2nthrt (problem 3.4.6)

Percentage Accurate: 53.4% → 87.1%

Time: 28.9s

Alternatives: 17

Speedup: 1.9×

• Could not converge on a ground truth

  1. x = -1.9859192395849068e-56
  2. n = 3.681344216598277e-31

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 53.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 87.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 110:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right)}{n} + \frac{\frac{\mathsf{fma}\left(\frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{3} - {\log x}^{3}}{n}, 0.16666666666666666, 0.5 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}\right)\right)}{n} - \log x}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{{x}^{\left({n}^{-1}\right)}}{n}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 110.0)
   (+
    (/ (log1p x) n)
    (/
     (-
      (/
       (fma
        (/ (- (pow (log1p x) 3.0) (pow (log x) 3.0)) n)
        0.16666666666666666
        (* 0.5 (- (pow (log1p x) 2.0) (pow (log x) 2.0))))
       n)
      (log x))
     n))
   (/ (/ (pow x (pow n -1.0)) n) x)))

C

double code(double x, double n) {
	double tmp;
	if (x <= 110.0) {
		tmp = (log1p(x) / n) + (((fma(((pow(log1p(x), 3.0) - pow(log(x), 3.0)) / n), 0.16666666666666666, (0.5 * (pow(log1p(x), 2.0) - pow(log(x), 2.0)))) / n) - log(x)) / n);
	} else {
		tmp = (pow(x, pow(n, -1.0)) / n) / x;
	}
	return tmp;
}

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 110.0)
		tmp = Float64(Float64(log1p(x) / n) + Float64(Float64(Float64(fma(Float64(Float64((log1p(x) ^ 3.0) - (log(x) ^ 3.0)) / n), 0.16666666666666666, Float64(0.5 * Float64((log1p(x) ^ 2.0) - (log(x) ^ 2.0)))) / n) - log(x)) / n));
	else
		tmp = Float64(Float64((x ^ (n ^ -1.0)) / n) / x);
	end
	return tmp
end

Wolfram

code[x_, n_] := If[LessEqual[x, 110.0], N[(N[(N[Log[1 + x], $MachinePrecision] / n), $MachinePrecision] + N[(N[(N[(N[(N[(N[(N[Power[N[Log[1 + x], $MachinePrecision], 3.0], $MachinePrecision] - N[Power[N[Log[x], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision] * 0.16666666666666666 + N[(0.5 * N[(N[Power[N[Log[1 + x], $MachinePrecision], 2.0], $MachinePrecision] - N[Power[N[Log[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision] - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision], N[(N[(N[Power[x, N[Power[n, -1.0], $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 110

   1. Initial program 45.4%

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

   3. Taylor expanded in n around -inf

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

   5. Applied rewrites 76.8%

      \[\leadsto \color{blue}{\frac{\left(\left(-\mathsf{log1p}\left(x\right)\right) - \frac{\mathsf{fma}\left(0.16666666666666666, \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{3} - {\log x}^{3}}{n}, \left({\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}\right) \cdot 0.5\right)}{n}\right) + \log x}{-n}} \]
   6. Step-by-step derivation

   7. Applied rewrites 76.9%

      \[\leadsto \frac{\mathsf{log1p}\left(x\right)}{n} - \color{blue}{\frac{\frac{\mathsf{fma}\left(\frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{3} - {\log x}^{3}}{n}, 0.16666666666666666, 0.5 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}\right)\right)}{n} - \log x}{-n}} \]

   if 110 < x

   1. Initial program 60.3%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. log-rec N/A

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

      3. mul-1-neg N/A

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

      4. associate-*r/ N/A

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

      5. mul-1-neg N/A

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

      6. distribute-lft-neg-in N/A

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

      7. metadata-eval N/A

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

      8. *-lft-identity N/A

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

      9. lower-exp.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-log.f64 N/A

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

      12. lower-*.f64 98.2

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

   5. Applied rewrites 98.2%

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

   7. Applied rewrites 99.3%

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

   9. Applied rewrites 99.4%

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

4. Final simplification 86.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 110:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right)}{n} + \frac{\frac{\mathsf{fma}\left(\frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{3} - {\log x}^{3}}{n}, 0.16666666666666666, 0.5 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}\right)\right)}{n} - \log x}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{{x}^{\left({n}^{-1}\right)}}{n}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 87.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 116:\\ \;\;\;\;\frac{\left(\mathsf{log1p}\left(x\right) + \frac{\mathsf{fma}\left(0.16666666666666666, \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{3} - {\log x}^{3}}{n}, \left({\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}\right) \cdot 0.5\right)}{n}\right) - \log x}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{{x}^{\left({n}^{-1}\right)}}{n}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 116.0)
   (/
    (-
     (+
      (log1p x)
      (/
       (fma
        0.16666666666666666
        (/ (- (pow (log1p x) 3.0) (pow (log x) 3.0)) n)
        (* (- (pow (log1p x) 2.0) (pow (log x) 2.0)) 0.5))
       n))
     (log x))
    n)
   (/ (/ (pow x (pow n -1.0)) n) x)))

C

double code(double x, double n) {
	double tmp;
	if (x <= 116.0) {
		tmp = ((log1p(x) + (fma(0.16666666666666666, ((pow(log1p(x), 3.0) - pow(log(x), 3.0)) / n), ((pow(log1p(x), 2.0) - pow(log(x), 2.0)) * 0.5)) / n)) - log(x)) / n;
	} else {
		tmp = (pow(x, pow(n, -1.0)) / n) / x;
	}
	return tmp;
}

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 116.0)
		tmp = Float64(Float64(Float64(log1p(x) + Float64(fma(0.16666666666666666, Float64(Float64((log1p(x) ^ 3.0) - (log(x) ^ 3.0)) / n), Float64(Float64((log1p(x) ^ 2.0) - (log(x) ^ 2.0)) * 0.5)) / n)) - log(x)) / n);
	else
		tmp = Float64(Float64((x ^ (n ^ -1.0)) / n) / x);
	end
	return tmp
end

Wolfram

code[x_, n_] := If[LessEqual[x, 116.0], N[(N[(N[(N[Log[1 + x], $MachinePrecision] + N[(N[(0.16666666666666666 * N[(N[(N[Power[N[Log[1 + x], $MachinePrecision], 3.0], $MachinePrecision] - N[Power[N[Log[x], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision] + N[(N[(N[Power[N[Log[1 + x], $MachinePrecision], 2.0], $MachinePrecision] - N[Power[N[Log[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision] - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], N[(N[(N[Power[x, N[Power[n, -1.0], $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 116

   1. Initial program 45.4%

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

   3. Taylor expanded in n around -inf

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

   5. Applied rewrites 76.8%

      \[\leadsto \color{blue}{\frac{\left(\left(-\mathsf{log1p}\left(x\right)\right) - \frac{\mathsf{fma}\left(0.16666666666666666, \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{3} - {\log x}^{3}}{n}, \left({\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}\right) \cdot 0.5\right)}{n}\right) + \log x}{-n}} \]

   if 116 < x

   1. Initial program 60.3%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. log-rec N/A

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

      3. mul-1-neg N/A

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

      4. associate-*r/ N/A

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

      5. mul-1-neg N/A

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

      6. distribute-lft-neg-in N/A

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

      7. metadata-eval N/A

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

      8. *-lft-identity N/A

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

      9. lower-exp.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-log.f64 N/A

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

      12. lower-*.f64 98.2

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

   5. Applied rewrites 98.2%

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

   7. Applied rewrites 99.3%

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

   9. Applied rewrites 99.4%

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

4. Final simplification 86.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 116:\\ \;\;\;\;\frac{\left(\mathsf{log1p}\left(x\right) + \frac{\mathsf{fma}\left(0.16666666666666666, \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{3} - {\log x}^{3}}{n}, \left({\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}\right) \cdot 0.5\right)}{n}\right) - \log x}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{{x}^{\left({n}^{-1}\right)}}{n}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 86.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.96:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{{\log x}^{3}}{n}, -0.16666666666666666, {\log x}^{2} \cdot -0.5\right)}{n}, -1, \log x\right)}{-n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{{x}^{\left({n}^{-1}\right)}}{n}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 0.96)
   (/
    (fma
     (/
      (fma
       (/ (pow (log x) 3.0) n)
       -0.16666666666666666
       (* (pow (log x) 2.0) -0.5))
      n)
     -1.0
     (log x))
    (- n))
   (/ (/ (pow x (pow n -1.0)) n) x)))

C

double code(double x, double n) {
	double tmp;
	if (x <= 0.96) {
		tmp = fma((fma((pow(log(x), 3.0) / n), -0.16666666666666666, (pow(log(x), 2.0) * -0.5)) / n), -1.0, log(x)) / -n;
	} else {
		tmp = (pow(x, pow(n, -1.0)) / n) / x;
	}
	return tmp;
}

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 0.96)
		tmp = Float64(fma(Float64(fma(Float64((log(x) ^ 3.0) / n), -0.16666666666666666, Float64((log(x) ^ 2.0) * -0.5)) / n), -1.0, log(x)) / Float64(-n));
	else
		tmp = Float64(Float64((x ^ (n ^ -1.0)) / n) / x);
	end
	return tmp
end

Wolfram

code[x_, n_] := If[LessEqual[x, 0.96], N[(N[(N[(N[(N[(N[Power[N[Log[x], $MachinePrecision], 3.0], $MachinePrecision] / n), $MachinePrecision] * -0.16666666666666666 + N[(N[Power[N[Log[x], $MachinePrecision], 2.0], $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision] * -1.0 + N[Log[x], $MachinePrecision]), $MachinePrecision] / (-n)), $MachinePrecision], N[(N[(N[Power[x, N[Power[n, -1.0], $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.95999999999999996

   1. Initial program 45.6%

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

   3. Taylor expanded in n around -inf

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

   5. Applied rewrites 76.7%

      \[\leadsto \color{blue}{\frac{\left(\left(-\mathsf{log1p}\left(x\right)\right) - \frac{\mathsf{fma}\left(0.16666666666666666, \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{3} - {\log x}^{3}}{n}, \left({\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}\right) \cdot 0.5\right)}{n}\right) + \log x}{-n}} \]

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 74.8%

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

   9. Taylor expanded in n around -inf

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

   11. Applied rewrites 75.5%

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

   if 0.95999999999999996 < x

   1. Initial program 59.8%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. log-rec N/A

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

      3. mul-1-neg N/A

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

      4. associate-*r/ N/A

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

      5. mul-1-neg N/A

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

      6. distribute-lft-neg-in N/A

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

      7. metadata-eval N/A

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

      8. *-lft-identity N/A

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

      9. lower-exp.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-log.f64 N/A

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

      12. lower-*.f64 97.5

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

   5. Applied rewrites 97.5%

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

   7. Applied rewrites 98.6%

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

   9. Applied rewrites 98.7%

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

Alternative 4: 82.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\log x}{n}\\ \mathbf{if}\;{n}^{-1} \leq -1 \cdot 10^{-30}:\\ \;\;\;\;\frac{\frac{e^{t\_0}}{x}}{n}\\ \mathbf{elif}\;{n}^{-1} \leq 0.002:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right)}{n} - t\_0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{0.5}{n} + -0.5, x, 1\right)}{n}, x, 1\right) - {x}^{\left({n}^{-1}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (/ (log x) n)))
   (if (<= (pow n -1.0) -1e-30)
     (/ (/ (exp t_0) x) n)
     (if (<= (pow n -1.0) 0.002)
       (- (/ (log1p x) n) t_0)
       (-
        (fma (/ (fma (+ (/ 0.5 n) -0.5) x 1.0) n) x 1.0)
        (pow x (pow n -1.0)))))))

C

double code(double x, double n) {
	double t_0 = log(x) / n;
	double tmp;
	if (pow(n, -1.0) <= -1e-30) {
		tmp = (exp(t_0) / x) / n;
	} else if (pow(n, -1.0) <= 0.002) {
		tmp = (log1p(x) / n) - t_0;
	} else {
		tmp = fma((fma(((0.5 / n) + -0.5), x, 1.0) / n), x, 1.0) - pow(x, pow(n, -1.0));
	}
	return tmp;
}

Julia

function code(x, n)
	t_0 = Float64(log(x) / n)
	tmp = 0.0
	if ((n ^ -1.0) <= -1e-30)
		tmp = Float64(Float64(exp(t_0) / x) / n);
	elseif ((n ^ -1.0) <= 0.002)
		tmp = Float64(Float64(log1p(x) / n) - t_0);
	else
		tmp = Float64(fma(Float64(fma(Float64(Float64(0.5 / n) + -0.5), x, 1.0) / n), x, 1.0) - (x ^ (n ^ -1.0)));
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(N[Log[x], $MachinePrecision] / n), $MachinePrecision]}, If[LessEqual[N[Power[n, -1.0], $MachinePrecision], -1e-30], N[(N[(N[Exp[t$95$0], $MachinePrecision] / x), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[Power[n, -1.0], $MachinePrecision], 0.002], N[(N[(N[Log[1 + x], $MachinePrecision] / n), $MachinePrecision] - t$95$0), $MachinePrecision], N[(N[(N[(N[(N[(N[(0.5 / n), $MachinePrecision] + -0.5), $MachinePrecision] * x + 1.0), $MachinePrecision] / n), $MachinePrecision] * x + 1.0), $MachinePrecision] - N[Power[x, N[Power[n, -1.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -1e-30

   1. Initial program 91.3%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. log-rec N/A

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

      3. mul-1-neg N/A

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

      4. associate-*r/ N/A

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

      5. mul-1-neg N/A

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

      6. distribute-lft-neg-in N/A

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

      7. metadata-eval N/A

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

      8. *-lft-identity N/A

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

      9. lower-exp.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-log.f64 N/A

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

      12. lower-*.f64 97.4

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

   5. Applied rewrites 97.4%

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

   7. Applied rewrites 97.4%

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

   if -1e-30 < (/.f64 #s(literal 1 binary64) n) < 2e-3

   1. Initial program 26.7%

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

   3. Taylor expanded in n around -inf

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

   5. Applied rewrites 74.8%

      \[\leadsto \color{blue}{\frac{\left(\left(-\mathsf{log1p}\left(x\right)\right) - \frac{\mathsf{fma}\left(0.16666666666666666, \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{3} - {\log x}^{3}}{n}, \left({\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}\right) \cdot 0.5\right)}{n}\right) + \log x}{-n}} \]
   6. Step-by-step derivation

   7. Applied rewrites 74.9%

      \[\leadsto \frac{\left(-\mathsf{log1p}\left(x\right)\right) - \frac{\mathsf{fma}\left(\frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{3} - {\log x}^{3}}{n}, 0.16666666666666666, 0.5 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}\right)\right)}{n}}{-n} + \color{blue}{\frac{-\log x}{n}} \]

   8. Taylor expanded in n around inf

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

   10. Applied rewrites 74.6%

       \[\leadsto \frac{\mathsf{log1p}\left(x\right)}{n} + \frac{\color{blue}{-\log x}}{n} \]

   if 2e-3 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 65.3%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

   5. Applied rewrites 55.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-0.5 + \frac{0.5}{n}}{n}, x, \frac{1}{n}\right), x, 1\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 80.5%

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

4. Final simplification 82.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{n}^{-1} \leq -1 \cdot 10^{-30}:\\ \;\;\;\;\frac{\frac{e^{\frac{\log x}{n}}}{x}}{n}\\ \mathbf{elif}\;{n}^{-1} \leq 0.002:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right)}{n} - \frac{\log x}{n}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{0.5}{n} + -0.5, x, 1\right)}{n}, x, 1\right) - {x}^{\left({n}^{-1}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 82.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;{n}^{-1} \leq -1 \cdot 10^{-30}:\\ \;\;\;\;\frac{\frac{{\left({x}^{\left(\frac{-1}{n}\right)}\right)}^{-1}}{n}}{x}\\ \mathbf{elif}\;{n}^{-1} \leq 0.002:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right)}{n} - \frac{\log x}{n}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{0.5}{n} + -0.5, x, 1\right)}{n}, x, 1\right) - {x}^{\left({n}^{-1}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= (pow n -1.0) -1e-30)
   (/ (/ (pow (pow x (/ -1.0 n)) -1.0) n) x)
   (if (<= (pow n -1.0) 0.002)
     (- (/ (log1p x) n) (/ (log x) n))
     (-
      (fma (/ (fma (+ (/ 0.5 n) -0.5) x 1.0) n) x 1.0)
      (pow x (pow n -1.0))))))

C

double code(double x, double n) {
	double tmp;
	if (pow(n, -1.0) <= -1e-30) {
		tmp = (pow(pow(x, (-1.0 / n)), -1.0) / n) / x;
	} else if (pow(n, -1.0) <= 0.002) {
		tmp = (log1p(x) / n) - (log(x) / n);
	} else {
		tmp = fma((fma(((0.5 / n) + -0.5), x, 1.0) / n), x, 1.0) - pow(x, pow(n, -1.0));
	}
	return tmp;
}

Julia

function code(x, n)
	tmp = 0.0
	if ((n ^ -1.0) <= -1e-30)
		tmp = Float64(Float64(((x ^ Float64(-1.0 / n)) ^ -1.0) / n) / x);
	elseif ((n ^ -1.0) <= 0.002)
		tmp = Float64(Float64(log1p(x) / n) - Float64(log(x) / n));
	else
		tmp = Float64(fma(Float64(fma(Float64(Float64(0.5 / n) + -0.5), x, 1.0) / n), x, 1.0) - (x ^ (n ^ -1.0)));
	end
	return tmp
end

Wolfram

code[x_, n_] := If[LessEqual[N[Power[n, -1.0], $MachinePrecision], -1e-30], N[(N[(N[Power[N[Power[x, N[(-1.0 / n), $MachinePrecision]], $MachinePrecision], -1.0], $MachinePrecision] / n), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[N[Power[n, -1.0], $MachinePrecision], 0.002], N[(N[(N[Log[1 + x], $MachinePrecision] / n), $MachinePrecision] - N[(N[Log[x], $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(N[(0.5 / n), $MachinePrecision] + -0.5), $MachinePrecision] * x + 1.0), $MachinePrecision] / n), $MachinePrecision] * x + 1.0), $MachinePrecision] - N[Power[x, N[Power[n, -1.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -1e-30

   1. Initial program 91.3%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. log-rec N/A

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

      3. mul-1-neg N/A

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

      4. associate-*r/ N/A

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

      5. mul-1-neg N/A

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

      6. distribute-lft-neg-in N/A

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

      7. metadata-eval N/A

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

      8. *-lft-identity N/A

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

      9. lower-exp.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-log.f64 N/A

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

      12. lower-*.f64 97.4

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

   5. Applied rewrites 97.4%

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

   7. Applied rewrites 97.4%

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

   9. Applied rewrites 97.4%

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

   11. Applied rewrites 97.4%

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

   if -1e-30 < (/.f64 #s(literal 1 binary64) n) < 2e-3

   1. Initial program 26.7%

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

   3. Taylor expanded in n around -inf

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

   5. Applied rewrites 74.8%

      \[\leadsto \color{blue}{\frac{\left(\left(-\mathsf{log1p}\left(x\right)\right) - \frac{\mathsf{fma}\left(0.16666666666666666, \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{3} - {\log x}^{3}}{n}, \left({\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}\right) \cdot 0.5\right)}{n}\right) + \log x}{-n}} \]
   6. Step-by-step derivation

   7. Applied rewrites 74.9%

      \[\leadsto \frac{\left(-\mathsf{log1p}\left(x\right)\right) - \frac{\mathsf{fma}\left(\frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{3} - {\log x}^{3}}{n}, 0.16666666666666666, 0.5 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}\right)\right)}{n}}{-n} + \color{blue}{\frac{-\log x}{n}} \]

   8. Taylor expanded in n around inf

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

   10. Applied rewrites 74.6%

       \[\leadsto \frac{\mathsf{log1p}\left(x\right)}{n} + \frac{\color{blue}{-\log x}}{n} \]

   if 2e-3 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 65.3%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

   5. Applied rewrites 55.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-0.5 + \frac{0.5}{n}}{n}, x, \frac{1}{n}\right), x, 1\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 80.5%

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

4. Final simplification 82.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{n}^{-1} \leq -1 \cdot 10^{-30}:\\ \;\;\;\;\frac{\frac{{\left({x}^{\left(\frac{-1}{n}\right)}\right)}^{-1}}{n}}{x}\\ \mathbf{elif}\;{n}^{-1} \leq 0.002:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right)}{n} - \frac{\log x}{n}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{0.5}{n} + -0.5, x, 1\right)}{n}, x, 1\right) - {x}^{\left({n}^{-1}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 82.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;{n}^{-1} \leq -1 \cdot 10^{-30}:\\ \;\;\;\;\frac{\frac{{\left({x}^{\left(\frac{-1}{n}\right)}\right)}^{-1}}{n}}{x}\\ \mathbf{elif}\;{n}^{-1} \leq 0.002:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right) - \log x}{n}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{0.5}{n} + -0.5, x, 1\right)}{n}, x, 1\right) - {x}^{\left({n}^{-1}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= (pow n -1.0) -1e-30)
   (/ (/ (pow (pow x (/ -1.0 n)) -1.0) n) x)
   (if (<= (pow n -1.0) 0.002)
     (/ (- (log1p x) (log x)) n)
     (-
      (fma (/ (fma (+ (/ 0.5 n) -0.5) x 1.0) n) x 1.0)
      (pow x (pow n -1.0))))))

C

double code(double x, double n) {
	double tmp;
	if (pow(n, -1.0) <= -1e-30) {
		tmp = (pow(pow(x, (-1.0 / n)), -1.0) / n) / x;
	} else if (pow(n, -1.0) <= 0.002) {
		tmp = (log1p(x) - log(x)) / n;
	} else {
		tmp = fma((fma(((0.5 / n) + -0.5), x, 1.0) / n), x, 1.0) - pow(x, pow(n, -1.0));
	}
	return tmp;
}

Julia

function code(x, n)
	tmp = 0.0
	if ((n ^ -1.0) <= -1e-30)
		tmp = Float64(Float64(((x ^ Float64(-1.0 / n)) ^ -1.0) / n) / x);
	elseif ((n ^ -1.0) <= 0.002)
		tmp = Float64(Float64(log1p(x) - log(x)) / n);
	else
		tmp = Float64(fma(Float64(fma(Float64(Float64(0.5 / n) + -0.5), x, 1.0) / n), x, 1.0) - (x ^ (n ^ -1.0)));
	end
	return tmp
end

Wolfram

code[x_, n_] := If[LessEqual[N[Power[n, -1.0], $MachinePrecision], -1e-30], N[(N[(N[Power[N[Power[x, N[(-1.0 / n), $MachinePrecision]], $MachinePrecision], -1.0], $MachinePrecision] / n), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[N[Power[n, -1.0], $MachinePrecision], 0.002], N[(N[(N[Log[1 + x], $MachinePrecision] - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], N[(N[(N[(N[(N[(N[(0.5 / n), $MachinePrecision] + -0.5), $MachinePrecision] * x + 1.0), $MachinePrecision] / n), $MachinePrecision] * x + 1.0), $MachinePrecision] - N[Power[x, N[Power[n, -1.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -1e-30

   1. Initial program 91.3%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. log-rec N/A

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

      3. mul-1-neg N/A

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

      4. associate-*r/ N/A

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

      5. mul-1-neg N/A

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

      6. distribute-lft-neg-in N/A

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

      7. metadata-eval N/A

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

      8. *-lft-identity N/A

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

      9. lower-exp.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-log.f64 N/A

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

      12. lower-*.f64 97.4

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

   5. Applied rewrites 97.4%

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

   7. Applied rewrites 97.4%

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

   9. Applied rewrites 97.4%

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

   11. Applied rewrites 97.4%

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

   if -1e-30 < (/.f64 #s(literal 1 binary64) n) < 2e-3

   1. Initial program 26.7%

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

   3. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-log1p.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(x\right)} - \log x}{n} \]

      4. lower-log.f64 74.6

         \[\leadsto \frac{\mathsf{log1p}\left(x\right) - \color{blue}{\log x}}{n} \]

   5. Applied rewrites 74.6%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) - \log x}{n}} \]

   if 2e-3 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 65.3%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

   5. Applied rewrites 55.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-0.5 + \frac{0.5}{n}}{n}, x, \frac{1}{n}\right), x, 1\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 80.5%

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

4. Final simplification 82.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{n}^{-1} \leq -1 \cdot 10^{-30}:\\ \;\;\;\;\frac{\frac{{\left({x}^{\left(\frac{-1}{n}\right)}\right)}^{-1}}{n}}{x}\\ \mathbf{elif}\;{n}^{-1} \leq 0.002:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right) - \log x}{n}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{0.5}{n} + -0.5, x, 1\right)}{n}, x, 1\right) - {x}^{\left({n}^{-1}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 71.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 8 \cdot 10^{-257}:\\ \;\;\;\;\left(\frac{x}{n} + 1\right) - {x}^{\left({n}^{-1}\right)}\\ \mathbf{elif}\;x \leq 0.15:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{{\left({x}^{\left(\frac{-1}{n}\right)}\right)}^{-1}}{n}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 8e-257)
   (- (+ (/ x n) 1.0) (pow x (pow n -1.0)))
   (if (<= x 0.15)
     (/ (- x (log x)) n)
     (/ (/ (pow (pow x (/ -1.0 n)) -1.0) n) x))))

C

double code(double x, double n) {
	double tmp;
	if (x <= 8e-257) {
		tmp = ((x / n) + 1.0) - pow(x, pow(n, -1.0));
	} else if (x <= 0.15) {
		tmp = (x - log(x)) / n;
	} else {
		tmp = (pow(pow(x, (-1.0 / n)), -1.0) / n) / x;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, n)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if (x <= 8d-257) then
        tmp = ((x / n) + 1.0d0) - (x ** (n ** (-1.0d0)))
    else if (x <= 0.15d0) then
        tmp = (x - log(x)) / n
    else
        tmp = (((x ** ((-1.0d0) / n)) ** (-1.0d0)) / n) / x
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (x <= 8e-257) {
		tmp = ((x / n) + 1.0) - Math.pow(x, Math.pow(n, -1.0));
	} else if (x <= 0.15) {
		tmp = (x - Math.log(x)) / n;
	} else {
		tmp = (Math.pow(Math.pow(x, (-1.0 / n)), -1.0) / n) / x;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 8e-257:
		tmp = ((x / n) + 1.0) - math.pow(x, math.pow(n, -1.0))
	elif x <= 0.15:
		tmp = (x - math.log(x)) / n
	else:
		tmp = (math.pow(math.pow(x, (-1.0 / n)), -1.0) / n) / x
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 8e-257)
		tmp = Float64(Float64(Float64(x / n) + 1.0) - (x ^ (n ^ -1.0)));
	elseif (x <= 0.15)
		tmp = Float64(Float64(x - log(x)) / n);
	else
		tmp = Float64(Float64(((x ^ Float64(-1.0 / n)) ^ -1.0) / n) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (x <= 8e-257)
		tmp = ((x / n) + 1.0) - (x ^ (n ^ -1.0));
	elseif (x <= 0.15)
		tmp = (x - log(x)) / n;
	else
		tmp = (((x ^ (-1.0 / n)) ^ -1.0) / n) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 8e-257], N[(N[(N[(x / n), $MachinePrecision] + 1.0), $MachinePrecision] - N[Power[x, N[Power[n, -1.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.15], N[(N[(x - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], N[(N[(N[Power[N[Power[x, N[(-1.0 / n), $MachinePrecision]], $MachinePrecision], -1.0], $MachinePrecision] / n), $MachinePrecision] / x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 7.9999999999999998e-257

   1. Initial program 67.1%

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

   3. Taylor expanded in x around 0

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

      1. *-rgt-identity N/A

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

      2. associate-*r/ N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. associate-*r/ N/A

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

      6. *-rgt-identity N/A

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

      7. lower-/.f64 68.2

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

   5. Applied rewrites 68.2%

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

   if 7.9999999999999998e-257 < x < 0.149999999999999994

   1. Initial program 41.1%

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

   3. Taylor expanded in n around -inf

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

   5. Applied rewrites 80.4%

      \[\leadsto \color{blue}{\frac{\left(\left(-\mathsf{log1p}\left(x\right)\right) - \frac{\mathsf{fma}\left(0.16666666666666666, \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{3} - {\log x}^{3}}{n}, \left({\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}\right) \cdot 0.5\right)}{n}\right) + \log x}{-n}} \]
   6. Step-by-step derivation

   7. Applied rewrites 80.4%

      \[\leadsto \frac{\mathsf{log1p}\left(x\right)}{n} - \color{blue}{\frac{\frac{\mathsf{fma}\left(\frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{3} - {\log x}^{3}}{n}, 0.16666666666666666, 0.5 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}\right)\right)}{n} - \log x}{-n}} \]

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 78.7%

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

   11. Taylor expanded in n around inf

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

   13. Applied rewrites 53.7%

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

   if 0.149999999999999994 < x

   1. Initial program 59.8%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. log-rec N/A

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

      3. mul-1-neg N/A

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

      4. associate-*r/ N/A

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

      5. mul-1-neg N/A

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

      6. distribute-lft-neg-in N/A

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

      7. metadata-eval N/A

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

      8. *-lft-identity N/A

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

      9. lower-exp.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-log.f64 N/A

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

      12. lower-*.f64 97.5

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

   5. Applied rewrites 97.5%

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

   7. Applied rewrites 98.6%

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

   9. Applied rewrites 98.7%

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

   11. Applied rewrites 98.7%

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

4. Final simplification 74.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 8 \cdot 10^{-257}:\\ \;\;\;\;\left(\frac{x}{n} + 1\right) - {x}^{\left({n}^{-1}\right)}\\ \mathbf{elif}\;x \leq 0.15:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{{\left({x}^{\left(\frac{-1}{n}\right)}\right)}^{-1}}{n}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 59.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 8 \cdot 10^{-257}:\\ \;\;\;\;\left(\frac{x}{n} + 1\right) - {x}^{\left({n}^{-1}\right)}\\ \mathbf{elif}\;x \leq 0.15:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{elif}\;x \leq 5.2 \cdot 10^{+205}:\\ \;\;\;\;\frac{\frac{\frac{\log x}{n} + 1}{x}}{n}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 8e-257)
   (- (+ (/ x n) 1.0) (pow x (pow n -1.0)))
   (if (<= x 0.15)
     (/ (- x (log x)) n)
     (if (<= x 5.2e+205) (/ (/ (+ (/ (log x) n) 1.0) x) n) 0.0))))

C

double code(double x, double n) {
	double tmp;
	if (x <= 8e-257) {
		tmp = ((x / n) + 1.0) - pow(x, pow(n, -1.0));
	} else if (x <= 0.15) {
		tmp = (x - log(x)) / n;
	} else if (x <= 5.2e+205) {
		tmp = (((log(x) / n) + 1.0) / x) / n;
	} else {
		tmp = 0.0;
	}
	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 <= 8d-257) then
        tmp = ((x / n) + 1.0d0) - (x ** (n ** (-1.0d0)))
    else if (x <= 0.15d0) then
        tmp = (x - log(x)) / n
    else if (x <= 5.2d+205) then
        tmp = (((log(x) / n) + 1.0d0) / x) / n
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (x <= 8e-257) {
		tmp = ((x / n) + 1.0) - Math.pow(x, Math.pow(n, -1.0));
	} else if (x <= 0.15) {
		tmp = (x - Math.log(x)) / n;
	} else if (x <= 5.2e+205) {
		tmp = (((Math.log(x) / n) + 1.0) / x) / n;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 8e-257:
		tmp = ((x / n) + 1.0) - math.pow(x, math.pow(n, -1.0))
	elif x <= 0.15:
		tmp = (x - math.log(x)) / n
	elif x <= 5.2e+205:
		tmp = (((math.log(x) / n) + 1.0) / x) / n
	else:
		tmp = 0.0
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 8e-257)
		tmp = Float64(Float64(Float64(x / n) + 1.0) - (x ^ (n ^ -1.0)));
	elseif (x <= 0.15)
		tmp = Float64(Float64(x - log(x)) / n);
	elseif (x <= 5.2e+205)
		tmp = Float64(Float64(Float64(Float64(log(x) / n) + 1.0) / x) / n);
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (x <= 8e-257)
		tmp = ((x / n) + 1.0) - (x ^ (n ^ -1.0));
	elseif (x <= 0.15)
		tmp = (x - log(x)) / n;
	elseif (x <= 5.2e+205)
		tmp = (((log(x) / n) + 1.0) / x) / n;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 8e-257], N[(N[(N[(x / n), $MachinePrecision] + 1.0), $MachinePrecision] - N[Power[x, N[Power[n, -1.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.15], N[(N[(x - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[x, 5.2e+205], N[(N[(N[(N[(N[Log[x], $MachinePrecision] / n), $MachinePrecision] + 1.0), $MachinePrecision] / x), $MachinePrecision] / n), $MachinePrecision], 0.0]]]

Derivation

1. Split input into 4 regimes

2. if x < 7.9999999999999998e-257

   1. Initial program 67.1%

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

   3. Taylor expanded in x around 0

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

      1. *-rgt-identity N/A

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

      2. associate-*r/ N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. associate-*r/ N/A

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

      6. *-rgt-identity N/A

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

      7. lower-/.f64 68.2

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

   5. Applied rewrites 68.2%

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

   if 7.9999999999999998e-257 < x < 0.149999999999999994

   1. Initial program 41.1%

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

   3. Taylor expanded in n around -inf

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

   5. Applied rewrites 80.4%

      \[\leadsto \color{blue}{\frac{\left(\left(-\mathsf{log1p}\left(x\right)\right) - \frac{\mathsf{fma}\left(0.16666666666666666, \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{3} - {\log x}^{3}}{n}, \left({\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}\right) \cdot 0.5\right)}{n}\right) + \log x}{-n}} \]
   6. Step-by-step derivation

   7. Applied rewrites 80.4%

      \[\leadsto \frac{\mathsf{log1p}\left(x\right)}{n} - \color{blue}{\frac{\frac{\mathsf{fma}\left(\frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{3} - {\log x}^{3}}{n}, 0.16666666666666666, 0.5 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}\right)\right)}{n} - \log x}{-n}} \]

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 78.7%

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

   11. Taylor expanded in n around inf

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

   13. Applied rewrites 53.7%

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

   if 0.149999999999999994 < x < 5.1999999999999998e205

   1. Initial program 44.9%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. log-rec N/A

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

      3. mul-1-neg N/A

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

      4. associate-*r/ N/A

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

      5. mul-1-neg N/A

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

      6. distribute-lft-neg-in N/A

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

      7. metadata-eval N/A

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

      8. *-lft-identity N/A

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

      9. lower-exp.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-log.f64 N/A

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

      12. lower-*.f64 96.3

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

   5. Applied rewrites 96.3%

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

   6. Taylor expanded in n around inf

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

   8. Applied rewrites 76.9%

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

   if 5.1999999999999998e205 < x

   1. Initial program 91.8%

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

   3. Taylor expanded in n around -inf

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

   5. Applied rewrites 91.8%

      \[\leadsto \color{blue}{\frac{\left(\left(-\mathsf{log1p}\left(x\right)\right) - \frac{\mathsf{fma}\left(0.16666666666666666, \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{3} - {\log x}^{3}}{n}, \left({\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}\right) \cdot 0.5\right)}{n}\right) + \log x}{-n}} \]
   6. Step-by-step derivation

   7. Applied rewrites 88.9%

      \[\leadsto \frac{\left(-\mathsf{log1p}\left(x\right)\right) - \frac{\mathsf{fma}\left(\frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{3} - {\log x}^{3}}{n}, 0.16666666666666666, 0.5 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}\right)\right)}{n}}{-n} + \color{blue}{\frac{-\log x}{n}} \]

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 88.9%

       \[\leadsto -0 \cdot \frac{\log x}{n} \]
3. Recombined 4 regimes into one program.

4. Final simplification 66.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 8 \cdot 10^{-257}:\\ \;\;\;\;\left(\frac{x}{n} + 1\right) - {x}^{\left({n}^{-1}\right)}\\ \mathbf{elif}\;x \leq 0.15:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{elif}\;x \leq 5.2 \cdot 10^{+205}:\\ \;\;\;\;\frac{\frac{\frac{\log x}{n} + 1}{x}}{n}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 60.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 8 \cdot 10^{-257}:\\ \;\;\;\;\left(\frac{x}{n} + 1\right) - {x}^{\left({n}^{-1}\right)}\\ \mathbf{elif}\;x \leq 0.95:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{elif}\;x \leq 5.2 \cdot 10^{+205}:\\ \;\;\;\;\frac{\frac{1 - \frac{0.5}{x}}{n}}{x}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 8e-257)
   (- (+ (/ x n) 1.0) (pow x (pow n -1.0)))
   (if (<= x 0.95)
     (/ (- x (log x)) n)
     (if (<= x 5.2e+205) (/ (/ (- 1.0 (/ 0.5 x)) n) x) 0.0))))

C

double code(double x, double n) {
	double tmp;
	if (x <= 8e-257) {
		tmp = ((x / n) + 1.0) - pow(x, pow(n, -1.0));
	} else if (x <= 0.95) {
		tmp = (x - log(x)) / n;
	} else if (x <= 5.2e+205) {
		tmp = ((1.0 - (0.5 / x)) / n) / x;
	} else {
		tmp = 0.0;
	}
	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 <= 8d-257) then
        tmp = ((x / n) + 1.0d0) - (x ** (n ** (-1.0d0)))
    else if (x <= 0.95d0) then
        tmp = (x - log(x)) / n
    else if (x <= 5.2d+205) then
        tmp = ((1.0d0 - (0.5d0 / x)) / n) / x
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (x <= 8e-257) {
		tmp = ((x / n) + 1.0) - Math.pow(x, Math.pow(n, -1.0));
	} else if (x <= 0.95) {
		tmp = (x - Math.log(x)) / n;
	} else if (x <= 5.2e+205) {
		tmp = ((1.0 - (0.5 / x)) / n) / x;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 8e-257:
		tmp = ((x / n) + 1.0) - math.pow(x, math.pow(n, -1.0))
	elif x <= 0.95:
		tmp = (x - math.log(x)) / n
	elif x <= 5.2e+205:
		tmp = ((1.0 - (0.5 / x)) / n) / x
	else:
		tmp = 0.0
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 8e-257)
		tmp = Float64(Float64(Float64(x / n) + 1.0) - (x ^ (n ^ -1.0)));
	elseif (x <= 0.95)
		tmp = Float64(Float64(x - log(x)) / n);
	elseif (x <= 5.2e+205)
		tmp = Float64(Float64(Float64(1.0 - Float64(0.5 / x)) / n) / x);
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (x <= 8e-257)
		tmp = ((x / n) + 1.0) - (x ^ (n ^ -1.0));
	elseif (x <= 0.95)
		tmp = (x - log(x)) / n;
	elseif (x <= 5.2e+205)
		tmp = ((1.0 - (0.5 / x)) / n) / x;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 8e-257], N[(N[(N[(x / n), $MachinePrecision] + 1.0), $MachinePrecision] - N[Power[x, N[Power[n, -1.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.95], N[(N[(x - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[x, 5.2e+205], N[(N[(N[(1.0 - N[(0.5 / x), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision] / x), $MachinePrecision], 0.0]]]

Derivation

1. Split input into 4 regimes

2. if x < 7.9999999999999998e-257

   1. Initial program 67.1%

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

   3. Taylor expanded in x around 0

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

      1. *-rgt-identity N/A

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

      2. associate-*r/ N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. associate-*r/ N/A

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

      6. *-rgt-identity N/A

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

      7. lower-/.f64 68.2

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

   5. Applied rewrites 68.2%

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

   if 7.9999999999999998e-257 < x < 0.94999999999999996

   1. Initial program 41.1%

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

   3. Taylor expanded in n around -inf

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

   5. Applied rewrites 80.4%

      \[\leadsto \color{blue}{\frac{\left(\left(-\mathsf{log1p}\left(x\right)\right) - \frac{\mathsf{fma}\left(0.16666666666666666, \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{3} - {\log x}^{3}}{n}, \left({\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}\right) \cdot 0.5\right)}{n}\right) + \log x}{-n}} \]
   6. Step-by-step derivation

   7. Applied rewrites 80.4%

      \[\leadsto \frac{\mathsf{log1p}\left(x\right)}{n} - \color{blue}{\frac{\frac{\mathsf{fma}\left(\frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{3} - {\log x}^{3}}{n}, 0.16666666666666666, 0.5 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}\right)\right)}{n} - \log x}{-n}} \]

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 78.7%

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

   11. Taylor expanded in n around inf

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

   13. Applied rewrites 53.7%

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

   if 0.94999999999999996 < x < 5.1999999999999998e205

   1. Initial program 44.9%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 87.8%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\frac{\frac{-0.5 + \frac{0.5}{n}}{n}}{x}, e^{\frac{\log x}{n}}, \frac{e^{\frac{\log x}{n}}}{n}\right)}{x}} \]

   6. Taylor expanded in n around inf

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

   8. Applied rewrites 76.0%

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

   if 5.1999999999999998e205 < x

   1. Initial program 91.8%

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

   3. Taylor expanded in n around -inf

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

   5. Applied rewrites 91.8%

      \[\leadsto \color{blue}{\frac{\left(\left(-\mathsf{log1p}\left(x\right)\right) - \frac{\mathsf{fma}\left(0.16666666666666666, \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{3} - {\log x}^{3}}{n}, \left({\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}\right) \cdot 0.5\right)}{n}\right) + \log x}{-n}} \]
   6. Step-by-step derivation

   7. Applied rewrites 88.9%

      \[\leadsto \frac{\left(-\mathsf{log1p}\left(x\right)\right) - \frac{\mathsf{fma}\left(\frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{3} - {\log x}^{3}}{n}, 0.16666666666666666, 0.5 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}\right)\right)}{n}}{-n} + \color{blue}{\frac{-\log x}{n}} \]

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 88.9%

       \[\leadsto -0 \cdot \frac{\log x}{n} \]
3. Recombined 4 regimes into one program.

4. Final simplification 66.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 8 \cdot 10^{-257}:\\ \;\;\;\;\left(\frac{x}{n} + 1\right) - {x}^{\left({n}^{-1}\right)}\\ \mathbf{elif}\;x \leq 0.95:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{elif}\;x \leq 5.2 \cdot 10^{+205}:\\ \;\;\;\;\frac{\frac{1 - \frac{0.5}{x}}{n}}{x}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 60.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 8 \cdot 10^{-257}:\\ \;\;\;\;1 - {x}^{\left({n}^{-1}\right)}\\ \mathbf{elif}\;x \leq 0.95:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{elif}\;x \leq 5.2 \cdot 10^{+205}:\\ \;\;\;\;\frac{\frac{1 - \frac{0.5}{x}}{n}}{x}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 8e-257)
   (- 1.0 (pow x (pow n -1.0)))
   (if (<= x 0.95)
     (/ (- x (log x)) n)
     (if (<= x 5.2e+205) (/ (/ (- 1.0 (/ 0.5 x)) n) x) 0.0))))

C

double code(double x, double n) {
	double tmp;
	if (x <= 8e-257) {
		tmp = 1.0 - pow(x, pow(n, -1.0));
	} else if (x <= 0.95) {
		tmp = (x - log(x)) / n;
	} else if (x <= 5.2e+205) {
		tmp = ((1.0 - (0.5 / x)) / n) / x;
	} else {
		tmp = 0.0;
	}
	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 <= 8d-257) then
        tmp = 1.0d0 - (x ** (n ** (-1.0d0)))
    else if (x <= 0.95d0) then
        tmp = (x - log(x)) / n
    else if (x <= 5.2d+205) then
        tmp = ((1.0d0 - (0.5d0 / x)) / n) / x
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (x <= 8e-257) {
		tmp = 1.0 - Math.pow(x, Math.pow(n, -1.0));
	} else if (x <= 0.95) {
		tmp = (x - Math.log(x)) / n;
	} else if (x <= 5.2e+205) {
		tmp = ((1.0 - (0.5 / x)) / n) / x;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 8e-257:
		tmp = 1.0 - math.pow(x, math.pow(n, -1.0))
	elif x <= 0.95:
		tmp = (x - math.log(x)) / n
	elif x <= 5.2e+205:
		tmp = ((1.0 - (0.5 / x)) / n) / x
	else:
		tmp = 0.0
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 8e-257)
		tmp = Float64(1.0 - (x ^ (n ^ -1.0)));
	elseif (x <= 0.95)
		tmp = Float64(Float64(x - log(x)) / n);
	elseif (x <= 5.2e+205)
		tmp = Float64(Float64(Float64(1.0 - Float64(0.5 / x)) / n) / x);
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (x <= 8e-257)
		tmp = 1.0 - (x ^ (n ^ -1.0));
	elseif (x <= 0.95)
		tmp = (x - log(x)) / n;
	elseif (x <= 5.2e+205)
		tmp = ((1.0 - (0.5 / x)) / n) / x;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 8e-257], N[(1.0 - N[Power[x, N[Power[n, -1.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.95], N[(N[(x - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[x, 5.2e+205], N[(N[(N[(1.0 - N[(0.5 / x), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision] / x), $MachinePrecision], 0.0]]]

Derivation

1. Split input into 4 regimes

2. if x < 7.9999999999999998e-257

   1. Initial program 67.1%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 67.1%

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

   if 7.9999999999999998e-257 < x < 0.94999999999999996

   1. Initial program 41.1%

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

   3. Taylor expanded in n around -inf

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

   5. Applied rewrites 80.4%

      \[\leadsto \color{blue}{\frac{\left(\left(-\mathsf{log1p}\left(x\right)\right) - \frac{\mathsf{fma}\left(0.16666666666666666, \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{3} - {\log x}^{3}}{n}, \left({\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}\right) \cdot 0.5\right)}{n}\right) + \log x}{-n}} \]
   6. Step-by-step derivation

   7. Applied rewrites 80.4%

      \[\leadsto \frac{\mathsf{log1p}\left(x\right)}{n} - \color{blue}{\frac{\frac{\mathsf{fma}\left(\frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{3} - {\log x}^{3}}{n}, 0.16666666666666666, 0.5 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}\right)\right)}{n} - \log x}{-n}} \]

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 78.7%

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

   11. Taylor expanded in n around inf

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

   13. Applied rewrites 53.7%

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

   if 0.94999999999999996 < x < 5.1999999999999998e205

   1. Initial program 44.9%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 87.8%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\frac{\frac{-0.5 + \frac{0.5}{n}}{n}}{x}, e^{\frac{\log x}{n}}, \frac{e^{\frac{\log x}{n}}}{n}\right)}{x}} \]

   6. Taylor expanded in n around inf

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

   8. Applied rewrites 76.0%

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

   if 5.1999999999999998e205 < x

   1. Initial program 91.8%

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

   3. Taylor expanded in n around -inf

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

   5. Applied rewrites 91.8%

      \[\leadsto \color{blue}{\frac{\left(\left(-\mathsf{log1p}\left(x\right)\right) - \frac{\mathsf{fma}\left(0.16666666666666666, \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{3} - {\log x}^{3}}{n}, \left({\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}\right) \cdot 0.5\right)}{n}\right) + \log x}{-n}} \]
   6. Step-by-step derivation

   7. Applied rewrites 88.9%

      \[\leadsto \frac{\left(-\mathsf{log1p}\left(x\right)\right) - \frac{\mathsf{fma}\left(\frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{3} - {\log x}^{3}}{n}, 0.16666666666666666, 0.5 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}\right)\right)}{n}}{-n} + \color{blue}{\frac{-\log x}{n}} \]

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 88.9%

       \[\leadsto -0 \cdot \frac{\log x}{n} \]
3. Recombined 4 regimes into one program.

4. Final simplification 66.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 8 \cdot 10^{-257}:\\ \;\;\;\;1 - {x}^{\left({n}^{-1}\right)}\\ \mathbf{elif}\;x \leq 0.95:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{elif}\;x \leq 5.2 \cdot 10^{+205}:\\ \;\;\;\;\frac{\frac{1 - \frac{0.5}{x}}{n}}{x}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 57.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 8 \cdot 10^{-257}:\\ \;\;\;\;1 - {x}^{\left({n}^{-1}\right)}\\ \mathbf{elif}\;x \leq 0.95:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{elif}\;x \leq 1.45 \cdot 10^{+239}:\\ \;\;\;\;\frac{\frac{1 - \frac{0.5}{x}}{n}}{x}\\ \mathbf{else}:\\ \;\;\;\;{\left(\left(n \cdot x\right) \cdot \left(n \cdot x\right)\right)}^{-0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 8e-257)
   (- 1.0 (pow x (pow n -1.0)))
   (if (<= x 0.95)
     (/ (- x (log x)) n)
     (if (<= x 1.45e+239)
       (/ (/ (- 1.0 (/ 0.5 x)) n) x)
       (pow (* (* n x) (* n x)) -0.5)))))

C

double code(double x, double n) {
	double tmp;
	if (x <= 8e-257) {
		tmp = 1.0 - pow(x, pow(n, -1.0));
	} else if (x <= 0.95) {
		tmp = (x - log(x)) / n;
	} else if (x <= 1.45e+239) {
		tmp = ((1.0 - (0.5 / x)) / n) / x;
	} else {
		tmp = pow(((n * x) * (n * x)), -0.5);
	}
	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 <= 8d-257) then
        tmp = 1.0d0 - (x ** (n ** (-1.0d0)))
    else if (x <= 0.95d0) then
        tmp = (x - log(x)) / n
    else if (x <= 1.45d+239) then
        tmp = ((1.0d0 - (0.5d0 / x)) / n) / x
    else
        tmp = ((n * x) * (n * x)) ** (-0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (x <= 8e-257) {
		tmp = 1.0 - Math.pow(x, Math.pow(n, -1.0));
	} else if (x <= 0.95) {
		tmp = (x - Math.log(x)) / n;
	} else if (x <= 1.45e+239) {
		tmp = ((1.0 - (0.5 / x)) / n) / x;
	} else {
		tmp = Math.pow(((n * x) * (n * x)), -0.5);
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 8e-257:
		tmp = 1.0 - math.pow(x, math.pow(n, -1.0))
	elif x <= 0.95:
		tmp = (x - math.log(x)) / n
	elif x <= 1.45e+239:
		tmp = ((1.0 - (0.5 / x)) / n) / x
	else:
		tmp = math.pow(((n * x) * (n * x)), -0.5)
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 8e-257)
		tmp = Float64(1.0 - (x ^ (n ^ -1.0)));
	elseif (x <= 0.95)
		tmp = Float64(Float64(x - log(x)) / n);
	elseif (x <= 1.45e+239)
		tmp = Float64(Float64(Float64(1.0 - Float64(0.5 / x)) / n) / x);
	else
		tmp = Float64(Float64(n * x) * Float64(n * x)) ^ -0.5;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (x <= 8e-257)
		tmp = 1.0 - (x ^ (n ^ -1.0));
	elseif (x <= 0.95)
		tmp = (x - log(x)) / n;
	elseif (x <= 1.45e+239)
		tmp = ((1.0 - (0.5 / x)) / n) / x;
	else
		tmp = ((n * x) * (n * x)) ^ -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 8e-257], N[(1.0 - N[Power[x, N[Power[n, -1.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.95], N[(N[(x - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[x, 1.45e+239], N[(N[(N[(1.0 - N[(0.5 / x), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision] / x), $MachinePrecision], N[Power[N[(N[(n * x), $MachinePrecision] * N[(n * x), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < 7.9999999999999998e-257

   1. Initial program 67.1%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 67.1%

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

   if 7.9999999999999998e-257 < x < 0.94999999999999996

   1. Initial program 41.1%

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

   3. Taylor expanded in n around -inf

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

   5. Applied rewrites 80.4%

      \[\leadsto \color{blue}{\frac{\left(\left(-\mathsf{log1p}\left(x\right)\right) - \frac{\mathsf{fma}\left(0.16666666666666666, \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{3} - {\log x}^{3}}{n}, \left({\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}\right) \cdot 0.5\right)}{n}\right) + \log x}{-n}} \]
   6. Step-by-step derivation

   7. Applied rewrites 80.4%

      \[\leadsto \frac{\mathsf{log1p}\left(x\right)}{n} - \color{blue}{\frac{\frac{\mathsf{fma}\left(\frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{3} - {\log x}^{3}}{n}, 0.16666666666666666, 0.5 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}\right)\right)}{n} - \log x}{-n}} \]

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 78.7%

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

   11. Taylor expanded in n around inf

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

   13. Applied rewrites 53.7%

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

   if 0.94999999999999996 < x < 1.4500000000000001e239

   1. Initial program 48.2%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 86.9%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\frac{\frac{-0.5 + \frac{0.5}{n}}{n}}{x}, e^{\frac{\log x}{n}}, \frac{e^{\frac{\log x}{n}}}{n}\right)}{x}} \]

   6. Taylor expanded in n around inf

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

   8. Applied rewrites 73.6%

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

   if 1.4500000000000001e239 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. log-rec N/A

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

      3. mul-1-neg N/A

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

      4. associate-*r/ N/A

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

      5. mul-1-neg N/A

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

      6. distribute-lft-neg-in N/A

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

      7. metadata-eval N/A

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

      8. *-lft-identity N/A

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

      9. lower-exp.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-log.f64 N/A

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

      12. lower-*.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in n around inf

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

   8. Applied rewrites 48.2%

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

   10. Applied rewrites 48.2%

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

   12. Applied rewrites 71.9%

       \[\leadsto {\left(\left(n \cdot x\right) \cdot \left(n \cdot x\right)\right)}^{-0.5} \]
3. Recombined 4 regimes into one program.

4. Final simplification 63.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 8 \cdot 10^{-257}:\\ \;\;\;\;1 - {x}^{\left({n}^{-1}\right)}\\ \mathbf{elif}\;x \leq 0.95:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{elif}\;x \leq 1.45 \cdot 10^{+239}:\\ \;\;\;\;\frac{\frac{1 - \frac{0.5}{x}}{n}}{x}\\ \mathbf{else}:\\ \;\;\;\;{\left(\left(n \cdot x\right) \cdot \left(n \cdot x\right)\right)}^{-0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 56.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 8 \cdot 10^{-257}:\\ \;\;\;\;1 - {x}^{\left({n}^{-1}\right)}\\ \mathbf{elif}\;x \leq 0.95:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 - \frac{0.5}{x}}{n}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 8e-257)
   (- 1.0 (pow x (pow n -1.0)))
   (if (<= x 0.95) (/ (- x (log x)) n) (/ (/ (- 1.0 (/ 0.5 x)) n) x))))

C

double code(double x, double n) {
	double tmp;
	if (x <= 8e-257) {
		tmp = 1.0 - pow(x, pow(n, -1.0));
	} else if (x <= 0.95) {
		tmp = (x - log(x)) / n;
	} else {
		tmp = ((1.0 - (0.5 / x)) / n) / x;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, n)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if (x <= 8d-257) then
        tmp = 1.0d0 - (x ** (n ** (-1.0d0)))
    else if (x <= 0.95d0) then
        tmp = (x - log(x)) / n
    else
        tmp = ((1.0d0 - (0.5d0 / x)) / n) / x
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (x <= 8e-257) {
		tmp = 1.0 - Math.pow(x, Math.pow(n, -1.0));
	} else if (x <= 0.95) {
		tmp = (x - Math.log(x)) / n;
	} else {
		tmp = ((1.0 - (0.5 / x)) / n) / x;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 8e-257:
		tmp = 1.0 - math.pow(x, math.pow(n, -1.0))
	elif x <= 0.95:
		tmp = (x - math.log(x)) / n
	else:
		tmp = ((1.0 - (0.5 / x)) / n) / x
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 8e-257)
		tmp = Float64(1.0 - (x ^ (n ^ -1.0)));
	elseif (x <= 0.95)
		tmp = Float64(Float64(x - log(x)) / n);
	else
		tmp = Float64(Float64(Float64(1.0 - Float64(0.5 / x)) / n) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (x <= 8e-257)
		tmp = 1.0 - (x ^ (n ^ -1.0));
	elseif (x <= 0.95)
		tmp = (x - log(x)) / n;
	else
		tmp = ((1.0 - (0.5 / x)) / n) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 8e-257], N[(1.0 - N[Power[x, N[Power[n, -1.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.95], N[(N[(x - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], N[(N[(N[(1.0 - N[(0.5 / x), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision] / x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 7.9999999999999998e-257

   1. Initial program 67.1%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 67.1%

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

   if 7.9999999999999998e-257 < x < 0.94999999999999996

   1. Initial program 41.1%

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

   3. Taylor expanded in n around -inf

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

   5. Applied rewrites 80.4%

      \[\leadsto \color{blue}{\frac{\left(\left(-\mathsf{log1p}\left(x\right)\right) - \frac{\mathsf{fma}\left(0.16666666666666666, \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{3} - {\log x}^{3}}{n}, \left({\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}\right) \cdot 0.5\right)}{n}\right) + \log x}{-n}} \]
   6. Step-by-step derivation

   7. Applied rewrites 80.4%

      \[\leadsto \frac{\mathsf{log1p}\left(x\right)}{n} - \color{blue}{\frac{\frac{\mathsf{fma}\left(\frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{3} - {\log x}^{3}}{n}, 0.16666666666666666, 0.5 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}\right)\right)}{n} - \log x}{-n}} \]

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 78.7%

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

   11. Taylor expanded in n around inf

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

   13. Applied rewrites 53.7%

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

   if 0.94999999999999996 < x

   1. Initial program 59.8%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 84.2%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\frac{\frac{-0.5 + \frac{0.5}{n}}{n}}{x}, e^{\frac{\log x}{n}}, \frac{e^{\frac{\log x}{n}}}{n}\right)}{x}} \]

   6. Taylor expanded in n around inf

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

   8. Applied rewrites 67.9%

      \[\leadsto \frac{\frac{1 - \frac{0.5}{x}}{n}}{x} \]
3. Recombined 3 regimes into one program.

4. Final simplification 61.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 8 \cdot 10^{-257}:\\ \;\;\;\;1 - {x}^{\left({n}^{-1}\right)}\\ \mathbf{elif}\;x \leq 0.95:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 - \frac{0.5}{x}}{n}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 57.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.95:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 - \frac{0.5}{x}}{n}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 0.95) (/ (- x (log x)) n) (/ (/ (- 1.0 (/ 0.5 x)) n) x)))

C

double code(double x, double n) {
	double tmp;
	if (x <= 0.95) {
		tmp = (x - log(x)) / n;
	} else {
		tmp = ((1.0 - (0.5 / x)) / n) / x;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, n)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if (x <= 0.95d0) then
        tmp = (x - log(x)) / n
    else
        tmp = ((1.0d0 - (0.5d0 / x)) / n) / x
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (x <= 0.95) {
		tmp = (x - Math.log(x)) / n;
	} else {
		tmp = ((1.0 - (0.5 / x)) / n) / x;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 0.95:
		tmp = (x - math.log(x)) / n
	else:
		tmp = ((1.0 - (0.5 / x)) / n) / x
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 0.95)
		tmp = Float64(Float64(x - log(x)) / n);
	else
		tmp = Float64(Float64(Float64(1.0 - Float64(0.5 / x)) / n) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (x <= 0.95)
		tmp = (x - log(x)) / n;
	else
		tmp = ((1.0 - (0.5 / x)) / n) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 0.95], N[(N[(x - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], N[(N[(N[(1.0 - N[(0.5 / x), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.94999999999999996

   1. Initial program 45.6%

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

   3. Taylor expanded in n around -inf

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

   5. Applied rewrites 76.7%

      \[\leadsto \color{blue}{\frac{\left(\left(-\mathsf{log1p}\left(x\right)\right) - \frac{\mathsf{fma}\left(0.16666666666666666, \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{3} - {\log x}^{3}}{n}, \left({\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}\right) \cdot 0.5\right)}{n}\right) + \log x}{-n}} \]
   6. Step-by-step derivation

   7. Applied rewrites 76.7%

      \[\leadsto \frac{\mathsf{log1p}\left(x\right)}{n} - \color{blue}{\frac{\frac{\mathsf{fma}\left(\frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{3} - {\log x}^{3}}{n}, 0.16666666666666666, 0.5 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}\right)\right)}{n} - \log x}{-n}} \]

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 75.3%

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

   11. Taylor expanded in n around inf

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

   13. Applied rewrites 50.9%

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

   if 0.94999999999999996 < x

   1. Initial program 59.8%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 84.2%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\frac{\frac{-0.5 + \frac{0.5}{n}}{n}}{x}, e^{\frac{\log x}{n}}, \frac{e^{\frac{\log x}{n}}}{n}\right)}{x}} \]

   6. Taylor expanded in n around inf

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

   8. Applied rewrites 67.9%

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

Alternative 14: 56.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.66:\\ \;\;\;\;\frac{\log x}{-n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 - \frac{0.5}{x}}{n}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 0.66) (/ (log x) (- n)) (/ (/ (- 1.0 (/ 0.5 x)) n) x)))

C

double code(double x, double n) {
	double tmp;
	if (x <= 0.66) {
		tmp = log(x) / -n;
	} else {
		tmp = ((1.0 - (0.5 / x)) / n) / x;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, n)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if (x <= 0.66d0) then
        tmp = log(x) / -n
    else
        tmp = ((1.0d0 - (0.5d0 / x)) / n) / x
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (x <= 0.66) {
		tmp = Math.log(x) / -n;
	} else {
		tmp = ((1.0 - (0.5 / x)) / n) / x;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 0.66:
		tmp = math.log(x) / -n
	else:
		tmp = ((1.0 - (0.5 / x)) / n) / x
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 0.66)
		tmp = Float64(log(x) / Float64(-n));
	else
		tmp = Float64(Float64(Float64(1.0 - Float64(0.5 / x)) / n) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (x <= 0.66)
		tmp = log(x) / -n;
	else
		tmp = ((1.0 - (0.5 / x)) / n) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 0.66], N[(N[Log[x], $MachinePrecision] / (-n)), $MachinePrecision], N[(N[(N[(1.0 - N[(0.5 / x), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.660000000000000031

   1. Initial program 45.6%

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

   3. Taylor expanded in n around -inf

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

   5. Applied rewrites 76.7%

      \[\leadsto \color{blue}{\frac{\left(\left(-\mathsf{log1p}\left(x\right)\right) - \frac{\mathsf{fma}\left(0.16666666666666666, \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{3} - {\log x}^{3}}{n}, \left({\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}\right) \cdot 0.5\right)}{n}\right) + \log x}{-n}} \]

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 74.8%

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

   9. Taylor expanded in n around inf

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

   11. Applied rewrites 50.5%

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

   if 0.660000000000000031 < x

   1. Initial program 59.8%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 84.2%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\frac{\frac{-0.5 + \frac{0.5}{n}}{n}}{x}, e^{\frac{\log x}{n}}, \frac{e^{\frac{\log x}{n}}}{n}\right)}{x}} \]

   6. Taylor expanded in n around inf

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

   8. Applied rewrites 67.9%

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

Alternative 15: 41.2% accurate, 2.0× speedup

Math

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

FPCore

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

C

double code(double x, double n) {
	return pow(n, -1.0) / 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 = (n ** (-1.0d0)) / x
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 51.5%

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

3. Taylor expanded in x around inf

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

   1. lower-/.f64 N/A

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

   2. log-rec N/A

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

   3. mul-1-neg N/A

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

   4. associate-*r/ N/A

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

   5. mul-1-neg N/A

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

   6. distribute-lft-neg-in N/A

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

   7. metadata-eval N/A

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

   8. *-lft-identity N/A

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

   9. lower-exp.f64 N/A

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

   10. lower-/.f64 N/A

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

   11. lower-log.f64 N/A

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

   12. lower-*.f64 57.1

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

5. Applied rewrites 57.1%

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

6. Taylor expanded in n around inf

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

8. Applied rewrites 39.0%

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

9. Final simplification 39.0%

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

Alternative 16: 40.6% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ {\left(n \cdot x\right)}^{-1} \end{array} \]

FPCore

(FPCore (x n) :precision binary64 (pow (* n x) -1.0))

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 51.5%

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

3. Taylor expanded in x around inf

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

   1. lower-/.f64 N/A

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

   2. log-rec N/A

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

   3. mul-1-neg N/A

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

   4. associate-*r/ N/A

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

   5. mul-1-neg N/A

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

   6. distribute-lft-neg-in N/A

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

   7. metadata-eval N/A

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

   8. *-lft-identity N/A

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

   9. lower-exp.f64 N/A

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

   10. lower-/.f64 N/A

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

   11. lower-log.f64 N/A

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

   12. lower-*.f64 57.1

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

5. Applied rewrites 57.1%

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

6. Taylor expanded in n around inf

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

8. Applied rewrites 39.0%

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

10. Applied rewrites 38.5%

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

11. Final simplification 38.5%

    \[\leadsto {\left(n \cdot x\right)}^{-1} \]
12. Add Preprocessing

Alternative 17: 4.5% accurate, 19.3× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, n)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    code = x / n
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 51.5%

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

3. Taylor expanded in n around -inf

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

5. Applied rewrites 70.1%

   \[\leadsto \color{blue}{\frac{\left(\left(-\mathsf{log1p}\left(x\right)\right) - \frac{\mathsf{fma}\left(0.16666666666666666, \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{3} - {\log x}^{3}}{n}, \left({\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}\right) \cdot 0.5\right)}{n}\right) + \log x}{-n}} \]
6. Step-by-step derivation

7. Applied rewrites 69.7%

   \[\leadsto \frac{\mathsf{log1p}\left(x\right)}{n} - \color{blue}{\frac{\frac{\mathsf{fma}\left(\frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{3} - {\log x}^{3}}{n}, 0.16666666666666666, 0.5 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}\right)\right)}{n} - \log x}{-n}} \]

8. Taylor expanded in x around 0

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

10. Applied rewrites 45.1%

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

11. Taylor expanded in x around inf

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

13. Applied rewrites 4.7%

    \[\leadsto \frac{x}{n} \]
14. Add Preprocessing

Reproduce

herbie shell --seed 2024352 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  :precision binary64
  (- (pow (+ x 1.0) (/ 1.0 n)) (pow x (/ 1.0 n))))