2nthrt (problem 3.4.6)

Percentage Accurate: 53.8% → 94.6%

Time: 19.7s

Alternatives: 18

Speedup: 3.0×

• Could not converge on a ground truth

  1. x = -3.94600001630843e-238
  2. n = 2.5284478234169664e-10

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: 53.8% 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: 94.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{-5}:\\ \;\;\;\;\frac{\frac{{x}^{\left(\frac{1}{n}\right)}}{x}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-15}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(\frac{1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{+219}:\\ \;\;\;\;{\left(x - -1\right)}^{\left(\frac{1}{n}\right)} \cdot \left(-\mathsf{expm1}\left(\frac{\log x}{n} - \frac{\log \left(x - -1\right)}{n}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-\frac{-1}{x}}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= (/ 1.0 n) -2e-5)
   (/ (/ (pow x (/ 1.0 n)) x) n)
   (if (<= (/ 1.0 n) 1e-15)
     (/ (log1p (/ 1.0 x)) n)
     (if (<= (/ 1.0 n) 5e+219)
       (*
        (pow (- x -1.0) (/ 1.0 n))
        (- (expm1 (- (/ (log x) n) (/ (log (- x -1.0)) n)))))
       (/ (- (/ -1.0 x)) n)))))

C

double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -2e-5) {
		tmp = (pow(x, (1.0 / n)) / x) / n;
	} else if ((1.0 / n) <= 1e-15) {
		tmp = log1p((1.0 / x)) / n;
	} else if ((1.0 / n) <= 5e+219) {
		tmp = pow((x - -1.0), (1.0 / n)) * -expm1(((log(x) / n) - (log((x - -1.0)) / n)));
	} else {
		tmp = -(-1.0 / x) / n;
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -2e-5) {
		tmp = (Math.pow(x, (1.0 / n)) / x) / n;
	} else if ((1.0 / n) <= 1e-15) {
		tmp = Math.log1p((1.0 / x)) / n;
	} else if ((1.0 / n) <= 5e+219) {
		tmp = Math.pow((x - -1.0), (1.0 / n)) * -Math.expm1(((Math.log(x) / n) - (Math.log((x - -1.0)) / n)));
	} else {
		tmp = -(-1.0 / x) / n;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if (1.0 / n) <= -2e-5:
		tmp = (math.pow(x, (1.0 / n)) / x) / n
	elif (1.0 / n) <= 1e-15:
		tmp = math.log1p((1.0 / x)) / n
	elif (1.0 / n) <= 5e+219:
		tmp = math.pow((x - -1.0), (1.0 / n)) * -math.expm1(((math.log(x) / n) - (math.log((x - -1.0)) / n)))
	else:
		tmp = -(-1.0 / x) / n
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -2e-5)
		tmp = Float64(Float64((x ^ Float64(1.0 / n)) / x) / n);
	elseif (Float64(1.0 / n) <= 1e-15)
		tmp = Float64(log1p(Float64(1.0 / x)) / n);
	elseif (Float64(1.0 / n) <= 5e+219)
		tmp = Float64((Float64(x - -1.0) ^ Float64(1.0 / n)) * Float64(-expm1(Float64(Float64(log(x) / n) - Float64(log(Float64(x - -1.0)) / n)))));
	else
		tmp = Float64(Float64(-Float64(-1.0 / x)) / n);
	end
	return tmp
end

Wolfram

code[x_, n_] := If[LessEqual[N[(1.0 / n), $MachinePrecision], -2e-5], N[(N[(N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision] / x), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 1e-15], N[(N[Log[1 + N[(1.0 / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e+219], N[(N[Power[N[(x - -1.0), $MachinePrecision], N[(1.0 / n), $MachinePrecision]], $MachinePrecision] * (-N[(Exp[N[(N[(N[Log[x], $MachinePrecision] / n), $MachinePrecision] - N[(N[Log[N[(x - -1.0), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision])), $MachinePrecision], N[((-N[(-1.0 / x), $MachinePrecision]) / n), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -2.00000000000000016e-5

   1. Initial program 53.8%

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

   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 57.5

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

   4. Applied rewrites 57.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

   6. Applied rewrites 58.3%

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

   if -2.00000000000000016e-5 < (/.f64 #s(literal 1 binary64) n) < 1.0000000000000001e-15

   1. Initial program 53.8%

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

   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-+.f64 N/A

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

      4. +-commutative N/A

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

      5. add-flip N/A

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

      6. metadata-eval N/A

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

      7. lift--.f64 N/A

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

      8. lift-log.f64 N/A

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

      9. diff-log N/A

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

      10. lift--.f64 N/A

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

      11. metadata-eval N/A

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

      12. add-flip N/A

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

      13. *-lft-identity N/A

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

      14. add-to-fraction N/A

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

      15. lift-/.f64 N/A

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

      16. lower-log1p.f64 57.2

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

   6. Applied rewrites 57.2%

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

   if 1.0000000000000001e-15 < (/.f64 #s(literal 1 binary64) n) < 5e219

   1. Initial program 53.8%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   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 78.9%

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

   if 5e219 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 53.8%

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

   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. add-flip N/A

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

      9. metadata-eval N/A

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

      10. lift--.f64 N/A

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

      11. diff-log N/A

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

      12. lower-log.f64 N/A

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

      13. lower-/.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 inf

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

      1. lower-/.f64 39.7

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

   9. Applied rewrites 39.7%

      \[\leadsto \frac{-\frac{-1}{x}}{n} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 2: 93.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{-5}:\\ \;\;\;\;\frac{\frac{t\_0}{x}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-12}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(\frac{1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{+219}:\\ \;\;\;\;{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{-\frac{-1}{x}}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -2e-5)
     (/ (/ t_0 x) n)
     (if (<= (/ 1.0 n) 1e-12)
       (/ (log1p (/ 1.0 x)) n)
       (if (<= (/ 1.0 n) 5e+219)
         (- (pow (+ x 1.0) (/ 1.0 n)) t_0)
         (/ (- (/ -1.0 x)) n))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -2e-5) {
		tmp = (t_0 / x) / n;
	} else if ((1.0 / n) <= 1e-12) {
		tmp = log1p((1.0 / x)) / n;
	} else if ((1.0 / n) <= 5e+219) {
		tmp = pow((x + 1.0), (1.0 / n)) - t_0;
	} else {
		tmp = -(-1.0 / x) / n;
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -2e-5) {
		tmp = (t_0 / x) / n;
	} else if ((1.0 / n) <= 1e-12) {
		tmp = Math.log1p((1.0 / x)) / n;
	} else if ((1.0 / n) <= 5e+219) {
		tmp = Math.pow((x + 1.0), (1.0 / n)) - t_0;
	} else {
		tmp = -(-1.0 / x) / n;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	tmp = 0
	if (1.0 / n) <= -2e-5:
		tmp = (t_0 / x) / n
	elif (1.0 / n) <= 1e-12:
		tmp = math.log1p((1.0 / x)) / n
	elif (1.0 / n) <= 5e+219:
		tmp = math.pow((x + 1.0), (1.0 / n)) - t_0
	else:
		tmp = -(-1.0 / x) / n
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -2e-5)
		tmp = Float64(Float64(t_0 / x) / n);
	elseif (Float64(1.0 / n) <= 1e-12)
		tmp = Float64(log1p(Float64(1.0 / x)) / n);
	elseif (Float64(1.0 / n) <= 5e+219)
		tmp = Float64((Float64(x + 1.0) ^ Float64(1.0 / n)) - t_0);
	else
		tmp = Float64(Float64(-Float64(-1.0 / x)) / n);
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -2e-5], N[(N[(t$95$0 / x), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 1e-12], N[(N[Log[1 + N[(1.0 / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e+219], N[(N[Power[N[(x + 1.0), $MachinePrecision], N[(1.0 / n), $MachinePrecision]], $MachinePrecision] - t$95$0), $MachinePrecision], N[((-N[(-1.0 / x), $MachinePrecision]) / n), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -2.00000000000000016e-5

   1. Initial program 53.8%

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

   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 57.5

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

   4. Applied rewrites 57.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

   6. Applied rewrites 58.3%

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

   if -2.00000000000000016e-5 < (/.f64 #s(literal 1 binary64) n) < 9.9999999999999998e-13

   1. Initial program 53.8%

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

   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-+.f64 N/A

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

      4. +-commutative N/A

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

      5. add-flip N/A

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

      6. metadata-eval N/A

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

      7. lift--.f64 N/A

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

      8. lift-log.f64 N/A

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

      9. diff-log N/A

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

      10. lift--.f64 N/A

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

      11. metadata-eval N/A

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

      12. add-flip N/A

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

      13. *-lft-identity N/A

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

      14. add-to-fraction N/A

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

      15. lift-/.f64 N/A

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

      16. lower-log1p.f64 57.2

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

   6. Applied rewrites 57.2%

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

   if 9.9999999999999998e-13 < (/.f64 #s(literal 1 binary64) n) < 5e219

   1. Initial program 53.8%

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

   if 5e219 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 53.8%

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

   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. add-flip N/A

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

      9. metadata-eval N/A

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

      10. lift--.f64 N/A

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

      11. diff-log N/A

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

      12. lower-log.f64 N/A

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

      13. lower-/.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 inf

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

      1. lower-/.f64 39.7

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

   9. Applied rewrites 39.7%

      \[\leadsto \frac{-\frac{-1}{x}}{n} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 3: 93.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{-5}:\\ \;\;\;\;\frac{\frac{t\_0}{x}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-12}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(\frac{1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{+219}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{-\frac{-1}{x}}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -2e-5)
     (/ (/ t_0 x) n)
     (if (<= (/ 1.0 n) 1e-12)
       (/ (log1p (/ 1.0 x)) n)
       (if (<= (/ 1.0 n) 5e+219)
         (- (+ 1.0 (/ x n)) t_0)
         (/ (- (/ -1.0 x)) n))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -2e-5) {
		tmp = (t_0 / x) / n;
	} else if ((1.0 / n) <= 1e-12) {
		tmp = log1p((1.0 / x)) / n;
	} else if ((1.0 / n) <= 5e+219) {
		tmp = (1.0 + (x / n)) - t_0;
	} else {
		tmp = -(-1.0 / x) / n;
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -2e-5) {
		tmp = (t_0 / x) / n;
	} else if ((1.0 / n) <= 1e-12) {
		tmp = Math.log1p((1.0 / x)) / n;
	} else if ((1.0 / n) <= 5e+219) {
		tmp = (1.0 + (x / n)) - t_0;
	} else {
		tmp = -(-1.0 / x) / n;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	tmp = 0
	if (1.0 / n) <= -2e-5:
		tmp = (t_0 / x) / n
	elif (1.0 / n) <= 1e-12:
		tmp = math.log1p((1.0 / x)) / n
	elif (1.0 / n) <= 5e+219:
		tmp = (1.0 + (x / n)) - t_0
	else:
		tmp = -(-1.0 / x) / n
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -2e-5)
		tmp = Float64(Float64(t_0 / x) / n);
	elseif (Float64(1.0 / n) <= 1e-12)
		tmp = Float64(log1p(Float64(1.0 / x)) / n);
	elseif (Float64(1.0 / n) <= 5e+219)
		tmp = Float64(Float64(1.0 + Float64(x / n)) - t_0);
	else
		tmp = Float64(Float64(-Float64(-1.0 / x)) / n);
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -2e-5], N[(N[(t$95$0 / x), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 1e-12], N[(N[Log[1 + N[(1.0 / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e+219], N[(N[(1.0 + N[(x / n), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], N[((-N[(-1.0 / x), $MachinePrecision]) / n), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -2.00000000000000016e-5

   1. Initial program 53.8%

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

   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 57.5

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

   4. Applied rewrites 57.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

   6. Applied rewrites 58.3%

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

   if -2.00000000000000016e-5 < (/.f64 #s(literal 1 binary64) n) < 9.9999999999999998e-13

   1. Initial program 53.8%

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

   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-+.f64 N/A

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

      4. +-commutative N/A

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

      5. add-flip N/A

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

      6. metadata-eval N/A

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

      7. lift--.f64 N/A

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

      8. lift-log.f64 N/A

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

      9. diff-log N/A

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

      10. lift--.f64 N/A

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

      11. metadata-eval N/A

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

      12. add-flip N/A

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

      13. *-lft-identity N/A

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

      14. add-to-fraction N/A

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

      15. lift-/.f64 N/A

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

      16. lower-log1p.f64 57.2

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

   6. Applied rewrites 57.2%

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

   if 9.9999999999999998e-13 < (/.f64 #s(literal 1 binary64) n) < 5e219

   1. Initial program 53.8%

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

   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 31.1

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

   4. Applied rewrites 31.1%

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

   if 5e219 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 53.8%

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

   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. add-flip N/A

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

      9. metadata-eval N/A

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

      10. lift--.f64 N/A

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

      11. diff-log N/A

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

      12. lower-log.f64 N/A

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

      13. lower-/.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 inf

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

      1. lower-/.f64 39.7

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

   9. Applied rewrites 39.7%

      \[\leadsto \frac{-\frac{-1}{x}}{n} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 4: 93.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{-5}:\\ \;\;\;\;\frac{\frac{{x}^{\left(\frac{1}{n}\right)}}{x}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-15}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(\frac{1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{+219}:\\ \;\;\;\;1 \cdot \left(-\mathsf{expm1}\left(\frac{\log x}{n}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-\frac{-1}{x}}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= (/ 1.0 n) -2e-5)
   (/ (/ (pow x (/ 1.0 n)) x) n)
   (if (<= (/ 1.0 n) 1e-15)
     (/ (log1p (/ 1.0 x)) n)
     (if (<= (/ 1.0 n) 5e+219)
       (* 1.0 (- (expm1 (/ (log x) n))))
       (/ (- (/ -1.0 x)) n)))))

C

double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -2e-5) {
		tmp = (pow(x, (1.0 / n)) / x) / n;
	} else if ((1.0 / n) <= 1e-15) {
		tmp = log1p((1.0 / x)) / n;
	} else if ((1.0 / n) <= 5e+219) {
		tmp = 1.0 * -expm1((log(x) / n));
	} else {
		tmp = -(-1.0 / x) / n;
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -2e-5) {
		tmp = (Math.pow(x, (1.0 / n)) / x) / n;
	} else if ((1.0 / n) <= 1e-15) {
		tmp = Math.log1p((1.0 / x)) / n;
	} else if ((1.0 / n) <= 5e+219) {
		tmp = 1.0 * -Math.expm1((Math.log(x) / n));
	} else {
		tmp = -(-1.0 / x) / n;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if (1.0 / n) <= -2e-5:
		tmp = (math.pow(x, (1.0 / n)) / x) / n
	elif (1.0 / n) <= 1e-15:
		tmp = math.log1p((1.0 / x)) / n
	elif (1.0 / n) <= 5e+219:
		tmp = 1.0 * -math.expm1((math.log(x) / n))
	else:
		tmp = -(-1.0 / x) / n
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -2e-5)
		tmp = Float64(Float64((x ^ Float64(1.0 / n)) / x) / n);
	elseif (Float64(1.0 / n) <= 1e-15)
		tmp = Float64(log1p(Float64(1.0 / x)) / n);
	elseif (Float64(1.0 / n) <= 5e+219)
		tmp = Float64(1.0 * Float64(-expm1(Float64(log(x) / n))));
	else
		tmp = Float64(Float64(-Float64(-1.0 / x)) / n);
	end
	return tmp
end

Wolfram

code[x_, n_] := If[LessEqual[N[(1.0 / n), $MachinePrecision], -2e-5], N[(N[(N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision] / x), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 1e-15], N[(N[Log[1 + N[(1.0 / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e+219], N[(1.0 * (-N[(Exp[N[(N[Log[x], $MachinePrecision] / n), $MachinePrecision]] - 1), $MachinePrecision])), $MachinePrecision], N[((-N[(-1.0 / x), $MachinePrecision]) / n), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -2.00000000000000016e-5

   1. Initial program 53.8%

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

   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 57.5

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

   4. Applied rewrites 57.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

   6. Applied rewrites 58.3%

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

   if -2.00000000000000016e-5 < (/.f64 #s(literal 1 binary64) n) < 1.0000000000000001e-15

   1. Initial program 53.8%

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

   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-+.f64 N/A

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

      4. +-commutative N/A

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

      5. add-flip N/A

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

      6. metadata-eval N/A

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

      7. lift--.f64 N/A

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

      8. lift-log.f64 N/A

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

      9. diff-log N/A

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

      10. lift--.f64 N/A

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

      11. metadata-eval N/A

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

      12. add-flip N/A

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

      13. *-lft-identity N/A

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

      14. add-to-fraction N/A

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

      15. lift-/.f64 N/A

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

      16. lower-log1p.f64 57.2

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

   6. Applied rewrites 57.2%

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

   if 1.0000000000000001e-15 < (/.f64 #s(literal 1 binary64) n) < 5e219

   1. Initial program 53.8%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   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 78.9%

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

   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 78.8%

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

   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 51.1

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

   9. Applied rewrites 51.1%

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

   if 5e219 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 53.8%

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

   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. add-flip N/A

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

      9. metadata-eval N/A

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

      10. lift--.f64 N/A

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

      11. diff-log N/A

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

      12. lower-log.f64 N/A

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

      13. lower-/.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 inf

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

      1. lower-/.f64 39.7

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

   9. Applied rewrites 39.7%

      \[\leadsto \frac{-\frac{-1}{x}}{n} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 5: 93.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{-5}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-15}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(\frac{1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{+219}:\\ \;\;\;\;1 \cdot \left(-\mathsf{expm1}\left(\frac{\log x}{n}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-\frac{-1}{x}}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= (/ 1.0 n) -2e-5)
   (/ (pow x (/ 1.0 n)) (* n x))
   (if (<= (/ 1.0 n) 1e-15)
     (/ (log1p (/ 1.0 x)) n)
     (if (<= (/ 1.0 n) 5e+219)
       (* 1.0 (- (expm1 (/ (log x) n))))
       (/ (- (/ -1.0 x)) n)))))

C

double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -2e-5) {
		tmp = pow(x, (1.0 / n)) / (n * x);
	} else if ((1.0 / n) <= 1e-15) {
		tmp = log1p((1.0 / x)) / n;
	} else if ((1.0 / n) <= 5e+219) {
		tmp = 1.0 * -expm1((log(x) / n));
	} else {
		tmp = -(-1.0 / x) / n;
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -2e-5) {
		tmp = Math.pow(x, (1.0 / n)) / (n * x);
	} else if ((1.0 / n) <= 1e-15) {
		tmp = Math.log1p((1.0 / x)) / n;
	} else if ((1.0 / n) <= 5e+219) {
		tmp = 1.0 * -Math.expm1((Math.log(x) / n));
	} else {
		tmp = -(-1.0 / x) / n;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if (1.0 / n) <= -2e-5:
		tmp = math.pow(x, (1.0 / n)) / (n * x)
	elif (1.0 / n) <= 1e-15:
		tmp = math.log1p((1.0 / x)) / n
	elif (1.0 / n) <= 5e+219:
		tmp = 1.0 * -math.expm1((math.log(x) / n))
	else:
		tmp = -(-1.0 / x) / n
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -2e-5)
		tmp = Float64((x ^ Float64(1.0 / n)) / Float64(n * x));
	elseif (Float64(1.0 / n) <= 1e-15)
		tmp = Float64(log1p(Float64(1.0 / x)) / n);
	elseif (Float64(1.0 / n) <= 5e+219)
		tmp = Float64(1.0 * Float64(-expm1(Float64(log(x) / n))));
	else
		tmp = Float64(Float64(-Float64(-1.0 / x)) / n);
	end
	return tmp
end

Wolfram

code[x_, n_] := If[LessEqual[N[(1.0 / n), $MachinePrecision], -2e-5], N[(N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision] / N[(n * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 1e-15], N[(N[Log[1 + N[(1.0 / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e+219], N[(1.0 * (-N[(Exp[N[(N[Log[x], $MachinePrecision] / n), $MachinePrecision]] - 1), $MachinePrecision])), $MachinePrecision], N[((-N[(-1.0 / x), $MachinePrecision]) / n), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -2.00000000000000016e-5

   1. Initial program 53.8%

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

   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 57.5

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

   4. Applied rewrites 57.5%

      \[\leadsto \color{blue}{\frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}}{n \cdot x}} \]
   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. lift-log.f64 N/A

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

      10. frac-2neg N/A

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

      11. mult-flip N/A

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

      12. lift-log.f64 N/A

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

      13. lift-/.f64 N/A

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

      14. pow-to-exp N/A

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

      15. lift-pow.f64 57.5

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

   6. Applied rewrites 57.5%

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

   if -2.00000000000000016e-5 < (/.f64 #s(literal 1 binary64) n) < 1.0000000000000001e-15

   1. Initial program 53.8%

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

   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-+.f64 N/A

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

      4. +-commutative N/A

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

      5. add-flip N/A

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

      6. metadata-eval N/A

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

      7. lift--.f64 N/A

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

      8. lift-log.f64 N/A

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

      9. diff-log N/A

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

      10. lift--.f64 N/A

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

      11. metadata-eval N/A

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

      12. add-flip N/A

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

      13. *-lft-identity N/A

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

      14. add-to-fraction N/A

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

      15. lift-/.f64 N/A

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

      16. lower-log1p.f64 57.2

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

   6. Applied rewrites 57.2%

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

   if 1.0000000000000001e-15 < (/.f64 #s(literal 1 binary64) n) < 5e219

   1. Initial program 53.8%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   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 78.9%

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

   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 78.8%

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

   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 51.1

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

   9. Applied rewrites 51.1%

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

   if 5e219 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 53.8%

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

   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. add-flip N/A

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

      9. metadata-eval N/A

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

      10. lift--.f64 N/A

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

      11. diff-log N/A

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

      12. lower-log.f64 N/A

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

      13. lower-/.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 inf

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

      1. lower-/.f64 39.7

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

   9. Applied rewrites 39.7%

      \[\leadsto \frac{-\frac{-1}{x}}{n} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 6: 93.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{-5}:\\ \;\;\;\;\frac{\frac{t\_0}{x}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-12}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(\frac{1}{x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + x \cdot \frac{1 + \mathsf{fma}\left(x, 0.3333333333333333 \cdot x - 0.5, \frac{x \cdot \left(0.5 + -0.5 \cdot x\right)}{n}\right)}{n}\right) - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -2e-5)
     (/ (/ t_0 x) n)
     (if (<= (/ 1.0 n) 1e-12)
       (/ (log1p (/ 1.0 x)) n)
       (-
        (+
         1.0
         (*
          x
          (/
           (+
            1.0
            (fma
             x
             (- (* 0.3333333333333333 x) 0.5)
             (/ (* x (+ 0.5 (* -0.5 x))) n)))
           n)))
        t_0)))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -2e-5) {
		tmp = (t_0 / x) / n;
	} else if ((1.0 / n) <= 1e-12) {
		tmp = log1p((1.0 / x)) / n;
	} else {
		tmp = (1.0 + (x * ((1.0 + fma(x, ((0.3333333333333333 * x) - 0.5), ((x * (0.5 + (-0.5 * x))) / n))) / n))) - t_0;
	}
	return tmp;
}

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -2e-5)
		tmp = Float64(Float64(t_0 / x) / n);
	elseif (Float64(1.0 / n) <= 1e-12)
		tmp = Float64(log1p(Float64(1.0 / x)) / n);
	else
		tmp = Float64(Float64(1.0 + Float64(x * Float64(Float64(1.0 + fma(x, Float64(Float64(0.3333333333333333 * x) - 0.5), Float64(Float64(x * Float64(0.5 + Float64(-0.5 * x))) / n))) / n))) - t_0);
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -2e-5], N[(N[(t$95$0 / x), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 1e-12], N[(N[Log[1 + N[(1.0 / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], N[(N[(1.0 + N[(x * N[(N[(1.0 + N[(x * N[(N[(0.3333333333333333 * x), $MachinePrecision] - 0.5), $MachinePrecision] + N[(N[(x * N[(0.5 + N[(-0.5 * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -2.00000000000000016e-5

   1. Initial program 53.8%

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

   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 57.5

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

   4. Applied rewrites 57.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

   6. Applied rewrites 58.3%

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

   if -2.00000000000000016e-5 < (/.f64 #s(literal 1 binary64) n) < 9.9999999999999998e-13

   1. Initial program 53.8%

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

   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-+.f64 N/A

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

      4. +-commutative N/A

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

      5. add-flip N/A

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

      6. metadata-eval N/A

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

      7. lift--.f64 N/A

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

      8. lift-log.f64 N/A

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

      9. diff-log N/A

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

      10. lift--.f64 N/A

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

      11. metadata-eval N/A

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

      12. add-flip N/A

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

      13. *-lft-identity N/A

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

      14. add-to-fraction N/A

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

      15. lift-/.f64 N/A

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

      16. lower-log1p.f64 57.2

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

   6. Applied rewrites 57.2%

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

   if 9.9999999999999998e-13 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 53.8%

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

   2. Taylor expanded in x around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-fma.f64 N/A

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

   4. Applied rewrites 16.3%

      \[\leadsto \color{blue}{\left(1 + x \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(0.5, \frac{1}{{n}^{2}}, x \cdot \left(\mathsf{fma}\left(0.16666666666666666, \frac{1}{{n}^{3}}, 0.3333333333333333 \cdot \frac{1}{n}\right) - 0.5 \cdot \frac{1}{{n}^{2}}\right)\right) - 0.5 \cdot \frac{1}{n}, \frac{1}{n}\right)\right)} - {x}^{\left(\frac{1}{n}\right)} \]

   5. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-+.f64 N/A

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

      9. lower-*.f64 25.5

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

   7. Applied rewrites 25.5%

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

Alternative 7: 83.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{+97}:\\ \;\;\;\;\frac{-\log \left(\frac{\frac{x}{\frac{x - -1}{x}}}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq -5 \cdot 10^{-17}:\\ \;\;\;\;1 \cdot \left(-\mathsf{expm1}\left(\frac{1}{\frac{n}{\log x}}\right)\right)\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-15}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(\frac{1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{+219}:\\ \;\;\;\;1 \cdot \left(-\mathsf{expm1}\left(\frac{\log x}{n}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-\frac{-1}{x}}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= (/ 1.0 n) -2e+97)
   (/ (- (log (/ (/ x (/ (- x -1.0) x)) x))) n)
   (if (<= (/ 1.0 n) -5e-17)
     (* 1.0 (- (expm1 (/ 1.0 (/ n (log x))))))
     (if (<= (/ 1.0 n) 1e-15)
       (/ (log1p (/ 1.0 x)) n)
       (if (<= (/ 1.0 n) 5e+219)
         (* 1.0 (- (expm1 (/ (log x) n))))
         (/ (- (/ -1.0 x)) n))))))

C

double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -2e+97) {
		tmp = -log(((x / ((x - -1.0) / x)) / x)) / n;
	} else if ((1.0 / n) <= -5e-17) {
		tmp = 1.0 * -expm1((1.0 / (n / log(x))));
	} else if ((1.0 / n) <= 1e-15) {
		tmp = log1p((1.0 / x)) / n;
	} else if ((1.0 / n) <= 5e+219) {
		tmp = 1.0 * -expm1((log(x) / n));
	} else {
		tmp = -(-1.0 / x) / n;
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -2e+97) {
		tmp = -Math.log(((x / ((x - -1.0) / x)) / x)) / n;
	} else if ((1.0 / n) <= -5e-17) {
		tmp = 1.0 * -Math.expm1((1.0 / (n / Math.log(x))));
	} else if ((1.0 / n) <= 1e-15) {
		tmp = Math.log1p((1.0 / x)) / n;
	} else if ((1.0 / n) <= 5e+219) {
		tmp = 1.0 * -Math.expm1((Math.log(x) / n));
	} else {
		tmp = -(-1.0 / x) / n;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if (1.0 / n) <= -2e+97:
		tmp = -math.log(((x / ((x - -1.0) / x)) / x)) / n
	elif (1.0 / n) <= -5e-17:
		tmp = 1.0 * -math.expm1((1.0 / (n / math.log(x))))
	elif (1.0 / n) <= 1e-15:
		tmp = math.log1p((1.0 / x)) / n
	elif (1.0 / n) <= 5e+219:
		tmp = 1.0 * -math.expm1((math.log(x) / n))
	else:
		tmp = -(-1.0 / x) / n
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -2e+97)
		tmp = Float64(Float64(-log(Float64(Float64(x / Float64(Float64(x - -1.0) / x)) / x))) / n);
	elseif (Float64(1.0 / n) <= -5e-17)
		tmp = Float64(1.0 * Float64(-expm1(Float64(1.0 / Float64(n / log(x))))));
	elseif (Float64(1.0 / n) <= 1e-15)
		tmp = Float64(log1p(Float64(1.0 / x)) / n);
	elseif (Float64(1.0 / n) <= 5e+219)
		tmp = Float64(1.0 * Float64(-expm1(Float64(log(x) / n))));
	else
		tmp = Float64(Float64(-Float64(-1.0 / x)) / n);
	end
	return tmp
end

Wolfram

code[x_, n_] := If[LessEqual[N[(1.0 / n), $MachinePrecision], -2e+97], N[((-N[Log[N[(N[(x / N[(N[(x - -1.0), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision]) / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], -5e-17], N[(1.0 * (-N[(Exp[N[(1.0 / N[(n / N[Log[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision])), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 1e-15], N[(N[Log[1 + N[(1.0 / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e+219], N[(1.0 * (-N[(Exp[N[(N[Log[x], $MachinePrecision] / n), $MachinePrecision]] - 1), $MachinePrecision])), $MachinePrecision], N[((-N[(-1.0 / x), $MachinePrecision]) / n), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -2.0000000000000001e97

   1. Initial program 53.8%

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

   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. add-flip N/A

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

      9. metadata-eval N/A

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

      10. lift--.f64 N/A

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

      11. diff-log N/A

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

      12. lower-log.f64 N/A

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

      13. lower-/.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. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. metadata-eval N/A

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

      4. add-flip N/A

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

      5. sum-to-mult N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

      8. add-to-fraction N/A

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

      9. *-lft-identity N/A

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

      10. add-flip N/A

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

      11. metadata-eval N/A

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

      12. lift--.f64 N/A

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

      13. lift-/.f64 N/A

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

      14. lower-/.f64 53.6

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

   8. Applied rewrites 53.6%

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

   if -2.0000000000000001e97 < (/.f64 #s(literal 1 binary64) n) < -4.9999999999999999e-17

   1. Initial program 53.8%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   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 78.9%

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

   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 78.8%

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

      1. lift--.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. sub-div N/A

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

      5. lift-log.f64 N/A

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

      6. lift--.f64 N/A

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

      7. metadata-eval N/A

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

      8. add-flip N/A

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

      9. +-commutative N/A

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

      10. div-flip N/A

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

      11. sub-negate-rev N/A

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

      12. lift-log.f64 N/A

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

      13. diff-log N/A

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

      14. +-commutative N/A

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

      15. add-flip N/A

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

      16. metadata-eval N/A

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

      17. lift--.f64 N/A

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

      18. lift-/.f64 N/A

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

      19. lift-log.f64 N/A

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

      20. distribute-neg-frac2 N/A

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

   8. Applied rewrites 79.0%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 51.1%

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

   if -4.9999999999999999e-17 < (/.f64 #s(literal 1 binary64) n) < 1.0000000000000001e-15

   1. Initial program 53.8%

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

   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-+.f64 N/A

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

      4. +-commutative N/A

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

      5. add-flip N/A

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

      6. metadata-eval N/A

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

      7. lift--.f64 N/A

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

      8. lift-log.f64 N/A

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

      9. diff-log N/A

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

      10. lift--.f64 N/A

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

      11. metadata-eval N/A

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

      12. add-flip N/A

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

      13. *-lft-identity N/A

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

      14. add-to-fraction N/A

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

      15. lift-/.f64 N/A

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

      16. lower-log1p.f64 57.2

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

   6. Applied rewrites 57.2%

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

   if 1.0000000000000001e-15 < (/.f64 #s(literal 1 binary64) n) < 5e219

   1. Initial program 53.8%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   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 78.9%

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

   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 78.8%

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

   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 51.1

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

   9. Applied rewrites 51.1%

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

   if 5e219 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 53.8%

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

   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. add-flip N/A

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

      9. metadata-eval N/A

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

      10. lift--.f64 N/A

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

      11. diff-log N/A

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

      12. lower-log.f64 N/A

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

      13. lower-/.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 inf

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

      1. lower-/.f64 39.7

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

   9. Applied rewrites 39.7%

      \[\leadsto \frac{-\frac{-1}{x}}{n} \]
3. Recombined 5 regimes into one program.
4. Add Preprocessing

Alternative 8: 83.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 \cdot \left(-\mathsf{expm1}\left(\frac{\log x}{n}\right)\right)\\ \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{+97}:\\ \;\;\;\;\frac{-\log \left(\frac{\frac{x}{\frac{x - -1}{x}}}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq -5 \cdot 10^{-17}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-15}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(\frac{1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{+219}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{-\frac{-1}{x}}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (* 1.0 (- (expm1 (/ (log x) n))))))
   (if (<= (/ 1.0 n) -2e+97)
     (/ (- (log (/ (/ x (/ (- x -1.0) x)) x))) n)
     (if (<= (/ 1.0 n) -5e-17)
       t_0
       (if (<= (/ 1.0 n) 1e-15)
         (/ (log1p (/ 1.0 x)) n)
         (if (<= (/ 1.0 n) 5e+219) t_0 (/ (- (/ -1.0 x)) n)))))))

C

double code(double x, double n) {
	double t_0 = 1.0 * -expm1((log(x) / n));
	double tmp;
	if ((1.0 / n) <= -2e+97) {
		tmp = -log(((x / ((x - -1.0) / x)) / x)) / n;
	} else if ((1.0 / n) <= -5e-17) {
		tmp = t_0;
	} else if ((1.0 / n) <= 1e-15) {
		tmp = log1p((1.0 / x)) / n;
	} else if ((1.0 / n) <= 5e+219) {
		tmp = t_0;
	} else {
		tmp = -(-1.0 / x) / 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+97) {
		tmp = -Math.log(((x / ((x - -1.0) / x)) / x)) / n;
	} else if ((1.0 / n) <= -5e-17) {
		tmp = t_0;
	} else if ((1.0 / n) <= 1e-15) {
		tmp = Math.log1p((1.0 / x)) / n;
	} else if ((1.0 / n) <= 5e+219) {
		tmp = t_0;
	} else {
		tmp = -(-1.0 / x) / 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+97:
		tmp = -math.log(((x / ((x - -1.0) / x)) / x)) / n
	elif (1.0 / n) <= -5e-17:
		tmp = t_0
	elif (1.0 / n) <= 1e-15:
		tmp = math.log1p((1.0 / x)) / n
	elif (1.0 / n) <= 5e+219:
		tmp = t_0
	else:
		tmp = -(-1.0 / x) / 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+97)
		tmp = Float64(Float64(-log(Float64(Float64(x / Float64(Float64(x - -1.0) / x)) / x))) / n);
	elseif (Float64(1.0 / n) <= -5e-17)
		tmp = t_0;
	elseif (Float64(1.0 / n) <= 1e-15)
		tmp = Float64(log1p(Float64(1.0 / x)) / n);
	elseif (Float64(1.0 / n) <= 5e+219)
		tmp = t_0;
	else
		tmp = Float64(Float64(-Float64(-1.0 / x)) / 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+97], N[((-N[Log[N[(N[(x / N[(N[(x - -1.0), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision]) / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], -5e-17], t$95$0, If[LessEqual[N[(1.0 / n), $MachinePrecision], 1e-15], N[(N[Log[1 + N[(1.0 / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e+219], t$95$0, N[((-N[(-1.0 / x), $MachinePrecision]) / n), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -2.0000000000000001e97

   1. Initial program 53.8%

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

   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. add-flip N/A

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

      9. metadata-eval N/A

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

      10. lift--.f64 N/A

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

      11. diff-log N/A

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

      12. lower-log.f64 N/A

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

      13. lower-/.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. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. metadata-eval N/A

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

      4. add-flip N/A

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

      5. sum-to-mult N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

      8. add-to-fraction N/A

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

      9. *-lft-identity N/A

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

      10. add-flip N/A

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

      11. metadata-eval N/A

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

      12. lift--.f64 N/A

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

      13. lift-/.f64 N/A

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

      14. lower-/.f64 53.6

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

   8. Applied rewrites 53.6%

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

   if -2.0000000000000001e97 < (/.f64 #s(literal 1 binary64) n) < -4.9999999999999999e-17 or 1.0000000000000001e-15 < (/.f64 #s(literal 1 binary64) n) < 5e219

   1. Initial program 53.8%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   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 78.9%

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

   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 78.8%

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

   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 51.1

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

   9. Applied rewrites 51.1%

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

   if -4.9999999999999999e-17 < (/.f64 #s(literal 1 binary64) n) < 1.0000000000000001e-15

   1. Initial program 53.8%

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

   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-+.f64 N/A

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

      4. +-commutative N/A

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

      5. add-flip N/A

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

      6. metadata-eval N/A

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

      7. lift--.f64 N/A

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

      8. lift-log.f64 N/A

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

      9. diff-log N/A

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

      10. lift--.f64 N/A

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

      11. metadata-eval N/A

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

      12. add-flip N/A

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

      13. *-lft-identity N/A

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

      14. add-to-fraction N/A

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

      15. lift-/.f64 N/A

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

      16. lower-log1p.f64 57.2

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

   6. Applied rewrites 57.2%

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

   if 5e219 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 53.8%

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

   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. add-flip N/A

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

      9. metadata-eval N/A

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

      10. lift--.f64 N/A

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

      11. diff-log N/A

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

      12. lower-log.f64 N/A

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

      13. lower-/.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 inf

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

      1. lower-/.f64 39.7

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

   9. Applied rewrites 39.7%

      \[\leadsto \frac{-\frac{-1}{x}}{n} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 9: 80.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.65 \cdot 10^{-15}:\\ \;\;\;\;1 \cdot \left(-\mathsf{expm1}\left(\frac{\log x}{n}\right)\right)\\ \mathbf{elif}\;x \leq 5.5 \cdot 10^{+123}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(\frac{1}{x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log \left(\frac{x - -1}{x}\right)}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 1.65e-15)
   (* 1.0 (- (expm1 (/ (log x) n))))
   (if (<= x 5.5e+123) (/ (log1p (/ 1.0 x)) n) (/ (log (/ (- x -1.0) x)) n))))

C

double code(double x, double n) {
	double tmp;
	if (x <= 1.65e-15) {
		tmp = 1.0 * -expm1((log(x) / n));
	} else if (x <= 5.5e+123) {
		tmp = log1p((1.0 / x)) / n;
	} else {
		tmp = log(((x - -1.0) / x)) / n;
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double tmp;
	if (x <= 1.65e-15) {
		tmp = 1.0 * -Math.expm1((Math.log(x) / n));
	} else if (x <= 5.5e+123) {
		tmp = Math.log1p((1.0 / x)) / n;
	} else {
		tmp = Math.log(((x - -1.0) / x)) / n;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 1.65e-15:
		tmp = 1.0 * -math.expm1((math.log(x) / n))
	elif x <= 5.5e+123:
		tmp = math.log1p((1.0 / x)) / n
	else:
		tmp = math.log(((x - -1.0) / x)) / n
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 1.65e-15)
		tmp = Float64(1.0 * Float64(-expm1(Float64(log(x) / n))));
	elseif (x <= 5.5e+123)
		tmp = Float64(log1p(Float64(1.0 / x)) / n);
	else
		tmp = Float64(log(Float64(Float64(x - -1.0) / x)) / n);
	end
	return tmp
end

Wolfram

code[x_, n_] := If[LessEqual[x, 1.65e-15], N[(1.0 * (-N[(Exp[N[(N[Log[x], $MachinePrecision] / n), $MachinePrecision]] - 1), $MachinePrecision])), $MachinePrecision], If[LessEqual[x, 5.5e+123], N[(N[Log[1 + N[(1.0 / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], N[(N[Log[N[(N[(x - -1.0), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 1.65e-15

   1. Initial program 53.8%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   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 78.9%

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

   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 78.8%

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

   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 51.1

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

   9. Applied rewrites 51.1%

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

   if 1.65e-15 < x < 5.5000000000000002e123

   1. Initial program 53.8%

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

   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-+.f64 N/A

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

      4. +-commutative N/A

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

      5. add-flip N/A

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

      6. metadata-eval N/A

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

      7. lift--.f64 N/A

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

      8. lift-log.f64 N/A

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

      9. diff-log N/A

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

      10. lift--.f64 N/A

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

      11. metadata-eval N/A

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

      12. add-flip N/A

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

      13. *-lft-identity N/A

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

      14. add-to-fraction N/A

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

      15. lift-/.f64 N/A

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

      16. lower-log1p.f64 57.2

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

   6. Applied rewrites 57.2%

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

   if 5.5000000000000002e123 < x

   1. Initial program 53.8%

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

   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. add-flip N/A

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

      13. metadata-eval N/A

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

      14. lift--.f64 N/A

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

      15. lower-/.f64 59.0

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

   6. Applied rewrites 59.0%

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

Alternative 10: 80.5% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.65 \cdot 10^{-15}:\\ \;\;\;\;1 \cdot \left(-\mathsf{expm1}\left(\frac{\log x}{n}\right)\right)\\ \mathbf{elif}\;x \leq 5.5 \cdot 10^{+123}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(\frac{1}{x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log \left(1 + x\right) - \log x}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 1.65e-15)
   (* 1.0 (- (expm1 (/ (log x) n))))
   (if (<= x 5.5e+123)
     (/ (log1p (/ 1.0 x)) n)
     (/ (- (log (+ 1.0 x)) (log x)) n))))

C

double code(double x, double n) {
	double tmp;
	if (x <= 1.65e-15) {
		tmp = 1.0 * -expm1((log(x) / n));
	} else if (x <= 5.5e+123) {
		tmp = log1p((1.0 / x)) / n;
	} else {
		tmp = (log((1.0 + x)) - log(x)) / n;
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double tmp;
	if (x <= 1.65e-15) {
		tmp = 1.0 * -Math.expm1((Math.log(x) / n));
	} else if (x <= 5.5e+123) {
		tmp = Math.log1p((1.0 / x)) / n;
	} else {
		tmp = (Math.log((1.0 + x)) - Math.log(x)) / n;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 1.65e-15:
		tmp = 1.0 * -math.expm1((math.log(x) / n))
	elif x <= 5.5e+123:
		tmp = math.log1p((1.0 / x)) / n
	else:
		tmp = (math.log((1.0 + x)) - math.log(x)) / n
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 1.65e-15)
		tmp = Float64(1.0 * Float64(-expm1(Float64(log(x) / n))));
	elseif (x <= 5.5e+123)
		tmp = Float64(log1p(Float64(1.0 / x)) / n);
	else
		tmp = Float64(Float64(log(Float64(1.0 + x)) - log(x)) / n);
	end
	return tmp
end

Wolfram

code[x_, n_] := If[LessEqual[x, 1.65e-15], N[(1.0 * (-N[(Exp[N[(N[Log[x], $MachinePrecision] / n), $MachinePrecision]] - 1), $MachinePrecision])), $MachinePrecision], If[LessEqual[x, 5.5e+123], N[(N[Log[1 + N[(1.0 / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], N[(N[(N[Log[N[(1.0 + x), $MachinePrecision]], $MachinePrecision] - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 1.65e-15

   1. Initial program 53.8%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   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 78.9%

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

   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 78.8%

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

   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 51.1

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

   9. Applied rewrites 51.1%

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

   if 1.65e-15 < x < 5.5000000000000002e123

   1. Initial program 53.8%

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

   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-+.f64 N/A

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

      4. +-commutative N/A

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

      5. add-flip N/A

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

      6. metadata-eval N/A

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

      7. lift--.f64 N/A

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

      8. lift-log.f64 N/A

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

      9. diff-log N/A

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

      10. lift--.f64 N/A

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

      11. metadata-eval N/A

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

      12. add-flip N/A

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

      13. *-lft-identity N/A

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

      14. add-to-fraction N/A

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

      15. lift-/.f64 N/A

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

      16. lower-log1p.f64 57.2

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

   6. Applied rewrites 57.2%

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

   if 5.5000000000000002e123 < x

   1. Initial program 53.8%

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

   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%

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

Alternative 11: 73.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -10000:\\ \;\;\;\;\frac{\log \left(\frac{x - -1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-12}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(\frac{1}{x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;\log \left(1 + \frac{1}{n \cdot x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= (/ 1.0 n) -10000.0)
   (/ (log (/ (- x -1.0) x)) n)
   (if (<= (/ 1.0 n) 1e-12)
     (/ (log1p (/ 1.0 x)) n)
     (log (+ 1.0 (/ 1.0 (* n x)))))))

C

double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -10000.0) {
		tmp = log(((x - -1.0) / x)) / n;
	} else if ((1.0 / n) <= 1e-12) {
		tmp = log1p((1.0 / x)) / n;
	} else {
		tmp = log((1.0 + (1.0 / (n * x))));
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -10000.0) {
		tmp = Math.log(((x - -1.0) / x)) / n;
	} else if ((1.0 / n) <= 1e-12) {
		tmp = Math.log1p((1.0 / x)) / n;
	} else {
		tmp = Math.log((1.0 + (1.0 / (n * x))));
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if (1.0 / n) <= -10000.0:
		tmp = math.log(((x - -1.0) / x)) / n
	elif (1.0 / n) <= 1e-12:
		tmp = math.log1p((1.0 / x)) / n
	else:
		tmp = math.log((1.0 + (1.0 / (n * x))))
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -10000.0)
		tmp = Float64(log(Float64(Float64(x - -1.0) / x)) / n);
	elseif (Float64(1.0 / n) <= 1e-12)
		tmp = Float64(log1p(Float64(1.0 / x)) / n);
	else
		tmp = log(Float64(1.0 + Float64(1.0 / Float64(n * x))));
	end
	return tmp
end

Wolfram

code[x_, n_] := If[LessEqual[N[(1.0 / n), $MachinePrecision], -10000.0], N[(N[Log[N[(N[(x - -1.0), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 1e-12], N[(N[Log[1 + N[(1.0 / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], N[Log[N[(1.0 + N[(1.0 / N[(n * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 53.8%

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

   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. add-flip N/A

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

      13. metadata-eval N/A

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

      14. lift--.f64 N/A

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

      15. lower-/.f64 59.0

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

   6. Applied rewrites 59.0%

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

   if -1e4 < (/.f64 #s(literal 1 binary64) n) < 9.9999999999999998e-13

   1. Initial program 53.8%

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

   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-+.f64 N/A

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

      4. +-commutative N/A

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

      5. add-flip N/A

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

      6. metadata-eval N/A

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

      7. lift--.f64 N/A

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

      8. lift-log.f64 N/A

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

      9. diff-log N/A

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

      10. lift--.f64 N/A

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

      11. metadata-eval N/A

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

      12. add-flip N/A

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

      13. *-lft-identity N/A

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

      14. add-to-fraction N/A

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

      15. lift-/.f64 N/A

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

      16. lower-log1p.f64 57.2

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

   6. Applied rewrites 57.2%

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

   if 9.9999999999999998e-13 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 53.8%

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

   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. add-flip N/A

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

      17. metadata-eval N/A

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

      18. lift--.f64 N/A

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

      19. lower-/.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

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

      2. lift-/.f64 N/A

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

      3. associate-/r/ N/A

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

      4. lift-log.f64 N/A

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

      5. log-pow-rev N/A

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

      6. lift-/.f64 N/A

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

      7. lift--.f64 N/A

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

      8. div-sub N/A

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

      9. *-inverses N/A

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

      10. lower-log.f64 N/A

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

      11. lower-pow.f64 N/A

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

      12. *-inverses N/A

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

      13. div-sub N/A

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

      14. lift--.f64 N/A

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

      15. lift-/.f64 N/A

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

      16. lower-/.f64 51.7

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

   8. Applied rewrites 51.7%

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

   9. Taylor expanded in x around inf

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

       1. lower-+.f64 N/A

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

       2. lower-/.f64 N/A

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

       3. lower-*.f64 26.3

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

   11. Applied rewrites 26.3%

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

Alternative 12: 68.5% accurate, 0.4× speedup

Math

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

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))
     (/ (- (/ -1.0 x)) n)
     (if (<= t_0 4e-10)
       (/ (- (log (/ x (- x -1.0)))) n)
       (log (+ 1.0 (/ 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 = -(-1.0 / x) / n;
	} else if (t_0 <= 4e-10) {
		tmp = -log((x / (x - -1.0))) / n;
	} else {
		tmp = log((1.0 + (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 = -(-1.0 / x) / n;
	} else if (t_0 <= 4e-10) {
		tmp = -Math.log((x / (x - -1.0))) / n;
	} else {
		tmp = Math.log((1.0 + (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 = -(-1.0 / x) / n
	elif t_0 <= 4e-10:
		tmp = -math.log((x / (x - -1.0))) / n
	else:
		tmp = math.log((1.0 + (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(-Float64(-1.0 / x)) / n);
	elseif (t_0 <= 4e-10)
		tmp = Float64(Float64(-log(Float64(x / Float64(x - -1.0)))) / n);
	else
		tmp = log(Float64(1.0 + 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 = -(-1.0 / x) / n;
	elseif (t_0 <= 4e-10)
		tmp = -log((x / (x - -1.0))) / n;
	else
		tmp = log((1.0 + (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[(-1.0 / x), $MachinePrecision]) / n), $MachinePrecision], If[LessEqual[t$95$0, 4e-10], N[((-N[Log[N[(x / N[(x - -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / n), $MachinePrecision], N[Log[N[(1.0 + N[(1.0 / N[(n * x), $MachinePrecision]), $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 53.8%

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

   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. add-flip N/A

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

      9. metadata-eval N/A

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

      10. lift--.f64 N/A

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

      11. diff-log N/A

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

      12. lower-log.f64 N/A

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

      13. lower-/.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 inf

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

      1. lower-/.f64 39.7

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

   9. Applied rewrites 39.7%

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

   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))) < 4.00000000000000015e-10

   1. Initial program 53.8%

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

   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. add-flip N/A

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

      9. metadata-eval N/A

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

      10. lift--.f64 N/A

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

      11. diff-log N/A

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

      12. lower-log.f64 N/A

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

      13. lower-/.f64 59.0

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

   6. Applied rewrites 59.0%

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

   if 4.00000000000000015e-10 < (-.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 53.8%

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

   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. add-flip N/A

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

      17. metadata-eval N/A

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

      18. lift--.f64 N/A

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

      19. lower-/.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

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

      2. lift-/.f64 N/A

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

      3. associate-/r/ N/A

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

      4. lift-log.f64 N/A

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

      5. log-pow-rev N/A

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

      6. lift-/.f64 N/A

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

      7. lift--.f64 N/A

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

      8. div-sub N/A

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

      9. *-inverses N/A

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

      10. lower-log.f64 N/A

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

      11. lower-pow.f64 N/A

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

      12. *-inverses N/A

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

      13. div-sub N/A

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

      14. lift--.f64 N/A

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

      15. lift-/.f64 N/A

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

      16. lower-/.f64 51.7

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

   8. Applied rewrites 51.7%

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

   9. Taylor expanded in x around inf

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

       1. lower-+.f64 N/A

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

       2. lower-/.f64 N/A

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

       3. lower-*.f64 26.3

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

   11. Applied rewrites 26.3%

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

Alternative 13: 68.5% accurate, 0.4× speedup

Math

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

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))
     (/ (- (/ -1.0 x)) n)
     (if (<= t_0 4e-10)
       (/ (log (/ (- x -1.0) x)) n)
       (log (+ 1.0 (/ 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 = -(-1.0 / x) / n;
	} else if (t_0 <= 4e-10) {
		tmp = log(((x - -1.0) / x)) / n;
	} else {
		tmp = log((1.0 + (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 = -(-1.0 / x) / n;
	} else if (t_0 <= 4e-10) {
		tmp = Math.log(((x - -1.0) / x)) / n;
	} else {
		tmp = Math.log((1.0 + (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 = -(-1.0 / x) / n
	elif t_0 <= 4e-10:
		tmp = math.log(((x - -1.0) / x)) / n
	else:
		tmp = math.log((1.0 + (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(-Float64(-1.0 / x)) / n);
	elseif (t_0 <= 4e-10)
		tmp = Float64(log(Float64(Float64(x - -1.0) / x)) / n);
	else
		tmp = log(Float64(1.0 + 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 = -(-1.0 / x) / n;
	elseif (t_0 <= 4e-10)
		tmp = log(((x - -1.0) / x)) / n;
	else
		tmp = log((1.0 + (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[(-1.0 / x), $MachinePrecision]) / n), $MachinePrecision], If[LessEqual[t$95$0, 4e-10], N[(N[Log[N[(N[(x - -1.0), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], N[Log[N[(1.0 + N[(1.0 / N[(n * x), $MachinePrecision]), $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 53.8%

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

   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. add-flip N/A

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

      9. metadata-eval N/A

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

      10. lift--.f64 N/A

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

      11. diff-log N/A

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

      12. lower-log.f64 N/A

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

      13. lower-/.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 inf

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

      1. lower-/.f64 39.7

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

   9. Applied rewrites 39.7%

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

   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))) < 4.00000000000000015e-10

   1. Initial program 53.8%

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

   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. add-flip N/A

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

      13. metadata-eval N/A

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

      14. lift--.f64 N/A

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

      15. lower-/.f64 59.0

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

   6. Applied rewrites 59.0%

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

   if 4.00000000000000015e-10 < (-.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 53.8%

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

   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. add-flip N/A

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

      17. metadata-eval N/A

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

      18. lift--.f64 N/A

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

      19. lower-/.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

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

      2. lift-/.f64 N/A

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

      3. associate-/r/ N/A

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

      4. lift-log.f64 N/A

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

      5. log-pow-rev N/A

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

      6. lift-/.f64 N/A

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

      7. lift--.f64 N/A

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

      8. div-sub N/A

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

      9. *-inverses N/A

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

      10. lower-log.f64 N/A

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

      11. lower-pow.f64 N/A

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

      12. *-inverses N/A

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

      13. div-sub N/A

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

      14. lift--.f64 N/A

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

      15. lift-/.f64 N/A

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

      16. lower-/.f64 51.7

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

   8. Applied rewrites 51.7%

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

   9. Taylor expanded in x around inf

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

       1. lower-+.f64 N/A

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

       2. lower-/.f64 N/A

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

       3. lower-*.f64 26.3

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

   11. Applied rewrites 26.3%

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

Alternative 14: 58.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{elif}\;x \leq 1.5 \cdot 10^{+162}:\\ \;\;\;\;\frac{-\frac{-1}{x}}{n}\\ \mathbf{else}:\\ \;\;\;\;\log \left(1 + \frac{1}{n \cdot x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 1.0)
   (/ (- x (log x)) n)
   (if (<= x 1.5e+162) (/ (- (/ -1.0 x)) n) (log (+ 1.0 (/ 1.0 (* n x)))))))

C

double code(double x, double n) {
	double tmp;
	if (x <= 1.0) {
		tmp = (x - log(x)) / n;
	} else if (x <= 1.5e+162) {
		tmp = -(-1.0 / x) / n;
	} else {
		tmp = log((1.0 + (1.0 / (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 <= 1.0d0) then
        tmp = (x - log(x)) / n
    else if (x <= 1.5d+162) then
        tmp = -((-1.0d0) / x) / n
    else
        tmp = log((1.0d0 + (1.0d0 / (n * x))))
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (x <= 1.0) {
		tmp = (x - Math.log(x)) / n;
	} else if (x <= 1.5e+162) {
		tmp = -(-1.0 / x) / n;
	} else {
		tmp = Math.log((1.0 + (1.0 / (n * x))));
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 1.0:
		tmp = (x - math.log(x)) / n
	elif x <= 1.5e+162:
		tmp = -(-1.0 / x) / n
	else:
		tmp = math.log((1.0 + (1.0 / (n * x))))
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 1.0)
		tmp = Float64(Float64(x - log(x)) / n);
	elseif (x <= 1.5e+162)
		tmp = Float64(Float64(-Float64(-1.0 / x)) / n);
	else
		tmp = log(Float64(1.0 + Float64(1.0 / Float64(n * x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (x <= 1.0)
		tmp = (x - log(x)) / n;
	elseif (x <= 1.5e+162)
		tmp = -(-1.0 / x) / n;
	else
		tmp = log((1.0 + (1.0 / (n * x))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 1.0], N[(N[(x - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[x, 1.5e+162], N[((-N[(-1.0 / x), $MachinePrecision]) / n), $MachinePrecision], N[Log[N[(1.0 + N[(1.0 / N[(n * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 1

   1. Initial program 53.8%

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

   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 1 < x < 1.4999999999999999e162

   1. Initial program 53.8%

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

   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. add-flip N/A

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

      9. metadata-eval N/A

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

      10. lift--.f64 N/A

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

      11. diff-log N/A

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

      12. lower-log.f64 N/A

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

      13. lower-/.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 inf

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

      1. lower-/.f64 39.7

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

   9. Applied rewrites 39.7%

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

   if 1.4999999999999999e162 < x

   1. Initial program 53.8%

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

   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. add-flip N/A

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

      17. metadata-eval N/A

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

      18. lift--.f64 N/A

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

      19. lower-/.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

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

      2. lift-/.f64 N/A

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

      3. associate-/r/ N/A

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

      4. lift-log.f64 N/A

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

      5. log-pow-rev N/A

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

      6. lift-/.f64 N/A

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

      7. lift--.f64 N/A

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

      8. div-sub N/A

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

      9. *-inverses N/A

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

      10. lower-log.f64 N/A

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

      11. lower-pow.f64 N/A

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

      12. *-inverses N/A

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

      13. div-sub N/A

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

      14. lift--.f64 N/A

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

      15. lift-/.f64 N/A

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

      16. lower-/.f64 51.7

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

   8. Applied rewrites 51.7%

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

   9. Taylor expanded in x around inf

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

       1. lower-+.f64 N/A

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

       2. lower-/.f64 N/A

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

       3. lower-*.f64 26.3

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

   11. Applied rewrites 26.3%

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

Alternative 15: 56.3% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\frac{-1}{x}}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 1.0) (/ (- x (log x)) n) (/ (- (/ -1.0 x)) n)))

C

double code(double x, double n) {
	double tmp;
	if (x <= 1.0) {
		tmp = (x - log(x)) / n;
	} else {
		tmp = -(-1.0 / x) / 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 <= 1.0d0) then
        tmp = (x - log(x)) / n
    else
        tmp = -((-1.0d0) / x) / n
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (x <= 1.0) {
		tmp = (x - Math.log(x)) / n;
	} else {
		tmp = -(-1.0 / x) / n;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 1.0:
		tmp = (x - math.log(x)) / n
	else:
		tmp = -(-1.0 / x) / n
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 1.0)
		tmp = Float64(Float64(x - log(x)) / n);
	else
		tmp = Float64(Float64(-Float64(-1.0 / x)) / n);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (x <= 1.0)
		tmp = (x - log(x)) / n;
	else
		tmp = -(-1.0 / x) / n;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 1.0], N[(N[(x - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], N[((-N[(-1.0 / x), $MachinePrecision]) / n), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1

   1. Initial program 53.8%

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

   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 1 < x

   1. Initial program 53.8%

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

   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. add-flip N/A

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

      9. metadata-eval N/A

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

      10. lift--.f64 N/A

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

      11. diff-log N/A

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

      12. lower-log.f64 N/A

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

      13. lower-/.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 inf

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

      1. lower-/.f64 39.7

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

   9. Applied rewrites 39.7%

      \[\leadsto \frac{-\frac{-1}{x}}{n} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 16: 56.1% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.55:\\ \;\;\;\;\frac{-\log x}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\frac{-1}{x}}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 0.55) (/ (- (log x)) n) (/ (- (/ -1.0 x)) n)))

C

double code(double x, double n) {
	double tmp;
	if (x <= 0.55) {
		tmp = -log(x) / n;
	} else {
		tmp = -(-1.0 / x) / n;
	}
	return tmp;
}

Fortran

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
        tmp = -((-1.0d0) / x) / n
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (x <= 0.55) {
		tmp = -Math.log(x) / n;
	} else {
		tmp = -(-1.0 / x) / n;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 0.55:
		tmp = -math.log(x) / n
	else:
		tmp = -(-1.0 / x) / n
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 0.55)
		tmp = Float64(Float64(-log(x)) / n);
	else
		tmp = Float64(Float64(-Float64(-1.0 / x)) / n);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (x <= 0.55)
		tmp = -log(x) / n;
	else
		tmp = -(-1.0 / x) / n;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 0.55], N[((-N[Log[x], $MachinePrecision]) / n), $MachinePrecision], N[((-N[(-1.0 / x), $MachinePrecision]) / n), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.55000000000000004

   1. Initial program 53.8%

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

   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. add-flip N/A

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

      9. metadata-eval N/A

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

      10. lift--.f64 N/A

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

      11. diff-log N/A

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

      12. lower-log.f64 N/A

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

      13. lower-/.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

   9. Applied rewrites 30.9%

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

   if 0.55000000000000004 < x

   1. Initial program 53.8%

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

   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. add-flip N/A

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

      9. metadata-eval N/A

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

      10. lift--.f64 N/A

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

      11. diff-log N/A

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

      12. lower-log.f64 N/A

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

      13. lower-/.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 inf

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

      1. lower-/.f64 39.7

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

   9. Applied rewrites 39.7%

      \[\leadsto \frac{-\frac{-1}{x}}{n} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 17: 39.7% accurate, 5.1× 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(-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 53.8%

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

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. add-flip N/A

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

   9. metadata-eval N/A

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

   10. lift--.f64 N/A

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

   11. diff-log N/A

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

   12. lower-log.f64 N/A

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

   13. lower-/.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 inf

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

   1. lower-/.f64 39.7

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

9. Applied rewrites 39.7%

   \[\leadsto \frac{-\frac{-1}{x}}{n} \]
10. Add Preprocessing

Alternative 18: 15.3% 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 53.8%

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

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 15.3

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

7. Applied rewrites 15.3%

   \[\leadsto \frac{\frac{-1}{x}}{n} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2025140 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  :precision binary64
  (- (pow (+ x 1.0) (/ 1.0 n)) (pow x (/ 1.0 n))))