.99999999999999991e158 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 53.3%

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

   2. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-log.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-log.f64 58.9

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

   4. Applied rewrites 58.9%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. div-sub N/A

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

      4. frac-sub N/A

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

      5. *-commutative N/A

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. add-flip N/A

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

      9. metadata-eval N/A

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

      10. lift--.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-/.f64 N/A

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

   6. Applied rewrites 48.1%

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

   7. Taylor expanded in x around inf

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

      1. lower-/.f64 40.5

         \[\leadsto \frac{\frac{n}{x}}{n \cdot n} \]

   9. Applied rewrites 40.5%

      \[\leadsto \frac{\frac{n}{x}}{\color{blue}{n} \cdot n} \]
3. Recombined 5 regimes into one program.
4. Add Preprocessing

Alternative 9: 83.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -20000:\\ \;\;\;\;\frac{1}{\frac{\frac{n}{\log \left(\frac{x - -1}{x}\right)} \cdot n}{n}}\\ \mathbf{elif}\;\frac{1}{n} \leq 2:\\ \;\;\;\;\frac{\mathsf{log1p}\left(\frac{1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{+158}:\\ \;\;\;\;1 \cdot \left(-\mathsf{expm1}\left(\frac{\log x}{n}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{n}{x}}{n \cdot n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= (/ 1.0 n) -20000.0)
   (/ 1.0 (/ (* (/ n (log (/ (- x -1.0) x))) n) n))
   (if (<= (/ 1.0 n) 2.0)
     (/ (log1p (/ 1.0 x)) n)
     (if (<= (/ 1.0 n) 2e+158)
       (* 1.0 (- (expm1 (/ (log x) n))))
       (/ (/ n x) (* n n))))))

C

double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -20000.0) {
		tmp = 1.0 / (((n / log(((x - -1.0) / x))) * n) / n);
	} else if ((1.0 / n) <= 2.0) {
		tmp = log1p((1.0 / x)) / n;
	} else if ((1.0 / n) <= 2e+158) {
		tmp = 1.0 * -expm1((log(x) / n));
	} else {
		tmp = (n / x) / (n * n);
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -20000.0) {
		tmp = 1.0 / (((n / Math.log(((x - -1.0) / x))) * n) / n);
	} else if ((1.0 / n) <= 2.0) {
		tmp = Math.log1p((1.0 / x)) / n;
	} else if ((1.0 / n) <= 2e+158) {
		tmp = 1.0 * -Math.expm1((Math.log(x) / n));
	} else {
		tmp = (n / x) / (n * n);
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if (1.0 / n) <= -20000.0:
		tmp = 1.0 / (((n / math.log(((x - -1.0) / x))) * n) / n)
	elif (1.0 / n) <= 2.0:
		tmp = math.log1p((1.0 / x)) / n
	elif (1.0 / n) <= 2e+158:
		tmp = 1.0 * -math.expm1((math.log(x) / n))
	else:
		tmp = (n / x) / (n * n)
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -20000.0)
		tmp = Float64(1.0 / Float64(Float64(Float64(n / log(Float64(Float64(x - -1.0) / x))) * n) / n));
	elseif (Float64(1.0 / n) <= 2.0)
		tmp = Float64(log1p(Float64(1.0 / x)) / n);
	elseif (Float64(1.0 / n) <= 2e+158)
		tmp = Float64(1.0 * Float64(-expm1(Float64(log(x) / n))));
	else
		tmp = Float64(Float64(n / x) / Float64(n * n));
	end
	return tmp
end

Wolfram

code[x_, n_] := If[LessEqual[N[(1.0 / n), $MachinePrecision], -20000.0], N[(1.0 / N[(N[(N[(n / N[Log[N[(N[(x - -1.0), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * n), $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2.0], N[(N[Log[1 + N[(1.0 / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e+158], N[(1.0 * (-N[(Exp[N[(N[Log[x], $MachinePrecision] / n), $MachinePrecision]] - 1), $MachinePrecision])), $MachinePrecision], N[(N[(n / x), $MachinePrecision] / N[(n * n), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -2e4

   1. Initial program 53.3%

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

   2. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-log.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-log.f64 58.9

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

   4. Applied rewrites 58.9%

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

      1. lift-/.f64 N/A

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

      2. div-flip N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 58.9

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

      5. lift--.f64 N/A

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

      6. lift-log.f64 N/A

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

      7. lift-log.f64 N/A

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

      8. diff-log N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. *-lft-identity N/A

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

      12. add-to-fraction N/A

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

      13. lower-log.f64 N/A

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

      14. add-to-fraction N/A

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

      15. *-lft-identity N/A

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

      16. lift-+.f64 N/A

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

      17. lower-/.f64 58.9

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

      18. lift-+.f64 N/A

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

      19. add-flip N/A

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

      20. metadata-eval N/A

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

      21. lift--.f64 58.9

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

   6. Applied rewrites 58.9%

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

      1. lift-/.f64 N/A

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

      2. div-flip N/A

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

      3. lift-log.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. log-div N/A

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

      6. lift-log.f64 N/A

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

      7. lift-log.f64 N/A

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

      8. sub-div N/A

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

      9. frac-sub N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. lift--.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. div-flip N/A

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

      16. lift-*.f64 N/A

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

      17. associate-/l* N/A

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

      18. lower-*.f64 N/A

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

   8. Applied rewrites 58.6%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/r* N/A

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

      6. associate-*l/ N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 55.5

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

   10. Applied rewrites 55.5%

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

   if -2e4 < (/.f64 #s(literal 1 binary64) n) < 2

   1. Initial program 53.3%

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

   2. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-log.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-log.f64 58.9

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

   4. Applied rewrites 58.9%

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

      1. lift--.f64 N/A

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

      2. lift-log.f64 N/A

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

      3. lift-log.f64 N/A

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

      4. diff-log N/A

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. *-lft-identity N/A

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

      8. add-to-fraction N/A

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

      9. lower-log1p.f64 N/A

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

      10. lower-/.f64 58.3

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

   6. Applied rewrites 58.3%

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

   if 2 < (/.f64 #s(literal 1 binary64) n) < 1.99999999999999991e158

   1. Initial program 53.3%

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

      1. lift--.f64 N/A

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

      2. sub-to-mult N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. add-flip N/A

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

      7. metadata-eval N/A

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

      8. lower--.f64 N/A

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

      9. sub-negate-rev N/A

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

      10. lower-neg.f64 N/A

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

      11. lift-pow.f64 N/A

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

      12. pow-to-exp N/A

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

      13. lift-pow.f64 N/A

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

      14. pow-to-exp N/A

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

      15. div-exp N/A

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

      16. lower-expm1.f64 N/A

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

      17. lower--.f64 N/A

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

   3. Applied rewrites 78.6%

      \[\leadsto \color{blue}{{\left(x - -1\right)}^{\left(\frac{1}{n}\right)} \cdot \left(-\mathsf{expm1}\left(\frac{\log x}{n} - \frac{\log \left(x - -1\right)}{n}\right)\right)} \]

   4. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1} \cdot \left(-\mathsf{expm1}\left(\frac{\log x}{n} - \frac{\log \left(x - -1\right)}{n}\right)\right) \]
   5. Step-by-step derivation

   6. Applied rewrites 78.6%

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

   7. Taylor expanded in x around 0

      \[\leadsto 1 \cdot \left(-\mathsf{expm1}\left(\color{blue}{\frac{\log x}{n}}\right)\right) \]
   8. Step-by-step derivation

      1. lower-/.f64 N/A

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

      2. lower-log.f64 50.9

         \[\leadsto 1 \cdot \left(-\mathsf{expm1}\left(\frac{\log x}{n}\right)\right) \]

   9. Applied rewrites 50.9%

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

   if 1.99999999999999991e158 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 53.3%

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

   2. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-log.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-log.f64 58.9

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

   4. Applied rewrites 58.9%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. div-sub N/A

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

      4. frac-sub N/A

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

      5. *-commutative N/A

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. add-flip N/A

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

      9. metadata-eval N/A

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

      10. lift--.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-/.f64 N/A

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

   6. Applied rewrites 48.1%

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

   7. Taylor expanded in x around inf

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

      1. lower-/.f64 40.5

         \[\leadsto \frac{\frac{n}{x}}{n \cdot n} \]

   9. Applied rewrites 40.5%

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

Alternative 10: 78.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{+142}:\\ \;\;\;\;\frac{1}{n \cdot \frac{n}{\frac{n}{x}}}\\ \mathbf{elif}\;\frac{1}{n} \leq -20000:\\ \;\;\;\;\frac{\log \left(\frac{x - -1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{+134}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(\frac{1}{x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) \cdot \frac{1}{n \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= (/ 1.0 n) -5e+142)
   (/ 1.0 (* n (/ n (/ n x))))
   (if (<= (/ 1.0 n) -20000.0)
     (/ (log (/ (- x -1.0) x)) n)
     (if (<= (/ 1.0 n) 1e+134)
       (/ (log1p (/ 1.0 x)) n)
       (* (+ 1.0 (/ x n)) (/ 1.0 (* n x)))))))

C

double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -5e+142) {
		tmp = 1.0 / (n * (n / (n / x)));
	} else if ((1.0 / n) <= -20000.0) {
		tmp = log(((x - -1.0) / x)) / n;
	} else if ((1.0 / n) <= 1e+134) {
		tmp = log1p((1.0 / x)) / n;
	} else {
		tmp = (1.0 + (x / n)) * (1.0 / (n * x));
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -5e+142) {
		tmp = 1.0 / (n * (n / (n / x)));
	} else if ((1.0 / n) <= -20000.0) {
		tmp = Math.log(((x - -1.0) / x)) / n;
	} else if ((1.0 / n) <= 1e+134) {
		tmp = Math.log1p((1.0 / x)) / n;
	} else {
		tmp = (1.0 + (x / n)) * (1.0 / (n * x));
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if (1.0 / n) <= -5e+142:
		tmp = 1.0 / (n * (n / (n / x)))
	elif (1.0 / n) <= -20000.0:
		tmp = math.log(((x - -1.0) / x)) / n
	elif (1.0 / n) <= 1e+134:
		tmp = math.log1p((1.0 / x)) / n
	else:
		tmp = (1.0 + (x / n)) * (1.0 / (n * x))
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -5e+142)
		tmp = Float64(1.0 / Float64(n * Float64(n / Float64(n / x))));
	elseif (Float64(1.0 / n) <= -20000.0)
		tmp = Float64(log(Float64(Float64(x - -1.0) / x)) / n);
	elseif (Float64(1.0 / n) <= 1e+134)
		tmp = Float64(log1p(Float64(1.0 / x)) / n);
	else
		tmp = Float64(Float64(1.0 + Float64(x / n)) * Float64(1.0 / Float64(n * x)));
	end
	return tmp
end

Wolfram

code[x_, n_] := If[LessEqual[N[(1.0 / n), $MachinePrecision], -5e+142], N[(1.0 / N[(n * N[(n / N[(n / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], -20000.0], N[(N[Log[N[(N[(x - -1.0), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 1e+134], N[(N[Log[1 + N[(1.0 / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], N[(N[(1.0 + N[(x / n), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(n * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -5.0000000000000001e142

   1. Initial program 53.3%

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

   2. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-log.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-log.f64 58.9

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

   4. Applied rewrites 58.9%

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

      1. lift-/.f64 N/A

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

      2. div-flip N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 58.9

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

      5. lift--.f64 N/A

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

      6. lift-log.f64 N/A

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

      7. lift-log.f64 N/A

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

      8. diff-log N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

          \[\leadsto \frac{1}{\frac{n}{\log \left(\frac{x + 1}{x}\right)}} \]

      11. *-lft-identity N/A

          \[\leadsto \frac{1}{\frac{n}{\log \left(\frac{1 \cdot x + 1}{x}\right)}} \]

      12. add-to-fraction N/A

          \[\leadsto \frac{1}{\frac{n}{\log \left(1 + \frac{1}{x}\right)}} \]

      13. lower-log.f64 N/A

          \[\leadsto \frac{1}{\frac{n}{\log \left(1 + \frac{1}{x}\right)}} \]

      14. add-to-fraction N/A

          \[\leadsto \frac{1}{\frac{n}{\log \left(\frac{1 \cdot x + 1}{x}\right)}} \]

      15. *-lft-identity N/A

          \[\leadsto \frac{1}{\frac{n}{\log \left(\frac{x + 1}{x}\right)}} \]

      16. lift-+.f64 N/A

          \[\leadsto \frac{1}{\frac{n}{\log \left(\frac{x + 1}{x}\right)}} \]

      17. lower-/.f64 58.9

          \[\leadsto \frac{1}{\frac{n}{\log \left(\frac{x + 1}{x}\right)}} \]

      18. lift-+.f64 N/A

          \[\leadsto \frac{1}{\frac{n}{\log \left(\frac{x + 1}{x}\right)}} \]

      19. add-flip N/A

          \[\leadsto \frac{1}{\frac{n}{\log \left(\frac{x - \left(\mathsf{neg}\left(1\right)\right)}{x}\right)}} \]

      20. metadata-eval N/A

          \[\leadsto \frac{1}{\frac{n}{\log \left(\frac{x - -1}{x}\right)}} \]

      21. lift--.f64 58.9

          \[\leadsto \frac{1}{\frac{n}{\log \left(\frac{x - -1}{x}\right)}} \]

   6. Applied rewrites 58.9%

      \[\leadsto \frac{1}{\color{blue}{\frac{n}{\log \left(\frac{x - -1}{x}\right)}}} \]
   7. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{1}{\frac{n}{\color{blue}{\log \left(\frac{x - -1}{x}\right)}}} \]

      2. div-flip N/A

         \[\leadsto \frac{1}{\frac{1}{\color{blue}{\frac{\log \left(\frac{x - -1}{x}\right)}{n}}}} \]

      3. lift-log.f64 N/A

         \[\leadsto \frac{1}{\frac{1}{\frac{\log \left(\frac{x - -1}{x}\right)}{n}}} \]

      4. lift-/.f64 N/A

         \[\leadsto \frac{1}{\frac{1}{\frac{\log \left(\frac{x - -1}{x}\right)}{n}}} \]

      5. log-div N/A

         \[\leadsto \frac{1}{\frac{1}{\frac{\log \left(x - -1\right) - \log x}{n}}} \]

      6. lift-log.f64 N/A

         \[\leadsto \frac{1}{\frac{1}{\frac{\log \left(x - -1\right) - \log x}{n}}} \]

      7. lift-log.f64 N/A

         \[\leadsto \frac{1}{\frac{1}{\frac{\log \left(x - -1\right) - \log x}{n}}} \]

      8. sub-div N/A

         \[\leadsto \frac{1}{\frac{1}{\frac{\log \left(x - -1\right)}{n} - \color{blue}{\frac{\log x}{n}}}} \]

      9. frac-sub N/A

         \[\leadsto \frac{1}{\frac{1}{\frac{\log \left(x - -1\right) \cdot n - n \cdot \log x}{\color{blue}{n \cdot n}}}} \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{1}{\frac{1}{\frac{\log \left(x - -1\right) \cdot n - n \cdot \log x}{n \cdot n}}} \]

      11. *-commutative N/A

          \[\leadsto \frac{1}{\frac{1}{\frac{\log \left(x - -1\right) \cdot n - \log x \cdot n}{n \cdot n}}} \]

      12. lift-*.f64 N/A

          \[\leadsto \frac{1}{\frac{1}{\frac{\log \left(x - -1\right) \cdot n - \log x \cdot n}{n \cdot n}}} \]

      13. lift--.f64 N/A

          \[\leadsto \frac{1}{\frac{1}{\frac{\log \left(x - -1\right) \cdot n - \log x \cdot n}{\color{blue}{n} \cdot n}}} \]

      14. lift-*.f64 N/A

          \[\leadsto \frac{1}{\frac{1}{\frac{\log \left(x - -1\right) \cdot n - \log x \cdot n}{n \cdot \color{blue}{n}}}} \]

      15. div-flip N/A

          \[\leadsto \frac{1}{\frac{n \cdot n}{\color{blue}{\log \left(x - -1\right) \cdot n - \log x \cdot n}}} \]

      16. lift-*.f64 N/A

          \[\leadsto \frac{1}{\frac{n \cdot n}{\color{blue}{\log \left(x - -1\right) \cdot n} - \log x \cdot n}} \]

      17. associate-/l* N/A

          \[\leadsto \frac{1}{n \cdot \color{blue}{\frac{n}{\log \left(x - -1\right) \cdot n - \log x \cdot n}}} \]

      18. lower-*.f64 N/A

          \[\leadsto \frac{1}{n \cdot \color{blue}{\frac{n}{\log \left(x - -1\right) \cdot n - \log x \cdot n}}} \]

   8. Applied rewrites 58.6%

      \[\leadsto \frac{1}{n \cdot \color{blue}{\frac{n}{\log \left(\frac{x - -1}{x}\right) \cdot n}}} \]

   9. Taylor expanded in x around inf

      \[\leadsto \frac{1}{n \cdot \frac{n}{\frac{n}{\color{blue}{x}}}} \]
   10. Step-by-step derivation

       1. lower-/.f64 46.2

          \[\leadsto \frac{1}{n \cdot \frac{n}{\frac{n}{x}}} \]

   11. Applied rewrites 46.2%

       \[\leadsto \frac{1}{n \cdot \frac{n}{\frac{n}{\color{blue}{x}}}} \]

   if -5.0000000000000001e142 < (/.f64 #s(literal 1 binary64) n) < -2e4

   1. Initial program 53.3%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]

   2. Taylor expanded in n around inf

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{\color{blue}{n}} \]

      2. lower--.f64 N/A

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{n} \]

      3. lower-log.f64 N/A

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{n} \]

      4. lower-+.f64 N/A

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{n} \]

      5. lower-log.f64 58.9

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{n} \]

   4. Applied rewrites 58.9%

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
   5. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{n} \]

      2. lift-log.f64 N/A

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{n} \]

      3. lift-log.f64 N/A

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{n} \]

      4. diff-log N/A

         \[\leadsto \frac{\log \left(\frac{1 + x}{x}\right)}{n} \]

      5. lift-+.f64 N/A

         \[\leadsto \frac{\log \left(\frac{1 + x}{x}\right)}{n} \]

      6. +-commutative N/A

         \[\leadsto \frac{\log \left(\frac{x + 1}{x}\right)}{n} \]

      7. *-lft-identity N/A

         \[\leadsto \frac{\log \left(\frac{1 \cdot x + 1}{x}\right)}{n} \]

      8. add-to-fraction N/A

         \[\leadsto \frac{\log \left(1 + \frac{1}{x}\right)}{n} \]

      9. lower-log.f64 N/A

         \[\leadsto \frac{\log \left(1 + \frac{1}{x}\right)}{n} \]

      10. add-to-fraction N/A

          \[\leadsto \frac{\log \left(\frac{1 \cdot x + 1}{x}\right)}{n} \]

      11. *-lft-identity N/A

          \[\leadsto \frac{\log \left(\frac{x + 1}{x}\right)}{n} \]

      12. lift-+.f64 N/A

          \[\leadsto \frac{\log \left(\frac{x + 1}{x}\right)}{n} \]

      13. lower-/.f64 59.0

          \[\leadsto \frac{\log \left(\frac{x + 1}{x}\right)}{n} \]

      14. lift-+.f64 N/A

          \[\leadsto \frac{\log \left(\frac{x + 1}{x}\right)}{n} \]

      15. add-flip N/A

          \[\leadsto \frac{\log \left(\frac{x - \left(\mathsf{neg}\left(1\right)\right)}{x}\right)}{n} \]

      16. metadata-eval N/A

          \[\leadsto \frac{\log \left(\frac{x - -1}{x}\right)}{n} \]

      17. lift--.f64 59.0

          \[\leadsto \frac{\log \left(\frac{x - -1}{x}\right)}{n} \]

   6. Applied rewrites 59.0%

      \[\leadsto \frac{\log \left(\frac{x - -1}{x}\right)}{\color{blue}{n}} \]

   if -2e4 < (/.f64 #s(literal 1 binary64) n) < 9.99999999999999921e133

   1. Initial program 53.3%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]

   2. Taylor expanded in n around inf

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{\color{blue}{n}} \]

      2. lower--.f64 N/A

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{n} \]

      3. lower-log.f64 N/A

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{n} \]

      4. lower-+.f64 N/A

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{n} \]

      5. lower-log.f64 58.9

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{n} \]

   4. Applied rewrites 58.9%

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
   5. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{n} \]

      2. lift-log.f64 N/A

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{n} \]

      3. lift-log.f64 N/A

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{n} \]

      4. diff-log N/A

         \[\leadsto \frac{\log \left(\frac{1 + x}{x}\right)}{n} \]

      5. lift-+.f64 N/A

         \[\leadsto \frac{\log \left(\frac{1 + x}{x}\right)}{n} \]

      6. +-commutative N/A

         \[\leadsto \frac{\log \left(\frac{x + 1}{x}\right)}{n} \]

      7. *-lft-identity N/A

         \[\leadsto \frac{\log \left(\frac{1 \cdot x + 1}{x}\right)}{n} \]

      8. add-to-fraction N/A

         \[\leadsto \frac{\log \left(1 + \frac{1}{x}\right)}{n} \]

      9. lower-log1p.f64 N/A

         \[\leadsto \frac{\mathsf{log1p}\left(\frac{1}{x}\right)}{n} \]

      10. lower-/.f64 58.3

          \[\leadsto \frac{\mathsf{log1p}\left(\frac{1}{x}\right)}{n} \]

   6. Applied rewrites 58.3%

      \[\leadsto \frac{\mathsf{log1p}\left(\frac{1}{x}\right)}{n} \]

   if 9.99999999999999921e133 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 53.3%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   2. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \color{blue}{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}} \]

      2. sub-to-mult N/A

         \[\leadsto \color{blue}{\left(1 - \frac{{x}^{\left(\frac{1}{n}\right)}}{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}\right) \cdot {\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} \]

      3. *-commutative N/A

         \[\leadsto \color{blue}{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} \cdot \left(1 - \frac{{x}^{\left(\frac{1}{n}\right)}}{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} \cdot \left(1 - \frac{{x}^{\left(\frac{1}{n}\right)}}{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}\right)} \]

      5. lift-+.f64 N/A

         \[\leadsto {\color{blue}{\left(x + 1\right)}}^{\left(\frac{1}{n}\right)} \cdot \left(1 - \frac{{x}^{\left(\frac{1}{n}\right)}}{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}\right) \]

      6. add-flip N/A

         \[\leadsto {\color{blue}{\left(x - \left(\mathsf{neg}\left(1\right)\right)\right)}}^{\left(\frac{1}{n}\right)} \cdot \left(1 - \frac{{x}^{\left(\frac{1}{n}\right)}}{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}\right) \]

      7. metadata-eval N/A

         \[\leadsto {\left(x - \color{blue}{-1}\right)}^{\left(\frac{1}{n}\right)} \cdot \left(1 - \frac{{x}^{\left(\frac{1}{n}\right)}}{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}\right) \]

      8. lower--.f64 N/A

         \[\leadsto {\color{blue}{\left(x - -1\right)}}^{\left(\frac{1}{n}\right)} \cdot \left(1 - \frac{{x}^{\left(\frac{1}{n}\right)}}{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}\right) \]

      9. sub-negate-rev N/A

         \[\leadsto {\left(x - -1\right)}^{\left(\frac{1}{n}\right)} \cdot \color{blue}{\left(\mathsf{neg}\left(\left(\frac{{x}^{\left(\frac{1}{n}\right)}}{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} - 1\right)\right)\right)} \]

      10. lower-neg.f64 N/A

          \[\leadsto {\left(x - -1\right)}^{\left(\frac{1}{n}\right)} \cdot \color{blue}{\left(-\left(\frac{{x}^{\left(\frac{1}{n}\right)}}{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} - 1\right)\right)} \]

      11. lift-pow.f64 N/A

          \[\leadsto {\left(x - -1\right)}^{\left(\frac{1}{n}\right)} \cdot \left(-\left(\frac{\color{blue}{{x}^{\left(\frac{1}{n}\right)}}}{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} - 1\right)\right) \]

      12. pow-to-exp N/A

          \[\leadsto {\left(x - -1\right)}^{\left(\frac{1}{n}\right)} \cdot \left(-\left(\frac{\color{blue}{e^{\log x \cdot \frac{1}{n}}}}{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} - 1\right)\right) \]

      13. lift-pow.f64 N/A

          \[\leadsto {\left(x - -1\right)}^{\left(\frac{1}{n}\right)} \cdot \left(-\left(\frac{e^{\log x \cdot \frac{1}{n}}}{\color{blue}{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}} - 1\right)\right) \]

      14. pow-to-exp N/A

          \[\leadsto {\left(x - -1\right)}^{\left(\frac{1}{n}\right)} \cdot \left(-\left(\frac{e^{\log x \cdot \frac{1}{n}}}{\color{blue}{e^{\log \left(x + 1\right) \cdot \frac{1}{n}}}} - 1\right)\right) \]

      15. div-exp N/A

          \[\leadsto {\left(x - -1\right)}^{\left(\frac{1}{n}\right)} \cdot \left(-\left(\color{blue}{e^{\log x \cdot \frac{1}{n} - \log \left(x + 1\right) \cdot \frac{1}{n}}} - 1\right)\right) \]

      16. lower-expm1.f64 N/A

          \[\leadsto {\left(x - -1\right)}^{\left(\frac{1}{n}\right)} \cdot \left(-\color{blue}{\mathsf{expm1}\left(\log x \cdot \frac{1}{n} - \log \left(x + 1\right) \cdot \frac{1}{n}\right)}\right) \]

      17. lower--.f64 N/A

          \[\leadsto {\left(x - -1\right)}^{\left(\frac{1}{n}\right)} \cdot \left(-\mathsf{expm1}\left(\color{blue}{\log x \cdot \frac{1}{n} - \log \left(x + 1\right) \cdot \frac{1}{n}}\right)\right) \]

   3. Applied rewrites 78.6%

      \[\leadsto \color{blue}{{\left(x - -1\right)}^{\left(\frac{1}{n}\right)} \cdot \left(-\mathsf{expm1}\left(\frac{\log x}{n} - \frac{\log \left(x - -1\right)}{n}\right)\right)} \]

   4. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1} \cdot \left(-\mathsf{expm1}\left(\frac{\log x}{n} - \frac{\log \left(x - -1\right)}{n}\right)\right) \]
   5. Step-by-step derivation

   6. Applied rewrites 78.6%

      \[\leadsto \color{blue}{1} \cdot \left(-\mathsf{expm1}\left(\frac{\log x}{n} - \frac{\log \left(x - -1\right)}{n}\right)\right) \]

   7. Taylor expanded in x around inf

      \[\leadsto 1 \cdot \color{blue}{\frac{1}{n \cdot x}} \]
   8. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto 1 \cdot \frac{1}{\color{blue}{n \cdot x}} \]

      2. lower-*.f64 40.0

         \[\leadsto 1 \cdot \frac{1}{n \cdot \color{blue}{x}} \]

   9. Applied rewrites 40.0%

      \[\leadsto 1 \cdot \color{blue}{\frac{1}{n \cdot x}} \]

   10. Taylor expanded in x around 0

       \[\leadsto \color{blue}{\left(1 + \frac{x}{n}\right)} \cdot \frac{1}{n \cdot x} \]
   11. Step-by-step derivation

       1. lower-+.f64 N/A

          \[\leadsto \left(1 + \color{blue}{\frac{x}{n}}\right) \cdot \frac{1}{n \cdot x} \]

       2. lower-/.f64 31.4

          \[\leadsto \left(1 + \frac{x}{\color{blue}{n}}\right) \cdot \frac{1}{n \cdot x} \]

   12. Applied rewrites 31.4%

       \[\leadsto \color{blue}{\left(1 + \frac{x}{n}\right)} \cdot \frac{1}{n \cdot x} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 11: 70.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\frac{\frac{n}{x}}{n \cdot n}\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\frac{\log \left(\frac{x - -1}{x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) \cdot \frac{1}{n \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (- (pow (+ x 1.0) (/ 1.0 n)) (pow x (/ 1.0 n)))))
   (if (<= t_0 (- INFINITY))
     (/ (/ n x) (* n n))
     (if (<= t_0 0.0)
       (/ (log (/ (- x -1.0) x)) n)
       (* (+ 1.0 (/ x n)) (/ 1.0 (* n x)))))))

C

double code(double x, double n) {
	double t_0 = pow((x + 1.0), (1.0 / n)) - pow(x, (1.0 / n));
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = (n / x) / (n * n);
	} else if (t_0 <= 0.0) {
		tmp = log(((x - -1.0) / x)) / n;
	} else {
		tmp = (1.0 + (x / n)) * (1.0 / (n * x));
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double t_0 = Math.pow((x + 1.0), (1.0 / n)) - Math.pow(x, (1.0 / n));
	double tmp;
	if (t_0 <= -Double.POSITIVE_INFINITY) {
		tmp = (n / x) / (n * n);
	} else if (t_0 <= 0.0) {
		tmp = Math.log(((x - -1.0) / x)) / n;
	} else {
		tmp = (1.0 + (x / n)) * (1.0 / (n * x));
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow((x + 1.0), (1.0 / n)) - math.pow(x, (1.0 / n))
	tmp = 0
	if t_0 <= -math.inf:
		tmp = (n / x) / (n * n)
	elif t_0 <= 0.0:
		tmp = math.log(((x - -1.0) / x)) / n
	else:
		tmp = (1.0 + (x / n)) * (1.0 / (n * x))
	return tmp

Julia

function code(x, n)
	t_0 = Float64((Float64(x + 1.0) ^ Float64(1.0 / n)) - (x ^ Float64(1.0 / n)))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(Float64(n / x) / Float64(n * n));
	elseif (t_0 <= 0.0)
		tmp = Float64(log(Float64(Float64(x - -1.0) / x)) / n);
	else
		tmp = Float64(Float64(1.0 + Float64(x / n)) * Float64(1.0 / Float64(n * x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = ((x + 1.0) ^ (1.0 / n)) - (x ^ (1.0 / n));
	tmp = 0.0;
	if (t_0 <= -Inf)
		tmp = (n / x) / (n * n);
	elseif (t_0 <= 0.0)
		tmp = log(((x - -1.0) / x)) / n;
	else
		tmp = (1.0 + (x / n)) * (1.0 / (n * x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(N[Power[N[(x + 1.0), $MachinePrecision], N[(1.0 / n), $MachinePrecision]], $MachinePrecision] - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[(n / x), $MachinePrecision] / N[(n * n), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.0], N[(N[Log[N[(N[(x - -1.0), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], N[(N[(1.0 + N[(x / n), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(n * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (pow.f64 (+.f64 x #s(literal 1 binary64)) (/.f64 #s(literal 1 binary64) n)) (pow.f64 x (/.f64 #s(literal 1 binary64) n))) < -inf.0

   1. Initial program 53.3%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]

   2. Taylor expanded in n around inf

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{\color{blue}{n}} \]

      2. lower--.f64 N/A

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{n} \]

      3. lower-log.f64 N/A

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{n} \]

      4. lower-+.f64 N/A

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{n} \]

      5. lower-log.f64 58.9

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{n} \]

   4. Applied rewrites 58.9%

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{\color{blue}{n}} \]

      2. lift--.f64 N/A

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{n} \]

      3. div-sub N/A

         \[\leadsto \frac{\log \left(1 + x\right)}{n} - \color{blue}{\frac{\log x}{n}} \]

      4. frac-sub N/A

         \[\leadsto \frac{\log \left(1 + x\right) \cdot n - n \cdot \log x}{\color{blue}{n \cdot n}} \]

      5. *-commutative N/A

         \[\leadsto \frac{n \cdot \log \left(1 + x\right) - n \cdot \log x}{n \cdot n} \]

      6. lift-+.f64 N/A

         \[\leadsto \frac{n \cdot \log \left(1 + x\right) - n \cdot \log x}{n \cdot n} \]

      7. +-commutative N/A

         \[\leadsto \frac{n \cdot \log \left(x + 1\right) - n \cdot \log x}{n \cdot n} \]

      8. add-flip N/A

         \[\leadsto \frac{n \cdot \log \left(x - \left(\mathsf{neg}\left(1\right)\right)\right) - n \cdot \log x}{n \cdot n} \]

      9. metadata-eval N/A

         \[\leadsto \frac{n \cdot \log \left(x - -1\right) - n \cdot \log x}{n \cdot n} \]

      10. lift--.f64 N/A

          \[\leadsto \frac{n \cdot \log \left(x - -1\right) - n \cdot \log x}{n \cdot n} \]

      11. *-commutative N/A

          \[\leadsto \frac{n \cdot \log \left(x - -1\right) - \log x \cdot n}{n \cdot n} \]

      12. lower-/.f64 N/A

          \[\leadsto \frac{n \cdot \log \left(x - -1\right) - \log x \cdot n}{\color{blue}{n \cdot n}} \]

   6. Applied rewrites 48.1%

      \[\leadsto \frac{\log \left(x - -1\right) \cdot n - \log x \cdot n}{\color{blue}{n \cdot n}} \]

   7. Taylor expanded in x around inf

      \[\leadsto \frac{\frac{n}{x}}{\color{blue}{n} \cdot n} \]
   8. Step-by-step derivation

      1. lower-/.f64 40.5

         \[\leadsto \frac{\frac{n}{x}}{n \cdot n} \]

   9. Applied rewrites 40.5%

      \[\leadsto \frac{\frac{n}{x}}{\color{blue}{n} \cdot n} \]

   if -inf.0 < (-.f64 (pow.f64 (+.f64 x #s(literal 1 binary64)) (/.f64 #s(literal 1 binary64) n)) (pow.f64 x (/.f64 #s(literal 1 binary64) n))) < 0.0

   1. Initial program 53.3%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]

   2. Taylor expanded in n around inf

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{\color{blue}{n}} \]

      2. lower--.f64 N/A

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{n} \]

      3. lower-log.f64 N/A

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{n} \]

      4. lower-+.f64 N/A

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{n} \]

      5. lower-log.f64 58.9

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{n} \]

   4. Applied rewrites 58.9%

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
   5. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{n} \]

      2. lift-log.f64 N/A

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{n} \]

      3. lift-log.f64 N/A

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{n} \]

      4. diff-log N/A

         \[\leadsto \frac{\log \left(\frac{1 + x}{x}\right)}{n} \]

      5. lift-+.f64 N/A

         \[\leadsto \frac{\log \left(\frac{1 + x}{x}\right)}{n} \]

      6. +-commutative N/A

         \[\leadsto \frac{\log \left(\frac{x + 1}{x}\right)}{n} \]

      7. *-lft-identity N/A

         \[\leadsto \frac{\log \left(\frac{1 \cdot x + 1}{x}\right)}{n} \]

      8. add-to-fraction N/A

         \[\leadsto \frac{\log \left(1 + \frac{1}{x}\right)}{n} \]

      9. lower-log.f64 N/A

         \[\leadsto \frac{\log \left(1 + \frac{1}{x}\right)}{n} \]

      10. add-to-fraction N/A

          \[\leadsto \frac{\log \left(\frac{1 \cdot x + 1}{x}\right)}{n} \]

      11. *-lft-identity N/A

          \[\leadsto \frac{\log \left(\frac{x + 1}{x}\right)}{n} \]

      12. lift-+.f64 N/A

          \[\leadsto \frac{\log \left(\frac{x + 1}{x}\right)}{n} \]

      13. lower-/.f64 59.0

          \[\leadsto \frac{\log \left(\frac{x + 1}{x}\right)}{n} \]

      14. lift-+.f64 N/A

          \[\leadsto \frac{\log \left(\frac{x + 1}{x}\right)}{n} \]

      15. add-flip N/A

          \[\leadsto \frac{\log \left(\frac{x - \left(\mathsf{neg}\left(1\right)\right)}{x}\right)}{n} \]

      16. metadata-eval N/A

          \[\leadsto \frac{\log \left(\frac{x - -1}{x}\right)}{n} \]

      17. lift--.f64 59.0

          \[\leadsto \frac{\log \left(\frac{x - -1}{x}\right)}{n} \]

   6. Applied rewrites 59.0%

      \[\leadsto \frac{\log \left(\frac{x - -1}{x}\right)}{\color{blue}{n}} \]

   if 0.0 < (-.f64 (pow.f64 (+.f64 x #s(literal 1 binary64)) (/.f64 #s(literal 1 binary64) n)) (pow.f64 x (/.f64 #s(literal 1 binary64) n)))

   1. Initial program 53.3%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   2. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \color{blue}{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}} \]

      2. sub-to-mult N/A

         \[\leadsto \color{blue}{\left(1 - \frac{{x}^{\left(\frac{1}{n}\right)}}{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}\right) \cdot {\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} \]

      3. *-commutative N/A

         \[\leadsto \color{blue}{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} \cdot \left(1 - \frac{{x}^{\left(\frac{1}{n}\right)}}{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} \cdot \left(1 - \frac{{x}^{\left(\frac{1}{n}\right)}}{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}\right)} \]

      5. lift-+.f64 N/A

         \[\leadsto {\color{blue}{\left(x + 1\right)}}^{\left(\frac{1}{n}\right)} \cdot \left(1 - \frac{{x}^{\left(\frac{1}{n}\right)}}{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}\right) \]

      6. add-flip N/A

         \[\leadsto {\color{blue}{\left(x - \left(\mathsf{neg}\left(1\right)\right)\right)}}^{\left(\frac{1}{n}\right)} \cdot \left(1 - \frac{{x}^{\left(\frac{1}{n}\right)}}{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}\right) \]

      7. metadata-eval N/A

         \[\leadsto {\left(x - \color{blue}{-1}\right)}^{\left(\frac{1}{n}\right)} \cdot \left(1 - \frac{{x}^{\left(\frac{1}{n}\right)}}{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}\right) \]

      8. lower--.f64 N/A

         \[\leadsto {\color{blue}{\left(x - -1\right)}}^{\left(\frac{1}{n}\right)} \cdot \left(1 - \frac{{x}^{\left(\frac{1}{n}\right)}}{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}\right) \]

      9. sub-negate-rev N/A

         \[\leadsto {\left(x - -1\right)}^{\left(\frac{1}{n}\right)} \cdot \color{blue}{\left(\mathsf{neg}\left(\left(\frac{{x}^{\left(\frac{1}{n}\right)}}{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} - 1\right)\right)\right)} \]

      10. lower-neg.f64 N/A

          \[\leadsto {\left(x - -1\right)}^{\left(\frac{1}{n}\right)} \cdot \color{blue}{\left(-\left(\frac{{x}^{\left(\frac{1}{n}\right)}}{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} - 1\right)\right)} \]

      11. lift-pow.f64 N/A

          \[\leadsto {\left(x - -1\right)}^{\left(\frac{1}{n}\right)} \cdot \left(-\left(\frac{\color{blue}{{x}^{\left(\frac{1}{n}\right)}}}{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} - 1\right)\right) \]

      12. pow-to-exp N/A

          \[\leadsto {\left(x - -1\right)}^{\left(\frac{1}{n}\right)} \cdot \left(-\left(\frac{\color{blue}{e^{\log x \cdot \frac{1}{n}}}}{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} - 1\right)\right) \]

      13. lift-pow.f64 N/A

          \[\leadsto {\left(x - -1\right)}^{\left(\frac{1}{n}\right)} \cdot \left(-\left(\frac{e^{\log x \cdot \frac{1}{n}}}{\color{blue}{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}} - 1\right)\right) \]

      14. pow-to-exp N/A

          \[\leadsto {\left(x - -1\right)}^{\left(\frac{1}{n}\right)} \cdot \left(-\left(\frac{e^{\log x \cdot \frac{1}{n}}}{\color{blue}{e^{\log \left(x + 1\right) \cdot \frac{1}{n}}}} - 1\right)\right) \]

      15. div-exp N/A

          \[\leadsto {\left(x - -1\right)}^{\left(\frac{1}{n}\right)} \cdot \left(-\left(\color{blue}{e^{\log x \cdot \frac{1}{n} - \log \left(x + 1\right) \cdot \frac{1}{n}}} - 1\right)\right) \]

      16. lower-expm1.f64 N/A

          \[\leadsto {\left(x - -1\right)}^{\left(\frac{1}{n}\right)} \cdot \left(-\color{blue}{\mathsf{expm1}\left(\log x \cdot \frac{1}{n} - \log \left(x + 1\right) \cdot \frac{1}{n}\right)}\right) \]

      17. lower--.f64 N/A

          \[\leadsto {\left(x - -1\right)}^{\left(\frac{1}{n}\right)} \cdot \left(-\mathsf{expm1}\left(\color{blue}{\log x \cdot \frac{1}{n} - \log \left(x + 1\right) \cdot \frac{1}{n}}\right)\right) \]

   3. Applied rewrites 78.6%

      \[\leadsto \color{blue}{{\left(x - -1\right)}^{\left(\frac{1}{n}\right)} \cdot \left(-\mathsf{expm1}\left(\frac{\log x}{n} - \frac{\log \left(x - -1\right)}{n}\right)\right)} \]

   4. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1} \cdot \left(-\mathsf{expm1}\left(\frac{\log x}{n} - \frac{\log \left(x - -1\right)}{n}\right)\right) \]
   5. Step-by-step derivation

   6. Applied rewrites 78.6%

      \[\leadsto \color{blue}{1} \cdot \left(-\mathsf{expm1}\left(\frac{\log x}{n} - \frac{\log \left(x - -1\right)}{n}\right)\right) \]

   7. Taylor expanded in x around inf

      \[\leadsto 1 \cdot \color{blue}{\frac{1}{n \cdot x}} \]
   8. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto 1 \cdot \frac{1}{\color{blue}{n \cdot x}} \]

      2. lower-*.f64 40.0

         \[\leadsto 1 \cdot \frac{1}{n \cdot \color{blue}{x}} \]

   9. Applied rewrites 40.0%

      \[\leadsto 1 \cdot \color{blue}{\frac{1}{n \cdot x}} \]

   10. Taylor expanded in x around 0

       \[\leadsto \color{blue}{\left(1 + \frac{x}{n}\right)} \cdot \frac{1}{n \cdot x} \]
   11. Step-by-step derivation

       1. lower-+.f64 N/A

          \[\leadsto \left(1 + \color{blue}{\frac{x}{n}}\right) \cdot \frac{1}{n \cdot x} \]

       2. lower-/.f64 31.4

          \[\leadsto \left(1 + \frac{x}{\color{blue}{n}}\right) \cdot \frac{1}{n \cdot x} \]

   12. Applied rewrites 31.4%

       \[\leadsto \color{blue}{\left(1 + \frac{x}{n}\right)} \cdot \frac{1}{n \cdot x} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 12: 70.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\\ t_1 := \frac{\frac{n}{x}}{n \cdot n}\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\frac{\log \left(\frac{x - -1}{x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (- (pow (+ x 1.0) (/ 1.0 n)) (pow x (/ 1.0 n))))
        (t_1 (/ (/ n x) (* n n))))
   (if (<= t_0 (- INFINITY))
     t_1
     (if (<= t_0 0.0) (/ (log (/ (- x -1.0) x)) n) t_1))))

C

double code(double x, double n) {
	double t_0 = pow((x + 1.0), (1.0 / n)) - pow(x, (1.0 / n));
	double t_1 = (n / x) / (n * n);
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = t_1;
	} else if (t_0 <= 0.0) {
		tmp = log(((x - -1.0) / x)) / n;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double t_0 = Math.pow((x + 1.0), (1.0 / n)) - Math.pow(x, (1.0 / n));
	double t_1 = (n / x) / (n * n);
	double tmp;
	if (t_0 <= -Double.POSITIVE_INFINITY) {
		tmp = t_1;
	} else if (t_0 <= 0.0) {
		tmp = Math.log(((x - -1.0) / x)) / n;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow((x + 1.0), (1.0 / n)) - math.pow(x, (1.0 / n))
	t_1 = (n / x) / (n * n)
	tmp = 0
	if t_0 <= -math.inf:
		tmp = t_1
	elif t_0 <= 0.0:
		tmp = math.log(((x - -1.0) / x)) / n
	else:
		tmp = t_1
	return tmp

Julia

function code(x, n)
	t_0 = Float64((Float64(x + 1.0) ^ Float64(1.0 / n)) - (x ^ Float64(1.0 / n)))
	t_1 = Float64(Float64(n / x) / Float64(n * n))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = t_1;
	elseif (t_0 <= 0.0)
		tmp = Float64(log(Float64(Float64(x - -1.0) / x)) / n);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = ((x + 1.0) ^ (1.0 / n)) - (x ^ (1.0 / n));
	t_1 = (n / x) / (n * n);
	tmp = 0.0;
	if (t_0 <= -Inf)
		tmp = t_1;
	elseif (t_0 <= 0.0)
		tmp = log(((x - -1.0) / x)) / n;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(N[Power[N[(x + 1.0), $MachinePrecision], N[(1.0 / n), $MachinePrecision]], $MachinePrecision] - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(n / x), $MachinePrecision] / N[(n * n), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], t$95$1, If[LessEqual[t$95$0, 0.0], N[(N[Log[N[(N[(x - -1.0), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (pow.f64 (+.f64 x #s(literal 1 binary64)) (/.f64 #s(literal 1 binary64) n)) (pow.f64 x (/.f64 #s(literal 1 binary64) n))) < -inf.0 or 0.0 < (-.f64 (pow.f64 (+.f64 x #s(literal 1 binary64)) (/.f64 #s(literal 1 binary64) n)) (pow.f64 x (/.f64 #s(literal 1 binary64) n)))

   1. Initial program 53.3%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]

   2. Taylor expanded in n around inf

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{\color{blue}{n}} \]

      2. lower--.f64 N/A

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{n} \]

      3. lower-log.f64 N/A

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{n} \]

      4. lower-+.f64 N/A

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{n} \]

      5. lower-log.f64 58.9

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{n} \]

   4. Applied rewrites 58.9%

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{\color{blue}{n}} \]

      2. lift--.f64 N/A

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{n} \]

      3. div-sub N/A

         \[\leadsto \frac{\log \left(1 + x\right)}{n} - \color{blue}{\frac{\log x}{n}} \]

      4. frac-sub N/A

         \[\leadsto \frac{\log \left(1 + x\right) \cdot n - n \cdot \log x}{\color{blue}{n \cdot n}} \]

      5. *-commutative N/A

         \[\leadsto \frac{n \cdot \log \left(1 + x\right) - n \cdot \log x}{n \cdot n} \]

      6. lift-+.f64 N/A

         \[\leadsto \frac{n \cdot \log \left(1 + x\right) - n \cdot \log x}{n \cdot n} \]

      7. +-commutative N/A

         \[\leadsto \frac{n \cdot \log \left(x + 1\right) - n \cdot \log x}{n \cdot n} \]

      8. add-flip N/A

         \[\leadsto \frac{n \cdot \log \left(x - \left(\mathsf{neg}\left(1\right)\right)\right) - n \cdot \log x}{n \cdot n} \]

      9. metadata-eval N/A

         \[\leadsto \frac{n \cdot \log \left(x - -1\right) - n \cdot \log x}{n \cdot n} \]

      10. lift--.f64 N/A

          \[\leadsto \frac{n \cdot \log \left(x - -1\right) - n \cdot \log x}{n \cdot n} \]

      11. *-commutative N/A

          \[\leadsto \frac{n \cdot \log \left(x - -1\right) - \log x \cdot n}{n \cdot n} \]

      12. lower-/.f64 N/A

          \[\leadsto \frac{n \cdot \log \left(x - -1\right) - \log x \cdot n}{\color{blue}{n \cdot n}} \]

   6. Applied rewrites 48.1%

      \[\leadsto \frac{\log \left(x - -1\right) \cdot n - \log x \cdot n}{\color{blue}{n \cdot n}} \]

   7. Taylor expanded in x around inf

      \[\leadsto \frac{\frac{n}{x}}{\color{blue}{n} \cdot n} \]
   8. Step-by-step derivation

      1. lower-/.f64 40.5

         \[\leadsto \frac{\frac{n}{x}}{n \cdot n} \]

   9. Applied rewrites 40.5%

      \[\leadsto \frac{\frac{n}{x}}{\color{blue}{n} \cdot n} \]

   if -inf.0 < (-.f64 (pow.f64 (+.f64 x #s(literal 1 binary64)) (/.f64 #s(literal 1 binary64) n)) (pow.f64 x (/.f64 #s(literal 1 binary64) n))) < 0.0

   1. Initial program 53.3%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]

   2. Taylor expanded in n around inf

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{\color{blue}{n}} \]

      2. lower--.f64 N/A

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{n} \]

      3. lower-log.f64 N/A

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{n} \]

      4. lower-+.f64 N/A

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{n} \]

      5. lower-log.f64 58.9

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{n} \]

   4. Applied rewrites 58.9%

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
   5. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{n} \]

      2. lift-log.f64 N/A

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{n} \]

      3. lift-log.f64 N/A

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{n} \]

      4. diff-log N/A

         \[\leadsto \frac{\log \left(\frac{1 + x}{x}\right)}{n} \]

      5. lift-+.f64 N/A

         \[\leadsto \frac{\log \left(\frac{1 + x}{x}\right)}{n} \]

      6. +-commutative N/A

         \[\leadsto \frac{\log \left(\frac{x + 1}{x}\right)}{n} \]

      7. *-lft-identity N/A

         \[\leadsto \frac{\log \left(\frac{1 \cdot x + 1}{x}\right)}{n} \]

      8. add-to-fraction N/A

         \[\leadsto \frac{\log \left(1 + \frac{1}{x}\right)}{n} \]

      9. lower-log.f64 N/A

         \[\leadsto \frac{\log \left(1 + \frac{1}{x}\right)}{n} \]

      10. add-to-fraction N/A

          \[\leadsto \frac{\log \left(\frac{1 \cdot x + 1}{x}\right)}{n} \]

      11. *-lft-identity N/A

          \[\leadsto \frac{\log \left(\frac{x + 1}{x}\right)}{n} \]

      12. lift-+.f64 N/A

          \[\leadsto \frac{\log \left(\frac{x + 1}{x}\right)}{n} \]

      13. lower-/.f64 59.0

          \[\leadsto \frac{\log \left(\frac{x + 1}{x}\right)}{n} \]

      14. lift-+.f64 N/A

          \[\leadsto \frac{\log \left(\frac{x + 1}{x}\right)}{n} \]

      15. add-flip N/A

          \[\leadsto \frac{\log \left(\frac{x - \left(\mathsf{neg}\left(1\right)\right)}{x}\right)}{n} \]

      16. metadata-eval N/A

          \[\leadsto \frac{\log \left(\frac{x - -1}{x}\right)}{n} \]

      17. lift--.f64 59.0

          \[\leadsto \frac{\log \left(\frac{x - -1}{x}\right)}{n} \]

   6. Applied rewrites 59.0%

      \[\leadsto \frac{\log \left(\frac{x - -1}{x}\right)}{\color{blue}{n}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 13: 63.6% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.98:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{n \cdot \frac{n}{\frac{n}{x}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 0.98) (/ (- x (log x)) n) (/ 1.0 (* n (/ n (/ n x))))))

C

double code(double x, double n) {
	double tmp;
	if (x <= 0.98) {
		tmp = (x - log(x)) / n;
	} else {
		tmp = 1.0 / (n * (n / (n / x)));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, n)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if (x <= 0.98d0) then
        tmp = (x - log(x)) / n
    else
        tmp = 1.0d0 / (n * (n / (n / x)))
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (x <= 0.98) {
		tmp = (x - Math.log(x)) / n;
	} else {
		tmp = 1.0 / (n * (n / (n / x)));
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 0.98:
		tmp = (x - math.log(x)) / n
	else:
		tmp = 1.0 / (n * (n / (n / x)))
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 0.98)
		tmp = Float64(Float64(x - log(x)) / n);
	else
		tmp = Float64(1.0 / Float64(n * Float64(n / Float64(n / x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (x <= 0.98)
		tmp = (x - log(x)) / n;
	else
		tmp = 1.0 / (n * (n / (n / x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 0.98], N[(N[(x - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], N[(1.0 / N[(n * N[(n / N[(n / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.97999999999999998

   1. Initial program 53.3%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]

   2. Taylor expanded in n around inf

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{\color{blue}{n}} \]

      2. lower--.f64 N/A

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{n} \]

      3. lower-log.f64 N/A

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{n} \]

      4. lower-+.f64 N/A

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{n} \]

      5. lower-log.f64 58.9

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{n} \]

   4. Applied rewrites 58.9%

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]

   5. Taylor expanded in x around 0

      \[\leadsto \frac{x - \log x}{n} \]
   6. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \frac{x - \log x}{n} \]

      2. lower-log.f64 31.0

         \[\leadsto \frac{x - \log x}{n} \]

   7. Applied rewrites 31.0%

      \[\leadsto \frac{x - \log x}{n} \]

   if 0.97999999999999998 < x

   1. Initial program 53.3%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]

   2. Taylor expanded in n around inf

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{\color{blue}{n}} \]

      2. lower--.f64 N/A

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{n} \]

      3. lower-log.f64 N/A

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{n} \]

      4. lower-+.f64 N/A

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{n} \]

      5. lower-log.f64 58.9

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{n} \]

   4. Applied rewrites 58.9%

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{\color{blue}{n}} \]

      2. div-flip N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{n}{\log \left(1 + x\right) - \log x}}} \]

      3. lower-/.f64 N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{n}{\log \left(1 + x\right) - \log x}}} \]

      4. lower-/.f64 58.9

         \[\leadsto \frac{1}{\frac{n}{\color{blue}{\log \left(1 + x\right) - \log x}}} \]

      5. lift--.f64 N/A

         \[\leadsto \frac{1}{\frac{n}{\log \left(1 + x\right) - \color{blue}{\log x}}} \]

      6. lift-log.f64 N/A

         \[\leadsto \frac{1}{\frac{n}{\log \left(1 + x\right) - \log \color{blue}{x}}} \]

      7. lift-log.f64 N/A

         \[\leadsto \frac{1}{\frac{n}{\log \left(1 + x\right) - \log x}} \]

      8. diff-log N/A

         \[\leadsto \frac{1}{\frac{n}{\log \left(\frac{1 + x}{x}\right)}} \]

      9. lift-+.f64 N/A

         \[\leadsto \frac{1}{\frac{n}{\log \left(\frac{1 + x}{x}\right)}} \]

      10. +-commutative N/A

          \[\leadsto \frac{1}{\frac{n}{\log \left(\frac{x + 1}{x}\right)}} \]

      11. *-lft-identity N/A

          \[\leadsto \frac{1}{\frac{n}{\log \left(\frac{1 \cdot x + 1}{x}\right)}} \]

      12. add-to-fraction N/A

          \[\leadsto \frac{1}{\frac{n}{\log \left(1 + \frac{1}{x}\right)}} \]

      13. lower-log.f64 N/A

          \[\leadsto \frac{1}{\frac{n}{\log \left(1 + \frac{1}{x}\right)}} \]

      14. add-to-fraction N/A

          \[\leadsto \frac{1}{\frac{n}{\log \left(\frac{1 \cdot x + 1}{x}\right)}} \]

      15. *-lft-identity N/A

          \[\leadsto \frac{1}{\frac{n}{\log \left(\frac{x + 1}{x}\right)}} \]

      16. lift-+.f64 N/A

          \[\leadsto \frac{1}{\frac{n}{\log \left(\frac{x + 1}{x}\right)}} \]

      17. lower-/.f64 58.9

          \[\leadsto \frac{1}{\frac{n}{\log \left(\frac{x + 1}{x}\right)}} \]

      18. lift-+.f64 N/A

          \[\leadsto \frac{1}{\frac{n}{\log \left(\frac{x + 1}{x}\right)}} \]

      19. add-flip N/A

          \[\leadsto \frac{1}{\frac{n}{\log \left(\frac{x - \left(\mathsf{neg}\left(1\right)\right)}{x}\right)}} \]

      20. metadata-eval N/A

          \[\leadsto \frac{1}{\frac{n}{\log \left(\frac{x - -1}{x}\right)}} \]

      21. lift--.f64 58.9

          \[\leadsto \frac{1}{\frac{n}{\log \left(\frac{x - -1}{x}\right)}} \]

   6. Applied rewrites 58.9%

      \[\leadsto \frac{1}{\color{blue}{\frac{n}{\log \left(\frac{x - -1}{x}\right)}}} \]
   7. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{1}{\frac{n}{\color{blue}{\log \left(\frac{x - -1}{x}\right)}}} \]

      2. div-flip N/A

         \[\leadsto \frac{1}{\frac{1}{\color{blue}{\frac{\log \left(\frac{x - -1}{x}\right)}{n}}}} \]

      3. lift-log.f64 N/A

         \[\leadsto \frac{1}{\frac{1}{\frac{\log \left(\frac{x - -1}{x}\right)}{n}}} \]

      4. lift-/.f64 N/A

         \[\leadsto \frac{1}{\frac{1}{\frac{\log \left(\frac{x - -1}{x}\right)}{n}}} \]

      5. log-div N/A

         \[\leadsto \frac{1}{\frac{1}{\frac{\log \left(x - -1\right) - \log x}{n}}} \]

      6. lift-log.f64 N/A

         \[\leadsto \frac{1}{\frac{1}{\frac{\log \left(x - -1\right) - \log x}{n}}} \]

      7. lift-log.f64 N/A

         \[\leadsto \frac{1}{\frac{1}{\frac{\log \left(x - -1\right) - \log x}{n}}} \]

      8. sub-div N/A

         \[\leadsto \frac{1}{\frac{1}{\frac{\log \left(x - -1\right)}{n} - \color{blue}{\frac{\log x}{n}}}} \]

      9. frac-sub N/A

         \[\leadsto \frac{1}{\frac{1}{\frac{\log \left(x - -1\right) \cdot n - n \cdot \log x}{\color{blue}{n \cdot n}}}} \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{1}{\frac{1}{\frac{\log \left(x - -1\right) \cdot n - n \cdot \log x}{n \cdot n}}} \]

      11. *-commutative N/A

          \[\leadsto \frac{1}{\frac{1}{\frac{\log \left(x - -1\right) \cdot n - \log x \cdot n}{n \cdot n}}} \]

      12. lift-*.f64 N/A

          \[\leadsto \frac{1}{\frac{1}{\frac{\log \left(x - -1\right) \cdot n - \log x \cdot n}{n \cdot n}}} \]

      13. lift--.f64 N/A

          \[\leadsto \frac{1}{\frac{1}{\frac{\log \left(x - -1\right) \cdot n - \log x \cdot n}{\color{blue}{n} \cdot n}}} \]

      14. lift-*.f64 N/A

          \[\leadsto \frac{1}{\frac{1}{\frac{\log \left(x - -1\right) \cdot n - \log x \cdot n}{n \cdot \color{blue}{n}}}} \]

      15. div-flip N/A

          \[\leadsto \frac{1}{\frac{n \cdot n}{\color{blue}{\log \left(x - -1\right) \cdot n - \log x \cdot n}}} \]

      16. lift-*.f64 N/A

          \[\leadsto \frac{1}{\frac{n \cdot n}{\color{blue}{\log \left(x - -1\right) \cdot n} - \log x \cdot n}} \]

      17. associate-/l* N/A

          \[\leadsto \frac{1}{n \cdot \color{blue}{\frac{n}{\log \left(x - -1\right) \cdot n - \log x \cdot n}}} \]

      18. lower-*.f64 N/A

          \[\leadsto \frac{1}{n \cdot \color{blue}{\frac{n}{\log \left(x - -1\right) \cdot n - \log x \cdot n}}} \]

   8. Applied rewrites 58.6%

      \[\leadsto \frac{1}{n \cdot \color{blue}{\frac{n}{\log \left(\frac{x - -1}{x}\right) \cdot n}}} \]

   9. Taylor expanded in x around inf

      \[\leadsto \frac{1}{n \cdot \frac{n}{\frac{n}{\color{blue}{x}}}} \]
   10. Step-by-step derivation

       1. lower-/.f64 46.2

          \[\leadsto \frac{1}{n \cdot \frac{n}{\frac{n}{x}}} \]

   11. Applied rewrites 46.2%

       \[\leadsto \frac{1}{n \cdot \frac{n}{\frac{n}{\color{blue}{x}}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 14: 58.7% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.98:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{elif}\;x \leq 4.4 \cdot 10^{+191}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{n}{x}}{n \cdot n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 0.98)
   (/ (- x (log x)) n)
   (if (<= x 4.4e+191) (/ (/ 1.0 x) n) (/ (/ n x) (* n n)))))

C

double code(double x, double n) {
	double tmp;
	if (x <= 0.98) {
		tmp = (x - log(x)) / n;
	} else if (x <= 4.4e+191) {
		tmp = (1.0 / x) / n;
	} else {
		tmp = (n / x) / (n * n);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, n)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if (x <= 0.98d0) then
        tmp = (x - log(x)) / n
    else if (x <= 4.4d+191) then
        tmp = (1.0d0 / x) / n
    else
        tmp = (n / x) / (n * n)
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (x <= 0.98) {
		tmp = (x - Math.log(x)) / n;
	} else if (x <= 4.4e+191) {
		tmp = (1.0 / x) / n;
	} else {
		tmp = (n / x) / (n * n);
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 0.98:
		tmp = (x - math.log(x)) / n
	elif x <= 4.4e+191:
		tmp = (1.0 / x) / n
	else:
		tmp = (n / x) / (n * n)
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 0.98)
		tmp = Float64(Float64(x - log(x)) / n);
	elseif (x <= 4.4e+191)
		tmp = Float64(Float64(1.0 / x) / n);
	else
		tmp = Float64(Float64(n / x) / Float64(n * n));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (x <= 0.98)
		tmp = (x - log(x)) / n;
	elseif (x <= 4.4e+191)
		tmp = (1.0 / x) / n;
	else
		tmp = (n / x) / (n * n);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 0.98], N[(N[(x - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[x, 4.4e+191], N[(N[(1.0 / x), $MachinePrecision] / n), $MachinePrecision], N[(N[(n / x), $MachinePrecision] / N[(n * n), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 0.97999999999999998

   1. Initial program 53.3%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]

   2. Taylor expanded in n around inf

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{\color{blue}{n}} \]

      2. lower--.f64 N/A

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{n} \]

      3. lower-log.f64 N/A

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{n} \]

      4. lower-+.f64 N/A

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{n} \]

      5. lower-log.f64 58.9

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{n} \]

   4. Applied rewrites 58.9%

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]

   5. Taylor expanded in x around 0

      \[\leadsto \frac{x - \log x}{n} \]
   6. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \frac{x - \log x}{n} \]

      2. lower-log.f64 31.0

         \[\leadsto \frac{x - \log x}{n} \]

   7. Applied rewrites 31.0%

      \[\leadsto \frac{x - \log x}{n} \]

   if 0.97999999999999998 < x < 4.4e191

   1. Initial program 53.3%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]

   2. Taylor expanded in n around inf

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{\color{blue}{n}} \]

      2. lower--.f64 N/A

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{n} \]

      3. lower-log.f64 N/A

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{n} \]

      4. lower-+.f64 N/A

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{n} \]

      5. lower-log.f64 58.9

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{n} \]

   4. Applied rewrites 58.9%

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]

   5. Taylor expanded in x around inf

      \[\leadsto \frac{\frac{1}{x}}{n} \]
   6. Step-by-step derivation

      1. lower-/.f64 40.4

         \[\leadsto \frac{\frac{1}{x}}{n} \]

   7. Applied rewrites 40.4%

      \[\leadsto \frac{\frac{1}{x}}{n} \]

   if 4.4e191 < x

   1. Initial program 53.3%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]

   2. Taylor expanded in n around inf

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{\color{blue}{n}} \]

      2. lower--.f64 N/A

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{n} \]

      3. lower-log.f64 N/A

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{n} \]

      4. lower-+.f64 N/A

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{n} \]

      5. lower-log.f64 58.9

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{n} \]

   4. Applied rewrites 58.9%

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{\color{blue}{n}} \]

      2. lift--.f64 N/A

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{n} \]

      3. div-sub N/A

         \[\leadsto \frac{\log \left(1 + x\right)}{n} - \color{blue}{\frac{\log x}{n}} \]

      4. frac-sub N/A

         \[\leadsto \frac{\log \left(1 + x\right) \cdot n - n \cdot \log x}{\color{blue}{n \cdot n}} \]

      5. *-commutative N/A

         \[\leadsto \frac{n \cdot \log \left(1 + x\right) - n \cdot \log x}{n \cdot n} \]

      6. lift-+.f64 N/A

         \[\leadsto \frac{n \cdot \log \left(1 + x\right) - n \cdot \log x}{n \cdot n} \]

      7. +-commutative N/A

         \[\leadsto \frac{n \cdot \log \left(x + 1\right) - n \cdot \log x}{n \cdot n} \]

      8. add-flip N/A

         \[\leadsto \frac{n \cdot \log \left(x - \left(\mathsf{neg}\left(1\right)\right)\right) - n \cdot \log x}{n \cdot n} \]

      9. metadata-eval N/A

         \[\leadsto \frac{n \cdot \log \left(x - -1\right) - n \cdot \log x}{n \cdot n} \]

      10. lift--.f64 N/A

          \[\leadsto \frac{n \cdot \log \left(x - -1\right) - n \cdot \log x}{n \cdot n} \]

      11. *-commutative N/A

          \[\leadsto \frac{n \cdot \log \left(x - -1\right) - \log x \cdot n}{n \cdot n} \]

      12. lower-/.f64 N/A

          \[\leadsto \frac{n \cdot \log \left(x - -1\right) - \log x \cdot n}{\color{blue}{n \cdot n}} \]

   6. Applied rewrites 48.1%

      \[\leadsto \frac{\log \left(x - -1\right) \cdot n - \log x \cdot n}{\color{blue}{n \cdot n}} \]

   7. Taylor expanded in x around inf

      \[\leadsto \frac{\frac{n}{x}}{\color{blue}{n} \cdot n} \]
   8. Step-by-step derivation

      1. lower-/.f64 40.5

         \[\leadsto \frac{\frac{n}{x}}{n \cdot n} \]

   9. Applied rewrites 40.5%

      \[\leadsto \frac{\frac{n}{x}}{\color{blue}{n} \cdot n} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 15: 58.5% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.55:\\ \;\;\;\;\frac{-\log x}{n}\\ \mathbf{elif}\;x \leq 4.4 \cdot 10^{+191}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{n}{x}}{n \cdot n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 0.55)
   (/ (- (log x)) n)
   (if (<= x 4.4e+191) (/ (/ 1.0 x) n) (/ (/ n x) (* n n)))))

C

double code(double x, double n) {
	double tmp;
	if (x <= 0.55) {
		tmp = -log(x) / n;
	} else if (x <= 4.4e+191) {
		tmp = (1.0 / x) / n;
	} else {
		tmp = (n / x) / (n * n);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, n)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if (x <= 0.55d0) then
        tmp = -log(x) / n
    else if (x <= 4.4d+191) then
        tmp = (1.0d0 / x) / n
    else
        tmp = (n / x) / (n * 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 if (x <= 4.4e+191) {
		tmp = (1.0 / x) / n;
	} else {
		tmp = (n / x) / (n * n);
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 0.55:
		tmp = -math.log(x) / n
	elif x <= 4.4e+191:
		tmp = (1.0 / x) / n
	else:
		tmp = (n / x) / (n * n)
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 0.55)
		tmp = Float64(Float64(-log(x)) / n);
	elseif (x <= 4.4e+191)
		tmp = Float64(Float64(1.0 / x) / n);
	else
		tmp = Float64(Float64(n / x) / Float64(n * n));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (x <= 0.55)
		tmp = -log(x) / n;
	elseif (x <= 4.4e+191)
		tmp = (1.0 / x) / n;
	else
		tmp = (n / x) / (n * n);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 0.55], N[((-N[Log[x], $MachinePrecision]) / n), $MachinePrecision], If[LessEqual[x, 4.4e+191], N[(N[(1.0 / x), $MachinePrecision] / n), $MachinePrecision], N[(N[(n / x), $MachinePrecision] / N[(n * n), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 0.55000000000000004

   1. Initial program 53.3%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]

   2. Taylor expanded in n around inf

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{\color{blue}{n}} \]

      2. lower--.f64 N/A

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{n} \]

      3. lower-log.f64 N/A

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{n} \]

      4. lower-+.f64 N/A

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{n} \]

      5. lower-log.f64 58.9

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{n} \]

   4. Applied rewrites 58.9%

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
   5. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{n} \]

      2. sub-negate-rev N/A

         \[\leadsto \frac{\mathsf{neg}\left(\left(\log x - \log \left(1 + x\right)\right)\right)}{n} \]

      3. lower-neg.f64 N/A

         \[\leadsto \frac{-\left(\log x - \log \left(1 + x\right)\right)}{n} \]

      4. lift-log.f64 N/A

         \[\leadsto \frac{-\left(\log x - \log \left(1 + x\right)\right)}{n} \]

      5. lift-log.f64 N/A

         \[\leadsto \frac{-\left(\log x - \log \left(1 + x\right)\right)}{n} \]

      6. lift-+.f64 N/A

         \[\leadsto \frac{-\left(\log x - \log \left(1 + x\right)\right)}{n} \]

      7. +-commutative N/A

         \[\leadsto \frac{-\left(\log x - \log \left(x + 1\right)\right)}{n} \]

      8. lift-+.f64 N/A

         \[\leadsto \frac{-\left(\log x - \log \left(x + 1\right)\right)}{n} \]

      9. diff-log N/A

         \[\leadsto \frac{-\log \left(\frac{x}{x + 1}\right)}{n} \]

      10. lower-log.f64 N/A

          \[\leadsto \frac{-\log \left(\frac{x}{x + 1}\right)}{n} \]

      11. lower-/.f64 59.0

          \[\leadsto \frac{-\log \left(\frac{x}{x + 1}\right)}{n} \]

      12. lift-+.f64 N/A

          \[\leadsto \frac{-\log \left(\frac{x}{x + 1}\right)}{n} \]

      13. add-flip N/A

          \[\leadsto \frac{-\log \left(\frac{x}{x - \left(\mathsf{neg}\left(1\right)\right)}\right)}{n} \]

      14. metadata-eval N/A

          \[\leadsto \frac{-\log \left(\frac{x}{x - -1}\right)}{n} \]

      15. lift--.f64 59.0

          \[\leadsto \frac{-\log \left(\frac{x}{x - -1}\right)}{n} \]

   6. Applied rewrites 59.0%

      \[\leadsto \frac{-\log \left(\frac{x}{x - -1}\right)}{n} \]

   7. Taylor expanded in x around 0

      \[\leadsto \frac{-\log x}{n} \]
   8. Step-by-step derivation

      1. lower-log.f64 31.0

         \[\leadsto \frac{-\log x}{n} \]

   9. Applied rewrites 31.0%

      \[\leadsto \frac{-\log x}{n} \]

   if 0.55000000000000004 < x < 4.4e191

   1. Initial program 53.3%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]

   2. Taylor expanded in n around inf

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{\color{blue}{n}} \]

      2. lower--.f64 N/A

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{n} \]

      3. lower-log.f64 N/A

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{n} \]

      4. lower-+.f64 N/A

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{n} \]

      5. lower-log.f64 58.9

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{n} \]

   4. Applied rewrites 58.9%

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]

   5. Taylor expanded in x around inf

      \[\leadsto \frac{\frac{1}{x}}{n} \]
   6. Step-by-step derivation

      1. lower-/.f64 40.4

         \[\leadsto \frac{\frac{1}{x}}{n} \]

   7. Applied rewrites 40.4%

      \[\leadsto \frac{\frac{1}{x}}{n} \]

   if 4.4e191 < x

   1. Initial program 53.3%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]

   2. Taylor expanded in n around inf

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{\color{blue}{n}} \]

      2. lower--.f64 N/A

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{n} \]

      3. lower-log.f64 N/A

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{n} \]

      4. lower-+.f64 N/A

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{n} \]

      5. lower-log.f64 58.9

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{n} \]

   4. Applied rewrites 58.9%

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{\color{blue}{n}} \]

      2. lift--.f64 N/A

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{n} \]

      3. div-sub N/A

         \[\leadsto \frac{\log \left(1 + x\right)}{n} - \color{blue}{\frac{\log x}{n}} \]

      4. frac-sub N/A

         \[\leadsto \frac{\log \left(1 + x\right) \cdot n - n \cdot \log x}{\color{blue}{n \cdot n}} \]

      5. *-commutative N/A

         \[\leadsto \frac{n \cdot \log \left(1 + x\right) - n \cdot \log x}{n \cdot n} \]

      6. lift-+.f64 N/A

         \[\leadsto \frac{n \cdot \log \left(1 + x\right) - n \cdot \log x}{n \cdot n} \]

      7. +-commutative N/A

         \[\leadsto \frac{n \cdot \log \left(x + 1\right) - n \cdot \log x}{n \cdot n} \]

      8. add-flip N/A

         \[\leadsto \frac{n \cdot \log \left(x - \left(\mathsf{neg}\left(1\right)\right)\right) - n \cdot \log x}{n \cdot n} \]

      9. metadata-eval N/A

         \[\leadsto \frac{n \cdot \log \left(x - -1\right) - n \cdot \log x}{n \cdot n} \]

      10. lift--.f64 N/A

          \[\leadsto \frac{n \cdot \log \left(x - -1\right) - n \cdot \log x}{n \cdot n} \]

      11. *-commutative N/A

          \[\leadsto \frac{n \cdot \log \left(x - -1\right) - \log x \cdot n}{n \cdot n} \]

      12. lower-/.f64 N/A

          \[\leadsto \frac{n \cdot \log \left(x - -1\right) - \log x \cdot n}{\color{blue}{n \cdot n}} \]

   6. Applied rewrites 48.1%

      \[\leadsto \frac{\log \left(x - -1\right) \cdot n - \log x \cdot n}{\color{blue}{n \cdot n}} \]

   7. Taylor expanded in x around inf

      \[\leadsto \frac{\frac{n}{x}}{\color{blue}{n} \cdot n} \]
   8. Step-by-step derivation

      1. lower-/.f64 40.5

         \[\leadsto \frac{\frac{n}{x}}{n \cdot n} \]

   9. Applied rewrites 40.5%

      \[\leadsto \frac{\frac{n}{x}}{\color{blue}{n} \cdot n} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 16: 40.5% accurate, 4.2× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{n}{x}}{n \cdot n} \end{array} \]

FPCore

(FPCore (x n) :precision binary64 (/ (/ n x) (* n n)))

C

double code(double x, double n) {
	return (n / x) / (n * n);
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, n)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    code = (n / x) / (n * n)
end function

Java

public static double code(double x, double n) {
	return (n / x) / (n * n);
}

Python

def code(x, n):
	return (n / x) / (n * n)

Julia

function code(x, n)
	return Float64(Float64(n / x) / Float64(n * n))
end

MATLAB

function tmp = code(x, n)
	tmp = (n / x) / (n * n);
end

Wolfram

code[x_, n_] := N[(N[(n / x), $MachinePrecision] / N[(n * n), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 53.3%

   \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]

2. Taylor expanded in n around inf

   \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
3. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \frac{\log \left(1 + x\right) - \log x}{\color{blue}{n}} \]

   2. lower--.f64 N/A

      \[\leadsto \frac{\log \left(1 + x\right) - \log x}{n} \]

   3. lower-log.f64 N/A

      \[\leadsto \frac{\log \left(1 + x\right) - \log x}{n} \]

   4. lower-+.f64 N/A

      \[\leadsto \frac{\log \left(1 + x\right) - \log x}{n} \]

   5. lower-log.f64 58.9

      \[\leadsto \frac{\log \left(1 + x\right) - \log x}{n} \]

4. Applied rewrites 58.9%

   \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
5. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \frac{\log \left(1 + x\right) - \log x}{\color{blue}{n}} \]

   2. lift--.f64 N/A

      \[\leadsto \frac{\log \left(1 + x\right) - \log x}{n} \]

   3. div-sub N/A

      \[\leadsto \frac{\log \left(1 + x\right)}{n} - \color{blue}{\frac{\log x}{n}} \]

   4. frac-sub N/A

      \[\leadsto \frac{\log \left(1 + x\right) \cdot n - n \cdot \log x}{\color{blue}{n \cdot n}} \]

   5. *-commutative N/A

      \[\leadsto \frac{n \cdot \log \left(1 + x\right) - n \cdot \log x}{n \cdot n} \]

   6. lift-+.f64 N/A

      \[\leadsto \frac{n \cdot \log \left(1 + x\right) - n \cdot \log x}{n \cdot n} \]

   7. +-commutative N/A

      \[\leadsto \frac{n \cdot \log \left(x + 1\right) - n \cdot \log x}{n \cdot n} \]

   8. add-flip N/A

      \[\leadsto \frac{n \cdot \log \left(x - \left(\mathsf{neg}\left(1\right)\right)\right) - n \cdot \log x}{n \cdot n} \]

   9. metadata-eval N/A

      \[\leadsto \frac{n \cdot \log \left(x - -1\right) - n \cdot \log x}{n \cdot n} \]

   10. lift--.f64 N/A

       \[\leadsto \frac{n \cdot \log \left(x - -1\right) - n \cdot \log x}{n \cdot n} \]

   11. *-commutative N/A

       \[\leadsto \frac{n \cdot \log \left(x - -1\right) - \log x \cdot n}{n \cdot n} \]

   12. lower-/.f64 N/A

       \[\leadsto \frac{n \cdot \log \left(x - -1\right) - \log x \cdot n}{\color{blue}{n \cdot n}} \]

6. Applied rewrites 48.1%

   \[\leadsto \frac{\log \left(x - -1\right) \cdot n - \log x \cdot n}{\color{blue}{n \cdot n}} \]

7. Taylor expanded in x around inf

   \[\leadsto \frac{\frac{n}{x}}{\color{blue}{n} \cdot n} \]
8. Step-by-step derivation

   1. lower-/.f64 40.5

      \[\leadsto \frac{\frac{n}{x}}{n \cdot n} \]

9. Applied rewrites 40.5%

   \[\leadsto \frac{\frac{n}{x}}{\color{blue}{n} \cdot n} \]
10. Add Preprocessing

Alternative 17: 40.4% accurate, 5.8× 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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, n)
use fmin_fmax_functions
    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 53.3%

   \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]

2. Taylor expanded in n around inf

   \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
3. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \frac{\log \left(1 + x\right) - \log x}{\color{blue}{n}} \]

   2. lower--.f64 N/A

      \[\leadsto \frac{\log \left(1 + x\right) - \log x}{n} \]

   3. lower-log.f64 N/A

      \[\leadsto \frac{\log \left(1 + x\right) - \log x}{n} \]

   4. lower-+.f64 N/A

      \[\leadsto \frac{\log \left(1 + x\right) - \log x}{n} \]

   5. lower-log.f64 58.9

      \[\leadsto \frac{\log \left(1 + x\right) - \log x}{n} \]

4. Applied rewrites 58.9%

   \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]

5. Taylor expanded in x around inf

   \[\leadsto \frac{\frac{1}{x}}{n} \]
6. Step-by-step derivation

   1. lower-/.f64 40.4

      \[\leadsto \frac{\frac{1}{x}}{n} \]

7. Applied rewrites 40.4%

   \[\leadsto \frac{\frac{1}{x}}{n} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2025142 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  :precision binary64
  (- (pow (+ x 1.0) (/ 1.0 n)) (pow x (/ 1.0 n))))