2nthrt (problem 3.4.6)

Percentage Accurate: 53.1% → 87.2%

Time: 28.6s

Alternatives: 16

Speedup: 1.9×

• Could not converge on a ground truth

  1. x = -1.1639501146720082e-172
  2. n = 3.5959560856390625e-256

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: 53.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 87.2% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 6500000:\\ \;\;\;\;\frac{\mathsf{fma}\left(1, \mathsf{log1p}\left(x\right), \frac{\mathsf{fma}\left(0.16666666666666666, \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{3} - {\log x}^{3}}{n}, \left({\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}\right) \cdot 0.5\right)}{n} - \log x\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\frac{\log x}{n}}}{n \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 6500000.0)
   (/
    (fma
     1.0
     (log1p x)
     (-
      (/
       (fma
        0.16666666666666666
        (/ (- (pow (log1p x) 3.0) (pow (log x) 3.0)) n)
        (* (- (pow (log1p x) 2.0) (pow (log x) 2.0)) 0.5))
       n)
      (log x)))
    n)
   (/ (exp (/ (log x) n)) (* n x))))

C

double code(double x, double n) {
	double tmp;
	if (x <= 6500000.0) {
		tmp = fma(1.0, log1p(x), ((fma(0.16666666666666666, ((pow(log1p(x), 3.0) - pow(log(x), 3.0)) / n), ((pow(log1p(x), 2.0) - pow(log(x), 2.0)) * 0.5)) / n) - log(x))) / n;
	} else {
		tmp = exp((log(x) / n)) / (n * x);
	}
	return tmp;
}

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 6500000.0)
		tmp = Float64(fma(1.0, log1p(x), Float64(Float64(fma(0.16666666666666666, Float64(Float64((log1p(x) ^ 3.0) - (log(x) ^ 3.0)) / n), Float64(Float64((log1p(x) ^ 2.0) - (log(x) ^ 2.0)) * 0.5)) / n) - log(x))) / n);
	else
		tmp = Float64(exp(Float64(log(x) / n)) / Float64(n * x));
	end
	return tmp
end

Wolfram

code[x_, n_] := If[LessEqual[x, 6500000.0], N[(N[(1.0 * N[Log[1 + x], $MachinePrecision] + N[(N[(N[(0.16666666666666666 * N[(N[(N[Power[N[Log[1 + x], $MachinePrecision], 3.0], $MachinePrecision] - N[Power[N[Log[x], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision] + N[(N[(N[Power[N[Log[1 + x], $MachinePrecision], 2.0], $MachinePrecision] - N[Power[N[Log[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision] - N[Log[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], N[(N[Exp[N[(N[Log[x], $MachinePrecision] / n), $MachinePrecision]], $MachinePrecision] / N[(n * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 6.5e6

   1. Initial program 41.1%

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

   3. Taylor expanded in n around -inf

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

   5. Applied rewrites 84.4%

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

   if 6.5e6 < x

   1. Initial program 72.2%

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

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 97.8%

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

Alternative 2: 82.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ t_1 := {\left(x - -1\right)}^{\left(\frac{1}{n}\right)} - t\_0\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{-8}:\\ \;\;\;\;1 - t\_0\\ \mathbf{elif}\;t\_1 \leq 10^{-11}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right) - \log x}{n}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(-0.3333333333333333, x, \frac{\mathsf{fma}\left(\frac{x}{n}, 0.16666666666666666, -0.5 \cdot x\right) - -0.5}{-n}\right) - -0.5}{-n}, x, \frac{1}{n}\right), x, 1\right) - t\_0\\ \end{array} \end{array} \]

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 -2e-8)
     (- 1.0 t_0)
     (if (<= t_1 1e-11)
       (/ (- (log1p x) (log x)) n)
       (-
        (fma
         (fma
          (/
           (-
            (fma
             -0.3333333333333333
             x
             (/ (- (fma (/ x n) 0.16666666666666666 (* -0.5 x)) -0.5) (- n)))
            -0.5)
           (- n))
          x
          (/ 1.0 n))
         x
         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 <= -2e-8) {
		tmp = 1.0 - t_0;
	} else if (t_1 <= 1e-11) {
		tmp = (log1p(x) - log(x)) / n;
	} else {
		tmp = fma(fma(((fma(-0.3333333333333333, x, ((fma((x / n), 0.16666666666666666, (-0.5 * x)) - -0.5) / -n)) - -0.5) / -n), x, (1.0 / n)), x, 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 <= -2e-8)
		tmp = Float64(1.0 - t_0);
	elseif (t_1 <= 1e-11)
		tmp = Float64(Float64(log1p(x) - log(x)) / n);
	else
		tmp = Float64(fma(fma(Float64(Float64(fma(-0.3333333333333333, x, Float64(Float64(fma(Float64(x / n), 0.16666666666666666, Float64(-0.5 * x)) - -0.5) / Float64(-n))) - -0.5) / Float64(-n)), x, Float64(1.0 / n)), x, 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, -2e-8], N[(1.0 - t$95$0), $MachinePrecision], If[LessEqual[t$95$1, 1e-11], N[(N[(N[Log[1 + x], $MachinePrecision] - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], N[(N[(N[(N[(N[(N[(-0.3333333333333333 * x + N[(N[(N[(N[(x / n), $MachinePrecision] * 0.16666666666666666 + N[(-0.5 * x), $MachinePrecision]), $MachinePrecision] - -0.5), $MachinePrecision] / (-n)), $MachinePrecision]), $MachinePrecision] - -0.5), $MachinePrecision] / (-n)), $MachinePrecision] * x + N[(1.0 / n), $MachinePrecision]), $MachinePrecision] * x + 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))) < -2e-8

   1. Initial program 98.3%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   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 98.3%

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

   if -2e-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))) < 9.99999999999999939e-12

   1. Initial program 46.1%

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

   3. Taylor expanded in n around inf

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

   5. Applied rewrites 82.6%

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

   if 9.99999999999999939e-12 < (-.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 33.5%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 24.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{0.16666666666666666}{{n}^{3}} - \frac{-0.3333333333333333 + \frac{0.5}{n}}{n}, x, \frac{-0.5 + \frac{0.5}{n}}{n}\right), x, \frac{1}{n}\right), x, 1\right)} - {x}^{\left(\frac{1}{n}\right)} \]

   6. Taylor expanded in n around -inf

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

   8. Applied rewrites 75.6%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-\frac{\mathsf{fma}\left(-0.3333333333333333, x, -\frac{\mathsf{fma}\left(\frac{x}{n}, 0.16666666666666666, -0.5 \cdot x\right) - -0.5}{n}\right) - -0.5}{n}, x, \frac{1}{n}\right), x, 1\right) - {x}^{\left(\frac{1}{n}\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 84.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(x - -1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \leq -2 \cdot 10^{-8}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;{\left(x - -1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \leq 10^{-11}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right) - \log x}{n}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(-0.3333333333333333, x, \frac{\mathsf{fma}\left(\frac{x}{n}, 0.16666666666666666, -0.5 \cdot x\right) - -0.5}{-n}\right) - -0.5}{-n}, x, \frac{1}{n}\right), x, 1\right) - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 55.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ t_1 := {\left(x - -1\right)}^{\left(\frac{1}{n}\right)} - t\_0\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{-16}:\\ \;\;\;\;1 - t\_0\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;1 - 1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\frac{-0.5 + \frac{0.5}{n}}{n}, x, \frac{1}{n}\right), x, 1\right) - t\_0\\ \end{array} \end{array} \]

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 -2e-16)
     (- 1.0 t_0)
     (if (<= t_1 0.0)
       (- 1.0 1.0)
       (- (fma (fma (/ (+ -0.5 (/ 0.5 n)) n) x (/ 1.0 n)) x 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 <= -2e-16) {
		tmp = 1.0 - t_0;
	} else if (t_1 <= 0.0) {
		tmp = 1.0 - 1.0;
	} else {
		tmp = fma(fma(((-0.5 + (0.5 / n)) / n), x, (1.0 / n)), x, 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 <= -2e-16)
		tmp = Float64(1.0 - t_0);
	elseif (t_1 <= 0.0)
		tmp = Float64(1.0 - 1.0);
	else
		tmp = Float64(fma(fma(Float64(Float64(-0.5 + Float64(0.5 / n)) / n), x, Float64(1.0 / n)), x, 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, -2e-16], N[(1.0 - t$95$0), $MachinePrecision], If[LessEqual[t$95$1, 0.0], N[(1.0 - 1.0), $MachinePrecision], N[(N[(N[(N[(N[(-0.5 + N[(0.5 / n), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision] * x + N[(1.0 / n), $MachinePrecision]), $MachinePrecision] * x + 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))) < -2e-16

   1. Initial program 96.7%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   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 96.7%

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

   if -2e-16 < (-.f64 (pow.f64 (+.f64 x #s(literal 1 binary64)) (/.f64 #s(literal 1 binary64) n)) (pow.f64 x (/.f64 #s(literal 1 binary64) n))) < 0.0

   1. Initial program 46.4%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 17.7%

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

   6. Taylor expanded in n around inf

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

   8. Applied rewrites 46.4%

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

   if 0.0 < (-.f64 (pow.f64 (+.f64 x #s(literal 1 binary64)) (/.f64 #s(literal 1 binary64) n)) (pow.f64 x (/.f64 #s(literal 1 binary64) n)))

   1. Initial program 33.7%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 71.2%

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

4. Final simplification 59.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(x - -1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \leq -2 \cdot 10^{-16}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;{\left(x - -1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \leq 0:\\ \;\;\;\;1 - 1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\frac{-0.5 + \frac{0.5}{n}}{n}, x, \frac{1}{n}\right), x, 1\right) - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 55.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ t_1 := {\left(x - -1\right)}^{\left(\frac{1}{n}\right)} - t\_0\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{-16}:\\ \;\;\;\;1 - t\_0\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;1 - 1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{n \cdot n} \cdot 0.5, x, 1\right) - t\_0\\ \end{array} \end{array} \]

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 -2e-16)
     (- 1.0 t_0)
     (if (<= t_1 0.0) (- 1.0 1.0) (- (fma (* (/ x (* n n)) 0.5) x 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 <= -2e-16) {
		tmp = 1.0 - t_0;
	} else if (t_1 <= 0.0) {
		tmp = 1.0 - 1.0;
	} else {
		tmp = fma(((x / (n * n)) * 0.5), x, 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 <= -2e-16)
		tmp = Float64(1.0 - t_0);
	elseif (t_1 <= 0.0)
		tmp = Float64(1.0 - 1.0);
	else
		tmp = Float64(fma(Float64(Float64(x / Float64(n * n)) * 0.5), x, 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, -2e-16], N[(1.0 - t$95$0), $MachinePrecision], If[LessEqual[t$95$1, 0.0], N[(1.0 - 1.0), $MachinePrecision], N[(N[(N[(N[(x / N[(n * n), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision] * x + 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))) < -2e-16

   1. Initial program 96.7%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   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 96.7%

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

   if -2e-16 < (-.f64 (pow.f64 (+.f64 x #s(literal 1 binary64)) (/.f64 #s(literal 1 binary64) n)) (pow.f64 x (/.f64 #s(literal 1 binary64) n))) < 0.0

   1. Initial program 46.4%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 17.7%

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

   6. Taylor expanded in n around inf

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

   8. Applied rewrites 46.4%

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

   if 0.0 < (-.f64 (pow.f64 (+.f64 x #s(literal 1 binary64)) (/.f64 #s(literal 1 binary64) n)) (pow.f64 x (/.f64 #s(literal 1 binary64) n)))

   1. Initial program 33.7%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 71.2%

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

   6. Taylor expanded in n around 0

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

   8. Applied rewrites 69.9%

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

4. Final simplification 59.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(x - -1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \leq -2 \cdot 10^{-16}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;{\left(x - -1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \leq 0:\\ \;\;\;\;1 - 1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{n \cdot n} \cdot 0.5, x, 1\right) - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 86.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\log x}{n}\\ \mathbf{if}\;x \leq 0.095:\\ \;\;\;\;\mathsf{fma}\left(\frac{-0.5}{n}, \frac{{\log x}^{2}}{n}, \mathsf{fma}\left({t\_0}^{3}, -0.16666666666666666, \frac{x}{n}\right)\right) - t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{t\_0}}{n \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (/ (log x) n)))
   (if (<= x 0.095)
     (-
      (fma
       (/ -0.5 n)
       (/ (pow (log x) 2.0) n)
       (fma (pow t_0 3.0) -0.16666666666666666 (/ x n)))
      t_0)
     (/ (exp t_0) (* n x)))))

C

double code(double x, double n) {
	double t_0 = log(x) / n;
	double tmp;
	if (x <= 0.095) {
		tmp = fma((-0.5 / n), (pow(log(x), 2.0) / n), fma(pow(t_0, 3.0), -0.16666666666666666, (x / n))) - t_0;
	} else {
		tmp = exp(t_0) / (n * x);
	}
	return tmp;
}

Julia

function code(x, n)
	t_0 = Float64(log(x) / n)
	tmp = 0.0
	if (x <= 0.095)
		tmp = Float64(fma(Float64(-0.5 / n), Float64((log(x) ^ 2.0) / n), fma((t_0 ^ 3.0), -0.16666666666666666, Float64(x / n))) - t_0);
	else
		tmp = Float64(exp(t_0) / Float64(n * x));
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(N[Log[x], $MachinePrecision] / n), $MachinePrecision]}, If[LessEqual[x, 0.095], N[(N[(N[(-0.5 / n), $MachinePrecision] * N[(N[Power[N[Log[x], $MachinePrecision], 2.0], $MachinePrecision] / n), $MachinePrecision] + N[(N[Power[t$95$0, 3.0], $MachinePrecision] * -0.16666666666666666 + N[(x / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], N[(N[Exp[t$95$0], $MachinePrecision] / N[(n * x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 0.095000000000000001

   1. Initial program 41.2%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   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 84.3%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 84.0%

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

   if 0.095000000000000001 < x

   1. Initial program 71.2%

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

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 96.7%

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

Alternative 6: 86.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.029:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{-0.16666666666666666}{n}, \frac{{\log x}^{3}}{n}, \frac{{\log x}^{2}}{n} \cdot -0.5\right) - \log x}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\frac{\log x}{n}}}{n \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 0.029)
   (/
    (-
     (fma
      (/ -0.16666666666666666 n)
      (/ (pow (log x) 3.0) n)
      (* (/ (pow (log x) 2.0) n) -0.5))
     (log x))
    n)
   (/ (exp (/ (log x) n)) (* n x))))

C

double code(double x, double n) {
	double tmp;
	if (x <= 0.029) {
		tmp = (fma((-0.16666666666666666 / n), (pow(log(x), 3.0) / n), ((pow(log(x), 2.0) / n) * -0.5)) - log(x)) / n;
	} else {
		tmp = exp((log(x) / n)) / (n * x);
	}
	return tmp;
}

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 0.029)
		tmp = Float64(Float64(fma(Float64(-0.16666666666666666 / n), Float64((log(x) ^ 3.0) / n), Float64(Float64((log(x) ^ 2.0) / n) * -0.5)) - log(x)) / n);
	else
		tmp = Float64(exp(Float64(log(x) / n)) / Float64(n * x));
	end
	return tmp
end

Wolfram

code[x_, n_] := If[LessEqual[x, 0.029], N[(N[(N[(N[(-0.16666666666666666 / n), $MachinePrecision] * N[(N[Power[N[Log[x], $MachinePrecision], 3.0], $MachinePrecision] / n), $MachinePrecision] + N[(N[(N[Power[N[Log[x], $MachinePrecision], 2.0], $MachinePrecision] / n), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision] - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], N[(N[Exp[N[(N[Log[x], $MachinePrecision] / n), $MachinePrecision]], $MachinePrecision] / N[(n * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.0290000000000000015

   1. Initial program 41.2%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   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 84.3%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 83.7%

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

   if 0.0290000000000000015 < x

   1. Initial program 71.2%

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

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 96.7%

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

Alternative 7: 85.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{-5}:\\ \;\;\;\;\frac{e^{\frac{\log x}{n}}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 1000000:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right) - \log x}{n}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= (/ 1.0 n) -5e-5)
   (/ (exp (/ (log x) n)) (* n x))
   (if (<= (/ 1.0 n) 1000000.0)
     (/ (- (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) <= -5e-5) {
		tmp = exp((log(x) / n)) / (n * x);
	} else if ((1.0 / n) <= 1000000.0) {
		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) <= -5e-5) {
		tmp = Math.exp((Math.log(x) / n)) / (n * x);
	} else if ((1.0 / n) <= 1000000.0) {
		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) <= -5e-5:
		tmp = math.exp((math.log(x) / n)) / (n * x)
	elif (1.0 / n) <= 1000000.0:
		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) <= -5e-5)
		tmp = Float64(exp(Float64(log(x) / n)) / Float64(n * x));
	elseif (Float64(1.0 / n) <= 1000000.0)
		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], -5e-5], N[(N[Exp[N[(N[Log[x], $MachinePrecision] / n), $MachinePrecision]], $MachinePrecision] / N[(n * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 1000000.0], 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) < -5.00000000000000024e-5

   1. Initial program 98.0%

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

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 100.0%

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

   if -5.00000000000000024e-5 < (/.f64 #s(literal 1 binary64) n) < 1e6

   1. Initial program 25.5%

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

   3. Taylor expanded in n around inf

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

   5. Applied rewrites 76.2%

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

   if 1e6 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 34.4%

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

   3. Taylor expanded in n around 0

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

   5. Applied rewrites 100.0%

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

Alternative 8: 82.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{-5}:\\ \;\;\;\;\frac{e^{\frac{\log x}{n}}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 1000000:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right) - \log x}{n}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(-0.3333333333333333, x, \frac{\mathsf{fma}\left(\frac{x}{n}, 0.16666666666666666, -0.5 \cdot x\right) - -0.5}{-n}\right) - -0.5}{-n}, x, \frac{1}{n}\right), x, 1\right) - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= (/ 1.0 n) -5e-5)
   (/ (exp (/ (log x) n)) (* n x))
   (if (<= (/ 1.0 n) 1000000.0)
     (/ (- (log1p x) (log x)) n)
     (-
      (fma
       (fma
        (/
         (-
          (fma
           -0.3333333333333333
           x
           (/ (- (fma (/ x n) 0.16666666666666666 (* -0.5 x)) -0.5) (- n)))
          -0.5)
         (- n))
        x
        (/ 1.0 n))
       x
       1.0)
      (pow x (/ 1.0 n))))))

C

double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -5e-5) {
		tmp = exp((log(x) / n)) / (n * x);
	} else if ((1.0 / n) <= 1000000.0) {
		tmp = (log1p(x) - log(x)) / n;
	} else {
		tmp = fma(fma(((fma(-0.3333333333333333, x, ((fma((x / n), 0.16666666666666666, (-0.5 * x)) - -0.5) / -n)) - -0.5) / -n), x, (1.0 / n)), x, 1.0) - pow(x, (1.0 / n));
	}
	return tmp;
}

Julia

function code(x, n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -5e-5)
		tmp = Float64(exp(Float64(log(x) / n)) / Float64(n * x));
	elseif (Float64(1.0 / n) <= 1000000.0)
		tmp = Float64(Float64(log1p(x) - log(x)) / n);
	else
		tmp = Float64(fma(fma(Float64(Float64(fma(-0.3333333333333333, x, Float64(Float64(fma(Float64(x / n), 0.16666666666666666, Float64(-0.5 * x)) - -0.5) / Float64(-n))) - -0.5) / Float64(-n)), x, Float64(1.0 / n)), x, 1.0) - (x ^ Float64(1.0 / n)));
	end
	return tmp
end

Wolfram

code[x_, n_] := If[LessEqual[N[(1.0 / n), $MachinePrecision], -5e-5], N[(N[Exp[N[(N[Log[x], $MachinePrecision] / n), $MachinePrecision]], $MachinePrecision] / N[(n * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 1000000.0], N[(N[(N[Log[1 + x], $MachinePrecision] - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], N[(N[(N[(N[(N[(N[(-0.3333333333333333 * x + N[(N[(N[(N[(x / n), $MachinePrecision] * 0.16666666666666666 + N[(-0.5 * x), $MachinePrecision]), $MachinePrecision] - -0.5), $MachinePrecision] / (-n)), $MachinePrecision]), $MachinePrecision] - -0.5), $MachinePrecision] / (-n)), $MachinePrecision] * x + N[(1.0 / n), $MachinePrecision]), $MachinePrecision] * x + 1.0), $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) < -5.00000000000000024e-5

   1. Initial program 98.0%

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

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 100.0%

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

   if -5.00000000000000024e-5 < (/.f64 #s(literal 1 binary64) n) < 1e6

   1. Initial program 25.5%

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

   3. Taylor expanded in n around inf

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

   5. Applied rewrites 76.2%

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

   if 1e6 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 34.4%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 25.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{0.16666666666666666}{{n}^{3}} - \frac{-0.3333333333333333 + \frac{0.5}{n}}{n}, x, \frac{-0.5 + \frac{0.5}{n}}{n}\right), x, \frac{1}{n}\right), x, 1\right)} - {x}^{\left(\frac{1}{n}\right)} \]

   6. Taylor expanded in n around -inf

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

   8. Applied rewrites 77.5%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-\frac{\mathsf{fma}\left(-0.3333333333333333, x, -\frac{\mathsf{fma}\left(\frac{x}{n}, 0.16666666666666666, -0.5 \cdot x\right) - -0.5}{n}\right) - -0.5}{n}, x, \frac{1}{n}\right), x, 1\right) - {x}^{\left(\frac{1}{n}\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 85.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{-5}:\\ \;\;\;\;\frac{e^{\frac{\log x}{n}}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 1000000:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right) - \log x}{n}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(-0.3333333333333333, x, \frac{\mathsf{fma}\left(\frac{x}{n}, 0.16666666666666666, -0.5 \cdot x\right) - -0.5}{-n}\right) - -0.5}{-n}, x, \frac{1}{n}\right), x, 1\right) - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 57.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ t_1 := \mathsf{fma}\left(\frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.3333333333333333, x, 0.5\right), x, \frac{\mathsf{fma}\left(\frac{0.16666666666666666}{n} - 0.5, x, 0.5\right) \cdot x}{-n}\right) - 1}{-n}, x, 1\right) - t\_0\\ t_2 := \frac{-\log x}{n}\\ \mathbf{if}\;x \leq 1.45 \cdot 10^{-136}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 2.9 \cdot 10^{-124}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;x \leq 5.6 \cdot 10^{-84}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(-0.3333333333333333, x, \frac{\mathsf{fma}\left(\frac{x}{n}, 0.16666666666666666, -0.5 \cdot x\right) - -0.5}{-n}\right) - -0.5}{-n}, x, \frac{1}{n}\right), x, 1\right) - t\_0\\ \mathbf{elif}\;x \leq 1.45 \cdot 10^{-44}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;1 - 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n)))
        (t_1
         (-
          (fma
           (/
            (-
             (fma
              (fma -0.3333333333333333 x 0.5)
              x
              (/ (* (fma (- (/ 0.16666666666666666 n) 0.5) x 0.5) x) (- n)))
             1.0)
            (- n))
           x
           1.0)
          t_0))
        (t_2 (/ (- (log x)) n)))
   (if (<= x 1.45e-136)
     t_1
     (if (<= x 2.9e-124)
       t_2
       (if (<= x 5.6e-84)
         (-
          (fma
           (fma
            (/
             (-
              (fma
               -0.3333333333333333
               x
               (/ (- (fma (/ x n) 0.16666666666666666 (* -0.5 x)) -0.5) (- n)))
              -0.5)
             (- n))
            x
            (/ 1.0 n))
           x
           1.0)
          t_0)
         (if (<= x 1.45e-44) t_2 (if (<= x 1.0) t_1 (- 1.0 1.0))))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double t_1 = fma(((fma(fma(-0.3333333333333333, x, 0.5), x, ((fma(((0.16666666666666666 / n) - 0.5), x, 0.5) * x) / -n)) - 1.0) / -n), x, 1.0) - t_0;
	double t_2 = -log(x) / n;
	double tmp;
	if (x <= 1.45e-136) {
		tmp = t_1;
	} else if (x <= 2.9e-124) {
		tmp = t_2;
	} else if (x <= 5.6e-84) {
		tmp = fma(fma(((fma(-0.3333333333333333, x, ((fma((x / n), 0.16666666666666666, (-0.5 * x)) - -0.5) / -n)) - -0.5) / -n), x, (1.0 / n)), x, 1.0) - t_0;
	} else if (x <= 1.45e-44) {
		tmp = t_2;
	} else if (x <= 1.0) {
		tmp = t_1;
	} else {
		tmp = 1.0 - 1.0;
	}
	return tmp;
}

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	t_1 = Float64(fma(Float64(Float64(fma(fma(-0.3333333333333333, x, 0.5), x, Float64(Float64(fma(Float64(Float64(0.16666666666666666 / n) - 0.5), x, 0.5) * x) / Float64(-n))) - 1.0) / Float64(-n)), x, 1.0) - t_0)
	t_2 = Float64(Float64(-log(x)) / n)
	tmp = 0.0
	if (x <= 1.45e-136)
		tmp = t_1;
	elseif (x <= 2.9e-124)
		tmp = t_2;
	elseif (x <= 5.6e-84)
		tmp = Float64(fma(fma(Float64(Float64(fma(-0.3333333333333333, x, Float64(Float64(fma(Float64(x / n), 0.16666666666666666, Float64(-0.5 * x)) - -0.5) / Float64(-n))) - -0.5) / Float64(-n)), x, Float64(1.0 / n)), x, 1.0) - t_0);
	elseif (x <= 1.45e-44)
		tmp = t_2;
	elseif (x <= 1.0)
		tmp = t_1;
	else
		tmp = Float64(1.0 - 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[(N[(N[(N[(N[(-0.3333333333333333 * x + 0.5), $MachinePrecision] * x + N[(N[(N[(N[(N[(0.16666666666666666 / n), $MachinePrecision] - 0.5), $MachinePrecision] * x + 0.5), $MachinePrecision] * x), $MachinePrecision] / (-n)), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision] / (-n)), $MachinePrecision] * x + 1.0), $MachinePrecision] - t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[((-N[Log[x], $MachinePrecision]) / n), $MachinePrecision]}, If[LessEqual[x, 1.45e-136], t$95$1, If[LessEqual[x, 2.9e-124], t$95$2, If[LessEqual[x, 5.6e-84], N[(N[(N[(N[(N[(N[(-0.3333333333333333 * x + N[(N[(N[(N[(x / n), $MachinePrecision] * 0.16666666666666666 + N[(-0.5 * x), $MachinePrecision]), $MachinePrecision] - -0.5), $MachinePrecision] / (-n)), $MachinePrecision]), $MachinePrecision] - -0.5), $MachinePrecision] / (-n)), $MachinePrecision] * x + N[(1.0 / n), $MachinePrecision]), $MachinePrecision] * x + 1.0), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[x, 1.45e-44], t$95$2, If[LessEqual[x, 1.0], t$95$1, N[(1.0 - 1.0), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if x < 1.44999999999999997e-136 or 1.4500000000000001e-44 < x < 1

   1. Initial program 47.6%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 21.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{0.16666666666666666}{{n}^{3}} - \frac{-0.3333333333333333 + \frac{0.5}{n}}{n}, x, \frac{-0.5 + \frac{0.5}{n}}{n}\right), x, \frac{1}{n}\right), x, 1\right)} - {x}^{\left(\frac{1}{n}\right)} \]

   6. Taylor expanded in n around -inf

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

   8. Applied rewrites 53.7%

      \[\leadsto \mathsf{fma}\left(-\frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.3333333333333333, x, 0.5\right), x, -\frac{\mathsf{fma}\left(\frac{x \cdot x}{n}, 0.16666666666666666, \mathsf{fma}\left(-0.5, x, 0.5\right) \cdot x\right)}{n}\right) - 1}{n}, x, 1\right) - {x}^{\left(\frac{1}{n}\right)} \]

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 58.5%

       \[\leadsto \mathsf{fma}\left(-\frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.3333333333333333, x, 0.5\right), x, -\frac{\mathsf{fma}\left(\frac{0.16666666666666666}{n} - 0.5, x, 0.5\right) \cdot x}{n}\right) - 1}{n}, x, 1\right) - {x}^{\left(\frac{1}{n}\right)} \]

   if 1.44999999999999997e-136 < x < 2.9000000000000002e-124 or 5.59999999999999964e-84 < x < 1.4500000000000001e-44

   1. Initial program 9.6%

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

   3. Taylor expanded in n around inf

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

   5. Applied rewrites 84.4%

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

   7. Applied rewrites 84.0%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 84.0%

       \[\leadsto \frac{\frac{0.25}{n} \cdot \frac{{\log x}^{4}}{n} - {\log x}^{2}}{\color{blue}{\mathsf{fma}\left(\frac{{\log x}^{2}}{n}, -0.5, \log x\right) \cdot n}} \]

   11. Taylor expanded in n around inf

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

   13. Applied rewrites 84.2%

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

   if 2.9000000000000002e-124 < x < 5.59999999999999964e-84

   1. Initial program 50.5%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 28.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{0.16666666666666666}{{n}^{3}} - \frac{-0.3333333333333333 + \frac{0.5}{n}}{n}, x, \frac{-0.5 + \frac{0.5}{n}}{n}\right), x, \frac{1}{n}\right), x, 1\right)} - {x}^{\left(\frac{1}{n}\right)} \]

   6. Taylor expanded in n around -inf

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

   8. Applied rewrites 66.7%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-\frac{\mathsf{fma}\left(-0.3333333333333333, x, -\frac{\mathsf{fma}\left(\frac{x}{n}, 0.16666666666666666, -0.5 \cdot x\right) - -0.5}{n}\right) - -0.5}{n}, x, \frac{1}{n}\right), x, 1\right) - {x}^{\left(\frac{1}{n}\right)} \]

   if 1 < x

   1. Initial program 70.9%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   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 25.1%

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

   6. Taylor expanded in n around inf

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

   8. Applied rewrites 70.9%

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

4. Final simplification 66.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.45 \cdot 10^{-136}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.3333333333333333, x, 0.5\right), x, \frac{\mathsf{fma}\left(\frac{0.16666666666666666}{n} - 0.5, x, 0.5\right) \cdot x}{-n}\right) - 1}{-n}, x, 1\right) - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;x \leq 2.9 \cdot 10^{-124}:\\ \;\;\;\;\frac{-\log x}{n}\\ \mathbf{elif}\;x \leq 5.6 \cdot 10^{-84}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(-0.3333333333333333, x, \frac{\mathsf{fma}\left(\frac{x}{n}, 0.16666666666666666, -0.5 \cdot x\right) - -0.5}{-n}\right) - -0.5}{-n}, x, \frac{1}{n}\right), x, 1\right) - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;x \leq 1.45 \cdot 10^{-44}:\\ \;\;\;\;\frac{-\log x}{n}\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;\mathsf{fma}\left(\frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.3333333333333333, x, 0.5\right), x, \frac{\mathsf{fma}\left(\frac{0.16666666666666666}{n} - 0.5, x, 0.5\right) \cdot x}{-n}\right) - 1}{-n}, x, 1\right) - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;1 - 1\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 57.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.3333333333333333, x, 0.5\right), x, \frac{\mathsf{fma}\left(\frac{0.16666666666666666}{n} - 0.5, x, 0.5\right) \cdot x}{-n}\right) - 1}{-n}, x, 1\right) - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;x \leq 1.45 \cdot 10^{-136}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 1.45 \cdot 10^{-44}:\\ \;\;\;\;\frac{-\log x}{n}\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;1 - 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0
         (-
          (fma
           (/
            (-
             (fma
              (fma -0.3333333333333333 x 0.5)
              x
              (/ (* (fma (- (/ 0.16666666666666666 n) 0.5) x 0.5) x) (- n)))
             1.0)
            (- n))
           x
           1.0)
          (pow x (/ 1.0 n)))))
   (if (<= x 1.45e-136)
     t_0
     (if (<= x 1.45e-44) (/ (- (log x)) n) (if (<= x 1.0) t_0 (- 1.0 1.0))))))

C

double code(double x, double n) {
	double t_0 = fma(((fma(fma(-0.3333333333333333, x, 0.5), x, ((fma(((0.16666666666666666 / n) - 0.5), x, 0.5) * x) / -n)) - 1.0) / -n), x, 1.0) - pow(x, (1.0 / n));
	double tmp;
	if (x <= 1.45e-136) {
		tmp = t_0;
	} else if (x <= 1.45e-44) {
		tmp = -log(x) / n;
	} else if (x <= 1.0) {
		tmp = t_0;
	} else {
		tmp = 1.0 - 1.0;
	}
	return tmp;
}

Julia

function code(x, n)
	t_0 = Float64(fma(Float64(Float64(fma(fma(-0.3333333333333333, x, 0.5), x, Float64(Float64(fma(Float64(Float64(0.16666666666666666 / n) - 0.5), x, 0.5) * x) / Float64(-n))) - 1.0) / Float64(-n)), x, 1.0) - (x ^ Float64(1.0 / n)))
	tmp = 0.0
	if (x <= 1.45e-136)
		tmp = t_0;
	elseif (x <= 1.45e-44)
		tmp = Float64(Float64(-log(x)) / n);
	elseif (x <= 1.0)
		tmp = t_0;
	else
		tmp = Float64(1.0 - 1.0);
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(N[(N[(N[(N[(N[(-0.3333333333333333 * x + 0.5), $MachinePrecision] * x + N[(N[(N[(N[(N[(0.16666666666666666 / n), $MachinePrecision] - 0.5), $MachinePrecision] * x + 0.5), $MachinePrecision] * x), $MachinePrecision] / (-n)), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision] / (-n)), $MachinePrecision] * x + 1.0), $MachinePrecision] - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 1.45e-136], t$95$0, If[LessEqual[x, 1.45e-44], N[((-N[Log[x], $MachinePrecision]) / n), $MachinePrecision], If[LessEqual[x, 1.0], t$95$0, N[(1.0 - 1.0), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if x < 1.44999999999999997e-136 or 1.4500000000000001e-44 < x < 1

   1. Initial program 47.6%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 21.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{0.16666666666666666}{{n}^{3}} - \frac{-0.3333333333333333 + \frac{0.5}{n}}{n}, x, \frac{-0.5 + \frac{0.5}{n}}{n}\right), x, \frac{1}{n}\right), x, 1\right)} - {x}^{\left(\frac{1}{n}\right)} \]

   6. Taylor expanded in n around -inf

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

   8. Applied rewrites 53.7%

      \[\leadsto \mathsf{fma}\left(-\frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.3333333333333333, x, 0.5\right), x, -\frac{\mathsf{fma}\left(\frac{x \cdot x}{n}, 0.16666666666666666, \mathsf{fma}\left(-0.5, x, 0.5\right) \cdot x\right)}{n}\right) - 1}{n}, x, 1\right) - {x}^{\left(\frac{1}{n}\right)} \]

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 58.5%

       \[\leadsto \mathsf{fma}\left(-\frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.3333333333333333, x, 0.5\right), x, -\frac{\mathsf{fma}\left(\frac{0.16666666666666666}{n} - 0.5, x, 0.5\right) \cdot x}{n}\right) - 1}{n}, x, 1\right) - {x}^{\left(\frac{1}{n}\right)} \]

   if 1.44999999999999997e-136 < x < 1.4500000000000001e-44

   1. Initial program 29.2%

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

   3. Taylor expanded in n around inf

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

   5. Applied rewrites 72.7%

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

   7. Applied rewrites 70.5%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 70.5%

       \[\leadsto \frac{\frac{0.25}{n} \cdot \frac{{\log x}^{4}}{n} - {\log x}^{2}}{\color{blue}{\mathsf{fma}\left(\frac{{\log x}^{2}}{n}, -0.5, \log x\right) \cdot n}} \]

   11. Taylor expanded in n around inf

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

   13. Applied rewrites 59.5%

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

   if 1 < x

   1. Initial program 70.9%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   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 25.1%

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

   6. Taylor expanded in n around inf

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

   8. Applied rewrites 70.9%

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

4. Final simplification 63.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.45 \cdot 10^{-136}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.3333333333333333, x, 0.5\right), x, \frac{\mathsf{fma}\left(\frac{0.16666666666666666}{n} - 0.5, x, 0.5\right) \cdot x}{-n}\right) - 1}{-n}, x, 1\right) - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;x \leq 1.45 \cdot 10^{-44}:\\ \;\;\;\;\frac{-\log x}{n}\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;\mathsf{fma}\left(\frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.3333333333333333, x, 0.5\right), x, \frac{\mathsf{fma}\left(\frac{0.16666666666666666}{n} - 0.5, x, 0.5\right) \cdot x}{-n}\right) - 1}{-n}, x, 1\right) - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;1 - 1\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 57.2% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 9.5 \cdot 10^{-216}:\\ \;\;\;\;\left(\frac{x}{n} - -1\right) - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;\frac{-\log x}{n}\\ \mathbf{else}:\\ \;\;\;\;1 - 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 9.5e-216)
   (- (- (/ x n) -1.0) (pow x (/ 1.0 n)))
   (if (<= x 1.0) (/ (- (log x)) n) (- 1.0 1.0))))

C

double code(double x, double n) {
	double tmp;
	if (x <= 9.5e-216) {
		tmp = ((x / n) - -1.0) - pow(x, (1.0 / n));
	} else if (x <= 1.0) {
		tmp = -log(x) / n;
	} else {
		tmp = 1.0 - 1.0;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, n)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if (x <= 9.5d-216) then
        tmp = ((x / n) - (-1.0d0)) - (x ** (1.0d0 / n))
    else if (x <= 1.0d0) then
        tmp = -log(x) / n
    else
        tmp = 1.0d0 - 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (x <= 9.5e-216) {
		tmp = ((x / n) - -1.0) - Math.pow(x, (1.0 / n));
	} else if (x <= 1.0) {
		tmp = -Math.log(x) / n;
	} else {
		tmp = 1.0 - 1.0;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 9.5e-216:
		tmp = ((x / n) - -1.0) - math.pow(x, (1.0 / n))
	elif x <= 1.0:
		tmp = -math.log(x) / n
	else:
		tmp = 1.0 - 1.0
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 9.5e-216)
		tmp = Float64(Float64(Float64(x / n) - -1.0) - (x ^ Float64(1.0 / n)));
	elseif (x <= 1.0)
		tmp = Float64(Float64(-log(x)) / n);
	else
		tmp = Float64(1.0 - 1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (x <= 9.5e-216)
		tmp = ((x / n) - -1.0) - (x ^ (1.0 / n));
	elseif (x <= 1.0)
		tmp = -log(x) / n;
	else
		tmp = 1.0 - 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 9.5e-216], N[(N[(N[(x / n), $MachinePrecision] - -1.0), $MachinePrecision] - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.0], N[((-N[Log[x], $MachinePrecision]) / n), $MachinePrecision], N[(1.0 - 1.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 9.49999999999999943e-216

   1. Initial program 56.0%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 56.0%

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

   if 9.49999999999999943e-216 < x < 1

   1. Initial program 35.9%

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

   3. Taylor expanded in n around inf

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

   5. Applied rewrites 65.7%

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

   7. Applied rewrites 64.7%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 63.8%

       \[\leadsto \frac{\frac{0.25}{n} \cdot \frac{{\log x}^{4}}{n} - {\log x}^{2}}{\color{blue}{\mathsf{fma}\left(\frac{{\log x}^{2}}{n}, -0.5, \log x\right) \cdot n}} \]

   11. Taylor expanded in n around inf

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

   13. Applied rewrites 49.1%

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

   if 1 < x

   1. Initial program 70.9%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   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 25.1%

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

   6. Taylor expanded in n around inf

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

   8. Applied rewrites 70.9%

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

4. Final simplification 59.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 9.5 \cdot 10^{-216}:\\ \;\;\;\;\left(\frac{x}{n} - -1\right) - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;\frac{-\log x}{n}\\ \mathbf{else}:\\ \;\;\;\;1 - 1\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 57.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 9.5 \cdot 10^{-216}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;\frac{-\log x}{n}\\ \mathbf{else}:\\ \;\;\;\;1 - 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 9.5e-216)
   (- 1.0 (pow x (/ 1.0 n)))
   (if (<= x 1.0) (/ (- (log x)) n) (- 1.0 1.0))))

C

double code(double x, double n) {
	double tmp;
	if (x <= 9.5e-216) {
		tmp = 1.0 - pow(x, (1.0 / n));
	} else if (x <= 1.0) {
		tmp = -log(x) / n;
	} else {
		tmp = 1.0 - 1.0;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, n)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if (x <= 9.5d-216) then
        tmp = 1.0d0 - (x ** (1.0d0 / n))
    else if (x <= 1.0d0) then
        tmp = -log(x) / n
    else
        tmp = 1.0d0 - 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (x <= 9.5e-216) {
		tmp = 1.0 - Math.pow(x, (1.0 / n));
	} else if (x <= 1.0) {
		tmp = -Math.log(x) / n;
	} else {
		tmp = 1.0 - 1.0;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 9.5e-216:
		tmp = 1.0 - math.pow(x, (1.0 / n))
	elif x <= 1.0:
		tmp = -math.log(x) / n
	else:
		tmp = 1.0 - 1.0
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 9.5e-216)
		tmp = Float64(1.0 - (x ^ Float64(1.0 / n)));
	elseif (x <= 1.0)
		tmp = Float64(Float64(-log(x)) / n);
	else
		tmp = Float64(1.0 - 1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (x <= 9.5e-216)
		tmp = 1.0 - (x ^ (1.0 / n));
	elseif (x <= 1.0)
		tmp = -log(x) / n;
	else
		tmp = 1.0 - 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 9.5e-216], N[(1.0 - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.0], N[((-N[Log[x], $MachinePrecision]) / n), $MachinePrecision], N[(1.0 - 1.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 9.49999999999999943e-216

   1. Initial program 56.0%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 56.0%

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

   if 9.49999999999999943e-216 < x < 1

   1. Initial program 35.9%

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

   3. Taylor expanded in n around inf

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

   5. Applied rewrites 65.7%

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

   7. Applied rewrites 64.7%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 63.8%

       \[\leadsto \frac{\frac{0.25}{n} \cdot \frac{{\log x}^{4}}{n} - {\log x}^{2}}{\color{blue}{\mathsf{fma}\left(\frac{{\log x}^{2}}{n}, -0.5, \log x\right) \cdot n}} \]

   11. Taylor expanded in n around inf

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

   13. Applied rewrites 49.1%

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

   if 1 < x

   1. Initial program 70.9%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   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 25.1%

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

   6. Taylor expanded in n around inf

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

   8. Applied rewrites 70.9%

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

Alternative 13: 58.5% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 1

   1. Initial program 41.6%

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

   3. Taylor expanded in n around inf

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

   5. Applied rewrites 63.1%

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

   7. Applied rewrites 61.7%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 61.1%

       \[\leadsto \frac{\frac{0.25}{n} \cdot \frac{{\log x}^{4}}{n} - {\log x}^{2}}{\color{blue}{\mathsf{fma}\left(\frac{{\log x}^{2}}{n}, -0.5, \log x\right) \cdot n}} \]

   11. Taylor expanded in n around inf

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

   13. Applied rewrites 46.8%

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

   if 1 < x

   1. Initial program 70.9%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   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 25.1%

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

   6. Taylor expanded in n around inf

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

   8. Applied rewrites 70.9%

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

Alternative 14: 49.3% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\frac{1}{x}}{n}\\ \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{+264}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\frac{1}{n} \leq -2000000:\\ \;\;\;\;1 - 1\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{+132}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\frac{-0.5 + \frac{0.5}{n}}{n}, x, \frac{1}{n}\right), x, 1\right) - 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (/ (/ 1.0 x) n)))
   (if (<= (/ 1.0 n) -5e+264)
     t_0
     (if (<= (/ 1.0 n) -2000000.0)
       (- 1.0 1.0)
       (if (<= (/ 1.0 n) 5e+132)
         t_0
         (- (fma (fma (/ (+ -0.5 (/ 0.5 n)) n) x (/ 1.0 n)) x 1.0) 1.0))))))

C

double code(double x, double n) {
	double t_0 = (1.0 / x) / n;
	double tmp;
	if ((1.0 / n) <= -5e+264) {
		tmp = t_0;
	} else if ((1.0 / n) <= -2000000.0) {
		tmp = 1.0 - 1.0;
	} else if ((1.0 / n) <= 5e+132) {
		tmp = t_0;
	} else {
		tmp = fma(fma(((-0.5 + (0.5 / n)) / n), x, (1.0 / n)), x, 1.0) - 1.0;
	}
	return tmp;
}

Julia

function code(x, n)
	t_0 = Float64(Float64(1.0 / x) / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -5e+264)
		tmp = t_0;
	elseif (Float64(1.0 / n) <= -2000000.0)
		tmp = Float64(1.0 - 1.0);
	elseif (Float64(1.0 / n) <= 5e+132)
		tmp = t_0;
	else
		tmp = Float64(fma(fma(Float64(Float64(-0.5 + Float64(0.5 / n)) / n), x, Float64(1.0 / n)), x, 1.0) - 1.0);
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(N[(1.0 / x), $MachinePrecision] / n), $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -5e+264], t$95$0, If[LessEqual[N[(1.0 / n), $MachinePrecision], -2000000.0], N[(1.0 - 1.0), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e+132], t$95$0, N[(N[(N[(N[(N[(-0.5 + N[(0.5 / n), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision] * x + N[(1.0 / n), $MachinePrecision]), $MachinePrecision] * x + 1.0), $MachinePrecision] - 1.0), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -5.00000000000000033e264 or -2e6 < (/.f64 #s(literal 1 binary64) n) < 5.0000000000000001e132

   1. Initial program 35.6%

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

   3. Taylor expanded in n around inf

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

   5. Applied rewrites 71.9%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 42.9%

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

   9. Taylor expanded in n around inf

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

   11. Applied rewrites 43.0%

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

   if -5.00000000000000033e264 < (/.f64 #s(literal 1 binary64) n) < -2e6

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 43.9%

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

   6. Taylor expanded in n around inf

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

   8. Applied rewrites 58.6%

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

   if 5.0000000000000001e132 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 24.5%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 81.1%

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

   6. Taylor expanded in n around inf

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

   8. Applied rewrites 76.5%

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

Alternative 15: 47.9% accurate, 6.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -0.66 \lor \neg \left(n \leq -1.86 \cdot 10^{-265}\right):\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \mathbf{else}:\\ \;\;\;\;1 - 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (or (<= n -0.66) (not (<= n -1.86e-265))) (/ (/ 1.0 x) n) (- 1.0 1.0)))

C

double code(double x, double n) {
	double tmp;
	if ((n <= -0.66) || !(n <= -1.86e-265)) {
		tmp = (1.0 / x) / n;
	} else {
		tmp = 1.0 - 1.0;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, n)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if ((n <= (-0.66d0)) .or. (.not. (n <= (-1.86d-265)))) then
        tmp = (1.0d0 / x) / n
    else
        tmp = 1.0d0 - 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if ((n <= -0.66) || !(n <= -1.86e-265)) {
		tmp = (1.0 / x) / n;
	} else {
		tmp = 1.0 - 1.0;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if (n <= -0.66) or not (n <= -1.86e-265):
		tmp = (1.0 / x) / n
	else:
		tmp = 1.0 - 1.0
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if ((n <= -0.66) || !(n <= -1.86e-265))
		tmp = Float64(Float64(1.0 / x) / n);
	else
		tmp = Float64(1.0 - 1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if ((n <= -0.66) || ~((n <= -1.86e-265)))
		tmp = (1.0 / x) / n;
	else
		tmp = 1.0 - 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[Or[LessEqual[n, -0.66], N[Not[LessEqual[n, -1.86e-265]], $MachinePrecision]], N[(N[(1.0 / x), $MachinePrecision] / n), $MachinePrecision], N[(1.0 - 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < -0.660000000000000031 or -1.86e-265 < n

   1. Initial program 33.8%

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

   3. Taylor expanded in n around inf

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

   5. Applied rewrites 60.7%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 36.3%

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

   9. Taylor expanded in n around inf

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

   11. Applied rewrites 45.1%

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

   if -0.660000000000000031 < n < -1.86e-265

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 43.9%

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

   6. Taylor expanded in n around inf

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

   8. Applied rewrites 58.6%

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

4. Final simplification 49.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -0.66 \lor \neg \left(n \leq -1.86 \cdot 10^{-265}\right):\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \mathbf{else}:\\ \;\;\;\;1 - 1\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 31.7% 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 34.5%

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

6. Taylor expanded in n around inf

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

8. Applied rewrites 30.7%

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

Reproduce

herbie shell --seed 2025019 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  :precision binary64
  (- (pow (+ x 1.0) (/ 1.0 n)) (pow x (/ 1.0 n))))