2nthrt (problem 3.4.6)

Percentage Accurate: 52.3% → 86.0%

Time: 27.1s

Alternatives: 14

Speedup: 5.1×

• could not determine a ground truth

  1. x = 5.485827254888333e+217
  2. n = 1.1348673731152441e-246

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 52.3% 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: 86.0% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 7200:\\ \;\;\;\;\frac{\mathsf{fma}\left(-1, \mathsf{log1p}\left(x\right) + \frac{\mathsf{fma}\left(\frac{-0.16666666666666666 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{3} - {\log x}^{3}\right)}{n}, -1, 0.5 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}\right)\right)}{n}, \log x\right)}{-n}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\frac{\log x}{n}}}{n \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 7200.0)
   (/
    (fma
     -1.0
     (+
      (log1p x)
      (/
       (fma
        (/
         (* -0.16666666666666666 (- (pow (log1p x) 3.0) (pow (log x) 3.0)))
         n)
        -1.0
        (* 0.5 (- (pow (log1p x) 2.0) (pow (log x) 2.0))))
       n))
     (log x))
    (- n))
   (/ (exp (/ (log x) n)) (* n x))))

C

double code(double x, double n) {
	double tmp;
	if (x <= 7200.0) {
		tmp = fma(-1.0, (log1p(x) + (fma(((-0.16666666666666666 * (pow(log1p(x), 3.0) - pow(log(x), 3.0))) / n), -1.0, (0.5 * (pow(log1p(x), 2.0) - pow(log(x), 2.0)))) / n)), log(x)) / -n;
	} else {
		tmp = exp((log(x) / n)) / (n * x);
	}
	return tmp;
}

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 7200.0)
		tmp = Float64(fma(-1.0, Float64(log1p(x) + Float64(fma(Float64(Float64(-0.16666666666666666 * Float64((log1p(x) ^ 3.0) - (log(x) ^ 3.0))) / n), -1.0, Float64(0.5 * Float64((log1p(x) ^ 2.0) - (log(x) ^ 2.0)))) / n)), log(x)) / Float64(-n));
	else
		tmp = Float64(exp(Float64(log(x) / n)) / Float64(n * x));
	end
	return tmp
end

Wolfram

code[x_, n_] := If[LessEqual[x, 7200.0], N[(N[(-1.0 * N[(N[Log[1 + x], $MachinePrecision] + N[(N[(N[(N[(-0.16666666666666666 * N[(N[Power[N[Log[1 + x], $MachinePrecision], 3.0], $MachinePrecision] - N[Power[N[Log[x], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision] * -1.0 + 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]), $MachinePrecision] + N[Log[x], $MachinePrecision]), $MachinePrecision] / (-n)), $MachinePrecision], N[(N[Exp[N[(N[Log[x], $MachinePrecision] / n), $MachinePrecision]], $MachinePrecision] / N[(n * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 7200

   1. Initial program 46.5%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   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 81.8%

      \[\leadsto \color{blue}{-\frac{\mathsf{fma}\left(-1, \mathsf{log1p}\left(x\right) + \frac{\mathsf{fma}\left(\frac{-0.16666666666666666 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{3} - {\log x}^{3}\right)}{n}, -1, 0.5 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}\right)\right)}{n}, \log x\right)}{n}} \]

   if 7200 < x

   1. Initial program 62.2%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower-exp.f64 N/A

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

      3. mul-1-neg N/A

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

      4. log-rec N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-log.f64 N/A

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

      11. lower-*.f64 99.7

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

   5. Applied rewrites 99.7%

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

   6. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. lift-log.f64 99.7

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

   8. Applied rewrites 99.7%

      \[\leadsto \frac{e^{\frac{\log x}{n}}}{n \cdot x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 88.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 7200:\\ \;\;\;\;\frac{\mathsf{fma}\left(-1, \mathsf{log1p}\left(x\right) + \frac{\mathsf{fma}\left(\frac{-0.16666666666666666 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{3} - {\log x}^{3}\right)}{n}, -1, 0.5 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}\right)\right)}{n}, \log x\right)}{-n}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\frac{\log x}{n}}}{n \cdot x}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 78.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ t_1 := {\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - t\_0\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{-8}:\\ \;\;\;\;1 - t\_0\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{-10}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{x}{n} + 1\right) - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))) (t_1 (- (pow (+ x 1.0) (/ 1.0 n)) t_0)))
   (if (<= t_1 -1e-8)
     (- 1.0 t_0)
     (if (<= t_1 5e-10) (/ (log (/ (+ 1.0 x) x)) n) (- (+ (/ x n) 1.0) t_0)))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double t_1 = pow((x + 1.0), (1.0 / n)) - t_0;
	double tmp;
	if (t_1 <= -1e-8) {
		tmp = 1.0 - t_0;
	} else if (t_1 <= 5e-10) {
		tmp = log(((1.0 + x) / x)) / n;
	} else {
		tmp = ((x / n) + 1.0) - t_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) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = x ** (1.0d0 / n)
    t_1 = ((x + 1.0d0) ** (1.0d0 / n)) - t_0
    if (t_1 <= (-1d-8)) then
        tmp = 1.0d0 - t_0
    else if (t_1 <= 5d-10) then
        tmp = log(((1.0d0 + x) / x)) / n
    else
        tmp = ((x / n) + 1.0d0) - t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, (1.0 / n));
	double t_1 = Math.pow((x + 1.0), (1.0 / n)) - t_0;
	double tmp;
	if (t_1 <= -1e-8) {
		tmp = 1.0 - t_0;
	} else if (t_1 <= 5e-10) {
		tmp = Math.log(((1.0 + x) / x)) / n;
	} else {
		tmp = ((x / n) + 1.0) - t_0;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	t_1 = math.pow((x + 1.0), (1.0 / n)) - t_0
	tmp = 0
	if t_1 <= -1e-8:
		tmp = 1.0 - t_0
	elif t_1 <= 5e-10:
		tmp = math.log(((1.0 + x) / x)) / n
	else:
		tmp = ((x / n) + 1.0) - t_0
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	t_1 = Float64((Float64(x + 1.0) ^ Float64(1.0 / n)) - t_0)
	tmp = 0.0
	if (t_1 <= -1e-8)
		tmp = Float64(1.0 - t_0);
	elseif (t_1 <= 5e-10)
		tmp = Float64(log(Float64(Float64(1.0 + x) / x)) / n);
	else
		tmp = Float64(Float64(Float64(x / n) + 1.0) - t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = x ^ (1.0 / n);
	t_1 = ((x + 1.0) ^ (1.0 / n)) - t_0;
	tmp = 0.0;
	if (t_1 <= -1e-8)
		tmp = 1.0 - t_0;
	elseif (t_1 <= 5e-10)
		tmp = log(((1.0 + x) / x)) / n;
	else
		tmp = ((x / n) + 1.0) - t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[Power[N[(x + 1.0), $MachinePrecision], N[(1.0 / n), $MachinePrecision]], $MachinePrecision] - t$95$0), $MachinePrecision]}, If[LessEqual[t$95$1, -1e-8], N[(1.0 - t$95$0), $MachinePrecision], If[LessEqual[t$95$1, 5e-10], N[(N[Log[N[(N[(1.0 + x), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], N[(N[(N[(x / n), $MachinePrecision] + 1.0), $MachinePrecision] - t$95$0), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (pow.f64 (+.f64 x #s(literal 1 binary64)) (/.f64 #s(literal 1 binary64) n)) (pow.f64 x (/.f64 #s(literal 1 binary64) n))) < -1e-8

   1. Initial program 99.0%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 99.0%

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

   if -1e-8 < (-.f64 (pow.f64 (+.f64 x #s(literal 1 binary64)) (/.f64 #s(literal 1 binary64) n)) (pow.f64 x (/.f64 #s(literal 1 binary64) n))) < 5.00000000000000031e-10

   1. Initial program 39.8%

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

   3. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-log1p.f64 N/A

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

      4. lower-log.f64 76.4

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

   5. Applied rewrites 76.4%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) - \log x}{n}} \]
   6. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. lift-log1p.f64 N/A

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

      3. lift-log.f64 N/A

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

      4. diff-log N/A

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

      5. lower-log.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-+.f64 76.4

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

   7. Applied rewrites 76.4%

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

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

   1. Initial program 56.2%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 57.1

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

   5. Applied rewrites 57.1%

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

Alternative 3: 78.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ t_1 := {\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - t\_0\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{-8} \lor \neg \left(t\_1 \leq 5 \cdot 10^{-10}\right):\\ \;\;\;\;1 - t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))) (t_1 (- (pow (+ x 1.0) (/ 1.0 n)) t_0)))
   (if (or (<= t_1 -1e-8) (not (<= t_1 5e-10)))
     (- 1.0 t_0)
     (/ (log (/ (+ 1.0 x) x)) n))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double t_1 = pow((x + 1.0), (1.0 / n)) - t_0;
	double tmp;
	if ((t_1 <= -1e-8) || !(t_1 <= 5e-10)) {
		tmp = 1.0 - t_0;
	} else {
		tmp = log(((1.0 + x) / x)) / n;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, n)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = x ** (1.0d0 / n)
    t_1 = ((x + 1.0d0) ** (1.0d0 / n)) - t_0
    if ((t_1 <= (-1d-8)) .or. (.not. (t_1 <= 5d-10))) then
        tmp = 1.0d0 - t_0
    else
        tmp = log(((1.0d0 + x) / x)) / n
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, (1.0 / n));
	double t_1 = Math.pow((x + 1.0), (1.0 / n)) - t_0;
	double tmp;
	if ((t_1 <= -1e-8) || !(t_1 <= 5e-10)) {
		tmp = 1.0 - t_0;
	} else {
		tmp = Math.log(((1.0 + x) / x)) / n;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	t_1 = math.pow((x + 1.0), (1.0 / n)) - t_0
	tmp = 0
	if (t_1 <= -1e-8) or not (t_1 <= 5e-10):
		tmp = 1.0 - t_0
	else:
		tmp = math.log(((1.0 + x) / x)) / n
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	t_1 = Float64((Float64(x + 1.0) ^ Float64(1.0 / n)) - t_0)
	tmp = 0.0
	if ((t_1 <= -1e-8) || !(t_1 <= 5e-10))
		tmp = Float64(1.0 - t_0);
	else
		tmp = Float64(log(Float64(Float64(1.0 + x) / x)) / n);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = x ^ (1.0 / n);
	t_1 = ((x + 1.0) ^ (1.0 / n)) - t_0;
	tmp = 0.0;
	if ((t_1 <= -1e-8) || ~((t_1 <= 5e-10)))
		tmp = 1.0 - t_0;
	else
		tmp = log(((1.0 + x) / x)) / n;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[Power[N[(x + 1.0), $MachinePrecision], N[(1.0 / n), $MachinePrecision]], $MachinePrecision] - t$95$0), $MachinePrecision]}, If[Or[LessEqual[t$95$1, -1e-8], N[Not[LessEqual[t$95$1, 5e-10]], $MachinePrecision]], N[(1.0 - t$95$0), $MachinePrecision], N[(N[Log[N[(N[(1.0 + x), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (pow.f64 (+.f64 x #s(literal 1 binary64)) (/.f64 #s(literal 1 binary64) n)) (pow.f64 x (/.f64 #s(literal 1 binary64) n))) < -1e-8 or 5.00000000000000031e-10 < (-.f64 (pow.f64 (+.f64 x #s(literal 1 binary64)) (/.f64 #s(literal 1 binary64) n)) (pow.f64 x (/.f64 #s(literal 1 binary64) n)))

   1. Initial program 77.3%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 77.3%

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

   if -1e-8 < (-.f64 (pow.f64 (+.f64 x #s(literal 1 binary64)) (/.f64 #s(literal 1 binary64) n)) (pow.f64 x (/.f64 #s(literal 1 binary64) n))) < 5.00000000000000031e-10

   1. Initial program 39.8%

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

   3. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-log1p.f64 N/A

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

      4. lower-log.f64 76.4

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

   5. Applied rewrites 76.4%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) - \log x}{n}} \]
   6. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. lift-log1p.f64 N/A

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

      3. lift-log.f64 N/A

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

      4. diff-log N/A

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

      5. lower-log.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-+.f64 76.4

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

   7. Applied rewrites 76.4%

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

4. Final simplification 76.7%

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

Alternative 4: 85.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.45 \cdot 10^{-5}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-1, \frac{\mathsf{fma}\left(-0.5, {\log x}^{2}, -0.16666666666666666 \cdot \frac{{\log x}^{3}}{n}\right)}{n}, \log x\right)}{-n}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\frac{\log x}{n}}}{n \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 1.45e-5)
   (/
    (fma
     -1.0
     (/
      (fma
       -0.5
       (pow (log x) 2.0)
       (* -0.16666666666666666 (/ (pow (log x) 3.0) n)))
      n)
     (log x))
    (- n))
   (/ (exp (/ (log x) n)) (* n x))))

C

double code(double x, double n) {
	double tmp;
	if (x <= 1.45e-5) {
		tmp = fma(-1.0, (fma(-0.5, pow(log(x), 2.0), (-0.16666666666666666 * (pow(log(x), 3.0) / n))) / n), log(x)) / -n;
	} else {
		tmp = exp((log(x) / n)) / (n * x);
	}
	return tmp;
}

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 1.45e-5)
		tmp = Float64(fma(-1.0, Float64(fma(-0.5, (log(x) ^ 2.0), Float64(-0.16666666666666666 * Float64((log(x) ^ 3.0) / n))) / n), log(x)) / Float64(-n));
	else
		tmp = Float64(exp(Float64(log(x) / n)) / Float64(n * x));
	end
	return tmp
end

Wolfram

code[x_, n_] := If[LessEqual[x, 1.45e-5], N[(N[(-1.0 * N[(N[(-0.5 * N[Power[N[Log[x], $MachinePrecision], 2.0], $MachinePrecision] + N[(-0.16666666666666666 * N[(N[Power[N[Log[x], $MachinePrecision], 3.0], $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision] + N[Log[x], $MachinePrecision]), $MachinePrecision] / (-n)), $MachinePrecision], N[(N[Exp[N[(N[Log[x], $MachinePrecision] / n), $MachinePrecision]], $MachinePrecision] / N[(n * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.45e-5

   1. Initial program 46.4%

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

   3. Taylor expanded in n around -inf

      \[\leadsto \color{blue}{-1 \cdot \frac{\left(-1 \cdot \log \left(1 + x\right) + -1 \cdot \frac{\left(-1 \cdot \frac{\left(-1 \cdot \frac{\frac{1}{24} \cdot {\log \left(1 + x\right)}^{4} - \frac{1}{24} \cdot {\log x}^{4}}{n} + \frac{-1}{6} \cdot {\log \left(1 + x\right)}^{3}\right) - \frac{-1}{6} \cdot {\log x}^{3}}{n} + \frac{1}{2} \cdot {\log \left(1 + x\right)}^{2}\right) - \frac{1}{2} \cdot {\log x}^{2}}{n}\right) - -1 \cdot \log x}{n}} \]

   5. Applied rewrites 70.4%

      \[\leadsto \color{blue}{-\frac{\mathsf{fma}\left(-1, \mathsf{log1p}\left(x\right) + \frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{0.041666666666666664 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{4} - {\log x}^{4}\right)}{n}, -1, -0.16666666666666666 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{3} - {\log x}^{3}\right)\right)}{n}, -1, 0.5 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}\right)\right)}{n}, \log x\right)}{n}} \]

   6. Taylor expanded in x around 0

      \[\leadsto -\frac{\mathsf{fma}\left(-1, -1 \cdot \frac{\frac{1}{24} \cdot \frac{{\log x}^{4}}{n} + \frac{1}{6} \cdot {\log x}^{3}}{{n}^{2}} + \frac{-1}{2} \cdot \frac{{\log x}^{2}}{n}, \log x\right)}{n} \]
   7. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto -\frac{\mathsf{fma}\left(-1, \mathsf{fma}\left(-1, \frac{\frac{1}{24} \cdot \frac{{\log x}^{4}}{n} + \frac{1}{6} \cdot {\log x}^{3}}{{n}^{2}}, \frac{-1}{2} \cdot \frac{{\log x}^{2}}{n}\right), \log x\right)}{n} \]

   8. Applied rewrites 69.8%

      \[\leadsto -\frac{\mathsf{fma}\left(-1, \mathsf{fma}\left(-1, \frac{\mathsf{fma}\left(0.041666666666666664, \frac{{\log x}^{4}}{n}, 0.16666666666666666 \cdot {\log x}^{3}\right)}{n \cdot n}, -0.5 \cdot \frac{{\log x}^{2}}{n}\right), \log x\right)}{n} \]

   9. Taylor expanded in n around inf

      \[\leadsto -\frac{\mathsf{fma}\left(-1, \frac{\frac{-1}{2} \cdot {\log x}^{2} + \frac{-1}{6} \cdot \frac{{\log x}^{3}}{n}}{n}, \log x\right)}{n} \]
   10. Step-by-step derivation

       1. lower-/.f64 N/A

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

       2. lower-fma.f64 N/A

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

       3. lift-pow.f64 N/A

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

       4. lift-log.f64 N/A

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

       5. lower-*.f64 N/A

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

       6. lower-/.f64 N/A

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

       7. lift-pow.f64 N/A

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

       8. lift-log.f64 81.1

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

   11. Applied rewrites 81.1%

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

   if 1.45e-5 < x

   1. Initial program 62.0%

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

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

      2. lower-exp.f64 N/A

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

      3. mul-1-neg N/A

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

      4. log-rec N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-log.f64 N/A

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

      11. lower-*.f64 99.1

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

   5. Applied rewrites 99.1%

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

   6. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. lift-log.f64 99.1

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

   8. Applied rewrites 99.1%

      \[\leadsto \frac{e^{\frac{\log x}{n}}}{n \cdot x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 88.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.45 \cdot 10^{-5}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-1, \frac{\mathsf{fma}\left(-0.5, {\log x}^{2}, -0.16666666666666666 \cdot \frac{{\log x}^{3}}{n}\right)}{n}, \log x\right)}{-n}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\frac{\log x}{n}}}{n \cdot x}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 85.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -4 \cdot 10^{-90}:\\ \;\;\;\;\frac{e^{\frac{\log x}{n}}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-9}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= (/ 1.0 n) -4e-90)
   (/ (exp (/ (log x) n)) (* n x))
   (if (<= (/ 1.0 n) 5e-9)
     (/ (log (/ (+ 1.0 x) x)) n)
     (- (exp (/ (log1p x) n)) (pow x (/ 1.0 n))))))

C

double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -4e-90) {
		tmp = exp((log(x) / n)) / (n * x);
	} else if ((1.0 / n) <= 5e-9) {
		tmp = log(((1.0 + x) / x)) / n;
	} else {
		tmp = exp((log1p(x) / n)) - pow(x, (1.0 / n));
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -4e-90) {
		tmp = Math.exp((Math.log(x) / n)) / (n * x);
	} else if ((1.0 / n) <= 5e-9) {
		tmp = Math.log(((1.0 + x) / x)) / n;
	} else {
		tmp = Math.exp((Math.log1p(x) / n)) - Math.pow(x, (1.0 / n));
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if (1.0 / n) <= -4e-90:
		tmp = math.exp((math.log(x) / n)) / (n * x)
	elif (1.0 / n) <= 5e-9:
		tmp = math.log(((1.0 + x) / x)) / n
	else:
		tmp = math.exp((math.log1p(x) / n)) - math.pow(x, (1.0 / n))
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -4e-90)
		tmp = Float64(exp(Float64(log(x) / n)) / Float64(n * x));
	elseif (Float64(1.0 / n) <= 5e-9)
		tmp = Float64(log(Float64(Float64(1.0 + x) / x)) / n);
	else
		tmp = Float64(exp(Float64(log1p(x) / n)) - (x ^ Float64(1.0 / n)));
	end
	return tmp
end

Wolfram

code[x_, n_] := If[LessEqual[N[(1.0 / n), $MachinePrecision], -4e-90], N[(N[Exp[N[(N[Log[x], $MachinePrecision] / n), $MachinePrecision]], $MachinePrecision] / N[(n * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e-9], N[(N[Log[N[(N[(1.0 + x), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], N[(N[Exp[N[(N[Log[1 + x], $MachinePrecision] / n), $MachinePrecision]], $MachinePrecision] - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -3.99999999999999998e-90

   1. Initial program 88.3%

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

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

      2. lower-exp.f64 N/A

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

      3. mul-1-neg N/A

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

      4. log-rec N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-log.f64 N/A

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

      11. lower-*.f64 94.4

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

   5. Applied rewrites 94.4%

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

   6. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. lift-log.f64 94.4

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

   8. Applied rewrites 94.4%

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

   if -3.99999999999999998e-90 < (/.f64 #s(literal 1 binary64) n) < 5.0000000000000001e-9

   1. Initial program 27.4%

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

   3. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-log1p.f64 N/A

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

      4. lower-log.f64 74.6

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

   5. Applied rewrites 74.6%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) - \log x}{n}} \]
   6. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. lift-log1p.f64 N/A

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

      3. lift-log.f64 N/A

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

      4. diff-log N/A

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

      5. lower-log.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-+.f64 74.6

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

   7. Applied rewrites 74.6%

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

   if 5.0000000000000001e-9 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 56.2%

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

   3. Taylor expanded in n around 0

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

      1. lower-exp.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-log1p.f64 98.6

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

   5. Applied rewrites 98.6%

      \[\leadsto \color{blue}{e^{\frac{\mathsf{log1p}\left(x\right)}{n}}} - {x}^{\left(\frac{1}{n}\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 6: 81.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -4 \cdot 10^{-90}:\\ \;\;\;\;\frac{e^{\frac{\log x}{n}}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-9}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{+164}:\\ \;\;\;\;\left(\frac{x}{n} + 1\right) - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(1 + \frac{0.3333333333333333}{x \cdot x}\right) - 0.5 \cdot {x}^{-1}}{x}}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= (/ 1.0 n) -4e-90)
   (/ (exp (/ (log x) n)) (* n x))
   (if (<= (/ 1.0 n) 5e-9)
     (/ (log (/ (+ 1.0 x) x)) n)
     (if (<= (/ 1.0 n) 1e+164)
       (- (+ (/ x n) 1.0) (pow x (/ 1.0 n)))
       (/
        (/ (- (+ 1.0 (/ 0.3333333333333333 (* x x))) (* 0.5 (pow x -1.0))) x)
        n)))))

C

double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -4e-90) {
		tmp = exp((log(x) / n)) / (n * x);
	} else if ((1.0 / n) <= 5e-9) {
		tmp = log(((1.0 + x) / x)) / n;
	} else if ((1.0 / n) <= 1e+164) {
		tmp = ((x / n) + 1.0) - pow(x, (1.0 / n));
	} else {
		tmp = (((1.0 + (0.3333333333333333 / (x * x))) - (0.5 * pow(x, -1.0))) / x) / n;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, n)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if ((1.0d0 / n) <= (-4d-90)) then
        tmp = exp((log(x) / n)) / (n * x)
    else if ((1.0d0 / n) <= 5d-9) then
        tmp = log(((1.0d0 + x) / x)) / n
    else if ((1.0d0 / n) <= 1d+164) then
        tmp = ((x / n) + 1.0d0) - (x ** (1.0d0 / n))
    else
        tmp = (((1.0d0 + (0.3333333333333333d0 / (x * x))) - (0.5d0 * (x ** (-1.0d0)))) / x) / n
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -4e-90) {
		tmp = Math.exp((Math.log(x) / n)) / (n * x);
	} else if ((1.0 / n) <= 5e-9) {
		tmp = Math.log(((1.0 + x) / x)) / n;
	} else if ((1.0 / n) <= 1e+164) {
		tmp = ((x / n) + 1.0) - Math.pow(x, (1.0 / n));
	} else {
		tmp = (((1.0 + (0.3333333333333333 / (x * x))) - (0.5 * Math.pow(x, -1.0))) / x) / n;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if (1.0 / n) <= -4e-90:
		tmp = math.exp((math.log(x) / n)) / (n * x)
	elif (1.0 / n) <= 5e-9:
		tmp = math.log(((1.0 + x) / x)) / n
	elif (1.0 / n) <= 1e+164:
		tmp = ((x / n) + 1.0) - math.pow(x, (1.0 / n))
	else:
		tmp = (((1.0 + (0.3333333333333333 / (x * x))) - (0.5 * math.pow(x, -1.0))) / x) / n
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -4e-90)
		tmp = Float64(exp(Float64(log(x) / n)) / Float64(n * x));
	elseif (Float64(1.0 / n) <= 5e-9)
		tmp = Float64(log(Float64(Float64(1.0 + x) / x)) / n);
	elseif (Float64(1.0 / n) <= 1e+164)
		tmp = Float64(Float64(Float64(x / n) + 1.0) - (x ^ Float64(1.0 / n)));
	else
		tmp = Float64(Float64(Float64(Float64(1.0 + Float64(0.3333333333333333 / Float64(x * x))) - Float64(0.5 * (x ^ -1.0))) / x) / n);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if ((1.0 / n) <= -4e-90)
		tmp = exp((log(x) / n)) / (n * x);
	elseif ((1.0 / n) <= 5e-9)
		tmp = log(((1.0 + x) / x)) / n;
	elseif ((1.0 / n) <= 1e+164)
		tmp = ((x / n) + 1.0) - (x ^ (1.0 / n));
	else
		tmp = (((1.0 + (0.3333333333333333 / (x * x))) - (0.5 * (x ^ -1.0))) / x) / n;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[N[(1.0 / n), $MachinePrecision], -4e-90], N[(N[Exp[N[(N[Log[x], $MachinePrecision] / n), $MachinePrecision]], $MachinePrecision] / N[(n * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e-9], N[(N[Log[N[(N[(1.0 + x), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 1e+164], N[(N[(N[(x / n), $MachinePrecision] + 1.0), $MachinePrecision] - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(1.0 + N[(0.3333333333333333 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(0.5 * N[Power[x, -1.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] / n), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -3.99999999999999998e-90

   1. Initial program 88.3%

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

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

      2. lower-exp.f64 N/A

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

      3. mul-1-neg N/A

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

      4. log-rec N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-log.f64 N/A

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

      11. lower-*.f64 94.4

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

   5. Applied rewrites 94.4%

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

   6. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. lift-log.f64 94.4

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

   8. Applied rewrites 94.4%

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

   if -3.99999999999999998e-90 < (/.f64 #s(literal 1 binary64) n) < 5.0000000000000001e-9

   1. Initial program 27.4%

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

   3. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-log1p.f64 N/A

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

      4. lower-log.f64 74.6

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

   5. Applied rewrites 74.6%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) - \log x}{n}} \]
   6. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. lift-log1p.f64 N/A

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

      3. lift-log.f64 N/A

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

      4. diff-log N/A

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

      5. lower-log.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-+.f64 74.6

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

   7. Applied rewrites 74.6%

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

   if 5.0000000000000001e-9 < (/.f64 #s(literal 1 binary64) n) < 1e164

   1. Initial program 88.3%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 89.4

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

   5. Applied rewrites 89.4%

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

   if 1e164 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 17.7%

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

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

      2. lower--.f64 N/A

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

      3. lower-log1p.f64 N/A

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

      4. lower-log.f64 6.6

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

   5. Applied rewrites 6.6%

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

   6. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. inv-pow N/A

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

      9. lower-pow.f64 85.5

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

   8. Applied rewrites 85.5%

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

Alternative 7: 60.7% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\frac{1}{x}}{n}\\ \mathbf{if}\;n \leq -8.5:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n \leq 4.3 \cdot 10^{-165}:\\ \;\;\;\;\frac{\frac{\frac{0.3333333333333333}{x \cdot x}}{x}}{n}\\ \mathbf{elif}\;n \leq 23000000000:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;n \leq 1.26 \cdot 10^{+73} \lor \neg \left(n \leq 6.2 \cdot 10^{+168}\right):\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (/ (/ 1.0 x) n)))
   (if (<= n -8.5)
     t_0
     (if (<= n 4.3e-165)
       (/ (/ (/ 0.3333333333333333 (* x x)) x) n)
       (if (<= n 23000000000.0)
         (- 1.0 (pow x (/ 1.0 n)))
         (if (or (<= n 1.26e+73) (not (<= n 6.2e+168)))
           (/ (- x (log x)) n)
           t_0))))))

C

double code(double x, double n) {
	double t_0 = (1.0 / x) / n;
	double tmp;
	if (n <= -8.5) {
		tmp = t_0;
	} else if (n <= 4.3e-165) {
		tmp = ((0.3333333333333333 / (x * x)) / x) / n;
	} else if (n <= 23000000000.0) {
		tmp = 1.0 - pow(x, (1.0 / n));
	} else if ((n <= 1.26e+73) || !(n <= 6.2e+168)) {
		tmp = (x - log(x)) / n;
	} else {
		tmp = t_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) :: t_0
    real(8) :: tmp
    t_0 = (1.0d0 / x) / n
    if (n <= (-8.5d0)) then
        tmp = t_0
    else if (n <= 4.3d-165) then
        tmp = ((0.3333333333333333d0 / (x * x)) / x) / n
    else if (n <= 23000000000.0d0) then
        tmp = 1.0d0 - (x ** (1.0d0 / n))
    else if ((n <= 1.26d+73) .or. (.not. (n <= 6.2d+168))) then
        tmp = (x - log(x)) / n
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = (1.0 / x) / n;
	double tmp;
	if (n <= -8.5) {
		tmp = t_0;
	} else if (n <= 4.3e-165) {
		tmp = ((0.3333333333333333 / (x * x)) / x) / n;
	} else if (n <= 23000000000.0) {
		tmp = 1.0 - Math.pow(x, (1.0 / n));
	} else if ((n <= 1.26e+73) || !(n <= 6.2e+168)) {
		tmp = (x - Math.log(x)) / n;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = (1.0 / x) / n
	tmp = 0
	if n <= -8.5:
		tmp = t_0
	elif n <= 4.3e-165:
		tmp = ((0.3333333333333333 / (x * x)) / x) / n
	elif n <= 23000000000.0:
		tmp = 1.0 - math.pow(x, (1.0 / n))
	elif (n <= 1.26e+73) or not (n <= 6.2e+168):
		tmp = (x - math.log(x)) / n
	else:
		tmp = t_0
	return tmp

Julia

function code(x, n)
	t_0 = Float64(Float64(1.0 / x) / n)
	tmp = 0.0
	if (n <= -8.5)
		tmp = t_0;
	elseif (n <= 4.3e-165)
		tmp = Float64(Float64(Float64(0.3333333333333333 / Float64(x * x)) / x) / n);
	elseif (n <= 23000000000.0)
		tmp = Float64(1.0 - (x ^ Float64(1.0 / n)));
	elseif ((n <= 1.26e+73) || !(n <= 6.2e+168))
		tmp = Float64(Float64(x - log(x)) / n);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = (1.0 / x) / n;
	tmp = 0.0;
	if (n <= -8.5)
		tmp = t_0;
	elseif (n <= 4.3e-165)
		tmp = ((0.3333333333333333 / (x * x)) / x) / n;
	elseif (n <= 23000000000.0)
		tmp = 1.0 - (x ^ (1.0 / n));
	elseif ((n <= 1.26e+73) || ~((n <= 6.2e+168)))
		tmp = (x - log(x)) / n;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(N[(1.0 / x), $MachinePrecision] / n), $MachinePrecision]}, If[LessEqual[n, -8.5], t$95$0, If[LessEqual[n, 4.3e-165], N[(N[(N[(0.3333333333333333 / N[(x * x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[n, 23000000000.0], N[(1.0 - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[n, 1.26e+73], N[Not[LessEqual[n, 6.2e+168]], $MachinePrecision]], N[(N[(x - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 4 regimes

2. if n < -8.5 or 1.25999999999999993e73 < n < 6.19999999999999993e168

   1. Initial program 29.9%

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

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

      2. lower--.f64 N/A

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

      3. lower-log1p.f64 N/A

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

      4. lower-log.f64 66.8

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

   5. Applied rewrites 66.8%

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

   6. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. inv-pow N/A

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

      9. lower-pow.f64 57.6

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

   8. Applied rewrites 57.6%

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

   9. Taylor expanded in x around inf

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

   11. Applied rewrites 58.5%

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

   if -8.5 < n < 4.30000000000000007e-165

   1. Initial program 81.9%

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

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

      2. lower--.f64 N/A

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

      3. lower-log1p.f64 N/A

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

      4. lower-log.f64 35.8

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

   5. Applied rewrites 35.8%

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

   6. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. inv-pow N/A

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

      9. lower-pow.f64 59.8

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

   8. Applied rewrites 59.8%

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

   9. Taylor expanded in x around 0

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

       1. pow2 N/A

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

       2. lift-/.f64 N/A

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

       3. lift-*.f64 80.0

          \[\leadsto \frac{\frac{\frac{0.3333333333333333}{x \cdot x}}{x}}{n} \]

   11. Applied rewrites 80.0%

       \[\leadsto \frac{\frac{\frac{0.3333333333333333}{x \cdot x}}{x}}{n} \]

   if 4.30000000000000007e-165 < n < 2.3e10

   1. Initial program 82.1%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   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 81.7%

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

   if 2.3e10 < n < 1.25999999999999993e73 or 6.19999999999999993e168 < n

   1. Initial program 24.2%

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

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

      2. lower--.f64 N/A

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

      3. lower-log1p.f64 N/A

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

      4. lower-log.f64 82.4

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

   5. Applied rewrites 82.4%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 65.6%

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

4. Final simplification 69.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -8.5:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \mathbf{elif}\;n \leq 4.3 \cdot 10^{-165}:\\ \;\;\;\;\frac{\frac{\frac{0.3333333333333333}{x \cdot x}}{x}}{n}\\ \mathbf{elif}\;n \leq 23000000000:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;n \leq 1.26 \cdot 10^{+73} \lor \neg \left(n \leq 6.2 \cdot 10^{+168}\right):\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 55.7% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\frac{1}{x}}{n}\\ \mathbf{if}\;n \leq -8.5:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n \leq 7 \cdot 10^{-102}:\\ \;\;\;\;\frac{\frac{\frac{0.3333333333333333}{x \cdot x}}{x}}{n}\\ \mathbf{elif}\;n \leq 1.26 \cdot 10^{+73} \lor \neg \left(n \leq 6.2 \cdot 10^{+168}\right):\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (/ (/ 1.0 x) n)))
   (if (<= n -8.5)
     t_0
     (if (<= n 7e-102)
       (/ (/ (/ 0.3333333333333333 (* x x)) x) n)
       (if (or (<= n 1.26e+73) (not (<= n 6.2e+168)))
         (/ (- x (log x)) n)
         t_0)))))

C

double code(double x, double n) {
	double t_0 = (1.0 / x) / n;
	double tmp;
	if (n <= -8.5) {
		tmp = t_0;
	} else if (n <= 7e-102) {
		tmp = ((0.3333333333333333 / (x * x)) / x) / n;
	} else if ((n <= 1.26e+73) || !(n <= 6.2e+168)) {
		tmp = (x - log(x)) / n;
	} else {
		tmp = t_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) :: t_0
    real(8) :: tmp
    t_0 = (1.0d0 / x) / n
    if (n <= (-8.5d0)) then
        tmp = t_0
    else if (n <= 7d-102) then
        tmp = ((0.3333333333333333d0 / (x * x)) / x) / n
    else if ((n <= 1.26d+73) .or. (.not. (n <= 6.2d+168))) then
        tmp = (x - log(x)) / n
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = (1.0 / x) / n;
	double tmp;
	if (n <= -8.5) {
		tmp = t_0;
	} else if (n <= 7e-102) {
		tmp = ((0.3333333333333333 / (x * x)) / x) / n;
	} else if ((n <= 1.26e+73) || !(n <= 6.2e+168)) {
		tmp = (x - Math.log(x)) / n;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = (1.0 / x) / n
	tmp = 0
	if n <= -8.5:
		tmp = t_0
	elif n <= 7e-102:
		tmp = ((0.3333333333333333 / (x * x)) / x) / n
	elif (n <= 1.26e+73) or not (n <= 6.2e+168):
		tmp = (x - math.log(x)) / n
	else:
		tmp = t_0
	return tmp

Julia

function code(x, n)
	t_0 = Float64(Float64(1.0 / x) / n)
	tmp = 0.0
	if (n <= -8.5)
		tmp = t_0;
	elseif (n <= 7e-102)
		tmp = Float64(Float64(Float64(0.3333333333333333 / Float64(x * x)) / x) / n);
	elseif ((n <= 1.26e+73) || !(n <= 6.2e+168))
		tmp = Float64(Float64(x - log(x)) / n);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = (1.0 / x) / n;
	tmp = 0.0;
	if (n <= -8.5)
		tmp = t_0;
	elseif (n <= 7e-102)
		tmp = ((0.3333333333333333 / (x * x)) / x) / n;
	elseif ((n <= 1.26e+73) || ~((n <= 6.2e+168)))
		tmp = (x - log(x)) / n;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(N[(1.0 / x), $MachinePrecision] / n), $MachinePrecision]}, If[LessEqual[n, -8.5], t$95$0, If[LessEqual[n, 7e-102], N[(N[(N[(0.3333333333333333 / N[(x * x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] / n), $MachinePrecision], If[Or[LessEqual[n, 1.26e+73], N[Not[LessEqual[n, 6.2e+168]], $MachinePrecision]], N[(N[(x - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if n < -8.5 or 1.25999999999999993e73 < n < 6.19999999999999993e168

   1. Initial program 29.9%

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

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

      2. lower--.f64 N/A

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

      3. lower-log1p.f64 N/A

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

      4. lower-log.f64 66.8

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

   5. Applied rewrites 66.8%

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

   6. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. inv-pow N/A

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

      9. lower-pow.f64 57.6

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

   8. Applied rewrites 57.6%

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

   9. Taylor expanded in x around inf

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

   11. Applied rewrites 58.5%

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

   if -8.5 < n < 6.99999999999999973e-102

   1. Initial program 80.8%

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

   3. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-log1p.f64 N/A

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

      4. lower-log.f64 33.9

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

   5. Applied rewrites 33.9%

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

   6. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. inv-pow N/A

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

      9. lower-pow.f64 58.3

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

   8. Applied rewrites 58.3%

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

   9. Taylor expanded in x around 0

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

       1. pow2 N/A

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

       2. lift-/.f64 N/A

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

       3. lift-*.f64 77.3

          \[\leadsto \frac{\frac{\frac{0.3333333333333333}{x \cdot x}}{x}}{n} \]

   11. Applied rewrites 77.3%

       \[\leadsto \frac{\frac{\frac{0.3333333333333333}{x \cdot x}}{x}}{n} \]

   if 6.99999999999999973e-102 < n < 1.25999999999999993e73 or 6.19999999999999993e168 < n

   1. Initial program 42.1%

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

   3. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-log1p.f64 N/A

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

      4. lower-log.f64 63.1

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

   5. Applied rewrites 63.1%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 50.9%

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

4. Final simplification 63.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -8.5:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \mathbf{elif}\;n \leq 7 \cdot 10^{-102}:\\ \;\;\;\;\frac{\frac{\frac{0.3333333333333333}{x \cdot x}}{x}}{n}\\ \mathbf{elif}\;n \leq 1.26 \cdot 10^{+73} \lor \neg \left(n \leq 6.2 \cdot 10^{+168}\right):\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 55.8% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\frac{1}{x}}{n}\\ \mathbf{if}\;n \leq -8.5:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n \leq 7 \cdot 10^{-102}:\\ \;\;\;\;\frac{\frac{\frac{0.3333333333333333}{x \cdot x}}{x}}{n}\\ \mathbf{elif}\;n \leq 1.26 \cdot 10^{+73} \lor \neg \left(n \leq 6.2 \cdot 10^{+168}\right):\\ \;\;\;\;\frac{-\log x}{n}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (/ (/ 1.0 x) n)))
   (if (<= n -8.5)
     t_0
     (if (<= n 7e-102)
       (/ (/ (/ 0.3333333333333333 (* x x)) x) n)
       (if (or (<= n 1.26e+73) (not (<= n 6.2e+168)))
         (/ (- (log x)) n)
         t_0)))))

C

double code(double x, double n) {
	double t_0 = (1.0 / x) / n;
	double tmp;
	if (n <= -8.5) {
		tmp = t_0;
	} else if (n <= 7e-102) {
		tmp = ((0.3333333333333333 / (x * x)) / x) / n;
	} else if ((n <= 1.26e+73) || !(n <= 6.2e+168)) {
		tmp = -log(x) / n;
	} else {
		tmp = t_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) :: t_0
    real(8) :: tmp
    t_0 = (1.0d0 / x) / n
    if (n <= (-8.5d0)) then
        tmp = t_0
    else if (n <= 7d-102) then
        tmp = ((0.3333333333333333d0 / (x * x)) / x) / n
    else if ((n <= 1.26d+73) .or. (.not. (n <= 6.2d+168))) then
        tmp = -log(x) / n
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = (1.0 / x) / n;
	double tmp;
	if (n <= -8.5) {
		tmp = t_0;
	} else if (n <= 7e-102) {
		tmp = ((0.3333333333333333 / (x * x)) / x) / n;
	} else if ((n <= 1.26e+73) || !(n <= 6.2e+168)) {
		tmp = -Math.log(x) / n;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = (1.0 / x) / n
	tmp = 0
	if n <= -8.5:
		tmp = t_0
	elif n <= 7e-102:
		tmp = ((0.3333333333333333 / (x * x)) / x) / n
	elif (n <= 1.26e+73) or not (n <= 6.2e+168):
		tmp = -math.log(x) / n
	else:
		tmp = t_0
	return tmp

Julia

function code(x, n)
	t_0 = Float64(Float64(1.0 / x) / n)
	tmp = 0.0
	if (n <= -8.5)
		tmp = t_0;
	elseif (n <= 7e-102)
		tmp = Float64(Float64(Float64(0.3333333333333333 / Float64(x * x)) / x) / n);
	elseif ((n <= 1.26e+73) || !(n <= 6.2e+168))
		tmp = Float64(Float64(-log(x)) / n);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = (1.0 / x) / n;
	tmp = 0.0;
	if (n <= -8.5)
		tmp = t_0;
	elseif (n <= 7e-102)
		tmp = ((0.3333333333333333 / (x * x)) / x) / n;
	elseif ((n <= 1.26e+73) || ~((n <= 6.2e+168)))
		tmp = -log(x) / n;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(N[(1.0 / x), $MachinePrecision] / n), $MachinePrecision]}, If[LessEqual[n, -8.5], t$95$0, If[LessEqual[n, 7e-102], N[(N[(N[(0.3333333333333333 / N[(x * x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] / n), $MachinePrecision], If[Or[LessEqual[n, 1.26e+73], N[Not[LessEqual[n, 6.2e+168]], $MachinePrecision]], N[((-N[Log[x], $MachinePrecision]) / n), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if n < -8.5 or 1.25999999999999993e73 < n < 6.19999999999999993e168

   1. Initial program 29.9%

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

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

      2. lower--.f64 N/A

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

      3. lower-log1p.f64 N/A

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

      4. lower-log.f64 66.8

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

   5. Applied rewrites 66.8%

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

   6. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. inv-pow N/A

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

      9. lower-pow.f64 57.6

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

   8. Applied rewrites 57.6%

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

   9. Taylor expanded in x around inf

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

   11. Applied rewrites 58.5%

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

   if -8.5 < n < 6.99999999999999973e-102

   1. Initial program 80.8%

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

   3. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-log1p.f64 N/A

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

      4. lower-log.f64 33.9

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

   5. Applied rewrites 33.9%

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

   6. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. inv-pow N/A

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

      9. lower-pow.f64 58.3

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

   8. Applied rewrites 58.3%

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

   9. Taylor expanded in x around 0

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

       1. pow2 N/A

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

       2. lift-/.f64 N/A

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

       3. lift-*.f64 77.3

          \[\leadsto \frac{\frac{\frac{0.3333333333333333}{x \cdot x}}{x}}{n} \]

   11. Applied rewrites 77.3%

       \[\leadsto \frac{\frac{\frac{0.3333333333333333}{x \cdot x}}{x}}{n} \]

   if 6.99999999999999973e-102 < n < 1.25999999999999993e73 or 6.19999999999999993e168 < n

   1. Initial program 42.1%

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

   3. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-log1p.f64 N/A

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

      4. lower-log.f64 63.1

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

   5. Applied rewrites 63.1%

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

   6. Taylor expanded in x around 0

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

      1. log-pow-rev N/A

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

      2. inv-pow N/A

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

      3. neg-log N/A

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

      4. lift-log.f64 N/A

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

      5. lift-neg.f64 50.0

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

   8. Applied rewrites 50.0%

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

4. Final simplification 63.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -8.5:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \mathbf{elif}\;n \leq 7 \cdot 10^{-102}:\\ \;\;\;\;\frac{\frac{\frac{0.3333333333333333}{x \cdot x}}{x}}{n}\\ \mathbf{elif}\;n \leq 1.26 \cdot 10^{+73} \lor \neg \left(n \leq 6.2 \cdot 10^{+168}\right):\\ \;\;\;\;\frac{-\log x}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 55.1% accurate, 4.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -8.5 \lor \neg \left(n \leq 3.3 \cdot 10^{-100}\right):\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{0.3333333333333333}{x \cdot x}}{x}}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (or (<= n -8.5) (not (<= n 3.3e-100)))
   (/ (/ 1.0 x) n)
   (/ (/ (/ 0.3333333333333333 (* x x)) x) n)))

C

double code(double x, double n) {
	double tmp;
	if ((n <= -8.5) || !(n <= 3.3e-100)) {
		tmp = (1.0 / x) / n;
	} else {
		tmp = ((0.3333333333333333 / (x * x)) / x) / n;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, n)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if ((n <= (-8.5d0)) .or. (.not. (n <= 3.3d-100))) then
        tmp = (1.0d0 / x) / n
    else
        tmp = ((0.3333333333333333d0 / (x * x)) / x) / n
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if ((n <= -8.5) || !(n <= 3.3e-100)) {
		tmp = (1.0 / x) / n;
	} else {
		tmp = ((0.3333333333333333 / (x * x)) / x) / n;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if (n <= -8.5) or not (n <= 3.3e-100):
		tmp = (1.0 / x) / n
	else:
		tmp = ((0.3333333333333333 / (x * x)) / x) / n
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if ((n <= -8.5) || !(n <= 3.3e-100))
		tmp = Float64(Float64(1.0 / x) / n);
	else
		tmp = Float64(Float64(Float64(0.3333333333333333 / Float64(x * x)) / x) / n);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if ((n <= -8.5) || ~((n <= 3.3e-100)))
		tmp = (1.0 / x) / n;
	else
		tmp = ((0.3333333333333333 / (x * x)) / x) / n;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[Or[LessEqual[n, -8.5], N[Not[LessEqual[n, 3.3e-100]], $MachinePrecision]], N[(N[(1.0 / x), $MachinePrecision] / n), $MachinePrecision], N[(N[(N[(0.3333333333333333 / N[(x * x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] / n), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < -8.5 or 3.29999999999999996e-100 < n

   1. Initial program 35.4%

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

   3. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-log1p.f64 N/A

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

      4. lower-log.f64 65.2

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

   5. Applied rewrites 65.2%

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

   6. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. inv-pow N/A

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

      9. lower-pow.f64 44.0

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

   8. Applied rewrites 44.0%

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

   9. Taylor expanded in x around inf

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

   11. Applied rewrites 44.8%

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

   if -8.5 < n < 3.29999999999999996e-100

   1. Initial program 80.8%

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

   3. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-log1p.f64 N/A

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

      4. lower-log.f64 33.9

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

   5. Applied rewrites 33.9%

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

   6. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. inv-pow N/A

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

      9. lower-pow.f64 58.3

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

   8. Applied rewrites 58.3%

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

   9. Taylor expanded in x around 0

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

       1. pow2 N/A

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

       2. lift-/.f64 N/A

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

       3. lift-*.f64 77.3

          \[\leadsto \frac{\frac{\frac{0.3333333333333333}{x \cdot x}}{x}}{n} \]

   11. Applied rewrites 77.3%

       \[\leadsto \frac{\frac{\frac{0.3333333333333333}{x \cdot x}}{x}}{n} \]
3. Recombined 2 regimes into one program.

4. Final simplification 57.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -8.5 \lor \neg \left(n \leq 3.3 \cdot 10^{-100}\right):\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{0.3333333333333333}{x \cdot x}}{x}}{n}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 53.7% accurate, 5.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -8.5 \lor \neg \left(n \leq 3.3 \cdot 10^{-100}\right):\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.3333333333333333}{n \cdot \left(x \cdot x\right)}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (or (<= n -8.5) (not (<= n 3.3e-100)))
   (/ (/ 1.0 x) n)
   (/ (/ 0.3333333333333333 (* n (* x x))) x)))

C

double code(double x, double n) {
	double tmp;
	if ((n <= -8.5) || !(n <= 3.3e-100)) {
		tmp = (1.0 / x) / n;
	} else {
		tmp = (0.3333333333333333 / (n * (x * x))) / x;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, n)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if ((n <= (-8.5d0)) .or. (.not. (n <= 3.3d-100))) then
        tmp = (1.0d0 / x) / n
    else
        tmp = (0.3333333333333333d0 / (n * (x * x))) / x
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if ((n <= -8.5) || !(n <= 3.3e-100)) {
		tmp = (1.0 / x) / n;
	} else {
		tmp = (0.3333333333333333 / (n * (x * x))) / x;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if (n <= -8.5) or not (n <= 3.3e-100):
		tmp = (1.0 / x) / n
	else:
		tmp = (0.3333333333333333 / (n * (x * x))) / x
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if ((n <= -8.5) || !(n <= 3.3e-100))
		tmp = Float64(Float64(1.0 / x) / n);
	else
		tmp = Float64(Float64(0.3333333333333333 / Float64(n * Float64(x * x))) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if ((n <= -8.5) || ~((n <= 3.3e-100)))
		tmp = (1.0 / x) / n;
	else
		tmp = (0.3333333333333333 / (n * (x * x))) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[Or[LessEqual[n, -8.5], N[Not[LessEqual[n, 3.3e-100]], $MachinePrecision]], N[(N[(1.0 / x), $MachinePrecision] / n), $MachinePrecision], N[(N[(0.3333333333333333 / N[(n * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < -8.5 or 3.29999999999999996e-100 < n

   1. Initial program 35.4%

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

   3. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-log1p.f64 N/A

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

      4. lower-log.f64 65.2

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

   5. Applied rewrites 65.2%

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

   6. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. inv-pow N/A

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

      9. lower-pow.f64 44.0

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

   8. Applied rewrites 44.0%

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

   9. Taylor expanded in x around inf

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

   11. Applied rewrites 44.8%

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

   if -8.5 < n < 3.29999999999999996e-100

   1. Initial program 80.8%

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

   3. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-log1p.f64 N/A

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

      4. lower-log.f64 33.9

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

   5. Applied rewrites 33.9%

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

   6. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. inv-pow N/A

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

      9. lower-pow.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift-*.f64 25.1

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

   8. Applied rewrites 25.1%

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

   9. Taylor expanded in x around 0

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

       1. pow2 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. lift-/.f64 72.4

          \[\leadsto \frac{\frac{0.3333333333333333}{n \cdot \left(x \cdot x\right)}}{x} \]

   11. Applied rewrites 72.4%

       \[\leadsto \frac{\frac{0.3333333333333333}{n \cdot \left(x \cdot x\right)}}{x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 55.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -8.5 \lor \neg \left(n \leq 3.3 \cdot 10^{-100}\right):\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.3333333333333333}{n \cdot \left(x \cdot x\right)}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 46.1% accurate, 6.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -0.36 \lor \neg \left(n \leq -2 \cdot 10^{-264}\right):\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \mathbf{else}:\\ \;\;\;\;1 - 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (or (<= n -0.36) (not (<= n -2e-264))) (/ (/ 1.0 x) n) (- 1.0 1.0)))

C

double code(double x, double n) {
	double tmp;
	if ((n <= -0.36) || !(n <= -2e-264)) {
		tmp = (1.0 / x) / n;
	} else {
		tmp = 1.0 - 1.0;
	}
	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 ((n <= (-0.36d0)) .or. (.not. (n <= (-2d-264)))) then
        tmp = (1.0d0 / x) / n
    else
        tmp = 1.0d0 - 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if ((n <= -0.36) || !(n <= -2e-264)) {
		tmp = (1.0 / x) / n;
	} else {
		tmp = 1.0 - 1.0;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if (n <= -0.36) or not (n <= -2e-264):
		tmp = (1.0 / x) / n
	else:
		tmp = 1.0 - 1.0
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if ((n <= -0.36) || !(n <= -2e-264))
		tmp = Float64(Float64(1.0 / x) / n);
	else
		tmp = Float64(1.0 - 1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if ((n <= -0.36) || ~((n <= -2e-264)))
		tmp = (1.0 / x) / n;
	else
		tmp = 1.0 - 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[Or[LessEqual[n, -0.36], N[Not[LessEqual[n, -2e-264]], $MachinePrecision]], N[(N[(1.0 / x), $MachinePrecision] / n), $MachinePrecision], N[(1.0 - 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < -0.35999999999999999 or -2e-264 < n

   1. Initial program 38.4%

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

   3. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-log1p.f64 N/A

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

      4. lower-log.f64 55.3

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

   5. Applied rewrites 55.3%

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

   6. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. inv-pow N/A

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

      9. lower-pow.f64 49.9

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

   8. Applied rewrites 49.9%

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

   9. Taylor expanded in x around inf

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

   11. Applied rewrites 47.2%

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

   if -0.35999999999999999 < n < -2e-264

   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 0

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

   5. Applied rewrites 55.7%

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

   6. Taylor expanded in n around inf

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

   8. Applied rewrites 46.6%

      \[\leadsto 1 - \color{blue}{1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 47.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -0.36 \lor \neg \left(n \leq -2 \cdot 10^{-264}\right):\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \mathbf{else}:\\ \;\;\;\;1 - 1\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 45.6% accurate, 8.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -0.36 \lor \neg \left(n \leq -2 \cdot 10^{-264}\right):\\ \;\;\;\;\frac{1}{n \cdot x}\\ \mathbf{else}:\\ \;\;\;\;1 - 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (or (<= n -0.36) (not (<= n -2e-264))) (/ 1.0 (* n x)) (- 1.0 1.0)))

C

double code(double x, double n) {
	double tmp;
	if ((n <= -0.36) || !(n <= -2e-264)) {
		tmp = 1.0 / (n * x);
	} else {
		tmp = 1.0 - 1.0;
	}
	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 ((n <= (-0.36d0)) .or. (.not. (n <= (-2d-264)))) then
        tmp = 1.0d0 / (n * x)
    else
        tmp = 1.0d0 - 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if ((n <= -0.36) || !(n <= -2e-264)) {
		tmp = 1.0 / (n * x);
	} else {
		tmp = 1.0 - 1.0;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if (n <= -0.36) or not (n <= -2e-264):
		tmp = 1.0 / (n * x)
	else:
		tmp = 1.0 - 1.0
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if ((n <= -0.36) || !(n <= -2e-264))
		tmp = Float64(1.0 / Float64(n * x));
	else
		tmp = Float64(1.0 - 1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if ((n <= -0.36) || ~((n <= -2e-264)))
		tmp = 1.0 / (n * x);
	else
		tmp = 1.0 - 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[Or[LessEqual[n, -0.36], N[Not[LessEqual[n, -2e-264]], $MachinePrecision]], N[(1.0 / N[(n * x), $MachinePrecision]), $MachinePrecision], N[(1.0 - 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < -0.35999999999999999 or -2e-264 < n

   1. Initial program 38.4%

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

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

      2. lower-exp.f64 N/A

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

      3. mul-1-neg N/A

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

      4. log-rec N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-log.f64 N/A

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

      11. lower-*.f64 43.4

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

   5. Applied rewrites 43.4%

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

   6. Taylor expanded in n around inf

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

   8. Applied rewrites 47.1%

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

   if -0.35999999999999999 < n < -2e-264

   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 0

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

   5. Applied rewrites 55.7%

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

   6. Taylor expanded in n around inf

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

   8. Applied rewrites 46.6%

      \[\leadsto 1 - \color{blue}{1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 47.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -0.36 \lor \neg \left(n \leq -2 \cdot 10^{-264}\right):\\ \;\;\;\;\frac{1}{n \cdot x}\\ \mathbf{else}:\\ \;\;\;\;1 - 1\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 29.2% accurate, 57.8× speedup

Math

\[\begin{array}{l} \\ 1 - 1 \end{array} \]

FPCore

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

C

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

Java

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

Python

def code(x, n):
	return 1.0 - 1.0

Julia

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

MATLAB

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

Wolfram

code[x_, n_] := N[(1.0 - 1.0), $MachinePrecision]

Derivation

1. Initial program 52.6%

   \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
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 41.1%

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

6. Taylor expanded in n around inf

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

8. Applied rewrites 26.1%

   \[\leadsto 1 - \color{blue}{1} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2025064 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  :precision binary64
  (- (pow (+ x 1.0) (/ 1.0 n)) (pow x (/ 1.0 n))))