2nthrt (problem 3.4.6)

Percentage Accurate: 53.6% → 83.1%

Time: 27.5s

Alternatives: 18

Speedup: 1.1×

• could not determine a ground truth

  1. x = -1.0142989665455385e-172
  2. n = 8.033394836880647e+92

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: 53.6% 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: 83.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\frac{\log x}{n}}\\ \mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{-13}:\\ \;\;\;\;\frac{t\_0}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 1.5 \cdot 10^{-182}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.5, \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}}{n}, \mathsf{log1p}\left(x\right)\right) - \log x}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 50:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{\frac{-0.5 + \frac{0.5}{n}}{n}}{x}, t\_0, \frac{t\_0}{n}\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{x}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (exp (/ (log x) n))))
   (if (<= (/ 1.0 n) -1e-13)
     (/ t_0 (* n x))
     (if (<= (/ 1.0 n) 1.5e-182)
       (/
        (-
         (fma 0.5 (/ (- (pow (log1p x) 2.0) (pow (log x) 2.0)) n) (log1p x))
         (log x))
        n)
       (if (<= (/ 1.0 n) 50.0)
         (/ (fma (/ (/ (+ -0.5 (/ 0.5 n)) n) x) t_0 (/ t_0 n)) x)
         (- (exp (/ x n)) (pow x (/ 1.0 n))))))))

C

double code(double x, double n) {
	double t_0 = exp((log(x) / n));
	double tmp;
	if ((1.0 / n) <= -1e-13) {
		tmp = t_0 / (n * x);
	} else if ((1.0 / n) <= 1.5e-182) {
		tmp = (fma(0.5, ((pow(log1p(x), 2.0) - pow(log(x), 2.0)) / n), log1p(x)) - log(x)) / n;
	} else if ((1.0 / n) <= 50.0) {
		tmp = fma((((-0.5 + (0.5 / n)) / n) / x), t_0, (t_0 / n)) / x;
	} else {
		tmp = exp((x / n)) - pow(x, (1.0 / n));
	}
	return tmp;
}

Julia

function code(x, n)
	t_0 = exp(Float64(log(x) / n))
	tmp = 0.0
	if (Float64(1.0 / n) <= -1e-13)
		tmp = Float64(t_0 / Float64(n * x));
	elseif (Float64(1.0 / n) <= 1.5e-182)
		tmp = Float64(Float64(fma(0.5, Float64(Float64((log1p(x) ^ 2.0) - (log(x) ^ 2.0)) / n), log1p(x)) - log(x)) / n);
	elseif (Float64(1.0 / n) <= 50.0)
		tmp = Float64(fma(Float64(Float64(Float64(-0.5 + Float64(0.5 / n)) / n) / x), t_0, Float64(t_0 / n)) / x);
	else
		tmp = Float64(exp(Float64(x / n)) - (x ^ Float64(1.0 / n)));
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Exp[N[(N[Log[x], $MachinePrecision] / n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -1e-13], N[(t$95$0 / N[(n * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 1.5e-182], N[(N[(N[(0.5 * N[(N[(N[Power[N[Log[1 + x], $MachinePrecision], 2.0], $MachinePrecision] - N[Power[N[Log[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision] + N[Log[1 + x], $MachinePrecision]), $MachinePrecision] - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 50.0], N[(N[(N[(N[(N[(-0.5 + N[(0.5 / n), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision] / x), $MachinePrecision] * t$95$0 + N[(t$95$0 / n), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], N[(N[Exp[N[(x / n), $MachinePrecision]], $MachinePrecision] - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -1e-13

   1. Initial program 96.5%

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

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 98.8%

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

   if -1e-13 < (/.f64 #s(literal 1 binary64) n) < 1.5000000000000001e-182

   1. Initial program 41.3%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   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 83.7%

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

   if 1.5000000000000001e-182 < (/.f64 #s(literal 1 binary64) n) < 50

   1. Initial program 20.0%

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

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 67.7%

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

   if 50 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 51.1%

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

   3. Taylor expanded in n around 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%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 100.0%

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

Alternative 2: 82.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ t_1 := {\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - t\_0\\ \mathbf{if}\;t\_1 \leq -2:\\ \;\;\;\;1 - t\_0\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right) - \log x}{n}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{0.5}{n} - 0.5, x, 1\right)}{n}, x, 1\right) - t\_0\\ \end{array} \end{array} \]

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 -2.0)
     (- 1.0 t_0)
     (if (<= t_1 0.0)
       (/ (- (log1p x) (log x)) n)
       (- (fma (/ (fma (- (/ 0.5 n) 0.5) 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 <= -2.0) {
		tmp = 1.0 - t_0;
	} else if (t_1 <= 0.0) {
		tmp = (log1p(x) - log(x)) / n;
	} else {
		tmp = fma((fma(((0.5 / n) - 0.5), 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 <= -2.0)
		tmp = Float64(1.0 - t_0);
	elseif (t_1 <= 0.0)
		tmp = Float64(Float64(log1p(x) - log(x)) / n);
	else
		tmp = Float64(fma(Float64(fma(Float64(Float64(0.5 / n) - 0.5), x, 1.0) / n), x, 1.0) - 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, -2.0], N[(1.0 - t$95$0), $MachinePrecision], If[LessEqual[t$95$1, 0.0], N[(N[(N[Log[1 + x], $MachinePrecision] - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], N[(N[(N[(N[(N[(N[(0.5 / n), $MachinePrecision] - 0.5), $MachinePrecision] * x + 1.0), $MachinePrecision] / n), $MachinePrecision] * x + 1.0), $MachinePrecision] - 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))) < -2

   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 100.0%

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

   if -2 < (-.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 50.7%

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

   3. Taylor expanded in n around inf

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

   5. Applied rewrites 79.9%

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

   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 50.1%

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

   3. Taylor expanded in n around 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 98.0%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 75.3%

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

Alternative 3: 81.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\frac{\log x}{n}}\\ \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{-122}:\\ \;\;\;\;\frac{t\_0}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 1.5 \cdot 10^{-182}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right) - \log x}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 50:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{\frac{-0.5 + \frac{0.5}{n}}{n}}{x}, t\_0, \frac{t\_0}{n}\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{x}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (exp (/ (log x) n))))
   (if (<= (/ 1.0 n) -5e-122)
     (/ t_0 (* n x))
     (if (<= (/ 1.0 n) 1.5e-182)
       (/ (- (log1p x) (log x)) n)
       (if (<= (/ 1.0 n) 50.0)
         (/ (fma (/ (/ (+ -0.5 (/ 0.5 n)) n) x) t_0 (/ t_0 n)) x)
         (- (exp (/ x n)) (pow x (/ 1.0 n))))))))

C

double code(double x, double n) {
	double t_0 = exp((log(x) / n));
	double tmp;
	if ((1.0 / n) <= -5e-122) {
		tmp = t_0 / (n * x);
	} else if ((1.0 / n) <= 1.5e-182) {
		tmp = (log1p(x) - log(x)) / n;
	} else if ((1.0 / n) <= 50.0) {
		tmp = fma((((-0.5 + (0.5 / n)) / n) / x), t_0, (t_0 / n)) / x;
	} else {
		tmp = exp((x / n)) - pow(x, (1.0 / n));
	}
	return tmp;
}

Julia

function code(x, n)
	t_0 = exp(Float64(log(x) / n))
	tmp = 0.0
	if (Float64(1.0 / n) <= -5e-122)
		tmp = Float64(t_0 / Float64(n * x));
	elseif (Float64(1.0 / n) <= 1.5e-182)
		tmp = Float64(Float64(log1p(x) - log(x)) / n);
	elseif (Float64(1.0 / n) <= 50.0)
		tmp = Float64(fma(Float64(Float64(Float64(-0.5 + Float64(0.5 / n)) / n) / x), t_0, Float64(t_0 / n)) / x);
	else
		tmp = Float64(exp(Float64(x / n)) - (x ^ Float64(1.0 / n)));
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Exp[N[(N[Log[x], $MachinePrecision] / n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -5e-122], N[(t$95$0 / N[(n * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 1.5e-182], N[(N[(N[Log[1 + x], $MachinePrecision] - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 50.0], N[(N[(N[(N[(N[(-0.5 + N[(0.5 / n), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision] / x), $MachinePrecision] * t$95$0 + N[(t$95$0 / n), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], N[(N[Exp[N[(x / n), $MachinePrecision]], $MachinePrecision] - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -4.9999999999999999e-122

   1. Initial program 81.4%

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

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 90.6%

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

   if -4.9999999999999999e-122 < (/.f64 #s(literal 1 binary64) n) < 1.5000000000000001e-182

   1. Initial program 46.7%

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

   3. Taylor expanded in n around inf

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

   5. Applied rewrites 91.1%

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

   if 1.5000000000000001e-182 < (/.f64 #s(literal 1 binary64) n) < 50

   1. Initial program 20.0%

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

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 67.7%

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

   if 50 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 51.1%

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

   3. Taylor expanded in n around 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%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 100.0%

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

Alternative 4: 81.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{e^{\frac{\log x}{n}}}{n \cdot x}\\ \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{-122}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-188}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right) - \log x}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 50:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{x}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (/ (exp (/ (log x) n)) (* n x))))
   (if (<= (/ 1.0 n) -5e-122)
     t_0
     (if (<= (/ 1.0 n) 2e-188)
       (/ (- (log1p x) (log x)) n)
       (if (<= (/ 1.0 n) 50.0) t_0 (- (exp (/ x n)) (pow x (/ 1.0 n))))))))

C

double code(double x, double n) {
	double t_0 = exp((log(x) / n)) / (n * x);
	double tmp;
	if ((1.0 / n) <= -5e-122) {
		tmp = t_0;
	} else if ((1.0 / n) <= 2e-188) {
		tmp = (log1p(x) - log(x)) / n;
	} else if ((1.0 / n) <= 50.0) {
		tmp = t_0;
	} else {
		tmp = exp((x / n)) - pow(x, (1.0 / n));
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double t_0 = Math.exp((Math.log(x) / n)) / (n * x);
	double tmp;
	if ((1.0 / n) <= -5e-122) {
		tmp = t_0;
	} else if ((1.0 / n) <= 2e-188) {
		tmp = (Math.log1p(x) - Math.log(x)) / n;
	} else if ((1.0 / n) <= 50.0) {
		tmp = t_0;
	} else {
		tmp = Math.exp((x / n)) - Math.pow(x, (1.0 / n));
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.exp((math.log(x) / n)) / (n * x)
	tmp = 0
	if (1.0 / n) <= -5e-122:
		tmp = t_0
	elif (1.0 / n) <= 2e-188:
		tmp = (math.log1p(x) - math.log(x)) / n
	elif (1.0 / n) <= 50.0:
		tmp = t_0
	else:
		tmp = math.exp((x / n)) - math.pow(x, (1.0 / n))
	return tmp

Julia

function code(x, n)
	t_0 = Float64(exp(Float64(log(x) / n)) / Float64(n * x))
	tmp = 0.0
	if (Float64(1.0 / n) <= -5e-122)
		tmp = t_0;
	elseif (Float64(1.0 / n) <= 2e-188)
		tmp = Float64(Float64(log1p(x) - log(x)) / n);
	elseif (Float64(1.0 / n) <= 50.0)
		tmp = t_0;
	else
		tmp = Float64(exp(Float64(x / n)) - (x ^ Float64(1.0 / n)));
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(N[Exp[N[(N[Log[x], $MachinePrecision] / n), $MachinePrecision]], $MachinePrecision] / N[(n * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -5e-122], t$95$0, If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-188], N[(N[(N[Log[1 + x], $MachinePrecision] - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 50.0], t$95$0, N[(N[Exp[N[(x / 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) < -4.9999999999999999e-122 or 1.9999999999999999e-188 < (/.f64 #s(literal 1 binary64) n) < 50

   1. Initial program 64.1%

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

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 84.0%

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

   if -4.9999999999999999e-122 < (/.f64 #s(literal 1 binary64) n) < 1.9999999999999999e-188

   1. Initial program 47.9%

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

   3. Taylor expanded in n around inf

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

   5. Applied rewrites 92.3%

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

   if 50 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 51.1%

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

   3. Taylor expanded in n around 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%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 100.0%

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

Alternative 5: 85.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -0.005 \lor \neg \left(\frac{1}{n} \leq 50\right):\\ \;\;\;\;e^{\frac{x}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right) - \log x}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (or (<= (/ 1.0 n) -0.005) (not (<= (/ 1.0 n) 50.0)))
   (- (exp (/ x n)) (pow x (/ 1.0 n)))
   (/ (- (log1p x) (log x)) n)))

C

double code(double x, double n) {
	double tmp;
	if (((1.0 / n) <= -0.005) || !((1.0 / n) <= 50.0)) {
		tmp = exp((x / n)) - pow(x, (1.0 / n));
	} else {
		tmp = (log1p(x) - log(x)) / n;
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double tmp;
	if (((1.0 / n) <= -0.005) || !((1.0 / n) <= 50.0)) {
		tmp = Math.exp((x / n)) - Math.pow(x, (1.0 / n));
	} else {
		tmp = (Math.log1p(x) - Math.log(x)) / n;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if ((1.0 / n) <= -0.005) or not ((1.0 / n) <= 50.0):
		tmp = math.exp((x / n)) - math.pow(x, (1.0 / n))
	else:
		tmp = (math.log1p(x) - math.log(x)) / n
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if ((Float64(1.0 / n) <= -0.005) || !(Float64(1.0 / n) <= 50.0))
		tmp = Float64(exp(Float64(x / n)) - (x ^ Float64(1.0 / n)));
	else
		tmp = Float64(Float64(log1p(x) - log(x)) / n);
	end
	return tmp
end

Wolfram

code[x_, n_] := If[Or[LessEqual[N[(1.0 / n), $MachinePrecision], -0.005], N[Not[LessEqual[N[(1.0 / n), $MachinePrecision], 50.0]], $MachinePrecision]], N[(N[Exp[N[(x / n), $MachinePrecision]], $MachinePrecision] - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(N[Log[1 + x], $MachinePrecision] - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -0.0050000000000000001 or 50 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 81.0%

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

   3. Taylor expanded in n around 0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 100.0%

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

   if -0.0050000000000000001 < (/.f64 #s(literal 1 binary64) n) < 50

   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 inf

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

   5. Applied rewrites 72.8%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) - \log x}{n}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 86.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -0.005 \lor \neg \left(\frac{1}{n} \leq 50\right):\\ \;\;\;\;e^{\frac{x}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right) - \log x}{n}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 54.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{+233}:\\ \;\;\;\;1 - t\_0\\ \mathbf{elif}\;\frac{1}{n} \leq -1 \cdot 10^{+124}:\\ \;\;\;\;1 - 1\\ \mathbf{elif}\;\frac{1}{n} \leq -4000:\\ \;\;\;\;\frac{\log x \cdot x + n \cdot x}{\left(\left(n \cdot x\right) \cdot x\right) \cdot n}\\ \mathbf{elif}\;\frac{1}{n} \leq 50:\\ \;\;\;\;\frac{\frac{\log x}{n \cdot x} - \frac{-1}{x}}{n}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{0.5}{n} - 0.5, x, 1\right)}{n}, x, 1\right) - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -1e+233)
     (- 1.0 t_0)
     (if (<= (/ 1.0 n) -1e+124)
       (- 1.0 1.0)
       (if (<= (/ 1.0 n) -4000.0)
         (/ (+ (* (log x) x) (* n x)) (* (* (* n x) x) n))
         (if (<= (/ 1.0 n) 50.0)
           (/ (- (/ (log x) (* n x)) (/ -1.0 x)) n)
           (- (fma (/ (fma (- (/ 0.5 n) 0.5) 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 tmp;
	if ((1.0 / n) <= -1e+233) {
		tmp = 1.0 - t_0;
	} else if ((1.0 / n) <= -1e+124) {
		tmp = 1.0 - 1.0;
	} else if ((1.0 / n) <= -4000.0) {
		tmp = ((log(x) * x) + (n * x)) / (((n * x) * x) * n);
	} else if ((1.0 / n) <= 50.0) {
		tmp = ((log(x) / (n * x)) - (-1.0 / x)) / n;
	} else {
		tmp = fma((fma(((0.5 / n) - 0.5), x, 1.0) / n), x, 1.0) - t_0;
	}
	return tmp;
}

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -1e+233)
		tmp = Float64(1.0 - t_0);
	elseif (Float64(1.0 / n) <= -1e+124)
		tmp = Float64(1.0 - 1.0);
	elseif (Float64(1.0 / n) <= -4000.0)
		tmp = Float64(Float64(Float64(log(x) * x) + Float64(n * x)) / Float64(Float64(Float64(n * x) * x) * n));
	elseif (Float64(1.0 / n) <= 50.0)
		tmp = Float64(Float64(Float64(log(x) / Float64(n * x)) - Float64(-1.0 / x)) / n);
	else
		tmp = Float64(fma(Float64(fma(Float64(Float64(0.5 / n) - 0.5), x, 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]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -1e+233], N[(1.0 - t$95$0), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], -1e+124], N[(1.0 - 1.0), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], -4000.0], N[(N[(N[(N[Log[x], $MachinePrecision] * x), $MachinePrecision] + N[(n * x), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(n * x), $MachinePrecision] * x), $MachinePrecision] * n), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 50.0], N[(N[(N[(N[Log[x], $MachinePrecision] / N[(n * x), $MachinePrecision]), $MachinePrecision] - N[(-1.0 / x), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], N[(N[(N[(N[(N[(N[(0.5 / n), $MachinePrecision] - 0.5), $MachinePrecision] * x + 1.0), $MachinePrecision] / n), $MachinePrecision] * x + 1.0), $MachinePrecision] - t$95$0), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -9.99999999999999974e232

   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 70.5%

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

   if -9.99999999999999974e232 < (/.f64 #s(literal 1 binary64) n) < -9.99999999999999948e123

   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 16.3%

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

   6. Taylor expanded in n around inf

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

   8. Applied rewrites 86.6%

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

   if -9.99999999999999948e123 < (/.f64 #s(literal 1 binary64) n) < -4e3

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in n around -inf

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

   8. Applied rewrites 29.4%

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

   10. Applied rewrites 68.9%

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

   if -4e3 < (/.f64 #s(literal 1 binary64) n) < 50

   1. Initial program 34.9%

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

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 59.3%

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

   6. Taylor expanded in n around -inf

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

   8. Applied rewrites 57.9%

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

   if 50 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 51.1%

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

   3. Taylor expanded in n around 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%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 76.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{0.5}{n} - 0.5, x, 1\right)}{n}, x, 1\right)} - {x}^{\left(\frac{1}{n}\right)} \]
3. Recombined 5 regimes into one program.

4. Final simplification 66.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{+233}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;\frac{1}{n} \leq -1 \cdot 10^{+124}:\\ \;\;\;\;1 - 1\\ \mathbf{elif}\;\frac{1}{n} \leq -4000:\\ \;\;\;\;\frac{\log x \cdot x + n \cdot x}{\left(\left(n \cdot x\right) \cdot x\right) \cdot n}\\ \mathbf{elif}\;\frac{1}{n} \leq 50:\\ \;\;\;\;\frac{\frac{\log x}{n \cdot x} - \frac{-1}{x}}{n}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{0.5}{n} - 0.5, x, 1\right)}{n}, x, 1\right) - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 53.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{+233}:\\ \;\;\;\;1 - t\_0\\ \mathbf{elif}\;\frac{1}{n} \leq -1 \cdot 10^{+124}:\\ \;\;\;\;1 - 1\\ \mathbf{elif}\;\frac{1}{n} \leq -4000:\\ \;\;\;\;\frac{\log x \cdot x + n \cdot x}{\left(\left(n \cdot x\right) \cdot x\right) \cdot n}\\ \mathbf{elif}\;\frac{1}{n} \leq 50:\\ \;\;\;\;\frac{\frac{\log x}{n \cdot x} - \frac{-1}{x}}{n}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\frac{0.5}{n} - 0.5\right) \cdot \frac{x}{n}, x, 1\right) - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -1e+233)
     (- 1.0 t_0)
     (if (<= (/ 1.0 n) -1e+124)
       (- 1.0 1.0)
       (if (<= (/ 1.0 n) -4000.0)
         (/ (+ (* (log x) x) (* n x)) (* (* (* n x) x) n))
         (if (<= (/ 1.0 n) 50.0)
           (/ (- (/ (log x) (* n x)) (/ -1.0 x)) n)
           (- (fma (* (- (/ 0.5 n) 0.5) (/ x n)) x 1.0) t_0)))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -1e+233) {
		tmp = 1.0 - t_0;
	} else if ((1.0 / n) <= -1e+124) {
		tmp = 1.0 - 1.0;
	} else if ((1.0 / n) <= -4000.0) {
		tmp = ((log(x) * x) + (n * x)) / (((n * x) * x) * n);
	} else if ((1.0 / n) <= 50.0) {
		tmp = ((log(x) / (n * x)) - (-1.0 / x)) / n;
	} else {
		tmp = fma((((0.5 / n) - 0.5) * (x / n)), x, 1.0) - t_0;
	}
	return tmp;
}

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -1e+233)
		tmp = Float64(1.0 - t_0);
	elseif (Float64(1.0 / n) <= -1e+124)
		tmp = Float64(1.0 - 1.0);
	elseif (Float64(1.0 / n) <= -4000.0)
		tmp = Float64(Float64(Float64(log(x) * x) + Float64(n * x)) / Float64(Float64(Float64(n * x) * x) * n));
	elseif (Float64(1.0 / n) <= 50.0)
		tmp = Float64(Float64(Float64(log(x) / Float64(n * x)) - Float64(-1.0 / x)) / n);
	else
		tmp = Float64(fma(Float64(Float64(Float64(0.5 / n) - 0.5) * Float64(x / 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]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -1e+233], N[(1.0 - t$95$0), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], -1e+124], N[(1.0 - 1.0), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], -4000.0], N[(N[(N[(N[Log[x], $MachinePrecision] * x), $MachinePrecision] + N[(n * x), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(n * x), $MachinePrecision] * x), $MachinePrecision] * n), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 50.0], N[(N[(N[(N[Log[x], $MachinePrecision] / N[(n * x), $MachinePrecision]), $MachinePrecision] - N[(-1.0 / x), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], N[(N[(N[(N[(N[(0.5 / n), $MachinePrecision] - 0.5), $MachinePrecision] * N[(x / n), $MachinePrecision]), $MachinePrecision] * x + 1.0), $MachinePrecision] - t$95$0), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -9.99999999999999974e232

   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 70.5%

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

   if -9.99999999999999974e232 < (/.f64 #s(literal 1 binary64) n) < -9.99999999999999948e123

   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 16.3%

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

   6. Taylor expanded in n around inf

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

   8. Applied rewrites 86.6%

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

   if -9.99999999999999948e123 < (/.f64 #s(literal 1 binary64) n) < -4e3

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in n around -inf

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

   8. Applied rewrites 29.4%

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

   10. Applied rewrites 68.9%

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

   if -4e3 < (/.f64 #s(literal 1 binary64) n) < 50

   1. Initial program 34.9%

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

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 59.3%

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

   6. Taylor expanded in n around -inf

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

   8. Applied rewrites 57.9%

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

   if 50 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 51.1%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 70.8%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 75.1%

      \[\leadsto \mathsf{fma}\left(\left(\frac{0.5}{n} - 0.5\right) \cdot \frac{x}{n}, x, 1\right) - {x}^{\left(\frac{1}{n}\right)} \]
3. Recombined 5 regimes into one program.

4. Final simplification 66.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{+233}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;\frac{1}{n} \leq -1 \cdot 10^{+124}:\\ \;\;\;\;1 - 1\\ \mathbf{elif}\;\frac{1}{n} \leq -4000:\\ \;\;\;\;\frac{\log x \cdot x + n \cdot x}{\left(\left(n \cdot x\right) \cdot x\right) \cdot n}\\ \mathbf{elif}\;\frac{1}{n} \leq 50:\\ \;\;\;\;\frac{\frac{\log x}{n \cdot x} - \frac{-1}{x}}{n}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\frac{0.5}{n} - 0.5\right) \cdot \frac{x}{n}, x, 1\right) - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 53.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{+233}:\\ \;\;\;\;1 - t\_0\\ \mathbf{elif}\;\frac{1}{n} \leq -1 \cdot 10^{+124}:\\ \;\;\;\;1 - 1\\ \mathbf{elif}\;\frac{1}{n} \leq -4000:\\ \;\;\;\;\frac{\log x \cdot x + n \cdot x}{\left(\left(n \cdot x\right) \cdot x\right) \cdot n}\\ \mathbf{elif}\;\frac{1}{n} \leq 50:\\ \;\;\;\;\frac{\frac{\log x}{n \cdot x} - \frac{-1}{x}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{+188}:\\ \;\;\;\;\left(\frac{x}{n} - -1\right) - t\_0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{n}{n \cdot n}, x, 1\right) - 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -1e+233)
     (- 1.0 t_0)
     (if (<= (/ 1.0 n) -1e+124)
       (- 1.0 1.0)
       (if (<= (/ 1.0 n) -4000.0)
         (/ (+ (* (log x) x) (* n x)) (* (* (* n x) x) n))
         (if (<= (/ 1.0 n) 50.0)
           (/ (- (/ (log x) (* n x)) (/ -1.0 x)) n)
           (if (<= (/ 1.0 n) 1e+188)
             (- (- (/ x n) -1.0) t_0)
             (- (fma (/ n (* n n)) x 1.0) 1.0))))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -1e+233) {
		tmp = 1.0 - t_0;
	} else if ((1.0 / n) <= -1e+124) {
		tmp = 1.0 - 1.0;
	} else if ((1.0 / n) <= -4000.0) {
		tmp = ((log(x) * x) + (n * x)) / (((n * x) * x) * n);
	} else if ((1.0 / n) <= 50.0) {
		tmp = ((log(x) / (n * x)) - (-1.0 / x)) / n;
	} else if ((1.0 / n) <= 1e+188) {
		tmp = ((x / n) - -1.0) - t_0;
	} else {
		tmp = fma((n / (n * n)), x, 1.0) - 1.0;
	}
	return tmp;
}

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -1e+233)
		tmp = Float64(1.0 - t_0);
	elseif (Float64(1.0 / n) <= -1e+124)
		tmp = Float64(1.0 - 1.0);
	elseif (Float64(1.0 / n) <= -4000.0)
		tmp = Float64(Float64(Float64(log(x) * x) + Float64(n * x)) / Float64(Float64(Float64(n * x) * x) * n));
	elseif (Float64(1.0 / n) <= 50.0)
		tmp = Float64(Float64(Float64(log(x) / Float64(n * x)) - Float64(-1.0 / x)) / n);
	elseif (Float64(1.0 / n) <= 1e+188)
		tmp = Float64(Float64(Float64(x / n) - -1.0) - t_0);
	else
		tmp = Float64(fma(Float64(n / Float64(n * n)), x, 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]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -1e+233], N[(1.0 - t$95$0), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], -1e+124], N[(1.0 - 1.0), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], -4000.0], N[(N[(N[(N[Log[x], $MachinePrecision] * x), $MachinePrecision] + N[(n * x), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(n * x), $MachinePrecision] * x), $MachinePrecision] * n), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 50.0], N[(N[(N[(N[Log[x], $MachinePrecision] / N[(n * x), $MachinePrecision]), $MachinePrecision] - N[(-1.0 / x), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 1e+188], N[(N[(N[(x / n), $MachinePrecision] - -1.0), $MachinePrecision] - t$95$0), $MachinePrecision], N[(N[(N[(n / N[(n * n), $MachinePrecision]), $MachinePrecision] * x + 1.0), $MachinePrecision] - 1.0), $MachinePrecision]]]]]]]

Derivation

1. Split input into 6 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -9.99999999999999974e232

   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 70.5%

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

   if -9.99999999999999974e232 < (/.f64 #s(literal 1 binary64) n) < -9.99999999999999948e123

   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 16.3%

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

   6. Taylor expanded in n around inf

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

   8. Applied rewrites 86.6%

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

   if -9.99999999999999948e123 < (/.f64 #s(literal 1 binary64) n) < -4e3

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in n around -inf

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

   8. Applied rewrites 29.4%

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

   10. Applied rewrites 68.9%

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

   if -4e3 < (/.f64 #s(literal 1 binary64) n) < 50

   1. Initial program 34.9%

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

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 59.3%

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

   6. Taylor expanded in n around -inf

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

   8. Applied rewrites 57.9%

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

   if 50 < (/.f64 #s(literal 1 binary64) n) < 1e188

   1. Initial program 72.7%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 72.1%

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

   if 1e188 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 13.9%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 89.2%

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

   6. Taylor expanded in n around 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 89.2%

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

   9. Taylor expanded in n around 0

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

   11. Applied rewrites 89.2%

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

   12. Taylor expanded in x around 0

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

   14. Applied rewrites 89.2%

       \[\leadsto \mathsf{fma}\left(\frac{n}{n \cdot n}, x, 1\right) - 1 \]
3. Recombined 6 regimes into one program.

4. Final simplification 66.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{+233}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;\frac{1}{n} \leq -1 \cdot 10^{+124}:\\ \;\;\;\;1 - 1\\ \mathbf{elif}\;\frac{1}{n} \leq -4000:\\ \;\;\;\;\frac{\log x \cdot x + n \cdot x}{\left(\left(n \cdot x\right) \cdot x\right) \cdot n}\\ \mathbf{elif}\;\frac{1}{n} \leq 50:\\ \;\;\;\;\frac{\frac{\log x}{n \cdot x} - \frac{-1}{x}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{+188}:\\ \;\;\;\;\left(\frac{x}{n} - -1\right) - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{n}{n \cdot n}, x, 1\right) - 1\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 52.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{+233}:\\ \;\;\;\;1 - t\_0\\ \mathbf{elif}\;\frac{1}{n} \leq -4000:\\ \;\;\;\;1 - 1\\ \mathbf{elif}\;\frac{1}{n} \leq 50:\\ \;\;\;\;\frac{\frac{1}{n}}{x}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{+188}:\\ \;\;\;\;\left(\frac{x}{n} - -1\right) - t\_0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{n}{n \cdot n}, x, 1\right) - 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -1e+233)
     (- 1.0 t_0)
     (if (<= (/ 1.0 n) -4000.0)
       (- 1.0 1.0)
       (if (<= (/ 1.0 n) 50.0)
         (/ (/ 1.0 n) x)
         (if (<= (/ 1.0 n) 1e+188)
           (- (- (/ x n) -1.0) t_0)
           (- (fma (/ n (* n n)) x 1.0) 1.0)))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -1e+233) {
		tmp = 1.0 - t_0;
	} else if ((1.0 / n) <= -4000.0) {
		tmp = 1.0 - 1.0;
	} else if ((1.0 / n) <= 50.0) {
		tmp = (1.0 / n) / x;
	} else if ((1.0 / n) <= 1e+188) {
		tmp = ((x / n) - -1.0) - t_0;
	} else {
		tmp = fma((n / (n * n)), x, 1.0) - 1.0;
	}
	return tmp;
}

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -1e+233)
		tmp = Float64(1.0 - t_0);
	elseif (Float64(1.0 / n) <= -4000.0)
		tmp = Float64(1.0 - 1.0);
	elseif (Float64(1.0 / n) <= 50.0)
		tmp = Float64(Float64(1.0 / n) / x);
	elseif (Float64(1.0 / n) <= 1e+188)
		tmp = Float64(Float64(Float64(x / n) - -1.0) - t_0);
	else
		tmp = Float64(fma(Float64(n / Float64(n * n)), x, 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]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -1e+233], N[(1.0 - t$95$0), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], -4000.0], N[(1.0 - 1.0), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 50.0], N[(N[(1.0 / n), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 1e+188], N[(N[(N[(x / n), $MachinePrecision] - -1.0), $MachinePrecision] - t$95$0), $MachinePrecision], N[(N[(N[(n / N[(n * n), $MachinePrecision]), $MachinePrecision] * x + 1.0), $MachinePrecision] - 1.0), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -9.99999999999999974e232

   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 70.5%

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

   if -9.99999999999999974e232 < (/.f64 #s(literal 1 binary64) n) < -4e3

   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 36.0%

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

   6. Taylor expanded in n around inf

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

   8. Applied rewrites 66.6%

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

   if -4e3 < (/.f64 #s(literal 1 binary64) n) < 50

   1. Initial program 34.9%

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

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 59.3%

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

   6. Taylor expanded in n around -inf

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

   8. Applied rewrites 57.9%

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

   9. Taylor expanded in n around inf

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

   11. Applied rewrites 57.3%

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

   if 50 < (/.f64 #s(literal 1 binary64) n) < 1e188

   1. Initial program 72.7%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 72.1%

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

   if 1e188 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 13.9%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 89.2%

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

   6. Taylor expanded in n around 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 89.2%

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

   9. Taylor expanded in n around 0

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

   11. Applied rewrites 89.2%

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

   12. Taylor expanded in x around 0

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

   14. Applied rewrites 89.2%

       \[\leadsto \mathsf{fma}\left(\frac{n}{n \cdot n}, x, 1\right) - 1 \]
3. Recombined 5 regimes into one program.
4. Add Preprocessing

Alternative 10: 52.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{+233}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\frac{1}{n} \leq -4000:\\ \;\;\;\;1 - 1\\ \mathbf{elif}\;\frac{1}{n} \leq 50:\\ \;\;\;\;\frac{\frac{1}{n}}{x}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{+188}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{n}{n \cdot n}, x, 1\right) - 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (- 1.0 (pow x (/ 1.0 n)))))
   (if (<= (/ 1.0 n) -1e+233)
     t_0
     (if (<= (/ 1.0 n) -4000.0)
       (- 1.0 1.0)
       (if (<= (/ 1.0 n) 50.0)
         (/ (/ 1.0 n) x)
         (if (<= (/ 1.0 n) 1e+188) t_0 (- (fma (/ n (* n n)) x 1.0) 1.0)))))))

C

double code(double x, double n) {
	double t_0 = 1.0 - pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -1e+233) {
		tmp = t_0;
	} else if ((1.0 / n) <= -4000.0) {
		tmp = 1.0 - 1.0;
	} else if ((1.0 / n) <= 50.0) {
		tmp = (1.0 / n) / x;
	} else if ((1.0 / n) <= 1e+188) {
		tmp = t_0;
	} else {
		tmp = fma((n / (n * n)), x, 1.0) - 1.0;
	}
	return tmp;
}

Julia

function code(x, n)
	t_0 = Float64(1.0 - (x ^ Float64(1.0 / n)))
	tmp = 0.0
	if (Float64(1.0 / n) <= -1e+233)
		tmp = t_0;
	elseif (Float64(1.0 / n) <= -4000.0)
		tmp = Float64(1.0 - 1.0);
	elseif (Float64(1.0 / n) <= 50.0)
		tmp = Float64(Float64(1.0 / n) / x);
	elseif (Float64(1.0 / n) <= 1e+188)
		tmp = t_0;
	else
		tmp = Float64(fma(Float64(n / Float64(n * n)), x, 1.0) - 1.0);
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(1.0 - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -1e+233], t$95$0, If[LessEqual[N[(1.0 / n), $MachinePrecision], -4000.0], N[(1.0 - 1.0), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 50.0], N[(N[(1.0 / n), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 1e+188], t$95$0, N[(N[(N[(n / N[(n * n), $MachinePrecision]), $MachinePrecision] * x + 1.0), $MachinePrecision] - 1.0), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -9.99999999999999974e232 or 50 < (/.f64 #s(literal 1 binary64) n) < 1e188

   1. Initial program 84.3%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   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 70.0%

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

   if -9.99999999999999974e232 < (/.f64 #s(literal 1 binary64) n) < -4e3

   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 36.0%

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

   6. Taylor expanded in n around inf

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

   8. Applied rewrites 66.6%

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

   if -4e3 < (/.f64 #s(literal 1 binary64) n) < 50

   1. Initial program 34.9%

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

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 59.3%

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

   6. Taylor expanded in n around -inf

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

   8. Applied rewrites 57.9%

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

   9. Taylor expanded in n around inf

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

   11. Applied rewrites 57.3%

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

   if 1e188 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 13.9%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 89.2%

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

   6. Taylor expanded in n around 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 89.2%

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

   9. Taylor expanded in n around 0

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

   11. Applied rewrites 89.2%

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

   12. Taylor expanded in x around 0

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

   14. Applied rewrites 89.2%

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

Alternative 11: 53.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -200:\\ \;\;\;\;\frac{\frac{\log x}{n} + 1}{n \cdot x}\\ \mathbf{elif}\;n \leq -4.6 \cdot 10^{-117}:\\ \;\;\;\;\frac{\log x \cdot x + n \cdot x}{\left(\left(n \cdot x\right) \cdot x\right) \cdot n}\\ \mathbf{elif}\;n \leq -1 \cdot 10^{-227}:\\ \;\;\;\;1 - 1\\ \mathbf{elif}\;n \leq 3.5 \cdot 10^{-190}:\\ \;\;\;\;\mathsf{fma}\left(\frac{n}{n \cdot n}, x, 1\right) - 1\\ \mathbf{elif}\;n \leq 0.03:\\ \;\;\;\;\left(\frac{x}{n} - -1\right) - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{n}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= n -200.0)
   (/ (+ (/ (log x) n) 1.0) (* n x))
   (if (<= n -4.6e-117)
     (/ (+ (* (log x) x) (* n x)) (* (* (* n x) x) n))
     (if (<= n -1e-227)
       (- 1.0 1.0)
       (if (<= n 3.5e-190)
         (- (fma (/ n (* n n)) x 1.0) 1.0)
         (if (<= n 0.03)
           (- (- (/ x n) -1.0) (pow x (/ 1.0 n)))
           (/ (/ 1.0 n) x)))))))

C

double code(double x, double n) {
	double tmp;
	if (n <= -200.0) {
		tmp = ((log(x) / n) + 1.0) / (n * x);
	} else if (n <= -4.6e-117) {
		tmp = ((log(x) * x) + (n * x)) / (((n * x) * x) * n);
	} else if (n <= -1e-227) {
		tmp = 1.0 - 1.0;
	} else if (n <= 3.5e-190) {
		tmp = fma((n / (n * n)), x, 1.0) - 1.0;
	} else if (n <= 0.03) {
		tmp = ((x / n) - -1.0) - pow(x, (1.0 / n));
	} else {
		tmp = (1.0 / n) / x;
	}
	return tmp;
}

Julia

function code(x, n)
	tmp = 0.0
	if (n <= -200.0)
		tmp = Float64(Float64(Float64(log(x) / n) + 1.0) / Float64(n * x));
	elseif (n <= -4.6e-117)
		tmp = Float64(Float64(Float64(log(x) * x) + Float64(n * x)) / Float64(Float64(Float64(n * x) * x) * n));
	elseif (n <= -1e-227)
		tmp = Float64(1.0 - 1.0);
	elseif (n <= 3.5e-190)
		tmp = Float64(fma(Float64(n / Float64(n * n)), x, 1.0) - 1.0);
	elseif (n <= 0.03)
		tmp = Float64(Float64(Float64(x / n) - -1.0) - (x ^ Float64(1.0 / n)));
	else
		tmp = Float64(Float64(1.0 / n) / x);
	end
	return tmp
end

Wolfram

code[x_, n_] := If[LessEqual[n, -200.0], N[(N[(N[(N[Log[x], $MachinePrecision] / n), $MachinePrecision] + 1.0), $MachinePrecision] / N[(n * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, -4.6e-117], N[(N[(N[(N[Log[x], $MachinePrecision] * x), $MachinePrecision] + N[(n * x), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(n * x), $MachinePrecision] * x), $MachinePrecision] * n), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, -1e-227], N[(1.0 - 1.0), $MachinePrecision], If[LessEqual[n, 3.5e-190], N[(N[(N[(n / N[(n * n), $MachinePrecision]), $MachinePrecision] * x + 1.0), $MachinePrecision] - 1.0), $MachinePrecision], If[LessEqual[n, 0.03], N[(N[(N[(x / n), $MachinePrecision] - -1.0), $MachinePrecision] - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / n), $MachinePrecision] / x), $MachinePrecision]]]]]]

Derivation

1. Split input into 6 regimes

2. if n < -200

   1. Initial program 31.7%

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

   3. Taylor expanded in x around 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 53.6%

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

   6. Taylor expanded in n around inf

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

   8. Applied rewrites 52.1%

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

   if -200 < n < -4.59999999999999989e-117

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 97.1%

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

   6. Taylor expanded in n around -inf

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

   8. Applied rewrites 28.7%

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

   10. Applied rewrites 67.0%

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

   if -4.59999999999999989e-117 < n < -9.99999999999999945e-228

   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 16.3%

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

   6. Taylor expanded in n around inf

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

   8. Applied rewrites 86.6%

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

   if -9.99999999999999945e-228 < n < 3.4999999999999999e-190

   1. Initial program 62.2%

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

   3. Taylor expanded in x around 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 39.4%

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

   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 39.4%

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

   9. Taylor expanded in n around 0

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

   11. Applied rewrites 44.3%

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

   12. Taylor expanded in x around 0

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

   14. Applied rewrites 78.5%

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

   if 3.4999999999999999e-190 < n < 0.029999999999999999

   1. Initial program 72.7%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 72.1%

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

   if 0.029999999999999999 < n

   1. Initial program 36.7%

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

   3. Taylor expanded in x around 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 64.8%

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

   6. Taylor expanded in n around -inf

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

   8. Applied rewrites 63.5%

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

   9. Taylor expanded in n around inf

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

   11. Applied rewrites 63.4%

       \[\leadsto \frac{\frac{1}{n}}{\color{blue}{x}} \]
3. Recombined 6 regimes into one program.

4. Final simplification 66.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -200:\\ \;\;\;\;\frac{\frac{\log x}{n} + 1}{n \cdot x}\\ \mathbf{elif}\;n \leq -4.6 \cdot 10^{-117}:\\ \;\;\;\;\frac{\log x \cdot x + n \cdot x}{\left(\left(n \cdot x\right) \cdot x\right) \cdot n}\\ \mathbf{elif}\;n \leq -1 \cdot 10^{-227}:\\ \;\;\;\;1 - 1\\ \mathbf{elif}\;n \leq 3.5 \cdot 10^{-190}:\\ \;\;\;\;\mathsf{fma}\left(\frac{n}{n \cdot n}, x, 1\right) - 1\\ \mathbf{elif}\;n \leq 0.03:\\ \;\;\;\;\left(\frac{x}{n} - -1\right) - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{n}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 52.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -15:\\ \;\;\;\;\frac{\frac{\log x}{n} + 1}{n \cdot x}\\ \mathbf{elif}\;n \leq -1 \cdot 10^{-227}:\\ \;\;\;\;1 - 1\\ \mathbf{elif}\;n \leq 3.5 \cdot 10^{-190}:\\ \;\;\;\;\mathsf{fma}\left(\frac{n}{n \cdot n}, x, 1\right) - 1\\ \mathbf{elif}\;n \leq 0.03:\\ \;\;\;\;\left(\frac{x}{n} - -1\right) - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{n}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= n -15.0)
   (/ (+ (/ (log x) n) 1.0) (* n x))
   (if (<= n -1e-227)
     (- 1.0 1.0)
     (if (<= n 3.5e-190)
       (- (fma (/ n (* n n)) x 1.0) 1.0)
       (if (<= n 0.03)
         (- (- (/ x n) -1.0) (pow x (/ 1.0 n)))
         (/ (/ 1.0 n) x))))))

C

double code(double x, double n) {
	double tmp;
	if (n <= -15.0) {
		tmp = ((log(x) / n) + 1.0) / (n * x);
	} else if (n <= -1e-227) {
		tmp = 1.0 - 1.0;
	} else if (n <= 3.5e-190) {
		tmp = fma((n / (n * n)), x, 1.0) - 1.0;
	} else if (n <= 0.03) {
		tmp = ((x / n) - -1.0) - pow(x, (1.0 / n));
	} else {
		tmp = (1.0 / n) / x;
	}
	return tmp;
}

Julia

function code(x, n)
	tmp = 0.0
	if (n <= -15.0)
		tmp = Float64(Float64(Float64(log(x) / n) + 1.0) / Float64(n * x));
	elseif (n <= -1e-227)
		tmp = Float64(1.0 - 1.0);
	elseif (n <= 3.5e-190)
		tmp = Float64(fma(Float64(n / Float64(n * n)), x, 1.0) - 1.0);
	elseif (n <= 0.03)
		tmp = Float64(Float64(Float64(x / n) - -1.0) - (x ^ Float64(1.0 / n)));
	else
		tmp = Float64(Float64(1.0 / n) / x);
	end
	return tmp
end

Wolfram

code[x_, n_] := If[LessEqual[n, -15.0], N[(N[(N[(N[Log[x], $MachinePrecision] / n), $MachinePrecision] + 1.0), $MachinePrecision] / N[(n * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, -1e-227], N[(1.0 - 1.0), $MachinePrecision], If[LessEqual[n, 3.5e-190], N[(N[(N[(n / N[(n * n), $MachinePrecision]), $MachinePrecision] * x + 1.0), $MachinePrecision] - 1.0), $MachinePrecision], If[LessEqual[n, 0.03], N[(N[(N[(x / n), $MachinePrecision] - -1.0), $MachinePrecision] - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / n), $MachinePrecision] / x), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if n < -15

   1. Initial program 32.8%

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

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 52.9%

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

   6. Taylor expanded in n around inf

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

   8. Applied rewrites 51.3%

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

   if -15 < n < -9.99999999999999945e-228

   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 36.0%

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

   6. Taylor expanded in n around inf

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

   8. Applied rewrites 66.6%

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

   if -9.99999999999999945e-228 < n < 3.4999999999999999e-190

   1. Initial program 62.2%

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

   3. Taylor expanded in x around 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 39.4%

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

   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 39.4%

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

   9. Taylor expanded in n around 0

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

   11. Applied rewrites 44.3%

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

   12. Taylor expanded in x around 0

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

   14. Applied rewrites 78.5%

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

   if 3.4999999999999999e-190 < n < 0.029999999999999999

   1. Initial program 72.7%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 72.1%

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

   if 0.029999999999999999 < n

   1. Initial program 36.7%

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

   3. Taylor expanded in x around 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 64.8%

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

   6. Taylor expanded in n around -inf

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

   8. Applied rewrites 63.5%

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

   9. Taylor expanded in n around inf

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

   11. Applied rewrites 63.4%

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

Alternative 13: 48.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{+233}:\\ \;\;\;\;\mathsf{fma}\left(\frac{n}{n \cdot n}, x, 1\right) - 1\\ \mathbf{elif}\;\frac{1}{n} \leq -4000:\\ \;\;\;\;1 - 1\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-5}:\\ \;\;\;\;\frac{\frac{1}{n}}{x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(-0.3333333333333333, x, \frac{\mathsf{fma}\left(0.16666666666666666, \frac{x}{n}, -0.5 \cdot x\right) - -0.5}{-n}\right)}{-n} - \frac{0.5}{n}, x, \frac{1}{n}\right), x, 1\right) - 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= (/ 1.0 n) -1e+233)
   (- (fma (/ n (* n n)) x 1.0) 1.0)
   (if (<= (/ 1.0 n) -4000.0)
     (- 1.0 1.0)
     (if (<= (/ 1.0 n) 1e-5)
       (/ (/ 1.0 n) x)
       (-
        (fma
         (fma
          (-
           (/
            (fma
             -0.3333333333333333
             x
             (/ (- (fma 0.16666666666666666 (/ x n) (* -0.5 x)) -0.5) (- n)))
            (- n))
           (/ 0.5 n))
          x
          (/ 1.0 n))
         x
         1.0)
        1.0)))))

C

double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -1e+233) {
		tmp = fma((n / (n * n)), x, 1.0) - 1.0;
	} else if ((1.0 / n) <= -4000.0) {
		tmp = 1.0 - 1.0;
	} else if ((1.0 / n) <= 1e-5) {
		tmp = (1.0 / n) / x;
	} else {
		tmp = fma(fma(((fma(-0.3333333333333333, x, ((fma(0.16666666666666666, (x / n), (-0.5 * x)) - -0.5) / -n)) / -n) - (0.5 / n)), x, (1.0 / n)), x, 1.0) - 1.0;
	}
	return tmp;
}

Julia

function code(x, n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -1e+233)
		tmp = Float64(fma(Float64(n / Float64(n * n)), x, 1.0) - 1.0);
	elseif (Float64(1.0 / n) <= -4000.0)
		tmp = Float64(1.0 - 1.0);
	elseif (Float64(1.0 / n) <= 1e-5)
		tmp = Float64(Float64(1.0 / n) / x);
	else
		tmp = Float64(fma(fma(Float64(Float64(fma(-0.3333333333333333, x, Float64(Float64(fma(0.16666666666666666, Float64(x / n), Float64(-0.5 * x)) - -0.5) / Float64(-n))) / Float64(-n)) - Float64(0.5 / n)), x, Float64(1.0 / n)), x, 1.0) - 1.0);
	end
	return tmp
end

Wolfram

code[x_, n_] := If[LessEqual[N[(1.0 / n), $MachinePrecision], -1e+233], N[(N[(N[(n / N[(n * n), $MachinePrecision]), $MachinePrecision] * x + 1.0), $MachinePrecision] - 1.0), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], -4000.0], N[(1.0 - 1.0), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 1e-5], N[(N[(1.0 / n), $MachinePrecision] / x), $MachinePrecision], N[(N[(N[(N[(N[(N[(-0.3333333333333333 * x + N[(N[(N[(0.16666666666666666 * N[(x / n), $MachinePrecision] + N[(-0.5 * x), $MachinePrecision]), $MachinePrecision] - -0.5), $MachinePrecision] / (-n)), $MachinePrecision]), $MachinePrecision] / (-n)), $MachinePrecision] - N[(0.5 / n), $MachinePrecision]), $MachinePrecision] * x + N[(1.0 / n), $MachinePrecision]), $MachinePrecision] * x + 1.0), $MachinePrecision] - 1.0), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -9.99999999999999974e232

   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

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

   5. Applied rewrites 0.5%

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

   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 0.5%

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

   9. Taylor expanded in n around 0

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

   11. Applied rewrites 9.2%

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

   12. Taylor expanded in x around 0

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

   14. Applied rewrites 70.0%

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

   if -9.99999999999999974e232 < (/.f64 #s(literal 1 binary64) n) < -4e3

   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 36.0%

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

   6. Taylor expanded in n around inf

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

   8. Applied rewrites 66.6%

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

   if -4e3 < (/.f64 #s(literal 1 binary64) n) < 1.00000000000000008e-5

   1. Initial program 35.2%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   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 59.0%

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

   6. Taylor expanded in n around -inf

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

   8. Applied rewrites 58.3%

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

   9. Taylor expanded in n around inf

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

   11. Applied rewrites 57.7%

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

   if 1.00000000000000008e-5 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 50.1%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 48.1%

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

   6. Taylor expanded in n around inf

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

   8. Applied rewrites 2.3%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 13.8%

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

   12. Taylor expanded in n around -inf

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

   14. Applied rewrites 42.7%

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\left(-\frac{\mathsf{fma}\left(-0.3333333333333333, x, -\frac{\mathsf{fma}\left(0.16666666666666666, \frac{x}{n}, -0.5 \cdot x\right) - -0.5}{n}\right)}{n}\right) - \frac{0.5}{n}, x, \frac{1}{n}\right), x, 1\right) - 1 \]
3. Recombined 4 regimes into one program.

4. Final simplification 57.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{+233}:\\ \;\;\;\;\mathsf{fma}\left(\frac{n}{n \cdot n}, x, 1\right) - 1\\ \mathbf{elif}\;\frac{1}{n} \leq -4000:\\ \;\;\;\;1 - 1\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-5}:\\ \;\;\;\;\frac{\frac{1}{n}}{x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(-0.3333333333333333, x, \frac{\mathsf{fma}\left(0.16666666666666666, \frac{x}{n}, -0.5 \cdot x\right) - -0.5}{-n}\right)}{-n} - \frac{0.5}{n}, x, \frac{1}{n}\right), x, 1\right) - 1\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 48.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -8.5:\\ \;\;\;\;\frac{1}{n \cdot x}\\ \mathbf{elif}\;n \leq -1 \cdot 10^{-227}:\\ \;\;\;\;1 - 1\\ \mathbf{elif}\;n \leq 0.5:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{x}{n}, -0.5, \mathsf{fma}\left(\frac{x}{n \cdot n}, 0.16666666666666666, \mathsf{fma}\left(0.3333333333333333, x, \frac{0.5}{n}\right)\right)\right)}{n} - \frac{0.5}{n}, x, \frac{1}{n}\right), x, 1\right) - 1\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{n}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= n -8.5)
   (/ 1.0 (* n x))
   (if (<= n -1e-227)
     (- 1.0 1.0)
     (if (<= n 0.5)
       (-
        (fma
         (fma
          (-
           (/
            (fma
             (/ x n)
             -0.5
             (fma
              (/ x (* n n))
              0.16666666666666666
              (fma 0.3333333333333333 x (/ 0.5 n))))
            n)
           (/ 0.5 n))
          x
          (/ 1.0 n))
         x
         1.0)
        1.0)
       (/ (/ 1.0 n) x)))))

C

double code(double x, double n) {
	double tmp;
	if (n <= -8.5) {
		tmp = 1.0 / (n * x);
	} else if (n <= -1e-227) {
		tmp = 1.0 - 1.0;
	} else if (n <= 0.5) {
		tmp = fma(fma(((fma((x / n), -0.5, fma((x / (n * n)), 0.16666666666666666, fma(0.3333333333333333, x, (0.5 / n)))) / n) - (0.5 / n)), x, (1.0 / n)), x, 1.0) - 1.0;
	} else {
		tmp = (1.0 / n) / x;
	}
	return tmp;
}

Julia

function code(x, n)
	tmp = 0.0
	if (n <= -8.5)
		tmp = Float64(1.0 / Float64(n * x));
	elseif (n <= -1e-227)
		tmp = Float64(1.0 - 1.0);
	elseif (n <= 0.5)
		tmp = Float64(fma(fma(Float64(Float64(fma(Float64(x / n), -0.5, fma(Float64(x / Float64(n * n)), 0.16666666666666666, fma(0.3333333333333333, x, Float64(0.5 / n)))) / n) - Float64(0.5 / n)), x, Float64(1.0 / n)), x, 1.0) - 1.0);
	else
		tmp = Float64(Float64(1.0 / n) / x);
	end
	return tmp
end

Wolfram

code[x_, n_] := If[LessEqual[n, -8.5], N[(1.0 / N[(n * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, -1e-227], N[(1.0 - 1.0), $MachinePrecision], If[LessEqual[n, 0.5], N[(N[(N[(N[(N[(N[(N[(x / n), $MachinePrecision] * -0.5 + N[(N[(x / N[(n * n), $MachinePrecision]), $MachinePrecision] * 0.16666666666666666 + N[(0.3333333333333333 * x + N[(0.5 / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision] - N[(0.5 / n), $MachinePrecision]), $MachinePrecision] * x + N[(1.0 / n), $MachinePrecision]), $MachinePrecision] * x + 1.0), $MachinePrecision] - 1.0), $MachinePrecision], N[(N[(1.0 / n), $MachinePrecision] / x), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if n < -8.5

   1. Initial program 32.8%

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

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 52.9%

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

   6. Taylor expanded in n around inf

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

   8. Applied rewrites 50.2%

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

   if -8.5 < n < -9.99999999999999945e-228

   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 36.0%

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

   6. Taylor expanded in n around inf

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

   8. Applied rewrites 66.6%

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

   if -9.99999999999999945e-228 < n < 0.5

   1. Initial program 65.8%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 55.2%

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

   6. Taylor expanded in n around inf

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

   8. Applied rewrites 11.5%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 9.5%

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

   12. Taylor expanded in n around inf

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

   14. Applied rewrites 51.3%

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

   if 0.5 < n

   1. Initial program 37.3%

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

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 64.3%

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

   6. Taylor expanded in n around -inf

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

   8. Applied rewrites 64.4%

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

   9. Taylor expanded in n around inf

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

   11. Applied rewrites 64.2%

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

Alternative 15: 46.3% accurate, 4.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{+233}:\\ \;\;\;\;\mathsf{fma}\left(\frac{n}{n \cdot n}, x, 1\right) - 1\\ \mathbf{elif}\;\frac{1}{n} \leq -4000:\\ \;\;\;\;1 - 1\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{n}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= (/ 1.0 n) -1e+233)
   (- (fma (/ n (* n n)) x 1.0) 1.0)
   (if (<= (/ 1.0 n) -4000.0) (- 1.0 1.0) (/ (/ 1.0 n) x))))

C

double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -1e+233) {
		tmp = fma((n / (n * n)), x, 1.0) - 1.0;
	} else if ((1.0 / n) <= -4000.0) {
		tmp = 1.0 - 1.0;
	} else {
		tmp = (1.0 / n) / x;
	}
	return tmp;
}

Julia

function code(x, n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -1e+233)
		tmp = Float64(fma(Float64(n / Float64(n * n)), x, 1.0) - 1.0);
	elseif (Float64(1.0 / n) <= -4000.0)
		tmp = Float64(1.0 - 1.0);
	else
		tmp = Float64(Float64(1.0 / n) / x);
	end
	return tmp
end

Wolfram

code[x_, n_] := If[LessEqual[N[(1.0 / n), $MachinePrecision], -1e+233], N[(N[(N[(n / N[(n * n), $MachinePrecision]), $MachinePrecision] * x + 1.0), $MachinePrecision] - 1.0), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], -4000.0], N[(1.0 - 1.0), $MachinePrecision], N[(N[(1.0 / n), $MachinePrecision] / x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -9.99999999999999974e232

   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

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

   5. Applied rewrites 0.5%

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

   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 0.5%

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

   9. Taylor expanded in n around 0

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

   11. Applied rewrites 9.2%

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

   12. Taylor expanded in x around 0

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

   14. Applied rewrites 70.0%

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

   if -9.99999999999999974e232 < (/.f64 #s(literal 1 binary64) n) < -4e3

   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 36.0%

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

   6. Taylor expanded in n around inf

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

   8. Applied rewrites 66.6%

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

   if -4e3 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 39.3%

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

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 43.5%

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

   6. Taylor expanded in n around -inf

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

   8. Applied rewrites 42.2%

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

   9. Taylor expanded in n around inf

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

   11. Applied rewrites 50.9%

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

Alternative 16: 45.8% accurate, 6.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -8.5:\\ \;\;\;\;\frac{1}{n \cdot x}\\ \mathbf{elif}\;n \leq -8.8 \cdot 10^{-230}:\\ \;\;\;\;1 - 1\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{n}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= n -8.5)
   (/ 1.0 (* n x))
   (if (<= n -8.8e-230) (- 1.0 1.0) (/ (/ 1.0 n) x))))

C

double code(double x, double n) {
	double tmp;
	if (n <= -8.5) {
		tmp = 1.0 / (n * x);
	} else if (n <= -8.8e-230) {
		tmp = 1.0 - 1.0;
	} else {
		tmp = (1.0 / n) / x;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, n)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if (n <= (-8.5d0)) then
        tmp = 1.0d0 / (n * x)
    else if (n <= (-8.8d-230)) then
        tmp = 1.0d0 - 1.0d0
    else
        tmp = (1.0d0 / n) / x
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (n <= -8.5) {
		tmp = 1.0 / (n * x);
	} else if (n <= -8.8e-230) {
		tmp = 1.0 - 1.0;
	} else {
		tmp = (1.0 / n) / x;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if n <= -8.5:
		tmp = 1.0 / (n * x)
	elif n <= -8.8e-230:
		tmp = 1.0 - 1.0
	else:
		tmp = (1.0 / n) / x
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (n <= -8.5)
		tmp = Float64(1.0 / Float64(n * x));
	elseif (n <= -8.8e-230)
		tmp = Float64(1.0 - 1.0);
	else
		tmp = Float64(Float64(1.0 / n) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (n <= -8.5)
		tmp = 1.0 / (n * x);
	elseif (n <= -8.8e-230)
		tmp = 1.0 - 1.0;
	else
		tmp = (1.0 / n) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[n, -8.5], N[(1.0 / N[(n * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, -8.8e-230], N[(1.0 - 1.0), $MachinePrecision], N[(N[(1.0 / n), $MachinePrecision] / x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < -8.5

   1. Initial program 32.8%

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

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 52.9%

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

   6. Taylor expanded in n around inf

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

   8. Applied rewrites 50.2%

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

   if -8.5 < n < -8.79999999999999922e-230

   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 36.0%

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

   6. Taylor expanded in n around inf

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

   8. Applied rewrites 66.6%

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

   if -8.79999999999999922e-230 < n

   1. Initial program 51.8%

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

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 48.7%

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

   6. Taylor expanded in n around -inf

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

   8. Applied rewrites 43.0%

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

   9. Taylor expanded in n around inf

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

   11. Applied rewrites 51.8%

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

Alternative 17: 45.5% accurate, 8.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -8.5 \lor \neg \left(n \leq -8.8 \cdot 10^{-230}\right):\\ \;\;\;\;\frac{1}{n \cdot x}\\ \mathbf{else}:\\ \;\;\;\;1 - 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (or (<= n -8.5) (not (<= n -8.8e-230))) (/ 1.0 (* n x)) (- 1.0 1.0)))

C

double code(double x, double n) {
	double tmp;
	if ((n <= -8.5) || !(n <= -8.8e-230)) {
		tmp = 1.0 / (n * x);
	} else {
		tmp = 1.0 - 1.0;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, n)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if ((n <= (-8.5d0)) .or. (.not. (n <= (-8.8d-230)))) then
        tmp = 1.0d0 / (n * x)
    else
        tmp = 1.0d0 - 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if ((n <= -8.5) || !(n <= -8.8e-230)) {
		tmp = 1.0 / (n * x);
	} else {
		tmp = 1.0 - 1.0;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if (n <= -8.5) or not (n <= -8.8e-230):
		tmp = 1.0 / (n * x)
	else:
		tmp = 1.0 - 1.0
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if ((n <= -8.5) || !(n <= -8.8e-230))
		tmp = Float64(1.0 / Float64(n * x));
	else
		tmp = Float64(1.0 - 1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if ((n <= -8.5) || ~((n <= -8.8e-230)))
		tmp = 1.0 / (n * x);
	else
		tmp = 1.0 - 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[Or[LessEqual[n, -8.5], N[Not[LessEqual[n, -8.8e-230]], $MachinePrecision]], N[(1.0 / N[(n * x), $MachinePrecision]), $MachinePrecision], N[(1.0 - 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < -8.5 or -8.79999999999999922e-230 < n

   1. Initial program 46.2%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   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 49.9%

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

   6. Taylor expanded in n around inf

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

   8. Applied rewrites 51.0%

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

   if -8.5 < n < -8.79999999999999922e-230

   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 36.0%

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

   6. Taylor expanded in n around inf

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

   8. Applied rewrites 66.6%

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

4. Final simplification 54.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -8.5 \lor \neg \left(n \leq -8.8 \cdot 10^{-230}\right):\\ \;\;\;\;\frac{1}{n \cdot x}\\ \mathbf{else}:\\ \;\;\;\;1 - 1\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 30.5% accurate, 57.8× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 57.3%

   \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
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 41.0%

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

6. Taylor expanded in n around inf

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

8. Applied rewrites 34.1%

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

Reproduce

herbie shell --seed 2025026 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  :precision binary64
  (- (pow (+ x 1.0) (/ 1.0 n)) (pow x (/ 1.0 n))))