2nthrt (problem 3.4.6)

Percentage Accurate: 53.5% → 88.5%

Time: 18.6s

Alternatives: 16

Speedup: 1.9×

• Could not converge on a ground truth

  1. x = -8.660537888369224e+135
  2. n = -7.417858440492442e-140

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 53.5% 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: 88.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -\log x\\ t_1 := \frac{\log x}{n}\\ \mathbf{if}\;x \leq 0.9:\\ \;\;\;\;x \cdot \mathsf{fma}\left(-1, \frac{t\_1 - \frac{\mathsf{fma}\left(-0.5, \frac{{t\_0}^{2}}{n}, 0.16666666666666666 \cdot \frac{{t\_0}^{3}}{n \cdot n}\right)}{n}}{x}, {n}^{-1}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{t\_1}}{n \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (- (log x))) (t_1 (/ (log x) n)))
   (if (<= x 0.9)
     (*
      x
      (fma
       -1.0
       (/
        (-
         t_1
         (/
          (fma
           -0.5
           (/ (pow t_0 2.0) n)
           (* 0.16666666666666666 (/ (pow t_0 3.0) (* n n))))
          n))
        x)
       (pow n -1.0)))
     (/ (exp t_1) (* n x)))))

C

double code(double x, double n) {
	double t_0 = -log(x);
	double t_1 = log(x) / n;
	double tmp;
	if (x <= 0.9) {
		tmp = x * fma(-1.0, ((t_1 - (fma(-0.5, (pow(t_0, 2.0) / n), (0.16666666666666666 * (pow(t_0, 3.0) / (n * n)))) / n)) / x), pow(n, -1.0));
	} else {
		tmp = exp(t_1) / (n * x);
	}
	return tmp;
}

Julia

function code(x, n)
	t_0 = Float64(-log(x))
	t_1 = Float64(log(x) / n)
	tmp = 0.0
	if (x <= 0.9)
		tmp = Float64(x * fma(-1.0, Float64(Float64(t_1 - Float64(fma(-0.5, Float64((t_0 ^ 2.0) / n), Float64(0.16666666666666666 * Float64((t_0 ^ 3.0) / Float64(n * n)))) / n)) / x), (n ^ -1.0)));
	else
		tmp = Float64(exp(t_1) / Float64(n * x));
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = (-N[Log[x], $MachinePrecision])}, Block[{t$95$1 = N[(N[Log[x], $MachinePrecision] / n), $MachinePrecision]}, If[LessEqual[x, 0.9], N[(x * N[(-1.0 * N[(N[(t$95$1 - N[(N[(-0.5 * N[(N[Power[t$95$0, 2.0], $MachinePrecision] / n), $MachinePrecision] + N[(0.16666666666666666 * N[(N[Power[t$95$0, 3.0], $MachinePrecision] / N[(n * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] + N[Power[n, -1.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Exp[t$95$1], $MachinePrecision] / N[(n * x), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if x < 0.900000000000000022

   1. Initial program 42.2%

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

   3. Taylor expanded in n around -inf

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

   5. Applied rewrites 79.1%

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

   6. Taylor expanded in x around 0

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

      1. lower--.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lower-fma.f64 N/A

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

   8. Applied rewrites 78.3%

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

   9. Taylor expanded in x around inf

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

   11. Applied rewrites 82.1%

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

   if 0.900000000000000022 < x

   1. Initial program 68.4%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   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 97.0

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

   5. Applied rewrites 97.0%

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

4. Final simplification 88.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.9:\\ \;\;\;\;x \cdot \mathsf{fma}\left(-1, \frac{\frac{\log x}{n} - \frac{\mathsf{fma}\left(-0.5, \frac{{\left(-\log x\right)}^{2}}{n}, 0.16666666666666666 \cdot \frac{{\left(-\log x\right)}^{3}}{n \cdot n}\right)}{n}}{x}, {n}^{-1}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\frac{\log x}{n}}}{n \cdot x}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 79.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ t_1 := {\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - t\_0\\ \mathbf{if}\;t\_1 \leq -4 \cdot 10^{-8}:\\ \;\;\;\;x \cdot \frac{1 + \frac{x}{n}}{x} - t\_0\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{-10}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right) - \log x}{n}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - 1\\ \end{array} \end{array} \]

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 -4e-8)
     (- (* x (/ (+ 1.0 (/ x n)) x)) t_0)
     (if (<= t_1 5e-10)
       (/ (- (log1p x) (log x)) n)
       (- (exp (/ (log1p x) n)) 1.0)))))

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 <= -4e-8) {
		tmp = (x * ((1.0 + (x / n)) / x)) - t_0;
	} else if (t_1 <= 5e-10) {
		tmp = (log1p(x) - log(x)) / n;
	} else {
		tmp = exp((log1p(x) / n)) - 1.0;
	}
	return tmp;
}

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 <= -4e-8) {
		tmp = (x * ((1.0 + (x / n)) / x)) - t_0;
	} else if (t_1 <= 5e-10) {
		tmp = (Math.log1p(x) - Math.log(x)) / n;
	} else {
		tmp = Math.exp((Math.log1p(x) / n)) - 1.0;
	}
	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 <= -4e-8:
		tmp = (x * ((1.0 + (x / n)) / x)) - t_0
	elif t_1 <= 5e-10:
		tmp = (math.log1p(x) - math.log(x)) / n
	else:
		tmp = math.exp((math.log1p(x) / n)) - 1.0
	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 <= -4e-8)
		tmp = Float64(Float64(x * Float64(Float64(1.0 + Float64(x / n)) / x)) - t_0);
	elseif (t_1 <= 5e-10)
		tmp = Float64(Float64(log1p(x) - log(x)) / n);
	else
		tmp = Float64(exp(Float64(log1p(x) / n)) - 1.0);
	end
	return 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, -4e-8], N[(N[(x * N[(N[(1.0 + N[(x / n), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[t$95$1, 5e-10], N[(N[(N[Log[1 + x], $MachinePrecision] - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], N[(N[Exp[N[(N[Log[1 + x], $MachinePrecision] / n), $MachinePrecision]], $MachinePrecision] - 1.0), $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))) < -4.0000000000000001e-8

   1. Initial program 99.6%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(1 + \frac{x}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   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 99.6

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

   5. Applied rewrites 99.6%

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

   6. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. inv-pow N/A

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

      4. lower-pow.f64 N/A

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

      5. inv-pow N/A

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

      6. lower-pow.f64 99.6

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

   8. Applied rewrites 99.6%

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

   9. Taylor expanded in x around 0

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

       1. lower-/.f64 N/A

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

       2. lower-+.f64 N/A

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

       3. lift-/.f64 99.6

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

   11. Applied rewrites 99.6%

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

   if -4.0000000000000001e-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 43.9%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   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 80.7

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

   5. Applied rewrites 80.7%

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

   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 54.9%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 50.2%

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

   6. Taylor expanded in n around inf

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

   8. Applied rewrites 2.4%

      \[\leadsto 1 - \color{blue}{1} \]

   9. Taylor expanded in n around 0

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

       1. lower-exp.f64 N/A

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

       2. lower-/.f64 N/A

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

       3. lift-log1p.f64 50.3

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

   11. Applied rewrites 50.3%

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

Alternative 3: 79.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ t_1 := {\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - t\_0\\ \mathbf{if}\;t\_1 \leq -4 \cdot 10^{-8}:\\ \;\;\;\;x \cdot \frac{1 + \frac{x}{n}}{x} - t\_0\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{-10}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right) - \log x}{n}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{x}{n} + 1\right) - t\_0\\ \end{array} \end{array} \]

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 -4e-8)
     (- (* x (/ (+ 1.0 (/ x n)) x)) t_0)
     (if (<= t_1 5e-10) (/ (- (log1p x) (log 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 <= -4e-8) {
		tmp = (x * ((1.0 + (x / n)) / x)) - t_0;
	} else if (t_1 <= 5e-10) {
		tmp = (log1p(x) - log(x)) / n;
	} else {
		tmp = ((x / n) + 1.0) - t_0;
	}
	return tmp;
}

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 <= -4e-8) {
		tmp = (x * ((1.0 + (x / n)) / x)) - t_0;
	} else if (t_1 <= 5e-10) {
		tmp = (Math.log1p(x) - Math.log(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 <= -4e-8:
		tmp = (x * ((1.0 + (x / n)) / x)) - t_0
	elif t_1 <= 5e-10:
		tmp = (math.log1p(x) - math.log(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 <= -4e-8)
		tmp = Float64(Float64(x * Float64(Float64(1.0 + Float64(x / n)) / x)) - t_0);
	elseif (t_1 <= 5e-10)
		tmp = Float64(Float64(log1p(x) - log(x)) / n);
	else
		tmp = Float64(Float64(Float64(x / n) + 1.0) - t_0);
	end
	return 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, -4e-8], N[(N[(x * N[(N[(1.0 + N[(x / n), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[t$95$1, 5e-10], N[(N[(N[Log[1 + x], $MachinePrecision] - N[Log[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))) < -4.0000000000000001e-8

   1. Initial program 99.6%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(1 + \frac{x}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   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 99.6

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

   5. Applied rewrites 99.6%

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

   6. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. inv-pow N/A

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

      4. lower-pow.f64 N/A

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

      5. inv-pow N/A

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

      6. lower-pow.f64 99.6

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

   8. Applied rewrites 99.6%

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

   9. Taylor expanded in x around 0

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

       1. lower-/.f64 N/A

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

       2. lower-+.f64 N/A

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

       3. lift-/.f64 99.6

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

   11. Applied rewrites 99.6%

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

   if -4.0000000000000001e-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 43.9%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   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 80.7

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

   5. Applied rewrites 80.7%

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

   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 54.9%

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

   3. Taylor expanded in x around 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 51.9

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

   5. Applied rewrites 51.9%

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

Alternative 4: 86.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -1.5 \cdot 10^{-16}:\\ \;\;\;\;\frac{e^{\frac{\log x}{n}}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-12}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-1, \mathsf{log1p}\left(x\right) + \frac{0.5 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}\right)}{n}, \log x\right)}{-n}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= (/ 1.0 n) -1.5e-16)
   (/ (exp (/ (log x) n)) (* n x))
   (if (<= (/ 1.0 n) 2e-12)
     (/
      (fma
       -1.0
       (+ (log1p x) (/ (* 0.5 (- (pow (log1p x) 2.0) (pow (log x) 2.0))) n))
       (log x))
      (- n))
     (- (exp (/ (log1p x) n)) (pow x (/ 1.0 n))))))

C

double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -1.5e-16) {
		tmp = exp((log(x) / n)) / (n * x);
	} else if ((1.0 / n) <= 2e-12) {
		tmp = fma(-1.0, (log1p(x) + ((0.5 * (pow(log1p(x), 2.0) - pow(log(x), 2.0))) / n)), log(x)) / -n;
	} else {
		tmp = exp((log1p(x) / n)) - pow(x, (1.0 / n));
	}
	return tmp;
}

Julia

function code(x, n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -1.5e-16)
		tmp = Float64(exp(Float64(log(x) / n)) / Float64(n * x));
	elseif (Float64(1.0 / n) <= 2e-12)
		tmp = Float64(fma(-1.0, Float64(log1p(x) + Float64(Float64(0.5 * Float64((log1p(x) ^ 2.0) - (log(x) ^ 2.0))) / n)), log(x)) / Float64(-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], -1.5e-16], N[(N[Exp[N[(N[Log[x], $MachinePrecision] / n), $MachinePrecision]], $MachinePrecision] / N[(n * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-12], N[(N[(-1.0 * N[(N[Log[1 + x], $MachinePrecision] + N[(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] / n), $MachinePrecision]), $MachinePrecision] + N[Log[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) < -1.49999999999999997e-16

   1. Initial program 97.3%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   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 97.9

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

   5. Applied rewrites 97.9%

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

   if -1.49999999999999997e-16 < (/.f64 #s(literal 1 binary64) n) < 1.99999999999999996e-12

   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

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-/.f64 N/A

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

   5. Applied rewrites 78.6%

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

   if 1.99999999999999996e-12 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 53.2%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   2. 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 95.2

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

   5. Applied rewrites 95.2%

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

4. Final simplification 86.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -1.5 \cdot 10^{-16}:\\ \;\;\;\;\frac{e^{\frac{\log x}{n}}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-12}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-1, \mathsf{log1p}\left(x\right) + \frac{0.5 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}\right)}{n}, \log x\right)}{-n}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 86.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\log x}^{2}\\ \mathbf{if}\;n \leq -6.6 \cdot 10^{+15}:\\ \;\;\;\;\frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2} - t\_0}{n \cdot \left(\log x + \mathsf{log1p}\left(x\right)\right)}\\ \mathbf{elif}\;n \leq 80000000:\\ \;\;\;\;e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-1, \mathsf{log1p}\left(x\right) + \frac{-0.5 \cdot t\_0}{n}, \log x\right)}{-n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow (log x) 2.0)))
   (if (<= n -6.6e+15)
     (/ (- (pow (log1p x) 2.0) t_0) (* n (+ (log x) (log1p x))))
     (if (<= n 80000000.0)
       (- (exp (/ (log1p x) n)) (pow x (/ 1.0 n)))
       (/ (fma -1.0 (+ (log1p x) (/ (* -0.5 t_0) n)) (log x)) (- n))))))

C

double code(double x, double n) {
	double t_0 = pow(log(x), 2.0);
	double tmp;
	if (n <= -6.6e+15) {
		tmp = (pow(log1p(x), 2.0) - t_0) / (n * (log(x) + log1p(x)));
	} else if (n <= 80000000.0) {
		tmp = exp((log1p(x) / n)) - pow(x, (1.0 / n));
	} else {
		tmp = fma(-1.0, (log1p(x) + ((-0.5 * t_0) / n)), log(x)) / -n;
	}
	return tmp;
}

Julia

function code(x, n)
	t_0 = log(x) ^ 2.0
	tmp = 0.0
	if (n <= -6.6e+15)
		tmp = Float64(Float64((log1p(x) ^ 2.0) - t_0) / Float64(n * Float64(log(x) + log1p(x))));
	elseif (n <= 80000000.0)
		tmp = Float64(exp(Float64(log1p(x) / n)) - (x ^ Float64(1.0 / n)));
	else
		tmp = Float64(fma(-1.0, Float64(log1p(x) + Float64(Float64(-0.5 * t_0) / n)), log(x)) / Float64(-n));
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[N[Log[x], $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[n, -6.6e+15], N[(N[(N[Power[N[Log[1 + x], $MachinePrecision], 2.0], $MachinePrecision] - t$95$0), $MachinePrecision] / N[(n * N[(N[Log[x], $MachinePrecision] + N[Log[1 + x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 80000000.0], N[(N[Exp[N[(N[Log[1 + x], $MachinePrecision] / n), $MachinePrecision]], $MachinePrecision] - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(-1.0 * N[(N[Log[1 + x], $MachinePrecision] + N[(N[(-0.5 * t$95$0), $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision] + N[Log[x], $MachinePrecision]), $MachinePrecision] / (-n)), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if n < -6.6e15

   1. Initial program 29.7%

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

   3. Taylor expanded in n around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-/.f64 N/A

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

   5. Applied rewrites 77.9%

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

   7. Applied rewrites 77.9%

      \[\leadsto -\frac{\frac{\left(-\left(\mathsf{log1p}\left(x\right) + \frac{0.5 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}\right)}{n}\right)\right) \cdot \left(-\left(\mathsf{log1p}\left(x\right) + \frac{0.5 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}\right)}{n}\right)\right) - {\log x}^{2}}{\left(-\left(\mathsf{log1p}\left(x\right) + \frac{0.5 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}\right)}{n}\right)\right) - \log x}}{n} \]

   8. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. lift-log1p.f64 N/A

         \[\leadsto \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}}{n \cdot \left(\log x + \log \left(1 + x\right)\right)} \]

      3. lift-pow.f64 N/A

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

      4. lift-pow.f64 N/A

         \[\leadsto \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}}{n \cdot \left(\log x + \log \left(1 + x\right)\right)} \]

      5. lift-log.f64 N/A

         \[\leadsto \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}}{n \cdot \left(\log x + \log \left(1 + x\right)\right)} \]

      6. lift--.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-+.f64 N/A

         \[\leadsto \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}}{n \cdot \left(\log x + \log \left(1 + x\right)\right)} \]

   10. Applied rewrites 77.1%

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

   if -6.6e15 < n < 8e7

   1. Initial program 83.0%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   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 97.1

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

   5. Applied rewrites 97.1%

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

   if 8e7 < n

   1. Initial program 30.1%

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

   3. Taylor expanded in n around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-/.f64 N/A

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

   5. Applied rewrites 78.9%

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

   6. Taylor expanded in x around 0

      \[\leadsto -\frac{\mathsf{fma}\left(-1, \mathsf{log1p}\left(x\right) + \frac{\frac{-1}{2} \cdot {\log x}^{2}}{n}, \log x\right)}{n} \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-log.f64 78.8

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

   8. Applied rewrites 78.8%

      \[\leadsto -\frac{\mathsf{fma}\left(-1, \mathsf{log1p}\left(x\right) + \frac{-0.5 \cdot {\log x}^{2}}{n}, \log x\right)}{n} \]
3. Recombined 3 regimes into one program.

4. Final simplification 86.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -6.6 \cdot 10^{+15}:\\ \;\;\;\;\frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}}{n \cdot \left(\log x + \mathsf{log1p}\left(x\right)\right)}\\ \mathbf{elif}\;n \leq 80000000:\\ \;\;\;\;e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-1, \mathsf{log1p}\left(x\right) + \frac{-0.5 \cdot {\log x}^{2}}{n}, \log x\right)}{-n}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 86.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -6.6 \cdot 10^{+15}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right) - \log x}{n}\\ \mathbf{elif}\;n \leq 80000000:\\ \;\;\;\;e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-1, \mathsf{log1p}\left(x\right) + \frac{-0.5 \cdot {\log x}^{2}}{n}, \log x\right)}{-n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= n -6.6e+15)
   (/ (- (log1p x) (log x)) n)
   (if (<= n 80000000.0)
     (- (exp (/ (log1p x) n)) (pow x (/ 1.0 n)))
     (/
      (fma -1.0 (+ (log1p x) (/ (* -0.5 (pow (log x) 2.0)) n)) (log x))
      (- n)))))

C

double code(double x, double n) {
	double tmp;
	if (n <= -6.6e+15) {
		tmp = (log1p(x) - log(x)) / n;
	} else if (n <= 80000000.0) {
		tmp = exp((log1p(x) / n)) - pow(x, (1.0 / n));
	} else {
		tmp = fma(-1.0, (log1p(x) + ((-0.5 * pow(log(x), 2.0)) / n)), log(x)) / -n;
	}
	return tmp;
}

Julia

function code(x, n)
	tmp = 0.0
	if (n <= -6.6e+15)
		tmp = Float64(Float64(log1p(x) - log(x)) / n);
	elseif (n <= 80000000.0)
		tmp = Float64(exp(Float64(log1p(x) / n)) - (x ^ Float64(1.0 / n)));
	else
		tmp = Float64(fma(-1.0, Float64(log1p(x) + Float64(Float64(-0.5 * (log(x) ^ 2.0)) / n)), log(x)) / Float64(-n));
	end
	return tmp
end

Wolfram

code[x_, n_] := If[LessEqual[n, -6.6e+15], N[(N[(N[Log[1 + x], $MachinePrecision] - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[n, 80000000.0], N[(N[Exp[N[(N[Log[1 + x], $MachinePrecision] / n), $MachinePrecision]], $MachinePrecision] - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(-1.0 * N[(N[Log[1 + x], $MachinePrecision] + N[(N[(-0.5 * N[Power[N[Log[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision] + N[Log[x], $MachinePrecision]), $MachinePrecision] / (-n)), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < -6.6e15

   1. Initial program 29.7%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   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 77.9

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

   5. Applied rewrites 77.9%

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

   if -6.6e15 < n < 8e7

   1. Initial program 83.0%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   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 97.1

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

   5. Applied rewrites 97.1%

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

   if 8e7 < n

   1. Initial program 30.1%

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

   3. Taylor expanded in n around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-/.f64 N/A

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

   5. Applied rewrites 78.9%

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

   6. Taylor expanded in x around 0

      \[\leadsto -\frac{\mathsf{fma}\left(-1, \mathsf{log1p}\left(x\right) + \frac{\frac{-1}{2} \cdot {\log x}^{2}}{n}, \log x\right)}{n} \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-log.f64 78.8

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

   8. Applied rewrites 78.8%

      \[\leadsto -\frac{\mathsf{fma}\left(-1, \mathsf{log1p}\left(x\right) + \frac{-0.5 \cdot {\log x}^{2}}{n}, \log x\right)}{n} \]
3. Recombined 3 regimes into one program.

4. Final simplification 86.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -6.6 \cdot 10^{+15}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right) - \log x}{n}\\ \mathbf{elif}\;n \leq 80000000:\\ \;\;\;\;e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-1, \mathsf{log1p}\left(x\right) + \frac{-0.5 \cdot {\log x}^{2}}{n}, \log x\right)}{-n}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 86.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -1.5 \cdot 10^{-16}:\\ \;\;\;\;\frac{e^{\frac{\log x}{n}}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-12}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right) - \log x}{n}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= (/ 1.0 n) -1.5e-16)
   (/ (exp (/ (log x) n)) (* n x))
   (if (<= (/ 1.0 n) 2e-12)
     (/ (- (log1p x) (log x)) n)
     (- (exp (/ (log1p x) n)) (pow x (/ 1.0 n))))))

C

double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -1.5e-16) {
		tmp = exp((log(x) / n)) / (n * x);
	} else if ((1.0 / n) <= 2e-12) {
		tmp = (log1p(x) - log(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) <= -1.5e-16) {
		tmp = Math.exp((Math.log(x) / n)) / (n * x);
	} else if ((1.0 / n) <= 2e-12) {
		tmp = (Math.log1p(x) - Math.log(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) <= -1.5e-16:
		tmp = math.exp((math.log(x) / n)) / (n * x)
	elif (1.0 / n) <= 2e-12:
		tmp = (math.log1p(x) - math.log(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) <= -1.5e-16)
		tmp = Float64(exp(Float64(log(x) / n)) / Float64(n * x));
	elseif (Float64(1.0 / n) <= 2e-12)
		tmp = Float64(Float64(log1p(x) - log(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], -1.5e-16], N[(N[Exp[N[(N[Log[x], $MachinePrecision] / n), $MachinePrecision]], $MachinePrecision] / N[(n * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-12], N[(N[(N[Log[1 + x], $MachinePrecision] - N[Log[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) < -1.49999999999999997e-16

   1. Initial program 97.3%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   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 97.9

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

   5. Applied rewrites 97.9%

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

   if -1.49999999999999997e-16 < (/.f64 #s(literal 1 binary64) n) < 1.99999999999999996e-12

   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 78.5

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

   5. Applied rewrites 78.5%

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

   if 1.99999999999999996e-12 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 53.2%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   2. 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 95.2

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

   5. Applied rewrites 95.2%

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

4. Final simplification 86.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -1.5 \cdot 10^{-16}:\\ \;\;\;\;\frac{e^{\frac{\log x}{n}}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-12}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right) - \log x}{n}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 83.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -1.5 \cdot 10^{-16}:\\ \;\;\;\;\frac{e^{\frac{\log x}{n}}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-12}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right) - \log x}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{+91}:\\ \;\;\;\;\left(\frac{x}{n} + 1\right) - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= (/ 1.0 n) -1.5e-16)
   (/ (exp (/ (log x) n)) (* n x))
   (if (<= (/ 1.0 n) 2e-12)
     (/ (- (log1p x) (log x)) n)
     (if (<= (/ 1.0 n) 5e+91)
       (- (+ (/ x n) 1.0) (pow x (/ 1.0 n)))
       (- (exp (/ (log1p x) n)) 1.0)))))

C

double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -1.5e-16) {
		tmp = exp((log(x) / n)) / (n * x);
	} else if ((1.0 / n) <= 2e-12) {
		tmp = (log1p(x) - log(x)) / n;
	} else if ((1.0 / n) <= 5e+91) {
		tmp = ((x / n) + 1.0) - pow(x, (1.0 / n));
	} else {
		tmp = exp((log1p(x) / n)) - 1.0;
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -1.5e-16) {
		tmp = Math.exp((Math.log(x) / n)) / (n * x);
	} else if ((1.0 / n) <= 2e-12) {
		tmp = (Math.log1p(x) - Math.log(x)) / n;
	} else if ((1.0 / n) <= 5e+91) {
		tmp = ((x / n) + 1.0) - Math.pow(x, (1.0 / n));
	} else {
		tmp = Math.exp((Math.log1p(x) / n)) - 1.0;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if (1.0 / n) <= -1.5e-16:
		tmp = math.exp((math.log(x) / n)) / (n * x)
	elif (1.0 / n) <= 2e-12:
		tmp = (math.log1p(x) - math.log(x)) / n
	elif (1.0 / n) <= 5e+91:
		tmp = ((x / n) + 1.0) - math.pow(x, (1.0 / n))
	else:
		tmp = math.exp((math.log1p(x) / n)) - 1.0
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -1.5e-16)
		tmp = Float64(exp(Float64(log(x) / n)) / Float64(n * x));
	elseif (Float64(1.0 / n) <= 2e-12)
		tmp = Float64(Float64(log1p(x) - log(x)) / n);
	elseif (Float64(1.0 / n) <= 5e+91)
		tmp = Float64(Float64(Float64(x / n) + 1.0) - (x ^ Float64(1.0 / n)));
	else
		tmp = Float64(exp(Float64(log1p(x) / n)) - 1.0);
	end
	return tmp
end

Wolfram

code[x_, n_] := If[LessEqual[N[(1.0 / n), $MachinePrecision], -1.5e-16], N[(N[Exp[N[(N[Log[x], $MachinePrecision] / n), $MachinePrecision]], $MachinePrecision] / N[(n * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-12], N[(N[(N[Log[1 + x], $MachinePrecision] - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e+91], N[(N[(N[(x / n), $MachinePrecision] + 1.0), $MachinePrecision] - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Exp[N[(N[Log[1 + x], $MachinePrecision] / n), $MachinePrecision]], $MachinePrecision] - 1.0), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -1.49999999999999997e-16

   1. Initial program 97.3%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   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 97.9

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

   5. Applied rewrites 97.9%

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

   if -1.49999999999999997e-16 < (/.f64 #s(literal 1 binary64) n) < 1.99999999999999996e-12

   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 78.5

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

   5. Applied rewrites 78.5%

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

   if 1.99999999999999996e-12 < (/.f64 #s(literal 1 binary64) n) < 5.0000000000000002e91

   1. Initial program 77.6%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(1 + \frac{x}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   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 75.9

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

   5. Applied rewrites 75.9%

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

   if 5.0000000000000002e91 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 40.4%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 34.7%

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

   6. Taylor expanded in n around inf

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

   8. Applied rewrites 2.1%

      \[\leadsto 1 - \color{blue}{1} \]

   9. Taylor expanded in n around 0

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

       1. lower-exp.f64 N/A

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

       2. lower-/.f64 N/A

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

       3. lift-log1p.f64 66.8

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

   11. Applied rewrites 66.8%

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

4. Final simplification 83.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -1.5 \cdot 10^{-16}:\\ \;\;\;\;\frac{e^{\frac{\log x}{n}}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-12}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right) - \log x}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{+91}:\\ \;\;\;\;\left(\frac{x}{n} + 1\right) - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - 1\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 82.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -4 \cdot 10^{-10}:\\ \;\;\;\;{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - t\_0\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-12}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right) - \log x}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{+91}:\\ \;\;\;\;\left(\frac{x}{n} + 1\right) - t\_0\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -4e-10)
     (- (pow (+ x 1.0) (/ 1.0 n)) t_0)
     (if (<= (/ 1.0 n) 2e-12)
       (/ (- (log1p x) (log x)) n)
       (if (<= (/ 1.0 n) 5e+91)
         (- (+ (/ x n) 1.0) t_0)
         (- (exp (/ (log1p x) n)) 1.0))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -4e-10) {
		tmp = pow((x + 1.0), (1.0 / n)) - t_0;
	} else if ((1.0 / n) <= 2e-12) {
		tmp = (log1p(x) - log(x)) / n;
	} else if ((1.0 / n) <= 5e+91) {
		tmp = ((x / n) + 1.0) - t_0;
	} else {
		tmp = exp((log1p(x) / n)) - 1.0;
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -4e-10) {
		tmp = Math.pow((x + 1.0), (1.0 / n)) - t_0;
	} else if ((1.0 / n) <= 2e-12) {
		tmp = (Math.log1p(x) - Math.log(x)) / n;
	} else if ((1.0 / n) <= 5e+91) {
		tmp = ((x / n) + 1.0) - t_0;
	} else {
		tmp = Math.exp((Math.log1p(x) / n)) - 1.0;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	tmp = 0
	if (1.0 / n) <= -4e-10:
		tmp = math.pow((x + 1.0), (1.0 / n)) - t_0
	elif (1.0 / n) <= 2e-12:
		tmp = (math.log1p(x) - math.log(x)) / n
	elif (1.0 / n) <= 5e+91:
		tmp = ((x / n) + 1.0) - t_0
	else:
		tmp = math.exp((math.log1p(x) / n)) - 1.0
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -4e-10)
		tmp = Float64((Float64(x + 1.0) ^ Float64(1.0 / n)) - t_0);
	elseif (Float64(1.0 / n) <= 2e-12)
		tmp = Float64(Float64(log1p(x) - log(x)) / n);
	elseif (Float64(1.0 / n) <= 5e+91)
		tmp = Float64(Float64(Float64(x / n) + 1.0) - t_0);
	else
		tmp = Float64(exp(Float64(log1p(x) / n)) - 1.0);
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -4e-10], N[(N[Power[N[(x + 1.0), $MachinePrecision], N[(1.0 / n), $MachinePrecision]], $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-12], N[(N[(N[Log[1 + x], $MachinePrecision] - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e+91], N[(N[(N[(x / n), $MachinePrecision] + 1.0), $MachinePrecision] - t$95$0), $MachinePrecision], N[(N[Exp[N[(N[Log[1 + x], $MachinePrecision] / n), $MachinePrecision]], $MachinePrecision] - 1.0), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -4.00000000000000015e-10

   1. Initial program 98.4%

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

   if -4.00000000000000015e-10 < (/.f64 #s(literal 1 binary64) n) < 1.99999999999999996e-12

   1. Initial program 29.8%

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

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

   5. Applied rewrites 78.1%

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

   if 1.99999999999999996e-12 < (/.f64 #s(literal 1 binary64) n) < 5.0000000000000002e91

   1. Initial program 77.6%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(1 + \frac{x}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   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 75.9

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

   5. Applied rewrites 75.9%

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

   if 5.0000000000000002e91 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 40.4%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 34.7%

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

   6. Taylor expanded in n around inf

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

   8. Applied rewrites 2.1%

      \[\leadsto 1 - \color{blue}{1} \]

   9. Taylor expanded in n around 0

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

       1. lower-exp.f64 N/A

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

       2. lower-/.f64 N/A

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

       3. lift-log1p.f64 66.8

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

   11. Applied rewrites 66.8%

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

Alternative 10: 59.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;x \leq 1.56 \cdot 10^{-280}:\\ \;\;\;\;1 - t\_0\\ \mathbf{elif}\;x \leq 8 \cdot 10^{-235}:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{elif}\;x \leq 1.95 \cdot 10^{-132}:\\ \;\;\;\;\left(\frac{x}{n} + 1\right) - t\_0\\ \mathbf{elif}\;x \leq 0.95:\\ \;\;\;\;\frac{x}{n} - \frac{\log x}{n}\\ \mathbf{elif}\;x \leq 1.12 \cdot 10^{+119}:\\ \;\;\;\;\frac{1 - 0.5 \cdot {x}^{-1}}{n \cdot x}\\ \mathbf{else}:\\ \;\;\;\;1 - 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= x 1.56e-280)
     (- 1.0 t_0)
     (if (<= x 8e-235)
       (/ (- x (log x)) n)
       (if (<= x 1.95e-132)
         (- (+ (/ x n) 1.0) t_0)
         (if (<= x 0.95)
           (- (/ x n) (/ (log x) n))
           (if (<= x 1.12e+119)
             (/ (- 1.0 (* 0.5 (pow x -1.0))) (* n x))
             (- 1.0 1.0))))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double tmp;
	if (x <= 1.56e-280) {
		tmp = 1.0 - t_0;
	} else if (x <= 8e-235) {
		tmp = (x - log(x)) / n;
	} else if (x <= 1.95e-132) {
		tmp = ((x / n) + 1.0) - t_0;
	} else if (x <= 0.95) {
		tmp = (x / n) - (log(x) / n);
	} else if (x <= 1.12e+119) {
		tmp = (1.0 - (0.5 * pow(x, -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) :: t_0
    real(8) :: tmp
    t_0 = x ** (1.0d0 / n)
    if (x <= 1.56d-280) then
        tmp = 1.0d0 - t_0
    else if (x <= 8d-235) then
        tmp = (x - log(x)) / n
    else if (x <= 1.95d-132) then
        tmp = ((x / n) + 1.0d0) - t_0
    else if (x <= 0.95d0) then
        tmp = (x / n) - (log(x) / n)
    else if (x <= 1.12d+119) then
        tmp = (1.0d0 - (0.5d0 * (x ** (-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 t_0 = Math.pow(x, (1.0 / n));
	double tmp;
	if (x <= 1.56e-280) {
		tmp = 1.0 - t_0;
	} else if (x <= 8e-235) {
		tmp = (x - Math.log(x)) / n;
	} else if (x <= 1.95e-132) {
		tmp = ((x / n) + 1.0) - t_0;
	} else if (x <= 0.95) {
		tmp = (x / n) - (Math.log(x) / n);
	} else if (x <= 1.12e+119) {
		tmp = (1.0 - (0.5 * Math.pow(x, -1.0))) / (n * x);
	} else {
		tmp = 1.0 - 1.0;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	tmp = 0
	if x <= 1.56e-280:
		tmp = 1.0 - t_0
	elif x <= 8e-235:
		tmp = (x - math.log(x)) / n
	elif x <= 1.95e-132:
		tmp = ((x / n) + 1.0) - t_0
	elif x <= 0.95:
		tmp = (x / n) - (math.log(x) / n)
	elif x <= 1.12e+119:
		tmp = (1.0 - (0.5 * math.pow(x, -1.0))) / (n * x)
	else:
		tmp = 1.0 - 1.0
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (x <= 1.56e-280)
		tmp = Float64(1.0 - t_0);
	elseif (x <= 8e-235)
		tmp = Float64(Float64(x - log(x)) / n);
	elseif (x <= 1.95e-132)
		tmp = Float64(Float64(Float64(x / n) + 1.0) - t_0);
	elseif (x <= 0.95)
		tmp = Float64(Float64(x / n) - Float64(log(x) / n));
	elseif (x <= 1.12e+119)
		tmp = Float64(Float64(1.0 - Float64(0.5 * (x ^ -1.0))) / Float64(n * x));
	else
		tmp = Float64(1.0 - 1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = x ^ (1.0 / n);
	tmp = 0.0;
	if (x <= 1.56e-280)
		tmp = 1.0 - t_0;
	elseif (x <= 8e-235)
		tmp = (x - log(x)) / n;
	elseif (x <= 1.95e-132)
		tmp = ((x / n) + 1.0) - t_0;
	elseif (x <= 0.95)
		tmp = (x / n) - (log(x) / n);
	elseif (x <= 1.12e+119)
		tmp = (1.0 - (0.5 * (x ^ -1.0))) / (n * x);
	else
		tmp = 1.0 - 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, 1.56e-280], N[(1.0 - t$95$0), $MachinePrecision], If[LessEqual[x, 8e-235], N[(N[(x - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[x, 1.95e-132], N[(N[(N[(x / n), $MachinePrecision] + 1.0), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[x, 0.95], N[(N[(x / n), $MachinePrecision] - N[(N[Log[x], $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.12e+119], N[(N[(1.0 - N[(0.5 * N[Power[x, -1.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(n * x), $MachinePrecision]), $MachinePrecision], N[(1.0 - 1.0), $MachinePrecision]]]]]]]

Derivation

1. Split input into 6 regimes

2. if x < 1.5600000000000001e-280

   1. Initial program 52.3%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   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 52.3%

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

   if 1.5600000000000001e-280 < x < 7.9999999999999997e-235

   1. Initial program 48.9%

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

   3. Taylor expanded in n around -inf

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

   5. Applied rewrites 72.7%

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

   6. Taylor expanded in x around 0

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

      1. lower--.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lower-fma.f64 N/A

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

   8. Applied rewrites 72.0%

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

   9. Taylor expanded in n around inf

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

       1. lower-/.f64 N/A

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

       2. lower--.f64 N/A

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

       3. lift-log.f64 52.0

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

   11. Applied rewrites 52.0%

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

   if 7.9999999999999997e-235 < x < 1.94999999999999991e-132

   1. Initial program 42.2%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   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 42.6

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

   5. Applied rewrites 42.6%

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

   if 1.94999999999999991e-132 < x < 0.94999999999999996

   1. Initial program 37.4%

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

   3. Taylor expanded in n around -inf

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

   5. Applied rewrites 84.4%

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

   6. Taylor expanded in x around 0

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

      1. lower--.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lower-fma.f64 N/A

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

   8. Applied rewrites 83.0%

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

   9. Taylor expanded in n around inf

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

       1. lift-log.f64 N/A

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

       2. lift-/.f64 52.1

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

   11. Applied rewrites 52.1%

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

   if 0.94999999999999996 < x < 1.11999999999999994e119

   1. Initial program 47.1%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 81.3%

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

   6. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. inv-pow N/A

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

      5. lower-pow.f64 N/A

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

      6. lift-*.f64 62.0

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

   8. Applied rewrites 62.0%

      \[\leadsto \frac{1 - 0.5 \cdot {x}^{-1}}{\color{blue}{n \cdot x}} \]

   if 1.11999999999999994e119 < x

   1. Initial program 81.2%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 47.7%

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

   6. Taylor expanded in n around inf

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

   8. Applied rewrites 81.2%

      \[\leadsto 1 - \color{blue}{1} \]
3. Recombined 6 regimes into one program.
4. Add Preprocessing

Alternative 11: 59.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;x \leq 1.56 \cdot 10^{-280}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 8 \cdot 10^{-235}:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{elif}\;x \leq 1.95 \cdot 10^{-132}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 0.95:\\ \;\;\;\;\frac{x}{n} - \frac{\log x}{n}\\ \mathbf{elif}\;x \leq 1.12 \cdot 10^{+119}:\\ \;\;\;\;\frac{1 - 0.5 \cdot {x}^{-1}}{n \cdot x}\\ \mathbf{else}:\\ \;\;\;\;1 - 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (- 1.0 (pow x (/ 1.0 n)))))
   (if (<= x 1.56e-280)
     t_0
     (if (<= x 8e-235)
       (/ (- x (log x)) n)
       (if (<= x 1.95e-132)
         t_0
         (if (<= x 0.95)
           (- (/ x n) (/ (log x) n))
           (if (<= x 1.12e+119)
             (/ (- 1.0 (* 0.5 (pow x -1.0))) (* n x))
             (- 1.0 1.0))))))))

C

double code(double x, double n) {
	double t_0 = 1.0 - pow(x, (1.0 / n));
	double tmp;
	if (x <= 1.56e-280) {
		tmp = t_0;
	} else if (x <= 8e-235) {
		tmp = (x - log(x)) / n;
	} else if (x <= 1.95e-132) {
		tmp = t_0;
	} else if (x <= 0.95) {
		tmp = (x / n) - (log(x) / n);
	} else if (x <= 1.12e+119) {
		tmp = (1.0 - (0.5 * pow(x, -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) :: t_0
    real(8) :: tmp
    t_0 = 1.0d0 - (x ** (1.0d0 / n))
    if (x <= 1.56d-280) then
        tmp = t_0
    else if (x <= 8d-235) then
        tmp = (x - log(x)) / n
    else if (x <= 1.95d-132) then
        tmp = t_0
    else if (x <= 0.95d0) then
        tmp = (x / n) - (log(x) / n)
    else if (x <= 1.12d+119) then
        tmp = (1.0d0 - (0.5d0 * (x ** (-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 t_0 = 1.0 - Math.pow(x, (1.0 / n));
	double tmp;
	if (x <= 1.56e-280) {
		tmp = t_0;
	} else if (x <= 8e-235) {
		tmp = (x - Math.log(x)) / n;
	} else if (x <= 1.95e-132) {
		tmp = t_0;
	} else if (x <= 0.95) {
		tmp = (x / n) - (Math.log(x) / n);
	} else if (x <= 1.12e+119) {
		tmp = (1.0 - (0.5 * Math.pow(x, -1.0))) / (n * x);
	} else {
		tmp = 1.0 - 1.0;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = 1.0 - math.pow(x, (1.0 / n))
	tmp = 0
	if x <= 1.56e-280:
		tmp = t_0
	elif x <= 8e-235:
		tmp = (x - math.log(x)) / n
	elif x <= 1.95e-132:
		tmp = t_0
	elif x <= 0.95:
		tmp = (x / n) - (math.log(x) / n)
	elif x <= 1.12e+119:
		tmp = (1.0 - (0.5 * math.pow(x, -1.0))) / (n * x)
	else:
		tmp = 1.0 - 1.0
	return tmp

Julia

function code(x, n)
	t_0 = Float64(1.0 - (x ^ Float64(1.0 / n)))
	tmp = 0.0
	if (x <= 1.56e-280)
		tmp = t_0;
	elseif (x <= 8e-235)
		tmp = Float64(Float64(x - log(x)) / n);
	elseif (x <= 1.95e-132)
		tmp = t_0;
	elseif (x <= 0.95)
		tmp = Float64(Float64(x / n) - Float64(log(x) / n));
	elseif (x <= 1.12e+119)
		tmp = Float64(Float64(1.0 - Float64(0.5 * (x ^ -1.0))) / Float64(n * x));
	else
		tmp = Float64(1.0 - 1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = 1.0 - (x ^ (1.0 / n));
	tmp = 0.0;
	if (x <= 1.56e-280)
		tmp = t_0;
	elseif (x <= 8e-235)
		tmp = (x - log(x)) / n;
	elseif (x <= 1.95e-132)
		tmp = t_0;
	elseif (x <= 0.95)
		tmp = (x / n) - (log(x) / n);
	elseif (x <= 1.12e+119)
		tmp = (1.0 - (0.5 * (x ^ -1.0))) / (n * x);
	else
		tmp = 1.0 - 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(1.0 - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 1.56e-280], t$95$0, If[LessEqual[x, 8e-235], N[(N[(x - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[x, 1.95e-132], t$95$0, If[LessEqual[x, 0.95], N[(N[(x / n), $MachinePrecision] - N[(N[Log[x], $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.12e+119], N[(N[(1.0 - N[(0.5 * N[Power[x, -1.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(n * x), $MachinePrecision]), $MachinePrecision], N[(1.0 - 1.0), $MachinePrecision]]]]]]]

Derivation

1. Split input into 5 regimes

2. if x < 1.5600000000000001e-280 or 7.9999999999999997e-235 < x < 1.94999999999999991e-132

   1. Initial program 44.4%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 44.4%

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

   if 1.5600000000000001e-280 < x < 7.9999999999999997e-235

   1. Initial program 48.9%

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

   3. Taylor expanded in n around -inf

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

   5. Applied rewrites 72.7%

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

   6. Taylor expanded in x around 0

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

      1. lower--.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lower-fma.f64 N/A

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

   8. Applied rewrites 72.0%

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

   9. Taylor expanded in n around inf

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

       1. lower-/.f64 N/A

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

       2. lower--.f64 N/A

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

       3. lift-log.f64 52.0

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

   11. Applied rewrites 52.0%

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

   if 1.94999999999999991e-132 < x < 0.94999999999999996

   1. Initial program 37.4%

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

   3. Taylor expanded in n around -inf

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

   5. Applied rewrites 84.4%

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

   6. Taylor expanded in x around 0

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

      1. lower--.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lower-fma.f64 N/A

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

   8. Applied rewrites 83.0%

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

   9. Taylor expanded in n around inf

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

       1. lift-log.f64 N/A

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

       2. lift-/.f64 52.1

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

   11. Applied rewrites 52.1%

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

   if 0.94999999999999996 < x < 1.11999999999999994e119

   1. Initial program 47.1%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 81.3%

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

   6. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. inv-pow N/A

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

      5. lower-pow.f64 N/A

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

      6. lift-*.f64 62.0

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

   8. Applied rewrites 62.0%

      \[\leadsto \frac{1 - 0.5 \cdot {x}^{-1}}{\color{blue}{n \cdot x}} \]

   if 1.11999999999999994e119 < x

   1. Initial program 81.2%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 47.7%

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

   6. Taylor expanded in n around inf

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

   8. Applied rewrites 81.2%

      \[\leadsto 1 - \color{blue}{1} \]
3. Recombined 5 regimes into one program.
4. Add Preprocessing

Alternative 12: 59.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;x \leq 1.56 \cdot 10^{-280}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 8 \cdot 10^{-235}:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{elif}\;x \leq 1.95 \cdot 10^{-132}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 0.95:\\ \;\;\;\;\frac{x}{n} - \frac{\log x}{n}\\ \mathbf{elif}\;x \leq 1.12 \cdot 10^{+119}:\\ \;\;\;\;\frac{\frac{\frac{x - 0.5}{x}}{n}}{x}\\ \mathbf{else}:\\ \;\;\;\;1 - 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (- 1.0 (pow x (/ 1.0 n)))))
   (if (<= x 1.56e-280)
     t_0
     (if (<= x 8e-235)
       (/ (- x (log x)) n)
       (if (<= x 1.95e-132)
         t_0
         (if (<= x 0.95)
           (- (/ x n) (/ (log x) n))
           (if (<= x 1.12e+119) (/ (/ (/ (- x 0.5) x) n) x) (- 1.0 1.0))))))))

C

double code(double x, double n) {
	double t_0 = 1.0 - pow(x, (1.0 / n));
	double tmp;
	if (x <= 1.56e-280) {
		tmp = t_0;
	} else if (x <= 8e-235) {
		tmp = (x - log(x)) / n;
	} else if (x <= 1.95e-132) {
		tmp = t_0;
	} else if (x <= 0.95) {
		tmp = (x / n) - (log(x) / n);
	} else if (x <= 1.12e+119) {
		tmp = (((x - 0.5) / x) / 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) :: t_0
    real(8) :: tmp
    t_0 = 1.0d0 - (x ** (1.0d0 / n))
    if (x <= 1.56d-280) then
        tmp = t_0
    else if (x <= 8d-235) then
        tmp = (x - log(x)) / n
    else if (x <= 1.95d-132) then
        tmp = t_0
    else if (x <= 0.95d0) then
        tmp = (x / n) - (log(x) / n)
    else if (x <= 1.12d+119) then
        tmp = (((x - 0.5d0) / x) / n) / x
    else
        tmp = 1.0d0 - 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = 1.0 - Math.pow(x, (1.0 / n));
	double tmp;
	if (x <= 1.56e-280) {
		tmp = t_0;
	} else if (x <= 8e-235) {
		tmp = (x - Math.log(x)) / n;
	} else if (x <= 1.95e-132) {
		tmp = t_0;
	} else if (x <= 0.95) {
		tmp = (x / n) - (Math.log(x) / n);
	} else if (x <= 1.12e+119) {
		tmp = (((x - 0.5) / x) / n) / x;
	} else {
		tmp = 1.0 - 1.0;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = 1.0 - math.pow(x, (1.0 / n))
	tmp = 0
	if x <= 1.56e-280:
		tmp = t_0
	elif x <= 8e-235:
		tmp = (x - math.log(x)) / n
	elif x <= 1.95e-132:
		tmp = t_0
	elif x <= 0.95:
		tmp = (x / n) - (math.log(x) / n)
	elif x <= 1.12e+119:
		tmp = (((x - 0.5) / x) / n) / x
	else:
		tmp = 1.0 - 1.0
	return tmp

Julia

function code(x, n)
	t_0 = Float64(1.0 - (x ^ Float64(1.0 / n)))
	tmp = 0.0
	if (x <= 1.56e-280)
		tmp = t_0;
	elseif (x <= 8e-235)
		tmp = Float64(Float64(x - log(x)) / n);
	elseif (x <= 1.95e-132)
		tmp = t_0;
	elseif (x <= 0.95)
		tmp = Float64(Float64(x / n) - Float64(log(x) / n));
	elseif (x <= 1.12e+119)
		tmp = Float64(Float64(Float64(Float64(x - 0.5) / x) / n) / x);
	else
		tmp = Float64(1.0 - 1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = 1.0 - (x ^ (1.0 / n));
	tmp = 0.0;
	if (x <= 1.56e-280)
		tmp = t_0;
	elseif (x <= 8e-235)
		tmp = (x - log(x)) / n;
	elseif (x <= 1.95e-132)
		tmp = t_0;
	elseif (x <= 0.95)
		tmp = (x / n) - (log(x) / n);
	elseif (x <= 1.12e+119)
		tmp = (((x - 0.5) / x) / n) / x;
	else
		tmp = 1.0 - 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(1.0 - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 1.56e-280], t$95$0, If[LessEqual[x, 8e-235], N[(N[(x - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[x, 1.95e-132], t$95$0, If[LessEqual[x, 0.95], N[(N[(x / n), $MachinePrecision] - N[(N[Log[x], $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.12e+119], N[(N[(N[(N[(x - 0.5), $MachinePrecision] / x), $MachinePrecision] / n), $MachinePrecision] / x), $MachinePrecision], N[(1.0 - 1.0), $MachinePrecision]]]]]]]

Derivation

1. Split input into 5 regimes

2. if x < 1.5600000000000001e-280 or 7.9999999999999997e-235 < x < 1.94999999999999991e-132

   1. Initial program 44.4%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 44.4%

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

   if 1.5600000000000001e-280 < x < 7.9999999999999997e-235

   1. Initial program 48.9%

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

   3. Taylor expanded in n around -inf

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

   5. Applied rewrites 72.7%

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

   6. Taylor expanded in x around 0

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

      1. lower--.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lower-fma.f64 N/A

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

   8. Applied rewrites 72.0%

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

   9. Taylor expanded in n around inf

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

       1. lower-/.f64 N/A

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

       2. lower--.f64 N/A

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

       3. lift-log.f64 52.0

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

   11. Applied rewrites 52.0%

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

   if 1.94999999999999991e-132 < x < 0.94999999999999996

   1. Initial program 37.4%

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

   3. Taylor expanded in n around -inf

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

   5. Applied rewrites 84.4%

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

   6. Taylor expanded in x around 0

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

      1. lower--.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lower-fma.f64 N/A

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

   8. Applied rewrites 83.0%

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

   9. Taylor expanded in n around inf

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

       1. lift-log.f64 N/A

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

       2. lift-/.f64 52.1

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

   11. Applied rewrites 52.1%

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

   if 0.94999999999999996 < x < 1.11999999999999994e119

   1. Initial program 47.1%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 81.3%

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

   6. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. inv-pow N/A

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

      5. lower-pow.f64 63.4

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

   8. Applied rewrites 63.4%

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

   9. Taylor expanded in x around 0

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

       1. lower-/.f64 N/A

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

       2. lower--.f64 63.4

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

   11. Applied rewrites 63.4%

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

   if 1.11999999999999994e119 < x

   1. Initial program 81.2%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 47.7%

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

   6. Taylor expanded in n around inf

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

   8. Applied rewrites 81.2%

      \[\leadsto 1 - \color{blue}{1} \]
3. Recombined 5 regimes into one program.
4. Add Preprocessing

Alternative 13: 59.8% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - {x}^{\left(\frac{1}{n}\right)}\\ t_1 := \frac{x - \log x}{n}\\ \mathbf{if}\;x \leq 1.56 \cdot 10^{-280}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 8 \cdot 10^{-235}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 1.95 \cdot 10^{-132}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 0.95:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 1.12 \cdot 10^{+119}:\\ \;\;\;\;\frac{\frac{\frac{x - 0.5}{x}}{n}}{x}\\ \mathbf{else}:\\ \;\;\;\;1 - 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (- 1.0 (pow x (/ 1.0 n)))) (t_1 (/ (- x (log x)) n)))
   (if (<= x 1.56e-280)
     t_0
     (if (<= x 8e-235)
       t_1
       (if (<= x 1.95e-132)
         t_0
         (if (<= x 0.95)
           t_1
           (if (<= x 1.12e+119) (/ (/ (/ (- x 0.5) x) n) x) (- 1.0 1.0))))))))

C

double code(double x, double n) {
	double t_0 = 1.0 - pow(x, (1.0 / n));
	double t_1 = (x - log(x)) / n;
	double tmp;
	if (x <= 1.56e-280) {
		tmp = t_0;
	} else if (x <= 8e-235) {
		tmp = t_1;
	} else if (x <= 1.95e-132) {
		tmp = t_0;
	} else if (x <= 0.95) {
		tmp = t_1;
	} else if (x <= 1.12e+119) {
		tmp = (((x - 0.5) / x) / 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) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = 1.0d0 - (x ** (1.0d0 / n))
    t_1 = (x - log(x)) / n
    if (x <= 1.56d-280) then
        tmp = t_0
    else if (x <= 8d-235) then
        tmp = t_1
    else if (x <= 1.95d-132) then
        tmp = t_0
    else if (x <= 0.95d0) then
        tmp = t_1
    else if (x <= 1.12d+119) then
        tmp = (((x - 0.5d0) / x) / n) / x
    else
        tmp = 1.0d0 - 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = 1.0 - Math.pow(x, (1.0 / n));
	double t_1 = (x - Math.log(x)) / n;
	double tmp;
	if (x <= 1.56e-280) {
		tmp = t_0;
	} else if (x <= 8e-235) {
		tmp = t_1;
	} else if (x <= 1.95e-132) {
		tmp = t_0;
	} else if (x <= 0.95) {
		tmp = t_1;
	} else if (x <= 1.12e+119) {
		tmp = (((x - 0.5) / x) / n) / x;
	} else {
		tmp = 1.0 - 1.0;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = 1.0 - math.pow(x, (1.0 / n))
	t_1 = (x - math.log(x)) / n
	tmp = 0
	if x <= 1.56e-280:
		tmp = t_0
	elif x <= 8e-235:
		tmp = t_1
	elif x <= 1.95e-132:
		tmp = t_0
	elif x <= 0.95:
		tmp = t_1
	elif x <= 1.12e+119:
		tmp = (((x - 0.5) / x) / n) / x
	else:
		tmp = 1.0 - 1.0
	return tmp

Julia

function code(x, n)
	t_0 = Float64(1.0 - (x ^ Float64(1.0 / n)))
	t_1 = Float64(Float64(x - log(x)) / n)
	tmp = 0.0
	if (x <= 1.56e-280)
		tmp = t_0;
	elseif (x <= 8e-235)
		tmp = t_1;
	elseif (x <= 1.95e-132)
		tmp = t_0;
	elseif (x <= 0.95)
		tmp = t_1;
	elseif (x <= 1.12e+119)
		tmp = Float64(Float64(Float64(Float64(x - 0.5) / x) / n) / x);
	else
		tmp = Float64(1.0 - 1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = 1.0 - (x ^ (1.0 / n));
	t_1 = (x - log(x)) / n;
	tmp = 0.0;
	if (x <= 1.56e-280)
		tmp = t_0;
	elseif (x <= 8e-235)
		tmp = t_1;
	elseif (x <= 1.95e-132)
		tmp = t_0;
	elseif (x <= 0.95)
		tmp = t_1;
	elseif (x <= 1.12e+119)
		tmp = (((x - 0.5) / x) / n) / x;
	else
		tmp = 1.0 - 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(1.0 - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision]}, If[LessEqual[x, 1.56e-280], t$95$0, If[LessEqual[x, 8e-235], t$95$1, If[LessEqual[x, 1.95e-132], t$95$0, If[LessEqual[x, 0.95], t$95$1, If[LessEqual[x, 1.12e+119], N[(N[(N[(N[(x - 0.5), $MachinePrecision] / x), $MachinePrecision] / n), $MachinePrecision] / x), $MachinePrecision], N[(1.0 - 1.0), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if x < 1.5600000000000001e-280 or 7.9999999999999997e-235 < x < 1.94999999999999991e-132

   1. Initial program 44.4%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 44.4%

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

   if 1.5600000000000001e-280 < x < 7.9999999999999997e-235 or 1.94999999999999991e-132 < x < 0.94999999999999996

   1. Initial program 40.5%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   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 81.3%

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

   6. Taylor expanded in x around 0

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

      1. lower--.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lower-fma.f64 N/A

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

   8. Applied rewrites 80.1%

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

   9. Taylor expanded in n around inf

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

       1. lower-/.f64 N/A

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

       2. lower--.f64 N/A

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

       3. lift-log.f64 52.0

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

   11. Applied rewrites 52.0%

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

   if 0.94999999999999996 < x < 1.11999999999999994e119

   1. Initial program 47.1%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 81.3%

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

   6. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. inv-pow N/A

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

      5. lower-pow.f64 63.4

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

   8. Applied rewrites 63.4%

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

   9. Taylor expanded in x around 0

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

       1. lower-/.f64 N/A

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

       2. lower--.f64 63.4

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

   11. Applied rewrites 63.4%

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

   if 1.11999999999999994e119 < x

   1. Initial program 81.2%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 47.7%

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

   6. Taylor expanded in n around inf

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

   8. Applied rewrites 81.2%

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

Alternative 14: 62.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.95:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{elif}\;x \leq 1.12 \cdot 10^{+119}:\\ \;\;\;\;\frac{\frac{\frac{x - 0.5}{x}}{n}}{x}\\ \mathbf{else}:\\ \;\;\;\;1 - 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 0.95)
   (/ (- x (log x)) n)
   (if (<= x 1.12e+119) (/ (/ (/ (- x 0.5) x) n) x) (- 1.0 1.0))))

C

double code(double x, double n) {
	double tmp;
	if (x <= 0.95) {
		tmp = (x - log(x)) / n;
	} else if (x <= 1.12e+119) {
		tmp = (((x - 0.5) / x) / 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 (x <= 0.95d0) then
        tmp = (x - log(x)) / n
    else if (x <= 1.12d+119) then
        tmp = (((x - 0.5d0) / x) / 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 (x <= 0.95) {
		tmp = (x - Math.log(x)) / n;
	} else if (x <= 1.12e+119) {
		tmp = (((x - 0.5) / x) / n) / x;
	} else {
		tmp = 1.0 - 1.0;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 0.95:
		tmp = (x - math.log(x)) / n
	elif x <= 1.12e+119:
		tmp = (((x - 0.5) / x) / n) / x
	else:
		tmp = 1.0 - 1.0
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 0.95)
		tmp = Float64(Float64(x - log(x)) / n);
	elseif (x <= 1.12e+119)
		tmp = Float64(Float64(Float64(Float64(x - 0.5) / x) / n) / x);
	else
		tmp = Float64(1.0 - 1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (x <= 0.95)
		tmp = (x - log(x)) / n;
	elseif (x <= 1.12e+119)
		tmp = (((x - 0.5) / x) / n) / x;
	else
		tmp = 1.0 - 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 0.95], N[(N[(x - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[x, 1.12e+119], N[(N[(N[(N[(x - 0.5), $MachinePrecision] / x), $MachinePrecision] / n), $MachinePrecision] / x), $MachinePrecision], N[(1.0 - 1.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 0.94999999999999996

   1. Initial program 42.2%

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

   3. Taylor expanded in n around -inf

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

   5. Applied rewrites 79.1%

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

   6. Taylor expanded in x around 0

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

      1. lower--.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lower-fma.f64 N/A

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

   8. Applied rewrites 78.3%

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

   9. Taylor expanded in n around inf

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

       1. lower-/.f64 N/A

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

       2. lower--.f64 N/A

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

       3. lift-log.f64 52.7

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

   11. Applied rewrites 52.7%

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

   if 0.94999999999999996 < x < 1.11999999999999994e119

   1. Initial program 47.1%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 81.3%

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

   6. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. inv-pow N/A

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

      5. lower-pow.f64 63.4

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

   8. Applied rewrites 63.4%

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

   9. Taylor expanded in x around 0

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

       1. lower-/.f64 N/A

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

       2. lower--.f64 63.4

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

   11. Applied rewrites 63.4%

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

   if 1.11999999999999994e119 < x

   1. Initial program 81.2%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 47.7%

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

   6. Taylor expanded in n around inf

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

   8. Applied rewrites 81.2%

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

Alternative 15: 44.7% accurate, 10.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.12 \cdot 10^{+119}:\\ \;\;\;\;\frac{1}{n \cdot x}\\ \mathbf{else}:\\ \;\;\;\;1 - 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 1.12e+119) (/ 1.0 (* n x)) (- 1.0 1.0)))

C

double code(double x, double n) {
	double tmp;
	if (x <= 1.12e+119) {
		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 (x <= 1.12d+119) 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 (x <= 1.12e+119) {
		tmp = 1.0 / (n * x);
	} else {
		tmp = 1.0 - 1.0;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 1.12e+119:
		tmp = 1.0 / (n * x)
	else:
		tmp = 1.0 - 1.0
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 1.12e+119)
		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 (x <= 1.12e+119)
		tmp = 1.0 / (n * x);
	else
		tmp = 1.0 - 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 1.12e+119], N[(1.0 / N[(n * x), $MachinePrecision]), $MachinePrecision], N[(1.0 - 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.11999999999999994e119

   1. Initial program 43.3%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   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 42.4

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

   5. Applied rewrites 42.4%

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

   6. Taylor expanded in n around inf

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

   8. Applied rewrites 31.3%

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

   if 1.11999999999999994e119 < x

   1. Initial program 81.2%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 47.7%

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

   6. Taylor expanded in n around inf

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

   8. Applied rewrites 81.2%

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

Alternative 16: 31.5% 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 53.5%

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

3. Taylor expanded in x around 0

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

5. Applied rewrites 38.5%

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

6. Taylor expanded in n around inf

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

8. Applied rewrites 31.5%

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

Reproduce

herbie shell --seed 2025086 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  :precision binary64
  (- (pow (+ x 1.0) (/ 1.0 n)) (pow x (/ 1.0 n))))