2nthrt (problem 3.4.6)

Percentage Accurate: 53.5% → 95.0%

Time: 19.8s

Alternatives: 19

Speedup: 2.5×

• could not determine a ground truth

  1. x = 2.9554634049907406e+53
  2. n = 2.0240213319632044e-14

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: 53.5% 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: 95.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{-18}:\\ \;\;\;\;\frac{\frac{1}{x}}{{x}^{\left(\frac{-1}{n}\right)} \cdot n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-14}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(\frac{1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{+147}:\\ \;\;\;\;{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\log x}{n} - -1}{\left(-n\right) \cdot x}\\ \end{array} \]

FPCore

(FPCore (x n)
  :precision binary64
  (if (<= (/ 1.0 n) -1e-18)
  (/ (/ 1.0 x) (* (pow x (/ -1.0 n)) n))
  (if (<= (/ 1.0 n) 1e-14)
    (/ (log1p (/ 1.0 x)) n)
    (if (<= (/ 1.0 n) 5e+147)
      (- (pow (+ x 1.0) (/ 1.0 n)) (pow x (/ 1.0 n)))
      (/ (- (/ (log x) n) -1.0) (* (- n) x))))))

C

double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -1e-18) {
		tmp = (1.0 / x) / (pow(x, (-1.0 / n)) * n);
	} else if ((1.0 / n) <= 1e-14) {
		tmp = log1p((1.0 / x)) / n;
	} else if ((1.0 / n) <= 5e+147) {
		tmp = pow((x + 1.0), (1.0 / n)) - pow(x, (1.0 / n));
	} else {
		tmp = ((log(x) / n) - -1.0) / (-n * x);
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -1e-18) {
		tmp = (1.0 / x) / (Math.pow(x, (-1.0 / n)) * n);
	} else if ((1.0 / n) <= 1e-14) {
		tmp = Math.log1p((1.0 / x)) / n;
	} else if ((1.0 / n) <= 5e+147) {
		tmp = Math.pow((x + 1.0), (1.0 / n)) - Math.pow(x, (1.0 / n));
	} else {
		tmp = ((Math.log(x) / n) - -1.0) / (-n * x);
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if (1.0 / n) <= -1e-18:
		tmp = (1.0 / x) / (math.pow(x, (-1.0 / n)) * n)
	elif (1.0 / n) <= 1e-14:
		tmp = math.log1p((1.0 / x)) / n
	elif (1.0 / n) <= 5e+147:
		tmp = math.pow((x + 1.0), (1.0 / n)) - math.pow(x, (1.0 / n))
	else:
		tmp = ((math.log(x) / n) - -1.0) / (-n * x)
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -1e-18)
		tmp = Float64(Float64(1.0 / x) / Float64((x ^ Float64(-1.0 / n)) * n));
	elseif (Float64(1.0 / n) <= 1e-14)
		tmp = Float64(log1p(Float64(1.0 / x)) / n);
	elseif (Float64(1.0 / n) <= 5e+147)
		tmp = Float64((Float64(x + 1.0) ^ Float64(1.0 / n)) - (x ^ Float64(1.0 / n)));
	else
		tmp = Float64(Float64(Float64(log(x) / n) - -1.0) / Float64(Float64(-n) * x));
	end
	return tmp
end

Wolfram

code[x_, n_] := If[LessEqual[N[(1.0 / n), $MachinePrecision], -1e-18], 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], 1e-14], N[(N[Log[1 + N[(1.0 / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e+147], 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[(N[(N[(N[Log[x], $MachinePrecision] / n), $MachinePrecision] - -1.0), $MachinePrecision] / N[((-n) * x), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -1.0000000000000001e-18

   1. Initial program 53.5%

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

   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 58.3%

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

   4. Applied rewrites 58.3%

      \[\leadsto \color{blue}{\frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}}{n \cdot x}} \]
   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 59.0%

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

   if -1.0000000000000001e-18 < (/.f64 #s(literal 1 binary64) n) < 1e-14

   1. Initial program 53.5%

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

   2. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-log.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-log.f64 58.9%

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

   4. Applied rewrites 58.9%

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

      1. lift--.f64 N/A

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

      2. sub-negate-rev N/A

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

      3. 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. lower-/.f64 58.9%

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

      16. lift-+.f64 N/A

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

      17. add-flip N/A

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

      18. metadata-eval N/A

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

      19. lower--.f64 58.9%

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

   6. Applied rewrites 58.9%

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

      1. lift-neg.f64 N/A

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

      2. lift-log.f64 N/A

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

      3. neg-log N/A

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

      4. lift-/.f64 N/A

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

      5. div-flip-rev N/A

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

      6. lift--.f64 N/A

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

      7. div-sub N/A

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

      8. *-inverses N/A

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

      9. metadata-eval N/A

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

      10. distribute-frac-neg N/A

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

      11. lift-/.f64 N/A

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

      12. add-flip-rev N/A

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

      13. lower-log1p.f64 58.4%

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

   8. Applied rewrites 58.4%

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

   if 1e-14 < (/.f64 #s(literal 1 binary64) n) < 5.0000000000000002e147

   1. Initial program 53.5%

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

   if 5.0000000000000002e147 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 53.5%

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

   2. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

   4. Applied rewrites 64.9%

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

   5. Taylor expanded in x around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-log.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 21.9%

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

   7. Applied rewrites 21.9%

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

      1. lift-*.f64 N/A

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

      2. mul-1-neg N/A

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

      3. lift-/.f64 N/A

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

      4. distribute-neg-frac2 N/A

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

      5. lower-/.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. add-flip N/A

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

      9. metadata-eval N/A

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

      10. lower--.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. mul-1-neg N/A

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

      13. lift-/.f64 N/A

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

      14. distribute-neg-frac2 N/A

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

      15. lift-log.f64 N/A

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

      16. lift-/.f64 N/A

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

      17. log-rec N/A

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

      18. lift-log.f64 N/A

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

      19. frac-2neg N/A

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

      20. lower-/.f64 N/A

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

   9. Applied rewrites 21.9%

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

Alternative 2: 94.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{-18}:\\ \;\;\;\;\frac{\frac{1}{x}}{{x}^{\left(\frac{-1}{n}\right)} \cdot n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-14}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(\frac{1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{+147}:\\ \;\;\;\;\frac{n + x}{n} - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\log x}{n} - -1}{\left(-n\right) \cdot x}\\ \end{array} \]

FPCore

(FPCore (x n)
  :precision binary64
  (if (<= (/ 1.0 n) -1e-18)
  (/ (/ 1.0 x) (* (pow x (/ -1.0 n)) n))
  (if (<= (/ 1.0 n) 1e-14)
    (/ (log1p (/ 1.0 x)) n)
    (if (<= (/ 1.0 n) 5e+147)
      (- (/ (+ n x) n) (pow x (/ 1.0 n)))
      (/ (- (/ (log x) n) -1.0) (* (- n) x))))))

C

double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -1e-18) {
		tmp = (1.0 / x) / (pow(x, (-1.0 / n)) * n);
	} else if ((1.0 / n) <= 1e-14) {
		tmp = log1p((1.0 / x)) / n;
	} else if ((1.0 / n) <= 5e+147) {
		tmp = ((n + x) / n) - pow(x, (1.0 / n));
	} else {
		tmp = ((log(x) / n) - -1.0) / (-n * x);
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -1e-18) {
		tmp = (1.0 / x) / (Math.pow(x, (-1.0 / n)) * n);
	} else if ((1.0 / n) <= 1e-14) {
		tmp = Math.log1p((1.0 / x)) / n;
	} else if ((1.0 / n) <= 5e+147) {
		tmp = ((n + x) / n) - Math.pow(x, (1.0 / n));
	} else {
		tmp = ((Math.log(x) / n) - -1.0) / (-n * x);
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if (1.0 / n) <= -1e-18:
		tmp = (1.0 / x) / (math.pow(x, (-1.0 / n)) * n)
	elif (1.0 / n) <= 1e-14:
		tmp = math.log1p((1.0 / x)) / n
	elif (1.0 / n) <= 5e+147:
		tmp = ((n + x) / n) - math.pow(x, (1.0 / n))
	else:
		tmp = ((math.log(x) / n) - -1.0) / (-n * x)
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -1e-18)
		tmp = Float64(Float64(1.0 / x) / Float64((x ^ Float64(-1.0 / n)) * n));
	elseif (Float64(1.0 / n) <= 1e-14)
		tmp = Float64(log1p(Float64(1.0 / x)) / n);
	elseif (Float64(1.0 / n) <= 5e+147)
		tmp = Float64(Float64(Float64(n + x) / n) - (x ^ Float64(1.0 / n)));
	else
		tmp = Float64(Float64(Float64(log(x) / n) - -1.0) / Float64(Float64(-n) * x));
	end
	return tmp
end

Wolfram

code[x_, n_] := If[LessEqual[N[(1.0 / n), $MachinePrecision], -1e-18], 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], 1e-14], N[(N[Log[1 + N[(1.0 / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e+147], N[(N[(N[(n + x), $MachinePrecision] / n), $MachinePrecision] - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Log[x], $MachinePrecision] / n), $MachinePrecision] - -1.0), $MachinePrecision] / N[((-n) * x), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -1.0000000000000001e-18

   1. Initial program 53.5%

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

   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 58.3%

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

   4. Applied rewrites 58.3%

      \[\leadsto \color{blue}{\frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}}{n \cdot x}} \]
   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 59.0%

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

   if -1.0000000000000001e-18 < (/.f64 #s(literal 1 binary64) n) < 1e-14

   1. Initial program 53.5%

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

   2. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-log.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-log.f64 58.9%

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

   4. Applied rewrites 58.9%

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

      1. lift--.f64 N/A

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

      2. sub-negate-rev N/A

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

      3. 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. lower-/.f64 58.9%

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

      16. lift-+.f64 N/A

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

      17. add-flip N/A

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

      18. metadata-eval N/A

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

      19. lower--.f64 58.9%

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

   6. Applied rewrites 58.9%

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

      1. lift-neg.f64 N/A

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

      2. lift-log.f64 N/A

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

      3. neg-log N/A

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

      4. lift-/.f64 N/A

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

      5. div-flip-rev N/A

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

      6. lift--.f64 N/A

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

      7. div-sub N/A

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

      8. *-inverses N/A

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

      9. metadata-eval N/A

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

      10. distribute-frac-neg N/A

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

      11. lift-/.f64 N/A

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

      12. add-flip-rev N/A

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

      13. lower-log1p.f64 58.4%

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

   8. Applied rewrites 58.4%

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

   if 1e-14 < (/.f64 #s(literal 1 binary64) n) < 5.0000000000000002e147

   1. Initial program 53.5%

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

   2. Taylor expanded in x around 0

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

      1. lower-+.f64 N/A

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

      2. lower-/.f64 30.9%

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

   4. Applied rewrites 30.9%

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

      1. lift-+.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. add-to-fraction N/A

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

      4. lower-/.f64 N/A

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

      5. *-lft-identity N/A

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

      6. lower-+.f64 30.9%

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

   6. Applied rewrites 30.9%

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

   if 5.0000000000000002e147 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 53.5%

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

   2. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

   4. Applied rewrites 64.9%

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

   5. Taylor expanded in x around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-log.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 21.9%

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

   7. Applied rewrites 21.9%

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

      1. lift-*.f64 N/A

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

      2. mul-1-neg N/A

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

      3. lift-/.f64 N/A

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

      4. distribute-neg-frac2 N/A

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

      5. lower-/.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. add-flip N/A

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

      9. metadata-eval N/A

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

      10. lower--.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. mul-1-neg N/A

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

      13. lift-/.f64 N/A

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

      14. distribute-neg-frac2 N/A

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

      15. lift-log.f64 N/A

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

      16. lift-/.f64 N/A

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

      17. log-rec N/A

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

      18. lift-log.f64 N/A

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

      19. frac-2neg N/A

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

      20. lower-/.f64 N/A

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

   9. Applied rewrites 21.9%

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

Alternative 3: 94.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -4 \cdot 10^{-26}:\\ \;\;\;\;\frac{t\_0}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-14}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(\frac{1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{+147}:\\ \;\;\;\;\frac{n + x}{n} - t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\log x}{n} - -1}{\left(-n\right) \cdot x}\\ \end{array} \]

FPCore

(FPCore (x n)
  :precision binary64
  (let* ((t_0 (pow x (/ 1.0 n))))
  (if (<= (/ 1.0 n) -4e-26)
    (/ t_0 (* n x))
    (if (<= (/ 1.0 n) 1e-14)
      (/ (log1p (/ 1.0 x)) n)
      (if (<= (/ 1.0 n) 5e+147)
        (- (/ (+ n x) n) t_0)
        (/ (- (/ (log x) n) -1.0) (* (- n) x)))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -4e-26) {
		tmp = t_0 / (n * x);
	} else if ((1.0 / n) <= 1e-14) {
		tmp = log1p((1.0 / x)) / n;
	} else if ((1.0 / n) <= 5e+147) {
		tmp = ((n + x) / n) - t_0;
	} else {
		tmp = ((log(x) / n) - -1.0) / (-n * x);
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -4e-26) {
		tmp = t_0 / (n * x);
	} else if ((1.0 / n) <= 1e-14) {
		tmp = Math.log1p((1.0 / x)) / n;
	} else if ((1.0 / n) <= 5e+147) {
		tmp = ((n + x) / n) - t_0;
	} else {
		tmp = ((Math.log(x) / n) - -1.0) / (-n * x);
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	tmp = 0
	if (1.0 / n) <= -4e-26:
		tmp = t_0 / (n * x)
	elif (1.0 / n) <= 1e-14:
		tmp = math.log1p((1.0 / x)) / n
	elif (1.0 / n) <= 5e+147:
		tmp = ((n + x) / n) - t_0
	else:
		tmp = ((math.log(x) / n) - -1.0) / (-n * x)
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -4e-26)
		tmp = Float64(t_0 / Float64(n * x));
	elseif (Float64(1.0 / n) <= 1e-14)
		tmp = Float64(log1p(Float64(1.0 / x)) / n);
	elseif (Float64(1.0 / n) <= 5e+147)
		tmp = Float64(Float64(Float64(n + x) / n) - t_0);
	else
		tmp = Float64(Float64(Float64(log(x) / n) - -1.0) / Float64(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], -4e-26], N[(t$95$0 / N[(n * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 1e-14], N[(N[Log[1 + N[(1.0 / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e+147], N[(N[(N[(n + x), $MachinePrecision] / n), $MachinePrecision] - t$95$0), $MachinePrecision], N[(N[(N[(N[Log[x], $MachinePrecision] / n), $MachinePrecision] - -1.0), $MachinePrecision] / N[((-n) * x), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -4.0000000000000002e-26

   1. Initial program 53.5%

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

   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 58.3%

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

   4. Applied rewrites 58.3%

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

      1. lift-exp.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. mult-flip N/A

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

      6. mul-1-neg N/A

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

      7. lift-log.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. log-rec N/A

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

      10. lift-log.f64 N/A

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

      11. remove-double-neg N/A

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

      12. lift-log.f64 N/A

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

      13. lift-/.f64 N/A

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

      14. exp-to-pow N/A

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

      15. lower-pow.f64 58.3%

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

   6. Applied rewrites 58.3%

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

   if -4.0000000000000002e-26 < (/.f64 #s(literal 1 binary64) n) < 1e-14

   1. Initial program 53.5%

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

   2. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-log.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-log.f64 58.9%

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

   4. Applied rewrites 58.9%

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

      1. lift--.f64 N/A

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

      2. sub-negate-rev N/A

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

      3. 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. lower-/.f64 58.9%

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

      16. lift-+.f64 N/A

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

      17. add-flip N/A

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

      18. metadata-eval N/A

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

      19. lower--.f64 58.9%

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

   6. Applied rewrites 58.9%

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

      1. lift-neg.f64 N/A

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

      2. lift-log.f64 N/A

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

      3. neg-log N/A

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

      4. lift-/.f64 N/A

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

      5. div-flip-rev N/A

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

      6. lift--.f64 N/A

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

      7. div-sub N/A

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

      8. *-inverses N/A

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

      9. metadata-eval N/A

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

      10. distribute-frac-neg N/A

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

      11. lift-/.f64 N/A

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

      12. add-flip-rev N/A

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

      13. lower-log1p.f64 58.4%

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

   8. Applied rewrites 58.4%

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

   if 1e-14 < (/.f64 #s(literal 1 binary64) n) < 5.0000000000000002e147

   1. Initial program 53.5%

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

   2. Taylor expanded in x around 0

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

      1. lower-+.f64 N/A

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

      2. lower-/.f64 30.9%

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

   4. Applied rewrites 30.9%

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

      1. lift-+.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. add-to-fraction N/A

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

      4. lower-/.f64 N/A

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

      5. *-lft-identity N/A

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

      6. lower-+.f64 30.9%

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

   6. Applied rewrites 30.9%

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

   if 5.0000000000000002e147 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 53.5%

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

   2. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

   4. Applied rewrites 64.9%

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

   5. Taylor expanded in x around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-log.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 21.9%

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

   7. Applied rewrites 21.9%

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

      1. lift-*.f64 N/A

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

      2. mul-1-neg N/A

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

      3. lift-/.f64 N/A

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

      4. distribute-neg-frac2 N/A

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

      5. lower-/.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. add-flip N/A

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

      9. metadata-eval N/A

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

      10. lower--.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. mul-1-neg N/A

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

      13. lift-/.f64 N/A

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

      14. distribute-neg-frac2 N/A

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

      15. lift-log.f64 N/A

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

      16. lift-/.f64 N/A

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

      17. log-rec N/A

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

      18. lift-log.f64 N/A

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

      19. frac-2neg N/A

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

      20. lower-/.f64 N/A

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

   9. Applied rewrites 21.9%

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

Alternative 4: 94.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} t_0 := \frac{\log x}{n}\\ \mathbf{if}\;\frac{1}{n} \leq -4 \cdot 10^{-26}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-14}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(\frac{1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{+147}:\\ \;\;\;\;1 - e^{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0 - -1}{\left(-n\right) \cdot x}\\ \end{array} \]

FPCore

(FPCore (x n)
  :precision binary64
  (let* ((t_0 (/ (log x) n)))
  (if (<= (/ 1.0 n) -4e-26)
    (/ (pow x (/ 1.0 n)) (* n x))
    (if (<= (/ 1.0 n) 1e-14)
      (/ (log1p (/ 1.0 x)) n)
      (if (<= (/ 1.0 n) 5e+147)
        (- 1.0 (exp t_0))
        (/ (- t_0 -1.0) (* (- n) x)))))))

C

double code(double x, double n) {
	double t_0 = log(x) / n;
	double tmp;
	if ((1.0 / n) <= -4e-26) {
		tmp = pow(x, (1.0 / n)) / (n * x);
	} else if ((1.0 / n) <= 1e-14) {
		tmp = log1p((1.0 / x)) / n;
	} else if ((1.0 / n) <= 5e+147) {
		tmp = 1.0 - exp(t_0);
	} else {
		tmp = (t_0 - -1.0) / (-n * x);
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double t_0 = Math.log(x) / n;
	double tmp;
	if ((1.0 / n) <= -4e-26) {
		tmp = Math.pow(x, (1.0 / n)) / (n * x);
	} else if ((1.0 / n) <= 1e-14) {
		tmp = Math.log1p((1.0 / x)) / n;
	} else if ((1.0 / n) <= 5e+147) {
		tmp = 1.0 - Math.exp(t_0);
	} else {
		tmp = (t_0 - -1.0) / (-n * x);
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.log(x) / n
	tmp = 0
	if (1.0 / n) <= -4e-26:
		tmp = math.pow(x, (1.0 / n)) / (n * x)
	elif (1.0 / n) <= 1e-14:
		tmp = math.log1p((1.0 / x)) / n
	elif (1.0 / n) <= 5e+147:
		tmp = 1.0 - math.exp(t_0)
	else:
		tmp = (t_0 - -1.0) / (-n * x)
	return tmp

Julia

function code(x, n)
	t_0 = Float64(log(x) / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -4e-26)
		tmp = Float64((x ^ Float64(1.0 / n)) / Float64(n * x));
	elseif (Float64(1.0 / n) <= 1e-14)
		tmp = Float64(log1p(Float64(1.0 / x)) / n);
	elseif (Float64(1.0 / n) <= 5e+147)
		tmp = Float64(1.0 - exp(t_0));
	else
		tmp = Float64(Float64(t_0 - -1.0) / Float64(Float64(-n) * x));
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(N[Log[x], $MachinePrecision] / n), $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -4e-26], N[(N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision] / N[(n * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 1e-14], N[(N[Log[1 + N[(1.0 / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e+147], N[(1.0 - N[Exp[t$95$0], $MachinePrecision]), $MachinePrecision], N[(N[(t$95$0 - -1.0), $MachinePrecision] / N[((-n) * x), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -4.0000000000000002e-26

   1. Initial program 53.5%

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

   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 58.3%

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

   4. Applied rewrites 58.3%

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

      1. lift-exp.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. mult-flip N/A

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

      6. mul-1-neg N/A

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

      7. lift-log.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. log-rec N/A

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

      10. lift-log.f64 N/A

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

      11. remove-double-neg N/A

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

      12. lift-log.f64 N/A

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

      13. lift-/.f64 N/A

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

      14. exp-to-pow N/A

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

      15. lower-pow.f64 58.3%

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

   6. Applied rewrites 58.3%

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

   if -4.0000000000000002e-26 < (/.f64 #s(literal 1 binary64) n) < 1e-14

   1. Initial program 53.5%

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

   2. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-log.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-log.f64 58.9%

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

   4. Applied rewrites 58.9%

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

      1. lift--.f64 N/A

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

      2. sub-negate-rev N/A

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

      3. 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. lower-/.f64 58.9%

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

      16. lift-+.f64 N/A

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

      17. add-flip N/A

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

      18. metadata-eval N/A

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

      19. lower--.f64 58.9%

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

   6. Applied rewrites 58.9%

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

      1. lift-neg.f64 N/A

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

      2. lift-log.f64 N/A

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

      3. neg-log N/A

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

      4. lift-/.f64 N/A

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

      5. div-flip-rev N/A

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

      6. lift--.f64 N/A

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

      7. div-sub N/A

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

      8. *-inverses N/A

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

      9. metadata-eval N/A

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

      10. distribute-frac-neg N/A

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

      11. lift-/.f64 N/A

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

      12. add-flip-rev N/A

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

      13. lower-log1p.f64 58.4%

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

   8. Applied rewrites 58.4%

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

   if 1e-14 < (/.f64 #s(literal 1 binary64) n) < 5.0000000000000002e147

   1. Initial program 53.5%

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

   2. Taylor expanded in x around 0

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

      1. lower--.f64 N/A

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

      2. lower-exp.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-log.f64 38.9%

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

   4. Applied rewrites 38.9%

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

   if 5.0000000000000002e147 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 53.5%

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

   2. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

   4. Applied rewrites 64.9%

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

   5. Taylor expanded in x around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-log.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 21.9%

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

   7. Applied rewrites 21.9%

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

      1. lift-*.f64 N/A

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

      2. mul-1-neg N/A

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

      3. lift-/.f64 N/A

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

      4. distribute-neg-frac2 N/A

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

      5. lower-/.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. add-flip N/A

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

      9. metadata-eval N/A

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

      10. lower--.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. mul-1-neg N/A

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

      13. lift-/.f64 N/A

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

      14. distribute-neg-frac2 N/A

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

      15. lift-log.f64 N/A

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

      16. lift-/.f64 N/A

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

      17. log-rec N/A

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

      18. lift-log.f64 N/A

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

      19. frac-2neg N/A

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

      20. lower-/.f64 N/A

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

   9. Applied rewrites 21.9%

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

Alternative 5: 94.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{-18}:\\ \;\;\;\;\frac{\frac{1}{x}}{{x}^{\left(\frac{-1}{n}\right)} \cdot n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-5}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(\frac{1}{x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + x \cdot \mathsf{fma}\left(x, 0.5 \cdot \frac{1}{{n}^{2}} - 0.5 \cdot \frac{1}{n}, \frac{1}{n}\right)\right) - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \]

FPCore

(FPCore (x n)
  :precision binary64
  (if (<= (/ 1.0 n) -1e-18)
  (/ (/ 1.0 x) (* (pow x (/ -1.0 n)) n))
  (if (<= (/ 1.0 n) 2e-5)
    (/ (log1p (/ 1.0 x)) n)
    (-
     (+
      1.0
      (*
       x
       (fma
        x
        (- (* 0.5 (/ 1.0 (pow n 2.0))) (* 0.5 (/ 1.0 n)))
        (/ 1.0 n))))
     (pow x (/ 1.0 n))))))

C

double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -1e-18) {
		tmp = (1.0 / x) / (pow(x, (-1.0 / n)) * n);
	} else if ((1.0 / n) <= 2e-5) {
		tmp = log1p((1.0 / x)) / n;
	} else {
		tmp = (1.0 + (x * fma(x, ((0.5 * (1.0 / pow(n, 2.0))) - (0.5 * (1.0 / n))), (1.0 / n)))) - pow(x, (1.0 / n));
	}
	return tmp;
}

Julia

function code(x, n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -1e-18)
		tmp = Float64(Float64(1.0 / x) / Float64((x ^ Float64(-1.0 / n)) * n));
	elseif (Float64(1.0 / n) <= 2e-5)
		tmp = Float64(log1p(Float64(1.0 / x)) / n);
	else
		tmp = Float64(Float64(1.0 + Float64(x * fma(x, Float64(Float64(0.5 * Float64(1.0 / (n ^ 2.0))) - Float64(0.5 * Float64(1.0 / n))), Float64(1.0 / n)))) - (x ^ Float64(1.0 / n)));
	end
	return tmp
end

Wolfram

code[x_, n_] := If[LessEqual[N[(1.0 / n), $MachinePrecision], -1e-18], 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], 2e-5], N[(N[Log[1 + N[(1.0 / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], N[(N[(1.0 + N[(x * N[(x * N[(N[(0.5 * N[(1.0 / N[Power[n, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(0.5 * N[(1.0 / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -1.0000000000000001e-18

   1. Initial program 53.5%

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

   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 58.3%

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

   4. Applied rewrites 58.3%

      \[\leadsto \color{blue}{\frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}}{n \cdot x}} \]
   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 59.0%

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

   if -1.0000000000000001e-18 < (/.f64 #s(literal 1 binary64) n) < 2.0000000000000002e-5

   1. Initial program 53.5%

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

   2. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-log.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-log.f64 58.9%

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

   4. Applied rewrites 58.9%

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

      1. lift--.f64 N/A

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

      2. sub-negate-rev N/A

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

      3. 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. lower-/.f64 58.9%

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

      16. lift-+.f64 N/A

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

      17. add-flip N/A

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

      18. metadata-eval N/A

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

      19. lower--.f64 58.9%

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

   6. Applied rewrites 58.9%

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

      1. lift-neg.f64 N/A

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

      2. lift-log.f64 N/A

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

      3. neg-log N/A

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

      4. lift-/.f64 N/A

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

      5. div-flip-rev N/A

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

      6. lift--.f64 N/A

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

      7. div-sub N/A

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

      8. *-inverses N/A

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

      9. metadata-eval N/A

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

      10. distribute-frac-neg N/A

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

      11. lift-/.f64 N/A

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

      12. add-flip-rev N/A

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

      13. lower-log1p.f64 58.4%

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

   8. Applied rewrites 58.4%

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

   if 2.0000000000000002e-5 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 53.5%

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

   2. Taylor expanded in x around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 22.8%

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

   4. Applied rewrites 22.8%

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

Alternative 6: 86.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} t_0 := \frac{\log x}{n}\\ t_1 := 1 - e^{t\_0}\\ \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{+29}:\\ \;\;\;\;\frac{x \cdot \log \left(\frac{x - -1}{x}\right)}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq -5000000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-14}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(\frac{1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{+147}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0 - -1}{\left(-n\right) \cdot x}\\ \end{array} \]

FPCore

(FPCore (x n)
  :precision binary64
  (let* ((t_0 (/ (log x) n)) (t_1 (- 1.0 (exp t_0))))
  (if (<= (/ 1.0 n) -5e+29)
    (/ (* x (log (/ (- x -1.0) x))) (* n x))
    (if (<= (/ 1.0 n) -5000000.0)
      t_1
      (if (<= (/ 1.0 n) 1e-14)
        (/ (log1p (/ 1.0 x)) n)
        (if (<= (/ 1.0 n) 5e+147)
          t_1
          (/ (- t_0 -1.0) (* (- n) x))))))))

C

double code(double x, double n) {
	double t_0 = log(x) / n;
	double t_1 = 1.0 - exp(t_0);
	double tmp;
	if ((1.0 / n) <= -5e+29) {
		tmp = (x * log(((x - -1.0) / x))) / (n * x);
	} else if ((1.0 / n) <= -5000000.0) {
		tmp = t_1;
	} else if ((1.0 / n) <= 1e-14) {
		tmp = log1p((1.0 / x)) / n;
	} else if ((1.0 / n) <= 5e+147) {
		tmp = t_1;
	} else {
		tmp = (t_0 - -1.0) / (-n * x);
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double t_0 = Math.log(x) / n;
	double t_1 = 1.0 - Math.exp(t_0);
	double tmp;
	if ((1.0 / n) <= -5e+29) {
		tmp = (x * Math.log(((x - -1.0) / x))) / (n * x);
	} else if ((1.0 / n) <= -5000000.0) {
		tmp = t_1;
	} else if ((1.0 / n) <= 1e-14) {
		tmp = Math.log1p((1.0 / x)) / n;
	} else if ((1.0 / n) <= 5e+147) {
		tmp = t_1;
	} else {
		tmp = (t_0 - -1.0) / (-n * x);
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.log(x) / n
	t_1 = 1.0 - math.exp(t_0)
	tmp = 0
	if (1.0 / n) <= -5e+29:
		tmp = (x * math.log(((x - -1.0) / x))) / (n * x)
	elif (1.0 / n) <= -5000000.0:
		tmp = t_1
	elif (1.0 / n) <= 1e-14:
		tmp = math.log1p((1.0 / x)) / n
	elif (1.0 / n) <= 5e+147:
		tmp = t_1
	else:
		tmp = (t_0 - -1.0) / (-n * x)
	return tmp

Julia

function code(x, n)
	t_0 = Float64(log(x) / n)
	t_1 = Float64(1.0 - exp(t_0))
	tmp = 0.0
	if (Float64(1.0 / n) <= -5e+29)
		tmp = Float64(Float64(x * log(Float64(Float64(x - -1.0) / x))) / Float64(n * x));
	elseif (Float64(1.0 / n) <= -5000000.0)
		tmp = t_1;
	elseif (Float64(1.0 / n) <= 1e-14)
		tmp = Float64(log1p(Float64(1.0 / x)) / n);
	elseif (Float64(1.0 / n) <= 5e+147)
		tmp = t_1;
	else
		tmp = Float64(Float64(t_0 - -1.0) / Float64(Float64(-n) * x));
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(N[Log[x], $MachinePrecision] / n), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 - N[Exp[t$95$0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -5e+29], N[(N[(x * N[Log[N[(N[(x - -1.0), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(n * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], -5000000.0], t$95$1, If[LessEqual[N[(1.0 / n), $MachinePrecision], 1e-14], N[(N[Log[1 + N[(1.0 / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e+147], t$95$1, N[(N[(t$95$0 - -1.0), $MachinePrecision] / N[((-n) * x), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -5.0000000000000001e29

   1. Initial program 53.5%

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

   2. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-log.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-log.f64 58.9%

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

   4. Applied rewrites 58.9%

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

      1. lift--.f64 N/A

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

      2. sub-negate-rev N/A

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

      3. 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. lower-/.f64 58.9%

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

      16. lift-+.f64 N/A

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

      17. add-flip N/A

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

      18. metadata-eval N/A

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

      19. lower--.f64 58.9%

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

   6. Applied rewrites 58.9%

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

      1. lift-neg.f64 N/A

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

      2. lift-log.f64 N/A

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

      3. neg-log N/A

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

      4. lift-/.f64 N/A

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

      5. div-flip-rev N/A

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

      6. lift-/.f64 N/A

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

      7. lift-log.f64 58.9%

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

      8. lower-/.f64 N/A

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

      9. *-lft-identity N/A

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

      10. associate-/l* N/A

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

      11. *-inverses N/A

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

      12. frac-times N/A

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

      13. *-commutative N/A

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

      14. lift-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lower-*.f64 67.5%

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

   8. Applied rewrites 67.5%

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

   if -5.0000000000000001e29 < (/.f64 #s(literal 1 binary64) n) < -5e6 or 1e-14 < (/.f64 #s(literal 1 binary64) n) < 5.0000000000000002e147

   1. Initial program 53.5%

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

   2. Taylor expanded in x around 0

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

      1. lower--.f64 N/A

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

      2. lower-exp.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-log.f64 38.9%

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

   4. Applied rewrites 38.9%

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

   if -5e6 < (/.f64 #s(literal 1 binary64) n) < 1e-14

   1. Initial program 53.5%

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

   2. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-log.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-log.f64 58.9%

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

   4. Applied rewrites 58.9%

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

      1. lift--.f64 N/A

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

      2. sub-negate-rev N/A

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

      3. 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. lower-/.f64 58.9%

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

      16. lift-+.f64 N/A

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

      17. add-flip N/A

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

      18. metadata-eval N/A

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

      19. lower--.f64 58.9%

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

   6. Applied rewrites 58.9%

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

      1. lift-neg.f64 N/A

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

      2. lift-log.f64 N/A

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

      3. neg-log N/A

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

      4. lift-/.f64 N/A

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

      5. div-flip-rev N/A

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

      6. lift--.f64 N/A

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

      7. div-sub N/A

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

      8. *-inverses N/A

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

      9. metadata-eval N/A

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

      10. distribute-frac-neg N/A

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

      11. lift-/.f64 N/A

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

      12. add-flip-rev N/A

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

      13. lower-log1p.f64 58.4%

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

   8. Applied rewrites 58.4%

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

   if 5.0000000000000002e147 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 53.5%

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

   2. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

   4. Applied rewrites 64.9%

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

   5. Taylor expanded in x around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-log.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 21.9%

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

   7. Applied rewrites 21.9%

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

      1. lift-*.f64 N/A

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

      2. mul-1-neg N/A

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

      3. lift-/.f64 N/A

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

      4. distribute-neg-frac2 N/A

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

      5. lower-/.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. add-flip N/A

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

      9. metadata-eval N/A

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

      10. lower--.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. mul-1-neg N/A

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

      13. lift-/.f64 N/A

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

      14. distribute-neg-frac2 N/A

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

      15. lift-log.f64 N/A

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

      16. lift-/.f64 N/A

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

      17. log-rec N/A

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

      18. lift-log.f64 N/A

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

      19. frac-2neg N/A

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

      20. lower-/.f64 N/A

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

   9. Applied rewrites 21.9%

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

Alternative 7: 81.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{+15}:\\ \;\;\;\;\frac{x \cdot \log \left(\frac{x - -1}{x}\right)}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{+139}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(\frac{1}{x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\log x}{n} - -1}{n}}{-x}\\ \end{array} \]

FPCore

(FPCore (x n)
  :precision binary64
  (if (<= (/ 1.0 n) -2e+15)
  (/ (* x (log (/ (- x -1.0) x))) (* n x))
  (if (<= (/ 1.0 n) 2e+139)
    (/ (log1p (/ 1.0 x)) n)
    (/ (/ (- (/ (log x) n) -1.0) n) (- x)))))

C

double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -2e+15) {
		tmp = (x * log(((x - -1.0) / x))) / (n * x);
	} else if ((1.0 / n) <= 2e+139) {
		tmp = log1p((1.0 / x)) / n;
	} else {
		tmp = (((log(x) / n) - -1.0) / n) / -x;
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -2e+15) {
		tmp = (x * Math.log(((x - -1.0) / x))) / (n * x);
	} else if ((1.0 / n) <= 2e+139) {
		tmp = Math.log1p((1.0 / x)) / n;
	} else {
		tmp = (((Math.log(x) / n) - -1.0) / n) / -x;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if (1.0 / n) <= -2e+15:
		tmp = (x * math.log(((x - -1.0) / x))) / (n * x)
	elif (1.0 / n) <= 2e+139:
		tmp = math.log1p((1.0 / x)) / n
	else:
		tmp = (((math.log(x) / n) - -1.0) / n) / -x
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -2e+15)
		tmp = Float64(Float64(x * log(Float64(Float64(x - -1.0) / x))) / Float64(n * x));
	elseif (Float64(1.0 / n) <= 2e+139)
		tmp = Float64(log1p(Float64(1.0 / x)) / n);
	else
		tmp = Float64(Float64(Float64(Float64(log(x) / n) - -1.0) / n) / Float64(-x));
	end
	return tmp
end

Wolfram

code[x_, n_] := If[LessEqual[N[(1.0 / n), $MachinePrecision], -2e+15], N[(N[(x * N[Log[N[(N[(x - -1.0), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(n * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e+139], N[(N[Log[1 + N[(1.0 / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], N[(N[(N[(N[(N[Log[x], $MachinePrecision] / n), $MachinePrecision] - -1.0), $MachinePrecision] / n), $MachinePrecision] / (-x)), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -2e15

   1. Initial program 53.5%

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

   2. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-log.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-log.f64 58.9%

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

   4. Applied rewrites 58.9%

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

      1. lift--.f64 N/A

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

      2. sub-negate-rev N/A

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

      3. 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. lower-/.f64 58.9%

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

      16. lift-+.f64 N/A

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

      17. add-flip N/A

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

      18. metadata-eval N/A

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

      19. lower--.f64 58.9%

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

   6. Applied rewrites 58.9%

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

      1. lift-neg.f64 N/A

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

      2. lift-log.f64 N/A

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

      3. neg-log N/A

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

      4. lift-/.f64 N/A

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

      5. div-flip-rev N/A

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

      6. lift-/.f64 N/A

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

      7. lift-log.f64 58.9%

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

      8. lower-/.f64 N/A

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

      9. *-lft-identity N/A

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

      10. associate-/l* N/A

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

      11. *-inverses N/A

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

      12. frac-times N/A

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

      13. *-commutative N/A

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

      14. lift-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lower-*.f64 67.5%

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

   8. Applied rewrites 67.5%

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

   if -2e15 < (/.f64 #s(literal 1 binary64) n) < 2.0000000000000001e139

   1. Initial program 53.5%

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

   2. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-log.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-log.f64 58.9%

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

   4. Applied rewrites 58.9%

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

      1. lift--.f64 N/A

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

      2. sub-negate-rev N/A

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

      3. 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. lower-/.f64 58.9%

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

      16. lift-+.f64 N/A

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

      17. add-flip N/A

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

      18. metadata-eval N/A

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

      19. lower--.f64 58.9%

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

   6. Applied rewrites 58.9%

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

      1. lift-neg.f64 N/A

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

      2. lift-log.f64 N/A

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

      3. neg-log N/A

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

      4. lift-/.f64 N/A

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

      5. div-flip-rev N/A

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

      6. lift--.f64 N/A

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

      7. div-sub N/A

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

      8. *-inverses N/A

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

      9. metadata-eval N/A

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

      10. distribute-frac-neg N/A

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

      11. lift-/.f64 N/A

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

      12. add-flip-rev N/A

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

      13. lower-log1p.f64 58.4%

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

   8. Applied rewrites 58.4%

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

   if 2.0000000000000001e139 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 53.5%

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

   2. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

   4. Applied rewrites 64.9%

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

   5. Taylor expanded in x around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-log.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 21.9%

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

   7. Applied rewrites 21.9%

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

      1. lift-*.f64 N/A

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

      2. mul-1-neg N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/r* N/A

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

      6. distribute-neg-frac2 N/A

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

      7. lower-/.f64 N/A

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

   9. Applied rewrites 21.9%

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

Alternative 8: 76.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{+150}:\\ \;\;\;\;\frac{\log \left(1 + x\right) - \log x}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq -5000000:\\ \;\;\;\;\frac{x \cdot 1}{x \cdot \left(n \cdot x\right)}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{+139}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(\frac{1}{x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\log x}{n} - -1}{n}}{-x}\\ \end{array} \]

FPCore

(FPCore (x n)
  :precision binary64
  (if (<= (/ 1.0 n) -1e+150)
  (/ (- (log (+ 1.0 x)) (log x)) n)
  (if (<= (/ 1.0 n) -5000000.0)
    (/ (* x 1.0) (* x (* n x)))
    (if (<= (/ 1.0 n) 2e+139)
      (/ (log1p (/ 1.0 x)) n)
      (/ (/ (- (/ (log x) n) -1.0) n) (- x))))))

C

double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -1e+150) {
		tmp = (log((1.0 + x)) - log(x)) / n;
	} else if ((1.0 / n) <= -5000000.0) {
		tmp = (x * 1.0) / (x * (n * x));
	} else if ((1.0 / n) <= 2e+139) {
		tmp = log1p((1.0 / x)) / n;
	} else {
		tmp = (((log(x) / n) - -1.0) / n) / -x;
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -1e+150) {
		tmp = (Math.log((1.0 + x)) - Math.log(x)) / n;
	} else if ((1.0 / n) <= -5000000.0) {
		tmp = (x * 1.0) / (x * (n * x));
	} else if ((1.0 / n) <= 2e+139) {
		tmp = Math.log1p((1.0 / x)) / n;
	} else {
		tmp = (((Math.log(x) / n) - -1.0) / n) / -x;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if (1.0 / n) <= -1e+150:
		tmp = (math.log((1.0 + x)) - math.log(x)) / n
	elif (1.0 / n) <= -5000000.0:
		tmp = (x * 1.0) / (x * (n * x))
	elif (1.0 / n) <= 2e+139:
		tmp = math.log1p((1.0 / x)) / n
	else:
		tmp = (((math.log(x) / n) - -1.0) / n) / -x
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -1e+150)
		tmp = Float64(Float64(log(Float64(1.0 + x)) - log(x)) / n);
	elseif (Float64(1.0 / n) <= -5000000.0)
		tmp = Float64(Float64(x * 1.0) / Float64(x * Float64(n * x)));
	elseif (Float64(1.0 / n) <= 2e+139)
		tmp = Float64(log1p(Float64(1.0 / x)) / n);
	else
		tmp = Float64(Float64(Float64(Float64(log(x) / n) - -1.0) / n) / Float64(-x));
	end
	return tmp
end

Wolfram

code[x_, n_] := If[LessEqual[N[(1.0 / n), $MachinePrecision], -1e+150], N[(N[(N[Log[N[(1.0 + x), $MachinePrecision]], $MachinePrecision] - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], -5000000.0], N[(N[(x * 1.0), $MachinePrecision] / N[(x * N[(n * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e+139], N[(N[Log[1 + N[(1.0 / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], N[(N[(N[(N[(N[Log[x], $MachinePrecision] / n), $MachinePrecision] - -1.0), $MachinePrecision] / n), $MachinePrecision] / (-x)), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -9.9999999999999998e149

   1. Initial program 53.5%

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

   2. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-log.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-log.f64 58.9%

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

   4. Applied rewrites 58.9%

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

   if -9.9999999999999998e149 < (/.f64 #s(literal 1 binary64) n) < -5e6

   1. Initial program 53.5%

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

   2. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-log.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-log.f64 58.9%

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

   4. Applied rewrites 58.9%

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

   5. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 40.7%

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

   7. Applied rewrites 40.7%

      \[\leadsto \frac{1}{\color{blue}{n \cdot x}} \]
   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 41.3%

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

   9. Applied rewrites 41.3%

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

       1. lift-/.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. associate-/l/ N/A

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

       4. lift-*.f64 N/A

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

       5. mult-flip N/A

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

       6. *-inverses N/A

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

       7. frac-times N/A

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

       8. lower-/.f64 N/A

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

       9. lower-*.f64 N/A

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

       10. lower-*.f64 41.5%

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

   11. Applied rewrites 41.5%

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

   if -5e6 < (/.f64 #s(literal 1 binary64) n) < 2.0000000000000001e139

   1. Initial program 53.5%

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

   2. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-log.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-log.f64 58.9%

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

   4. Applied rewrites 58.9%

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

      1. lift--.f64 N/A

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

      2. sub-negate-rev N/A

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

      3. 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. lower-/.f64 58.9%

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

      16. lift-+.f64 N/A

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

      17. add-flip N/A

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

      18. metadata-eval N/A

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

      19. lower--.f64 58.9%

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

   6. Applied rewrites 58.9%

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

      1. lift-neg.f64 N/A

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

      2. lift-log.f64 N/A

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

      3. neg-log N/A

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

      4. lift-/.f64 N/A

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

      5. div-flip-rev N/A

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

      6. lift--.f64 N/A

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

      7. div-sub N/A

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

      8. *-inverses N/A

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

      9. metadata-eval N/A

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

      10. distribute-frac-neg N/A

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

      11. lift-/.f64 N/A

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

      12. add-flip-rev N/A

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

      13. lower-log1p.f64 58.4%

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

   8. Applied rewrites 58.4%

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

   if 2.0000000000000001e139 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 53.5%

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

   2. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

   4. Applied rewrites 64.9%

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

   5. Taylor expanded in x around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-log.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 21.9%

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

   7. Applied rewrites 21.9%

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

      1. lift-*.f64 N/A

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

      2. mul-1-neg N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/r* N/A

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

      6. distribute-neg-frac2 N/A

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

      7. lower-/.f64 N/A

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

   9. Applied rewrites 21.9%

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

Alternative 9: 76.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{+150}:\\ \;\;\;\;\frac{\log \left(\frac{x - -1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq -5000000:\\ \;\;\;\;\frac{x \cdot 1}{x \cdot \left(n \cdot x\right)}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{+139}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(\frac{1}{x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\log x}{n} - -1}{\left(-n\right) \cdot x}\\ \end{array} \]

FPCore

(FPCore (x n)
  :precision binary64
  (if (<= (/ 1.0 n) -1e+150)
  (/ (log (/ (- x -1.0) x)) n)
  (if (<= (/ 1.0 n) -5000000.0)
    (/ (* x 1.0) (* x (* n x)))
    (if (<= (/ 1.0 n) 2e+139)
      (/ (log1p (/ 1.0 x)) n)
      (/ (- (/ (log x) n) -1.0) (* (- n) x))))))

C

double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -1e+150) {
		tmp = log(((x - -1.0) / x)) / n;
	} else if ((1.0 / n) <= -5000000.0) {
		tmp = (x * 1.0) / (x * (n * x));
	} else if ((1.0 / n) <= 2e+139) {
		tmp = log1p((1.0 / x)) / n;
	} else {
		tmp = ((log(x) / n) - -1.0) / (-n * x);
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -1e+150) {
		tmp = Math.log(((x - -1.0) / x)) / n;
	} else if ((1.0 / n) <= -5000000.0) {
		tmp = (x * 1.0) / (x * (n * x));
	} else if ((1.0 / n) <= 2e+139) {
		tmp = Math.log1p((1.0 / x)) / n;
	} else {
		tmp = ((Math.log(x) / n) - -1.0) / (-n * x);
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if (1.0 / n) <= -1e+150:
		tmp = math.log(((x - -1.0) / x)) / n
	elif (1.0 / n) <= -5000000.0:
		tmp = (x * 1.0) / (x * (n * x))
	elif (1.0 / n) <= 2e+139:
		tmp = math.log1p((1.0 / x)) / n
	else:
		tmp = ((math.log(x) / n) - -1.0) / (-n * x)
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -1e+150)
		tmp = Float64(log(Float64(Float64(x - -1.0) / x)) / n);
	elseif (Float64(1.0 / n) <= -5000000.0)
		tmp = Float64(Float64(x * 1.0) / Float64(x * Float64(n * x)));
	elseif (Float64(1.0 / n) <= 2e+139)
		tmp = Float64(log1p(Float64(1.0 / x)) / n);
	else
		tmp = Float64(Float64(Float64(log(x) / n) - -1.0) / Float64(Float64(-n) * x));
	end
	return tmp
end

Wolfram

code[x_, n_] := If[LessEqual[N[(1.0 / n), $MachinePrecision], -1e+150], N[(N[Log[N[(N[(x - -1.0), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], -5000000.0], N[(N[(x * 1.0), $MachinePrecision] / N[(x * N[(n * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e+139], N[(N[Log[1 + N[(1.0 / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], N[(N[(N[(N[Log[x], $MachinePrecision] / n), $MachinePrecision] - -1.0), $MachinePrecision] / N[((-n) * x), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -9.9999999999999998e149

   1. Initial program 53.5%

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

   2. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-log.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-log.f64 58.9%

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

   4. Applied rewrites 58.9%

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

      1. lift--.f64 N/A

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

      2. lift-log.f64 N/A

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

      3. lift-log.f64 N/A

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

      4. diff-log N/A

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

      5. lower-log.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. lift-+.f64 N/A

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

      9. lower-/.f64 58.9%

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

      10. lift-+.f64 N/A

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

      11. add-flip N/A

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

      12. metadata-eval N/A

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

      13. lower--.f64 58.9%

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

   6. Applied rewrites 58.9%

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

   if -9.9999999999999998e149 < (/.f64 #s(literal 1 binary64) n) < -5e6

   1. Initial program 53.5%

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

   2. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-log.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-log.f64 58.9%

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

   4. Applied rewrites 58.9%

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

   5. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 40.7%

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

   7. Applied rewrites 40.7%

      \[\leadsto \frac{1}{\color{blue}{n \cdot x}} \]
   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 41.3%

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

   9. Applied rewrites 41.3%

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

       1. lift-/.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. associate-/l/ N/A

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

       4. lift-*.f64 N/A

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

       5. mult-flip N/A

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

       6. *-inverses N/A

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

       7. frac-times N/A

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

       8. lower-/.f64 N/A

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

       9. lower-*.f64 N/A

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

       10. lower-*.f64 41.5%

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

   11. Applied rewrites 41.5%

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

   if -5e6 < (/.f64 #s(literal 1 binary64) n) < 2.0000000000000001e139

   1. Initial program 53.5%

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

   2. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-log.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-log.f64 58.9%

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

   4. Applied rewrites 58.9%

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

      1. lift--.f64 N/A

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

      2. sub-negate-rev N/A

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

      3. 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. lower-/.f64 58.9%

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

      16. lift-+.f64 N/A

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

      17. add-flip N/A

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

      18. metadata-eval N/A

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

      19. lower--.f64 58.9%

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

   6. Applied rewrites 58.9%

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

      1. lift-neg.f64 N/A

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

      2. lift-log.f64 N/A

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

      3. neg-log N/A

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

      4. lift-/.f64 N/A

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

      5. div-flip-rev N/A

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

      6. lift--.f64 N/A

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

      7. div-sub N/A

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

      8. *-inverses N/A

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

      9. metadata-eval N/A

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

      10. distribute-frac-neg N/A

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

      11. lift-/.f64 N/A

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

      12. add-flip-rev N/A

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

      13. lower-log1p.f64 58.4%

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

   8. Applied rewrites 58.4%

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

   if 2.0000000000000001e139 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 53.5%

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

   2. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

   4. Applied rewrites 64.9%

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

   5. Taylor expanded in x around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-log.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 21.9%

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

   7. Applied rewrites 21.9%

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

      1. lift-*.f64 N/A

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

      2. mul-1-neg N/A

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

      3. lift-/.f64 N/A

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

      4. distribute-neg-frac2 N/A

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

      5. lower-/.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. add-flip N/A

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

      9. metadata-eval N/A

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

      10. lower--.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. mul-1-neg N/A

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

      13. lift-/.f64 N/A

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

      14. distribute-neg-frac2 N/A

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

      15. lift-log.f64 N/A

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

      16. lift-/.f64 N/A

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

      17. log-rec N/A

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

      18. lift-log.f64 N/A

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

      19. frac-2neg N/A

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

      20. lower-/.f64 N/A

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

   9. Applied rewrites 21.9%

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

Alternative 10: 76.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{+150}:\\ \;\;\;\;\frac{\log \left(\frac{x - -1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq -5000000:\\ \;\;\;\;\frac{x \cdot 1}{x \cdot \left(n \cdot x\right)}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{+139}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(\frac{1}{x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\log x}{n} - -1}{n}}{-x}\\ \end{array} \]

FPCore

(FPCore (x n)
  :precision binary64
  (if (<= (/ 1.0 n) -1e+150)
  (/ (log (/ (- x -1.0) x)) n)
  (if (<= (/ 1.0 n) -5000000.0)
    (/ (* x 1.0) (* x (* n x)))
    (if (<= (/ 1.0 n) 2e+139)
      (/ (log1p (/ 1.0 x)) n)
      (/ (/ (- (/ (log x) n) -1.0) n) (- x))))))

C

double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -1e+150) {
		tmp = log(((x - -1.0) / x)) / n;
	} else if ((1.0 / n) <= -5000000.0) {
		tmp = (x * 1.0) / (x * (n * x));
	} else if ((1.0 / n) <= 2e+139) {
		tmp = log1p((1.0 / x)) / n;
	} else {
		tmp = (((log(x) / n) - -1.0) / n) / -x;
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -1e+150) {
		tmp = Math.log(((x - -1.0) / x)) / n;
	} else if ((1.0 / n) <= -5000000.0) {
		tmp = (x * 1.0) / (x * (n * x));
	} else if ((1.0 / n) <= 2e+139) {
		tmp = Math.log1p((1.0 / x)) / n;
	} else {
		tmp = (((Math.log(x) / n) - -1.0) / n) / -x;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if (1.0 / n) <= -1e+150:
		tmp = math.log(((x - -1.0) / x)) / n
	elif (1.0 / n) <= -5000000.0:
		tmp = (x * 1.0) / (x * (n * x))
	elif (1.0 / n) <= 2e+139:
		tmp = math.log1p((1.0 / x)) / n
	else:
		tmp = (((math.log(x) / n) - -1.0) / n) / -x
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -1e+150)
		tmp = Float64(log(Float64(Float64(x - -1.0) / x)) / n);
	elseif (Float64(1.0 / n) <= -5000000.0)
		tmp = Float64(Float64(x * 1.0) / Float64(x * Float64(n * x)));
	elseif (Float64(1.0 / n) <= 2e+139)
		tmp = Float64(log1p(Float64(1.0 / x)) / n);
	else
		tmp = Float64(Float64(Float64(Float64(log(x) / n) - -1.0) / n) / Float64(-x));
	end
	return tmp
end

Wolfram

code[x_, n_] := If[LessEqual[N[(1.0 / n), $MachinePrecision], -1e+150], N[(N[Log[N[(N[(x - -1.0), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], -5000000.0], N[(N[(x * 1.0), $MachinePrecision] / N[(x * N[(n * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e+139], N[(N[Log[1 + N[(1.0 / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], N[(N[(N[(N[(N[Log[x], $MachinePrecision] / n), $MachinePrecision] - -1.0), $MachinePrecision] / n), $MachinePrecision] / (-x)), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -9.9999999999999998e149

   1. Initial program 53.5%

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

   2. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-log.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-log.f64 58.9%

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

   4. Applied rewrites 58.9%

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

      1. lift--.f64 N/A

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

      2. lift-log.f64 N/A

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

      3. lift-log.f64 N/A

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

      4. diff-log N/A

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

      5. lower-log.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. lift-+.f64 N/A

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

      9. lower-/.f64 58.9%

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

      10. lift-+.f64 N/A

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

      11. add-flip N/A

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

      12. metadata-eval N/A

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

      13. lower--.f64 58.9%

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

   6. Applied rewrites 58.9%

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

   if -9.9999999999999998e149 < (/.f64 #s(literal 1 binary64) n) < -5e6

   1. Initial program 53.5%

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

   2. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-log.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-log.f64 58.9%

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

   4. Applied rewrites 58.9%

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

   5. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 40.7%

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

   7. Applied rewrites 40.7%

      \[\leadsto \frac{1}{\color{blue}{n \cdot x}} \]
   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 41.3%

         \[\leadsto \frac{\frac{1}{n}}{x} \]

   9. Applied rewrites 41.3%

      \[\leadsto \frac{\frac{1}{n}}{x} \]
   10. Step-by-step derivation

       1. lift-/.f64 N/A

          \[\leadsto \frac{\frac{1}{n}}{x} \]

       2. lift-/.f64 N/A

          \[\leadsto \frac{\frac{1}{n}}{x} \]

       3. associate-/l/ N/A

          \[\leadsto \frac{1}{n \cdot \color{blue}{x}} \]

       4. lift-*.f64 N/A

          \[\leadsto \frac{1}{n \cdot x} \]

       5. mult-flip N/A

          \[\leadsto 1 \cdot \frac{1}{\color{blue}{n \cdot x}} \]

       6. *-inverses N/A

          \[\leadsto \frac{x}{x} \cdot \frac{1}{\color{blue}{n} \cdot x} \]

       7. frac-times N/A

          \[\leadsto \frac{x \cdot 1}{x \cdot \color{blue}{\left(n \cdot x\right)}} \]

       8. lower-/.f64 N/A

          \[\leadsto \frac{x \cdot 1}{x \cdot \color{blue}{\left(n \cdot x\right)}} \]

       9. lower-*.f64 N/A

          \[\leadsto \frac{x \cdot 1}{x \cdot \left(\color{blue}{n} \cdot x\right)} \]

       10. lower-*.f64 41.5%

           \[\leadsto \frac{x \cdot 1}{x \cdot \left(n \cdot \color{blue}{x}\right)} \]

   11. Applied rewrites 41.5%

       \[\leadsto \frac{x \cdot 1}{x \cdot \color{blue}{\left(n \cdot x\right)}} \]

   if -5e6 < (/.f64 #s(literal 1 binary64) n) < 2.0000000000000001e139

   1. Initial program 53.5%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]

   2. Taylor expanded in n around inf

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{\color{blue}{n}} \]

      2. lower--.f64 N/A

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{n} \]

      3. lower-log.f64 N/A

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{n} \]

      4. lower-+.f64 N/A

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{n} \]

      5. lower-log.f64 58.9%

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{n} \]

   4. Applied rewrites 58.9%

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
   5. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{n} \]

      2. sub-negate-rev N/A

         \[\leadsto \frac{\mathsf{neg}\left(\left(\log x - \log \left(1 + x\right)\right)\right)}{n} \]

      3. 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. lower-/.f64 58.9%

          \[\leadsto \frac{-\log \left(\frac{x}{x + 1}\right)}{n} \]

      16. lift-+.f64 N/A

          \[\leadsto \frac{-\log \left(\frac{x}{x + 1}\right)}{n} \]

      17. add-flip N/A

          \[\leadsto \frac{-\log \left(\frac{x}{x - \left(\mathsf{neg}\left(1\right)\right)}\right)}{n} \]

      18. metadata-eval N/A

          \[\leadsto \frac{-\log \left(\frac{x}{x - -1}\right)}{n} \]

      19. lower--.f64 58.9%

          \[\leadsto \frac{-\log \left(\frac{x}{x - -1}\right)}{n} \]

   6. Applied rewrites 58.9%

      \[\leadsto \frac{-\log \left(\frac{x}{x - -1}\right)}{n} \]
   7. Step-by-step derivation

      1. lift-neg.f64 N/A

         \[\leadsto \frac{\mathsf{neg}\left(\log \left(\frac{x}{x - -1}\right)\right)}{n} \]

      2. lift-log.f64 N/A

         \[\leadsto \frac{\mathsf{neg}\left(\log \left(\frac{x}{x - -1}\right)\right)}{n} \]

      3. neg-log N/A

         \[\leadsto \frac{\log \left(\frac{1}{\frac{x}{x - -1}}\right)}{n} \]

      4. lift-/.f64 N/A

         \[\leadsto \frac{\log \left(\frac{1}{\frac{x}{x - -1}}\right)}{n} \]

      5. div-flip-rev N/A

         \[\leadsto \frac{\log \left(\frac{x - -1}{x}\right)}{n} \]

      6. lift--.f64 N/A

         \[\leadsto \frac{\log \left(\frac{x - -1}{x}\right)}{n} \]

      7. div-sub N/A

         \[\leadsto \frac{\log \left(\frac{x}{x} - \frac{-1}{x}\right)}{n} \]

      8. *-inverses N/A

         \[\leadsto \frac{\log \left(1 - \frac{-1}{x}\right)}{n} \]

      9. metadata-eval N/A

         \[\leadsto \frac{\log \left(1 - \frac{\mathsf{neg}\left(1\right)}{x}\right)}{n} \]

      10. distribute-frac-neg N/A

          \[\leadsto \frac{\log \left(1 - \left(\mathsf{neg}\left(\frac{1}{x}\right)\right)\right)}{n} \]

      11. lift-/.f64 N/A

          \[\leadsto \frac{\log \left(1 - \left(\mathsf{neg}\left(\frac{1}{x}\right)\right)\right)}{n} \]

      12. add-flip-rev N/A

          \[\leadsto \frac{\log \left(1 + \frac{1}{x}\right)}{n} \]

      13. lower-log1p.f64 58.4%

          \[\leadsto \frac{\mathsf{log1p}\left(\frac{1}{x}\right)}{n} \]

   8. Applied rewrites 58.4%

      \[\leadsto \frac{\mathsf{log1p}\left(\frac{1}{x}\right)}{n} \]

   if 2.0000000000000001e139 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 53.5%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]

   2. Taylor expanded in n around inf

      \[\leadsto \color{blue}{\frac{\left(\log \left(1 + x\right) + \frac{1}{2} \cdot \frac{{\log \left(1 + x\right)}^{2}}{n}\right) - \left(\log x + \frac{1}{2} \cdot \frac{{\log x}^{2}}{n}\right)}{n}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\left(\log \left(1 + x\right) + \frac{1}{2} \cdot \frac{{\log \left(1 + x\right)}^{2}}{n}\right) - \left(\log x + \frac{1}{2} \cdot \frac{{\log x}^{2}}{n}\right)}{\color{blue}{n}} \]

   4. Applied rewrites 64.9%

      \[\leadsto \color{blue}{\frac{\left(\log \left(1 + x\right) + 0.5 \cdot \frac{{\log \left(1 + x\right)}^{2}}{n}\right) - \left(\log x + 0.5 \cdot \frac{{\log x}^{2}}{n}\right)}{n}} \]

   5. Taylor expanded in x around -inf

      \[\leadsto -1 \cdot \color{blue}{\frac{1 + -1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}{n \cdot x}} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto -1 \cdot \frac{1 + -1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}{\color{blue}{n \cdot x}} \]

      2. lower-/.f64 N/A

         \[\leadsto -1 \cdot \frac{1 + -1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}{n \cdot \color{blue}{x}} \]

      3. lower-+.f64 N/A

         \[\leadsto -1 \cdot \frac{1 + -1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}{n \cdot x} \]

      4. lower-*.f64 N/A

         \[\leadsto -1 \cdot \frac{1 + -1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}{n \cdot x} \]

      5. lower-/.f64 N/A

         \[\leadsto -1 \cdot \frac{1 + -1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}{n \cdot x} \]

      6. lower-log.f64 N/A

         \[\leadsto -1 \cdot \frac{1 + -1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}{n \cdot x} \]

      7. lower-/.f64 N/A

         \[\leadsto -1 \cdot \frac{1 + -1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}{n \cdot x} \]

      8. lower-*.f64 21.9%

         \[\leadsto -1 \cdot \frac{1 + -1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}{n \cdot x} \]

   7. Applied rewrites 21.9%

      \[\leadsto -1 \cdot \color{blue}{\frac{1 + -1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}{n \cdot x}} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto -1 \cdot \frac{1 + -1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}{\color{blue}{n \cdot x}} \]

      2. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{1 + -1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}{n \cdot x}\right) \]

      3. lift-/.f64 N/A

         \[\leadsto \mathsf{neg}\left(\frac{1 + -1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}{n \cdot x}\right) \]

      4. lift-*.f64 N/A

         \[\leadsto \mathsf{neg}\left(\frac{1 + -1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}{n \cdot x}\right) \]

      5. associate-/r* N/A

         \[\leadsto \mathsf{neg}\left(\frac{\frac{1 + -1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}{n}}{x}\right) \]

      6. distribute-neg-frac2 N/A

         \[\leadsto \frac{\frac{1 + -1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}{n}}{\mathsf{neg}\left(x\right)} \]

      7. lower-/.f64 N/A

         \[\leadsto \frac{\frac{1 + -1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}{n}}{\mathsf{neg}\left(x\right)} \]

   9. Applied rewrites 21.9%

      \[\leadsto \frac{\frac{\frac{\log x}{n} - -1}{n}}{-x} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 11: 74.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{+150}:\\ \;\;\;\;\frac{\log \left(\frac{x - -1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq -5000000:\\ \;\;\;\;\frac{x \cdot 1}{x \cdot \left(n \cdot x\right)}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{+147}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(\frac{1}{x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{n \cdot x}\\ \end{array} \]

FPCore

(FPCore (x n)
  :precision binary64
  (if (<= (/ 1.0 n) -1e+150)
  (/ (log (/ (- x -1.0) x)) n)
  (if (<= (/ 1.0 n) -5000000.0)
    (/ (* x 1.0) (* x (* n x)))
    (if (<= (/ 1.0 n) 5e+147)
      (/ (log1p (/ 1.0 x)) n)
      (/ 1.0 (* n x))))))

C

double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -1e+150) {
		tmp = log(((x - -1.0) / x)) / n;
	} else if ((1.0 / n) <= -5000000.0) {
		tmp = (x * 1.0) / (x * (n * x));
	} else if ((1.0 / n) <= 5e+147) {
		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) <= -1e+150) {
		tmp = Math.log(((x - -1.0) / x)) / n;
	} else if ((1.0 / n) <= -5000000.0) {
		tmp = (x * 1.0) / (x * (n * x));
	} else if ((1.0 / n) <= 5e+147) {
		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) <= -1e+150:
		tmp = math.log(((x - -1.0) / x)) / n
	elif (1.0 / n) <= -5000000.0:
		tmp = (x * 1.0) / (x * (n * x))
	elif (1.0 / n) <= 5e+147:
		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) <= -1e+150)
		tmp = Float64(log(Float64(Float64(x - -1.0) / x)) / n);
	elseif (Float64(1.0 / n) <= -5000000.0)
		tmp = Float64(Float64(x * 1.0) / Float64(x * Float64(n * x)));
	elseif (Float64(1.0 / n) <= 5e+147)
		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], -1e+150], N[(N[Log[N[(N[(x - -1.0), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], -5000000.0], N[(N[(x * 1.0), $MachinePrecision] / N[(x * N[(n * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e+147], 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 4 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -9.9999999999999998e149

   1. Initial program 53.5%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]

   2. Taylor expanded in n around inf

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{\color{blue}{n}} \]

      2. lower--.f64 N/A

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{n} \]

      3. lower-log.f64 N/A

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{n} \]

      4. lower-+.f64 N/A

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{n} \]

      5. lower-log.f64 58.9%

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{n} \]

   4. Applied rewrites 58.9%

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
   5. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{n} \]

      2. lift-log.f64 N/A

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{n} \]

      3. lift-log.f64 N/A

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{n} \]

      4. diff-log N/A

         \[\leadsto \frac{\log \left(\frac{1 + x}{x}\right)}{n} \]

      5. lower-log.f64 N/A

         \[\leadsto \frac{\log \left(\frac{1 + x}{x}\right)}{n} \]

      6. lift-+.f64 N/A

         \[\leadsto \frac{\log \left(\frac{1 + x}{x}\right)}{n} \]

      7. +-commutative N/A

         \[\leadsto \frac{\log \left(\frac{x + 1}{x}\right)}{n} \]

      8. lift-+.f64 N/A

         \[\leadsto \frac{\log \left(\frac{x + 1}{x}\right)}{n} \]

      9. lower-/.f64 58.9%

         \[\leadsto \frac{\log \left(\frac{x + 1}{x}\right)}{n} \]

      10. lift-+.f64 N/A

          \[\leadsto \frac{\log \left(\frac{x + 1}{x}\right)}{n} \]

      11. add-flip N/A

          \[\leadsto \frac{\log \left(\frac{x - \left(\mathsf{neg}\left(1\right)\right)}{x}\right)}{n} \]

      12. metadata-eval N/A

          \[\leadsto \frac{\log \left(\frac{x - -1}{x}\right)}{n} \]

      13. lower--.f64 58.9%

          \[\leadsto \frac{\log \left(\frac{x - -1}{x}\right)}{n} \]

   6. Applied rewrites 58.9%

      \[\leadsto \frac{\log \left(\frac{x - -1}{x}\right)}{n} \]

   if -9.9999999999999998e149 < (/.f64 #s(literal 1 binary64) n) < -5e6

   1. Initial program 53.5%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]

   2. Taylor expanded in n around inf

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{\color{blue}{n}} \]

      2. lower--.f64 N/A

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{n} \]

      3. lower-log.f64 N/A

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{n} \]

      4. lower-+.f64 N/A

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{n} \]

      5. lower-log.f64 58.9%

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{n} \]

   4. Applied rewrites 58.9%

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]

   5. Taylor expanded in x around inf

      \[\leadsto \frac{1}{\color{blue}{n \cdot x}} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{1}{n \cdot \color{blue}{x}} \]

      2. lower-*.f64 40.7%

         \[\leadsto \frac{1}{n \cdot x} \]

   7. Applied rewrites 40.7%

      \[\leadsto \frac{1}{\color{blue}{n \cdot x}} \]
   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 41.3%

         \[\leadsto \frac{\frac{1}{n}}{x} \]

   9. Applied rewrites 41.3%

      \[\leadsto \frac{\frac{1}{n}}{x} \]
   10. Step-by-step derivation

       1. lift-/.f64 N/A

          \[\leadsto \frac{\frac{1}{n}}{x} \]

       2. lift-/.f64 N/A

          \[\leadsto \frac{\frac{1}{n}}{x} \]

       3. associate-/l/ N/A

          \[\leadsto \frac{1}{n \cdot \color{blue}{x}} \]

       4. lift-*.f64 N/A

          \[\leadsto \frac{1}{n \cdot x} \]

       5. mult-flip N/A

          \[\leadsto 1 \cdot \frac{1}{\color{blue}{n \cdot x}} \]

       6. *-inverses N/A

          \[\leadsto \frac{x}{x} \cdot \frac{1}{\color{blue}{n} \cdot x} \]

       7. frac-times N/A

          \[\leadsto \frac{x \cdot 1}{x \cdot \color{blue}{\left(n \cdot x\right)}} \]

       8. lower-/.f64 N/A

          \[\leadsto \frac{x \cdot 1}{x \cdot \color{blue}{\left(n \cdot x\right)}} \]

       9. lower-*.f64 N/A

          \[\leadsto \frac{x \cdot 1}{x \cdot \left(\color{blue}{n} \cdot x\right)} \]

       10. lower-*.f64 41.5%

           \[\leadsto \frac{x \cdot 1}{x \cdot \left(n \cdot \color{blue}{x}\right)} \]

   11. Applied rewrites 41.5%

       \[\leadsto \frac{x \cdot 1}{x \cdot \color{blue}{\left(n \cdot x\right)}} \]

   if -5e6 < (/.f64 #s(literal 1 binary64) n) < 5.0000000000000002e147

   1. Initial program 53.5%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]

   2. Taylor expanded in n around inf

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{\color{blue}{n}} \]

      2. lower--.f64 N/A

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{n} \]

      3. lower-log.f64 N/A

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{n} \]

      4. lower-+.f64 N/A

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{n} \]

      5. lower-log.f64 58.9%

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{n} \]

   4. Applied rewrites 58.9%

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
   5. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{n} \]

      2. sub-negate-rev N/A

         \[\leadsto \frac{\mathsf{neg}\left(\left(\log x - \log \left(1 + x\right)\right)\right)}{n} \]

      3. 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. lower-/.f64 58.9%

          \[\leadsto \frac{-\log \left(\frac{x}{x + 1}\right)}{n} \]

      16. lift-+.f64 N/A

          \[\leadsto \frac{-\log \left(\frac{x}{x + 1}\right)}{n} \]

      17. add-flip N/A

          \[\leadsto \frac{-\log \left(\frac{x}{x - \left(\mathsf{neg}\left(1\right)\right)}\right)}{n} \]

      18. metadata-eval N/A

          \[\leadsto \frac{-\log \left(\frac{x}{x - -1}\right)}{n} \]

      19. lower--.f64 58.9%

          \[\leadsto \frac{-\log \left(\frac{x}{x - -1}\right)}{n} \]

   6. Applied rewrites 58.9%

      \[\leadsto \frac{-\log \left(\frac{x}{x - -1}\right)}{n} \]
   7. Step-by-step derivation

      1. lift-neg.f64 N/A

         \[\leadsto \frac{\mathsf{neg}\left(\log \left(\frac{x}{x - -1}\right)\right)}{n} \]

      2. lift-log.f64 N/A

         \[\leadsto \frac{\mathsf{neg}\left(\log \left(\frac{x}{x - -1}\right)\right)}{n} \]

      3. neg-log N/A

         \[\leadsto \frac{\log \left(\frac{1}{\frac{x}{x - -1}}\right)}{n} \]

      4. lift-/.f64 N/A

         \[\leadsto \frac{\log \left(\frac{1}{\frac{x}{x - -1}}\right)}{n} \]

      5. div-flip-rev N/A

         \[\leadsto \frac{\log \left(\frac{x - -1}{x}\right)}{n} \]

      6. lift--.f64 N/A

         \[\leadsto \frac{\log \left(\frac{x - -1}{x}\right)}{n} \]

      7. div-sub N/A

         \[\leadsto \frac{\log \left(\frac{x}{x} - \frac{-1}{x}\right)}{n} \]

      8. *-inverses N/A

         \[\leadsto \frac{\log \left(1 - \frac{-1}{x}\right)}{n} \]

      9. metadata-eval N/A

         \[\leadsto \frac{\log \left(1 - \frac{\mathsf{neg}\left(1\right)}{x}\right)}{n} \]

      10. distribute-frac-neg N/A

          \[\leadsto \frac{\log \left(1 - \left(\mathsf{neg}\left(\frac{1}{x}\right)\right)\right)}{n} \]

      11. lift-/.f64 N/A

          \[\leadsto \frac{\log \left(1 - \left(\mathsf{neg}\left(\frac{1}{x}\right)\right)\right)}{n} \]

      12. add-flip-rev N/A

          \[\leadsto \frac{\log \left(1 + \frac{1}{x}\right)}{n} \]

      13. lower-log1p.f64 58.4%

          \[\leadsto \frac{\mathsf{log1p}\left(\frac{1}{x}\right)}{n} \]

   8. Applied rewrites 58.4%

      \[\leadsto \frac{\mathsf{log1p}\left(\frac{1}{x}\right)}{n} \]

   if 5.0000000000000002e147 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 53.5%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]

   2. Taylor expanded in n around inf

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{\color{blue}{n}} \]

      2. lower--.f64 N/A

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{n} \]

      3. lower-log.f64 N/A

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{n} \]

      4. lower-+.f64 N/A

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{n} \]

      5. lower-log.f64 58.9%

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{n} \]

   4. Applied rewrites 58.9%

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]

   5. Taylor expanded in x around inf

      \[\leadsto \frac{1}{\color{blue}{n \cdot x}} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{1}{n \cdot \color{blue}{x}} \]

      2. lower-*.f64 40.7%

         \[\leadsto \frac{1}{n \cdot x} \]

   7. Applied rewrites 40.7%

      \[\leadsto \frac{1}{\color{blue}{n \cdot x}} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 12: 71.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} t_0 := {\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\frac{x \cdot 1}{x \cdot \left(n \cdot x\right)}\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{-6}:\\ \;\;\;\;\frac{-\log \left(\frac{x}{x - -1}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{n \cdot x}\\ \end{array} \]

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))
    (/ (* x 1.0) (* x (* n x)))
    (if (<= t_0 2e-6)
      (/ (- (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 = (x * 1.0) / (x * (n * x));
	} else if (t_0 <= 2e-6) {
		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 = (x * 1.0) / (x * (n * x));
	} else if (t_0 <= 2e-6) {
		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 = (x * 1.0) / (x * (n * x))
	elif t_0 <= 2e-6:
		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(Float64(x * 1.0) / Float64(x * Float64(n * x)));
	elseif (t_0 <= 2e-6)
		tmp = Float64(Float64(-log(Float64(x / Float64(x - -1.0)))) / n);
	else
		tmp = Float64(1.0 / Float64(n * x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = ((x + 1.0) ^ (1.0 / n)) - (x ^ (1.0 / n));
	tmp = 0.0;
	if (t_0 <= -Inf)
		tmp = (x * 1.0) / (x * (n * x));
	elseif (t_0 <= 2e-6)
		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[(N[(x * 1.0), $MachinePrecision] / N[(x * N[(n * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 2e-6], N[((-N[Log[N[(x / N[(x - -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / n), $MachinePrecision], N[(1.0 / N[(n * x), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (pow.f64 (+.f64 x #s(literal 1 binary64)) (/.f64 #s(literal 1 binary64) n)) (pow.f64 x (/.f64 #s(literal 1 binary64) n))) < -inf.0

   1. Initial program 53.5%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]

   2. Taylor expanded in n around inf

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{\color{blue}{n}} \]

      2. lower--.f64 N/A

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{n} \]

      3. lower-log.f64 N/A

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{n} \]

      4. lower-+.f64 N/A

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{n} \]

      5. lower-log.f64 58.9%

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{n} \]

   4. Applied rewrites 58.9%

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]

   5. Taylor expanded in x around inf

      \[\leadsto \frac{1}{\color{blue}{n \cdot x}} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{1}{n \cdot \color{blue}{x}} \]

      2. lower-*.f64 40.7%

         \[\leadsto \frac{1}{n \cdot x} \]

   7. Applied rewrites 40.7%

      \[\leadsto \frac{1}{\color{blue}{n \cdot x}} \]
   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 41.3%

         \[\leadsto \frac{\frac{1}{n}}{x} \]

   9. Applied rewrites 41.3%

      \[\leadsto \frac{\frac{1}{n}}{x} \]
   10. Step-by-step derivation

       1. lift-/.f64 N/A

          \[\leadsto \frac{\frac{1}{n}}{x} \]

       2. lift-/.f64 N/A

          \[\leadsto \frac{\frac{1}{n}}{x} \]

       3. associate-/l/ N/A

          \[\leadsto \frac{1}{n \cdot \color{blue}{x}} \]

       4. lift-*.f64 N/A

          \[\leadsto \frac{1}{n \cdot x} \]

       5. mult-flip N/A

          \[\leadsto 1 \cdot \frac{1}{\color{blue}{n \cdot x}} \]

       6. *-inverses N/A

          \[\leadsto \frac{x}{x} \cdot \frac{1}{\color{blue}{n} \cdot x} \]

       7. frac-times N/A

          \[\leadsto \frac{x \cdot 1}{x \cdot \color{blue}{\left(n \cdot x\right)}} \]

       8. lower-/.f64 N/A

          \[\leadsto \frac{x \cdot 1}{x \cdot \color{blue}{\left(n \cdot x\right)}} \]

       9. lower-*.f64 N/A

          \[\leadsto \frac{x \cdot 1}{x \cdot \left(\color{blue}{n} \cdot x\right)} \]

       10. lower-*.f64 41.5%

           \[\leadsto \frac{x \cdot 1}{x \cdot \left(n \cdot \color{blue}{x}\right)} \]

   11. Applied rewrites 41.5%

       \[\leadsto \frac{x \cdot 1}{x \cdot \color{blue}{\left(n \cdot x\right)}} \]

   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))) < 1.9999999999999999e-6

   1. Initial program 53.5%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]

   2. Taylor expanded in n around inf

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{\color{blue}{n}} \]

      2. lower--.f64 N/A

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{n} \]

      3. lower-log.f64 N/A

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{n} \]

      4. lower-+.f64 N/A

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{n} \]

      5. lower-log.f64 58.9%

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{n} \]

   4. Applied rewrites 58.9%

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
   5. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{n} \]

      2. sub-negate-rev N/A

         \[\leadsto \frac{\mathsf{neg}\left(\left(\log x - \log \left(1 + x\right)\right)\right)}{n} \]

      3. 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. lower-/.f64 58.9%

          \[\leadsto \frac{-\log \left(\frac{x}{x + 1}\right)}{n} \]

      16. lift-+.f64 N/A

          \[\leadsto \frac{-\log \left(\frac{x}{x + 1}\right)}{n} \]

      17. add-flip N/A

          \[\leadsto \frac{-\log \left(\frac{x}{x - \left(\mathsf{neg}\left(1\right)\right)}\right)}{n} \]

      18. metadata-eval N/A

          \[\leadsto \frac{-\log \left(\frac{x}{x - -1}\right)}{n} \]

      19. lower--.f64 58.9%

          \[\leadsto \frac{-\log \left(\frac{x}{x - -1}\right)}{n} \]

   6. Applied rewrites 58.9%

      \[\leadsto \frac{-\log \left(\frac{x}{x - -1}\right)}{n} \]

   if 1.9999999999999999e-6 < (-.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 53.5%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]

   2. Taylor expanded in n around inf

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{\color{blue}{n}} \]

      2. lower--.f64 N/A

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{n} \]

      3. lower-log.f64 N/A

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{n} \]

      4. lower-+.f64 N/A

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{n} \]

      5. lower-log.f64 58.9%

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{n} \]

   4. Applied rewrites 58.9%

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]

   5. Taylor expanded in x around inf

      \[\leadsto \frac{1}{\color{blue}{n \cdot x}} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{1}{n \cdot \color{blue}{x}} \]

      2. lower-*.f64 40.7%

         \[\leadsto \frac{1}{n \cdot x} \]

   7. Applied rewrites 40.7%

      \[\leadsto \frac{1}{\color{blue}{n \cdot x}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 13: 71.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} t_0 := {\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\frac{x \cdot 1}{x \cdot \left(n \cdot x\right)}\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{-6}:\\ \;\;\;\;\frac{\log \left(\frac{x - -1}{x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{n \cdot x}\\ \end{array} \]

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))
    (/ (* x 1.0) (* x (* n x)))
    (if (<= t_0 2e-6) (/ (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 = (x * 1.0) / (x * (n * x));
	} else if (t_0 <= 2e-6) {
		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 = (x * 1.0) / (x * (n * x));
	} else if (t_0 <= 2e-6) {
		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 = (x * 1.0) / (x * (n * x))
	elif t_0 <= 2e-6:
		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(Float64(x * 1.0) / Float64(x * Float64(n * x)));
	elseif (t_0 <= 2e-6)
		tmp = Float64(log(Float64(Float64(x - -1.0) / x)) / n);
	else
		tmp = Float64(1.0 / Float64(n * x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = ((x + 1.0) ^ (1.0 / n)) - (x ^ (1.0 / n));
	tmp = 0.0;
	if (t_0 <= -Inf)
		tmp = (x * 1.0) / (x * (n * x));
	elseif (t_0 <= 2e-6)
		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[(N[(x * 1.0), $MachinePrecision] / N[(x * N[(n * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 2e-6], N[(N[Log[N[(N[(x - -1.0), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], N[(1.0 / N[(n * x), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (pow.f64 (+.f64 x #s(literal 1 binary64)) (/.f64 #s(literal 1 binary64) n)) (pow.f64 x (/.f64 #s(literal 1 binary64) n))) < -inf.0

   1. Initial program 53.5%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]

   2. Taylor expanded in n around inf

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{\color{blue}{n}} \]

      2. lower--.f64 N/A

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{n} \]

      3. lower-log.f64 N/A

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{n} \]

      4. lower-+.f64 N/A

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{n} \]

      5. lower-log.f64 58.9%

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{n} \]

   4. Applied rewrites 58.9%

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]

   5. Taylor expanded in x around inf

      \[\leadsto \frac{1}{\color{blue}{n \cdot x}} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{1}{n \cdot \color{blue}{x}} \]

      2. lower-*.f64 40.7%

         \[\leadsto \frac{1}{n \cdot x} \]

   7. Applied rewrites 40.7%

      \[\leadsto \frac{1}{\color{blue}{n \cdot x}} \]
   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 41.3%

         \[\leadsto \frac{\frac{1}{n}}{x} \]

   9. Applied rewrites 41.3%

      \[\leadsto \frac{\frac{1}{n}}{x} \]
   10. Step-by-step derivation

       1. lift-/.f64 N/A

          \[\leadsto \frac{\frac{1}{n}}{x} \]

       2. lift-/.f64 N/A

          \[\leadsto \frac{\frac{1}{n}}{x} \]

       3. associate-/l/ N/A

          \[\leadsto \frac{1}{n \cdot \color{blue}{x}} \]

       4. lift-*.f64 N/A

          \[\leadsto \frac{1}{n \cdot x} \]

       5. mult-flip N/A

          \[\leadsto 1 \cdot \frac{1}{\color{blue}{n \cdot x}} \]

       6. *-inverses N/A

          \[\leadsto \frac{x}{x} \cdot \frac{1}{\color{blue}{n} \cdot x} \]

       7. frac-times N/A

          \[\leadsto \frac{x \cdot 1}{x \cdot \color{blue}{\left(n \cdot x\right)}} \]

       8. lower-/.f64 N/A

          \[\leadsto \frac{x \cdot 1}{x \cdot \color{blue}{\left(n \cdot x\right)}} \]

       9. lower-*.f64 N/A

          \[\leadsto \frac{x \cdot 1}{x \cdot \left(\color{blue}{n} \cdot x\right)} \]

       10. lower-*.f64 41.5%

           \[\leadsto \frac{x \cdot 1}{x \cdot \left(n \cdot \color{blue}{x}\right)} \]

   11. Applied rewrites 41.5%

       \[\leadsto \frac{x \cdot 1}{x \cdot \color{blue}{\left(n \cdot x\right)}} \]

   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))) < 1.9999999999999999e-6

   1. Initial program 53.5%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]

   2. Taylor expanded in n around inf

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{\color{blue}{n}} \]

      2. lower--.f64 N/A

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{n} \]

      3. lower-log.f64 N/A

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{n} \]

      4. lower-+.f64 N/A

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{n} \]

      5. lower-log.f64 58.9%

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{n} \]

   4. Applied rewrites 58.9%

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
   5. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{n} \]

      2. lift-log.f64 N/A

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{n} \]

      3. lift-log.f64 N/A

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{n} \]

      4. diff-log N/A

         \[\leadsto \frac{\log \left(\frac{1 + x}{x}\right)}{n} \]

      5. lower-log.f64 N/A

         \[\leadsto \frac{\log \left(\frac{1 + x}{x}\right)}{n} \]

      6. lift-+.f64 N/A

         \[\leadsto \frac{\log \left(\frac{1 + x}{x}\right)}{n} \]

      7. +-commutative N/A

         \[\leadsto \frac{\log \left(\frac{x + 1}{x}\right)}{n} \]

      8. lift-+.f64 N/A

         \[\leadsto \frac{\log \left(\frac{x + 1}{x}\right)}{n} \]

      9. lower-/.f64 58.9%

         \[\leadsto \frac{\log \left(\frac{x + 1}{x}\right)}{n} \]

      10. lift-+.f64 N/A

          \[\leadsto \frac{\log \left(\frac{x + 1}{x}\right)}{n} \]

      11. add-flip N/A

          \[\leadsto \frac{\log \left(\frac{x - \left(\mathsf{neg}\left(1\right)\right)}{x}\right)}{n} \]

      12. metadata-eval N/A

          \[\leadsto \frac{\log \left(\frac{x - -1}{x}\right)}{n} \]

      13. lower--.f64 58.9%

          \[\leadsto \frac{\log \left(\frac{x - -1}{x}\right)}{n} \]

   6. Applied rewrites 58.9%

      \[\leadsto \frac{\log \left(\frac{x - -1}{x}\right)}{n} \]

   if 1.9999999999999999e-6 < (-.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 53.5%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]

   2. Taylor expanded in n around inf

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{\color{blue}{n}} \]

      2. lower--.f64 N/A

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{n} \]

      3. lower-log.f64 N/A

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{n} \]

      4. lower-+.f64 N/A

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{n} \]

      5. lower-log.f64 58.9%

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{n} \]

   4. Applied rewrites 58.9%

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]

   5. Taylor expanded in x around inf

      \[\leadsto \frac{1}{\color{blue}{n \cdot x}} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{1}{n \cdot \color{blue}{x}} \]

      2. lower-*.f64 40.7%

         \[\leadsto \frac{1}{n \cdot x} \]

   7. Applied rewrites 40.7%

      \[\leadsto \frac{1}{\color{blue}{n \cdot x}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 14: 58.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} \mathbf{if}\;x \leq 11.5:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{elif}\;x \leq 7.5 \cdot 10^{+279}:\\ \;\;\;\;\frac{1 - 0.5 \cdot \frac{1}{x}}{n \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-0.5}{n \cdot x}}{x}\\ \end{array} \]

FPCore

(FPCore (x n)
  :precision binary64
  (if (<= x 11.5)
  (/ (- x (log x)) n)
  (if (<= x 7.5e+279)
    (/ (- 1.0 (* 0.5 (/ 1.0 x))) (* n x))
    (/ (/ -0.5 (* n x)) x))))

C

double code(double x, double n) {
	double tmp;
	if (x <= 11.5) {
		tmp = (x - log(x)) / n;
	} else if (x <= 7.5e+279) {
		tmp = (1.0 - (0.5 * (1.0 / x))) / (n * x);
	} 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 <= 11.5d0) then
        tmp = (x - log(x)) / n
    else if (x <= 7.5d+279) then
        tmp = (1.0d0 - (0.5d0 * (1.0d0 / x))) / (n * x)
    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 <= 11.5) {
		tmp = (x - Math.log(x)) / n;
	} else if (x <= 7.5e+279) {
		tmp = (1.0 - (0.5 * (1.0 / x))) / (n * x);
	} else {
		tmp = (-0.5 / (n * x)) / x;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 11.5:
		tmp = (x - math.log(x)) / n
	elif x <= 7.5e+279:
		tmp = (1.0 - (0.5 * (1.0 / x))) / (n * x)
	else:
		tmp = (-0.5 / (n * x)) / x
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 11.5)
		tmp = Float64(Float64(x - log(x)) / n);
	elseif (x <= 7.5e+279)
		tmp = Float64(Float64(1.0 - Float64(0.5 * Float64(1.0 / x))) / Float64(n * x));
	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 <= 11.5)
		tmp = (x - log(x)) / n;
	elseif (x <= 7.5e+279)
		tmp = (1.0 - (0.5 * (1.0 / x))) / (n * x);
	else
		tmp = (-0.5 / (n * x)) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 11.5], N[(N[(x - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[x, 7.5e+279], N[(N[(1.0 - N[(0.5 * N[(1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(n * x), $MachinePrecision]), $MachinePrecision], N[(N[(-0.5 / N[(n * x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 11.5

   1. Initial program 53.5%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]

   2. Taylor expanded in n around inf

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{\color{blue}{n}} \]

      2. lower--.f64 N/A

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{n} \]

      3. lower-log.f64 N/A

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{n} \]

      4. lower-+.f64 N/A

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{n} \]

      5. lower-log.f64 58.9%

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{n} \]

   4. Applied rewrites 58.9%

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]

   5. Taylor expanded in x around 0

      \[\leadsto \frac{x - \log x}{n} \]
   6. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \frac{x - \log x}{n} \]

      2. lower-log.f64 30.8%

         \[\leadsto \frac{x - \log x}{n} \]

   7. Applied rewrites 30.8%

      \[\leadsto \frac{x - \log x}{n} \]

   if 11.5 < x < 7.5000000000000002e279

   1. Initial program 53.5%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]

   2. Taylor expanded in n around inf

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{\color{blue}{n}} \]

      2. lower--.f64 N/A

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{n} \]

      3. lower-log.f64 N/A

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{n} \]

      4. lower-+.f64 N/A

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{n} \]

      5. lower-log.f64 58.9%

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{n} \]

   4. Applied rewrites 58.9%

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]

   5. Taylor expanded in x around inf

      \[\leadsto \frac{\frac{1}{n} - \frac{1}{2} \cdot \frac{1}{n \cdot x}}{\color{blue}{x}} \]
   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 29.1%

         \[\leadsto \frac{\frac{1}{n} - 0.5 \cdot \frac{1}{n \cdot x}}{x} \]

   7. Applied rewrites 29.1%

      \[\leadsto \frac{\frac{1}{n} - 0.5 \cdot \frac{1}{n \cdot x}}{\color{blue}{x}} \]

   8. Taylor expanded in n around 0

      \[\leadsto \frac{1 - \frac{1}{2} \cdot \frac{1}{x}}{n \cdot \color{blue}{x}} \]
   9. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{1 - \frac{1}{2} \cdot \frac{1}{x}}{n \cdot x} \]

      2. lower--.f64 N/A

         \[\leadsto \frac{1 - \frac{1}{2} \cdot \frac{1}{x}}{n \cdot x} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{1 - \frac{1}{2} \cdot \frac{1}{x}}{n \cdot x} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{1 - \frac{1}{2} \cdot \frac{1}{x}}{n \cdot x} \]

      5. lower-*.f64 28.6%

         \[\leadsto \frac{1 - 0.5 \cdot \frac{1}{x}}{n \cdot x} \]

   10. Applied rewrites 28.6%

       \[\leadsto \frac{1 - 0.5 \cdot \frac{1}{x}}{n \cdot \color{blue}{x}} \]

   if 7.5000000000000002e279 < x

   1. Initial program 53.5%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]

   2. Taylor expanded in n around inf

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{\color{blue}{n}} \]

      2. lower--.f64 N/A

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{n} \]

      3. lower-log.f64 N/A

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{n} \]

      4. lower-+.f64 N/A

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{n} \]

      5. lower-log.f64 58.9%

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{n} \]

   4. Applied rewrites 58.9%

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]

   5. Taylor expanded in x around inf

      \[\leadsto \frac{\frac{1}{n} - \frac{1}{2} \cdot \frac{1}{n \cdot x}}{\color{blue}{x}} \]
   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 29.1%

         \[\leadsto \frac{\frac{1}{n} - 0.5 \cdot \frac{1}{n \cdot x}}{x} \]

   7. Applied rewrites 29.1%

      \[\leadsto \frac{\frac{1}{n} - 0.5 \cdot \frac{1}{n \cdot x}}{\color{blue}{x}} \]

   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 19.3%

         \[\leadsto \frac{\frac{-0.5}{n \cdot x}}{x} \]

   10. Applied rewrites 19.3%

       \[\leadsto \frac{\frac{-0.5}{n \cdot x}}{x} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 15: 58.3% accurate, 2.5× speedup

Math

\[\begin{array}{l} \mathbf{if}\;x \leq 0.42:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{elif}\;x \leq 7.5 \cdot 10^{+279}:\\ \;\;\;\;\frac{\frac{1}{n}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-0.5}{n \cdot x}}{x}\\ \end{array} \]

FPCore

(FPCore (x n)
  :precision binary64
  (if (<= x 0.42)
  (/ (- x (log x)) n)
  (if (<= x 7.5e+279) (/ (/ 1.0 n) x) (/ (/ -0.5 (* n x)) x))))

C

double code(double x, double n) {
	double tmp;
	if (x <= 0.42) {
		tmp = (x - log(x)) / n;
	} else if (x <= 7.5e+279) {
		tmp = (1.0 / n) / x;
	} 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 <= 0.42d0) then
        tmp = (x - log(x)) / n
    else if (x <= 7.5d+279) then
        tmp = (1.0d0 / n) / x
    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 <= 0.42) {
		tmp = (x - Math.log(x)) / n;
	} else if (x <= 7.5e+279) {
		tmp = (1.0 / n) / x;
	} else {
		tmp = (-0.5 / (n * x)) / x;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 0.42:
		tmp = (x - math.log(x)) / n
	elif x <= 7.5e+279:
		tmp = (1.0 / n) / x
	else:
		tmp = (-0.5 / (n * x)) / x
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 0.42)
		tmp = Float64(Float64(x - log(x)) / n);
	elseif (x <= 7.5e+279)
		tmp = Float64(Float64(1.0 / n) / x);
	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 <= 0.42)
		tmp = (x - log(x)) / n;
	elseif (x <= 7.5e+279)
		tmp = (1.0 / n) / x;
	else
		tmp = (-0.5 / (n * x)) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 0.42], N[(N[(x - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[x, 7.5e+279], N[(N[(1.0 / n), $MachinePrecision] / x), $MachinePrecision], N[(N[(-0.5 / N[(n * x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 0.41999999999999998

   1. Initial program 53.5%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]

   2. Taylor expanded in n around inf

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{\color{blue}{n}} \]

      2. lower--.f64 N/A

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{n} \]

      3. lower-log.f64 N/A

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{n} \]

      4. lower-+.f64 N/A

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{n} \]

      5. lower-log.f64 58.9%

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{n} \]

   4. Applied rewrites 58.9%

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]

   5. Taylor expanded in x around 0

      \[\leadsto \frac{x - \log x}{n} \]
   6. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \frac{x - \log x}{n} \]

      2. lower-log.f64 30.8%

         \[\leadsto \frac{x - \log x}{n} \]

   7. Applied rewrites 30.8%

      \[\leadsto \frac{x - \log x}{n} \]

   if 0.41999999999999998 < x < 7.5000000000000002e279

   1. Initial program 53.5%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]

   2. Taylor expanded in n around inf

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{\color{blue}{n}} \]

      2. lower--.f64 N/A

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{n} \]

      3. lower-log.f64 N/A

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{n} \]

      4. lower-+.f64 N/A

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{n} \]

      5. lower-log.f64 58.9%

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{n} \]

   4. Applied rewrites 58.9%

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]

   5. Taylor expanded in x around inf

      \[\leadsto \frac{1}{\color{blue}{n \cdot x}} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{1}{n \cdot \color{blue}{x}} \]

      2. lower-*.f64 40.7%

         \[\leadsto \frac{1}{n \cdot x} \]

   7. Applied rewrites 40.7%

      \[\leadsto \frac{1}{\color{blue}{n \cdot x}} \]
   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 41.3%

         \[\leadsto \frac{\frac{1}{n}}{x} \]

   9. Applied rewrites 41.3%

      \[\leadsto \frac{\frac{1}{n}}{x} \]

   if 7.5000000000000002e279 < x

   1. Initial program 53.5%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]

   2. Taylor expanded in n around inf

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{\color{blue}{n}} \]

      2. lower--.f64 N/A

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{n} \]

      3. lower-log.f64 N/A

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{n} \]

      4. lower-+.f64 N/A

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{n} \]

      5. lower-log.f64 58.9%

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{n} \]

   4. Applied rewrites 58.9%

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]

   5. Taylor expanded in x around inf

      \[\leadsto \frac{\frac{1}{n} - \frac{1}{2} \cdot \frac{1}{n \cdot x}}{\color{blue}{x}} \]
   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 29.1%

         \[\leadsto \frac{\frac{1}{n} - 0.5 \cdot \frac{1}{n \cdot x}}{x} \]

   7. Applied rewrites 29.1%

      \[\leadsto \frac{\frac{1}{n} - 0.5 \cdot \frac{1}{n \cdot x}}{\color{blue}{x}} \]

   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 19.3%

         \[\leadsto \frac{\frac{-0.5}{n \cdot x}}{x} \]

   10. Applied rewrites 19.3%

       \[\leadsto \frac{\frac{-0.5}{n \cdot x}}{x} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 16: 58.3% accurate, 2.5× speedup

Math

\[\begin{array}{l} \mathbf{if}\;x \leq 0.42:\\ \;\;\;\;\frac{-\log x}{n}\\ \mathbf{elif}\;x \leq 7.5 \cdot 10^{+279}:\\ \;\;\;\;\frac{\frac{1}{n}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-0.5}{n \cdot x}}{x}\\ \end{array} \]

FPCore

(FPCore (x n)
  :precision binary64
  (if (<= x 0.42)
  (/ (- (log x)) n)
  (if (<= x 7.5e+279) (/ (/ 1.0 n) x) (/ (/ -0.5 (* n x)) x))))

C

double code(double x, double n) {
	double tmp;
	if (x <= 0.42) {
		tmp = -log(x) / n;
	} else if (x <= 7.5e+279) {
		tmp = (1.0 / n) / x;
	} 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 <= 0.42d0) then
        tmp = -log(x) / n
    else if (x <= 7.5d+279) then
        tmp = (1.0d0 / n) / x
    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 <= 0.42) {
		tmp = -Math.log(x) / n;
	} else if (x <= 7.5e+279) {
		tmp = (1.0 / n) / x;
	} else {
		tmp = (-0.5 / (n * x)) / x;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 0.42:
		tmp = -math.log(x) / n
	elif x <= 7.5e+279:
		tmp = (1.0 / n) / x
	else:
		tmp = (-0.5 / (n * x)) / x
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 0.42)
		tmp = Float64(Float64(-log(x)) / n);
	elseif (x <= 7.5e+279)
		tmp = Float64(Float64(1.0 / n) / x);
	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 <= 0.42)
		tmp = -log(x) / n;
	elseif (x <= 7.5e+279)
		tmp = (1.0 / n) / x;
	else
		tmp = (-0.5 / (n * x)) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 0.42], N[((-N[Log[x], $MachinePrecision]) / n), $MachinePrecision], If[LessEqual[x, 7.5e+279], N[(N[(1.0 / n), $MachinePrecision] / x), $MachinePrecision], N[(N[(-0.5 / N[(n * x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 0.41999999999999998

   1. Initial program 53.5%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]

   2. Taylor expanded in n around inf

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{\color{blue}{n}} \]

      2. lower--.f64 N/A

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{n} \]

      3. lower-log.f64 N/A

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{n} \]

      4. lower-+.f64 N/A

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{n} \]

      5. lower-log.f64 58.9%

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{n} \]

   4. Applied rewrites 58.9%

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
   5. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{n} \]

      2. sub-negate-rev N/A

         \[\leadsto \frac{\mathsf{neg}\left(\left(\log x - \log \left(1 + x\right)\right)\right)}{n} \]

      3. 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. lower-/.f64 58.9%

          \[\leadsto \frac{-\log \left(\frac{x}{x + 1}\right)}{n} \]

      16. lift-+.f64 N/A

          \[\leadsto \frac{-\log \left(\frac{x}{x + 1}\right)}{n} \]

      17. add-flip N/A

          \[\leadsto \frac{-\log \left(\frac{x}{x - \left(\mathsf{neg}\left(1\right)\right)}\right)}{n} \]

      18. metadata-eval N/A

          \[\leadsto \frac{-\log \left(\frac{x}{x - -1}\right)}{n} \]

      19. lower--.f64 58.9%

          \[\leadsto \frac{-\log \left(\frac{x}{x - -1}\right)}{n} \]

   6. Applied rewrites 58.9%

      \[\leadsto \frac{-\log \left(\frac{x}{x - -1}\right)}{n} \]

   7. Taylor expanded in x around 0

      \[\leadsto \frac{-\log x}{n} \]
   8. Step-by-step derivation

      1. lower-log.f64 30.8%

         \[\leadsto \frac{-\log x}{n} \]

   9. Applied rewrites 30.8%

      \[\leadsto \frac{-\log x}{n} \]

   if 0.41999999999999998 < x < 7.5000000000000002e279

   1. Initial program 53.5%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]

   2. Taylor expanded in n around inf

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{\color{blue}{n}} \]

      2. lower--.f64 N/A

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{n} \]

      3. lower-log.f64 N/A

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{n} \]

      4. lower-+.f64 N/A

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{n} \]

      5. lower-log.f64 58.9%

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{n} \]

   4. Applied rewrites 58.9%

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]

   5. Taylor expanded in x around inf

      \[\leadsto \frac{1}{\color{blue}{n \cdot x}} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{1}{n \cdot \color{blue}{x}} \]

      2. lower-*.f64 40.7%

         \[\leadsto \frac{1}{n \cdot x} \]

   7. Applied rewrites 40.7%

      \[\leadsto \frac{1}{\color{blue}{n \cdot x}} \]
   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 41.3%

         \[\leadsto \frac{\frac{1}{n}}{x} \]

   9. Applied rewrites 41.3%

      \[\leadsto \frac{\frac{1}{n}}{x} \]

   if 7.5000000000000002e279 < x

   1. Initial program 53.5%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]

   2. Taylor expanded in n around inf

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{\color{blue}{n}} \]

      2. lower--.f64 N/A

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{n} \]

      3. lower-log.f64 N/A

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{n} \]

      4. lower-+.f64 N/A

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{n} \]

      5. lower-log.f64 58.9%

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{n} \]

   4. Applied rewrites 58.9%

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]

   5. Taylor expanded in x around inf

      \[\leadsto \frac{\frac{1}{n} - \frac{1}{2} \cdot \frac{1}{n \cdot x}}{\color{blue}{x}} \]
   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 29.1%

         \[\leadsto \frac{\frac{1}{n} - 0.5 \cdot \frac{1}{n \cdot x}}{x} \]

   7. Applied rewrites 29.1%

      \[\leadsto \frac{\frac{1}{n} - 0.5 \cdot \frac{1}{n \cdot x}}{\color{blue}{x}} \]

   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 19.3%

         \[\leadsto \frac{\frac{-0.5}{n \cdot x}}{x} \]

   10. Applied rewrites 19.3%

       \[\leadsto \frac{\frac{-0.5}{n \cdot x}}{x} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 17: 42.2% accurate, 3.1× speedup

Math

\[\begin{array}{l} \mathbf{if}\;x \leq 7.5 \cdot 10^{+279}:\\ \;\;\;\;\frac{\frac{1}{n}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-0.5}{n \cdot x}}{x}\\ \end{array} \]

FPCore

(FPCore (x n)
  :precision binary64
  (if (<= x 7.5e+279) (/ (/ 1.0 n) x) (/ (/ -0.5 (* n x)) x)))

C

double code(double x, double n) {
	double tmp;
	if (x <= 7.5e+279) {
		tmp = (1.0 / n) / x;
	} 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 <= 7.5d+279) then
        tmp = (1.0d0 / n) / x
    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 <= 7.5e+279) {
		tmp = (1.0 / n) / x;
	} else {
		tmp = (-0.5 / (n * x)) / x;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 7.5e+279:
		tmp = (1.0 / n) / x
	else:
		tmp = (-0.5 / (n * x)) / x
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 7.5e+279)
		tmp = Float64(Float64(1.0 / n) / x);
	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 <= 7.5e+279)
		tmp = (1.0 / n) / x;
	else
		tmp = (-0.5 / (n * x)) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 7.5e+279], N[(N[(1.0 / n), $MachinePrecision] / x), $MachinePrecision], N[(N[(-0.5 / N[(n * x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 7.5000000000000002e279

   1. Initial program 53.5%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]

   2. Taylor expanded in n around inf

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{\color{blue}{n}} \]

      2. lower--.f64 N/A

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{n} \]

      3. lower-log.f64 N/A

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{n} \]

      4. lower-+.f64 N/A

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{n} \]

      5. lower-log.f64 58.9%

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{n} \]

   4. Applied rewrites 58.9%

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]

   5. Taylor expanded in x around inf

      \[\leadsto \frac{1}{\color{blue}{n \cdot x}} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{1}{n \cdot \color{blue}{x}} \]

      2. lower-*.f64 40.7%

         \[\leadsto \frac{1}{n \cdot x} \]

   7. Applied rewrites 40.7%

      \[\leadsto \frac{1}{\color{blue}{n \cdot x}} \]
   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 41.3%

         \[\leadsto \frac{\frac{1}{n}}{x} \]

   9. Applied rewrites 41.3%

      \[\leadsto \frac{\frac{1}{n}}{x} \]

   if 7.5000000000000002e279 < x

   1. Initial program 53.5%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]

   2. Taylor expanded in n around inf

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{\color{blue}{n}} \]

      2. lower--.f64 N/A

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{n} \]

      3. lower-log.f64 N/A

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{n} \]

      4. lower-+.f64 N/A

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{n} \]

      5. lower-log.f64 58.9%

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{n} \]

   4. Applied rewrites 58.9%

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]

   5. Taylor expanded in x around inf

      \[\leadsto \frac{\frac{1}{n} - \frac{1}{2} \cdot \frac{1}{n \cdot x}}{\color{blue}{x}} \]
   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 29.1%

         \[\leadsto \frac{\frac{1}{n} - 0.5 \cdot \frac{1}{n \cdot x}}{x} \]

   7. Applied rewrites 29.1%

      \[\leadsto \frac{\frac{1}{n} - 0.5 \cdot \frac{1}{n \cdot x}}{\color{blue}{x}} \]

   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 19.3%

         \[\leadsto \frac{\frac{-0.5}{n \cdot x}}{x} \]

   10. Applied rewrites 19.3%

       \[\leadsto \frac{\frac{-0.5}{n \cdot x}}{x} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 18: 41.3% accurate, 5.6× 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 53.5%

   \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]

2. Taylor expanded in n around inf

   \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
3. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \frac{\log \left(1 + x\right) - \log x}{\color{blue}{n}} \]

   2. lower--.f64 N/A

      \[\leadsto \frac{\log \left(1 + x\right) - \log x}{n} \]

   3. lower-log.f64 N/A

      \[\leadsto \frac{\log \left(1 + x\right) - \log x}{n} \]

   4. lower-+.f64 N/A

      \[\leadsto \frac{\log \left(1 + x\right) - \log x}{n} \]

   5. lower-log.f64 58.9%

      \[\leadsto \frac{\log \left(1 + x\right) - \log x}{n} \]

4. Applied rewrites 58.9%

   \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]

5. Taylor expanded in x around inf

   \[\leadsto \frac{1}{\color{blue}{n \cdot x}} \]
6. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \frac{1}{n \cdot \color{blue}{x}} \]

   2. lower-*.f64 40.7%

      \[\leadsto \frac{1}{n \cdot x} \]

7. Applied rewrites 40.7%

   \[\leadsto \frac{1}{\color{blue}{n \cdot x}} \]
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 41.3%

      \[\leadsto \frac{\frac{1}{n}}{x} \]

9. Applied rewrites 41.3%

   \[\leadsto \frac{\frac{1}{n}}{x} \]
10. Add Preprocessing

Alternative 19: 40.7% 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 53.5%

   \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]

2. Taylor expanded in n around inf

   \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
3. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \frac{\log \left(1 + x\right) - \log x}{\color{blue}{n}} \]

   2. lower--.f64 N/A

      \[\leadsto \frac{\log \left(1 + x\right) - \log x}{n} \]

   3. lower-log.f64 N/A

      \[\leadsto \frac{\log \left(1 + x\right) - \log x}{n} \]

   4. lower-+.f64 N/A

      \[\leadsto \frac{\log \left(1 + x\right) - \log x}{n} \]

   5. lower-log.f64 58.9%

      \[\leadsto \frac{\log \left(1 + x\right) - \log x}{n} \]

4. Applied rewrites 58.9%

   \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]

5. Taylor expanded in x around inf

   \[\leadsto \frac{1}{\color{blue}{n \cdot x}} \]
6. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \frac{1}{n \cdot \color{blue}{x}} \]

   2. lower-*.f64 40.7%

      \[\leadsto \frac{1}{n \cdot x} \]

7. Applied rewrites 40.7%

   \[\leadsto \frac{1}{\color{blue}{n \cdot x}} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2025212 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  :precision binary64
  (- (pow (+ x 1.0) (/ 1.0 n)) (pow x (/ 1.0 n))))