2nthrt (problem 3.4.6)

Percentage Accurate: 52.8% → 94.4%

Time: 19.7s

Alternatives: 18

Speedup: 2.7×

• could not determine a ground truth

  1. x = -4.76744043185336e-33
  2. n = -2.3216342246633717e+22

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: 52.8% 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: 94.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{-10}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-15}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(\frac{1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{+150}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\log \left(x - -1\right) \cdot n - \log x \cdot n}{n \cdot n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (- (pow (+ x 1.0) (/ 1.0 n)) (pow x (/ 1.0 n)))))
   (if (<= (/ 1.0 n) -5e-10)
     t_0
     (if (<= (/ 1.0 n) 5e-15)
       (/ (log1p (/ 1.0 x)) n)
       (if (<= (/ 1.0 n) 1e+150)
         t_0
         (/ (- (* (log (- x -1.0)) n) (* (log x) n)) (* n n)))))))

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 ((1.0 / n) <= -5e-10) {
		tmp = t_0;
	} else if ((1.0 / n) <= 5e-15) {
		tmp = log1p((1.0 / x)) / n;
	} else if ((1.0 / n) <= 1e+150) {
		tmp = t_0;
	} else {
		tmp = ((log((x - -1.0)) * n) - (log(x) * n)) / (n * n);
	}
	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 ((1.0 / n) <= -5e-10) {
		tmp = t_0;
	} else if ((1.0 / n) <= 5e-15) {
		tmp = Math.log1p((1.0 / x)) / n;
	} else if ((1.0 / n) <= 1e+150) {
		tmp = t_0;
	} else {
		tmp = ((Math.log((x - -1.0)) * n) - (Math.log(x) * n)) / (n * n);
	}
	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 (1.0 / n) <= -5e-10:
		tmp = t_0
	elif (1.0 / n) <= 5e-15:
		tmp = math.log1p((1.0 / x)) / n
	elif (1.0 / n) <= 1e+150:
		tmp = t_0
	else:
		tmp = ((math.log((x - -1.0)) * n) - (math.log(x) * n)) / (n * n)
	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 (Float64(1.0 / n) <= -5e-10)
		tmp = t_0;
	elseif (Float64(1.0 / n) <= 5e-15)
		tmp = Float64(log1p(Float64(1.0 / x)) / n);
	elseif (Float64(1.0 / n) <= 1e+150)
		tmp = t_0;
	else
		tmp = Float64(Float64(Float64(log(Float64(x - -1.0)) * n) - Float64(log(x) * n)) / Float64(n * n));
	end
	return 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[N[(1.0 / n), $MachinePrecision], -5e-10], t$95$0, If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e-15], N[(N[Log[1 + N[(1.0 / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 1e+150], t$95$0, N[(N[(N[(N[Log[N[(x - -1.0), $MachinePrecision]], $MachinePrecision] * n), $MachinePrecision] - N[(N[Log[x], $MachinePrecision] * n), $MachinePrecision]), $MachinePrecision] / N[(n * n), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -5.00000000000000031e-10 or 4.99999999999999999e-15 < (/.f64 #s(literal 1 binary64) n) < 9.99999999999999981e149

   1. Initial program 52.8%

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

   if -5.00000000000000031e-10 < (/.f64 #s(literal 1 binary64) n) < 4.99999999999999999e-15

   1. Initial program 52.8%

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

   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 58.7

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

   4. Applied rewrites 58.7%

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

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

      10. lower-log1p.f64 57.7

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

   6. Applied rewrites 57.7%

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

   if 9.99999999999999981e149 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 52.8%

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

   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 58.7

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

   4. Applied rewrites 58.7%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. div-sub N/A

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

      4. frac-sub N/A

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

      5. *-commutative N/A

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. add-flip N/A

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

      9. metadata-eval N/A

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

      10. lift--.f64 N/A

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

      11. *-commutative N/A

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

      12. unpow2 N/A

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

      13. lift-pow.f64 N/A

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

      14. lower-/.f64 N/A

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

   6. Applied rewrites 49.2%

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

Alternative 2: 94.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{-10}:\\ \;\;\;\;{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - t\_0\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-15}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(\frac{1}{x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + x \cdot \mathsf{fma}\left(x, 0.5 \cdot \frac{1}{{n}^{2}} - 0.5 \cdot \frac{1}{n}, \frac{1}{n}\right)\right) - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -5e-10)
     (- (pow (+ x 1.0) (/ 1.0 n)) t_0)
     (if (<= (/ 1.0 n) 5e-15)
       (/ (log1p (/ 1.0 x)) n)
       (-
        (+
         1.0
         (*
          x
          (fma x (- (* 0.5 (/ 1.0 (pow n 2.0))) (* 0.5 (/ 1.0 n))) (/ 1.0 n))))
        t_0)))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -5e-10) {
		tmp = pow((x + 1.0), (1.0 / n)) - t_0;
	} else if ((1.0 / n) <= 5e-15) {
		tmp = log1p((1.0 / x)) / n;
	} else {
		tmp = (1.0 + (x * fma(x, ((0.5 * (1.0 / pow(n, 2.0))) - (0.5 * (1.0 / n))), (1.0 / n)))) - t_0;
	}
	return tmp;
}

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -5e-10)
		tmp = Float64((Float64(x + 1.0) ^ Float64(1.0 / n)) - t_0);
	elseif (Float64(1.0 / n) <= 5e-15)
		tmp = Float64(log1p(Float64(1.0 / x)) / n);
	else
		tmp = Float64(Float64(1.0 + Float64(x * fma(x, Float64(Float64(0.5 * Float64(1.0 / (n ^ 2.0))) - Float64(0.5 * Float64(1.0 / n))), Float64(1.0 / n)))) - 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], -5e-10], N[(N[Power[N[(x + 1.0), $MachinePrecision], N[(1.0 / n), $MachinePrecision]], $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e-15], N[(N[Log[1 + N[(1.0 / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], N[(N[(1.0 + N[(x * N[(x * N[(N[(0.5 * N[(1.0 / N[Power[n, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(0.5 * N[(1.0 / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -5.00000000000000031e-10

   1. Initial program 52.8%

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

   if -5.00000000000000031e-10 < (/.f64 #s(literal 1 binary64) n) < 4.99999999999999999e-15

   1. Initial program 52.8%

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

   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 58.7

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

   4. Applied rewrites 58.7%

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

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

      10. lower-log1p.f64 57.7

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

   6. Applied rewrites 57.7%

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

   if 4.99999999999999999e-15 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 52.8%

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

   2. Taylor expanded in x around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 22.4

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

   4. Applied rewrites 22.4%

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

Alternative 3: 94.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \log \left(x - -1\right)\\ \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{-15}:\\ \;\;\;\;1 \cdot \left(-\mathsf{expm1}\left(\frac{\log x}{n} - \frac{t\_0}{n}\right)\right)\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-15}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(\frac{1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{+150}:\\ \;\;\;\;\left(\frac{x}{n} - -1\right) - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0 \cdot n - \log x \cdot n}{n \cdot n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (log (- x -1.0))))
   (if (<= (/ 1.0 n) -2e-15)
     (* 1.0 (- (expm1 (- (/ (log x) n) (/ t_0 n)))))
     (if (<= (/ 1.0 n) 5e-15)
       (/ (log1p (/ 1.0 x)) n)
       (if (<= (/ 1.0 n) 1e+150)
         (- (- (/ x n) -1.0) (pow x (/ 1.0 n)))
         (/ (- (* t_0 n) (* (log x) n)) (* n n)))))))

C

double code(double x, double n) {
	double t_0 = log((x - -1.0));
	double tmp;
	if ((1.0 / n) <= -2e-15) {
		tmp = 1.0 * -expm1(((log(x) / n) - (t_0 / n)));
	} else if ((1.0 / n) <= 5e-15) {
		tmp = log1p((1.0 / x)) / n;
	} else if ((1.0 / n) <= 1e+150) {
		tmp = ((x / n) - -1.0) - pow(x, (1.0 / n));
	} else {
		tmp = ((t_0 * n) - (log(x) * n)) / (n * n);
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double t_0 = Math.log((x - -1.0));
	double tmp;
	if ((1.0 / n) <= -2e-15) {
		tmp = 1.0 * -Math.expm1(((Math.log(x) / n) - (t_0 / n)));
	} else if ((1.0 / n) <= 5e-15) {
		tmp = Math.log1p((1.0 / x)) / n;
	} else if ((1.0 / n) <= 1e+150) {
		tmp = ((x / n) - -1.0) - Math.pow(x, (1.0 / n));
	} else {
		tmp = ((t_0 * n) - (Math.log(x) * n)) / (n * n);
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.log((x - -1.0))
	tmp = 0
	if (1.0 / n) <= -2e-15:
		tmp = 1.0 * -math.expm1(((math.log(x) / n) - (t_0 / n)))
	elif (1.0 / n) <= 5e-15:
		tmp = math.log1p((1.0 / x)) / n
	elif (1.0 / n) <= 1e+150:
		tmp = ((x / n) - -1.0) - math.pow(x, (1.0 / n))
	else:
		tmp = ((t_0 * n) - (math.log(x) * n)) / (n * n)
	return tmp

Julia

function code(x, n)
	t_0 = log(Float64(x - -1.0))
	tmp = 0.0
	if (Float64(1.0 / n) <= -2e-15)
		tmp = Float64(1.0 * Float64(-expm1(Float64(Float64(log(x) / n) - Float64(t_0 / n)))));
	elseif (Float64(1.0 / n) <= 5e-15)
		tmp = Float64(log1p(Float64(1.0 / x)) / n);
	elseif (Float64(1.0 / n) <= 1e+150)
		tmp = Float64(Float64(Float64(x / n) - -1.0) - (x ^ Float64(1.0 / n)));
	else
		tmp = Float64(Float64(Float64(t_0 * n) - Float64(log(x) * n)) / Float64(n * n));
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Log[N[(x - -1.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -2e-15], N[(1.0 * (-N[(Exp[N[(N[(N[Log[x], $MachinePrecision] / n), $MachinePrecision] - N[(t$95$0 / n), $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision])), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e-15], N[(N[Log[1 + N[(1.0 / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 1e+150], N[(N[(N[(x / n), $MachinePrecision] - -1.0), $MachinePrecision] - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(N[(t$95$0 * n), $MachinePrecision] - N[(N[Log[x], $MachinePrecision] * n), $MachinePrecision]), $MachinePrecision] / N[(n * n), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -2.0000000000000002e-15

   1. Initial program 52.8%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   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 78.1%

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

   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 78.2%

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

   if -2.0000000000000002e-15 < (/.f64 #s(literal 1 binary64) n) < 4.99999999999999999e-15

   1. Initial program 52.8%

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

   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 58.7

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

   4. Applied rewrites 58.7%

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

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

      10. lower-log1p.f64 57.7

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

   6. Applied rewrites 57.7%

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

   if 4.99999999999999999e-15 < (/.f64 #s(literal 1 binary64) n) < 9.99999999999999981e149

   1. Initial program 52.8%

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

   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 30.9

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

   4. Applied rewrites 30.9%

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

      1. add-flip 30.9

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

      2. metadata-eval 30.9

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

      3. lift-+.f64 N/A

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

      4. +-commutative N/A

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

      5. add-flip N/A

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

      6. metadata-eval N/A

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

      7. lower--.f64 30.9

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

   6. Applied rewrites 30.9%

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

   if 9.99999999999999981e149 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 52.8%

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

   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 58.7

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

   4. Applied rewrites 58.7%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. div-sub N/A

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

      4. frac-sub N/A

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

      5. *-commutative N/A

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. add-flip N/A

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

      9. metadata-eval N/A

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

      10. lift--.f64 N/A

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

      11. *-commutative N/A

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

      12. unpow2 N/A

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

      13. lift-pow.f64 N/A

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

      14. lower-/.f64 N/A

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

   6. Applied rewrites 49.2%

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

Alternative 4: 93.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{-15}:\\ \;\;\;\;1 \cdot \left(-\mathsf{expm1}\left(\frac{1}{\frac{n}{\log \left(\frac{x}{x - -1}\right)}}\right)\right)\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-15}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(\frac{1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{+150}:\\ \;\;\;\;\left(\frac{x}{n} - -1\right) - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log \left(x - -1\right) \cdot n - \log x \cdot n}{n \cdot n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= (/ 1.0 n) -2e-15)
   (* 1.0 (- (expm1 (/ 1.0 (/ n (log (/ x (- x -1.0))))))))
   (if (<= (/ 1.0 n) 5e-15)
     (/ (log1p (/ 1.0 x)) n)
     (if (<= (/ 1.0 n) 1e+150)
       (- (- (/ x n) -1.0) (pow x (/ 1.0 n)))
       (/ (- (* (log (- x -1.0)) n) (* (log x) n)) (* n n))))))

C

double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -2e-15) {
		tmp = 1.0 * -expm1((1.0 / (n / log((x / (x - -1.0))))));
	} else if ((1.0 / n) <= 5e-15) {
		tmp = log1p((1.0 / x)) / n;
	} else if ((1.0 / n) <= 1e+150) {
		tmp = ((x / n) - -1.0) - pow(x, (1.0 / n));
	} else {
		tmp = ((log((x - -1.0)) * n) - (log(x) * n)) / (n * n);
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -2e-15) {
		tmp = 1.0 * -Math.expm1((1.0 / (n / Math.log((x / (x - -1.0))))));
	} else if ((1.0 / n) <= 5e-15) {
		tmp = Math.log1p((1.0 / x)) / n;
	} else if ((1.0 / n) <= 1e+150) {
		tmp = ((x / n) - -1.0) - Math.pow(x, (1.0 / n));
	} else {
		tmp = ((Math.log((x - -1.0)) * n) - (Math.log(x) * n)) / (n * n);
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if (1.0 / n) <= -2e-15:
		tmp = 1.0 * -math.expm1((1.0 / (n / math.log((x / (x - -1.0))))))
	elif (1.0 / n) <= 5e-15:
		tmp = math.log1p((1.0 / x)) / n
	elif (1.0 / n) <= 1e+150:
		tmp = ((x / n) - -1.0) - math.pow(x, (1.0 / n))
	else:
		tmp = ((math.log((x - -1.0)) * n) - (math.log(x) * n)) / (n * n)
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -2e-15)
		tmp = Float64(1.0 * Float64(-expm1(Float64(1.0 / Float64(n / log(Float64(x / Float64(x - -1.0))))))));
	elseif (Float64(1.0 / n) <= 5e-15)
		tmp = Float64(log1p(Float64(1.0 / x)) / n);
	elseif (Float64(1.0 / n) <= 1e+150)
		tmp = Float64(Float64(Float64(x / n) - -1.0) - (x ^ Float64(1.0 / n)));
	else
		tmp = Float64(Float64(Float64(log(Float64(x - -1.0)) * n) - Float64(log(x) * n)) / Float64(n * n));
	end
	return tmp
end

Wolfram

code[x_, n_] := If[LessEqual[N[(1.0 / n), $MachinePrecision], -2e-15], N[(1.0 * (-N[(Exp[N[(1.0 / N[(n / N[Log[N[(x / N[(x - -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision])), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e-15], N[(N[Log[1 + N[(1.0 / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 1e+150], N[(N[(N[(x / n), $MachinePrecision] - -1.0), $MachinePrecision] - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Log[N[(x - -1.0), $MachinePrecision]], $MachinePrecision] * n), $MachinePrecision] - N[(N[Log[x], $MachinePrecision] * n), $MachinePrecision]), $MachinePrecision] / N[(n * n), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -2.0000000000000002e-15

   1. Initial program 52.8%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   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 78.1%

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

      1. lift--.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. sub-div N/A

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

      5. lift--.f64 N/A

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

      6. metadata-eval N/A

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

      7. add-flip N/A

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

      8. +-commutative N/A

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

      9. lift-+.f64 N/A

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

      10. div-flip N/A

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

      11. lower-/.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lift-log.f64 N/A

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

      14. lift-log.f64 N/A

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

      15. lift-+.f64 N/A

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

      16. +-commutative N/A

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

      17. add-flip N/A

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

      18. metadata-eval N/A

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

      19. lift--.f64 N/A

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

      20. diff-log N/A

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

      21. lower-log.f64 N/A

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

      22. lower-/.f64 78.2

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

   5. Applied rewrites 78.2%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 78.3%

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

   if -2.0000000000000002e-15 < (/.f64 #s(literal 1 binary64) n) < 4.99999999999999999e-15

   1. Initial program 52.8%

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

   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 58.7

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

   4. Applied rewrites 58.7%

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

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

      10. lower-log1p.f64 57.7

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

   6. Applied rewrites 57.7%

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

   if 4.99999999999999999e-15 < (/.f64 #s(literal 1 binary64) n) < 9.99999999999999981e149

   1. Initial program 52.8%

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

   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 30.9

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

   4. Applied rewrites 30.9%

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

      1. add-flip 30.9

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

      2. metadata-eval 30.9

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

      3. lift-+.f64 N/A

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

      4. +-commutative N/A

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

      5. add-flip N/A

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

      6. metadata-eval N/A

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

      7. lower--.f64 30.9

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

   6. Applied rewrites 30.9%

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

   if 9.99999999999999981e149 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 52.8%

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

   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 58.7

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

   4. Applied rewrites 58.7%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. div-sub N/A

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

      4. frac-sub N/A

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

      5. *-commutative N/A

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. add-flip N/A

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

      9. metadata-eval N/A

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

      10. lift--.f64 N/A

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

      11. *-commutative N/A

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

      12. unpow2 N/A

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

      13. lift-pow.f64 N/A

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

      14. lower-/.f64 N/A

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

   6. Applied rewrites 49.2%

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

Alternative 5: 93.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{-5}:\\ \;\;\;\;t\_0 \cdot \frac{1}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-15}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(\frac{1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{+150}:\\ \;\;\;\;\left(\frac{x}{n} - -1\right) - t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\log \left(x - -1\right) \cdot n - \log x \cdot n}{n \cdot n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -2e-5)
     (* t_0 (/ 1.0 (* n x)))
     (if (<= (/ 1.0 n) 5e-15)
       (/ (log1p (/ 1.0 x)) n)
       (if (<= (/ 1.0 n) 1e+150)
         (- (- (/ x n) -1.0) t_0)
         (/ (- (* (log (- x -1.0)) n) (* (log x) n)) (* n 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 * (1.0 / (n * x));
	} else if ((1.0 / n) <= 5e-15) {
		tmp = log1p((1.0 / x)) / n;
	} else if ((1.0 / n) <= 1e+150) {
		tmp = ((x / n) - -1.0) - t_0;
	} else {
		tmp = ((log((x - -1.0)) * n) - (log(x) * n)) / (n * 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 * (1.0 / (n * x));
	} else if ((1.0 / n) <= 5e-15) {
		tmp = Math.log1p((1.0 / x)) / n;
	} else if ((1.0 / n) <= 1e+150) {
		tmp = ((x / n) - -1.0) - t_0;
	} else {
		tmp = ((Math.log((x - -1.0)) * n) - (Math.log(x) * n)) / (n * 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 * (1.0 / (n * x))
	elif (1.0 / n) <= 5e-15:
		tmp = math.log1p((1.0 / x)) / n
	elif (1.0 / n) <= 1e+150:
		tmp = ((x / n) - -1.0) - t_0
	else:
		tmp = ((math.log((x - -1.0)) * n) - (math.log(x) * n)) / (n * 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(t_0 * Float64(1.0 / Float64(n * x)));
	elseif (Float64(1.0 / n) <= 5e-15)
		tmp = Float64(log1p(Float64(1.0 / x)) / n);
	elseif (Float64(1.0 / n) <= 1e+150)
		tmp = Float64(Float64(Float64(x / n) - -1.0) - t_0);
	else
		tmp = Float64(Float64(Float64(log(Float64(x - -1.0)) * n) - Float64(log(x) * n)) / Float64(n * 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[(t$95$0 * N[(1.0 / N[(n * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e-15], N[(N[Log[1 + N[(1.0 / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 1e+150], N[(N[(N[(x / n), $MachinePrecision] - -1.0), $MachinePrecision] - t$95$0), $MachinePrecision], N[(N[(N[(N[Log[N[(x - -1.0), $MachinePrecision]], $MachinePrecision] * n), $MachinePrecision] - N[(N[Log[x], $MachinePrecision] * n), $MachinePrecision]), $MachinePrecision] / N[(n * n), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

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

   1. Initial program 52.8%

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

   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 57.4

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

   4. Applied rewrites 57.4%

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

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

      3. lower-*.f64 N/A

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

   6. Applied rewrites 57.4%

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

   if -2.00000000000000016e-5 < (/.f64 #s(literal 1 binary64) n) < 4.99999999999999999e-15

   1. Initial program 52.8%

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

   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 58.7

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

   4. Applied rewrites 58.7%

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

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

      10. lower-log1p.f64 57.7

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

   6. Applied rewrites 57.7%

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

   if 4.99999999999999999e-15 < (/.f64 #s(literal 1 binary64) n) < 9.99999999999999981e149

   1. Initial program 52.8%

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

   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 30.9

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

   4. Applied rewrites 30.9%

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

      1. add-flip 30.9

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

      2. metadata-eval 30.9

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

      3. lift-+.f64 N/A

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

      4. +-commutative N/A

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

      5. add-flip N/A

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

      6. metadata-eval N/A

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

      7. lower--.f64 30.9

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

   6. Applied rewrites 30.9%

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

   if 9.99999999999999981e149 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 52.8%

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

   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 58.7

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

   4. Applied rewrites 58.7%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. div-sub N/A

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

      4. frac-sub N/A

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

      5. *-commutative N/A

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. add-flip N/A

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

      9. metadata-eval N/A

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

      10. lift--.f64 N/A

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

      11. *-commutative N/A

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

      12. unpow2 N/A

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

      13. lift-pow.f64 N/A

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

      14. lower-/.f64 N/A

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

   6. Applied rewrites 49.2%

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

Alternative 6: 93.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{-5}:\\ \;\;\;\;t\_0 \cdot \frac{1}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-15}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(\frac{1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{+150}:\\ \;\;\;\;\left(\frac{x}{n} - -1\right) - t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{-1 \cdot \frac{-1 \cdot \frac{0.3333333333333333 \cdot \frac{1}{x} - 0.5}{x} - 1}{x}}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -2e-5)
     (* t_0 (/ 1.0 (* n x)))
     (if (<= (/ 1.0 n) 5e-15)
       (/ (log1p (/ 1.0 x)) n)
       (if (<= (/ 1.0 n) 1e+150)
         (- (- (/ x n) -1.0) t_0)
         (/
          (*
           -1.0
           (/
            (- (* -1.0 (/ (- (* 0.3333333333333333 (/ 1.0 x)) 0.5) x)) 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 * (1.0 / (n * x));
	} else if ((1.0 / n) <= 5e-15) {
		tmp = log1p((1.0 / x)) / n;
	} else if ((1.0 / n) <= 1e+150) {
		tmp = ((x / n) - -1.0) - t_0;
	} else {
		tmp = (-1.0 * (((-1.0 * (((0.3333333333333333 * (1.0 / x)) - 0.5) / x)) - 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 * (1.0 / (n * x));
	} else if ((1.0 / n) <= 5e-15) {
		tmp = Math.log1p((1.0 / x)) / n;
	} else if ((1.0 / n) <= 1e+150) {
		tmp = ((x / n) - -1.0) - t_0;
	} else {
		tmp = (-1.0 * (((-1.0 * (((0.3333333333333333 * (1.0 / x)) - 0.5) / x)) - 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 * (1.0 / (n * x))
	elif (1.0 / n) <= 5e-15:
		tmp = math.log1p((1.0 / x)) / n
	elif (1.0 / n) <= 1e+150:
		tmp = ((x / n) - -1.0) - t_0
	else:
		tmp = (-1.0 * (((-1.0 * (((0.3333333333333333 * (1.0 / x)) - 0.5) / x)) - 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(t_0 * Float64(1.0 / Float64(n * x)));
	elseif (Float64(1.0 / n) <= 5e-15)
		tmp = Float64(log1p(Float64(1.0 / x)) / n);
	elseif (Float64(1.0 / n) <= 1e+150)
		tmp = Float64(Float64(Float64(x / n) - -1.0) - t_0);
	else
		tmp = Float64(Float64(-1.0 * Float64(Float64(Float64(-1.0 * Float64(Float64(Float64(0.3333333333333333 * Float64(1.0 / x)) - 0.5) / x)) - 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[(t$95$0 * N[(1.0 / N[(n * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e-15], N[(N[Log[1 + N[(1.0 / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 1e+150], N[(N[(N[(x / n), $MachinePrecision] - -1.0), $MachinePrecision] - t$95$0), $MachinePrecision], N[(N[(-1.0 * N[(N[(N[(-1.0 * N[(N[(N[(0.3333333333333333 * N[(1.0 / x), $MachinePrecision]), $MachinePrecision] - 0.5), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

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

   1. Initial program 52.8%

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

   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 57.4

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

   4. Applied rewrites 57.4%

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

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

      3. lower-*.f64 N/A

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

   6. Applied rewrites 57.4%

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

   if -2.00000000000000016e-5 < (/.f64 #s(literal 1 binary64) n) < 4.99999999999999999e-15

   1. Initial program 52.8%

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

   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 58.7

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

   4. Applied rewrites 58.7%

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

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

      10. lower-log1p.f64 57.7

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

   6. Applied rewrites 57.7%

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

   if 4.99999999999999999e-15 < (/.f64 #s(literal 1 binary64) n) < 9.99999999999999981e149

   1. Initial program 52.8%

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

   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 30.9

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

   4. Applied rewrites 30.9%

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

      1. add-flip 30.9

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

      2. metadata-eval 30.9

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

      3. lift-+.f64 N/A

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

      4. +-commutative N/A

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

      5. add-flip N/A

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

      6. metadata-eval N/A

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

      7. lower--.f64 30.9

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

   6. Applied rewrites 30.9%

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

   if 9.99999999999999981e149 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 52.8%

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

   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 58.7

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

   4. Applied rewrites 58.7%

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

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

      10. lower-log1p.f64 57.7

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

   6. Applied rewrites 57.7%

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

   7. Taylor expanded in x around -inf

      \[\leadsto \frac{-1 \cdot \frac{-1 \cdot \frac{\frac{1}{3} \cdot \frac{1}{x} - \frac{1}{2}}{x} - 1}{x}}{n} \]
   8. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{-1 \cdot \frac{-1 \cdot \frac{\frac{1}{3} \cdot \frac{1}{x} - \frac{1}{2}}{x} - 1}{x}}{n} \]

      2. lower-/.f64 N/A

         \[\leadsto \frac{-1 \cdot \frac{-1 \cdot \frac{\frac{1}{3} \cdot \frac{1}{x} - \frac{1}{2}}{x} - 1}{x}}{n} \]

      3. lower--.f64 N/A

         \[\leadsto \frac{-1 \cdot \frac{-1 \cdot \frac{\frac{1}{3} \cdot \frac{1}{x} - \frac{1}{2}}{x} - 1}{x}}{n} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{-1 \cdot \frac{-1 \cdot \frac{\frac{1}{3} \cdot \frac{1}{x} - \frac{1}{2}}{x} - 1}{x}}{n} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{-1 \cdot \frac{-1 \cdot \frac{\frac{1}{3} \cdot \frac{1}{x} - \frac{1}{2}}{x} - 1}{x}}{n} \]

      6. lower--.f64 N/A

         \[\leadsto \frac{-1 \cdot \frac{-1 \cdot \frac{\frac{1}{3} \cdot \frac{1}{x} - \frac{1}{2}}{x} - 1}{x}}{n} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{-1 \cdot \frac{-1 \cdot \frac{\frac{1}{3} \cdot \frac{1}{x} - \frac{1}{2}}{x} - 1}{x}}{n} \]

      8. lower-/.f64 46.1

         \[\leadsto \frac{-1 \cdot \frac{-1 \cdot \frac{0.3333333333333333 \cdot \frac{1}{x} - 0.5}{x} - 1}{x}}{n} \]

   9. Applied rewrites 46.1%

      \[\leadsto \frac{-1 \cdot \frac{-1 \cdot \frac{0.3333333333333333 \cdot \frac{1}{x} - 0.5}{x} - 1}{x}}{n} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 7: 93.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{-5}:\\ \;\;\;\;\frac{t\_0}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-15}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(\frac{1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{+150}:\\ \;\;\;\;\left(\frac{x}{n} - -1\right) - t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{-1 \cdot \frac{-1 \cdot \frac{0.3333333333333333 \cdot \frac{1}{x} - 0.5}{x} - 1}{x}}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -2e-5)
     (/ t_0 (* n x))
     (if (<= (/ 1.0 n) 5e-15)
       (/ (log1p (/ 1.0 x)) n)
       (if (<= (/ 1.0 n) 1e+150)
         (- (- (/ x n) -1.0) t_0)
         (/
          (*
           -1.0
           (/
            (- (* -1.0 (/ (- (* 0.3333333333333333 (/ 1.0 x)) 0.5) x)) 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 / (n * x);
	} else if ((1.0 / n) <= 5e-15) {
		tmp = log1p((1.0 / x)) / n;
	} else if ((1.0 / n) <= 1e+150) {
		tmp = ((x / n) - -1.0) - t_0;
	} else {
		tmp = (-1.0 * (((-1.0 * (((0.3333333333333333 * (1.0 / x)) - 0.5) / x)) - 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 / (n * x);
	} else if ((1.0 / n) <= 5e-15) {
		tmp = Math.log1p((1.0 / x)) / n;
	} else if ((1.0 / n) <= 1e+150) {
		tmp = ((x / n) - -1.0) - t_0;
	} else {
		tmp = (-1.0 * (((-1.0 * (((0.3333333333333333 * (1.0 / x)) - 0.5) / x)) - 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 / (n * x)
	elif (1.0 / n) <= 5e-15:
		tmp = math.log1p((1.0 / x)) / n
	elif (1.0 / n) <= 1e+150:
		tmp = ((x / n) - -1.0) - t_0
	else:
		tmp = (-1.0 * (((-1.0 * (((0.3333333333333333 * (1.0 / x)) - 0.5) / x)) - 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(t_0 / Float64(n * x));
	elseif (Float64(1.0 / n) <= 5e-15)
		tmp = Float64(log1p(Float64(1.0 / x)) / n);
	elseif (Float64(1.0 / n) <= 1e+150)
		tmp = Float64(Float64(Float64(x / n) - -1.0) - t_0);
	else
		tmp = Float64(Float64(-1.0 * Float64(Float64(Float64(-1.0 * Float64(Float64(Float64(0.3333333333333333 * Float64(1.0 / x)) - 0.5) / x)) - 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[(t$95$0 / N[(n * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e-15], N[(N[Log[1 + N[(1.0 / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 1e+150], N[(N[(N[(x / n), $MachinePrecision] - -1.0), $MachinePrecision] - t$95$0), $MachinePrecision], N[(N[(-1.0 * N[(N[(N[(-1.0 * N[(N[(N[(0.3333333333333333 * N[(1.0 / x), $MachinePrecision]), $MachinePrecision] - 0.5), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

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

   1. Initial program 52.8%

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

   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 57.4

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

   4. Applied rewrites 57.4%

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

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

      6. distribute-neg-frac N/A

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

      7. lift-log.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. log-rec N/A

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

      10. lift-log.f64 N/A

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

      11. distribute-frac-neg2 N/A

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

      12. frac-2neg N/A

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

      13. div-flip N/A

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

      14. mult-flip N/A

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

      15. lift-/.f64 N/A

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

      16. lift-log.f64 N/A

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

      17. pow-to-exp N/A

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

      18. lower-pow.f64 57.4

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

   6. Applied rewrites 57.4%

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

   if -2.00000000000000016e-5 < (/.f64 #s(literal 1 binary64) n) < 4.99999999999999999e-15

   1. Initial program 52.8%

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

   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 58.7

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

   4. Applied rewrites 58.7%

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

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

      10. lower-log1p.f64 57.7

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

   6. Applied rewrites 57.7%

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

   if 4.99999999999999999e-15 < (/.f64 #s(literal 1 binary64) n) < 9.99999999999999981e149

   1. Initial program 52.8%

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

   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 30.9

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

   4. Applied rewrites 30.9%

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

      1. add-flip 30.9

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

      2. metadata-eval 30.9

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

      3. lift-+.f64 N/A

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

      4. +-commutative N/A

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

      5. add-flip N/A

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

      6. metadata-eval N/A

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

      7. lower--.f64 30.9

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

   6. Applied rewrites 30.9%

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

   if 9.99999999999999981e149 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 52.8%

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

   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 58.7

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

   4. Applied rewrites 58.7%

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

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

      10. lower-log1p.f64 57.7

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

   6. Applied rewrites 57.7%

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

   7. Taylor expanded in x around -inf

      \[\leadsto \frac{-1 \cdot \frac{-1 \cdot \frac{\frac{1}{3} \cdot \frac{1}{x} - \frac{1}{2}}{x} - 1}{x}}{n} \]
   8. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{-1 \cdot \frac{-1 \cdot \frac{\frac{1}{3} \cdot \frac{1}{x} - \frac{1}{2}}{x} - 1}{x}}{n} \]

      2. lower-/.f64 N/A

         \[\leadsto \frac{-1 \cdot \frac{-1 \cdot \frac{\frac{1}{3} \cdot \frac{1}{x} - \frac{1}{2}}{x} - 1}{x}}{n} \]

      3. lower--.f64 N/A

         \[\leadsto \frac{-1 \cdot \frac{-1 \cdot \frac{\frac{1}{3} \cdot \frac{1}{x} - \frac{1}{2}}{x} - 1}{x}}{n} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{-1 \cdot \frac{-1 \cdot \frac{\frac{1}{3} \cdot \frac{1}{x} - \frac{1}{2}}{x} - 1}{x}}{n} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{-1 \cdot \frac{-1 \cdot \frac{\frac{1}{3} \cdot \frac{1}{x} - \frac{1}{2}}{x} - 1}{x}}{n} \]

      6. lower--.f64 N/A

         \[\leadsto \frac{-1 \cdot \frac{-1 \cdot \frac{\frac{1}{3} \cdot \frac{1}{x} - \frac{1}{2}}{x} - 1}{x}}{n} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{-1 \cdot \frac{-1 \cdot \frac{\frac{1}{3} \cdot \frac{1}{x} - \frac{1}{2}}{x} - 1}{x}}{n} \]

      8. lower-/.f64 46.1

         \[\leadsto \frac{-1 \cdot \frac{-1 \cdot \frac{0.3333333333333333 \cdot \frac{1}{x} - 0.5}{x} - 1}{x}}{n} \]

   9. Applied rewrites 46.1%

      \[\leadsto \frac{-1 \cdot \frac{-1 \cdot \frac{0.3333333333333333 \cdot \frac{1}{x} - 0.5}{x} - 1}{x}}{n} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 8: 93.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{-5}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-15}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(\frac{1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{+150}:\\ \;\;\;\;1 \cdot \left(-\mathsf{expm1}\left(\frac{\log x}{n}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-1 \cdot \frac{-1 \cdot \frac{0.3333333333333333 \cdot \frac{1}{x} - 0.5}{x} - 1}{x}}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= (/ 1.0 n) -2e-5)
   (/ (pow x (/ 1.0 n)) (* n x))
   (if (<= (/ 1.0 n) 5e-15)
     (/ (log1p (/ 1.0 x)) n)
     (if (<= (/ 1.0 n) 1e+150)
       (* 1.0 (- (expm1 (/ (log x) n))))
       (/
        (*
         -1.0
         (/ (- (* -1.0 (/ (- (* 0.3333333333333333 (/ 1.0 x)) 0.5) x)) 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) <= 5e-15) {
		tmp = log1p((1.0 / x)) / n;
	} else if ((1.0 / n) <= 1e+150) {
		tmp = 1.0 * -expm1((log(x) / n));
	} else {
		tmp = (-1.0 * (((-1.0 * (((0.3333333333333333 * (1.0 / x)) - 0.5) / x)) - 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) <= 5e-15) {
		tmp = Math.log1p((1.0 / x)) / n;
	} else if ((1.0 / n) <= 1e+150) {
		tmp = 1.0 * -Math.expm1((Math.log(x) / n));
	} else {
		tmp = (-1.0 * (((-1.0 * (((0.3333333333333333 * (1.0 / x)) - 0.5) / x)) - 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) <= 5e-15:
		tmp = math.log1p((1.0 / x)) / n
	elif (1.0 / n) <= 1e+150:
		tmp = 1.0 * -math.expm1((math.log(x) / n))
	else:
		tmp = (-1.0 * (((-1.0 * (((0.3333333333333333 * (1.0 / x)) - 0.5) / x)) - 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) <= 5e-15)
		tmp = Float64(log1p(Float64(1.0 / x)) / n);
	elseif (Float64(1.0 / n) <= 1e+150)
		tmp = Float64(1.0 * Float64(-expm1(Float64(log(x) / n))));
	else
		tmp = Float64(Float64(-1.0 * Float64(Float64(Float64(-1.0 * Float64(Float64(Float64(0.3333333333333333 * Float64(1.0 / x)) - 0.5) / x)) - 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], 5e-15], N[(N[Log[1 + N[(1.0 / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 1e+150], N[(1.0 * (-N[(Exp[N[(N[Log[x], $MachinePrecision] / n), $MachinePrecision]] - 1), $MachinePrecision])), $MachinePrecision], N[(N[(-1.0 * N[(N[(N[(-1.0 * N[(N[(N[(0.3333333333333333 * N[(1.0 / x), $MachinePrecision]), $MachinePrecision] - 0.5), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

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

   1. Initial program 52.8%

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

   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 57.4

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

   4. Applied rewrites 57.4%

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

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

      6. distribute-neg-frac N/A

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

      7. lift-log.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. log-rec N/A

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

      10. lift-log.f64 N/A

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

      11. distribute-frac-neg2 N/A

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

      12. frac-2neg N/A

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

      13. div-flip N/A

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

      14. mult-flip N/A

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

      15. lift-/.f64 N/A

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

      16. lift-log.f64 N/A

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

      17. pow-to-exp N/A

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

      18. lower-pow.f64 57.4

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

   6. Applied rewrites 57.4%

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

   if -2.00000000000000016e-5 < (/.f64 #s(literal 1 binary64) n) < 4.99999999999999999e-15

   1. Initial program 52.8%

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

   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 58.7

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

   4. Applied rewrites 58.7%

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

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

      10. lower-log1p.f64 57.7

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

   6. Applied rewrites 57.7%

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

   if 4.99999999999999999e-15 < (/.f64 #s(literal 1 binary64) n) < 9.99999999999999981e149

   1. Initial program 52.8%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   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 78.1%

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

   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 78.2%

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

   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 51.0

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

   9. Applied rewrites 51.0%

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

   if 9.99999999999999981e149 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 52.8%

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

   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 58.7

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

   4. Applied rewrites 58.7%

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

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

      10. lower-log1p.f64 57.7

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

   6. Applied rewrites 57.7%

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

   7. Taylor expanded in x around -inf

      \[\leadsto \frac{-1 \cdot \frac{-1 \cdot \frac{\frac{1}{3} \cdot \frac{1}{x} - \frac{1}{2}}{x} - 1}{x}}{n} \]
   8. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{-1 \cdot \frac{-1 \cdot \frac{\frac{1}{3} \cdot \frac{1}{x} - \frac{1}{2}}{x} - 1}{x}}{n} \]

      2. lower-/.f64 N/A

         \[\leadsto \frac{-1 \cdot \frac{-1 \cdot \frac{\frac{1}{3} \cdot \frac{1}{x} - \frac{1}{2}}{x} - 1}{x}}{n} \]

      3. lower--.f64 N/A

         \[\leadsto \frac{-1 \cdot \frac{-1 \cdot \frac{\frac{1}{3} \cdot \frac{1}{x} - \frac{1}{2}}{x} - 1}{x}}{n} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{-1 \cdot \frac{-1 \cdot \frac{\frac{1}{3} \cdot \frac{1}{x} - \frac{1}{2}}{x} - 1}{x}}{n} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{-1 \cdot \frac{-1 \cdot \frac{\frac{1}{3} \cdot \frac{1}{x} - \frac{1}{2}}{x} - 1}{x}}{n} \]

      6. lower--.f64 N/A

         \[\leadsto \frac{-1 \cdot \frac{-1 \cdot \frac{\frac{1}{3} \cdot \frac{1}{x} - \frac{1}{2}}{x} - 1}{x}}{n} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{-1 \cdot \frac{-1 \cdot \frac{\frac{1}{3} \cdot \frac{1}{x} - \frac{1}{2}}{x} - 1}{x}}{n} \]

      8. lower-/.f64 46.1

         \[\leadsto \frac{-1 \cdot \frac{-1 \cdot \frac{0.3333333333333333 \cdot \frac{1}{x} - 0.5}{x} - 1}{x}}{n} \]

   9. Applied rewrites 46.1%

      \[\leadsto \frac{-1 \cdot \frac{-1 \cdot \frac{0.3333333333333333 \cdot \frac{1}{x} - 0.5}{x} - 1}{x}}{n} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 9: 93.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{-5}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-15}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(\frac{1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{+183}:\\ \;\;\;\;1 \cdot \left(-\mathsf{expm1}\left(\frac{\log x}{n}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \frac{1}{n \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= (/ 1.0 n) -2e-5)
   (/ (pow x (/ 1.0 n)) (* n x))
   (if (<= (/ 1.0 n) 5e-15)
     (/ (log1p (/ 1.0 x)) n)
     (if (<= (/ 1.0 n) 5e+183)
       (* 1.0 (- (expm1 (/ (log x) n))))
       (* 1.0 (/ 1.0 (* n x)))))))

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) <= 5e-15) {
		tmp = log1p((1.0 / x)) / n;
	} else if ((1.0 / n) <= 5e+183) {
		tmp = 1.0 * -expm1((log(x) / n));
	} else {
		tmp = 1.0 * (1.0 / (n * x));
	}
	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) <= 5e-15) {
		tmp = Math.log1p((1.0 / x)) / n;
	} else if ((1.0 / n) <= 5e+183) {
		tmp = 1.0 * -Math.expm1((Math.log(x) / n));
	} else {
		tmp = 1.0 * (1.0 / (n * x));
	}
	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) <= 5e-15:
		tmp = math.log1p((1.0 / x)) / n
	elif (1.0 / n) <= 5e+183:
		tmp = 1.0 * -math.expm1((math.log(x) / n))
	else:
		tmp = 1.0 * (1.0 / (n * x))
	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) <= 5e-15)
		tmp = Float64(log1p(Float64(1.0 / x)) / n);
	elseif (Float64(1.0 / n) <= 5e+183)
		tmp = Float64(1.0 * Float64(-expm1(Float64(log(x) / n))));
	else
		tmp = Float64(1.0 * Float64(1.0 / Float64(n * x)));
	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], 5e-15], N[(N[Log[1 + N[(1.0 / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e+183], N[(1.0 * (-N[(Exp[N[(N[Log[x], $MachinePrecision] / n), $MachinePrecision]] - 1), $MachinePrecision])), $MachinePrecision], N[(1.0 * N[(1.0 / N[(n * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

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

   1. Initial program 52.8%

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

   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 57.4

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

   4. Applied rewrites 57.4%

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

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

      6. distribute-neg-frac N/A

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

      7. lift-log.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. log-rec N/A

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

      10. lift-log.f64 N/A

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

      11. distribute-frac-neg2 N/A

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

      12. frac-2neg N/A

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

      13. div-flip N/A

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

      14. mult-flip N/A

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

      15. lift-/.f64 N/A

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

      16. lift-log.f64 N/A

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

      17. pow-to-exp N/A

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

      18. lower-pow.f64 57.4

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

   6. Applied rewrites 57.4%

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

   if -2.00000000000000016e-5 < (/.f64 #s(literal 1 binary64) n) < 4.99999999999999999e-15

   1. Initial program 52.8%

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

   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 58.7

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

   4. Applied rewrites 58.7%

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

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

      10. lower-log1p.f64 57.7

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

   6. Applied rewrites 57.7%

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

   if 4.99999999999999999e-15 < (/.f64 #s(literal 1 binary64) n) < 5.00000000000000009e183

   1. Initial program 52.8%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   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 78.1%

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

   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 78.2%

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

   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 51.0

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

   9. Applied rewrites 51.0%

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

   if 5.00000000000000009e183 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 52.8%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   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 78.1%

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

   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 78.2%

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

   7. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 40.3

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

   9. Applied rewrites 40.3%

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

Alternative 10: 91.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.9:\\ \;\;\;\;1 \cdot \left(-\mathsf{expm1}\left(\frac{\log x}{n}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{{x}^{\left(\frac{1}{n}\right)}}{n}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 0.9)
   (* 1.0 (- (expm1 (/ (log x) n))))
   (/ (/ (pow x (/ 1.0 n)) n) x)))

C

double code(double x, double n) {
	double tmp;
	if (x <= 0.9) {
		tmp = 1.0 * -expm1((log(x) / n));
	} else {
		tmp = (pow(x, (1.0 / n)) / n) / x;
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double tmp;
	if (x <= 0.9) {
		tmp = 1.0 * -Math.expm1((Math.log(x) / n));
	} else {
		tmp = (Math.pow(x, (1.0 / n)) / n) / x;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 0.9:
		tmp = 1.0 * -math.expm1((math.log(x) / n))
	else:
		tmp = (math.pow(x, (1.0 / n)) / n) / x
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 0.9)
		tmp = Float64(1.0 * Float64(-expm1(Float64(log(x) / n))));
	else
		tmp = Float64(Float64((x ^ Float64(1.0 / n)) / n) / x);
	end
	return tmp
end

Wolfram

code[x_, n_] := If[LessEqual[x, 0.9], N[(1.0 * (-N[(Exp[N[(N[Log[x], $MachinePrecision] / n), $MachinePrecision]] - 1), $MachinePrecision])), $MachinePrecision], N[(N[(N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.900000000000000022

   1. Initial program 52.8%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   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 78.1%

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

   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 78.2%

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

   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 51.0

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

   9. Applied rewrites 51.0%

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

   if 0.900000000000000022 < x

   1. Initial program 52.8%

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

   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 57.4

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

   4. Applied rewrites 57.4%

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

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

      4. lift-/.f64 N/A

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

      5. lower-/.f64 58.1

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

   6. Applied rewrites 58.1%

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

Alternative 11: 80.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.55 \cdot 10^{-8}:\\ \;\;\;\;1 \cdot \left(-\mathsf{expm1}\left(\frac{\log x}{n}\right)\right)\\ \mathbf{elif}\;x \leq 7.6 \cdot 10^{+101}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(\frac{1}{x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log \left(\frac{x - -1}{x}\right)}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 1.55e-8)
   (* 1.0 (- (expm1 (/ (log x) n))))
   (if (<= x 7.6e+101) (/ (log1p (/ 1.0 x)) n) (/ (log (/ (- x -1.0) x)) n))))

C

double code(double x, double n) {
	double tmp;
	if (x <= 1.55e-8) {
		tmp = 1.0 * -expm1((log(x) / n));
	} else if (x <= 7.6e+101) {
		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 <= 1.55e-8) {
		tmp = 1.0 * -Math.expm1((Math.log(x) / n));
	} else if (x <= 7.6e+101) {
		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 <= 1.55e-8:
		tmp = 1.0 * -math.expm1((math.log(x) / n))
	elif x <= 7.6e+101:
		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 <= 1.55e-8)
		tmp = Float64(1.0 * Float64(-expm1(Float64(log(x) / n))));
	elseif (x <= 7.6e+101)
		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, 1.55e-8], N[(1.0 * (-N[(Exp[N[(N[Log[x], $MachinePrecision] / n), $MachinePrecision]] - 1), $MachinePrecision])), $MachinePrecision], If[LessEqual[x, 7.6e+101], 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 < 1.55e-8

   1. Initial program 52.8%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   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 78.1%

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

   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 78.2%

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

   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 51.0

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

   9. Applied rewrites 51.0%

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

   if 1.55e-8 < x < 7.5999999999999996e101

   1. Initial program 52.8%

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

   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 58.7

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

   4. Applied rewrites 58.7%

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

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

      10. lower-log1p.f64 57.7

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

   6. Applied rewrites 57.7%

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

   if 7.5999999999999996e101 < x

   1. Initial program 52.8%

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

   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 58.7

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

   4. Applied rewrites 58.7%

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
   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. lower-log.f64 N/A

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

      16. add-to-fraction N/A

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

      17. *-lft-identity N/A

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

      18. add-flip N/A

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

      19. metadata-eval N/A

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

      20. lift--.f64 N/A

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

      21. lower-/.f64 58.7

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

   6. Applied rewrites 58.7%

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

Alternative 12: 74.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\frac{\frac{1 + \frac{\log x}{n}}{n}}{x}\\ \mathbf{elif}\;t\_0 \leq 0.9962437956983377:\\ \;\;\;\;\frac{-\log \left(\frac{x}{x - -1}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \frac{1}{n \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (- (pow (+ x 1.0) (/ 1.0 n)) (pow x (/ 1.0 n)))))
   (if (<= t_0 (- INFINITY))
     (/ (/ (+ 1.0 (/ (log x) n)) n) x)
     (if (<= t_0 0.9962437956983377)
       (/ (- (log (/ x (- x -1.0)))) n)
       (* 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 + (log(x) / n)) / n) / x;
	} else if (t_0 <= 0.9962437956983377) {
		tmp = -log((x / (x - -1.0))) / n;
	} else {
		tmp = 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 + (Math.log(x) / n)) / n) / x;
	} else if (t_0 <= 0.9962437956983377) {
		tmp = -Math.log((x / (x - -1.0))) / n;
	} else {
		tmp = 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 + (math.log(x) / n)) / n) / x
	elif t_0 <= 0.9962437956983377:
		tmp = -math.log((x / (x - -1.0))) / n
	else:
		tmp = 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 + Float64(log(x) / n)) / n) / x);
	elseif (t_0 <= 0.9962437956983377)
		tmp = Float64(Float64(-log(Float64(x / Float64(x - -1.0)))) / n);
	else
		tmp = 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 + (log(x) / n)) / n) / x;
	elseif (t_0 <= 0.9962437956983377)
		tmp = -log((x / (x - -1.0))) / n;
	else
		tmp = 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[(N[(1.0 + N[(N[Log[x], $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[t$95$0, 0.9962437956983377], N[((-N[Log[N[(x / N[(x - -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / n), $MachinePrecision], N[(1.0 * N[(1.0 / N[(n * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (pow.f64 (+.f64 x #s(literal 1 binary64)) (/.f64 #s(literal 1 binary64) n)) (pow.f64 x (/.f64 #s(literal 1 binary64) n))) < -inf.0

   1. Initial program 52.8%

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

   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 57.4

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

   4. Applied rewrites 57.4%

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

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

      4. lift-/.f64 N/A

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

      5. lower-/.f64 58.1

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

   6. Applied rewrites 58.1%

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

   7. Taylor expanded in n around inf

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

      1. lower-+.f64 N/A

         \[\leadsto \frac{\frac{1 + \frac{\log x}{n}}{n}}{x} \]

      2. lower-/.f64 N/A

         \[\leadsto \frac{\frac{1 + \frac{\log x}{n}}{n}}{x} \]

      3. lower-log.f64 40.2

         \[\leadsto \frac{\frac{1 + \frac{\log x}{n}}{n}}{x} \]

   9. Applied rewrites 40.2%

      \[\leadsto \frac{\frac{1 + \frac{\log x}{n}}{n}}{x} \]

   if -inf.0 < (-.f64 (pow.f64 (+.f64 x #s(literal 1 binary64)) (/.f64 #s(literal 1 binary64) n)) (pow.f64 x (/.f64 #s(literal 1 binary64) n))) < 0.99624379569833765

   1. Initial program 52.8%

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

   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 58.7

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

   4. Applied rewrites 58.7%

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
   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 58.8

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

   6. Applied rewrites 58.8%

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

   if 0.99624379569833765 < (-.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 52.8%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   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 78.1%

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

   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 78.2%

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

   7. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 40.3

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

   9. Applied rewrites 40.3%

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

Alternative 13: 72.4% accurate, 1.4× speedup

Math

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

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= (/ 1.0 n) -20000.0)
   (/ (log (/ (- x -1.0) x)) n)
   (if (<= (/ 1.0 n) 1e+150) (/ (log1p (/ 1.0 x)) n) (* 1.0 (/ 1.0 (* n x))))))

C

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

Java

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

Python

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

Julia

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

Wolfram

code[x_, n_] := If[LessEqual[N[(1.0 / n), $MachinePrecision], -20000.0], N[(N[Log[N[(N[(x - -1.0), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 1e+150], N[(N[Log[1 + N[(1.0 / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], N[(1.0 * N[(1.0 / N[(n * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 52.8%

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

   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 58.7

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

   4. Applied rewrites 58.7%

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
   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. lower-log.f64 N/A

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

      16. add-to-fraction N/A

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

      17. *-lft-identity N/A

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

      18. add-flip N/A

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

      19. metadata-eval N/A

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

      20. lift--.f64 N/A

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

      21. lower-/.f64 58.7

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

   6. Applied rewrites 58.7%

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

   if -2e4 < (/.f64 #s(literal 1 binary64) n) < 9.99999999999999981e149

   1. Initial program 52.8%

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

   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 58.7

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

   4. Applied rewrites 58.7%

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

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

      10. lower-log1p.f64 57.7

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

   6. Applied rewrites 57.7%

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

   if 9.99999999999999981e149 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 52.8%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   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 78.1%

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

   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 78.2%

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

   7. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 40.3

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

   9. Applied rewrites 40.3%

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

Alternative 14: 68.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\\ t_1 := 1 \cdot \frac{1}{n \cdot x}\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 0.9962437956983377:\\ \;\;\;\;\frac{-\log \left(\frac{x}{x - -1}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (- (pow (+ x 1.0) (/ 1.0 n)) (pow x (/ 1.0 n))))
        (t_1 (* 1.0 (/ 1.0 (* n x)))))
   (if (<= t_0 (- INFINITY))
     t_1
     (if (<= t_0 0.9962437956983377) (/ (- (log (/ x (- x -1.0)))) n) t_1))))

C

double code(double x, double n) {
	double t_0 = pow((x + 1.0), (1.0 / n)) - pow(x, (1.0 / n));
	double t_1 = 1.0 * (1.0 / (n * x));
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = t_1;
	} else if (t_0 <= 0.9962437956983377) {
		tmp = -log((x / (x - -1.0))) / n;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double t_0 = Math.pow((x + 1.0), (1.0 / n)) - Math.pow(x, (1.0 / n));
	double t_1 = 1.0 * (1.0 / (n * x));
	double tmp;
	if (t_0 <= -Double.POSITIVE_INFINITY) {
		tmp = t_1;
	} else if (t_0 <= 0.9962437956983377) {
		tmp = -Math.log((x / (x - -1.0))) / n;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow((x + 1.0), (1.0 / n)) - math.pow(x, (1.0 / n))
	t_1 = 1.0 * (1.0 / (n * x))
	tmp = 0
	if t_0 <= -math.inf:
		tmp = t_1
	elif t_0 <= 0.9962437956983377:
		tmp = -math.log((x / (x - -1.0))) / n
	else:
		tmp = t_1
	return tmp

Julia

function code(x, n)
	t_0 = Float64((Float64(x + 1.0) ^ Float64(1.0 / n)) - (x ^ Float64(1.0 / n)))
	t_1 = Float64(1.0 * Float64(1.0 / Float64(n * x)))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = t_1;
	elseif (t_0 <= 0.9962437956983377)
		tmp = Float64(Float64(-log(Float64(x / Float64(x - -1.0)))) / n);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = ((x + 1.0) ^ (1.0 / n)) - (x ^ (1.0 / n));
	t_1 = 1.0 * (1.0 / (n * x));
	tmp = 0.0;
	if (t_0 <= -Inf)
		tmp = t_1;
	elseif (t_0 <= 0.9962437956983377)
		tmp = -log((x / (x - -1.0))) / n;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(N[Power[N[(x + 1.0), $MachinePrecision], N[(1.0 / n), $MachinePrecision]], $MachinePrecision] - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 * N[(1.0 / N[(n * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], t$95$1, If[LessEqual[t$95$0, 0.9962437956983377], N[((-N[Log[N[(x / N[(x - -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / n), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (pow.f64 (+.f64 x #s(literal 1 binary64)) (/.f64 #s(literal 1 binary64) n)) (pow.f64 x (/.f64 #s(literal 1 binary64) n))) < -inf.0 or 0.99624379569833765 < (-.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 52.8%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   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 78.1%

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

   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 78.2%

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

   7. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 40.3

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

   9. Applied rewrites 40.3%

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

   if -inf.0 < (-.f64 (pow.f64 (+.f64 x #s(literal 1 binary64)) (/.f64 #s(literal 1 binary64) n)) (pow.f64 x (/.f64 #s(literal 1 binary64) n))) < 0.99624379569833765

   1. Initial program 52.8%

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

   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 58.7

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

   4. Applied rewrites 58.7%

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
   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 58.8

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

   6. Applied rewrites 58.8%

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

Alternative 15: 68.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\\ t_1 := 1 \cdot \frac{1}{n \cdot x}\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 0.9962437956983377:\\ \;\;\;\;\frac{\log \left(\frac{x - -1}{x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (- (pow (+ x 1.0) (/ 1.0 n)) (pow x (/ 1.0 n))))
        (t_1 (* 1.0 (/ 1.0 (* n x)))))
   (if (<= t_0 (- INFINITY))
     t_1
     (if (<= t_0 0.9962437956983377) (/ (log (/ (- x -1.0) x)) n) t_1))))

C

double code(double x, double n) {
	double t_0 = pow((x + 1.0), (1.0 / n)) - pow(x, (1.0 / n));
	double t_1 = 1.0 * (1.0 / (n * x));
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = t_1;
	} else if (t_0 <= 0.9962437956983377) {
		tmp = log(((x - -1.0) / x)) / n;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double t_0 = Math.pow((x + 1.0), (1.0 / n)) - Math.pow(x, (1.0 / n));
	double t_1 = 1.0 * (1.0 / (n * x));
	double tmp;
	if (t_0 <= -Double.POSITIVE_INFINITY) {
		tmp = t_1;
	} else if (t_0 <= 0.9962437956983377) {
		tmp = Math.log(((x - -1.0) / x)) / n;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow((x + 1.0), (1.0 / n)) - math.pow(x, (1.0 / n))
	t_1 = 1.0 * (1.0 / (n * x))
	tmp = 0
	if t_0 <= -math.inf:
		tmp = t_1
	elif t_0 <= 0.9962437956983377:
		tmp = math.log(((x - -1.0) / x)) / n
	else:
		tmp = t_1
	return tmp

Julia

function code(x, n)
	t_0 = Float64((Float64(x + 1.0) ^ Float64(1.0 / n)) - (x ^ Float64(1.0 / n)))
	t_1 = Float64(1.0 * Float64(1.0 / Float64(n * x)))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = t_1;
	elseif (t_0 <= 0.9962437956983377)
		tmp = Float64(log(Float64(Float64(x - -1.0) / x)) / n);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = ((x + 1.0) ^ (1.0 / n)) - (x ^ (1.0 / n));
	t_1 = 1.0 * (1.0 / (n * x));
	tmp = 0.0;
	if (t_0 <= -Inf)
		tmp = t_1;
	elseif (t_0 <= 0.9962437956983377)
		tmp = log(((x - -1.0) / x)) / n;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(N[Power[N[(x + 1.0), $MachinePrecision], N[(1.0 / n), $MachinePrecision]], $MachinePrecision] - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 * N[(1.0 / N[(n * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], t$95$1, If[LessEqual[t$95$0, 0.9962437956983377], N[(N[Log[N[(N[(x - -1.0), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (pow.f64 (+.f64 x #s(literal 1 binary64)) (/.f64 #s(literal 1 binary64) n)) (pow.f64 x (/.f64 #s(literal 1 binary64) n))) < -inf.0 or 0.99624379569833765 < (-.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 52.8%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   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 78.1%

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

   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 78.2%

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

   7. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 40.3

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

   9. Applied rewrites 40.3%

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

   if -inf.0 < (-.f64 (pow.f64 (+.f64 x #s(literal 1 binary64)) (/.f64 #s(literal 1 binary64) n)) (pow.f64 x (/.f64 #s(literal 1 binary64) n))) < 0.99624379569833765

   1. Initial program 52.8%

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

   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 58.7

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

   4. Applied rewrites 58.7%

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
   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. lower-log.f64 N/A

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

      16. add-to-fraction N/A

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

      17. *-lft-identity N/A

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

      18. add-flip N/A

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

      19. metadata-eval N/A

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

      20. lift--.f64 N/A

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

      21. lower-/.f64 58.7

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

   6. Applied rewrites 58.7%

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

Alternative 16: 56.5% accurate, 2.7× speedup

Math

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

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 1.0) (/ (- x (log x)) n) (* 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 {
		tmp = 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
        tmp = 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 {
		tmp = 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
	else:
		tmp = 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);
	else
		tmp = 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;
	else
		tmp = 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], N[(1.0 * N[(1.0 / N[(n * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1

   1. Initial program 52.8%

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

   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 58.7

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

   4. Applied rewrites 58.7%

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

   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 31.3

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

   7. Applied rewrites 31.3%

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

   if 1 < x

   1. Initial program 52.8%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   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 78.1%

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

   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 78.2%

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

   7. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 40.3

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

   9. Applied rewrites 40.3%

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

Alternative 17: 40.3% accurate, 4.4× speedup

Math

\[\begin{array}{l} \\ 1 \cdot \frac{1}{n \cdot x} \end{array} \]

FPCore

(FPCore (x n) :precision binary64 (* 1.0 (/ 1.0 (* n x))))

C

double code(double x, double n) {
	return 1.0 * (1.0 / (n * x));
}

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 * (1.0d0 / (n * x))
end function

Java

public static double code(double x, double n) {
	return 1.0 * (1.0 / (n * x));
}

Python

def code(x, n):
	return 1.0 * (1.0 / (n * x))

Julia

function code(x, n)
	return Float64(1.0 * Float64(1.0 / Float64(n * x)))
end

MATLAB

function tmp = code(x, n)
	tmp = 1.0 * (1.0 / (n * x));
end

Wolfram

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

Derivation

1. Initial program 52.8%

   \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
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 78.1%

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

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 78.2%

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

7. Taylor expanded in x around inf

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

   1. lower-/.f64 N/A

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

   2. lower-*.f64 40.3

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

9. Applied rewrites 40.3%

   \[\leadsto 1 \cdot \color{blue}{\frac{1}{n \cdot x}} \]
10. Add Preprocessing

Alternative 18: 15.2% 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 52.8%

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

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 58.7

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

4. Applied rewrites 58.7%

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

5. Taylor expanded in x around -inf

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

   1. lower-/.f64 15.2

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

7. Applied rewrites 15.2%

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

Reproduce

herbie shell --seed 2025156 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  :precision binary64
  (- (pow (+ x 1.0) (/ 1.0 n)) (pow x (/ 1.0 n))))