2nthrt (problem 3.4.6)

Percentage Accurate: 53.3% → 85.5%

Time: 28.7s

Alternatives: 22

Speedup: 1.9×

• could not determine a ground truth

  1. x = -1.2577864472700774e-116
  2. n = 2.503533927747953e-219

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

real(8) function code(x, n)
    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: 53.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, n)
    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: 85.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{-70}:\\ \;\;\;\;\frac{t\_0}{x \cdot n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-42}:\\ \;\;\;\;\frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-25}:\\ \;\;\;\;\frac{1}{x \cdot n}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{x}{n}} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -5e-70)
     (/ t_0 (* x n))
     (if (<= (/ 1.0 n) 2e-42)
       (/ (log (/ (+ x 1.0) x)) n)
       (if (<= (/ 1.0 n) 5e-25) (/ 1.0 (* x n)) (- (exp (/ x n)) t_0))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -5e-70) {
		tmp = t_0 / (x * n);
	} else if ((1.0 / n) <= 2e-42) {
		tmp = log(((x + 1.0) / x)) / n;
	} else if ((1.0 / n) <= 5e-25) {
		tmp = 1.0 / (x * n);
	} else {
		tmp = exp((x / n)) - t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x ** (1.0d0 / n)
    if ((1.0d0 / n) <= (-5d-70)) then
        tmp = t_0 / (x * n)
    else if ((1.0d0 / n) <= 2d-42) then
        tmp = log(((x + 1.0d0) / x)) / n
    else if ((1.0d0 / n) <= 5d-25) then
        tmp = 1.0d0 / (x * n)
    else
        tmp = exp((x / n)) - t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -5e-70) {
		tmp = t_0 / (x * n);
	} else if ((1.0 / n) <= 2e-42) {
		tmp = Math.log(((x + 1.0) / x)) / n;
	} else if ((1.0 / n) <= 5e-25) {
		tmp = 1.0 / (x * n);
	} else {
		tmp = Math.exp((x / n)) - t_0;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	tmp = 0
	if (1.0 / n) <= -5e-70:
		tmp = t_0 / (x * n)
	elif (1.0 / n) <= 2e-42:
		tmp = math.log(((x + 1.0) / x)) / n
	elif (1.0 / n) <= 5e-25:
		tmp = 1.0 / (x * n)
	else:
		tmp = math.exp((x / n)) - t_0
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -5e-70)
		tmp = Float64(t_0 / Float64(x * n));
	elseif (Float64(1.0 / n) <= 2e-42)
		tmp = Float64(log(Float64(Float64(x + 1.0) / x)) / n);
	elseif (Float64(1.0 / n) <= 5e-25)
		tmp = Float64(1.0 / Float64(x * n));
	else
		tmp = Float64(exp(Float64(x / n)) - t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = x ^ (1.0 / n);
	tmp = 0.0;
	if ((1.0 / n) <= -5e-70)
		tmp = t_0 / (x * n);
	elseif ((1.0 / n) <= 2e-42)
		tmp = log(((x + 1.0) / x)) / n;
	elseif ((1.0 / n) <= 5e-25)
		tmp = 1.0 / (x * n);
	else
		tmp = exp((x / n)) - t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -5e-70], N[(t$95$0 / N[(x * n), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-42], N[(N[Log[N[(N[(x + 1.0), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e-25], N[(1.0 / N[(x * n), $MachinePrecision]), $MachinePrecision], N[(N[Exp[N[(x / n), $MachinePrecision]], $MachinePrecision] - t$95$0), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -4.9999999999999998e-70

   1. Initial program 83.1%

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

      1. lower-/.f64 N/A

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

      2. log-rec N/A

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

      3. mul-1-neg N/A

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

      4. associate-*r/ N/A

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

      5. associate-*r* N/A

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

      6. metadata-eval N/A

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

      9. exp-to-pow N/A

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

      10. lower-pow.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 95.2

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

   5. Applied rewrites 95.2%

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

   if -4.9999999999999998e-70 < (/.f64 #s(literal 1 binary64) n) < 2.00000000000000008e-42

   1. Initial program 30.8%

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

   3. Taylor expanded in n around -inf

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

   5. Applied rewrites 84.2%

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

   7. Applied rewrites 64.4%

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

   8. Taylor expanded in n around inf

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

   10. Applied rewrites 84.4%

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

   12. Applied rewrites 84.4%

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

   if 2.00000000000000008e-42 < (/.f64 #s(literal 1 binary64) n) < 4.99999999999999962e-25

   1. Initial program 5.8%

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

   3. Taylor expanded in n around -inf

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

   5. Applied rewrites 5.8%

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

   7. Applied rewrites 1.5%

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

   8. Taylor expanded in n around inf

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

   10. Applied rewrites 5.8%

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

   11. Taylor expanded in x around inf

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

   13. Applied rewrites 99.7%

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

   if 4.99999999999999962e-25 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 45.4%

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

      1. lift-pow.f64 N/A

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

      2. pow-to-exp N/A

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

      3. lower-exp.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-log1p.f64 93.4

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

   4. Applied rewrites 93.4%

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

   5. Taylor expanded in x around 0

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

      1. lower-/.f64 93.4

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

   7. Applied rewrites 93.4%

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

Alternative 2: 86.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 40:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(0.5, {\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}, \frac{0.16666666666666666 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{3} - {\log x}^{3}\right)}{n}\right)}{n} + \left(\mathsf{log1p}\left(x\right) - \log x\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{x \cdot n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 40.0)
   (/
    (+
     (/
      (fma
       0.5
       (- (pow (log1p x) 2.0) (pow (log x) 2.0))
       (/ (* 0.16666666666666666 (- (pow (log1p x) 3.0) (pow (log x) 3.0))) n))
      n)
     (- (log1p x) (log x)))
    n)
   (/ (pow x (/ 1.0 n)) (* x n))))

C

double code(double x, double n) {
	double tmp;
	if (x <= 40.0) {
		tmp = ((fma(0.5, (pow(log1p(x), 2.0) - pow(log(x), 2.0)), ((0.16666666666666666 * (pow(log1p(x), 3.0) - pow(log(x), 3.0))) / n)) / n) + (log1p(x) - log(x))) / n;
	} else {
		tmp = pow(x, (1.0 / n)) / (x * n);
	}
	return tmp;
}

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 40.0)
		tmp = Float64(Float64(Float64(fma(0.5, Float64((log1p(x) ^ 2.0) - (log(x) ^ 2.0)), Float64(Float64(0.16666666666666666 * Float64((log1p(x) ^ 3.0) - (log(x) ^ 3.0))) / n)) / n) + Float64(log1p(x) - log(x))) / n);
	else
		tmp = Float64((x ^ Float64(1.0 / n)) / Float64(x * n));
	end
	return tmp
end

Wolfram

code[x_, n_] := If[LessEqual[x, 40.0], N[(N[(N[(N[(0.5 * N[(N[Power[N[Log[1 + x], $MachinePrecision], 2.0], $MachinePrecision] - N[Power[N[Log[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + N[(N[(0.16666666666666666 * N[(N[Power[N[Log[1 + x], $MachinePrecision], 3.0], $MachinePrecision] - N[Power[N[Log[x], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision] + N[(N[Log[1 + x], $MachinePrecision] - N[Log[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], N[(N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision] / N[(x * n), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 40

   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 n around -inf

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

   5. Applied rewrites 81.5%

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

   if 40 < x

   1. Initial program 63.1%

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

      1. lower-/.f64 N/A

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

      2. log-rec N/A

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

      3. mul-1-neg N/A

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

      4. associate-*r/ N/A

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

      5. associate-*r* N/A

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

      6. metadata-eval N/A

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

      9. exp-to-pow N/A

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

      10. lower-pow.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 96.9

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

   5. Applied rewrites 96.9%

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

4. Final simplification 87.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 40:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(0.5, {\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}, \frac{0.16666666666666666 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{3} - {\log x}^{3}\right)}{n}\right)}{n} + \left(\mathsf{log1p}\left(x\right) - \log x\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{x \cdot n}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 86.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 40:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right) + \frac{\mathsf{fma}\left(0.5, \log \left(x \cdot \left(x + 1\right)\right) \cdot \log \left(\frac{x + 1}{x}\right), \left({\left(\mathsf{log1p}\left(x\right)\right)}^{3} - {\log x}^{3}\right) \cdot \frac{0.16666666666666666}{n}\right)}{n}}{n} - \frac{\log x}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{x \cdot n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 40.0)
   (-
    (/
     (+
      (log1p x)
      (/
       (fma
        0.5
        (* (log (* x (+ x 1.0))) (log (/ (+ x 1.0) x)))
        (*
         (- (pow (log1p x) 3.0) (pow (log x) 3.0))
         (/ 0.16666666666666666 n)))
       n))
     n)
    (/ (log x) n))
   (/ (pow x (/ 1.0 n)) (* x n))))

C

double code(double x, double n) {
	double tmp;
	if (x <= 40.0) {
		tmp = ((log1p(x) + (fma(0.5, (log((x * (x + 1.0))) * log(((x + 1.0) / x))), ((pow(log1p(x), 3.0) - pow(log(x), 3.0)) * (0.16666666666666666 / n))) / n)) / n) - (log(x) / n);
	} else {
		tmp = pow(x, (1.0 / n)) / (x * n);
	}
	return tmp;
}

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 40.0)
		tmp = Float64(Float64(Float64(log1p(x) + Float64(fma(0.5, Float64(log(Float64(x * Float64(x + 1.0))) * log(Float64(Float64(x + 1.0) / x))), Float64(Float64((log1p(x) ^ 3.0) - (log(x) ^ 3.0)) * Float64(0.16666666666666666 / n))) / n)) / n) - Float64(log(x) / n));
	else
		tmp = Float64((x ^ Float64(1.0 / n)) / Float64(x * n));
	end
	return tmp
end

Wolfram

code[x_, n_] := If[LessEqual[x, 40.0], N[(N[(N[(N[Log[1 + x], $MachinePrecision] + N[(N[(0.5 * N[(N[Log[N[(x * N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Log[N[(N[(x + 1.0), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(N[(N[Power[N[Log[1 + x], $MachinePrecision], 3.0], $MachinePrecision] - N[Power[N[Log[x], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision] * N[(0.16666666666666666 / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision] - N[(N[Log[x], $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision], N[(N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision] / N[(x * n), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 40

   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 n around -inf

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

   5. Applied rewrites 81.5%

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

   7. Applied rewrites 81.5%

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

   if 40 < x

   1. Initial program 63.1%

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

      1. lower-/.f64 N/A

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

      2. log-rec N/A

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

      3. mul-1-neg N/A

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

      4. associate-*r/ N/A

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

      5. associate-*r* N/A

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

      6. metadata-eval N/A

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

      9. exp-to-pow N/A

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

      10. lower-pow.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 96.9

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

   5. Applied rewrites 96.9%

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

4. Final simplification 87.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 40:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right) + \frac{\mathsf{fma}\left(0.5, \log \left(x \cdot \left(x + 1\right)\right) \cdot \log \left(\frac{x + 1}{x}\right), \left({\left(\mathsf{log1p}\left(x\right)\right)}^{3} - {\log x}^{3}\right) \cdot \frac{0.16666666666666666}{n}\right)}{n}}{n} - \frac{\log x}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{x \cdot n}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 86.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 40:\\ \;\;\;\;\frac{1}{\frac{n}{\frac{\mathsf{fma}\left({\left(\mathsf{log1p}\left(x\right)\right)}^{3} - {\log x}^{3}, \frac{0.16666666666666666}{n}, \log \left(\mathsf{fma}\left(x, x, x\right)\right) \cdot \left(0.5 \cdot \log \left(\frac{x + 1}{x}\right)\right)\right)}{n} - \log \left(\frac{x}{x + 1}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{x \cdot n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 40.0)
   (/
    1.0
    (/
     n
     (-
      (/
       (fma
        (- (pow (log1p x) 3.0) (pow (log x) 3.0))
        (/ 0.16666666666666666 n)
        (* (log (fma x x x)) (* 0.5 (log (/ (+ x 1.0) x)))))
       n)
      (log (/ x (+ x 1.0))))))
   (/ (pow x (/ 1.0 n)) (* x n))))

C

double code(double x, double n) {
	double tmp;
	if (x <= 40.0) {
		tmp = 1.0 / (n / ((fma((pow(log1p(x), 3.0) - pow(log(x), 3.0)), (0.16666666666666666 / n), (log(fma(x, x, x)) * (0.5 * log(((x + 1.0) / x))))) / n) - log((x / (x + 1.0)))));
	} else {
		tmp = pow(x, (1.0 / n)) / (x * n);
	}
	return tmp;
}

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 40.0)
		tmp = Float64(1.0 / Float64(n / Float64(Float64(fma(Float64((log1p(x) ^ 3.0) - (log(x) ^ 3.0)), Float64(0.16666666666666666 / n), Float64(log(fma(x, x, x)) * Float64(0.5 * log(Float64(Float64(x + 1.0) / x))))) / n) - log(Float64(x / Float64(x + 1.0))))));
	else
		tmp = Float64((x ^ Float64(1.0 / n)) / Float64(x * n));
	end
	return tmp
end

Wolfram

code[x_, n_] := If[LessEqual[x, 40.0], N[(1.0 / N[(n / N[(N[(N[(N[(N[Power[N[Log[1 + x], $MachinePrecision], 3.0], $MachinePrecision] - N[Power[N[Log[x], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision] * N[(0.16666666666666666 / n), $MachinePrecision] + N[(N[Log[N[(x * x + x), $MachinePrecision]], $MachinePrecision] * N[(0.5 * N[Log[N[(N[(x + 1.0), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision] - N[Log[N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision] / N[(x * n), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 40

   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 n around -inf

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

   5. Applied rewrites 81.5%

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

   7. Applied rewrites 81.4%

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

   9. Applied rewrites 81.5%

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

   if 40 < x

   1. Initial program 63.1%

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

      1. lower-/.f64 N/A

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

      2. log-rec N/A

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

      3. mul-1-neg N/A

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

      4. associate-*r/ N/A

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

      5. associate-*r* N/A

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

      6. metadata-eval N/A

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

      9. exp-to-pow N/A

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

      10. lower-pow.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 96.9

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

   5. Applied rewrites 96.9%

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

4. Final simplification 87.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 40:\\ \;\;\;\;\frac{1}{\frac{n}{\frac{\mathsf{fma}\left({\left(\mathsf{log1p}\left(x\right)\right)}^{3} - {\log x}^{3}, \frac{0.16666666666666666}{n}, \log \left(\mathsf{fma}\left(x, x, x\right)\right) \cdot \left(0.5 \cdot \log \left(\frac{x + 1}{x}\right)\right)\right)}{n} - \log \left(\frac{x}{x + 1}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{x \cdot n}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 86.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 40:\\ \;\;\;\;\frac{1}{n} \cdot \left(\frac{\mathsf{fma}\left(0.5, \log \left(x \cdot \left(x + 1\right)\right) \cdot \log \left(\frac{x + 1}{x}\right), \left({\left(\mathsf{log1p}\left(x\right)\right)}^{3} - {\log x}^{3}\right) \cdot \frac{0.16666666666666666}{n}\right)}{n} - \log \left(\frac{x}{x + 1}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{x \cdot n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 40.0)
   (*
    (/ 1.0 n)
    (-
     (/
      (fma
       0.5
       (* (log (* x (+ x 1.0))) (log (/ (+ x 1.0) x)))
       (* (- (pow (log1p x) 3.0) (pow (log x) 3.0)) (/ 0.16666666666666666 n)))
      n)
     (log (/ x (+ x 1.0)))))
   (/ (pow x (/ 1.0 n)) (* x n))))

C

double code(double x, double n) {
	double tmp;
	if (x <= 40.0) {
		tmp = (1.0 / n) * ((fma(0.5, (log((x * (x + 1.0))) * log(((x + 1.0) / x))), ((pow(log1p(x), 3.0) - pow(log(x), 3.0)) * (0.16666666666666666 / n))) / n) - log((x / (x + 1.0))));
	} else {
		tmp = pow(x, (1.0 / n)) / (x * n);
	}
	return tmp;
}

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 40.0)
		tmp = Float64(Float64(1.0 / n) * Float64(Float64(fma(0.5, Float64(log(Float64(x * Float64(x + 1.0))) * log(Float64(Float64(x + 1.0) / x))), Float64(Float64((log1p(x) ^ 3.0) - (log(x) ^ 3.0)) * Float64(0.16666666666666666 / n))) / n) - log(Float64(x / Float64(x + 1.0)))));
	else
		tmp = Float64((x ^ Float64(1.0 / n)) / Float64(x * n));
	end
	return tmp
end

Wolfram

code[x_, n_] := If[LessEqual[x, 40.0], N[(N[(1.0 / n), $MachinePrecision] * N[(N[(N[(0.5 * N[(N[Log[N[(x * N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Log[N[(N[(x + 1.0), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(N[(N[Power[N[Log[1 + x], $MachinePrecision], 3.0], $MachinePrecision] - N[Power[N[Log[x], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision] * N[(0.16666666666666666 / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision] - N[Log[N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision] / N[(x * n), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 40

   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 n around -inf

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

   5. Applied rewrites 81.5%

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

   7. Applied rewrites 81.4%

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

   if 40 < x

   1. Initial program 63.1%

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

      1. lower-/.f64 N/A

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

      2. log-rec N/A

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

      3. mul-1-neg N/A

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

      4. associate-*r/ N/A

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

      5. associate-*r* N/A

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

      6. metadata-eval N/A

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

      9. exp-to-pow N/A

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

      10. lower-pow.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 96.9

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

   5. Applied rewrites 96.9%

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

4. Final simplification 87.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 40:\\ \;\;\;\;\frac{1}{n} \cdot \left(\frac{\mathsf{fma}\left(0.5, \log \left(x \cdot \left(x + 1\right)\right) \cdot \log \left(\frac{x + 1}{x}\right), \left({\left(\mathsf{log1p}\left(x\right)\right)}^{3} - {\log x}^{3}\right) \cdot \frac{0.16666666666666666}{n}\right)}{n} - \log \left(\frac{x}{x + 1}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{x \cdot n}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 86.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 40:\\ \;\;\;\;\frac{1}{n} \cdot \left(\frac{\mathsf{fma}\left(0.5, \log \left(x \cdot \left(x + 1\right)\right) \cdot \log \left(\frac{x + 1}{x}\right), \frac{0.16666666666666666}{n} \cdot \left(-{\log x}^{3}\right)\right)}{n} - \log \left(\frac{x}{x + 1}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{x \cdot n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 40.0)
   (*
    (/ 1.0 n)
    (-
     (/
      (fma
       0.5
       (* (log (* x (+ x 1.0))) (log (/ (+ x 1.0) x)))
       (* (/ 0.16666666666666666 n) (- (pow (log x) 3.0))))
      n)
     (log (/ x (+ x 1.0)))))
   (/ (pow x (/ 1.0 n)) (* x n))))

C

double code(double x, double n) {
	double tmp;
	if (x <= 40.0) {
		tmp = (1.0 / n) * ((fma(0.5, (log((x * (x + 1.0))) * log(((x + 1.0) / x))), ((0.16666666666666666 / n) * -pow(log(x), 3.0))) / n) - log((x / (x + 1.0))));
	} else {
		tmp = pow(x, (1.0 / n)) / (x * n);
	}
	return tmp;
}

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 40.0)
		tmp = Float64(Float64(1.0 / n) * Float64(Float64(fma(0.5, Float64(log(Float64(x * Float64(x + 1.0))) * log(Float64(Float64(x + 1.0) / x))), Float64(Float64(0.16666666666666666 / n) * Float64(-(log(x) ^ 3.0)))) / n) - log(Float64(x / Float64(x + 1.0)))));
	else
		tmp = Float64((x ^ Float64(1.0 / n)) / Float64(x * n));
	end
	return tmp
end

Wolfram

code[x_, n_] := If[LessEqual[x, 40.0], N[(N[(1.0 / n), $MachinePrecision] * N[(N[(N[(0.5 * N[(N[Log[N[(x * N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Log[N[(N[(x + 1.0), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(N[(0.16666666666666666 / n), $MachinePrecision] * (-N[Power[N[Log[x], $MachinePrecision], 3.0], $MachinePrecision])), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision] - N[Log[N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision] / N[(x * n), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 40

   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 n around -inf

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

   5. Applied rewrites 81.5%

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

   7. Applied rewrites 81.4%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 81.4%

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

   if 40 < x

   1. Initial program 63.1%

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

      1. lower-/.f64 N/A

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

      2. log-rec N/A

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

      3. mul-1-neg N/A

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

      4. associate-*r/ N/A

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

      5. associate-*r* N/A

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

      6. metadata-eval N/A

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

      9. exp-to-pow N/A

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

      10. lower-pow.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 96.9

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

   5. Applied rewrites 96.9%

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

4. Final simplification 87.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 40:\\ \;\;\;\;\frac{1}{n} \cdot \left(\frac{\mathsf{fma}\left(0.5, \log \left(x \cdot \left(x + 1\right)\right) \cdot \log \left(\frac{x + 1}{x}\right), \frac{0.16666666666666666}{n} \cdot \left(-{\log x}^{3}\right)\right)}{n} - \log \left(\frac{x}{x + 1}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{x \cdot n}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 76.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ t_1 := {\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - t\_0\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;1 - t\_0\\ \mathbf{elif}\;t\_1 \leq 0.05:\\ \;\;\;\;\frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{0.5}{n \cdot n} + \frac{-0.5}{n}, \frac{1}{n}\right), 1\right) - 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))) (t_1 (- (pow (+ x 1.0) (/ 1.0 n)) t_0)))
   (if (<= t_1 (- INFINITY))
     (- 1.0 t_0)
     (if (<= t_1 0.05)
       (/ (log (/ (+ x 1.0) x)) n)
       (- (fma x (fma x (+ (/ 0.5 (* n n)) (/ -0.5 n)) (/ 1.0 n)) 1.0) 1.0)))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double t_1 = pow((x + 1.0), (1.0 / n)) - t_0;
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = 1.0 - t_0;
	} else if (t_1 <= 0.05) {
		tmp = log(((x + 1.0) / x)) / n;
	} else {
		tmp = fma(x, fma(x, ((0.5 / (n * n)) + (-0.5 / n)), (1.0 / n)), 1.0) - 1.0;
	}
	return tmp;
}

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	t_1 = Float64((Float64(x + 1.0) ^ Float64(1.0 / n)) - t_0)
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(1.0 - t_0);
	elseif (t_1 <= 0.05)
		tmp = Float64(log(Float64(Float64(x + 1.0) / x)) / n);
	else
		tmp = Float64(fma(x, fma(x, Float64(Float64(0.5 / Float64(n * n)) + Float64(-0.5 / n)), Float64(1.0 / n)), 1.0) - 1.0);
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[Power[N[(x + 1.0), $MachinePrecision], N[(1.0 / n), $MachinePrecision]], $MachinePrecision] - t$95$0), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(1.0 - t$95$0), $MachinePrecision], If[LessEqual[t$95$1, 0.05], N[(N[Log[N[(N[(x + 1.0), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], N[(N[(x * N[(x * N[(N[(0.5 / N[(n * n), $MachinePrecision]), $MachinePrecision] + N[(-0.5 / n), $MachinePrecision]), $MachinePrecision] + N[(1.0 / n), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] - 1.0), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (pow.f64 (+.f64 x #s(literal 1 binary64)) (/.f64 #s(literal 1 binary64) n)) (pow.f64 x (/.f64 #s(literal 1 binary64) n))) < -inf.0

   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 -inf.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))) < 0.050000000000000003

   1. Initial program 39.5%

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

   3. Taylor expanded in n around -inf

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

   5. Applied rewrites 79.3%

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

   7. Applied rewrites 57.3%

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

   8. Taylor expanded in n around inf

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

   10. Applied rewrites 78.7%

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

   12. Applied rewrites 78.7%

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

   if 0.050000000000000003 < (-.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 42.9%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 39.8%

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

   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

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

       1. +-commutative N/A

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

       2. lower-fma.f64 N/A

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

       3. lower-fma.f64 N/A

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

       4. sub-neg N/A

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

       5. lower-+.f64 N/A

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

       6. associate-*r/ N/A

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

       7. metadata-eval N/A

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

       8. lower-/.f64 N/A

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

       9. unpow2 N/A

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

       10. lower-*.f64 N/A

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

       11. associate-*r/ N/A

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

       12. metadata-eval N/A

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

       13. distribute-neg-frac N/A

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

       14. metadata-eval N/A

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

       15. lower-/.f64 N/A

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

       16. lower-/.f64 46.3

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

   11. Applied rewrites 46.3%

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

Alternative 8: 86.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.32:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.5, \frac{{\log x}^{2}}{n}, \frac{{\log x}^{3} \cdot -0.16666666666666666}{n \cdot n}\right) - \log x}{n} + \frac{x}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{x \cdot n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 0.32)
   (+
    (/
     (-
      (fma
       -0.5
       (/ (pow (log x) 2.0) n)
       (/ (* (pow (log x) 3.0) -0.16666666666666666) (* n n)))
      (log x))
     n)
    (/ x n))
   (/ (pow x (/ 1.0 n)) (* x n))))

C

double code(double x, double n) {
	double tmp;
	if (x <= 0.32) {
		tmp = ((fma(-0.5, (pow(log(x), 2.0) / n), ((pow(log(x), 3.0) * -0.16666666666666666) / (n * n))) - log(x)) / n) + (x / n);
	} else {
		tmp = pow(x, (1.0 / n)) / (x * n);
	}
	return tmp;
}

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 0.32)
		tmp = Float64(Float64(Float64(fma(-0.5, Float64((log(x) ^ 2.0) / n), Float64(Float64((log(x) ^ 3.0) * -0.16666666666666666) / Float64(n * n))) - log(x)) / n) + Float64(x / n));
	else
		tmp = Float64((x ^ Float64(1.0 / n)) / Float64(x * n));
	end
	return tmp
end

Wolfram

code[x_, n_] := If[LessEqual[x, 0.32], N[(N[(N[(N[(-0.5 * N[(N[Power[N[Log[x], $MachinePrecision], 2.0], $MachinePrecision] / n), $MachinePrecision] + N[(N[(N[Power[N[Log[x], $MachinePrecision], 3.0], $MachinePrecision] * -0.16666666666666666), $MachinePrecision] / N[(n * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision] + N[(x / n), $MachinePrecision]), $MachinePrecision], N[(N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision] / N[(x * n), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.320000000000000007

   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 n around -inf

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

   5. Applied rewrites 81.5%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 80.0%

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

   if 0.320000000000000007 < x

   1. Initial program 63.1%

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

      1. lower-/.f64 N/A

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

      2. log-rec N/A

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

      3. mul-1-neg N/A

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

      4. associate-*r/ N/A

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

      5. associate-*r* N/A

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

      6. metadata-eval N/A

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

      9. exp-to-pow N/A

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

      10. lower-pow.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 96.9

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

   5. Applied rewrites 96.9%

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

4. Final simplification 87.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.32:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.5, \frac{{\log x}^{2}}{n}, \frac{{\log x}^{3} \cdot -0.16666666666666666}{n \cdot n}\right) - \log x}{n} + \frac{x}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{x \cdot n}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 85.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.25 \cdot 10^{-6}:\\ \;\;\;\;\frac{\log x - \mathsf{fma}\left(-0.5, \frac{{\log x}^{2}}{n}, \frac{{\log x}^{3} \cdot -0.16666666666666666}{n \cdot n}\right)}{-n}\\ \mathbf{else}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{x \cdot n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 1.25e-6)
   (/
    (-
     (log x)
     (fma
      -0.5
      (/ (pow (log x) 2.0) n)
      (/ (* (pow (log x) 3.0) -0.16666666666666666) (* n n))))
    (- n))
   (/ (pow x (/ 1.0 n)) (* x n))))

C

double code(double x, double n) {
	double tmp;
	if (x <= 1.25e-6) {
		tmp = (log(x) - fma(-0.5, (pow(log(x), 2.0) / n), ((pow(log(x), 3.0) * -0.16666666666666666) / (n * n)))) / -n;
	} else {
		tmp = pow(x, (1.0 / n)) / (x * n);
	}
	return tmp;
}

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 1.25e-6)
		tmp = Float64(Float64(log(x) - fma(-0.5, Float64((log(x) ^ 2.0) / n), Float64(Float64((log(x) ^ 3.0) * -0.16666666666666666) / Float64(n * n)))) / Float64(-n));
	else
		tmp = Float64((x ^ Float64(1.0 / n)) / Float64(x * n));
	end
	return tmp
end

Wolfram

code[x_, n_] := If[LessEqual[x, 1.25e-6], N[(N[(N[Log[x], $MachinePrecision] - N[(-0.5 * N[(N[Power[N[Log[x], $MachinePrecision], 2.0], $MachinePrecision] / n), $MachinePrecision] + N[(N[(N[Power[N[Log[x], $MachinePrecision], 3.0], $MachinePrecision] * -0.16666666666666666), $MachinePrecision] / N[(n * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / (-n)), $MachinePrecision], N[(N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision] / N[(x * n), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.2500000000000001e-6

   1. Initial program 35.6%

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

   3. Taylor expanded in n around -inf

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

   5. Applied rewrites 81.9%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 81.1%

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

   if 1.2500000000000001e-6 < x

   1. Initial program 62.8%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. log-rec N/A

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

      3. mul-1-neg N/A

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

      4. associate-*r/ N/A

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

      5. associate-*r* N/A

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

      6. metadata-eval N/A

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

      9. exp-to-pow N/A

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

      10. lower-pow.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 94.0

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

   5. Applied rewrites 94.0%

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

4. Final simplification 86.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.25 \cdot 10^{-6}:\\ \;\;\;\;\frac{\log x - \mathsf{fma}\left(-0.5, \frac{{\log x}^{2}}{n}, \frac{{\log x}^{3} \cdot -0.16666666666666666}{n \cdot n}\right)}{-n}\\ \mathbf{else}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{x \cdot n}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 82.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{-70}:\\ \;\;\;\;\frac{t\_0}{x \cdot n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-42}:\\ \;\;\;\;\frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-25}:\\ \;\;\;\;\frac{1}{x \cdot n}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \frac{\frac{\mathsf{fma}\left(0.16666666666666666, \frac{x \cdot x}{n}, x \cdot \mathsf{fma}\left(x, -0.5, 0.5\right)\right)}{n} - \mathsf{fma}\left(x, \mathsf{fma}\left(x, -0.3333333333333333, 0.5\right), -1\right)}{n}, 1\right) - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -5e-70)
     (/ t_0 (* x n))
     (if (<= (/ 1.0 n) 2e-42)
       (/ (log (/ (+ x 1.0) x)) n)
       (if (<= (/ 1.0 n) 5e-25)
         (/ 1.0 (* x n))
         (-
          (fma
           x
           (/
            (-
             (/
              (fma 0.16666666666666666 (/ (* x x) n) (* x (fma x -0.5 0.5)))
              n)
             (fma x (fma x -0.3333333333333333 0.5) -1.0))
            n)
           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) <= -5e-70) {
		tmp = t_0 / (x * n);
	} else if ((1.0 / n) <= 2e-42) {
		tmp = log(((x + 1.0) / x)) / n;
	} else if ((1.0 / n) <= 5e-25) {
		tmp = 1.0 / (x * n);
	} else {
		tmp = fma(x, (((fma(0.16666666666666666, ((x * x) / n), (x * fma(x, -0.5, 0.5))) / n) - fma(x, fma(x, -0.3333333333333333, 0.5), -1.0)) / n), 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) <= -5e-70)
		tmp = Float64(t_0 / Float64(x * n));
	elseif (Float64(1.0 / n) <= 2e-42)
		tmp = Float64(log(Float64(Float64(x + 1.0) / x)) / n);
	elseif (Float64(1.0 / n) <= 5e-25)
		tmp = Float64(1.0 / Float64(x * n));
	else
		tmp = Float64(fma(x, Float64(Float64(Float64(fma(0.16666666666666666, Float64(Float64(x * x) / n), Float64(x * fma(x, -0.5, 0.5))) / n) - fma(x, fma(x, -0.3333333333333333, 0.5), -1.0)) / n), 1.0) - t_0);
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -5e-70], N[(t$95$0 / N[(x * n), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-42], N[(N[Log[N[(N[(x + 1.0), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e-25], N[(1.0 / N[(x * n), $MachinePrecision]), $MachinePrecision], N[(N[(x * N[(N[(N[(N[(0.16666666666666666 * N[(N[(x * x), $MachinePrecision] / n), $MachinePrecision] + N[(x * N[(x * -0.5 + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision] - N[(x * N[(x * -0.3333333333333333 + 0.5), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision] + 1.0), $MachinePrecision] - t$95$0), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -4.9999999999999998e-70

   1. Initial program 83.1%

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

      1. lower-/.f64 N/A

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

      2. log-rec N/A

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

      3. mul-1-neg N/A

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

      4. associate-*r/ N/A

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

      5. associate-*r* N/A

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

      6. metadata-eval N/A

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

      9. exp-to-pow N/A

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

      10. lower-pow.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 95.2

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

   5. Applied rewrites 95.2%

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

   if -4.9999999999999998e-70 < (/.f64 #s(literal 1 binary64) n) < 2.00000000000000008e-42

   1. Initial program 30.8%

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

   3. Taylor expanded in n around -inf

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

   5. Applied rewrites 84.2%

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

   7. Applied rewrites 64.4%

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

   8. Taylor expanded in n around inf

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

   10. Applied rewrites 84.4%

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

   12. Applied rewrites 84.4%

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

   if 2.00000000000000008e-42 < (/.f64 #s(literal 1 binary64) n) < 4.99999999999999962e-25

   1. Initial program 5.8%

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

   3. Taylor expanded in n around -inf

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

   5. Applied rewrites 5.8%

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

   7. Applied rewrites 1.5%

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

   8. Taylor expanded in n around inf

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

   10. Applied rewrites 5.8%

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

   11. Taylor expanded in x around inf

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

   13. Applied rewrites 99.7%

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

   if 4.99999999999999962e-25 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 45.4%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

   5. Applied rewrites 25.7%

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

   6. Taylor expanded in n around -inf

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

   8. Applied rewrites 67.9%

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

4. Final simplification 85.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{-70}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{x \cdot n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-42}:\\ \;\;\;\;\frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-25}:\\ \;\;\;\;\frac{1}{x \cdot n}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \frac{\frac{\mathsf{fma}\left(0.16666666666666666, \frac{x \cdot x}{n}, x \cdot \mathsf{fma}\left(x, -0.5, 0.5\right)\right)}{n} - \mathsf{fma}\left(x, \mathsf{fma}\left(x, -0.3333333333333333, 0.5\right), -1\right)}{n}, 1\right) - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 81.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{-70}:\\ \;\;\;\;\frac{t\_0}{x \cdot n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-42}:\\ \;\;\;\;\frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-25}:\\ \;\;\;\;\frac{1}{x \cdot n}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{0.5}{n \cdot n} + \frac{-0.5}{n}, \frac{1}{n}\right), 1\right) - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -5e-70)
     (/ t_0 (* x n))
     (if (<= (/ 1.0 n) 2e-42)
       (/ (log (/ (+ x 1.0) x)) n)
       (if (<= (/ 1.0 n) 5e-25)
         (/ 1.0 (* x n))
         (-
          (fma x (fma x (+ (/ 0.5 (* n n)) (/ -0.5 n)) (/ 1.0 n)) 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) <= -5e-70) {
		tmp = t_0 / (x * n);
	} else if ((1.0 / n) <= 2e-42) {
		tmp = log(((x + 1.0) / x)) / n;
	} else if ((1.0 / n) <= 5e-25) {
		tmp = 1.0 / (x * n);
	} else {
		tmp = fma(x, fma(x, ((0.5 / (n * n)) + (-0.5 / n)), (1.0 / n)), 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) <= -5e-70)
		tmp = Float64(t_0 / Float64(x * n));
	elseif (Float64(1.0 / n) <= 2e-42)
		tmp = Float64(log(Float64(Float64(x + 1.0) / x)) / n);
	elseif (Float64(1.0 / n) <= 5e-25)
		tmp = Float64(1.0 / Float64(x * n));
	else
		tmp = Float64(fma(x, fma(x, Float64(Float64(0.5 / Float64(n * n)) + Float64(-0.5 / n)), Float64(1.0 / n)), 1.0) - t_0);
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -5e-70], N[(t$95$0 / N[(x * n), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-42], N[(N[Log[N[(N[(x + 1.0), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e-25], N[(1.0 / N[(x * n), $MachinePrecision]), $MachinePrecision], N[(N[(x * N[(x * N[(N[(0.5 / N[(n * n), $MachinePrecision]), $MachinePrecision] + N[(-0.5 / n), $MachinePrecision]), $MachinePrecision] + N[(1.0 / n), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] - t$95$0), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -4.9999999999999998e-70

   1. Initial program 83.1%

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

      1. lower-/.f64 N/A

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

      2. log-rec N/A

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

      3. mul-1-neg N/A

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

      4. associate-*r/ N/A

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

      5. associate-*r* N/A

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

      6. metadata-eval N/A

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

      9. exp-to-pow N/A

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

      10. lower-pow.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 95.2

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

   5. Applied rewrites 95.2%

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

   if -4.9999999999999998e-70 < (/.f64 #s(literal 1 binary64) n) < 2.00000000000000008e-42

   1. Initial program 30.8%

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

   3. Taylor expanded in n around -inf

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

   5. Applied rewrites 84.2%

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

   7. Applied rewrites 64.4%

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

   8. Taylor expanded in n around inf

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

   10. Applied rewrites 84.4%

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

   12. Applied rewrites 84.4%

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

   if 2.00000000000000008e-42 < (/.f64 #s(literal 1 binary64) n) < 4.99999999999999962e-25

   1. Initial program 5.8%

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

   3. Taylor expanded in n around -inf

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

   5. Applied rewrites 5.8%

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

   7. Applied rewrites 1.5%

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

   8. Taylor expanded in n around inf

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

   10. Applied rewrites 5.8%

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

   11. Taylor expanded in x around inf

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

   13. Applied rewrites 99.7%

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

   if 4.99999999999999962e-25 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 45.4%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-fma.f64 N/A

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

      4. sub-neg N/A

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

      5. lower-+.f64 N/A

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

      6. associate-*r/ N/A

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

      7. metadata-eval N/A

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

      8. lower-/.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. associate-*r/ N/A

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

      12. metadata-eval N/A

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

      13. distribute-neg-frac N/A

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

      14. metadata-eval N/A

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

      15. lower-/.f64 N/A

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

      16. lower-/.f64 66.6

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

   5. Applied rewrites 66.6%

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

Alternative 12: 81.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{-70}:\\ \;\;\;\;\frac{t\_0}{x \cdot n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-42}:\\ \;\;\;\;\frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-25}:\\ \;\;\;\;\frac{1}{x \cdot n}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \frac{1 + \mathsf{fma}\left(x, -0.5, 0.5 \cdot \frac{x}{n}\right)}{n}, 1\right) - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -5e-70)
     (/ t_0 (* x n))
     (if (<= (/ 1.0 n) 2e-42)
       (/ (log (/ (+ x 1.0) x)) n)
       (if (<= (/ 1.0 n) 5e-25)
         (/ 1.0 (* x n))
         (- (fma x (/ (+ 1.0 (fma x -0.5 (* 0.5 (/ x n)))) n) 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) <= -5e-70) {
		tmp = t_0 / (x * n);
	} else if ((1.0 / n) <= 2e-42) {
		tmp = log(((x + 1.0) / x)) / n;
	} else if ((1.0 / n) <= 5e-25) {
		tmp = 1.0 / (x * n);
	} else {
		tmp = fma(x, ((1.0 + fma(x, -0.5, (0.5 * (x / n)))) / n), 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) <= -5e-70)
		tmp = Float64(t_0 / Float64(x * n));
	elseif (Float64(1.0 / n) <= 2e-42)
		tmp = Float64(log(Float64(Float64(x + 1.0) / x)) / n);
	elseif (Float64(1.0 / n) <= 5e-25)
		tmp = Float64(1.0 / Float64(x * n));
	else
		tmp = Float64(fma(x, Float64(Float64(1.0 + fma(x, -0.5, Float64(0.5 * Float64(x / n)))) / n), 1.0) - t_0);
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -5e-70], N[(t$95$0 / N[(x * n), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-42], N[(N[Log[N[(N[(x + 1.0), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e-25], N[(1.0 / N[(x * n), $MachinePrecision]), $MachinePrecision], N[(N[(x * N[(N[(1.0 + N[(x * -0.5 + N[(0.5 * N[(x / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision] + 1.0), $MachinePrecision] - t$95$0), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -4.9999999999999998e-70

   1. Initial program 83.1%

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

      1. lower-/.f64 N/A

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

      2. log-rec N/A

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

      3. mul-1-neg N/A

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

      4. associate-*r/ N/A

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

      5. associate-*r* N/A

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

      6. metadata-eval N/A

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

      9. exp-to-pow N/A

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

      10. lower-pow.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 95.2

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

   5. Applied rewrites 95.2%

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

   if -4.9999999999999998e-70 < (/.f64 #s(literal 1 binary64) n) < 2.00000000000000008e-42

   1. Initial program 30.8%

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

   3. Taylor expanded in n around -inf

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

   5. Applied rewrites 84.2%

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

   7. Applied rewrites 64.4%

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

   8. Taylor expanded in n around inf

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

   10. Applied rewrites 84.4%

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

   12. Applied rewrites 84.4%

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

   if 2.00000000000000008e-42 < (/.f64 #s(literal 1 binary64) n) < 4.99999999999999962e-25

   1. Initial program 5.8%

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

   3. Taylor expanded in n around -inf

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

   5. Applied rewrites 5.8%

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

   7. Applied rewrites 1.5%

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

   8. Taylor expanded in n around inf

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

   10. Applied rewrites 5.8%

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

   11. Taylor expanded in x around inf

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

   13. Applied rewrites 99.7%

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

   if 4.99999999999999962e-25 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 45.4%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 42.8%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. associate-*r/ N/A

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

      6. metadata-eval N/A

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

      7. lower-/.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. associate-*r/ N/A

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

      11. metadata-eval N/A

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

      12. lower-/.f64 N/A

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

      13. lower-/.f64 66.6

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

   8. Applied rewrites 66.6%

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

   9. Taylor expanded in n around inf

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

   11. Applied rewrites 62.4%

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

Alternative 13: 81.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{-70}:\\ \;\;\;\;\frac{t\_0}{x \cdot n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-42}:\\ \;\;\;\;\frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-25}:\\ \;\;\;\;\frac{1}{x \cdot n}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \frac{\mathsf{fma}\left(x, \mathsf{fma}\left(n, -0.5, 0.5\right), n\right)}{n \cdot n}, 1\right) - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -5e-70)
     (/ t_0 (* x n))
     (if (<= (/ 1.0 n) 2e-42)
       (/ (log (/ (+ x 1.0) x)) n)
       (if (<= (/ 1.0 n) 5e-25)
         (/ 1.0 (* x n))
         (- (fma x (/ (fma x (fma n -0.5 0.5) n) (* n n)) 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) <= -5e-70) {
		tmp = t_0 / (x * n);
	} else if ((1.0 / n) <= 2e-42) {
		tmp = log(((x + 1.0) / x)) / n;
	} else if ((1.0 / n) <= 5e-25) {
		tmp = 1.0 / (x * n);
	} else {
		tmp = fma(x, (fma(x, fma(n, -0.5, 0.5), n) / (n * n)), 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) <= -5e-70)
		tmp = Float64(t_0 / Float64(x * n));
	elseif (Float64(1.0 / n) <= 2e-42)
		tmp = Float64(log(Float64(Float64(x + 1.0) / x)) / n);
	elseif (Float64(1.0 / n) <= 5e-25)
		tmp = Float64(1.0 / Float64(x * n));
	else
		tmp = Float64(fma(x, Float64(fma(x, fma(n, -0.5, 0.5), n) / Float64(n * n)), 1.0) - t_0);
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -5e-70], N[(t$95$0 / N[(x * n), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-42], N[(N[Log[N[(N[(x + 1.0), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e-25], N[(1.0 / N[(x * n), $MachinePrecision]), $MachinePrecision], N[(N[(x * N[(N[(x * N[(n * -0.5 + 0.5), $MachinePrecision] + n), $MachinePrecision] / N[(n * n), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] - t$95$0), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -4.9999999999999998e-70

   1. Initial program 83.1%

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

      1. lower-/.f64 N/A

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

      2. log-rec N/A

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

      3. mul-1-neg N/A

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

      4. associate-*r/ N/A

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

      5. associate-*r* N/A

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

      6. metadata-eval N/A

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

      9. exp-to-pow N/A

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

      10. lower-pow.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 95.2

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

   5. Applied rewrites 95.2%

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

   if -4.9999999999999998e-70 < (/.f64 #s(literal 1 binary64) n) < 2.00000000000000008e-42

   1. Initial program 30.8%

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

   3. Taylor expanded in n around -inf

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

   5. Applied rewrites 84.2%

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

   7. Applied rewrites 64.4%

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

   8. Taylor expanded in n around inf

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

   10. Applied rewrites 84.4%

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

   12. Applied rewrites 84.4%

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

   if 2.00000000000000008e-42 < (/.f64 #s(literal 1 binary64) n) < 4.99999999999999962e-25

   1. Initial program 5.8%

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

   3. Taylor expanded in n around -inf

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

   5. Applied rewrites 5.8%

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

   7. Applied rewrites 1.5%

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

   8. Taylor expanded in n around inf

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

   10. Applied rewrites 5.8%

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

   11. Taylor expanded in x around inf

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

   13. Applied rewrites 99.7%

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

   if 4.99999999999999962e-25 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 45.4%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 42.8%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. associate-*r/ N/A

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

      6. metadata-eval N/A

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

      7. lower-/.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. associate-*r/ N/A

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

      11. metadata-eval N/A

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

      12. lower-/.f64 N/A

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

      13. lower-/.f64 66.6

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

   8. Applied rewrites 66.6%

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

   9. Taylor expanded in n around 0

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

   11. Applied rewrites 59.0%

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

   12. Taylor expanded in x around 0

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

   14. Applied rewrites 59.0%

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

Alternative 14: 81.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{-70}:\\ \;\;\;\;\frac{t\_0}{x \cdot n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-42}:\\ \;\;\;\;\frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-25}:\\ \;\;\;\;\frac{1}{x \cdot n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{+151}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - t\_0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{0.5}{n \cdot n} + \frac{-0.5}{n}, \frac{1}{n}\right), 1\right) - 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -5e-70)
     (/ t_0 (* x n))
     (if (<= (/ 1.0 n) 2e-42)
       (/ (log (/ (+ x 1.0) x)) n)
       (if (<= (/ 1.0 n) 5e-25)
         (/ 1.0 (* x n))
         (if (<= (/ 1.0 n) 5e+151)
           (- (+ 1.0 (/ x n)) t_0)
           (-
            (fma x (fma x (+ (/ 0.5 (* n n)) (/ -0.5 n)) (/ 1.0 n)) 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) <= -5e-70) {
		tmp = t_0 / (x * n);
	} else if ((1.0 / n) <= 2e-42) {
		tmp = log(((x + 1.0) / x)) / n;
	} else if ((1.0 / n) <= 5e-25) {
		tmp = 1.0 / (x * n);
	} else if ((1.0 / n) <= 5e+151) {
		tmp = (1.0 + (x / n)) - t_0;
	} else {
		tmp = fma(x, fma(x, ((0.5 / (n * n)) + (-0.5 / n)), (1.0 / n)), 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) <= -5e-70)
		tmp = Float64(t_0 / Float64(x * n));
	elseif (Float64(1.0 / n) <= 2e-42)
		tmp = Float64(log(Float64(Float64(x + 1.0) / x)) / n);
	elseif (Float64(1.0 / n) <= 5e-25)
		tmp = Float64(1.0 / Float64(x * n));
	elseif (Float64(1.0 / n) <= 5e+151)
		tmp = Float64(Float64(1.0 + Float64(x / n)) - t_0);
	else
		tmp = Float64(fma(x, fma(x, Float64(Float64(0.5 / Float64(n * n)) + Float64(-0.5 / n)), Float64(1.0 / n)), 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], -5e-70], N[(t$95$0 / N[(x * n), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-42], N[(N[Log[N[(N[(x + 1.0), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e-25], N[(1.0 / N[(x * n), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e+151], N[(N[(1.0 + N[(x / n), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], N[(N[(x * N[(x * N[(N[(0.5 / N[(n * n), $MachinePrecision]), $MachinePrecision] + N[(-0.5 / n), $MachinePrecision]), $MachinePrecision] + N[(1.0 / n), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] - 1.0), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -4.9999999999999998e-70

   1. Initial program 83.1%

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

      1. lower-/.f64 N/A

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

      2. log-rec N/A

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

      3. mul-1-neg N/A

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

      4. associate-*r/ N/A

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

      5. associate-*r* N/A

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

      6. metadata-eval N/A

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

      9. exp-to-pow N/A

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

      10. lower-pow.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 95.2

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

   5. Applied rewrites 95.2%

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

   if -4.9999999999999998e-70 < (/.f64 #s(literal 1 binary64) n) < 2.00000000000000008e-42

   1. Initial program 30.8%

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

   3. Taylor expanded in n around -inf

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

   5. Applied rewrites 84.2%

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

   7. Applied rewrites 64.4%

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

   8. Taylor expanded in n around inf

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

   10. Applied rewrites 84.4%

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

   12. Applied rewrites 84.4%

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

   if 2.00000000000000008e-42 < (/.f64 #s(literal 1 binary64) n) < 4.99999999999999962e-25

   1. Initial program 5.8%

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

   3. Taylor expanded in n around -inf

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

   5. Applied rewrites 5.8%

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

   7. Applied rewrites 1.5%

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

   8. Taylor expanded in n around inf

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

   10. Applied rewrites 5.8%

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

   11. Taylor expanded in x around inf

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

   13. Applied rewrites 99.7%

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

   if 4.99999999999999962e-25 < (/.f64 #s(literal 1 binary64) n) < 5.0000000000000002e151

   1. Initial program 67.8%

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

   3. Taylor expanded in x around 0

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

      1. *-rgt-identity N/A

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

      2. associate-*r/ N/A

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

      3. lower-+.f64 N/A

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

      4. associate-*r/ N/A

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

      5. *-rgt-identity N/A

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

      6. lower-/.f64 64.5

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

   5. Applied rewrites 64.5%

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

   if 5.0000000000000002e151 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 27.3%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   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 27.3%

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

   6. Taylor expanded in n around inf

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

   8. Applied rewrites 2.1%

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

   9. Taylor expanded in x around 0

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

       1. +-commutative N/A

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

       2. lower-fma.f64 N/A

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

       3. lower-fma.f64 N/A

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

       4. sub-neg N/A

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

       5. lower-+.f64 N/A

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

       6. associate-*r/ N/A

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

       7. metadata-eval N/A

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

       8. lower-/.f64 N/A

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

       9. unpow2 N/A

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

       10. lower-*.f64 N/A

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

       11. associate-*r/ N/A

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

       12. metadata-eval N/A

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

       13. distribute-neg-frac N/A

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

       14. metadata-eval N/A

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

       15. lower-/.f64 N/A

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

       16. lower-/.f64 67.7

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

   11. Applied rewrites 67.7%

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

Alternative 15: 81.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{-70}:\\ \;\;\;\;\frac{t\_0}{x \cdot n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-42}:\\ \;\;\;\;\frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-25}:\\ \;\;\;\;\frac{1}{x \cdot n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{+151}:\\ \;\;\;\;1 - t\_0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{0.5}{n \cdot n} + \frac{-0.5}{n}, \frac{1}{n}\right), 1\right) - 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -5e-70)
     (/ t_0 (* x n))
     (if (<= (/ 1.0 n) 2e-42)
       (/ (log (/ (+ x 1.0) x)) n)
       (if (<= (/ 1.0 n) 5e-25)
         (/ 1.0 (* x n))
         (if (<= (/ 1.0 n) 5e+151)
           (- 1.0 t_0)
           (-
            (fma x (fma x (+ (/ 0.5 (* n n)) (/ -0.5 n)) (/ 1.0 n)) 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) <= -5e-70) {
		tmp = t_0 / (x * n);
	} else if ((1.0 / n) <= 2e-42) {
		tmp = log(((x + 1.0) / x)) / n;
	} else if ((1.0 / n) <= 5e-25) {
		tmp = 1.0 / (x * n);
	} else if ((1.0 / n) <= 5e+151) {
		tmp = 1.0 - t_0;
	} else {
		tmp = fma(x, fma(x, ((0.5 / (n * n)) + (-0.5 / n)), (1.0 / n)), 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) <= -5e-70)
		tmp = Float64(t_0 / Float64(x * n));
	elseif (Float64(1.0 / n) <= 2e-42)
		tmp = Float64(log(Float64(Float64(x + 1.0) / x)) / n);
	elseif (Float64(1.0 / n) <= 5e-25)
		tmp = Float64(1.0 / Float64(x * n));
	elseif (Float64(1.0 / n) <= 5e+151)
		tmp = Float64(1.0 - t_0);
	else
		tmp = Float64(fma(x, fma(x, Float64(Float64(0.5 / Float64(n * n)) + Float64(-0.5 / n)), Float64(1.0 / n)), 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], -5e-70], N[(t$95$0 / N[(x * n), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-42], N[(N[Log[N[(N[(x + 1.0), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e-25], N[(1.0 / N[(x * n), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e+151], N[(1.0 - t$95$0), $MachinePrecision], N[(N[(x * N[(x * N[(N[(0.5 / N[(n * n), $MachinePrecision]), $MachinePrecision] + N[(-0.5 / n), $MachinePrecision]), $MachinePrecision] + N[(1.0 / n), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] - 1.0), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -4.9999999999999998e-70

   1. Initial program 83.1%

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

      1. lower-/.f64 N/A

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

      2. log-rec N/A

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

      3. mul-1-neg N/A

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

      4. associate-*r/ N/A

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

      5. associate-*r* N/A

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

      6. metadata-eval N/A

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

      9. exp-to-pow N/A

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

      10. lower-pow.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 95.2

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

   5. Applied rewrites 95.2%

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

   if -4.9999999999999998e-70 < (/.f64 #s(literal 1 binary64) n) < 2.00000000000000008e-42

   1. Initial program 30.8%

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

   3. Taylor expanded in n around -inf

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

   5. Applied rewrites 84.2%

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

   7. Applied rewrites 64.4%

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

   8. Taylor expanded in n around inf

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

   10. Applied rewrites 84.4%

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

   12. Applied rewrites 84.4%

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

   if 2.00000000000000008e-42 < (/.f64 #s(literal 1 binary64) n) < 4.99999999999999962e-25

   1. Initial program 5.8%

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

   3. Taylor expanded in n around -inf

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

   5. Applied rewrites 5.8%

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

   7. Applied rewrites 1.5%

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

   8. Taylor expanded in n around inf

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

   10. Applied rewrites 5.8%

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

   11. Taylor expanded in x around inf

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

   13. Applied rewrites 99.7%

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

   if 4.99999999999999962e-25 < (/.f64 #s(literal 1 binary64) n) < 5.0000000000000002e151

   1. Initial program 67.8%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   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 62.1%

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

   if 5.0000000000000002e151 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 27.3%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   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 27.3%

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

   6. Taylor expanded in n around inf

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

   8. Applied rewrites 2.1%

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

   9. Taylor expanded in x around 0

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

       1. +-commutative N/A

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

       2. lower-fma.f64 N/A

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

       3. lower-fma.f64 N/A

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

       4. sub-neg N/A

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

       5. lower-+.f64 N/A

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

       6. associate-*r/ N/A

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

       7. metadata-eval N/A

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

       8. lower-/.f64 N/A

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

       9. unpow2 N/A

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

       10. lower-*.f64 N/A

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

       11. associate-*r/ N/A

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

       12. metadata-eval N/A

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

       13. distribute-neg-frac N/A

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

       14. metadata-eval N/A

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

       15. lower-/.f64 N/A

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

       16. lower-/.f64 67.7

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

   11. Applied rewrites 67.7%

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

Alternative 16: 61.2% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.88:\\ \;\;\;\;\frac{x}{n} - \frac{\log x}{n}\\ \mathbf{elif}\;x \leq 1.02 \cdot 10^{+170}:\\ \;\;\;\;\frac{\frac{-1 + \frac{0.5 - \frac{0.3333333333333333 - \frac{0.25}{x}}{x}}{x}}{x}}{-n}\\ \mathbf{else}:\\ \;\;\;\;1 - 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 0.88)
   (- (/ x n) (/ (log x) n))
   (if (<= x 1.02e+170)
     (/
      (/ (+ -1.0 (/ (- 0.5 (/ (- 0.3333333333333333 (/ 0.25 x)) x)) x)) x)
      (- n))
     (- 1.0 1.0))))

C

double code(double x, double n) {
	double tmp;
	if (x <= 0.88) {
		tmp = (x / n) - (log(x) / n);
	} else if (x <= 1.02e+170) {
		tmp = ((-1.0 + ((0.5 - ((0.3333333333333333 - (0.25 / x)) / x)) / x)) / x) / -n;
	} else {
		tmp = 1.0 - 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if (x <= 0.88d0) then
        tmp = (x / n) - (log(x) / n)
    else if (x <= 1.02d+170) then
        tmp = (((-1.0d0) + ((0.5d0 - ((0.3333333333333333d0 - (0.25d0 / x)) / x)) / x)) / x) / -n
    else
        tmp = 1.0d0 - 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (x <= 0.88) {
		tmp = (x / n) - (Math.log(x) / n);
	} else if (x <= 1.02e+170) {
		tmp = ((-1.0 + ((0.5 - ((0.3333333333333333 - (0.25 / x)) / x)) / x)) / x) / -n;
	} else {
		tmp = 1.0 - 1.0;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 0.88:
		tmp = (x / n) - (math.log(x) / n)
	elif x <= 1.02e+170:
		tmp = ((-1.0 + ((0.5 - ((0.3333333333333333 - (0.25 / x)) / x)) / x)) / x) / -n
	else:
		tmp = 1.0 - 1.0
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 0.88)
		tmp = Float64(Float64(x / n) - Float64(log(x) / n));
	elseif (x <= 1.02e+170)
		tmp = Float64(Float64(Float64(-1.0 + Float64(Float64(0.5 - Float64(Float64(0.3333333333333333 - Float64(0.25 / x)) / x)) / x)) / x) / Float64(-n));
	else
		tmp = Float64(1.0 - 1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (x <= 0.88)
		tmp = (x / n) - (log(x) / n);
	elseif (x <= 1.02e+170)
		tmp = ((-1.0 + ((0.5 - ((0.3333333333333333 - (0.25 / x)) / x)) / x)) / x) / -n;
	else
		tmp = 1.0 - 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 0.88], N[(N[(x / n), $MachinePrecision] - N[(N[Log[x], $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.02e+170], N[(N[(N[(-1.0 + N[(N[(0.5 - N[(N[(0.3333333333333333 - N[(0.25 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] / (-n)), $MachinePrecision], N[(1.0 - 1.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 0.880000000000000004

   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 n around -inf

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

   5. Applied rewrites 81.5%

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

   7. Applied rewrites 81.4%

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

   8. Taylor expanded in n around inf

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

   10. Applied rewrites 57.7%

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

   11. Taylor expanded in x around 0

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

   13. Applied rewrites 56.8%

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

   if 0.880000000000000004 < x < 1.02000000000000002e170

   1. Initial program 46.5%

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

   3. Taylor expanded in n around -inf

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

   5. Applied rewrites 48.1%

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

   7. Applied rewrites 44.8%

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

   8. Taylor expanded in n around inf

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

   10. Applied rewrites 48.8%

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

   11. Taylor expanded in x around -inf

       \[\leadsto \mathsf{neg}\left(\frac{-1 \cdot \frac{1 + -1 \cdot \frac{\frac{1}{2} + -1 \cdot \frac{\frac{1}{3} - \frac{1}{4} \cdot \frac{1}{x}}{x}}{x}}{x}}{n}\right) \]
   12. Step-by-step derivation

   13. Applied rewrites 76.7%

       \[\leadsto -\frac{-\frac{1 + \left(-\frac{0.5 + \left(-\frac{0.3333333333333333 - \frac{0.25}{x}}{x}\right)}{x}\right)}{x}}{n} \]

   if 1.02000000000000002e170 < x

   1. Initial program 83.4%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 55.1%

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

   6. Taylor expanded in n around inf

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

   8. Applied rewrites 83.4%

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

4. Final simplification 66.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.88:\\ \;\;\;\;\frac{x}{n} - \frac{\log x}{n}\\ \mathbf{elif}\;x \leq 1.02 \cdot 10^{+170}:\\ \;\;\;\;\frac{\frac{-1 + \frac{0.5 - \frac{0.3333333333333333 - \frac{0.25}{x}}{x}}{x}}{x}}{-n}\\ \mathbf{else}:\\ \;\;\;\;1 - 1\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 61.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.88:\\ \;\;\;\;\frac{\log x - x}{-n}\\ \mathbf{elif}\;x \leq 1.02 \cdot 10^{+170}:\\ \;\;\;\;\frac{\frac{-1 + \frac{0.5 - \frac{0.3333333333333333 - \frac{0.25}{x}}{x}}{x}}{x}}{-n}\\ \mathbf{else}:\\ \;\;\;\;1 - 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 0.88)
   (/ (- (log x) x) (- n))
   (if (<= x 1.02e+170)
     (/
      (/ (+ -1.0 (/ (- 0.5 (/ (- 0.3333333333333333 (/ 0.25 x)) x)) x)) x)
      (- n))
     (- 1.0 1.0))))

C

double code(double x, double n) {
	double tmp;
	if (x <= 0.88) {
		tmp = (log(x) - x) / -n;
	} else if (x <= 1.02e+170) {
		tmp = ((-1.0 + ((0.5 - ((0.3333333333333333 - (0.25 / x)) / x)) / x)) / x) / -n;
	} else {
		tmp = 1.0 - 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if (x <= 0.88d0) then
        tmp = (log(x) - x) / -n
    else if (x <= 1.02d+170) then
        tmp = (((-1.0d0) + ((0.5d0 - ((0.3333333333333333d0 - (0.25d0 / x)) / x)) / x)) / x) / -n
    else
        tmp = 1.0d0 - 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (x <= 0.88) {
		tmp = (Math.log(x) - x) / -n;
	} else if (x <= 1.02e+170) {
		tmp = ((-1.0 + ((0.5 - ((0.3333333333333333 - (0.25 / x)) / x)) / x)) / x) / -n;
	} else {
		tmp = 1.0 - 1.0;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 0.88:
		tmp = (math.log(x) - x) / -n
	elif x <= 1.02e+170:
		tmp = ((-1.0 + ((0.5 - ((0.3333333333333333 - (0.25 / x)) / x)) / x)) / x) / -n
	else:
		tmp = 1.0 - 1.0
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 0.88)
		tmp = Float64(Float64(log(x) - x) / Float64(-n));
	elseif (x <= 1.02e+170)
		tmp = Float64(Float64(Float64(-1.0 + Float64(Float64(0.5 - Float64(Float64(0.3333333333333333 - Float64(0.25 / x)) / x)) / x)) / x) / Float64(-n));
	else
		tmp = Float64(1.0 - 1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (x <= 0.88)
		tmp = (log(x) - x) / -n;
	elseif (x <= 1.02e+170)
		tmp = ((-1.0 + ((0.5 - ((0.3333333333333333 - (0.25 / x)) / x)) / x)) / x) / -n;
	else
		tmp = 1.0 - 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 0.88], N[(N[(N[Log[x], $MachinePrecision] - x), $MachinePrecision] / (-n)), $MachinePrecision], If[LessEqual[x, 1.02e+170], N[(N[(N[(-1.0 + N[(N[(0.5 - N[(N[(0.3333333333333333 - N[(0.25 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] / (-n)), $MachinePrecision], N[(1.0 - 1.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 0.880000000000000004

   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 n around -inf

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

   5. Applied rewrites 81.5%

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

   7. Applied rewrites 81.4%

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

   8. Taylor expanded in n around inf

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

   10. Applied rewrites 57.7%

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

   11. Taylor expanded in x around 0

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

   13. Applied rewrites 56.8%

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

   if 0.880000000000000004 < x < 1.02000000000000002e170

   1. Initial program 46.5%

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

   3. Taylor expanded in n around -inf

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

   5. Applied rewrites 48.1%

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

   7. Applied rewrites 44.8%

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

   8. Taylor expanded in n around inf

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

   10. Applied rewrites 48.8%

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

   11. Taylor expanded in x around -inf

       \[\leadsto \mathsf{neg}\left(\frac{-1 \cdot \frac{1 + -1 \cdot \frac{\frac{1}{2} + -1 \cdot \frac{\frac{1}{3} - \frac{1}{4} \cdot \frac{1}{x}}{x}}{x}}{x}}{n}\right) \]
   12. Step-by-step derivation

   13. Applied rewrites 76.7%

       \[\leadsto -\frac{-\frac{1 + \left(-\frac{0.5 + \left(-\frac{0.3333333333333333 - \frac{0.25}{x}}{x}\right)}{x}\right)}{x}}{n} \]

   if 1.02000000000000002e170 < x

   1. Initial program 83.4%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 55.1%

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

   6. Taylor expanded in n around inf

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

   8. Applied rewrites 83.4%

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

4. Final simplification 66.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.88:\\ \;\;\;\;\frac{\log x - x}{-n}\\ \mathbf{elif}\;x \leq 1.02 \cdot 10^{+170}:\\ \;\;\;\;\frac{\frac{-1 + \frac{0.5 - \frac{0.3333333333333333 - \frac{0.25}{x}}{x}}{x}}{x}}{-n}\\ \mathbf{else}:\\ \;\;\;\;1 - 1\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 60.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.7:\\ \;\;\;\;\frac{\log x}{-n}\\ \mathbf{elif}\;x \leq 1.02 \cdot 10^{+170}:\\ \;\;\;\;\frac{\frac{-1 + \frac{0.5 - \frac{0.3333333333333333 - \frac{0.25}{x}}{x}}{x}}{x}}{-n}\\ \mathbf{else}:\\ \;\;\;\;1 - 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 0.7)
   (/ (log x) (- n))
   (if (<= x 1.02e+170)
     (/
      (/ (+ -1.0 (/ (- 0.5 (/ (- 0.3333333333333333 (/ 0.25 x)) x)) x)) x)
      (- n))
     (- 1.0 1.0))))

C

double code(double x, double n) {
	double tmp;
	if (x <= 0.7) {
		tmp = log(x) / -n;
	} else if (x <= 1.02e+170) {
		tmp = ((-1.0 + ((0.5 - ((0.3333333333333333 - (0.25 / x)) / x)) / x)) / x) / -n;
	} else {
		tmp = 1.0 - 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if (x <= 0.7d0) then
        tmp = log(x) / -n
    else if (x <= 1.02d+170) then
        tmp = (((-1.0d0) + ((0.5d0 - ((0.3333333333333333d0 - (0.25d0 / x)) / x)) / x)) / x) / -n
    else
        tmp = 1.0d0 - 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (x <= 0.7) {
		tmp = Math.log(x) / -n;
	} else if (x <= 1.02e+170) {
		tmp = ((-1.0 + ((0.5 - ((0.3333333333333333 - (0.25 / x)) / x)) / x)) / x) / -n;
	} else {
		tmp = 1.0 - 1.0;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 0.7:
		tmp = math.log(x) / -n
	elif x <= 1.02e+170:
		tmp = ((-1.0 + ((0.5 - ((0.3333333333333333 - (0.25 / x)) / x)) / x)) / x) / -n
	else:
		tmp = 1.0 - 1.0
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 0.7)
		tmp = Float64(log(x) / Float64(-n));
	elseif (x <= 1.02e+170)
		tmp = Float64(Float64(Float64(-1.0 + Float64(Float64(0.5 - Float64(Float64(0.3333333333333333 - Float64(0.25 / x)) / x)) / x)) / x) / Float64(-n));
	else
		tmp = Float64(1.0 - 1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (x <= 0.7)
		tmp = log(x) / -n;
	elseif (x <= 1.02e+170)
		tmp = ((-1.0 + ((0.5 - ((0.3333333333333333 - (0.25 / x)) / x)) / x)) / x) / -n;
	else
		tmp = 1.0 - 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 0.7], N[(N[Log[x], $MachinePrecision] / (-n)), $MachinePrecision], If[LessEqual[x, 1.02e+170], N[(N[(N[(-1.0 + N[(N[(0.5 - N[(N[(0.3333333333333333 - N[(0.25 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] / (-n)), $MachinePrecision], N[(1.0 - 1.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 0.69999999999999996

   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 n around -inf

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

   5. Applied rewrites 81.5%

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

   7. Applied rewrites 81.4%

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

   8. Taylor expanded in n around inf

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

   10. Applied rewrites 57.7%

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

   11. Taylor expanded in x around 0

       \[\leadsto \mathsf{neg}\left(\frac{\log x}{n}\right) \]
   12. Step-by-step derivation

   13. Applied rewrites 56.4%

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

   if 0.69999999999999996 < x < 1.02000000000000002e170

   1. Initial program 46.5%

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

   3. Taylor expanded in n around -inf

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

   5. Applied rewrites 48.1%

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

   7. Applied rewrites 44.8%

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

   8. Taylor expanded in n around inf

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

   10. Applied rewrites 48.8%

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

   11. Taylor expanded in x around -inf

       \[\leadsto \mathsf{neg}\left(\frac{-1 \cdot \frac{1 + -1 \cdot \frac{\frac{1}{2} + -1 \cdot \frac{\frac{1}{3} - \frac{1}{4} \cdot \frac{1}{x}}{x}}{x}}{x}}{n}\right) \]
   12. Step-by-step derivation

   13. Applied rewrites 76.7%

       \[\leadsto -\frac{-\frac{1 + \left(-\frac{0.5 + \left(-\frac{0.3333333333333333 - \frac{0.25}{x}}{x}\right)}{x}\right)}{x}}{n} \]

   if 1.02000000000000002e170 < x

   1. Initial program 83.4%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 55.1%

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

   6. Taylor expanded in n around inf

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

   8. Applied rewrites 83.4%

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

4. Final simplification 66.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.7:\\ \;\;\;\;\frac{\log x}{-n}\\ \mathbf{elif}\;x \leq 1.02 \cdot 10^{+170}:\\ \;\;\;\;\frac{\frac{-1 + \frac{0.5 - \frac{0.3333333333333333 - \frac{0.25}{x}}{x}}{x}}{x}}{-n}\\ \mathbf{else}:\\ \;\;\;\;1 - 1\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 50.5% accurate, 3.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.02 \cdot 10^{+170}:\\ \;\;\;\;\frac{\frac{-1 + \frac{0.5 - \frac{0.3333333333333333}{x}}{x}}{x}}{-n}\\ \mathbf{else}:\\ \;\;\;\;1 - 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 1.02e+170)
   (/ (/ (+ -1.0 (/ (- 0.5 (/ 0.3333333333333333 x)) x)) x) (- n))
   (- 1.0 1.0)))

C

double code(double x, double n) {
	double tmp;
	if (x <= 1.02e+170) {
		tmp = ((-1.0 + ((0.5 - (0.3333333333333333 / x)) / x)) / x) / -n;
	} else {
		tmp = 1.0 - 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if (x <= 1.02d+170) then
        tmp = (((-1.0d0) + ((0.5d0 - (0.3333333333333333d0 / x)) / x)) / x) / -n
    else
        tmp = 1.0d0 - 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (x <= 1.02e+170) {
		tmp = ((-1.0 + ((0.5 - (0.3333333333333333 / x)) / x)) / x) / -n;
	} else {
		tmp = 1.0 - 1.0;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 1.02e+170:
		tmp = ((-1.0 + ((0.5 - (0.3333333333333333 / x)) / x)) / x) / -n
	else:
		tmp = 1.0 - 1.0
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 1.02e+170)
		tmp = Float64(Float64(Float64(-1.0 + Float64(Float64(0.5 - Float64(0.3333333333333333 / x)) / x)) / x) / Float64(-n));
	else
		tmp = Float64(1.0 - 1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (x <= 1.02e+170)
		tmp = ((-1.0 + ((0.5 - (0.3333333333333333 / x)) / x)) / x) / -n;
	else
		tmp = 1.0 - 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 1.02e+170], N[(N[(N[(-1.0 + N[(N[(0.5 - N[(0.3333333333333333 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] / (-n)), $MachinePrecision], N[(1.0 - 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.02000000000000002e170

   1. Initial program 39.5%

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

   3. Taylor expanded in n around -inf

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

   5. Applied rewrites 72.0%

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

   7. Applied rewrites 71.1%

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

   8. Taylor expanded in n around inf

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

   10. Applied rewrites 55.2%

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

   11. Taylor expanded in x around -inf

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

   13. Applied rewrites 42.4%

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

   if 1.02000000000000002e170 < x

   1. Initial program 83.4%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 55.1%

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

   6. Taylor expanded in n around inf

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

   8. Applied rewrites 83.4%

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

4. Final simplification 50.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.02 \cdot 10^{+170}:\\ \;\;\;\;\frac{\frac{-1 + \frac{0.5 - \frac{0.3333333333333333}{x}}{x}}{x}}{-n}\\ \mathbf{else}:\\ \;\;\;\;1 - 1\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 47.6% accurate, 5.5× speedup

Math

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

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= (/ 1.0 n) -10.0) (- 1.0 1.0) (/ (/ -1.0 x) (- n))))

C

double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -10.0) {
		tmp = 1.0 - 1.0;
	} else {
		tmp = (-1.0 / x) / -n;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if ((1.0d0 / n) <= (-10.0d0)) then
        tmp = 1.0d0 - 1.0d0
    else
        tmp = ((-1.0d0) / x) / -n
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -10.0) {
		tmp = 1.0 - 1.0;
	} else {
		tmp = (-1.0 / x) / -n;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if (1.0 / n) <= -10.0:
		tmp = 1.0 - 1.0
	else:
		tmp = (-1.0 / x) / -n
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -10.0)
		tmp = Float64(1.0 - 1.0);
	else
		tmp = Float64(Float64(-1.0 / x) / Float64(-n));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if ((1.0 / n) <= -10.0)
		tmp = 1.0 - 1.0;
	else
		tmp = (-1.0 / x) / -n;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[N[(1.0 / n), $MachinePrecision], -10.0], N[(1.0 - 1.0), $MachinePrecision], N[(N[(-1.0 / x), $MachinePrecision] / (-n)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 55.5%

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

   6. Taylor expanded in n around inf

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

   8. Applied rewrites 46.7%

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

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

   1. Initial program 31.3%

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

   3. Taylor expanded in n around -inf

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

   5. Applied rewrites 72.4%

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

   7. Applied rewrites 58.0%

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

   8. Taylor expanded in n around inf

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

   10. Applied rewrites 65.1%

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

   11. Taylor expanded in x around inf

       \[\leadsto \mathsf{neg}\left(\frac{\frac{-1}{x}}{n}\right) \]
   12. Step-by-step derivation

   13. Applied rewrites 46.1%

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

4. Final simplification 46.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -10:\\ \;\;\;\;1 - 1\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-1}{x}}{-n}\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 47.0% accurate, 6.8× speedup

Math

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

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= (/ 1.0 n) -10.0) (- 1.0 1.0) (/ 1.0 (* x n))))

C

double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -10.0) {
		tmp = 1.0 - 1.0;
	} else {
		tmp = 1.0 / (x * n);
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if ((1.0d0 / n) <= (-10.0d0)) then
        tmp = 1.0d0 - 1.0d0
    else
        tmp = 1.0d0 / (x * n)
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -10.0) {
		tmp = 1.0 - 1.0;
	} else {
		tmp = 1.0 / (x * n);
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if (1.0 / n) <= -10.0:
		tmp = 1.0 - 1.0
	else:
		tmp = 1.0 / (x * n)
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -10.0)
		tmp = Float64(1.0 - 1.0);
	else
		tmp = Float64(1.0 / Float64(x * n));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if ((1.0 / n) <= -10.0)
		tmp = 1.0 - 1.0;
	else
		tmp = 1.0 / (x * n);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[N[(1.0 / n), $MachinePrecision], -10.0], N[(1.0 - 1.0), $MachinePrecision], N[(1.0 / N[(x * n), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 55.5%

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

   6. Taylor expanded in n around inf

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

   8. Applied rewrites 46.7%

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

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

   1. Initial program 31.3%

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

   3. Taylor expanded in n around -inf

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

   5. Applied rewrites 72.4%

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

   7. Applied rewrites 58.0%

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

   8. Taylor expanded in n around inf

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

   10. Applied rewrites 65.1%

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

   11. Taylor expanded in x around inf

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

   13. Applied rewrites 45.6%

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

Alternative 22: 31.4% 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

real(8) function code(x, n)
    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 47.7%

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

3. Taylor expanded in x around 0

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

5. Applied rewrites 36.7%

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

6. Taylor expanded in n around inf

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

8. Applied rewrites 28.6%

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

Reproduce

herbie shell --seed 2024227 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  :precision binary64
  (- (pow (+ x 1.0) (/ 1.0 n)) (pow x (/ 1.0 n))))