2nthrt (problem 3.4.6)

Percentage Accurate: 53.2% → 94.9%

Time: 16.6s

Alternatives: 19

Speedup: 2.4×

• could not determine a ground truth

  1. x = -5.950374667046695e+214
  2. n = 3.2841364917295657e+115

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 53.2% accurate, 1.0× speedup

Math

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

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: 94.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{-6}:\\ \;\;\;\;\frac{\frac{1}{x}}{{x}^{\left(\frac{-1}{n}\right)} \cdot n}\\ \mathbf{elif}\;\frac{1}{n} \leq 0.02:\\ \;\;\;\;\frac{\mathsf{log1p}\left(\frac{1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{+117}:\\ \;\;\;\;{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\log \left({\left(\frac{x - -1}{x}\right)}^{\left(\frac{1}{n}\right)}\right)\\ \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= (/ 1.0 n) -2e-6)
   (/ (/ 1.0 x) (* (pow x (/ -1.0 n)) n))
   (if (<= (/ 1.0 n) 0.02)
     (/ (log1p (/ 1.0 x)) n)
     (if (<= (/ 1.0 n) 1e+117)
       (- (pow (+ x 1.0) (/ 1.0 n)) (pow x (/ 1.0 n)))
       (log (pow (/ (- x -1.0) x) (/ 1.0 n)))))))

C

double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -2e-6) {
		tmp = (1.0 / x) / (pow(x, (-1.0 / n)) * n);
	} else if ((1.0 / n) <= 0.02) {
		tmp = log1p((1.0 / x)) / n;
	} else if ((1.0 / n) <= 1e+117) {
		tmp = pow((x + 1.0), (1.0 / n)) - pow(x, (1.0 / n));
	} else {
		tmp = log(pow(((x - -1.0) / x), (1.0 / n)));
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -2e-6) {
		tmp = (1.0 / x) / (Math.pow(x, (-1.0 / n)) * n);
	} else if ((1.0 / n) <= 0.02) {
		tmp = Math.log1p((1.0 / x)) / n;
	} else if ((1.0 / n) <= 1e+117) {
		tmp = Math.pow((x + 1.0), (1.0 / n)) - Math.pow(x, (1.0 / n));
	} else {
		tmp = Math.log(Math.pow(((x - -1.0) / x), (1.0 / n)));
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if (1.0 / n) <= -2e-6:
		tmp = (1.0 / x) / (math.pow(x, (-1.0 / n)) * n)
	elif (1.0 / n) <= 0.02:
		tmp = math.log1p((1.0 / x)) / n
	elif (1.0 / n) <= 1e+117:
		tmp = math.pow((x + 1.0), (1.0 / n)) - math.pow(x, (1.0 / n))
	else:
		tmp = math.log(math.pow(((x - -1.0) / x), (1.0 / n)))
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -2e-6)
		tmp = Float64(Float64(1.0 / x) / Float64((x ^ Float64(-1.0 / n)) * n));
	elseif (Float64(1.0 / n) <= 0.02)
		tmp = Float64(log1p(Float64(1.0 / x)) / n);
	elseif (Float64(1.0 / n) <= 1e+117)
		tmp = Float64((Float64(x + 1.0) ^ Float64(1.0 / n)) - (x ^ Float64(1.0 / n)));
	else
		tmp = log((Float64(Float64(x - -1.0) / x) ^ Float64(1.0 / n)));
	end
	return tmp
end

Wolfram

code[x_, n_] := If[LessEqual[N[(1.0 / n), $MachinePrecision], -2e-6], N[(N[(1.0 / x), $MachinePrecision] / N[(N[Power[x, N[(-1.0 / n), $MachinePrecision]], $MachinePrecision] * n), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 0.02], N[(N[Log[1 + N[(1.0 / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 1e+117], N[(N[Power[N[(x + 1.0), $MachinePrecision], N[(1.0 / n), $MachinePrecision]], $MachinePrecision] - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Log[N[Power[N[(N[(x - -1.0), $MachinePrecision] / x), $MachinePrecision], N[(1.0 / n), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -1.99999999999999991e-6

   1. Initial program 53.2%

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

   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 57.6

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

   4. Applied rewrites 57.6%

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

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

      4. lift-*.f64 N/A

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

      5. mul-1-neg N/A

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

      6. exp-neg N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. associate-/r* N/A

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

      10. lift-/.f64 N/A

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

      11. frac-times N/A

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

      12. *-lft-identity N/A

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

      13. lower-/.f64 N/A

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

      14. lower-*.f64 N/A

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

   6. Applied rewrites 58.4%

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

   if -1.99999999999999991e-6 < (/.f64 #s(literal 1 binary64) n) < 0.0200000000000000004

   1. Initial program 53.2%

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

   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 59.4

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

   4. Applied rewrites 59.4%

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

      1. lift--.f64 N/A

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

      2. sub-flip N/A

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

      3. mul-1-neg N/A

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

      4. lift-*.f64 N/A

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

      5. +-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lift-log.f64 N/A

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

      9. log-rec N/A

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

      10. lift-/.f64 N/A

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

      11. lift-log.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. +-commutative N/A

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

      14. lift-+.f64 N/A

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

      15. sum-log N/A

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

      16. lower-log.f64 N/A

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

      17. lower-*.f64 55.2

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

      18. lift-+.f64 N/A

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

      19. add-flip N/A

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

      20. unpow1 N/A

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

      21. metadata-eval N/A

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

      22. pow-flip N/A

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

      23. inv-pow N/A

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

      24. lift-/.f64 N/A

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

      25. metadata-eval N/A

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

      26. lower--.f64 N/A

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

      27. lift-/.f64 N/A

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

      28. inv-pow N/A

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

      29. pow-flip N/A

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

      30. metadata-eval N/A

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

      31. unpow1 55.2

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

   6. Applied rewrites 55.2%

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

      1. lift-log.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. log-prod N/A

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

      4. lift--.f64 N/A

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

      5. metadata-eval N/A

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

      6. add-flip N/A

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

      7. log-prod N/A

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

      8. distribute-lft-in N/A

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

      9. lift-/.f64 N/A

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

      10. inv-pow N/A

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

      11. pow-plus N/A

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

      12. metadata-eval N/A

          \[\leadsto \frac{\log \left({x}^{0} + \frac{1}{x} \cdot 1\right)}{n} \]

      13. metadata-eval N/A

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

      14. *-commutative N/A

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

      15. *-lft-identity N/A

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

      16. lower-log1p.f64 58.2

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

   8. Applied rewrites 58.2%

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

   if 0.0200000000000000004 < (/.f64 #s(literal 1 binary64) n) < 1.00000000000000005e117

   1. Initial program 53.2%

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

   if 1.00000000000000005e117 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 53.2%

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

   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 59.4

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

   4. Applied rewrites 59.4%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. *-commutative N/A

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

      4. lift--.f64 N/A

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

      5. lift-log.f64 N/A

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

      6. lift-log.f64 N/A

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

      7. diff-log N/A

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

      8. log-pow-rev N/A

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

      9. lower-log.f64 N/A

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

      10. lower-pow.f64 N/A

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

   6. Applied rewrites 52.4%

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

Alternative 2: 94.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{-6}:\\ \;\;\;\;\frac{\frac{1}{x}}{{x}^{\left(\frac{-1}{n}\right)} \cdot n}\\ \mathbf{elif}\;\frac{1}{n} \leq 0.02:\\ \;\;\;\;\frac{\mathsf{log1p}\left(\frac{1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{+117}:\\ \;\;\;\;{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{-1 \cdot \frac{-0.16666666666666666 \cdot \frac{{\log x}^{3}}{n} - 0.5 \cdot {\log x}^{2}}{n} - -1 \cdot \log x}{n}\\ \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= (/ 1.0 n) -2e-6)
   (/ (/ 1.0 x) (* (pow x (/ -1.0 n)) n))
   (if (<= (/ 1.0 n) 0.02)
     (/ (log1p (/ 1.0 x)) n)
     (if (<= (/ 1.0 n) 1e+117)
       (- (pow (+ x 1.0) (/ 1.0 n)) (pow x (/ 1.0 n)))
       (*
        -1.0
        (/
         (-
          (*
           -1.0
           (/
            (-
             (* -0.16666666666666666 (/ (pow (log x) 3.0) n))
             (* 0.5 (pow (log x) 2.0)))
            n))
          (* -1.0 (log x)))
         n))))))

C

double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -2e-6) {
		tmp = (1.0 / x) / (pow(x, (-1.0 / n)) * n);
	} else if ((1.0 / n) <= 0.02) {
		tmp = log1p((1.0 / x)) / n;
	} else if ((1.0 / n) <= 1e+117) {
		tmp = pow((x + 1.0), (1.0 / n)) - pow(x, (1.0 / n));
	} else {
		tmp = -1.0 * (((-1.0 * (((-0.16666666666666666 * (pow(log(x), 3.0) / n)) - (0.5 * pow(log(x), 2.0))) / n)) - (-1.0 * log(x))) / n);
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -2e-6) {
		tmp = (1.0 / x) / (Math.pow(x, (-1.0 / n)) * n);
	} else if ((1.0 / n) <= 0.02) {
		tmp = Math.log1p((1.0 / x)) / n;
	} else if ((1.0 / n) <= 1e+117) {
		tmp = Math.pow((x + 1.0), (1.0 / n)) - Math.pow(x, (1.0 / n));
	} else {
		tmp = -1.0 * (((-1.0 * (((-0.16666666666666666 * (Math.pow(Math.log(x), 3.0) / n)) - (0.5 * Math.pow(Math.log(x), 2.0))) / n)) - (-1.0 * Math.log(x))) / n);
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if (1.0 / n) <= -2e-6:
		tmp = (1.0 / x) / (math.pow(x, (-1.0 / n)) * n)
	elif (1.0 / n) <= 0.02:
		tmp = math.log1p((1.0 / x)) / n
	elif (1.0 / n) <= 1e+117:
		tmp = math.pow((x + 1.0), (1.0 / n)) - math.pow(x, (1.0 / n))
	else:
		tmp = -1.0 * (((-1.0 * (((-0.16666666666666666 * (math.pow(math.log(x), 3.0) / n)) - (0.5 * math.pow(math.log(x), 2.0))) / n)) - (-1.0 * math.log(x))) / n)
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -2e-6)
		tmp = Float64(Float64(1.0 / x) / Float64((x ^ Float64(-1.0 / n)) * n));
	elseif (Float64(1.0 / n) <= 0.02)
		tmp = Float64(log1p(Float64(1.0 / x)) / n);
	elseif (Float64(1.0 / n) <= 1e+117)
		tmp = Float64((Float64(x + 1.0) ^ Float64(1.0 / n)) - (x ^ Float64(1.0 / n)));
	else
		tmp = Float64(-1.0 * Float64(Float64(Float64(-1.0 * Float64(Float64(Float64(-0.16666666666666666 * Float64((log(x) ^ 3.0) / n)) - Float64(0.5 * (log(x) ^ 2.0))) / n)) - Float64(-1.0 * log(x))) / n));
	end
	return tmp
end

Wolfram

code[x_, n_] := If[LessEqual[N[(1.0 / n), $MachinePrecision], -2e-6], N[(N[(1.0 / x), $MachinePrecision] / N[(N[Power[x, N[(-1.0 / n), $MachinePrecision]], $MachinePrecision] * n), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 0.02], N[(N[Log[1 + N[(1.0 / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 1e+117], N[(N[Power[N[(x + 1.0), $MachinePrecision], N[(1.0 / n), $MachinePrecision]], $MachinePrecision] - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(-1.0 * N[(N[(N[(-1.0 * N[(N[(N[(-0.16666666666666666 * N[(N[Power[N[Log[x], $MachinePrecision], 3.0], $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision] - N[(0.5 * N[Power[N[Log[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision] - N[(-1.0 * N[Log[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -1.99999999999999991e-6

   1. Initial program 53.2%

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

   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 57.6

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

   4. Applied rewrites 57.6%

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

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

      4. lift-*.f64 N/A

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

      5. mul-1-neg N/A

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

      6. exp-neg N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. associate-/r* N/A

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

      10. lift-/.f64 N/A

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

      11. frac-times N/A

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

      12. *-lft-identity N/A

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

      13. lower-/.f64 N/A

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

      14. lower-*.f64 N/A

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

   6. Applied rewrites 58.4%

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

   if -1.99999999999999991e-6 < (/.f64 #s(literal 1 binary64) n) < 0.0200000000000000004

   1. Initial program 53.2%

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

   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 59.4

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

   4. Applied rewrites 59.4%

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

      1. lift--.f64 N/A

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

      2. sub-flip N/A

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

      3. mul-1-neg N/A

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

      4. lift-*.f64 N/A

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

      5. +-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lift-log.f64 N/A

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

      9. log-rec N/A

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

      10. lift-/.f64 N/A

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

      11. lift-log.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. +-commutative N/A

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

      14. lift-+.f64 N/A

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

      15. sum-log N/A

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

      16. lower-log.f64 N/A

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

      17. lower-*.f64 55.2

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

      18. lift-+.f64 N/A

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

      19. add-flip N/A

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

      20. unpow1 N/A

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

      21. metadata-eval N/A

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

      22. pow-flip N/A

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

      23. inv-pow N/A

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

      24. lift-/.f64 N/A

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

      25. metadata-eval N/A

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

      26. lower--.f64 N/A

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

      27. lift-/.f64 N/A

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

      28. inv-pow N/A

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

      29. pow-flip N/A

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

      30. metadata-eval N/A

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

      31. unpow1 55.2

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

   6. Applied rewrites 55.2%

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

      1. lift-log.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. log-prod N/A

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

      4. lift--.f64 N/A

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

      5. metadata-eval N/A

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

      6. add-flip N/A

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

      7. log-prod N/A

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

      8. distribute-lft-in N/A

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

      9. lift-/.f64 N/A

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

      10. inv-pow N/A

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

      11. pow-plus N/A

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

      12. metadata-eval N/A

          \[\leadsto \frac{\log \left({x}^{0} + \frac{1}{x} \cdot 1\right)}{n} \]

      13. metadata-eval N/A

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

      14. *-commutative N/A

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

      15. *-lft-identity N/A

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

      16. lower-log1p.f64 58.2

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

   8. Applied rewrites 58.2%

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

   if 0.0200000000000000004 < (/.f64 #s(literal 1 binary64) n) < 1.00000000000000005e117

   1. Initial program 53.2%

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

   if 1.00000000000000005e117 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 53.2%

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

   2. Taylor expanded in n around -inf

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

   4. Applied rewrites 74.6%

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

   5. Taylor expanded in x around 0

      \[\leadsto -1 \cdot \frac{-1 \cdot \frac{\frac{-1}{6} \cdot \frac{{\log x}^{3}}{n} - \frac{1}{2} \cdot {\log x}^{2}}{n} - -1 \cdot \log x}{n} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto -1 \cdot \frac{-1 \cdot \frac{\frac{-1}{6} \cdot \frac{{\log x}^{3}}{n} - \frac{1}{2} \cdot {\log x}^{2}}{n} - -1 \cdot \log x}{n} \]

      2. lower-/.f64 N/A

         \[\leadsto -1 \cdot \frac{-1 \cdot \frac{\frac{-1}{6} \cdot \frac{{\log x}^{3}}{n} - \frac{1}{2} \cdot {\log x}^{2}}{n} - -1 \cdot \log x}{n} \]

   7. Applied rewrites 46.3%

      \[\leadsto -1 \cdot \frac{-1 \cdot \frac{-0.16666666666666666 \cdot \frac{{\log x}^{3}}{n} - 0.5 \cdot {\log x}^{2}}{n} - -1 \cdot \log x}{n} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 3: 94.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{-6}:\\ \;\;\;\;\frac{\frac{1}{x}}{{x}^{\left(\frac{-1}{n}\right)} \cdot n}\\ \mathbf{elif}\;\frac{1}{n} \leq 0.02:\\ \;\;\;\;\frac{\mathsf{log1p}\left(\frac{1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{+117}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\log \left({\left(\frac{x - -1}{x}\right)}^{\left(\frac{1}{n}\right)}\right)\\ \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= (/ 1.0 n) -2e-6)
   (/ (/ 1.0 x) (* (pow x (/ -1.0 n)) n))
   (if (<= (/ 1.0 n) 0.02)
     (/ (log1p (/ 1.0 x)) n)
     (if (<= (/ 1.0 n) 1e+117)
       (- (+ 1.0 (/ x n)) (pow x (/ 1.0 n)))
       (log (pow (/ (- x -1.0) x) (/ 1.0 n)))))))

C

double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -2e-6) {
		tmp = (1.0 / x) / (pow(x, (-1.0 / n)) * n);
	} else if ((1.0 / n) <= 0.02) {
		tmp = log1p((1.0 / x)) / n;
	} else if ((1.0 / n) <= 1e+117) {
		tmp = (1.0 + (x / n)) - pow(x, (1.0 / n));
	} else {
		tmp = log(pow(((x - -1.0) / x), (1.0 / n)));
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -2e-6) {
		tmp = (1.0 / x) / (Math.pow(x, (-1.0 / n)) * n);
	} else if ((1.0 / n) <= 0.02) {
		tmp = Math.log1p((1.0 / x)) / n;
	} else if ((1.0 / n) <= 1e+117) {
		tmp = (1.0 + (x / n)) - Math.pow(x, (1.0 / n));
	} else {
		tmp = Math.log(Math.pow(((x - -1.0) / x), (1.0 / n)));
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if (1.0 / n) <= -2e-6:
		tmp = (1.0 / x) / (math.pow(x, (-1.0 / n)) * n)
	elif (1.0 / n) <= 0.02:
		tmp = math.log1p((1.0 / x)) / n
	elif (1.0 / n) <= 1e+117:
		tmp = (1.0 + (x / n)) - math.pow(x, (1.0 / n))
	else:
		tmp = math.log(math.pow(((x - -1.0) / x), (1.0 / n)))
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -2e-6)
		tmp = Float64(Float64(1.0 / x) / Float64((x ^ Float64(-1.0 / n)) * n));
	elseif (Float64(1.0 / n) <= 0.02)
		tmp = Float64(log1p(Float64(1.0 / x)) / n);
	elseif (Float64(1.0 / n) <= 1e+117)
		tmp = Float64(Float64(1.0 + Float64(x / n)) - (x ^ Float64(1.0 / n)));
	else
		tmp = log((Float64(Float64(x - -1.0) / x) ^ Float64(1.0 / n)));
	end
	return tmp
end

Wolfram

code[x_, n_] := If[LessEqual[N[(1.0 / n), $MachinePrecision], -2e-6], N[(N[(1.0 / x), $MachinePrecision] / N[(N[Power[x, N[(-1.0 / n), $MachinePrecision]], $MachinePrecision] * n), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 0.02], N[(N[Log[1 + N[(1.0 / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 1e+117], N[(N[(1.0 + N[(x / n), $MachinePrecision]), $MachinePrecision] - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Log[N[Power[N[(N[(x - -1.0), $MachinePrecision] / x), $MachinePrecision], N[(1.0 / n), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -1.99999999999999991e-6

   1. Initial program 53.2%

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

   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 57.6

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

   4. Applied rewrites 57.6%

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

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

      4. lift-*.f64 N/A

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

      5. mul-1-neg N/A

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

      6. exp-neg N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. associate-/r* N/A

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

      10. lift-/.f64 N/A

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

      11. frac-times N/A

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

      12. *-lft-identity N/A

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

      13. lower-/.f64 N/A

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

      14. lower-*.f64 N/A

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

   6. Applied rewrites 58.4%

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

   if -1.99999999999999991e-6 < (/.f64 #s(literal 1 binary64) n) < 0.0200000000000000004

   1. Initial program 53.2%

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

   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 59.4

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

   4. Applied rewrites 59.4%

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

      1. lift--.f64 N/A

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

      2. sub-flip N/A

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

      3. mul-1-neg N/A

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

      4. lift-*.f64 N/A

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

      5. +-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lift-log.f64 N/A

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

      9. log-rec N/A

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

      10. lift-/.f64 N/A

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

      11. lift-log.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. +-commutative N/A

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

      14. lift-+.f64 N/A

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

      15. sum-log N/A

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

      16. lower-log.f64 N/A

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

      17. lower-*.f64 55.2

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

      18. lift-+.f64 N/A

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

      19. add-flip N/A

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

      20. unpow1 N/A

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

      21. metadata-eval N/A

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

      22. pow-flip N/A

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

      23. inv-pow N/A

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

      24. lift-/.f64 N/A

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

      25. metadata-eval N/A

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

      26. lower--.f64 N/A

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

      27. lift-/.f64 N/A

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

      28. inv-pow N/A

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

      29. pow-flip N/A

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

      30. metadata-eval N/A

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

      31. unpow1 55.2

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

   6. Applied rewrites 55.2%

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

      1. lift-log.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. log-prod N/A

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

      4. lift--.f64 N/A

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

      5. metadata-eval N/A

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

      6. add-flip N/A

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

      7. log-prod N/A

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

      8. distribute-lft-in N/A

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

      9. lift-/.f64 N/A

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

      10. inv-pow N/A

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

      11. pow-plus N/A

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

      12. metadata-eval N/A

          \[\leadsto \frac{\log \left({x}^{0} + \frac{1}{x} \cdot 1\right)}{n} \]

      13. metadata-eval N/A

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

      14. *-commutative N/A

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

      15. *-lft-identity N/A

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

      16. lower-log1p.f64 58.2

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

   8. Applied rewrites 58.2%

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

   if 0.0200000000000000004 < (/.f64 #s(literal 1 binary64) n) < 1.00000000000000005e117

   1. Initial program 53.2%

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

   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 30.6

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

   4. Applied rewrites 30.6%

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

   if 1.00000000000000005e117 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 53.2%

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

   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 59.4

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

   4. Applied rewrites 59.4%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. *-commutative N/A

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

      4. lift--.f64 N/A

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

      5. lift-log.f64 N/A

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

      6. lift-log.f64 N/A

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

      7. diff-log N/A

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

      8. log-pow-rev N/A

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

      9. lower-log.f64 N/A

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

      10. lower-pow.f64 N/A

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

   6. Applied rewrites 52.4%

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

Alternative 4: 94.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{-6}:\\ \;\;\;\;\frac{\frac{1}{x}}{{x}^{\left(\frac{-1}{n}\right)} \cdot n}\\ \mathbf{elif}\;\frac{1}{n} \leq 0.02:\\ \;\;\;\;\frac{\mathsf{log1p}\left(\frac{1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{+117}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1 - \frac{\log x}{n}}{n \cdot x}\\ \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= (/ 1.0 n) -2e-6)
   (/ (/ 1.0 x) (* (pow x (/ -1.0 n)) n))
   (if (<= (/ 1.0 n) 0.02)
     (/ (log1p (/ 1.0 x)) n)
     (if (<= (/ 1.0 n) 1e+117)
       (- (+ 1.0 (/ x n)) (pow x (/ 1.0 n)))
       (/ (- -1.0 (/ (log x) n)) (* n x))))))

C

double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -2e-6) {
		tmp = (1.0 / x) / (pow(x, (-1.0 / n)) * n);
	} else if ((1.0 / n) <= 0.02) {
		tmp = log1p((1.0 / x)) / n;
	} else if ((1.0 / n) <= 1e+117) {
		tmp = (1.0 + (x / n)) - pow(x, (1.0 / n));
	} else {
		tmp = (-1.0 - (log(x) / n)) / (n * x);
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -2e-6) {
		tmp = (1.0 / x) / (Math.pow(x, (-1.0 / n)) * n);
	} else if ((1.0 / n) <= 0.02) {
		tmp = Math.log1p((1.0 / x)) / n;
	} else if ((1.0 / n) <= 1e+117) {
		tmp = (1.0 + (x / n)) - Math.pow(x, (1.0 / n));
	} else {
		tmp = (-1.0 - (Math.log(x) / n)) / (n * x);
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if (1.0 / n) <= -2e-6:
		tmp = (1.0 / x) / (math.pow(x, (-1.0 / n)) * n)
	elif (1.0 / n) <= 0.02:
		tmp = math.log1p((1.0 / x)) / n
	elif (1.0 / n) <= 1e+117:
		tmp = (1.0 + (x / n)) - math.pow(x, (1.0 / n))
	else:
		tmp = (-1.0 - (math.log(x) / n)) / (n * x)
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -2e-6)
		tmp = Float64(Float64(1.0 / x) / Float64((x ^ Float64(-1.0 / n)) * n));
	elseif (Float64(1.0 / n) <= 0.02)
		tmp = Float64(log1p(Float64(1.0 / x)) / n);
	elseif (Float64(1.0 / n) <= 1e+117)
		tmp = Float64(Float64(1.0 + Float64(x / n)) - (x ^ Float64(1.0 / n)));
	else
		tmp = Float64(Float64(-1.0 - Float64(log(x) / n)) / Float64(n * x));
	end
	return tmp
end

Wolfram

code[x_, n_] := If[LessEqual[N[(1.0 / n), $MachinePrecision], -2e-6], N[(N[(1.0 / x), $MachinePrecision] / N[(N[Power[x, N[(-1.0 / n), $MachinePrecision]], $MachinePrecision] * n), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 0.02], N[(N[Log[1 + N[(1.0 / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 1e+117], N[(N[(1.0 + N[(x / n), $MachinePrecision]), $MachinePrecision] - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(-1.0 - N[(N[Log[x], $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision] / N[(n * x), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -1.99999999999999991e-6

   1. Initial program 53.2%

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

   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 57.6

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

   4. Applied rewrites 57.6%

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

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

      4. lift-*.f64 N/A

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

      5. mul-1-neg N/A

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

      6. exp-neg N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. associate-/r* N/A

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

      10. lift-/.f64 N/A

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

      11. frac-times N/A

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

      12. *-lft-identity N/A

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

      13. lower-/.f64 N/A

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

      14. lower-*.f64 N/A

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

   6. Applied rewrites 58.4%

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

   if -1.99999999999999991e-6 < (/.f64 #s(literal 1 binary64) n) < 0.0200000000000000004

   1. Initial program 53.2%

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

   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 59.4

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

   4. Applied rewrites 59.4%

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

      1. lift--.f64 N/A

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

      2. sub-flip N/A

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

      3. mul-1-neg N/A

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

      4. lift-*.f64 N/A

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

      5. +-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lift-log.f64 N/A

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

      9. log-rec N/A

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

      10. lift-/.f64 N/A

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

      11. lift-log.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. +-commutative N/A

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

      14. lift-+.f64 N/A

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

      15. sum-log N/A

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

      16. lower-log.f64 N/A

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

      17. lower-*.f64 55.2

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

      18. lift-+.f64 N/A

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

      19. add-flip N/A

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

      20. unpow1 N/A

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

      21. metadata-eval N/A

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

      22. pow-flip N/A

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

      23. inv-pow N/A

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

      24. lift-/.f64 N/A

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

      25. metadata-eval N/A

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

      26. lower--.f64 N/A

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

      27. lift-/.f64 N/A

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

      28. inv-pow N/A

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

      29. pow-flip N/A

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

      30. metadata-eval N/A

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

      31. unpow1 55.2

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

   6. Applied rewrites 55.2%

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

      1. lift-log.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. log-prod N/A

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

      4. lift--.f64 N/A

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

      5. metadata-eval N/A

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

      6. add-flip N/A

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

      7. log-prod N/A

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

      8. distribute-lft-in N/A

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

      9. lift-/.f64 N/A

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

      10. inv-pow N/A

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

      11. pow-plus N/A

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

      12. metadata-eval N/A

          \[\leadsto \frac{\log \left({x}^{0} + \frac{1}{x} \cdot 1\right)}{n} \]

      13. metadata-eval N/A

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

      14. *-commutative N/A

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

      15. *-lft-identity N/A

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

      16. lower-log1p.f64 58.2

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

   8. Applied rewrites 58.2%

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

   if 0.0200000000000000004 < (/.f64 #s(literal 1 binary64) n) < 1.00000000000000005e117

   1. Initial program 53.2%

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

   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 30.6

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

   4. Applied rewrites 30.6%

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

   if 1.00000000000000005e117 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 53.2%

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

   2. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

   4. Applied rewrites 65.4%

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

   5. Taylor expanded in x around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-log.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 22.5

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

   7. Applied rewrites 22.5%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. mul-1-neg N/A

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

      5. lift-/.f64 N/A

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

      6. lift-log.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. neg-log N/A

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

      9. lift-log.f64 N/A

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

      10. lift-neg.f64 N/A

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

      11. distribute-neg-frac2 N/A

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

      12. lift-neg.f64 N/A

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

      13. frac-2neg N/A

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

      14. mult-flip N/A

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

      15. lift-/.f64 N/A

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

      16. lower-fma.f64 22.5

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

   9. Applied rewrites 22.5%

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

       1. lift-*.f64 N/A

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

       2. mul-1-neg N/A

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

       3. lift-/.f64 N/A

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

       4. distribute-neg-frac N/A

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

       5. lower-/.f64 N/A

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

       6. lift-fma.f64 N/A

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

       7. add-flip N/A

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

       8. metadata-eval N/A

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

       9. sub-negate N/A

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

       10. lower--.f64 N/A

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

       11. lift-/.f64 N/A

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

       12. mult-flip-rev N/A

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

       13. lower-/.f64 22.5

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

   11. Applied rewrites 22.5%

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

Alternative 5: 94.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{-6}:\\ \;\;\;\;\frac{t\_0}{x \cdot n}\\ \mathbf{elif}\;\frac{1}{n} \leq 0.02:\\ \;\;\;\;\frac{\mathsf{log1p}\left(\frac{1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{+117}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{-1 - \frac{\log x}{n}}{n \cdot x}\\ \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -2e-6)
     (/ t_0 (* x n))
     (if (<= (/ 1.0 n) 0.02)
       (/ (log1p (/ 1.0 x)) n)
       (if (<= (/ 1.0 n) 1e+117)
         (- (+ 1.0 (/ x n)) t_0)
         (/ (- -1.0 (/ (log x) n)) (* n x)))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -2e-6) {
		tmp = t_0 / (x * n);
	} else if ((1.0 / n) <= 0.02) {
		tmp = log1p((1.0 / x)) / n;
	} else if ((1.0 / n) <= 1e+117) {
		tmp = (1.0 + (x / n)) - t_0;
	} else {
		tmp = (-1.0 - (log(x) / n)) / (n * x);
	}
	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-6) {
		tmp = t_0 / (x * n);
	} else if ((1.0 / n) <= 0.02) {
		tmp = Math.log1p((1.0 / x)) / n;
	} else if ((1.0 / n) <= 1e+117) {
		tmp = (1.0 + (x / n)) - t_0;
	} else {
		tmp = (-1.0 - (Math.log(x) / n)) / (n * x);
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	tmp = 0
	if (1.0 / n) <= -2e-6:
		tmp = t_0 / (x * n)
	elif (1.0 / n) <= 0.02:
		tmp = math.log1p((1.0 / x)) / n
	elif (1.0 / n) <= 1e+117:
		tmp = (1.0 + (x / n)) - t_0
	else:
		tmp = (-1.0 - (math.log(x) / n)) / (n * x)
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -2e-6)
		tmp = Float64(t_0 / Float64(x * n));
	elseif (Float64(1.0 / n) <= 0.02)
		tmp = Float64(log1p(Float64(1.0 / x)) / n);
	elseif (Float64(1.0 / n) <= 1e+117)
		tmp = Float64(Float64(1.0 + Float64(x / n)) - t_0);
	else
		tmp = Float64(Float64(-1.0 - Float64(log(x) / n)) / Float64(n * x));
	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-6], N[(t$95$0 / N[(x * n), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 0.02], N[(N[Log[1 + N[(1.0 / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 1e+117], N[(N[(1.0 + N[(x / n), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], N[(N[(-1.0 - N[(N[Log[x], $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision] / N[(n * x), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -1.99999999999999991e-6

   1. Initial program 53.2%

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

   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 57.6

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

   4. Applied rewrites 57.6%

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

   6. Applied rewrites 57.6%

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

   if -1.99999999999999991e-6 < (/.f64 #s(literal 1 binary64) n) < 0.0200000000000000004

   1. Initial program 53.2%

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

   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 59.4

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

   4. Applied rewrites 59.4%

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

      1. lift--.f64 N/A

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

      2. sub-flip N/A

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

      3. mul-1-neg N/A

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

      4. lift-*.f64 N/A

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

      5. +-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lift-log.f64 N/A

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

      9. log-rec N/A

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

      10. lift-/.f64 N/A

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

      11. lift-log.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. +-commutative N/A

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

      14. lift-+.f64 N/A

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

      15. sum-log N/A

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

      16. lower-log.f64 N/A

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

      17. lower-*.f64 55.2

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

      18. lift-+.f64 N/A

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

      19. add-flip N/A

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

      20. unpow1 N/A

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

      21. metadata-eval N/A

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

      22. pow-flip N/A

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

      23. inv-pow N/A

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

      24. lift-/.f64 N/A

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

      25. metadata-eval N/A

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

      26. lower--.f64 N/A

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

      27. lift-/.f64 N/A

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

      28. inv-pow N/A

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

      29. pow-flip N/A

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

      30. metadata-eval N/A

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

      31. unpow1 55.2

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

   6. Applied rewrites 55.2%

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

      1. lift-log.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. log-prod N/A

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

      4. lift--.f64 N/A

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

      5. metadata-eval N/A

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

      6. add-flip N/A

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

      7. log-prod N/A

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

      8. distribute-lft-in N/A

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

      9. lift-/.f64 N/A

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

      10. inv-pow N/A

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

      11. pow-plus N/A

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

      12. metadata-eval N/A

          \[\leadsto \frac{\log \left({x}^{0} + \frac{1}{x} \cdot 1\right)}{n} \]

      13. metadata-eval N/A

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

      14. *-commutative N/A

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

      15. *-lft-identity N/A

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

      16. lower-log1p.f64 58.2

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

   8. Applied rewrites 58.2%

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

   if 0.0200000000000000004 < (/.f64 #s(literal 1 binary64) n) < 1.00000000000000005e117

   1. Initial program 53.2%

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

   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 30.6

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

   4. Applied rewrites 30.6%

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

   if 1.00000000000000005e117 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 53.2%

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

   2. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

   4. Applied rewrites 65.4%

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

   5. Taylor expanded in x around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-log.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 22.5

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

   7. Applied rewrites 22.5%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. mul-1-neg N/A

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

      5. lift-/.f64 N/A

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

      6. lift-log.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. neg-log N/A

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

      9. lift-log.f64 N/A

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

      10. lift-neg.f64 N/A

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

      11. distribute-neg-frac2 N/A

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

      12. lift-neg.f64 N/A

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

      13. frac-2neg N/A

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

      14. mult-flip N/A

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

      15. lift-/.f64 N/A

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

      16. lower-fma.f64 22.5

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

   9. Applied rewrites 22.5%

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

       1. lift-*.f64 N/A

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

       2. mul-1-neg N/A

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

       3. lift-/.f64 N/A

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

       4. distribute-neg-frac N/A

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

       5. lower-/.f64 N/A

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

       6. lift-fma.f64 N/A

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

       7. add-flip N/A

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

       8. metadata-eval N/A

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

       9. sub-negate N/A

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

       10. lower--.f64 N/A

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

       11. lift-/.f64 N/A

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

       12. mult-flip-rev N/A

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

       13. lower-/.f64 22.5

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

   11. Applied rewrites 22.5%

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

Alternative 6: 94.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{-6}:\\ \;\;\;\;\frac{t\_0}{x \cdot n}\\ \mathbf{elif}\;\frac{1}{n} \leq 0.02:\\ \;\;\;\;\frac{\mathsf{log1p}\left(\frac{1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{+117}:\\ \;\;\;\;1 - t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{-1 - \frac{\log x}{n}}{n \cdot x}\\ \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -2e-6)
     (/ t_0 (* x n))
     (if (<= (/ 1.0 n) 0.02)
       (/ (log1p (/ 1.0 x)) n)
       (if (<= (/ 1.0 n) 1e+117)
         (- 1.0 t_0)
         (/ (- -1.0 (/ (log x) n)) (* n x)))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -2e-6) {
		tmp = t_0 / (x * n);
	} else if ((1.0 / n) <= 0.02) {
		tmp = log1p((1.0 / x)) / n;
	} else if ((1.0 / n) <= 1e+117) {
		tmp = 1.0 - t_0;
	} else {
		tmp = (-1.0 - (log(x) / n)) / (n * x);
	}
	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-6) {
		tmp = t_0 / (x * n);
	} else if ((1.0 / n) <= 0.02) {
		tmp = Math.log1p((1.0 / x)) / n;
	} else if ((1.0 / n) <= 1e+117) {
		tmp = 1.0 - t_0;
	} else {
		tmp = (-1.0 - (Math.log(x) / n)) / (n * x);
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	tmp = 0
	if (1.0 / n) <= -2e-6:
		tmp = t_0 / (x * n)
	elif (1.0 / n) <= 0.02:
		tmp = math.log1p((1.0 / x)) / n
	elif (1.0 / n) <= 1e+117:
		tmp = 1.0 - t_0
	else:
		tmp = (-1.0 - (math.log(x) / n)) / (n * x)
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -2e-6)
		tmp = Float64(t_0 / Float64(x * n));
	elseif (Float64(1.0 / n) <= 0.02)
		tmp = Float64(log1p(Float64(1.0 / x)) / n);
	elseif (Float64(1.0 / n) <= 1e+117)
		tmp = Float64(1.0 - t_0);
	else
		tmp = Float64(Float64(-1.0 - Float64(log(x) / n)) / Float64(n * x));
	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-6], N[(t$95$0 / N[(x * n), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 0.02], N[(N[Log[1 + N[(1.0 / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 1e+117], N[(1.0 - t$95$0), $MachinePrecision], N[(N[(-1.0 - N[(N[Log[x], $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision] / N[(n * x), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -1.99999999999999991e-6

   1. Initial program 53.2%

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

   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 57.6

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

   4. Applied rewrites 57.6%

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

   6. Applied rewrites 57.6%

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

   if -1.99999999999999991e-6 < (/.f64 #s(literal 1 binary64) n) < 0.0200000000000000004

   1. Initial program 53.2%

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

   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 59.4

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

   4. Applied rewrites 59.4%

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

      1. lift--.f64 N/A

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

      2. sub-flip N/A

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

      3. mul-1-neg N/A

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

      4. lift-*.f64 N/A

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

      5. +-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lift-log.f64 N/A

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

      9. log-rec N/A

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

      10. lift-/.f64 N/A

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

      11. lift-log.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. +-commutative N/A

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

      14. lift-+.f64 N/A

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

      15. sum-log N/A

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

      16. lower-log.f64 N/A

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

      17. lower-*.f64 55.2

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

      18. lift-+.f64 N/A

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

      19. add-flip N/A

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

      20. unpow1 N/A

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

      21. metadata-eval N/A

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

      22. pow-flip N/A

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

      23. inv-pow N/A

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

      24. lift-/.f64 N/A

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

      25. metadata-eval N/A

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

      26. lower--.f64 N/A

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

      27. lift-/.f64 N/A

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

      28. inv-pow N/A

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

      29. pow-flip N/A

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

      30. metadata-eval N/A

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

      31. unpow1 55.2

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

   6. Applied rewrites 55.2%

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

      1. lift-log.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. log-prod N/A

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

      4. lift--.f64 N/A

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

      5. metadata-eval N/A

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

      6. add-flip N/A

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

      7. log-prod N/A

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

      8. distribute-lft-in N/A

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

      9. lift-/.f64 N/A

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

      10. inv-pow N/A

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

      11. pow-plus N/A

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

      12. metadata-eval N/A

          \[\leadsto \frac{\log \left({x}^{0} + \frac{1}{x} \cdot 1\right)}{n} \]

      13. metadata-eval N/A

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

      14. *-commutative N/A

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

      15. *-lft-identity N/A

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

      16. lower-log1p.f64 58.2

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

   8. Applied rewrites 58.2%

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

   if 0.0200000000000000004 < (/.f64 #s(literal 1 binary64) n) < 1.00000000000000005e117

   1. Initial program 53.2%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 38.7%

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

   if 1.00000000000000005e117 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 53.2%

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

   2. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

   4. Applied rewrites 65.4%

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

   5. Taylor expanded in x around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-log.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 22.5

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

   7. Applied rewrites 22.5%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. mul-1-neg N/A

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

      5. lift-/.f64 N/A

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

      6. lift-log.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. neg-log N/A

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

      9. lift-log.f64 N/A

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

      10. lift-neg.f64 N/A

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

      11. distribute-neg-frac2 N/A

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

      12. lift-neg.f64 N/A

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

      13. frac-2neg N/A

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

      14. mult-flip N/A

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

      15. lift-/.f64 N/A

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

      16. lower-fma.f64 22.5

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

   9. Applied rewrites 22.5%

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

       1. lift-*.f64 N/A

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

       2. mul-1-neg N/A

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

       3. lift-/.f64 N/A

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

       4. distribute-neg-frac N/A

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

       5. lower-/.f64 N/A

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

       6. lift-fma.f64 N/A

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

       7. add-flip N/A

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

       8. metadata-eval N/A

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

       9. sub-negate N/A

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

       10. lower--.f64 N/A

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

       11. lift-/.f64 N/A

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

       12. mult-flip-rev N/A

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

       13. lower-/.f64 22.5

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

   11. Applied rewrites 22.5%

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

Alternative 7: 94.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{-6}:\\ \;\;\;\;\frac{{x}^{\left(-1 - \frac{-1}{n}\right)}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 0.02:\\ \;\;\;\;\frac{\mathsf{log1p}\left(\frac{1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{+117}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1 - \frac{\log x}{n}}{n \cdot x}\\ \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= (/ 1.0 n) -2e-6)
   (/ (pow x (- -1.0 (/ -1.0 n))) n)
   (if (<= (/ 1.0 n) 0.02)
     (/ (log1p (/ 1.0 x)) n)
     (if (<= (/ 1.0 n) 1e+117)
       (- 1.0 (pow x (/ 1.0 n)))
       (/ (- -1.0 (/ (log x) n)) (* n x))))))

C

double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -2e-6) {
		tmp = pow(x, (-1.0 - (-1.0 / n))) / n;
	} else if ((1.0 / n) <= 0.02) {
		tmp = log1p((1.0 / x)) / n;
	} else if ((1.0 / n) <= 1e+117) {
		tmp = 1.0 - pow(x, (1.0 / n));
	} else {
		tmp = (-1.0 - (log(x) / n)) / (n * x);
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -2e-6) {
		tmp = Math.pow(x, (-1.0 - (-1.0 / n))) / n;
	} else if ((1.0 / n) <= 0.02) {
		tmp = Math.log1p((1.0 / x)) / n;
	} else if ((1.0 / n) <= 1e+117) {
		tmp = 1.0 - Math.pow(x, (1.0 / n));
	} else {
		tmp = (-1.0 - (Math.log(x) / n)) / (n * x);
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if (1.0 / n) <= -2e-6:
		tmp = math.pow(x, (-1.0 - (-1.0 / n))) / n
	elif (1.0 / n) <= 0.02:
		tmp = math.log1p((1.0 / x)) / n
	elif (1.0 / n) <= 1e+117:
		tmp = 1.0 - math.pow(x, (1.0 / n))
	else:
		tmp = (-1.0 - (math.log(x) / n)) / (n * x)
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -2e-6)
		tmp = Float64((x ^ Float64(-1.0 - Float64(-1.0 / n))) / n);
	elseif (Float64(1.0 / n) <= 0.02)
		tmp = Float64(log1p(Float64(1.0 / x)) / n);
	elseif (Float64(1.0 / n) <= 1e+117)
		tmp = Float64(1.0 - (x ^ Float64(1.0 / n)));
	else
		tmp = Float64(Float64(-1.0 - Float64(log(x) / n)) / Float64(n * x));
	end
	return tmp
end

Wolfram

code[x_, n_] := If[LessEqual[N[(1.0 / n), $MachinePrecision], -2e-6], N[(N[Power[x, N[(-1.0 - N[(-1.0 / n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 0.02], N[(N[Log[1 + N[(1.0 / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 1e+117], N[(1.0 - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(-1.0 - N[(N[Log[x], $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision] / N[(n * x), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -1.99999999999999991e-6

   1. Initial program 53.2%

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

   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 57.6

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

   4. Applied rewrites 57.6%

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

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

      4. lift-*.f64 N/A

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

      5. mul-1-neg N/A

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

      6. exp-neg N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. associate-/r* N/A

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

      10. lift-/.f64 N/A

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

      11. frac-times N/A

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

      12. *-lft-identity N/A

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

      13. lower-/.f64 N/A

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

      14. lower-*.f64 N/A

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

   6. Applied rewrites 58.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. inv-pow N/A

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

      7. lift-pow.f64 N/A

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

      8. pow-div N/A

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

      9. lower-pow.f64 N/A

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

      10. lower--.f64 58.3

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

   8. Applied rewrites 58.3%

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

   if -1.99999999999999991e-6 < (/.f64 #s(literal 1 binary64) n) < 0.0200000000000000004

   1. Initial program 53.2%

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

   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 59.4

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

   4. Applied rewrites 59.4%

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

      1. lift--.f64 N/A

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

      2. sub-flip N/A

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

      3. mul-1-neg N/A

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

      4. lift-*.f64 N/A

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

      5. +-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lift-log.f64 N/A

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

      9. log-rec N/A

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

      10. lift-/.f64 N/A

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

      11. lift-log.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. +-commutative N/A

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

      14. lift-+.f64 N/A

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

      15. sum-log N/A

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

      16. lower-log.f64 N/A

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

      17. lower-*.f64 55.2

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

      18. lift-+.f64 N/A

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

      19. add-flip N/A

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

      20. unpow1 N/A

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

      21. metadata-eval N/A

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

      22. pow-flip N/A

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

      23. inv-pow N/A

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

      24. lift-/.f64 N/A

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

      25. metadata-eval N/A

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

      26. lower--.f64 N/A

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

      27. lift-/.f64 N/A

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

      28. inv-pow N/A

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

      29. pow-flip N/A

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

      30. metadata-eval N/A

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

      31. unpow1 55.2

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

   6. Applied rewrites 55.2%

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

      1. lift-log.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. log-prod N/A

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

      4. lift--.f64 N/A

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

      5. metadata-eval N/A

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

      6. add-flip N/A

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

      7. log-prod N/A

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

      8. distribute-lft-in N/A

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

      9. lift-/.f64 N/A

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

      10. inv-pow N/A

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

      11. pow-plus N/A

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

      12. metadata-eval N/A

          \[\leadsto \frac{\log \left({x}^{0} + \frac{1}{x} \cdot 1\right)}{n} \]

      13. metadata-eval N/A

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

      14. *-commutative N/A

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

      15. *-lft-identity N/A

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

      16. lower-log1p.f64 58.2

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

   8. Applied rewrites 58.2%

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

   if 0.0200000000000000004 < (/.f64 #s(literal 1 binary64) n) < 1.00000000000000005e117

   1. Initial program 53.2%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 38.7%

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

   if 1.00000000000000005e117 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 53.2%

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

   2. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

   4. Applied rewrites 65.4%

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

   5. Taylor expanded in x around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-log.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 22.5

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

   7. Applied rewrites 22.5%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. mul-1-neg N/A

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

      5. lift-/.f64 N/A

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

      6. lift-log.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. neg-log N/A

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

      9. lift-log.f64 N/A

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

      10. lift-neg.f64 N/A

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

      11. distribute-neg-frac2 N/A

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

      12. lift-neg.f64 N/A

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

      13. frac-2neg N/A

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

      14. mult-flip N/A

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

      15. lift-/.f64 N/A

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

      16. lower-fma.f64 22.5

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

   9. Applied rewrites 22.5%

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

       1. lift-*.f64 N/A

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

       2. mul-1-neg N/A

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

       3. lift-/.f64 N/A

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

       4. distribute-neg-frac N/A

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

       5. lower-/.f64 N/A

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

       6. lift-fma.f64 N/A

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

       7. add-flip N/A

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

       8. metadata-eval N/A

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

       9. sub-negate N/A

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

       10. lower--.f64 N/A

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

       11. lift-/.f64 N/A

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

       12. mult-flip-rev N/A

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

       13. lower-/.f64 22.5

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

   11. Applied rewrites 22.5%

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

Alternative 8: 80.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} t_0 := 1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -1000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\frac{1}{n} \leq 0.02:\\ \;\;\;\;\frac{\mathsf{log1p}\left(\frac{1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{+117}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{-1 - \frac{\log x}{n}}{n \cdot x}\\ \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (- 1.0 (pow x (/ 1.0 n)))))
   (if (<= (/ 1.0 n) -1000.0)
     t_0
     (if (<= (/ 1.0 n) 0.02)
       (/ (log1p (/ 1.0 x)) n)
       (if (<= (/ 1.0 n) 1e+117) t_0 (/ (- -1.0 (/ (log x) n)) (* n x)))))))

C

double code(double x, double n) {
	double t_0 = 1.0 - pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -1000.0) {
		tmp = t_0;
	} else if ((1.0 / n) <= 0.02) {
		tmp = log1p((1.0 / x)) / n;
	} else if ((1.0 / n) <= 1e+117) {
		tmp = t_0;
	} else {
		tmp = (-1.0 - (log(x) / n)) / (n * x);
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double t_0 = 1.0 - Math.pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -1000.0) {
		tmp = t_0;
	} else if ((1.0 / n) <= 0.02) {
		tmp = Math.log1p((1.0 / x)) / n;
	} else if ((1.0 / n) <= 1e+117) {
		tmp = t_0;
	} else {
		tmp = (-1.0 - (Math.log(x) / n)) / (n * x);
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = 1.0 - math.pow(x, (1.0 / n))
	tmp = 0
	if (1.0 / n) <= -1000.0:
		tmp = t_0
	elif (1.0 / n) <= 0.02:
		tmp = math.log1p((1.0 / x)) / n
	elif (1.0 / n) <= 1e+117:
		tmp = t_0
	else:
		tmp = (-1.0 - (math.log(x) / n)) / (n * x)
	return tmp

Julia

function code(x, n)
	t_0 = Float64(1.0 - (x ^ Float64(1.0 / n)))
	tmp = 0.0
	if (Float64(1.0 / n) <= -1000.0)
		tmp = t_0;
	elseif (Float64(1.0 / n) <= 0.02)
		tmp = Float64(log1p(Float64(1.0 / x)) / n);
	elseif (Float64(1.0 / n) <= 1e+117)
		tmp = t_0;
	else
		tmp = Float64(Float64(-1.0 - Float64(log(x) / n)) / Float64(n * x));
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(1.0 - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -1000.0], t$95$0, If[LessEqual[N[(1.0 / n), $MachinePrecision], 0.02], N[(N[Log[1 + N[(1.0 / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 1e+117], t$95$0, N[(N[(-1.0 - N[(N[Log[x], $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision] / N[(n * x), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -1e3 or 0.0200000000000000004 < (/.f64 #s(literal 1 binary64) n) < 1.00000000000000005e117

   1. Initial program 53.2%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 38.7%

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

   if -1e3 < (/.f64 #s(literal 1 binary64) n) < 0.0200000000000000004

   1. Initial program 53.2%

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

   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 59.4

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

   4. Applied rewrites 59.4%

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

      1. lift--.f64 N/A

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

      2. sub-flip N/A

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

      3. mul-1-neg N/A

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

      4. lift-*.f64 N/A

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

      5. +-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lift-log.f64 N/A

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

      9. log-rec N/A

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

      10. lift-/.f64 N/A

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

      11. lift-log.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. +-commutative N/A

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

      14. lift-+.f64 N/A

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

      15. sum-log N/A

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

      16. lower-log.f64 N/A

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

      17. lower-*.f64 55.2

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

      18. lift-+.f64 N/A

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

      19. add-flip N/A

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

      20. unpow1 N/A

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

      21. metadata-eval N/A

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

      22. pow-flip N/A

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

      23. inv-pow N/A

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

      24. lift-/.f64 N/A

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

      25. metadata-eval N/A

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

      26. lower--.f64 N/A

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

      27. lift-/.f64 N/A

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

      28. inv-pow N/A

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

      29. pow-flip N/A

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

      30. metadata-eval N/A

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

      31. unpow1 55.2

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

   6. Applied rewrites 55.2%

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

      1. lift-log.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. log-prod N/A

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

      4. lift--.f64 N/A

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

      5. metadata-eval N/A

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

      6. add-flip N/A

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

      7. log-prod N/A

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

      8. distribute-lft-in N/A

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

      9. lift-/.f64 N/A

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

      10. inv-pow N/A

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

      11. pow-plus N/A

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

      12. metadata-eval N/A

          \[\leadsto \frac{\log \left({x}^{0} + \frac{1}{x} \cdot 1\right)}{n} \]

      13. metadata-eval N/A

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

      14. *-commutative N/A

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

      15. *-lft-identity N/A

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

      16. lower-log1p.f64 58.2

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

   8. Applied rewrites 58.2%

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

   if 1.00000000000000005e117 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 53.2%

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

   2. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

   4. Applied rewrites 65.4%

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

   5. Taylor expanded in x around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-log.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 22.5

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

   7. Applied rewrites 22.5%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. mul-1-neg N/A

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

      5. lift-/.f64 N/A

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

      6. lift-log.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. neg-log N/A

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

      9. lift-log.f64 N/A

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

      10. lift-neg.f64 N/A

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

      11. distribute-neg-frac2 N/A

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

      12. lift-neg.f64 N/A

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

      13. frac-2neg N/A

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

      14. mult-flip N/A

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

      15. lift-/.f64 N/A

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

      16. lower-fma.f64 22.5

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

   9. Applied rewrites 22.5%

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

       1. lift-*.f64 N/A

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

       2. mul-1-neg N/A

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

       3. lift-/.f64 N/A

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

       4. distribute-neg-frac N/A

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

       5. lower-/.f64 N/A

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

       6. lift-fma.f64 N/A

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

       7. add-flip N/A

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

       8. metadata-eval N/A

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

       9. sub-negate N/A

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

       10. lower--.f64 N/A

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

       11. lift-/.f64 N/A

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

       12. mult-flip-rev N/A

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

       13. lower-/.f64 22.5

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

   11. Applied rewrites 22.5%

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

Alternative 9: 76.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -100:\\ \;\;\;\;\frac{\log \left(x - -1\right) - \log x}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{+117}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(\frac{1}{x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1 - \frac{\log x}{n}}{n \cdot x}\\ \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= (/ 1.0 n) -100.0)
   (/ (- (log (- x -1.0)) (log x)) n)
   (if (<= (/ 1.0 n) 1e+117)
     (/ (log1p (/ 1.0 x)) n)
     (/ (- -1.0 (/ (log x) n)) (* n x)))))

C

double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -100.0) {
		tmp = (log((x - -1.0)) - log(x)) / n;
	} else if ((1.0 / n) <= 1e+117) {
		tmp = log1p((1.0 / x)) / n;
	} else {
		tmp = (-1.0 - (log(x) / n)) / (n * x);
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -100.0) {
		tmp = (Math.log((x - -1.0)) - Math.log(x)) / n;
	} else if ((1.0 / n) <= 1e+117) {
		tmp = Math.log1p((1.0 / x)) / n;
	} else {
		tmp = (-1.0 - (Math.log(x) / n)) / (n * x);
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if (1.0 / n) <= -100.0:
		tmp = (math.log((x - -1.0)) - math.log(x)) / n
	elif (1.0 / n) <= 1e+117:
		tmp = math.log1p((1.0 / x)) / n
	else:
		tmp = (-1.0 - (math.log(x) / n)) / (n * x)
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -100.0)
		tmp = Float64(Float64(log(Float64(x - -1.0)) - log(x)) / n);
	elseif (Float64(1.0 / n) <= 1e+117)
		tmp = Float64(log1p(Float64(1.0 / x)) / n);
	else
		tmp = Float64(Float64(-1.0 - Float64(log(x) / n)) / Float64(n * x));
	end
	return tmp
end

Wolfram

code[x_, n_] := If[LessEqual[N[(1.0 / n), $MachinePrecision], -100.0], N[(N[(N[Log[N[(x - -1.0), $MachinePrecision]], $MachinePrecision] - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 1e+117], N[(N[Log[1 + N[(1.0 / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], N[(N[(-1.0 - N[(N[Log[x], $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision] / N[(n * x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -100

   1. Initial program 53.2%

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

   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 59.4

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

   4. Applied rewrites 59.4%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. add-flip N/A

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

      4. unpow1 N/A

         \[\leadsto \frac{\log \left({x}^{1} - \left(\mathsf{neg}\left(1\right)\right)\right) - \log x}{n} \]

      5. metadata-eval N/A

         \[\leadsto \frac{\log \left({x}^{\left(\mathsf{neg}\left(-1\right)\right)} - \left(\mathsf{neg}\left(1\right)\right)\right) - \log x}{n} \]

      6. pow-flip N/A

         \[\leadsto \frac{\log \left(\frac{1}{{x}^{-1}} - \left(\mathsf{neg}\left(1\right)\right)\right) - \log x}{n} \]

      7. inv-pow N/A

         \[\leadsto \frac{\log \left(\frac{1}{\frac{1}{x}} - \left(\mathsf{neg}\left(1\right)\right)\right) - \log x}{n} \]

      8. lift-/.f64 N/A

         \[\leadsto \frac{\log \left(\frac{1}{\frac{1}{x}} - \left(\mathsf{neg}\left(1\right)\right)\right) - \log x}{n} \]

      9. metadata-eval N/A

         \[\leadsto \frac{\log \left(\frac{1}{\frac{1}{x}} - -1\right) - \log x}{n} \]

      10. lower--.f64 N/A

          \[\leadsto \frac{\log \left(\frac{1}{\frac{1}{x}} - -1\right) - \log x}{n} \]

      11. lift-/.f64 N/A

          \[\leadsto \frac{\log \left(\frac{1}{\frac{1}{x}} - -1\right) - \log x}{n} \]

      12. inv-pow N/A

          \[\leadsto \frac{\log \left(\frac{1}{{x}^{-1}} - -1\right) - \log x}{n} \]

      13. pow-flip N/A

          \[\leadsto \frac{\log \left({x}^{\left(\mathsf{neg}\left(-1\right)\right)} - -1\right) - \log x}{n} \]

      14. metadata-eval N/A

          \[\leadsto \frac{\log \left({x}^{1} - -1\right) - \log x}{n} \]

      15. unpow1 59.4

          \[\leadsto \frac{\log \left(x - -1\right) - \log x}{n} \]

   6. Applied rewrites 59.4%

      \[\leadsto \frac{\log \left(x - -1\right) - \log x}{n} \]

   if -100 < (/.f64 #s(literal 1 binary64) n) < 1.00000000000000005e117

   1. Initial program 53.2%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]

   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 59.4

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{n} \]

   4. Applied rewrites 59.4%

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
   5. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{n} \]

      2. sub-flip N/A

         \[\leadsto \frac{\log \left(1 + x\right) + \left(\mathsf{neg}\left(\log x\right)\right)}{n} \]

      3. mul-1-neg N/A

         \[\leadsto \frac{\log \left(1 + x\right) + -1 \cdot \log x}{n} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\log \left(1 + x\right) + -1 \cdot \log x}{n} \]

      5. +-commutative N/A

         \[\leadsto \frac{-1 \cdot \log x + \log \left(1 + x\right)}{n} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{-1 \cdot \log x + \log \left(1 + x\right)}{n} \]

      7. mul-1-neg N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(\log x\right)\right) + \log \left(1 + x\right)}{n} \]

      8. lift-log.f64 N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(\log x\right)\right) + \log \left(1 + x\right)}{n} \]

      9. log-rec N/A

         \[\leadsto \frac{\log \left(\frac{1}{x}\right) + \log \left(1 + x\right)}{n} \]

      10. lift-/.f64 N/A

          \[\leadsto \frac{\log \left(\frac{1}{x}\right) + \log \left(1 + x\right)}{n} \]

      11. lift-log.f64 N/A

          \[\leadsto \frac{\log \left(\frac{1}{x}\right) + \log \left(1 + x\right)}{n} \]

      12. lift-+.f64 N/A

          \[\leadsto \frac{\log \left(\frac{1}{x}\right) + \log \left(1 + x\right)}{n} \]

      13. +-commutative N/A

          \[\leadsto \frac{\log \left(\frac{1}{x}\right) + \log \left(x + 1\right)}{n} \]

      14. lift-+.f64 N/A

          \[\leadsto \frac{\log \left(\frac{1}{x}\right) + \log \left(x + 1\right)}{n} \]

      15. sum-log N/A

          \[\leadsto \frac{\log \left(\frac{1}{x} \cdot \left(x + 1\right)\right)}{n} \]

      16. lower-log.f64 N/A

          \[\leadsto \frac{\log \left(\frac{1}{x} \cdot \left(x + 1\right)\right)}{n} \]

      17. lower-*.f64 55.2

          \[\leadsto \frac{\log \left(\frac{1}{x} \cdot \left(x + 1\right)\right)}{n} \]

      18. lift-+.f64 N/A

          \[\leadsto \frac{\log \left(\frac{1}{x} \cdot \left(x + 1\right)\right)}{n} \]

      19. add-flip N/A

          \[\leadsto \frac{\log \left(\frac{1}{x} \cdot \left(x - \left(\mathsf{neg}\left(1\right)\right)\right)\right)}{n} \]

      20. unpow1 N/A

          \[\leadsto \frac{\log \left(\frac{1}{x} \cdot \left({x}^{1} - \left(\mathsf{neg}\left(1\right)\right)\right)\right)}{n} \]

      21. metadata-eval N/A

          \[\leadsto \frac{\log \left(\frac{1}{x} \cdot \left({x}^{\left(\mathsf{neg}\left(-1\right)\right)} - \left(\mathsf{neg}\left(1\right)\right)\right)\right)}{n} \]

      22. pow-flip N/A

          \[\leadsto \frac{\log \left(\frac{1}{x} \cdot \left(\frac{1}{{x}^{-1}} - \left(\mathsf{neg}\left(1\right)\right)\right)\right)}{n} \]

      23. inv-pow N/A

          \[\leadsto \frac{\log \left(\frac{1}{x} \cdot \left(\frac{1}{\frac{1}{x}} - \left(\mathsf{neg}\left(1\right)\right)\right)\right)}{n} \]

      24. lift-/.f64 N/A

          \[\leadsto \frac{\log \left(\frac{1}{x} \cdot \left(\frac{1}{\frac{1}{x}} - \left(\mathsf{neg}\left(1\right)\right)\right)\right)}{n} \]

      25. metadata-eval N/A

          \[\leadsto \frac{\log \left(\frac{1}{x} \cdot \left(\frac{1}{\frac{1}{x}} - -1\right)\right)}{n} \]

      26. lower--.f64 N/A

          \[\leadsto \frac{\log \left(\frac{1}{x} \cdot \left(\frac{1}{\frac{1}{x}} - -1\right)\right)}{n} \]

      27. lift-/.f64 N/A

          \[\leadsto \frac{\log \left(\frac{1}{x} \cdot \left(\frac{1}{\frac{1}{x}} - -1\right)\right)}{n} \]

      28. inv-pow N/A

          \[\leadsto \frac{\log \left(\frac{1}{x} \cdot \left(\frac{1}{{x}^{-1}} - -1\right)\right)}{n} \]

      29. pow-flip N/A

          \[\leadsto \frac{\log \left(\frac{1}{x} \cdot \left({x}^{\left(\mathsf{neg}\left(-1\right)\right)} - -1\right)\right)}{n} \]

      30. metadata-eval N/A

          \[\leadsto \frac{\log \left(\frac{1}{x} \cdot \left({x}^{1} - -1\right)\right)}{n} \]

      31. unpow1 55.2

          \[\leadsto \frac{\log \left(\frac{1}{x} \cdot \left(x - -1\right)\right)}{n} \]

   6. Applied rewrites 55.2%

      \[\leadsto \frac{\log \left(\frac{1}{x} \cdot \left(x - -1\right)\right)}{n} \]
   7. Step-by-step derivation

      1. lift-log.f64 N/A

         \[\leadsto \frac{\log \left(\frac{1}{x} \cdot \left(x - -1\right)\right)}{n} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\log \left(\frac{1}{x} \cdot \left(x - -1\right)\right)}{n} \]

      3. log-prod N/A

         \[\leadsto \frac{\log \left(\frac{1}{x}\right) + \log \left(x - -1\right)}{n} \]

      4. lift--.f64 N/A

         \[\leadsto \frac{\log \left(\frac{1}{x}\right) + \log \left(x - -1\right)}{n} \]

      5. metadata-eval N/A

         \[\leadsto \frac{\log \left(\frac{1}{x}\right) + \log \left(x - \left(\mathsf{neg}\left(1\right)\right)\right)}{n} \]

      6. add-flip N/A

         \[\leadsto \frac{\log \left(\frac{1}{x}\right) + \log \left(x + 1\right)}{n} \]

      7. log-prod N/A

         \[\leadsto \frac{\log \left(\frac{1}{x} \cdot \left(x + 1\right)\right)}{n} \]

      8. distribute-lft-in N/A

         \[\leadsto \frac{\log \left(\frac{1}{x} \cdot x + \frac{1}{x} \cdot 1\right)}{n} \]

      9. lift-/.f64 N/A

         \[\leadsto \frac{\log \left(\frac{1}{x} \cdot x + \frac{1}{x} \cdot 1\right)}{n} \]

      10. inv-pow N/A

          \[\leadsto \frac{\log \left({x}^{-1} \cdot x + \frac{1}{x} \cdot 1\right)}{n} \]

      11. pow-plus N/A

          \[\leadsto \frac{\log \left({x}^{\left(-1 + 1\right)} + \frac{1}{x} \cdot 1\right)}{n} \]

      12. metadata-eval N/A

          \[\leadsto \frac{\log \left({x}^{0} + \frac{1}{x} \cdot 1\right)}{n} \]

      13. metadata-eval N/A

          \[\leadsto \frac{\log \left(1 + \frac{1}{x} \cdot 1\right)}{n} \]

      14. *-commutative N/A

          \[\leadsto \frac{\log \left(1 + 1 \cdot \frac{1}{x}\right)}{n} \]

      15. *-lft-identity N/A

          \[\leadsto \frac{\log \left(1 + \frac{1}{x}\right)}{n} \]

      16. lower-log1p.f64 58.2

          \[\leadsto \frac{\mathsf{log1p}\left(\frac{1}{x}\right)}{n} \]

   8. Applied rewrites 58.2%

      \[\leadsto \frac{\mathsf{log1p}\left(\frac{1}{x}\right)}{n} \]

   if 1.00000000000000005e117 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 53.2%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]

   2. Taylor expanded in n around inf

      \[\leadsto \color{blue}{\frac{\left(\log \left(1 + x\right) + \frac{1}{2} \cdot \frac{{\log \left(1 + x\right)}^{2}}{n}\right) - \left(\log x + \frac{1}{2} \cdot \frac{{\log x}^{2}}{n}\right)}{n}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\left(\log \left(1 + x\right) + \frac{1}{2} \cdot \frac{{\log \left(1 + x\right)}^{2}}{n}\right) - \left(\log x + \frac{1}{2} \cdot \frac{{\log x}^{2}}{n}\right)}{\color{blue}{n}} \]

   4. Applied rewrites 65.4%

      \[\leadsto \color{blue}{\frac{\left(\log \left(1 + x\right) + 0.5 \cdot \frac{{\log \left(1 + x\right)}^{2}}{n}\right) - \left(\log x + 0.5 \cdot \frac{{\log x}^{2}}{n}\right)}{n}} \]

   5. Taylor expanded in x around -inf

      \[\leadsto -1 \cdot \color{blue}{\frac{1 + -1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}{n \cdot x}} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto -1 \cdot \frac{1 + -1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}{\color{blue}{n \cdot x}} \]

      2. lower-/.f64 N/A

         \[\leadsto -1 \cdot \frac{1 + -1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}{n \cdot \color{blue}{x}} \]

      3. lower-+.f64 N/A

         \[\leadsto -1 \cdot \frac{1 + -1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}{n \cdot x} \]

      4. lower-*.f64 N/A

         \[\leadsto -1 \cdot \frac{1 + -1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}{n \cdot x} \]

      5. lower-/.f64 N/A

         \[\leadsto -1 \cdot \frac{1 + -1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}{n \cdot x} \]

      6. lower-log.f64 N/A

         \[\leadsto -1 \cdot \frac{1 + -1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}{n \cdot x} \]

      7. lower-/.f64 N/A

         \[\leadsto -1 \cdot \frac{1 + -1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}{n \cdot x} \]

      8. lower-*.f64 22.5

         \[\leadsto -1 \cdot \frac{1 + -1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}{n \cdot x} \]

   7. Applied rewrites 22.5%

      \[\leadsto -1 \cdot \color{blue}{\frac{1 + -1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}{n \cdot x}} \]
   8. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto -1 \cdot \frac{1 + -1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}{n \cdot x} \]

      2. +-commutative N/A

         \[\leadsto -1 \cdot \frac{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n} + 1}{n \cdot x} \]

      3. lift-*.f64 N/A

         \[\leadsto -1 \cdot \frac{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n} + 1}{n \cdot x} \]

      4. mul-1-neg N/A

         \[\leadsto -1 \cdot \frac{\left(\mathsf{neg}\left(\frac{\log \left(\frac{1}{x}\right)}{n}\right)\right) + 1}{n \cdot x} \]

      5. lift-/.f64 N/A

         \[\leadsto -1 \cdot \frac{\left(\mathsf{neg}\left(\frac{\log \left(\frac{1}{x}\right)}{n}\right)\right) + 1}{n \cdot x} \]

      6. lift-log.f64 N/A

         \[\leadsto -1 \cdot \frac{\left(\mathsf{neg}\left(\frac{\log \left(\frac{1}{x}\right)}{n}\right)\right) + 1}{n \cdot x} \]

      7. lift-/.f64 N/A

         \[\leadsto -1 \cdot \frac{\left(\mathsf{neg}\left(\frac{\log \left(\frac{1}{x}\right)}{n}\right)\right) + 1}{n \cdot x} \]

      8. neg-log N/A

         \[\leadsto -1 \cdot \frac{\left(\mathsf{neg}\left(\frac{\mathsf{neg}\left(\log x\right)}{n}\right)\right) + 1}{n \cdot x} \]

      9. lift-log.f64 N/A

         \[\leadsto -1 \cdot \frac{\left(\mathsf{neg}\left(\frac{\mathsf{neg}\left(\log x\right)}{n}\right)\right) + 1}{n \cdot x} \]

      10. lift-neg.f64 N/A

          \[\leadsto -1 \cdot \frac{\left(\mathsf{neg}\left(\frac{-\log x}{n}\right)\right) + 1}{n \cdot x} \]

      11. distribute-neg-frac2 N/A

          \[\leadsto -1 \cdot \frac{\frac{-\log x}{\mathsf{neg}\left(n\right)} + 1}{n \cdot x} \]

      12. lift-neg.f64 N/A

          \[\leadsto -1 \cdot \frac{\frac{\mathsf{neg}\left(\log x\right)}{\mathsf{neg}\left(n\right)} + 1}{n \cdot x} \]

      13. frac-2neg N/A

          \[\leadsto -1 \cdot \frac{\frac{\log x}{n} + 1}{n \cdot x} \]

      14. mult-flip N/A

          \[\leadsto -1 \cdot \frac{\log x \cdot \frac{1}{n} + 1}{n \cdot x} \]

      15. lift-/.f64 N/A

          \[\leadsto -1 \cdot \frac{\log x \cdot \frac{1}{n} + 1}{n \cdot x} \]

      16. lower-fma.f64 22.5

          \[\leadsto -1 \cdot \frac{\mathsf{fma}\left(\log x, \frac{1}{n}, 1\right)}{n \cdot x} \]

   9. Applied rewrites 22.5%

      \[\leadsto -1 \cdot \frac{\mathsf{fma}\left(\log x, \frac{1}{n}, 1\right)}{n \cdot x} \]
   10. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto -1 \cdot \frac{\mathsf{fma}\left(\log x, \frac{1}{n}, 1\right)}{\color{blue}{n \cdot x}} \]

       2. mul-1-neg N/A

          \[\leadsto \mathsf{neg}\left(\frac{\mathsf{fma}\left(\log x, \frac{1}{n}, 1\right)}{n \cdot x}\right) \]

       3. lift-/.f64 N/A

          \[\leadsto \mathsf{neg}\left(\frac{\mathsf{fma}\left(\log x, \frac{1}{n}, 1\right)}{n \cdot x}\right) \]

       4. distribute-neg-frac N/A

          \[\leadsto \frac{\mathsf{neg}\left(\mathsf{fma}\left(\log x, \frac{1}{n}, 1\right)\right)}{n \cdot \color{blue}{x}} \]

       5. lower-/.f64 N/A

          \[\leadsto \frac{\mathsf{neg}\left(\mathsf{fma}\left(\log x, \frac{1}{n}, 1\right)\right)}{n \cdot \color{blue}{x}} \]

       6. lift-fma.f64 N/A

          \[\leadsto \frac{\mathsf{neg}\left(\left(\log x \cdot \frac{1}{n} + 1\right)\right)}{n \cdot x} \]

       7. add-flip N/A

          \[\leadsto \frac{\mathsf{neg}\left(\left(\log x \cdot \frac{1}{n} - \left(\mathsf{neg}\left(1\right)\right)\right)\right)}{n \cdot x} \]

       8. metadata-eval N/A

          \[\leadsto \frac{\mathsf{neg}\left(\left(\log x \cdot \frac{1}{n} - -1\right)\right)}{n \cdot x} \]

       9. sub-negate N/A

          \[\leadsto \frac{-1 - \log x \cdot \frac{1}{n}}{n \cdot x} \]

       10. lower--.f64 N/A

           \[\leadsto \frac{-1 - \log x \cdot \frac{1}{n}}{n \cdot x} \]

       11. lift-/.f64 N/A

           \[\leadsto \frac{-1 - \log x \cdot \frac{1}{n}}{n \cdot x} \]

       12. mult-flip-rev N/A

           \[\leadsto \frac{-1 - \frac{\log x}{n}}{n \cdot x} \]

       13. lower-/.f64 22.5

           \[\leadsto \frac{-1 - \frac{\log x}{n}}{n \cdot x} \]

   11. Applied rewrites 22.5%

       \[\leadsto \frac{-1 - \frac{\log x}{n}}{\color{blue}{n \cdot x}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 10: 76.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -100:\\ \;\;\;\;\frac{\log \left(\frac{x - -1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{+117}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(\frac{1}{x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1 - \frac{\log x}{n}}{n \cdot x}\\ \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= (/ 1.0 n) -100.0)
   (/ (log (/ (- x -1.0) x)) n)
   (if (<= (/ 1.0 n) 1e+117)
     (/ (log1p (/ 1.0 x)) n)
     (/ (- -1.0 (/ (log x) n)) (* n x)))))

C

double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -100.0) {
		tmp = log(((x - -1.0) / x)) / n;
	} else if ((1.0 / n) <= 1e+117) {
		tmp = log1p((1.0 / x)) / n;
	} else {
		tmp = (-1.0 - (log(x) / n)) / (n * x);
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -100.0) {
		tmp = Math.log(((x - -1.0) / x)) / n;
	} else if ((1.0 / n) <= 1e+117) {
		tmp = Math.log1p((1.0 / x)) / n;
	} else {
		tmp = (-1.0 - (Math.log(x) / n)) / (n * x);
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if (1.0 / n) <= -100.0:
		tmp = math.log(((x - -1.0) / x)) / n
	elif (1.0 / n) <= 1e+117:
		tmp = math.log1p((1.0 / x)) / n
	else:
		tmp = (-1.0 - (math.log(x) / n)) / (n * x)
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -100.0)
		tmp = Float64(log(Float64(Float64(x - -1.0) / x)) / n);
	elseif (Float64(1.0 / n) <= 1e+117)
		tmp = Float64(log1p(Float64(1.0 / x)) / n);
	else
		tmp = Float64(Float64(-1.0 - Float64(log(x) / n)) / Float64(n * x));
	end
	return tmp
end

Wolfram

code[x_, n_] := If[LessEqual[N[(1.0 / n), $MachinePrecision], -100.0], N[(N[Log[N[(N[(x - -1.0), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 1e+117], N[(N[Log[1 + N[(1.0 / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], N[(N[(-1.0 - N[(N[Log[x], $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision] / N[(n * x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -100

   1. Initial program 53.2%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]

   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 59.4

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{n} \]

   4. Applied rewrites 59.4%

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
   5. Step-by-step derivation

   6. Applied rewrites 59.5%

      \[\leadsto \color{blue}{\frac{\log \left(\frac{x - -1}{x}\right)}{n}} \]

   if -100 < (/.f64 #s(literal 1 binary64) n) < 1.00000000000000005e117

   1. Initial program 53.2%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]

   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 59.4

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{n} \]

   4. Applied rewrites 59.4%

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
   5. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{n} \]

      2. sub-flip N/A

         \[\leadsto \frac{\log \left(1 + x\right) + \left(\mathsf{neg}\left(\log x\right)\right)}{n} \]

      3. mul-1-neg N/A

         \[\leadsto \frac{\log \left(1 + x\right) + -1 \cdot \log x}{n} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\log \left(1 + x\right) + -1 \cdot \log x}{n} \]

      5. +-commutative N/A

         \[\leadsto \frac{-1 \cdot \log x + \log \left(1 + x\right)}{n} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{-1 \cdot \log x + \log \left(1 + x\right)}{n} \]

      7. mul-1-neg N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(\log x\right)\right) + \log \left(1 + x\right)}{n} \]

      8. lift-log.f64 N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(\log x\right)\right) + \log \left(1 + x\right)}{n} \]

      9. log-rec N/A

         \[\leadsto \frac{\log \left(\frac{1}{x}\right) + \log \left(1 + x\right)}{n} \]

      10. lift-/.f64 N/A

          \[\leadsto \frac{\log \left(\frac{1}{x}\right) + \log \left(1 + x\right)}{n} \]

      11. lift-log.f64 N/A

          \[\leadsto \frac{\log \left(\frac{1}{x}\right) + \log \left(1 + x\right)}{n} \]

      12. lift-+.f64 N/A

          \[\leadsto \frac{\log \left(\frac{1}{x}\right) + \log \left(1 + x\right)}{n} \]

      13. +-commutative N/A

          \[\leadsto \frac{\log \left(\frac{1}{x}\right) + \log \left(x + 1\right)}{n} \]

      14. lift-+.f64 N/A

          \[\leadsto \frac{\log \left(\frac{1}{x}\right) + \log \left(x + 1\right)}{n} \]

      15. sum-log N/A

          \[\leadsto \frac{\log \left(\frac{1}{x} \cdot \left(x + 1\right)\right)}{n} \]

      16. lower-log.f64 N/A

          \[\leadsto \frac{\log \left(\frac{1}{x} \cdot \left(x + 1\right)\right)}{n} \]

      17. lower-*.f64 55.2

          \[\leadsto \frac{\log \left(\frac{1}{x} \cdot \left(x + 1\right)\right)}{n} \]

      18. lift-+.f64 N/A

          \[\leadsto \frac{\log \left(\frac{1}{x} \cdot \left(x + 1\right)\right)}{n} \]

      19. add-flip N/A

          \[\leadsto \frac{\log \left(\frac{1}{x} \cdot \left(x - \left(\mathsf{neg}\left(1\right)\right)\right)\right)}{n} \]

      20. unpow1 N/A

          \[\leadsto \frac{\log \left(\frac{1}{x} \cdot \left({x}^{1} - \left(\mathsf{neg}\left(1\right)\right)\right)\right)}{n} \]

      21. metadata-eval N/A

          \[\leadsto \frac{\log \left(\frac{1}{x} \cdot \left({x}^{\left(\mathsf{neg}\left(-1\right)\right)} - \left(\mathsf{neg}\left(1\right)\right)\right)\right)}{n} \]

      22. pow-flip N/A

          \[\leadsto \frac{\log \left(\frac{1}{x} \cdot \left(\frac{1}{{x}^{-1}} - \left(\mathsf{neg}\left(1\right)\right)\right)\right)}{n} \]

      23. inv-pow N/A

          \[\leadsto \frac{\log \left(\frac{1}{x} \cdot \left(\frac{1}{\frac{1}{x}} - \left(\mathsf{neg}\left(1\right)\right)\right)\right)}{n} \]

      24. lift-/.f64 N/A

          \[\leadsto \frac{\log \left(\frac{1}{x} \cdot \left(\frac{1}{\frac{1}{x}} - \left(\mathsf{neg}\left(1\right)\right)\right)\right)}{n} \]

      25. metadata-eval N/A

          \[\leadsto \frac{\log \left(\frac{1}{x} \cdot \left(\frac{1}{\frac{1}{x}} - -1\right)\right)}{n} \]

      26. lower--.f64 N/A

          \[\leadsto \frac{\log \left(\frac{1}{x} \cdot \left(\frac{1}{\frac{1}{x}} - -1\right)\right)}{n} \]

      27. lift-/.f64 N/A

          \[\leadsto \frac{\log \left(\frac{1}{x} \cdot \left(\frac{1}{\frac{1}{x}} - -1\right)\right)}{n} \]

      28. inv-pow N/A

          \[\leadsto \frac{\log \left(\frac{1}{x} \cdot \left(\frac{1}{{x}^{-1}} - -1\right)\right)}{n} \]

      29. pow-flip N/A

          \[\leadsto \frac{\log \left(\frac{1}{x} \cdot \left({x}^{\left(\mathsf{neg}\left(-1\right)\right)} - -1\right)\right)}{n} \]

      30. metadata-eval N/A

          \[\leadsto \frac{\log \left(\frac{1}{x} \cdot \left({x}^{1} - -1\right)\right)}{n} \]

      31. unpow1 55.2

          \[\leadsto \frac{\log \left(\frac{1}{x} \cdot \left(x - -1\right)\right)}{n} \]

   6. Applied rewrites 55.2%

      \[\leadsto \frac{\log \left(\frac{1}{x} \cdot \left(x - -1\right)\right)}{n} \]
   7. Step-by-step derivation

      1. lift-log.f64 N/A

         \[\leadsto \frac{\log \left(\frac{1}{x} \cdot \left(x - -1\right)\right)}{n} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\log \left(\frac{1}{x} \cdot \left(x - -1\right)\right)}{n} \]

      3. log-prod N/A

         \[\leadsto \frac{\log \left(\frac{1}{x}\right) + \log \left(x - -1\right)}{n} \]

      4. lift--.f64 N/A

         \[\leadsto \frac{\log \left(\frac{1}{x}\right) + \log \left(x - -1\right)}{n} \]

      5. metadata-eval N/A

         \[\leadsto \frac{\log \left(\frac{1}{x}\right) + \log \left(x - \left(\mathsf{neg}\left(1\right)\right)\right)}{n} \]

      6. add-flip N/A

         \[\leadsto \frac{\log \left(\frac{1}{x}\right) + \log \left(x + 1\right)}{n} \]

      7. log-prod N/A

         \[\leadsto \frac{\log \left(\frac{1}{x} \cdot \left(x + 1\right)\right)}{n} \]

      8. distribute-lft-in N/A

         \[\leadsto \frac{\log \left(\frac{1}{x} \cdot x + \frac{1}{x} \cdot 1\right)}{n} \]

      9. lift-/.f64 N/A

         \[\leadsto \frac{\log \left(\frac{1}{x} \cdot x + \frac{1}{x} \cdot 1\right)}{n} \]

      10. inv-pow N/A

          \[\leadsto \frac{\log \left({x}^{-1} \cdot x + \frac{1}{x} \cdot 1\right)}{n} \]

      11. pow-plus N/A

          \[\leadsto \frac{\log \left({x}^{\left(-1 + 1\right)} + \frac{1}{x} \cdot 1\right)}{n} \]

      12. metadata-eval N/A

          \[\leadsto \frac{\log \left({x}^{0} + \frac{1}{x} \cdot 1\right)}{n} \]

      13. metadata-eval N/A

          \[\leadsto \frac{\log \left(1 + \frac{1}{x} \cdot 1\right)}{n} \]

      14. *-commutative N/A

          \[\leadsto \frac{\log \left(1 + 1 \cdot \frac{1}{x}\right)}{n} \]

      15. *-lft-identity N/A

          \[\leadsto \frac{\log \left(1 + \frac{1}{x}\right)}{n} \]

      16. lower-log1p.f64 58.2

          \[\leadsto \frac{\mathsf{log1p}\left(\frac{1}{x}\right)}{n} \]

   8. Applied rewrites 58.2%

      \[\leadsto \frac{\mathsf{log1p}\left(\frac{1}{x}\right)}{n} \]

   if 1.00000000000000005e117 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 53.2%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]

   2. Taylor expanded in n around inf

      \[\leadsto \color{blue}{\frac{\left(\log \left(1 + x\right) + \frac{1}{2} \cdot \frac{{\log \left(1 + x\right)}^{2}}{n}\right) - \left(\log x + \frac{1}{2} \cdot \frac{{\log x}^{2}}{n}\right)}{n}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\left(\log \left(1 + x\right) + \frac{1}{2} \cdot \frac{{\log \left(1 + x\right)}^{2}}{n}\right) - \left(\log x + \frac{1}{2} \cdot \frac{{\log x}^{2}}{n}\right)}{\color{blue}{n}} \]

   4. Applied rewrites 65.4%

      \[\leadsto \color{blue}{\frac{\left(\log \left(1 + x\right) + 0.5 \cdot \frac{{\log \left(1 + x\right)}^{2}}{n}\right) - \left(\log x + 0.5 \cdot \frac{{\log x}^{2}}{n}\right)}{n}} \]

   5. Taylor expanded in x around -inf

      \[\leadsto -1 \cdot \color{blue}{\frac{1 + -1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}{n \cdot x}} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto -1 \cdot \frac{1 + -1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}{\color{blue}{n \cdot x}} \]

      2. lower-/.f64 N/A

         \[\leadsto -1 \cdot \frac{1 + -1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}{n \cdot \color{blue}{x}} \]

      3. lower-+.f64 N/A

         \[\leadsto -1 \cdot \frac{1 + -1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}{n \cdot x} \]

      4. lower-*.f64 N/A

         \[\leadsto -1 \cdot \frac{1 + -1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}{n \cdot x} \]

      5. lower-/.f64 N/A

         \[\leadsto -1 \cdot \frac{1 + -1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}{n \cdot x} \]

      6. lower-log.f64 N/A

         \[\leadsto -1 \cdot \frac{1 + -1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}{n \cdot x} \]

      7. lower-/.f64 N/A

         \[\leadsto -1 \cdot \frac{1 + -1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}{n \cdot x} \]

      8. lower-*.f64 22.5

         \[\leadsto -1 \cdot \frac{1 + -1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}{n \cdot x} \]

   7. Applied rewrites 22.5%

      \[\leadsto -1 \cdot \color{blue}{\frac{1 + -1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}{n \cdot x}} \]
   8. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto -1 \cdot \frac{1 + -1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}{n \cdot x} \]

      2. +-commutative N/A

         \[\leadsto -1 \cdot \frac{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n} + 1}{n \cdot x} \]

      3. lift-*.f64 N/A

         \[\leadsto -1 \cdot \frac{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n} + 1}{n \cdot x} \]

      4. mul-1-neg N/A

         \[\leadsto -1 \cdot \frac{\left(\mathsf{neg}\left(\frac{\log \left(\frac{1}{x}\right)}{n}\right)\right) + 1}{n \cdot x} \]

      5. lift-/.f64 N/A

         \[\leadsto -1 \cdot \frac{\left(\mathsf{neg}\left(\frac{\log \left(\frac{1}{x}\right)}{n}\right)\right) + 1}{n \cdot x} \]

      6. lift-log.f64 N/A

         \[\leadsto -1 \cdot \frac{\left(\mathsf{neg}\left(\frac{\log \left(\frac{1}{x}\right)}{n}\right)\right) + 1}{n \cdot x} \]

      7. lift-/.f64 N/A

         \[\leadsto -1 \cdot \frac{\left(\mathsf{neg}\left(\frac{\log \left(\frac{1}{x}\right)}{n}\right)\right) + 1}{n \cdot x} \]

      8. neg-log N/A

         \[\leadsto -1 \cdot \frac{\left(\mathsf{neg}\left(\frac{\mathsf{neg}\left(\log x\right)}{n}\right)\right) + 1}{n \cdot x} \]

      9. lift-log.f64 N/A

         \[\leadsto -1 \cdot \frac{\left(\mathsf{neg}\left(\frac{\mathsf{neg}\left(\log x\right)}{n}\right)\right) + 1}{n \cdot x} \]

      10. lift-neg.f64 N/A

          \[\leadsto -1 \cdot \frac{\left(\mathsf{neg}\left(\frac{-\log x}{n}\right)\right) + 1}{n \cdot x} \]

      11. distribute-neg-frac2 N/A

          \[\leadsto -1 \cdot \frac{\frac{-\log x}{\mathsf{neg}\left(n\right)} + 1}{n \cdot x} \]

      12. lift-neg.f64 N/A

          \[\leadsto -1 \cdot \frac{\frac{\mathsf{neg}\left(\log x\right)}{\mathsf{neg}\left(n\right)} + 1}{n \cdot x} \]

      13. frac-2neg N/A

          \[\leadsto -1 \cdot \frac{\frac{\log x}{n} + 1}{n \cdot x} \]

      14. mult-flip N/A

          \[\leadsto -1 \cdot \frac{\log x \cdot \frac{1}{n} + 1}{n \cdot x} \]

      15. lift-/.f64 N/A

          \[\leadsto -1 \cdot \frac{\log x \cdot \frac{1}{n} + 1}{n \cdot x} \]

      16. lower-fma.f64 22.5

          \[\leadsto -1 \cdot \frac{\mathsf{fma}\left(\log x, \frac{1}{n}, 1\right)}{n \cdot x} \]

   9. Applied rewrites 22.5%

      \[\leadsto -1 \cdot \frac{\mathsf{fma}\left(\log x, \frac{1}{n}, 1\right)}{n \cdot x} \]
   10. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto -1 \cdot \frac{\mathsf{fma}\left(\log x, \frac{1}{n}, 1\right)}{\color{blue}{n \cdot x}} \]

       2. mul-1-neg N/A

          \[\leadsto \mathsf{neg}\left(\frac{\mathsf{fma}\left(\log x, \frac{1}{n}, 1\right)}{n \cdot x}\right) \]

       3. lift-/.f64 N/A

          \[\leadsto \mathsf{neg}\left(\frac{\mathsf{fma}\left(\log x, \frac{1}{n}, 1\right)}{n \cdot x}\right) \]

       4. distribute-neg-frac N/A

          \[\leadsto \frac{\mathsf{neg}\left(\mathsf{fma}\left(\log x, \frac{1}{n}, 1\right)\right)}{n \cdot \color{blue}{x}} \]

       5. lower-/.f64 N/A

          \[\leadsto \frac{\mathsf{neg}\left(\mathsf{fma}\left(\log x, \frac{1}{n}, 1\right)\right)}{n \cdot \color{blue}{x}} \]

       6. lift-fma.f64 N/A

          \[\leadsto \frac{\mathsf{neg}\left(\left(\log x \cdot \frac{1}{n} + 1\right)\right)}{n \cdot x} \]

       7. add-flip N/A

          \[\leadsto \frac{\mathsf{neg}\left(\left(\log x \cdot \frac{1}{n} - \left(\mathsf{neg}\left(1\right)\right)\right)\right)}{n \cdot x} \]

       8. metadata-eval N/A

          \[\leadsto \frac{\mathsf{neg}\left(\left(\log x \cdot \frac{1}{n} - -1\right)\right)}{n \cdot x} \]

       9. sub-negate N/A

          \[\leadsto \frac{-1 - \log x \cdot \frac{1}{n}}{n \cdot x} \]

       10. lower--.f64 N/A

           \[\leadsto \frac{-1 - \log x \cdot \frac{1}{n}}{n \cdot x} \]

       11. lift-/.f64 N/A

           \[\leadsto \frac{-1 - \log x \cdot \frac{1}{n}}{n \cdot x} \]

       12. mult-flip-rev N/A

           \[\leadsto \frac{-1 - \frac{\log x}{n}}{n \cdot x} \]

       13. lower-/.f64 22.5

           \[\leadsto \frac{-1 - \frac{\log x}{n}}{n \cdot x} \]

   11. Applied rewrites 22.5%

       \[\leadsto \frac{-1 - \frac{\log x}{n}}{\color{blue}{n \cdot x}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 11: 74.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -100:\\ \;\;\;\;\frac{\log \left(\frac{x - -1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{+117}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(\frac{1}{x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{n \cdot x}\\ \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= (/ 1.0 n) -100.0)
   (/ (log (/ (- x -1.0) x)) n)
   (if (<= (/ 1.0 n) 1e+117) (/ (log1p (/ 1.0 x)) n) (/ 1.0 (* n x)))))

C

double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -100.0) {
		tmp = log(((x - -1.0) / x)) / n;
	} else if ((1.0 / n) <= 1e+117) {
		tmp = log1p((1.0 / x)) / n;
	} else {
		tmp = 1.0 / (n * x);
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -100.0) {
		tmp = Math.log(((x - -1.0) / x)) / n;
	} else if ((1.0 / n) <= 1e+117) {
		tmp = Math.log1p((1.0 / x)) / n;
	} else {
		tmp = 1.0 / (n * x);
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if (1.0 / n) <= -100.0:
		tmp = math.log(((x - -1.0) / x)) / n
	elif (1.0 / n) <= 1e+117:
		tmp = math.log1p((1.0 / x)) / n
	else:
		tmp = 1.0 / (n * x)
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -100.0)
		tmp = Float64(log(Float64(Float64(x - -1.0) / x)) / n);
	elseif (Float64(1.0 / n) <= 1e+117)
		tmp = Float64(log1p(Float64(1.0 / x)) / n);
	else
		tmp = Float64(1.0 / Float64(n * x));
	end
	return tmp
end

Wolfram

code[x_, n_] := If[LessEqual[N[(1.0 / n), $MachinePrecision], -100.0], N[(N[Log[N[(N[(x - -1.0), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 1e+117], N[(N[Log[1 + N[(1.0 / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], N[(1.0 / N[(n * x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -100

   1. Initial program 53.2%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]

   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 59.4

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{n} \]

   4. Applied rewrites 59.4%

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
   5. Step-by-step derivation

   6. Applied rewrites 59.5%

      \[\leadsto \color{blue}{\frac{\log \left(\frac{x - -1}{x}\right)}{n}} \]

   if -100 < (/.f64 #s(literal 1 binary64) n) < 1.00000000000000005e117

   1. Initial program 53.2%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]

   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 59.4

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{n} \]

   4. Applied rewrites 59.4%

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
   5. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{n} \]

      2. sub-flip N/A

         \[\leadsto \frac{\log \left(1 + x\right) + \left(\mathsf{neg}\left(\log x\right)\right)}{n} \]

      3. mul-1-neg N/A

         \[\leadsto \frac{\log \left(1 + x\right) + -1 \cdot \log x}{n} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\log \left(1 + x\right) + -1 \cdot \log x}{n} \]

      5. +-commutative N/A

         \[\leadsto \frac{-1 \cdot \log x + \log \left(1 + x\right)}{n} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{-1 \cdot \log x + \log \left(1 + x\right)}{n} \]

      7. mul-1-neg N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(\log x\right)\right) + \log \left(1 + x\right)}{n} \]

      8. lift-log.f64 N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(\log x\right)\right) + \log \left(1 + x\right)}{n} \]

      9. log-rec N/A

         \[\leadsto \frac{\log \left(\frac{1}{x}\right) + \log \left(1 + x\right)}{n} \]

      10. lift-/.f64 N/A

          \[\leadsto \frac{\log \left(\frac{1}{x}\right) + \log \left(1 + x\right)}{n} \]

      11. lift-log.f64 N/A

          \[\leadsto \frac{\log \left(\frac{1}{x}\right) + \log \left(1 + x\right)}{n} \]

      12. lift-+.f64 N/A

          \[\leadsto \frac{\log \left(\frac{1}{x}\right) + \log \left(1 + x\right)}{n} \]

      13. +-commutative N/A

          \[\leadsto \frac{\log \left(\frac{1}{x}\right) + \log \left(x + 1\right)}{n} \]

      14. lift-+.f64 N/A

          \[\leadsto \frac{\log \left(\frac{1}{x}\right) + \log \left(x + 1\right)}{n} \]

      15. sum-log N/A

          \[\leadsto \frac{\log \left(\frac{1}{x} \cdot \left(x + 1\right)\right)}{n} \]

      16. lower-log.f64 N/A

          \[\leadsto \frac{\log \left(\frac{1}{x} \cdot \left(x + 1\right)\right)}{n} \]

      17. lower-*.f64 55.2

          \[\leadsto \frac{\log \left(\frac{1}{x} \cdot \left(x + 1\right)\right)}{n} \]

      18. lift-+.f64 N/A

          \[\leadsto \frac{\log \left(\frac{1}{x} \cdot \left(x + 1\right)\right)}{n} \]

      19. add-flip N/A

          \[\leadsto \frac{\log \left(\frac{1}{x} \cdot \left(x - \left(\mathsf{neg}\left(1\right)\right)\right)\right)}{n} \]

      20. unpow1 N/A

          \[\leadsto \frac{\log \left(\frac{1}{x} \cdot \left({x}^{1} - \left(\mathsf{neg}\left(1\right)\right)\right)\right)}{n} \]

      21. metadata-eval N/A

          \[\leadsto \frac{\log \left(\frac{1}{x} \cdot \left({x}^{\left(\mathsf{neg}\left(-1\right)\right)} - \left(\mathsf{neg}\left(1\right)\right)\right)\right)}{n} \]

      22. pow-flip N/A

          \[\leadsto \frac{\log \left(\frac{1}{x} \cdot \left(\frac{1}{{x}^{-1}} - \left(\mathsf{neg}\left(1\right)\right)\right)\right)}{n} \]

      23. inv-pow N/A

          \[\leadsto \frac{\log \left(\frac{1}{x} \cdot \left(\frac{1}{\frac{1}{x}} - \left(\mathsf{neg}\left(1\right)\right)\right)\right)}{n} \]

      24. lift-/.f64 N/A

          \[\leadsto \frac{\log \left(\frac{1}{x} \cdot \left(\frac{1}{\frac{1}{x}} - \left(\mathsf{neg}\left(1\right)\right)\right)\right)}{n} \]

      25. metadata-eval N/A

          \[\leadsto \frac{\log \left(\frac{1}{x} \cdot \left(\frac{1}{\frac{1}{x}} - -1\right)\right)}{n} \]

      26. lower--.f64 N/A

          \[\leadsto \frac{\log \left(\frac{1}{x} \cdot \left(\frac{1}{\frac{1}{x}} - -1\right)\right)}{n} \]

      27. lift-/.f64 N/A

          \[\leadsto \frac{\log \left(\frac{1}{x} \cdot \left(\frac{1}{\frac{1}{x}} - -1\right)\right)}{n} \]

      28. inv-pow N/A

          \[\leadsto \frac{\log \left(\frac{1}{x} \cdot \left(\frac{1}{{x}^{-1}} - -1\right)\right)}{n} \]

      29. pow-flip N/A

          \[\leadsto \frac{\log \left(\frac{1}{x} \cdot \left({x}^{\left(\mathsf{neg}\left(-1\right)\right)} - -1\right)\right)}{n} \]

      30. metadata-eval N/A

          \[\leadsto \frac{\log \left(\frac{1}{x} \cdot \left({x}^{1} - -1\right)\right)}{n} \]

      31. unpow1 55.2

          \[\leadsto \frac{\log \left(\frac{1}{x} \cdot \left(x - -1\right)\right)}{n} \]

   6. Applied rewrites 55.2%

      \[\leadsto \frac{\log \left(\frac{1}{x} \cdot \left(x - -1\right)\right)}{n} \]
   7. Step-by-step derivation

      1. lift-log.f64 N/A

         \[\leadsto \frac{\log \left(\frac{1}{x} \cdot \left(x - -1\right)\right)}{n} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\log \left(\frac{1}{x} \cdot \left(x - -1\right)\right)}{n} \]

      3. log-prod N/A

         \[\leadsto \frac{\log \left(\frac{1}{x}\right) + \log \left(x - -1\right)}{n} \]

      4. lift--.f64 N/A

         \[\leadsto \frac{\log \left(\frac{1}{x}\right) + \log \left(x - -1\right)}{n} \]

      5. metadata-eval N/A

         \[\leadsto \frac{\log \left(\frac{1}{x}\right) + \log \left(x - \left(\mathsf{neg}\left(1\right)\right)\right)}{n} \]

      6. add-flip N/A

         \[\leadsto \frac{\log \left(\frac{1}{x}\right) + \log \left(x + 1\right)}{n} \]

      7. log-prod N/A

         \[\leadsto \frac{\log \left(\frac{1}{x} \cdot \left(x + 1\right)\right)}{n} \]

      8. distribute-lft-in N/A

         \[\leadsto \frac{\log \left(\frac{1}{x} \cdot x + \frac{1}{x} \cdot 1\right)}{n} \]

      9. lift-/.f64 N/A

         \[\leadsto \frac{\log \left(\frac{1}{x} \cdot x + \frac{1}{x} \cdot 1\right)}{n} \]

      10. inv-pow N/A

          \[\leadsto \frac{\log \left({x}^{-1} \cdot x + \frac{1}{x} \cdot 1\right)}{n} \]

      11. pow-plus N/A

          \[\leadsto \frac{\log \left({x}^{\left(-1 + 1\right)} + \frac{1}{x} \cdot 1\right)}{n} \]

      12. metadata-eval N/A

          \[\leadsto \frac{\log \left({x}^{0} + \frac{1}{x} \cdot 1\right)}{n} \]

      13. metadata-eval N/A

          \[\leadsto \frac{\log \left(1 + \frac{1}{x} \cdot 1\right)}{n} \]

      14. *-commutative N/A

          \[\leadsto \frac{\log \left(1 + 1 \cdot \frac{1}{x}\right)}{n} \]

      15. *-lft-identity N/A

          \[\leadsto \frac{\log \left(1 + \frac{1}{x}\right)}{n} \]

      16. lower-log1p.f64 58.2

          \[\leadsto \frac{\mathsf{log1p}\left(\frac{1}{x}\right)}{n} \]

   8. Applied rewrites 58.2%

      \[\leadsto \frac{\mathsf{log1p}\left(\frac{1}{x}\right)}{n} \]

   if 1.00000000000000005e117 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 53.2%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]

   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 59.4

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{n} \]

   4. Applied rewrites 59.4%

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]

   5. Taylor expanded in x around inf

      \[\leadsto \frac{1}{\color{blue}{n \cdot x}} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{1}{n \cdot \color{blue}{x}} \]

      2. lower-*.f64 40.4

         \[\leadsto \frac{1}{n \cdot x} \]

   7. Applied rewrites 40.4%

      \[\leadsto \frac{1}{\color{blue}{n \cdot x}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 12: 69.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} t_0 := {\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\frac{1}{n \cdot x}\\ \mathbf{elif}\;t\_0 \leq 10^{-12}:\\ \;\;\;\;\frac{-\log \left(\frac{x}{x - -1}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{n}}{x}\\ \end{array} \]

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 (* n x))
     (if (<= t_0 1e-12) (/ (- (log (/ x (- x -1.0)))) n) (/ (/ 1.0 n) x)))))

C

double code(double x, double n) {
	double t_0 = pow((x + 1.0), (1.0 / n)) - pow(x, (1.0 / n));
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = 1.0 / (n * x);
	} else if (t_0 <= 1e-12) {
		tmp = -log((x / (x - -1.0))) / n;
	} else {
		tmp = (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 / (n * x);
	} else if (t_0 <= 1e-12) {
		tmp = -Math.log((x / (x - -1.0))) / n;
	} else {
		tmp = (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 / (n * x)
	elif t_0 <= 1e-12:
		tmp = -math.log((x / (x - -1.0))) / n
	else:
		tmp = (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(1.0 / Float64(n * x));
	elseif (t_0 <= 1e-12)
		tmp = Float64(Float64(-log(Float64(x / Float64(x - -1.0)))) / n);
	else
		tmp = Float64(Float64(1.0 / 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 / (n * x);
	elseif (t_0 <= 1e-12)
		tmp = -log((x / (x - -1.0))) / n;
	else
		tmp = (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[(1.0 / N[(n * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1e-12], N[((-N[Log[N[(x / N[(x - -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / n), $MachinePrecision], N[(N[(1.0 / n), $MachinePrecision] / x), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (pow.f64 (+.f64 x #s(literal 1 binary64)) (/.f64 #s(literal 1 binary64) n)) (pow.f64 x (/.f64 #s(literal 1 binary64) n))) < -inf.0

   1. Initial program 53.2%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]

   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 59.4

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{n} \]

   4. Applied rewrites 59.4%

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]

   5. Taylor expanded in x around inf

      \[\leadsto \frac{1}{\color{blue}{n \cdot x}} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{1}{n \cdot \color{blue}{x}} \]

      2. lower-*.f64 40.4

         \[\leadsto \frac{1}{n \cdot x} \]

   7. Applied rewrites 40.4%

      \[\leadsto \frac{1}{\color{blue}{n \cdot x}} \]

   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))) < 9.9999999999999998e-13

   1. Initial program 53.2%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]

   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 59.4

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{n} \]

   4. Applied rewrites 59.4%

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
   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. sub-negate-rev N/A

         \[\leadsto \frac{\mathsf{neg}\left(\left(\mathsf{neg}\left(\left(\log \left(1 + x\right) - \log x\right)\right)\right)\right)}{n} \]

      4. lift--.f64 N/A

         \[\leadsto \frac{\mathsf{neg}\left(\left(\mathsf{neg}\left(\left(\log \left(1 + x\right) - \log x\right)\right)\right)\right)}{n} \]

      5. lower-neg.f64 N/A

         \[\leadsto \frac{-\left(\mathsf{neg}\left(\left(\log \left(1 + x\right) - \log x\right)\right)\right)}{n} \]

      6. lift--.f64 N/A

         \[\leadsto \frac{-\left(\mathsf{neg}\left(\left(\log \left(1 + x\right) - \log x\right)\right)\right)}{n} \]

      7. sub-negate-rev N/A

         \[\leadsto \frac{-\left(\log x - \log \left(1 + x\right)\right)}{n} \]

      8. lift-log.f64 N/A

         \[\leadsto \frac{-\left(\log x - \log \left(1 + x\right)\right)}{n} \]

      9. lift-log.f64 N/A

         \[\leadsto \frac{-\left(\log x - \log \left(1 + x\right)\right)}{n} \]

      10. lift-+.f64 N/A

          \[\leadsto \frac{-\left(\log x - \log \left(1 + x\right)\right)}{n} \]

      11. +-commutative N/A

          \[\leadsto \frac{-\left(\log x - \log \left(x + 1\right)\right)}{n} \]

      12. lift-+.f64 N/A

          \[\leadsto \frac{-\left(\log x - \log \left(x + 1\right)\right)}{n} \]

      13. diff-log N/A

          \[\leadsto \frac{-\log \left(\frac{x}{x + 1}\right)}{n} \]

      14. lower-log.f64 N/A

          \[\leadsto \frac{-\log \left(\frac{x}{x + 1}\right)}{n} \]

      15. unpow1 N/A

          \[\leadsto \frac{-\log \left(\frac{{x}^{1}}{x + 1}\right)}{n} \]

      16. metadata-eval N/A

          \[\leadsto \frac{-\log \left(\frac{{x}^{\left(\mathsf{neg}\left(-1\right)\right)}}{x + 1}\right)}{n} \]

      17. pow-flip N/A

          \[\leadsto \frac{-\log \left(\frac{\frac{1}{{x}^{-1}}}{x + 1}\right)}{n} \]

      18. inv-pow N/A

          \[\leadsto \frac{-\log \left(\frac{\frac{1}{\frac{1}{x}}}{x + 1}\right)}{n} \]

      19. lift-/.f64 N/A

          \[\leadsto \frac{-\log \left(\frac{\frac{1}{\frac{1}{x}}}{x + 1}\right)}{n} \]

      20. lower-/.f64 N/A

          \[\leadsto \frac{-\log \left(\frac{\frac{1}{\frac{1}{x}}}{x + 1}\right)}{n} \]

      21. lift-/.f64 N/A

          \[\leadsto \frac{-\log \left(\frac{\frac{1}{\frac{1}{x}}}{x + 1}\right)}{n} \]

      22. inv-pow N/A

          \[\leadsto \frac{-\log \left(\frac{\frac{1}{{x}^{-1}}}{x + 1}\right)}{n} \]

      23. pow-flip N/A

          \[\leadsto \frac{-\log \left(\frac{{x}^{\left(\mathsf{neg}\left(-1\right)\right)}}{x + 1}\right)}{n} \]

      24. metadata-eval N/A

          \[\leadsto \frac{-\log \left(\frac{{x}^{1}}{x + 1}\right)}{n} \]

      25. unpow1 59.5

          \[\leadsto \frac{-\log \left(\frac{x}{x + 1}\right)}{n} \]

      26. lift-+.f64 N/A

          \[\leadsto \frac{-\log \left(\frac{x}{x + 1}\right)}{n} \]

      27. add-flip N/A

          \[\leadsto \frac{-\log \left(\frac{x}{x - \left(\mathsf{neg}\left(1\right)\right)}\right)}{n} \]

   6. Applied rewrites 59.5%

      \[\leadsto \frac{-\log \left(\frac{x}{x - -1}\right)}{n} \]

   if 9.9999999999999998e-13 < (-.f64 (pow.f64 (+.f64 x #s(literal 1 binary64)) (/.f64 #s(literal 1 binary64) n)) (pow.f64 x (/.f64 #s(literal 1 binary64) n)))

   1. Initial program 53.2%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]

   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 59.4

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{n} \]

   4. Applied rewrites 59.4%

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]

   5. Taylor expanded in x around inf

      \[\leadsto \frac{1}{\color{blue}{n \cdot x}} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{1}{n \cdot \color{blue}{x}} \]

      2. lower-*.f64 40.4

         \[\leadsto \frac{1}{n \cdot x} \]

   7. Applied rewrites 40.4%

      \[\leadsto \frac{1}{\color{blue}{n \cdot x}} \]
   8. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{1}{n \cdot \color{blue}{x}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{1}{n \cdot x} \]

      3. associate-/r* N/A

         \[\leadsto \frac{\frac{1}{n}}{x} \]

      4. lift-/.f64 N/A

         \[\leadsto \frac{\frac{1}{n}}{x} \]

      5. lower-/.f64 41.0

         \[\leadsto \frac{\frac{1}{n}}{x} \]

   9. Applied rewrites 41.0%

      \[\leadsto \frac{\frac{1}{n}}{x} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 13: 68.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} t_0 := {\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\frac{1}{n \cdot x}\\ \mathbf{elif}\;t\_0 \leq 10^{-12}:\\ \;\;\;\;\frac{\log \left(\frac{1}{x} + 1\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{n}}{x}\\ \end{array} \]

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 (* n x))
     (if (<= t_0 1e-12) (/ (log (+ (/ 1.0 x) 1.0)) n) (/ (/ 1.0 n) x)))))

C

double code(double x, double n) {
	double t_0 = pow((x + 1.0), (1.0 / n)) - pow(x, (1.0 / n));
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = 1.0 / (n * x);
	} else if (t_0 <= 1e-12) {
		tmp = log(((1.0 / x) + 1.0)) / n;
	} else {
		tmp = (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 / (n * x);
	} else if (t_0 <= 1e-12) {
		tmp = Math.log(((1.0 / x) + 1.0)) / n;
	} else {
		tmp = (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 / (n * x)
	elif t_0 <= 1e-12:
		tmp = math.log(((1.0 / x) + 1.0)) / n
	else:
		tmp = (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(1.0 / Float64(n * x));
	elseif (t_0 <= 1e-12)
		tmp = Float64(log(Float64(Float64(1.0 / x) + 1.0)) / n);
	else
		tmp = Float64(Float64(1.0 / 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 / (n * x);
	elseif (t_0 <= 1e-12)
		tmp = log(((1.0 / x) + 1.0)) / n;
	else
		tmp = (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[(1.0 / N[(n * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1e-12], N[(N[Log[N[(N[(1.0 / x), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], N[(N[(1.0 / n), $MachinePrecision] / x), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (pow.f64 (+.f64 x #s(literal 1 binary64)) (/.f64 #s(literal 1 binary64) n)) (pow.f64 x (/.f64 #s(literal 1 binary64) n))) < -inf.0

   1. Initial program 53.2%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]

   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 59.4

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{n} \]

   4. Applied rewrites 59.4%

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]

   5. Taylor expanded in x around inf

      \[\leadsto \frac{1}{\color{blue}{n \cdot x}} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{1}{n \cdot \color{blue}{x}} \]

      2. lower-*.f64 40.4

         \[\leadsto \frac{1}{n \cdot x} \]

   7. Applied rewrites 40.4%

      \[\leadsto \frac{1}{\color{blue}{n \cdot x}} \]

   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))) < 9.9999999999999998e-13

   1. Initial program 53.2%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]

   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 59.4

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{n} \]

   4. Applied rewrites 59.4%

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
   5. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{n} \]

      2. sub-flip N/A

         \[\leadsto \frac{\log \left(1 + x\right) + \left(\mathsf{neg}\left(\log x\right)\right)}{n} \]

      3. mul-1-neg N/A

         \[\leadsto \frac{\log \left(1 + x\right) + -1 \cdot \log x}{n} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\log \left(1 + x\right) + -1 \cdot \log x}{n} \]

      5. +-commutative N/A

         \[\leadsto \frac{-1 \cdot \log x + \log \left(1 + x\right)}{n} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{-1 \cdot \log x + \log \left(1 + x\right)}{n} \]

      7. mul-1-neg N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(\log x\right)\right) + \log \left(1 + x\right)}{n} \]

      8. lift-log.f64 N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(\log x\right)\right) + \log \left(1 + x\right)}{n} \]

      9. log-rec N/A

         \[\leadsto \frac{\log \left(\frac{1}{x}\right) + \log \left(1 + x\right)}{n} \]

      10. lift-/.f64 N/A

          \[\leadsto \frac{\log \left(\frac{1}{x}\right) + \log \left(1 + x\right)}{n} \]

      11. lift-log.f64 N/A

          \[\leadsto \frac{\log \left(\frac{1}{x}\right) + \log \left(1 + x\right)}{n} \]

      12. lift-+.f64 N/A

          \[\leadsto \frac{\log \left(\frac{1}{x}\right) + \log \left(1 + x\right)}{n} \]

      13. +-commutative N/A

          \[\leadsto \frac{\log \left(\frac{1}{x}\right) + \log \left(x + 1\right)}{n} \]

      14. lift-+.f64 N/A

          \[\leadsto \frac{\log \left(\frac{1}{x}\right) + \log \left(x + 1\right)}{n} \]

      15. sum-log N/A

          \[\leadsto \frac{\log \left(\frac{1}{x} \cdot \left(x + 1\right)\right)}{n} \]

      16. lower-log.f64 N/A

          \[\leadsto \frac{\log \left(\frac{1}{x} \cdot \left(x + 1\right)\right)}{n} \]

      17. lower-*.f64 55.2

          \[\leadsto \frac{\log \left(\frac{1}{x} \cdot \left(x + 1\right)\right)}{n} \]

      18. lift-+.f64 N/A

          \[\leadsto \frac{\log \left(\frac{1}{x} \cdot \left(x + 1\right)\right)}{n} \]

      19. add-flip N/A

          \[\leadsto \frac{\log \left(\frac{1}{x} \cdot \left(x - \left(\mathsf{neg}\left(1\right)\right)\right)\right)}{n} \]

      20. unpow1 N/A

          \[\leadsto \frac{\log \left(\frac{1}{x} \cdot \left({x}^{1} - \left(\mathsf{neg}\left(1\right)\right)\right)\right)}{n} \]

      21. metadata-eval N/A

          \[\leadsto \frac{\log \left(\frac{1}{x} \cdot \left({x}^{\left(\mathsf{neg}\left(-1\right)\right)} - \left(\mathsf{neg}\left(1\right)\right)\right)\right)}{n} \]

      22. pow-flip N/A

          \[\leadsto \frac{\log \left(\frac{1}{x} \cdot \left(\frac{1}{{x}^{-1}} - \left(\mathsf{neg}\left(1\right)\right)\right)\right)}{n} \]

      23. inv-pow N/A

          \[\leadsto \frac{\log \left(\frac{1}{x} \cdot \left(\frac{1}{\frac{1}{x}} - \left(\mathsf{neg}\left(1\right)\right)\right)\right)}{n} \]

      24. lift-/.f64 N/A

          \[\leadsto \frac{\log \left(\frac{1}{x} \cdot \left(\frac{1}{\frac{1}{x}} - \left(\mathsf{neg}\left(1\right)\right)\right)\right)}{n} \]

      25. metadata-eval N/A

          \[\leadsto \frac{\log \left(\frac{1}{x} \cdot \left(\frac{1}{\frac{1}{x}} - -1\right)\right)}{n} \]

      26. lower--.f64 N/A

          \[\leadsto \frac{\log \left(\frac{1}{x} \cdot \left(\frac{1}{\frac{1}{x}} - -1\right)\right)}{n} \]

      27. lift-/.f64 N/A

          \[\leadsto \frac{\log \left(\frac{1}{x} \cdot \left(\frac{1}{\frac{1}{x}} - -1\right)\right)}{n} \]

      28. inv-pow N/A

          \[\leadsto \frac{\log \left(\frac{1}{x} \cdot \left(\frac{1}{{x}^{-1}} - -1\right)\right)}{n} \]

      29. pow-flip N/A

          \[\leadsto \frac{\log \left(\frac{1}{x} \cdot \left({x}^{\left(\mathsf{neg}\left(-1\right)\right)} - -1\right)\right)}{n} \]

      30. metadata-eval N/A

          \[\leadsto \frac{\log \left(\frac{1}{x} \cdot \left({x}^{1} - -1\right)\right)}{n} \]

      31. unpow1 55.2

          \[\leadsto \frac{\log \left(\frac{1}{x} \cdot \left(x - -1\right)\right)}{n} \]

   6. Applied rewrites 55.2%

      \[\leadsto \frac{\log \left(\frac{1}{x} \cdot \left(x - -1\right)\right)}{n} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\log \left(\frac{1}{x} \cdot \left(x - -1\right)\right)}{n} \]

      2. lift--.f64 N/A

         \[\leadsto \frac{\log \left(\frac{1}{x} \cdot \left(x - -1\right)\right)}{n} \]

      3. metadata-eval N/A

         \[\leadsto \frac{\log \left(\frac{1}{x} \cdot \left(x - \left(\mathsf{neg}\left(1\right)\right)\right)\right)}{n} \]

      4. add-flip N/A

         \[\leadsto \frac{\log \left(\frac{1}{x} \cdot \left(x + 1\right)\right)}{n} \]

      5. +-commutative N/A

         \[\leadsto \frac{\log \left(\frac{1}{x} \cdot \left(1 + x\right)\right)}{n} \]

      6. distribute-lft-in N/A

         \[\leadsto \frac{\log \left(\frac{1}{x} \cdot 1 + \frac{1}{x} \cdot x\right)}{n} \]

      7. *-commutative N/A

         \[\leadsto \frac{\log \left(1 \cdot \frac{1}{x} + \frac{1}{x} \cdot x\right)}{n} \]

      8. *-lft-identity N/A

         \[\leadsto \frac{\log \left(\frac{1}{x} + \frac{1}{x} \cdot x\right)}{n} \]

      9. lift-/.f64 N/A

         \[\leadsto \frac{\log \left(\frac{1}{x} + \frac{1}{x} \cdot x\right)}{n} \]

      10. inv-pow N/A

          \[\leadsto \frac{\log \left(\frac{1}{x} + {x}^{-1} \cdot x\right)}{n} \]

      11. pow-plus N/A

          \[\leadsto \frac{\log \left(\frac{1}{x} + {x}^{\left(-1 + 1\right)}\right)}{n} \]

      12. metadata-eval N/A

          \[\leadsto \frac{\log \left(\frac{1}{x} + {x}^{0}\right)}{n} \]

      13. metadata-eval N/A

          \[\leadsto \frac{\log \left(\frac{1}{x} + 1\right)}{n} \]

      14. lower-+.f64 59.5

          \[\leadsto \frac{\log \left(\frac{1}{x} + 1\right)}{n} \]

   8. Applied rewrites 59.5%

      \[\leadsto \frac{\log \left(\frac{1}{x} + 1\right)}{n} \]

   if 9.9999999999999998e-13 < (-.f64 (pow.f64 (+.f64 x #s(literal 1 binary64)) (/.f64 #s(literal 1 binary64) n)) (pow.f64 x (/.f64 #s(literal 1 binary64) n)))

   1. Initial program 53.2%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]

   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 59.4

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{n} \]

   4. Applied rewrites 59.4%

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]

   5. Taylor expanded in x around inf

      \[\leadsto \frac{1}{\color{blue}{n \cdot x}} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{1}{n \cdot \color{blue}{x}} \]

      2. lower-*.f64 40.4

         \[\leadsto \frac{1}{n \cdot x} \]

   7. Applied rewrites 40.4%

      \[\leadsto \frac{1}{\color{blue}{n \cdot x}} \]
   8. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{1}{n \cdot \color{blue}{x}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{1}{n \cdot x} \]

      3. associate-/r* N/A

         \[\leadsto \frac{\frac{1}{n}}{x} \]

      4. lift-/.f64 N/A

         \[\leadsto \frac{\frac{1}{n}}{x} \]

      5. lower-/.f64 41.0

         \[\leadsto \frac{\frac{1}{n}}{x} \]

   9. Applied rewrites 41.0%

      \[\leadsto \frac{\frac{1}{n}}{x} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 14: 68.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} t_0 := {\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\frac{1}{n \cdot x}\\ \mathbf{elif}\;t\_0 \leq 10^{-12}:\\ \;\;\;\;\frac{\log \left(\frac{x - -1}{x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{n}}{x}\\ \end{array} \]

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 (* n x))
     (if (<= t_0 1e-12) (/ (log (/ (- x -1.0) x)) n) (/ (/ 1.0 n) x)))))

C

double code(double x, double n) {
	double t_0 = pow((x + 1.0), (1.0 / n)) - pow(x, (1.0 / n));
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = 1.0 / (n * x);
	} else if (t_0 <= 1e-12) {
		tmp = log(((x - -1.0) / x)) / n;
	} else {
		tmp = (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 / (n * x);
	} else if (t_0 <= 1e-12) {
		tmp = Math.log(((x - -1.0) / x)) / n;
	} else {
		tmp = (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 / (n * x)
	elif t_0 <= 1e-12:
		tmp = math.log(((x - -1.0) / x)) / n
	else:
		tmp = (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(1.0 / Float64(n * x));
	elseif (t_0 <= 1e-12)
		tmp = Float64(log(Float64(Float64(x - -1.0) / x)) / n);
	else
		tmp = Float64(Float64(1.0 / 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 / (n * x);
	elseif (t_0 <= 1e-12)
		tmp = log(((x - -1.0) / x)) / n;
	else
		tmp = (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[(1.0 / N[(n * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1e-12], N[(N[Log[N[(N[(x - -1.0), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], N[(N[(1.0 / n), $MachinePrecision] / x), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (pow.f64 (+.f64 x #s(literal 1 binary64)) (/.f64 #s(literal 1 binary64) n)) (pow.f64 x (/.f64 #s(literal 1 binary64) n))) < -inf.0

   1. Initial program 53.2%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]

   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 59.4

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{n} \]

   4. Applied rewrites 59.4%

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]

   5. Taylor expanded in x around inf

      \[\leadsto \frac{1}{\color{blue}{n \cdot x}} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{1}{n \cdot \color{blue}{x}} \]

      2. lower-*.f64 40.4

         \[\leadsto \frac{1}{n \cdot x} \]

   7. Applied rewrites 40.4%

      \[\leadsto \frac{1}{\color{blue}{n \cdot x}} \]

   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))) < 9.9999999999999998e-13

   1. Initial program 53.2%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]

   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 59.4

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{n} \]

   4. Applied rewrites 59.4%

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
   5. Step-by-step derivation

   6. Applied rewrites 59.5%

      \[\leadsto \color{blue}{\frac{\log \left(\frac{x - -1}{x}\right)}{n}} \]

   if 9.9999999999999998e-13 < (-.f64 (pow.f64 (+.f64 x #s(literal 1 binary64)) (/.f64 #s(literal 1 binary64) n)) (pow.f64 x (/.f64 #s(literal 1 binary64) n)))

   1. Initial program 53.2%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]

   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 59.4

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{n} \]

   4. Applied rewrites 59.4%

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]

   5. Taylor expanded in x around inf

      \[\leadsto \frac{1}{\color{blue}{n \cdot x}} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{1}{n \cdot \color{blue}{x}} \]

      2. lower-*.f64 40.4

         \[\leadsto \frac{1}{n \cdot x} \]

   7. Applied rewrites 40.4%

      \[\leadsto \frac{1}{\color{blue}{n \cdot x}} \]
   8. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{1}{n \cdot \color{blue}{x}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{1}{n \cdot x} \]

      3. associate-/r* N/A

         \[\leadsto \frac{\frac{1}{n}}{x} \]

      4. lift-/.f64 N/A

         \[\leadsto \frac{\frac{1}{n}}{x} \]

      5. lower-/.f64 41.0

         \[\leadsto \frac{\frac{1}{n}}{x} \]

   9. Applied rewrites 41.0%

      \[\leadsto \frac{\frac{1}{n}}{x} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 15: 58.8% accurate, 2.4× speedup

Math

\[\begin{array}{l} \mathbf{if}\;x \leq 3.7:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{elif}\;x \leq 7 \cdot 10^{+186}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-0.5}{n \cdot x}}{x}\\ \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 3.7)
   (/ (- x (log x)) n)
   (if (<= x 7e+186) (/ (/ 1.0 x) n) (/ (/ -0.5 (* n x)) x))))

C

double code(double x, double n) {
	double tmp;
	if (x <= 3.7) {
		tmp = (x - log(x)) / n;
	} else if (x <= 7e+186) {
		tmp = (1.0 / x) / n;
	} else {
		tmp = (-0.5 / (n * x)) / 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 <= 3.7d0) then
        tmp = (x - log(x)) / n
    else if (x <= 7d+186) then
        tmp = (1.0d0 / x) / n
    else
        tmp = ((-0.5d0) / (n * x)) / x
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (x <= 3.7) {
		tmp = (x - Math.log(x)) / n;
	} else if (x <= 7e+186) {
		tmp = (1.0 / x) / n;
	} else {
		tmp = (-0.5 / (n * x)) / x;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 3.7:
		tmp = (x - math.log(x)) / n
	elif x <= 7e+186:
		tmp = (1.0 / x) / n
	else:
		tmp = (-0.5 / (n * x)) / x
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 3.7)
		tmp = Float64(Float64(x - log(x)) / n);
	elseif (x <= 7e+186)
		tmp = Float64(Float64(1.0 / x) / n);
	else
		tmp = Float64(Float64(-0.5 / Float64(n * x)) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (x <= 3.7)
		tmp = (x - log(x)) / n;
	elseif (x <= 7e+186)
		tmp = (1.0 / x) / n;
	else
		tmp = (-0.5 / (n * x)) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 3.7], N[(N[(x - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[x, 7e+186], N[(N[(1.0 / x), $MachinePrecision] / n), $MachinePrecision], N[(N[(-0.5 / N[(n * x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 3.7000000000000002

   1. Initial program 53.2%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]

   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 59.4

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{n} \]

   4. Applied rewrites 59.4%

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]

   5. Taylor expanded in x around 0

      \[\leadsto \frac{x - \log x}{n} \]
   6. Step-by-step derivation

   7. Applied rewrites 31.1%

      \[\leadsto \frac{x - \log x}{n} \]

   if 3.7000000000000002 < x < 6.99999999999999974e186

   1. Initial program 53.2%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]

   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 59.4

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{n} \]

   4. Applied rewrites 59.4%

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]

   5. Taylor expanded in x around inf

      \[\leadsto \frac{1}{\color{blue}{n \cdot x}} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{1}{n \cdot \color{blue}{x}} \]

      2. lower-*.f64 40.4

         \[\leadsto \frac{1}{n \cdot x} \]

   7. Applied rewrites 40.4%

      \[\leadsto \frac{1}{\color{blue}{n \cdot x}} \]
   8. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{1}{n \cdot \color{blue}{x}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{1}{n \cdot x} \]

      3. *-commutative N/A

         \[\leadsto \frac{1}{x \cdot n} \]

      4. associate-/r* N/A

         \[\leadsto \frac{\frac{1}{x}}{n} \]

      5. lift-/.f64 N/A

         \[\leadsto \frac{\frac{1}{x}}{n} \]

      6. lower-/.f64 41.0

         \[\leadsto \frac{\frac{1}{x}}{n} \]

   9. Applied rewrites 41.0%

      \[\leadsto \frac{\frac{1}{x}}{n} \]

   if 6.99999999999999974e186 < x

   1. Initial program 53.2%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]

   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 59.4

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{n} \]

   4. Applied rewrites 59.4%

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]

   5. Taylor expanded in x around inf

      \[\leadsto \frac{\frac{1}{n} - \frac{1}{2} \cdot \frac{1}{n \cdot x}}{\color{blue}{x}} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\frac{1}{n} - \frac{1}{2} \cdot \frac{1}{n \cdot x}}{x} \]

      2. lower--.f64 N/A

         \[\leadsto \frac{\frac{1}{n} - \frac{1}{2} \cdot \frac{1}{n \cdot x}}{x} \]

      3. lower-/.f64 N/A

         \[\leadsto \frac{\frac{1}{n} - \frac{1}{2} \cdot \frac{1}{n \cdot x}}{x} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\frac{1}{n} - \frac{1}{2} \cdot \frac{1}{n \cdot x}}{x} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{\frac{1}{n} - \frac{1}{2} \cdot \frac{1}{n \cdot x}}{x} \]

      6. lower-*.f64 28.6

         \[\leadsto \frac{\frac{1}{n} - 0.5 \cdot \frac{1}{n \cdot x}}{x} \]

   7. Applied rewrites 28.6%

      \[\leadsto \frac{\frac{1}{n} - 0.5 \cdot \frac{1}{n \cdot x}}{\color{blue}{x}} \]

   8. Taylor expanded in x around 0

      \[\leadsto \frac{\frac{\frac{-1}{2}}{n \cdot x}}{x} \]
   9. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\frac{\frac{-1}{2}}{n \cdot x}}{x} \]

      2. lower-*.f64 19.5

         \[\leadsto \frac{\frac{-0.5}{n \cdot x}}{x} \]

   10. Applied rewrites 19.5%

       \[\leadsto \frac{\frac{-0.5}{n \cdot x}}{x} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 16: 42.5% accurate, 3.1× speedup

Math

\[\begin{array}{l} \mathbf{if}\;x \leq 7 \cdot 10^{+186}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-0.5}{n \cdot x}}{x}\\ \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 7e+186) (/ (/ 1.0 x) n) (/ (/ -0.5 (* n x)) x)))

C

double code(double x, double n) {
	double tmp;
	if (x <= 7e+186) {
		tmp = (1.0 / x) / n;
	} else {
		tmp = (-0.5 / (n * x)) / 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 <= 7d+186) then
        tmp = (1.0d0 / x) / n
    else
        tmp = ((-0.5d0) / (n * x)) / x
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (x <= 7e+186) {
		tmp = (1.0 / x) / n;
	} else {
		tmp = (-0.5 / (n * x)) / x;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 7e+186:
		tmp = (1.0 / x) / n
	else:
		tmp = (-0.5 / (n * x)) / x
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 7e+186)
		tmp = Float64(Float64(1.0 / x) / n);
	else
		tmp = Float64(Float64(-0.5 / Float64(n * x)) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (x <= 7e+186)
		tmp = (1.0 / x) / n;
	else
		tmp = (-0.5 / (n * x)) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 7e+186], N[(N[(1.0 / x), $MachinePrecision] / n), $MachinePrecision], N[(N[(-0.5 / N[(n * x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 6.99999999999999974e186

   1. Initial program 53.2%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]

   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 59.4

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{n} \]

   4. Applied rewrites 59.4%

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]

   5. Taylor expanded in x around inf

      \[\leadsto \frac{1}{\color{blue}{n \cdot x}} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{1}{n \cdot \color{blue}{x}} \]

      2. lower-*.f64 40.4

         \[\leadsto \frac{1}{n \cdot x} \]

   7. Applied rewrites 40.4%

      \[\leadsto \frac{1}{\color{blue}{n \cdot x}} \]
   8. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{1}{n \cdot \color{blue}{x}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{1}{n \cdot x} \]

      3. *-commutative N/A

         \[\leadsto \frac{1}{x \cdot n} \]

      4. associate-/r* N/A

         \[\leadsto \frac{\frac{1}{x}}{n} \]

      5. lift-/.f64 N/A

         \[\leadsto \frac{\frac{1}{x}}{n} \]

      6. lower-/.f64 41.0

         \[\leadsto \frac{\frac{1}{x}}{n} \]

   9. Applied rewrites 41.0%

      \[\leadsto \frac{\frac{1}{x}}{n} \]

   if 6.99999999999999974e186 < x

   1. Initial program 53.2%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]

   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 59.4

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{n} \]

   4. Applied rewrites 59.4%

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]

   5. Taylor expanded in x around inf

      \[\leadsto \frac{\frac{1}{n} - \frac{1}{2} \cdot \frac{1}{n \cdot x}}{\color{blue}{x}} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\frac{1}{n} - \frac{1}{2} \cdot \frac{1}{n \cdot x}}{x} \]

      2. lower--.f64 N/A

         \[\leadsto \frac{\frac{1}{n} - \frac{1}{2} \cdot \frac{1}{n \cdot x}}{x} \]

      3. lower-/.f64 N/A

         \[\leadsto \frac{\frac{1}{n} - \frac{1}{2} \cdot \frac{1}{n \cdot x}}{x} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\frac{1}{n} - \frac{1}{2} \cdot \frac{1}{n \cdot x}}{x} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{\frac{1}{n} - \frac{1}{2} \cdot \frac{1}{n \cdot x}}{x} \]

      6. lower-*.f64 28.6

         \[\leadsto \frac{\frac{1}{n} - 0.5 \cdot \frac{1}{n \cdot x}}{x} \]

   7. Applied rewrites 28.6%

      \[\leadsto \frac{\frac{1}{n} - 0.5 \cdot \frac{1}{n \cdot x}}{\color{blue}{x}} \]

   8. Taylor expanded in x around 0

      \[\leadsto \frac{\frac{\frac{-1}{2}}{n \cdot x}}{x} \]
   9. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\frac{\frac{-1}{2}}{n \cdot x}}{x} \]

      2. lower-*.f64 19.5

         \[\leadsto \frac{\frac{-0.5}{n \cdot x}}{x} \]

   10. Applied rewrites 19.5%

       \[\leadsto \frac{\frac{-0.5}{n \cdot x}}{x} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 17: 41.0% accurate, 5.8× speedup

Math

\[\frac{\frac{1}{x}}{n} \]

FPCore

(FPCore (x n) :precision binary64 (/ (/ 1.0 x) n))

C

double code(double x, double n) {
	return (1.0 / x) / n;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, n)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    code = (1.0d0 / x) / n
end function

Java

public static double code(double x, double n) {
	return (1.0 / x) / n;
}

Python

def code(x, n):
	return (1.0 / x) / n

Julia

function code(x, n)
	return Float64(Float64(1.0 / x) / n)
end

MATLAB

function tmp = code(x, n)
	tmp = (1.0 / x) / n;
end

Wolfram

code[x_, n_] := N[(N[(1.0 / x), $MachinePrecision] / n), $MachinePrecision]

Derivation

1. Initial program 53.2%

   \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]

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 59.4

      \[\leadsto \frac{\log \left(1 + x\right) - \log x}{n} \]

4. Applied rewrites 59.4%

   \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]

5. Taylor expanded in x around inf

   \[\leadsto \frac{1}{\color{blue}{n \cdot x}} \]
6. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \frac{1}{n \cdot \color{blue}{x}} \]

   2. lower-*.f64 40.4

      \[\leadsto \frac{1}{n \cdot x} \]

7. Applied rewrites 40.4%

   \[\leadsto \frac{1}{\color{blue}{n \cdot x}} \]
8. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \frac{1}{n \cdot \color{blue}{x}} \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{1}{n \cdot x} \]

   3. *-commutative N/A

      \[\leadsto \frac{1}{x \cdot n} \]

   4. associate-/r* N/A

      \[\leadsto \frac{\frac{1}{x}}{n} \]

   5. lift-/.f64 N/A

      \[\leadsto \frac{\frac{1}{x}}{n} \]

   6. lower-/.f64 41.0

      \[\leadsto \frac{\frac{1}{x}}{n} \]

9. Applied rewrites 41.0%

   \[\leadsto \frac{\frac{1}{x}}{n} \]
10. Add Preprocessing

Alternative 18: 41.0% accurate, 5.8× speedup

Math

\[\frac{\frac{1}{n}}{x} \]

FPCore

(FPCore (x n) :precision binary64 (/ (/ 1.0 n) x))

C

double code(double x, double n) {
	return (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 / n) / x
end function

Java

public static double code(double x, double n) {
	return (1.0 / n) / x;
}

Python

def code(x, n):
	return (1.0 / n) / x

Julia

function code(x, n)
	return Float64(Float64(1.0 / n) / x)
end

MATLAB

function tmp = code(x, n)
	tmp = (1.0 / n) / x;
end

Wolfram

code[x_, n_] := N[(N[(1.0 / n), $MachinePrecision] / x), $MachinePrecision]

Derivation

1. Initial program 53.2%

   \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]

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 59.4

      \[\leadsto \frac{\log \left(1 + x\right) - \log x}{n} \]

4. Applied rewrites 59.4%

   \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]

5. Taylor expanded in x around inf

   \[\leadsto \frac{1}{\color{blue}{n \cdot x}} \]
6. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \frac{1}{n \cdot \color{blue}{x}} \]

   2. lower-*.f64 40.4

      \[\leadsto \frac{1}{n \cdot x} \]

7. Applied rewrites 40.4%

   \[\leadsto \frac{1}{\color{blue}{n \cdot x}} \]
8. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \frac{1}{n \cdot \color{blue}{x}} \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{1}{n \cdot x} \]

   3. associate-/r* N/A

      \[\leadsto \frac{\frac{1}{n}}{x} \]

   4. lift-/.f64 N/A

      \[\leadsto \frac{\frac{1}{n}}{x} \]

   5. lower-/.f64 41.0

      \[\leadsto \frac{\frac{1}{n}}{x} \]

9. Applied rewrites 41.0%

   \[\leadsto \frac{\frac{1}{n}}{x} \]
10. Add Preprocessing

Alternative 19: 40.4% accurate, 6.1× speedup

Math

\[\frac{1}{n \cdot x} \]

FPCore

(FPCore (x n) :precision binary64 (/ 1.0 (* n x)))

C

double code(double x, double n) {
	return 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 / (n * x)
end function

Java

public static double code(double x, double n) {
	return 1.0 / (n * x);
}

Python

def code(x, n):
	return 1.0 / (n * x)

Julia

function code(x, n)
	return Float64(1.0 / Float64(n * x))
end

MATLAB

function tmp = code(x, n)
	tmp = 1.0 / (n * x);
end

Wolfram

code[x_, n_] := N[(1.0 / N[(n * x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 53.2%

   \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]

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 59.4

      \[\leadsto \frac{\log \left(1 + x\right) - \log x}{n} \]

4. Applied rewrites 59.4%

   \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]

5. Taylor expanded in x around inf

   \[\leadsto \frac{1}{\color{blue}{n \cdot x}} \]
6. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \frac{1}{n \cdot \color{blue}{x}} \]

   2. lower-*.f64 40.4

      \[\leadsto \frac{1}{n \cdot x} \]

7. Applied rewrites 40.4%

   \[\leadsto \frac{1}{\color{blue}{n \cdot x}} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2025173 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  :precision binary64
  (- (pow (+ x 1.0) (/ 1.0 n)) (pow x (/ 1.0 n))))