2nthrt (problem 3.4.6)

Percentage Accurate: 52.6% → 87.1%

Time: 24.0s

Alternatives: 14

Speedup: 1.5×

• could not determine a ground truth

  1. x = -2.24782133833469e-64
  2. n = 8.431507889344372e+73

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: 52.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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: 87.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\log x}{n}\\ \mathbf{if}\;x \leq 2.65 \cdot 10^{-173}:\\ \;\;\;\;\left(1 - \cosh t\_0\right) - \sinh t\_0\\ \mathbf{elif}\;x \leq 7.7:\\ \;\;\;\;\frac{\left(-\log x\right) + \left(\frac{\mathsf{fma}\left(0.16666666666666666, \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{3} - {\log x}^{3}}{n}, \left({\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}\right) \cdot 0.5\right)}{n} + \mathsf{log1p}\left(x\right)\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;{\left({x}^{\left(\frac{-1}{n}\right)} \cdot \left(x \cdot n\right)\right)}^{-1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (/ (log x) n)))
   (if (<= x 2.65e-173)
     (- (- 1.0 (cosh t_0)) (sinh t_0))
     (if (<= x 7.7)
       (/
        (+
         (- (log x))
         (+
          (/
           (fma
            0.16666666666666666
            (/ (- (pow (log1p x) 3.0) (pow (log x) 3.0)) n)
            (* (- (pow (log1p x) 2.0) (pow (log x) 2.0)) 0.5))
           n)
          (log1p x)))
        n)
       (pow (* (pow x (/ -1.0 n)) (* x n)) -1.0)))))

C

double code(double x, double n) {
	double t_0 = log(x) / n;
	double tmp;
	if (x <= 2.65e-173) {
		tmp = (1.0 - cosh(t_0)) - sinh(t_0);
	} else if (x <= 7.7) {
		tmp = (-log(x) + ((fma(0.16666666666666666, ((pow(log1p(x), 3.0) - pow(log(x), 3.0)) / n), ((pow(log1p(x), 2.0) - pow(log(x), 2.0)) * 0.5)) / n) + log1p(x))) / n;
	} else {
		tmp = pow((pow(x, (-1.0 / n)) * (x * n)), -1.0);
	}
	return tmp;
}

Julia

function code(x, n)
	t_0 = Float64(log(x) / n)
	tmp = 0.0
	if (x <= 2.65e-173)
		tmp = Float64(Float64(1.0 - cosh(t_0)) - sinh(t_0));
	elseif (x <= 7.7)
		tmp = Float64(Float64(Float64(-log(x)) + Float64(Float64(fma(0.16666666666666666, Float64(Float64((log1p(x) ^ 3.0) - (log(x) ^ 3.0)) / n), Float64(Float64((log1p(x) ^ 2.0) - (log(x) ^ 2.0)) * 0.5)) / n) + log1p(x))) / n);
	else
		tmp = Float64((x ^ Float64(-1.0 / n)) * Float64(x * n)) ^ -1.0;
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(N[Log[x], $MachinePrecision] / n), $MachinePrecision]}, If[LessEqual[x, 2.65e-173], N[(N[(1.0 - N[Cosh[t$95$0], $MachinePrecision]), $MachinePrecision] - N[Sinh[t$95$0], $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 7.7], N[(N[((-N[Log[x], $MachinePrecision]) + N[(N[(N[(0.16666666666666666 * N[(N[(N[Power[N[Log[1 + x], $MachinePrecision], 3.0], $MachinePrecision] - N[Power[N[Log[x], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision] + N[(N[(N[Power[N[Log[1 + x], $MachinePrecision], 2.0], $MachinePrecision] - N[Power[N[Log[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision] + N[Log[1 + x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], N[Power[N[(N[Power[x, N[(-1.0 / n), $MachinePrecision]], $MachinePrecision] * N[(x * n), $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < 2.64999999999999982e-173

   1. Initial program 46.0%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   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 46.0%

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

      1. lift--.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. pow-to-exp N/A

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

      5. lift-log.f64 N/A

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

      6. div-inv N/A

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

      7. lift-/.f64 N/A

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

      8. sinh-+-cosh-rev N/A

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

      9. associate--r+ N/A

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

      10. lower--.f64 N/A

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

      11. lower--.f64 N/A

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

      12. lower-cosh.f64 N/A

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

      13. lower-sinh.f64 85.5

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

   7. Applied rewrites 85.5%

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

   if 2.64999999999999982e-173 < x < 7.70000000000000018

   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 89.4%

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

   if 7.70000000000000018 < x

   1. Initial program 71.8%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   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. mul-1-neg N/A

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

      3. log-rec N/A

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

      4. mul-1-neg N/A

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

      5. distribute-neg-frac N/A

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

      6. mul-1-neg N/A

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

      7. remove-double-neg N/A

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

      8. lower-exp.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-log.f64 N/A

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

      11. lower-*.f64 96.1

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

   5. Applied rewrites 96.1%

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

   7. Applied rewrites 96.1%

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

   9. Applied rewrites 96.2%

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

4. Final simplification 91.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.65 \cdot 10^{-173}:\\ \;\;\;\;\left(1 - \cosh \left(\frac{\log x}{n}\right)\right) - \sinh \left(\frac{\log x}{n}\right)\\ \mathbf{elif}\;x \leq 7.7:\\ \;\;\;\;\frac{\left(-\log x\right) + \left(\frac{\mathsf{fma}\left(0.16666666666666666, \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{3} - {\log x}^{3}}{n}, \left({\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}\right) \cdot 0.5\right)}{n} + \mathsf{log1p}\left(x\right)\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;{\left({x}^{\left(\frac{-1}{n}\right)} \cdot \left(x \cdot n\right)\right)}^{-1}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 85.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{0.5}{n}\right)}\\ \mathbf{if}\;{n}^{-1} \leq -1 \cdot 10^{-7}:\\ \;\;\;\;{\left(x + 1\right)}^{\left({n}^{-1}\right)} - {x}^{\left({n}^{-1}\right)}\\ \mathbf{elif}\;{n}^{-1} \leq 5 \cdot 10^{-8}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right) - \log x}{n}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t\_0, -t\_0, e^{\frac{\mathsf{log1p}\left(x\right)}{n}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 0.5 n))))
   (if (<= (pow n -1.0) -1e-7)
     (- (pow (+ x 1.0) (pow n -1.0)) (pow x (pow n -1.0)))
     (if (<= (pow n -1.0) 5e-8)
       (/ (- (log1p x) (log x)) n)
       (fma t_0 (- t_0) (exp (/ (log1p x) n)))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (0.5 / n));
	double tmp;
	if (pow(n, -1.0) <= -1e-7) {
		tmp = pow((x + 1.0), pow(n, -1.0)) - pow(x, pow(n, -1.0));
	} else if (pow(n, -1.0) <= 5e-8) {
		tmp = (log1p(x) - log(x)) / n;
	} else {
		tmp = fma(t_0, -t_0, exp((log1p(x) / n)));
	}
	return tmp;
}

Julia

function code(x, n)
	t_0 = x ^ Float64(0.5 / n)
	tmp = 0.0
	if ((n ^ -1.0) <= -1e-7)
		tmp = Float64((Float64(x + 1.0) ^ (n ^ -1.0)) - (x ^ (n ^ -1.0)));
	elseif ((n ^ -1.0) <= 5e-8)
		tmp = Float64(Float64(log1p(x) - log(x)) / n);
	else
		tmp = fma(t_0, Float64(-t_0), exp(Float64(log1p(x) / n)));
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(0.5 / n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[Power[n, -1.0], $MachinePrecision], -1e-7], N[(N[Power[N[(x + 1.0), $MachinePrecision], N[Power[n, -1.0], $MachinePrecision]], $MachinePrecision] - N[Power[x, N[Power[n, -1.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Power[n, -1.0], $MachinePrecision], 5e-8], N[(N[(N[Log[1 + x], $MachinePrecision] - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], N[(t$95$0 * (-t$95$0) + N[Exp[N[(N[Log[1 + x], $MachinePrecision] / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -9.9999999999999995e-8

   1. Initial program 99.8%

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

   if -9.9999999999999995e-8 < (/.f64 #s(literal 1 binary64) n) < 4.9999999999999998e-8

   1. Initial program 28.9%

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

   3. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-log1p.f64 N/A

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

      4. lower-log.f64 79.5

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

   5. Applied rewrites 79.5%

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

   if 4.9999999999999998e-8 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 33.5%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-pow.f64 N/A

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

      5. sqr-pow N/A

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

      6. distribute-rgt-neg-in N/A

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

      7. lower-fma.f64 N/A

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

   4. Applied rewrites 99.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{\left(\frac{0.5}{n}\right)}, -{x}^{\left(\frac{0.5}{n}\right)}, e^{\frac{\mathsf{log1p}\left(x\right)}{n}}\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 88.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{n}^{-1} \leq -1 \cdot 10^{-7}:\\ \;\;\;\;{\left(x + 1\right)}^{\left({n}^{-1}\right)} - {x}^{\left({n}^{-1}\right)}\\ \mathbf{elif}\;{n}^{-1} \leq 5 \cdot 10^{-8}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right) - \log x}{n}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left({x}^{\left(\frac{0.5}{n}\right)}, -{x}^{\left(\frac{0.5}{n}\right)}, e^{\frac{\mathsf{log1p}\left(x\right)}{n}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 85.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left({n}^{-1}\right)}\\ \mathbf{if}\;{n}^{-1} \leq -1 \cdot 10^{-7}:\\ \;\;\;\;{\left(x + 1\right)}^{\left({n}^{-1}\right)} - t\_0\\ \mathbf{elif}\;{n}^{-1} \leq 10^{-12}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right) - \log x}{n}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (pow n -1.0))))
   (if (<= (pow n -1.0) -1e-7)
     (- (pow (+ x 1.0) (pow n -1.0)) t_0)
     (if (<= (pow n -1.0) 1e-12)
       (/ (- (log1p x) (log x)) n)
       (- (exp (/ (log1p x) n)) t_0)))))

C

double code(double x, double n) {
	double t_0 = pow(x, pow(n, -1.0));
	double tmp;
	if (pow(n, -1.0) <= -1e-7) {
		tmp = pow((x + 1.0), pow(n, -1.0)) - t_0;
	} else if (pow(n, -1.0) <= 1e-12) {
		tmp = (log1p(x) - log(x)) / n;
	} else {
		tmp = exp((log1p(x) / n)) - t_0;
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, Math.pow(n, -1.0));
	double tmp;
	if (Math.pow(n, -1.0) <= -1e-7) {
		tmp = Math.pow((x + 1.0), Math.pow(n, -1.0)) - t_0;
	} else if (Math.pow(n, -1.0) <= 1e-12) {
		tmp = (Math.log1p(x) - Math.log(x)) / n;
	} else {
		tmp = Math.exp((Math.log1p(x) / n)) - t_0;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, math.pow(n, -1.0))
	tmp = 0
	if math.pow(n, -1.0) <= -1e-7:
		tmp = math.pow((x + 1.0), math.pow(n, -1.0)) - t_0
	elif math.pow(n, -1.0) <= 1e-12:
		tmp = (math.log1p(x) - math.log(x)) / n
	else:
		tmp = math.exp((math.log1p(x) / n)) - t_0
	return tmp

Julia

function code(x, n)
	t_0 = x ^ (n ^ -1.0)
	tmp = 0.0
	if ((n ^ -1.0) <= -1e-7)
		tmp = Float64((Float64(x + 1.0) ^ (n ^ -1.0)) - t_0);
	elseif ((n ^ -1.0) <= 1e-12)
		tmp = Float64(Float64(log1p(x) - log(x)) / n);
	else
		tmp = Float64(exp(Float64(log1p(x) / n)) - t_0);
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[Power[n, -1.0], $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[Power[n, -1.0], $MachinePrecision], -1e-7], N[(N[Power[N[(x + 1.0), $MachinePrecision], N[Power[n, -1.0], $MachinePrecision]], $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[N[Power[n, -1.0], $MachinePrecision], 1e-12], N[(N[(N[Log[1 + x], $MachinePrecision] - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], N[(N[Exp[N[(N[Log[1 + x], $MachinePrecision] / n), $MachinePrecision]], $MachinePrecision] - t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -9.9999999999999995e-8

   1. Initial program 99.8%

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

   if -9.9999999999999995e-8 < (/.f64 #s(literal 1 binary64) n) < 9.9999999999999998e-13

   1. Initial program 29.2%

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

   3. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-log1p.f64 N/A

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

      4. lower-log.f64 80.5

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

   5. Applied rewrites 80.5%

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

   if 9.9999999999999998e-13 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 31.9%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   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 93.5

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

   4. Applied rewrites 93.5%

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

4. Final simplification 88.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{n}^{-1} \leq -1 \cdot 10^{-7}:\\ \;\;\;\;{\left(x + 1\right)}^{\left({n}^{-1}\right)} - {x}^{\left({n}^{-1}\right)}\\ \mathbf{elif}\;{n}^{-1} \leq 10^{-12}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right) - \log x}{n}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left({n}^{-1}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 86.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\log x}{n}\\ \mathbf{if}\;x \leq 2.65 \cdot 10^{-173}:\\ \;\;\;\;\left(1 - \cosh t\_0\right) - \sinh t\_0\\ \mathbf{elif}\;x \leq 0.205:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.5, \frac{{\log x}^{2}}{n}, \log x\right) - \frac{-0.16666666666666666}{n} \cdot \frac{{\log x}^{3}}{n}}{-n}\\ \mathbf{else}:\\ \;\;\;\;{\left({x}^{\left(\frac{-1}{n}\right)} \cdot \left(x \cdot n\right)\right)}^{-1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (/ (log x) n)))
   (if (<= x 2.65e-173)
     (- (- 1.0 (cosh t_0)) (sinh t_0))
     (if (<= x 0.205)
       (/
        (-
         (fma 0.5 (/ (pow (log x) 2.0) n) (log x))
         (* (/ -0.16666666666666666 n) (/ (pow (log x) 3.0) n)))
        (- n))
       (pow (* (pow x (/ -1.0 n)) (* x n)) -1.0)))))

C

double code(double x, double n) {
	double t_0 = log(x) / n;
	double tmp;
	if (x <= 2.65e-173) {
		tmp = (1.0 - cosh(t_0)) - sinh(t_0);
	} else if (x <= 0.205) {
		tmp = (fma(0.5, (pow(log(x), 2.0) / n), log(x)) - ((-0.16666666666666666 / n) * (pow(log(x), 3.0) / n))) / -n;
	} else {
		tmp = pow((pow(x, (-1.0 / n)) * (x * n)), -1.0);
	}
	return tmp;
}

Julia

function code(x, n)
	t_0 = Float64(log(x) / n)
	tmp = 0.0
	if (x <= 2.65e-173)
		tmp = Float64(Float64(1.0 - cosh(t_0)) - sinh(t_0));
	elseif (x <= 0.205)
		tmp = Float64(Float64(fma(0.5, Float64((log(x) ^ 2.0) / n), log(x)) - Float64(Float64(-0.16666666666666666 / n) * Float64((log(x) ^ 3.0) / n))) / Float64(-n));
	else
		tmp = Float64((x ^ Float64(-1.0 / n)) * Float64(x * n)) ^ -1.0;
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(N[Log[x], $MachinePrecision] / n), $MachinePrecision]}, If[LessEqual[x, 2.65e-173], N[(N[(1.0 - N[Cosh[t$95$0], $MachinePrecision]), $MachinePrecision] - N[Sinh[t$95$0], $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.205], N[(N[(N[(0.5 * N[(N[Power[N[Log[x], $MachinePrecision], 2.0], $MachinePrecision] / n), $MachinePrecision] + N[Log[x], $MachinePrecision]), $MachinePrecision] - N[(N[(-0.16666666666666666 / n), $MachinePrecision] * N[(N[Power[N[Log[x], $MachinePrecision], 3.0], $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / (-n)), $MachinePrecision], N[Power[N[(N[Power[x, N[(-1.0 / n), $MachinePrecision]], $MachinePrecision] * N[(x * n), $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < 2.64999999999999982e-173

   1. Initial program 46.0%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   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 46.0%

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

      1. lift--.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. pow-to-exp N/A

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

      5. lift-log.f64 N/A

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

      6. div-inv N/A

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

      7. lift-/.f64 N/A

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

      8. sinh-+-cosh-rev N/A

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

      9. associate--r+ N/A

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

      10. lower--.f64 N/A

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

      11. lower--.f64 N/A

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

      12. lower-cosh.f64 N/A

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

      13. lower-sinh.f64 85.5

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

   7. Applied rewrites 85.5%

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

   if 2.64999999999999982e-173 < x < 0.204999999999999988

   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 89.4%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 85.0%

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

   if 0.204999999999999988 < x

   1. Initial program 71.8%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   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. mul-1-neg N/A

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

      3. log-rec N/A

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

      4. mul-1-neg N/A

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

      5. distribute-neg-frac N/A

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

      6. mul-1-neg N/A

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

      7. remove-double-neg N/A

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

      8. lower-exp.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-log.f64 N/A

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

      11. lower-*.f64 96.1

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

   5. Applied rewrites 96.1%

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

   7. Applied rewrites 96.1%

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

   9. Applied rewrites 96.2%

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

4. Final simplification 89.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.65 \cdot 10^{-173}:\\ \;\;\;\;\left(1 - \cosh \left(\frac{\log x}{n}\right)\right) - \sinh \left(\frac{\log x}{n}\right)\\ \mathbf{elif}\;x \leq 0.205:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.5, \frac{{\log x}^{2}}{n}, \log x\right) - \frac{-0.16666666666666666}{n} \cdot \frac{{\log x}^{3}}{n}}{-n}\\ \mathbf{else}:\\ \;\;\;\;{\left({x}^{\left(\frac{-1}{n}\right)} \cdot \left(x \cdot n\right)\right)}^{-1}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 81.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left({n}^{-1}\right)}\\ \mathbf{if}\;{n}^{-1} \leq -1 \cdot 10^{-7}:\\ \;\;\;\;{\left(x + 1\right)}^{\left({n}^{-1}\right)} - t\_0\\ \mathbf{elif}\;{n}^{-1} \leq 5 \cdot 10^{-8}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right) - \log x}{n}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{n \cdot n} \cdot 0.5, x, 1\right) - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (pow n -1.0))))
   (if (<= (pow n -1.0) -1e-7)
     (- (pow (+ x 1.0) (pow n -1.0)) t_0)
     (if (<= (pow n -1.0) 5e-8)
       (/ (- (log1p x) (log x)) n)
       (- (fma (* (/ x (* n n)) 0.5) x 1.0) t_0)))))

C

double code(double x, double n) {
	double t_0 = pow(x, pow(n, -1.0));
	double tmp;
	if (pow(n, -1.0) <= -1e-7) {
		tmp = pow((x + 1.0), pow(n, -1.0)) - t_0;
	} else if (pow(n, -1.0) <= 5e-8) {
		tmp = (log1p(x) - log(x)) / n;
	} else {
		tmp = fma(((x / (n * n)) * 0.5), x, 1.0) - t_0;
	}
	return tmp;
}

Julia

function code(x, n)
	t_0 = x ^ (n ^ -1.0)
	tmp = 0.0
	if ((n ^ -1.0) <= -1e-7)
		tmp = Float64((Float64(x + 1.0) ^ (n ^ -1.0)) - t_0);
	elseif ((n ^ -1.0) <= 5e-8)
		tmp = Float64(Float64(log1p(x) - log(x)) / n);
	else
		tmp = Float64(fma(Float64(Float64(x / Float64(n * n)) * 0.5), x, 1.0) - t_0);
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[Power[n, -1.0], $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[Power[n, -1.0], $MachinePrecision], -1e-7], N[(N[Power[N[(x + 1.0), $MachinePrecision], N[Power[n, -1.0], $MachinePrecision]], $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[N[Power[n, -1.0], $MachinePrecision], 5e-8], N[(N[(N[Log[1 + x], $MachinePrecision] - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], N[(N[(N[(N[(x / N[(n * n), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision] * x + 1.0), $MachinePrecision] - t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -9.9999999999999995e-8

   1. Initial program 99.8%

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

   if -9.9999999999999995e-8 < (/.f64 #s(literal 1 binary64) n) < 4.9999999999999998e-8

   1. Initial program 28.9%

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

   3. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-log1p.f64 N/A

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

      4. lower-log.f64 79.5

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

   5. Applied rewrites 79.5%

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

   if 4.9999999999999998e-8 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 33.5%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(1 + x \cdot \left(x \cdot \left(\frac{1}{2} \cdot \frac{1}{{n}^{2}} - \frac{1}{2} \cdot \frac{1}{n}\right) + \frac{1}{n}\right)\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   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. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

   5. Applied rewrites 74.4%

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

   6. Taylor expanded in n around 0

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

   8. Applied rewrites 74.4%

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

4. Final simplification 85.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{n}^{-1} \leq -1 \cdot 10^{-7}:\\ \;\;\;\;{\left(x + 1\right)}^{\left({n}^{-1}\right)} - {x}^{\left({n}^{-1}\right)}\\ \mathbf{elif}\;{n}^{-1} \leq 5 \cdot 10^{-8}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right) - \log x}{n}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{n \cdot n} \cdot 0.5, x, 1\right) - {x}^{\left({n}^{-1}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 81.6% accurate, 0.5× speedup

Math

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

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= (pow n -1.0) -10.0)
   (pow (* (pow x (/ -1.0 n)) (* x n)) -1.0)
   (if (<= (pow n -1.0) 5e-8)
     (/ (- (log1p x) (log x)) n)
     (- (fma (* (/ x (* n n)) 0.5) x 1.0) (pow x (pow n -1.0))))))

C

double code(double x, double n) {
	double tmp;
	if (pow(n, -1.0) <= -10.0) {
		tmp = pow((pow(x, (-1.0 / n)) * (x * n)), -1.0);
	} else if (pow(n, -1.0) <= 5e-8) {
		tmp = (log1p(x) - log(x)) / n;
	} else {
		tmp = fma(((x / (n * n)) * 0.5), x, 1.0) - pow(x, pow(n, -1.0));
	}
	return tmp;
}

Julia

function code(x, n)
	tmp = 0.0
	if ((n ^ -1.0) <= -10.0)
		tmp = Float64((x ^ Float64(-1.0 / n)) * Float64(x * n)) ^ -1.0;
	elseif ((n ^ -1.0) <= 5e-8)
		tmp = Float64(Float64(log1p(x) - log(x)) / n);
	else
		tmp = Float64(fma(Float64(Float64(x / Float64(n * n)) * 0.5), x, 1.0) - (x ^ (n ^ -1.0)));
	end
	return tmp
end

Wolfram

code[x_, n_] := If[LessEqual[N[Power[n, -1.0], $MachinePrecision], -10.0], N[Power[N[(N[Power[x, N[(-1.0 / n), $MachinePrecision]], $MachinePrecision] * N[(x * n), $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision], If[LessEqual[N[Power[n, -1.0], $MachinePrecision], 5e-8], N[(N[(N[Log[1 + x], $MachinePrecision] - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], N[(N[(N[(N[(x / N[(n * n), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision] * x + 1.0), $MachinePrecision] - N[Power[x, N[Power[n, -1.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 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 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. mul-1-neg N/A

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

      3. log-rec N/A

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

      4. mul-1-neg N/A

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

      5. distribute-neg-frac N/A

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

      6. mul-1-neg N/A

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

      7. remove-double-neg N/A

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

      8. lower-exp.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-log.f64 N/A

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

      11. lower-*.f64 100.0

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

   5. Applied rewrites 100.0%

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

   7. Applied rewrites 100.0%

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

   9. Applied rewrites 100.0%

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

   if -10 < (/.f64 #s(literal 1 binary64) n) < 4.9999999999999998e-8

   1. Initial program 29.3%

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

   3. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-log1p.f64 N/A

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

      4. lower-log.f64 79.2

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

   5. Applied rewrites 79.2%

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

   if 4.9999999999999998e-8 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 33.5%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(1 + x \cdot \left(x \cdot \left(\frac{1}{2} \cdot \frac{1}{{n}^{2}} - \frac{1}{2} \cdot \frac{1}{n}\right) + \frac{1}{n}\right)\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   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. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

   5. Applied rewrites 74.4%

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

   6. Taylor expanded in n around 0

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

   8. Applied rewrites 74.4%

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

4. Final simplification 85.1%

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

Alternative 7: 70.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.3333333333333333, x, 0.5\right), x, -1\right), x, \log x\right)}{-n}\\ \mathbf{if}\;x \leq 2.05 \cdot 10^{-194}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 3.8 \cdot 10^{-129}:\\ \;\;\;\;\frac{\left(\frac{0.3333333333333333}{x \cdot x} + 1\right) - \frac{0.5}{x}}{n \cdot x}\\ \mathbf{elif}\;x \leq 0.00044:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;{\left({x}^{\left(\frac{-1}{n}\right)} \cdot \left(x \cdot n\right)\right)}^{-1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0
         (/
          (fma (fma (fma -0.3333333333333333 x 0.5) x -1.0) x (log x))
          (- n))))
   (if (<= x 2.05e-194)
     t_0
     (if (<= x 3.8e-129)
       (/ (- (+ (/ 0.3333333333333333 (* x x)) 1.0) (/ 0.5 x)) (* n x))
       (if (<= x 0.00044) t_0 (pow (* (pow x (/ -1.0 n)) (* x n)) -1.0))))))

C

double code(double x, double n) {
	double t_0 = fma(fma(fma(-0.3333333333333333, x, 0.5), x, -1.0), x, log(x)) / -n;
	double tmp;
	if (x <= 2.05e-194) {
		tmp = t_0;
	} else if (x <= 3.8e-129) {
		tmp = (((0.3333333333333333 / (x * x)) + 1.0) - (0.5 / x)) / (n * x);
	} else if (x <= 0.00044) {
		tmp = t_0;
	} else {
		tmp = pow((pow(x, (-1.0 / n)) * (x * n)), -1.0);
	}
	return tmp;
}

Julia

function code(x, n)
	t_0 = Float64(fma(fma(fma(-0.3333333333333333, x, 0.5), x, -1.0), x, log(x)) / Float64(-n))
	tmp = 0.0
	if (x <= 2.05e-194)
		tmp = t_0;
	elseif (x <= 3.8e-129)
		tmp = Float64(Float64(Float64(Float64(0.3333333333333333 / Float64(x * x)) + 1.0) - Float64(0.5 / x)) / Float64(n * x));
	elseif (x <= 0.00044)
		tmp = t_0;
	else
		tmp = Float64((x ^ Float64(-1.0 / n)) * Float64(x * n)) ^ -1.0;
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(N[(N[(N[(-0.3333333333333333 * x + 0.5), $MachinePrecision] * x + -1.0), $MachinePrecision] * x + N[Log[x], $MachinePrecision]), $MachinePrecision] / (-n)), $MachinePrecision]}, If[LessEqual[x, 2.05e-194], t$95$0, If[LessEqual[x, 3.8e-129], N[(N[(N[(N[(0.3333333333333333 / N[(x * x), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] - N[(0.5 / x), $MachinePrecision]), $MachinePrecision] / N[(n * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.00044], t$95$0, N[Power[N[(N[Power[x, N[(-1.0 / n), $MachinePrecision]], $MachinePrecision] * N[(x * n), $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if x < 2.0500000000000001e-194 or 3.79999999999999985e-129 < x < 4.40000000000000016e-4

   1. Initial program 33.6%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   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 85.8%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 67.1%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-\mathsf{fma}\left(\frac{0.5}{n \cdot n} - \left(\frac{0.3333333333333333}{n} + \frac{0.16666666666666666}{{n}^{3}}\right), x, \frac{0.5}{n} - \frac{0.5}{n \cdot n}\right), x, \frac{1}{n}\right), \color{blue}{x}, \frac{-\left(\mathsf{fma}\left(0.5, \frac{{\log x}^{2}}{n}, \log x\right) - \frac{-0.16666666666666666}{n} \cdot \frac{{\log x}^{3}}{n}\right)}{n}\right) \]

   9. Taylor expanded in n around -inf

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

   11. Applied rewrites 59.6%

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

   if 2.0500000000000001e-194 < x < 3.79999999999999985e-129

   1. Initial program 59.3%

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

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 35.5%

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

   6. Taylor expanded in n around inf

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

   8. Applied rewrites 74.6%

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

   if 4.40000000000000016e-4 < x

   1. Initial program 70.8%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   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. mul-1-neg N/A

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

      3. log-rec N/A

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

      4. mul-1-neg N/A

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

      5. distribute-neg-frac N/A

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

      6. mul-1-neg N/A

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

      7. remove-double-neg N/A

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

      8. lower-exp.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-log.f64 N/A

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

      11. lower-*.f64 94.7

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

   5. Applied rewrites 94.7%

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

   7. Applied rewrites 94.7%

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

   9. Applied rewrites 94.7%

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

4. Final simplification 76.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.05 \cdot 10^{-194}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.3333333333333333, x, 0.5\right), x, -1\right), x, \log x\right)}{-n}\\ \mathbf{elif}\;x \leq 3.8 \cdot 10^{-129}:\\ \;\;\;\;\frac{\left(\frac{0.3333333333333333}{x \cdot x} + 1\right) - \frac{0.5}{x}}{n \cdot x}\\ \mathbf{elif}\;x \leq 0.00044:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.3333333333333333, x, 0.5\right), x, -1\right), x, \log x\right)}{-n}\\ \mathbf{else}:\\ \;\;\;\;{\left({x}^{\left(\frac{-1}{n}\right)} \cdot \left(x \cdot n\right)\right)}^{-1}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 50.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.35 \cdot 10^{-169}:\\ \;\;\;\;\left(\frac{x}{n} + 1\right) - {x}^{\left({n}^{-1}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(\frac{0.3333333333333333}{x \cdot x} + 1\right) - \frac{0.5}{x}}{n}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 1.35e-169)
   (- (+ (/ x n) 1.0) (pow x (pow n -1.0)))
   (/ (/ (- (+ (/ 0.3333333333333333 (* x x)) 1.0) (/ 0.5 x)) n) x)))

C

double code(double x, double n) {
	double tmp;
	if (x <= 1.35e-169) {
		tmp = ((x / n) + 1.0) - pow(x, pow(n, -1.0));
	} else {
		tmp = ((((0.3333333333333333 / (x * x)) + 1.0) - (0.5 / x)) / n) / x;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if (x <= 1.35d-169) then
        tmp = ((x / n) + 1.0d0) - (x ** (n ** (-1.0d0)))
    else
        tmp = ((((0.3333333333333333d0 / (x * x)) + 1.0d0) - (0.5d0 / x)) / n) / x
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (x <= 1.35e-169) {
		tmp = ((x / n) + 1.0) - Math.pow(x, Math.pow(n, -1.0));
	} else {
		tmp = ((((0.3333333333333333 / (x * x)) + 1.0) - (0.5 / x)) / n) / x;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 1.35e-169:
		tmp = ((x / n) + 1.0) - math.pow(x, math.pow(n, -1.0))
	else:
		tmp = ((((0.3333333333333333 / (x * x)) + 1.0) - (0.5 / x)) / n) / x
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 1.35e-169)
		tmp = Float64(Float64(Float64(x / n) + 1.0) - (x ^ (n ^ -1.0)));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(0.3333333333333333 / Float64(x * x)) + 1.0) - Float64(0.5 / x)) / n) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (x <= 1.35e-169)
		tmp = ((x / n) + 1.0) - (x ^ (n ^ -1.0));
	else
		tmp = ((((0.3333333333333333 / (x * x)) + 1.0) - (0.5 / x)) / n) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 1.35e-169], N[(N[(N[(x / n), $MachinePrecision] + 1.0), $MachinePrecision] - N[Power[x, N[Power[n, -1.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(0.3333333333333333 / N[(x * x), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] - N[(0.5 / x), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.3500000000000001e-169

   1. Initial program 45.8%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   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. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. associate-*r/ N/A

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

      6. *-rgt-identity N/A

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

      7. lower-/.f64 45.8

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

   5. Applied rewrites 45.8%

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

   if 1.3500000000000001e-169 < x

   1. Initial program 54.1%

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

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 50.0%

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

   6. Taylor expanded in n around inf

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

   8. Applied rewrites 52.2%

      \[\leadsto \frac{\frac{\left(\frac{0.3333333333333333}{x \cdot x} + 1\right) - \frac{0.5}{x}}{n}}{x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 50.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.35 \cdot 10^{-169}:\\ \;\;\;\;\left(\frac{x}{n} + 1\right) - {x}^{\left({n}^{-1}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(\frac{0.3333333333333333}{x \cdot x} + 1\right) - \frac{0.5}{x}}{n}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 50.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.35 \cdot 10^{-169}:\\ \;\;\;\;1 - {x}^{\left({n}^{-1}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(\frac{0.3333333333333333}{x \cdot x} + 1\right) - \frac{0.5}{x}}{n}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 1.35e-169)
   (- 1.0 (pow x (pow n -1.0)))
   (/ (/ (- (+ (/ 0.3333333333333333 (* x x)) 1.0) (/ 0.5 x)) n) x)))

C

double code(double x, double n) {
	double tmp;
	if (x <= 1.35e-169) {
		tmp = 1.0 - pow(x, pow(n, -1.0));
	} else {
		tmp = ((((0.3333333333333333 / (x * x)) + 1.0) - (0.5 / x)) / n) / x;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if (x <= 1.35d-169) then
        tmp = 1.0d0 - (x ** (n ** (-1.0d0)))
    else
        tmp = ((((0.3333333333333333d0 / (x * x)) + 1.0d0) - (0.5d0 / x)) / n) / x
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (x <= 1.35e-169) {
		tmp = 1.0 - Math.pow(x, Math.pow(n, -1.0));
	} else {
		tmp = ((((0.3333333333333333 / (x * x)) + 1.0) - (0.5 / x)) / n) / x;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 1.35e-169:
		tmp = 1.0 - math.pow(x, math.pow(n, -1.0))
	else:
		tmp = ((((0.3333333333333333 / (x * x)) + 1.0) - (0.5 / x)) / n) / x
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 1.35e-169)
		tmp = Float64(1.0 - (x ^ (n ^ -1.0)));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(0.3333333333333333 / Float64(x * x)) + 1.0) - Float64(0.5 / x)) / n) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (x <= 1.35e-169)
		tmp = 1.0 - (x ^ (n ^ -1.0));
	else
		tmp = ((((0.3333333333333333 / (x * x)) + 1.0) - (0.5 / x)) / n) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 1.35e-169], N[(1.0 - N[Power[x, N[Power[n, -1.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(0.3333333333333333 / N[(x * x), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] - N[(0.5 / x), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.3500000000000001e-169

   1. Initial program 45.8%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   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 45.8%

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

   if 1.3500000000000001e-169 < x

   1. Initial program 54.1%

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

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 50.0%

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

   6. Taylor expanded in n around inf

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

   8. Applied rewrites 52.2%

      \[\leadsto \frac{\frac{\left(\frac{0.3333333333333333}{x \cdot x} + 1\right) - \frac{0.5}{x}}{n}}{x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 50.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.35 \cdot 10^{-169}:\\ \;\;\;\;1 - {x}^{\left({n}^{-1}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(\frac{0.3333333333333333}{x \cdot x} + 1\right) - \frac{0.5}{x}}{n}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 57.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\frac{0.3333333333333333}{x \cdot x} + 1\right) - \frac{0.5}{x}\\ t_1 := \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.3333333333333333, x, 0.5\right), x, -1\right), x, \log x\right)}{-n}\\ \mathbf{if}\;x \leq 2.05 \cdot 10^{-194}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 3.8 \cdot 10^{-129}:\\ \;\;\;\;\frac{t\_0}{n \cdot x}\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{t\_0}{n}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (- (+ (/ 0.3333333333333333 (* x x)) 1.0) (/ 0.5 x)))
        (t_1
         (/
          (fma (fma (fma -0.3333333333333333 x 0.5) x -1.0) x (log x))
          (- n))))
   (if (<= x 2.05e-194)
     t_1
     (if (<= x 3.8e-129)
       (/ t_0 (* n x))
       (if (<= x 1.0) t_1 (/ (/ t_0 n) x))))))

C

double code(double x, double n) {
	double t_0 = ((0.3333333333333333 / (x * x)) + 1.0) - (0.5 / x);
	double t_1 = fma(fma(fma(-0.3333333333333333, x, 0.5), x, -1.0), x, log(x)) / -n;
	double tmp;
	if (x <= 2.05e-194) {
		tmp = t_1;
	} else if (x <= 3.8e-129) {
		tmp = t_0 / (n * x);
	} else if (x <= 1.0) {
		tmp = t_1;
	} else {
		tmp = (t_0 / n) / x;
	}
	return tmp;
}

Julia

function code(x, n)
	t_0 = Float64(Float64(Float64(0.3333333333333333 / Float64(x * x)) + 1.0) - Float64(0.5 / x))
	t_1 = Float64(fma(fma(fma(-0.3333333333333333, x, 0.5), x, -1.0), x, log(x)) / Float64(-n))
	tmp = 0.0
	if (x <= 2.05e-194)
		tmp = t_1;
	elseif (x <= 3.8e-129)
		tmp = Float64(t_0 / Float64(n * x));
	elseif (x <= 1.0)
		tmp = t_1;
	else
		tmp = Float64(Float64(t_0 / n) / x);
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(N[(N[(0.3333333333333333 / N[(x * x), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] - N[(0.5 / x), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[(-0.3333333333333333 * x + 0.5), $MachinePrecision] * x + -1.0), $MachinePrecision] * x + N[Log[x], $MachinePrecision]), $MachinePrecision] / (-n)), $MachinePrecision]}, If[LessEqual[x, 2.05e-194], t$95$1, If[LessEqual[x, 3.8e-129], N[(t$95$0 / N[(n * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.0], t$95$1, N[(N[(t$95$0 / n), $MachinePrecision] / x), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < 2.0500000000000001e-194 or 3.79999999999999985e-129 < x < 1

   1. Initial program 33.6%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   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 86.1%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 66.5%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-\mathsf{fma}\left(\frac{0.5}{n \cdot n} - \left(\frac{0.3333333333333333}{n} + \frac{0.16666666666666666}{{n}^{3}}\right), x, \frac{0.5}{n} - \frac{0.5}{n \cdot n}\right), x, \frac{1}{n}\right), \color{blue}{x}, \frac{-\left(\mathsf{fma}\left(0.5, \frac{{\log x}^{2}}{n}, \log x\right) - \frac{-0.16666666666666666}{n} \cdot \frac{{\log x}^{3}}{n}\right)}{n}\right) \]

   9. Taylor expanded in n around -inf

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

   11. Applied rewrites 59.2%

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

   if 2.0500000000000001e-194 < x < 3.79999999999999985e-129

   1. Initial program 59.3%

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

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 35.5%

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

   6. Taylor expanded in n around inf

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

   8. Applied rewrites 74.6%

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

   if 1 < x

   1. Initial program 71.8%

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

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 75.6%

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

   6. Taylor expanded in n around inf

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

   8. Applied rewrites 62.0%

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

4. Final simplification 61.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.05 \cdot 10^{-194}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.3333333333333333, x, 0.5\right), x, -1\right), x, \log x\right)}{-n}\\ \mathbf{elif}\;x \leq 3.8 \cdot 10^{-129}:\\ \;\;\;\;\frac{\left(\frac{0.3333333333333333}{x \cdot x} + 1\right) - \frac{0.5}{x}}{n \cdot x}\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.3333333333333333, x, 0.5\right), x, -1\right), x, \log x\right)}{-n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(\frac{0.3333333333333333}{x \cdot x} + 1\right) - \frac{0.5}{x}}{n}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 41.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \frac{{x}^{-1}}{n} \end{array} \]

FPCore

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

C

double code(double x, double n) {
	return 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)) / n
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 51.9%

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

3. Taylor expanded in x around inf

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

   1. lower-/.f64 N/A

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

   2. mul-1-neg N/A

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

   3. log-rec N/A

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

   4. mul-1-neg N/A

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

   5. distribute-neg-frac N/A

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

   6. mul-1-neg N/A

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

   7. remove-double-neg N/A

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

   8. lower-exp.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. lower-log.f64 N/A

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

   11. lower-*.f64 57.7

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

5. Applied rewrites 57.7%

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

6. Taylor expanded in n around inf

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

8. Applied rewrites 39.7%

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

9. Taylor expanded in n around inf

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

11. Applied rewrites 39.9%

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

12. Final simplification 39.9%

    \[\leadsto \frac{{x}^{-1}}{n} \]
13. Add Preprocessing

Alternative 12: 40.7% accurate, 2.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 51.9%

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

3. Taylor expanded in x around inf

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

   1. lower-/.f64 N/A

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

   2. mul-1-neg N/A

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

   3. log-rec N/A

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

   4. mul-1-neg N/A

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

   5. distribute-neg-frac N/A

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

   6. mul-1-neg N/A

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

   7. remove-double-neg N/A

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

   8. lower-exp.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. lower-log.f64 N/A

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

   11. lower-*.f64 57.7

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

5. Applied rewrites 57.7%

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

6. Taylor expanded in n around inf

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

8. Applied rewrites 39.9%

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

10. Applied rewrites 38.9%

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

11. Final simplification 38.9%

    \[\leadsto {\left(x \cdot n\right)}^{-1} \]
12. Add Preprocessing

Alternative 13: 46.7% accurate, 4.1× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{\left(\frac{0.3333333333333333}{x \cdot x} + 1\right) - \frac{0.5}{x}}{n}}{x} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (/ (/ (- (+ (/ 0.3333333333333333 (* x x)) 1.0) (/ 0.5 x)) n) x))

C

double code(double x, double n) {
	return ((((0.3333333333333333 / (x * x)) + 1.0) - (0.5 / x)) / n) / x;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    code = ((((0.3333333333333333d0 / (x * x)) + 1.0d0) - (0.5d0 / x)) / n) / x
end function

Java

public static double code(double x, double n) {
	return ((((0.3333333333333333 / (x * x)) + 1.0) - (0.5 / x)) / n) / x;
}

Python

def code(x, n):
	return ((((0.3333333333333333 / (x * x)) + 1.0) - (0.5 / x)) / n) / x

Julia

function code(x, n)
	return Float64(Float64(Float64(Float64(Float64(0.3333333333333333 / Float64(x * x)) + 1.0) - Float64(0.5 / x)) / n) / x)
end

MATLAB

function tmp = code(x, n)
	tmp = ((((0.3333333333333333 / (x * x)) + 1.0) - (0.5 / x)) / n) / x;
end

Wolfram

code[x_, n_] := N[(N[(N[(N[(N[(0.3333333333333333 / N[(x * x), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] - N[(0.5 / x), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision] / x), $MachinePrecision]

Derivation

1. Initial program 51.9%

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

3. Taylor expanded in x around inf

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

5. Applied rewrites 41.0%

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

6. Taylor expanded in n around inf

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

8. Applied rewrites 47.0%

   \[\leadsto \frac{\frac{\left(\frac{0.3333333333333333}{x \cdot x} + 1\right) - \frac{0.5}{x}}{n}}{x} \]
9. Add Preprocessing

Alternative 14: 46.2% accurate, 4.6× speedup

Math

\[\begin{array}{l} \\ \frac{\left(\frac{0.3333333333333333}{x \cdot x} + 1\right) - \frac{0.5}{x}}{n \cdot x} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (/ (- (+ (/ 0.3333333333333333 (* x x)) 1.0) (/ 0.5 x)) (* n x)))

C

double code(double x, double n) {
	return (((0.3333333333333333 / (x * x)) + 1.0) - (0.5 / x)) / (n * x);
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    code = (((0.3333333333333333d0 / (x * x)) + 1.0d0) - (0.5d0 / x)) / (n * x)
end function

Java

public static double code(double x, double n) {
	return (((0.3333333333333333 / (x * x)) + 1.0) - (0.5 / x)) / (n * x);
}

Python

def code(x, n):
	return (((0.3333333333333333 / (x * x)) + 1.0) - (0.5 / x)) / (n * x)

Julia

function code(x, n)
	return Float64(Float64(Float64(Float64(0.3333333333333333 / Float64(x * x)) + 1.0) - Float64(0.5 / x)) / Float64(n * x))
end

MATLAB

function tmp = code(x, n)
	tmp = (((0.3333333333333333 / (x * x)) + 1.0) - (0.5 / x)) / (n * x);
end

Wolfram

code[x_, n_] := N[(N[(N[(N[(0.3333333333333333 / N[(x * x), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] - N[(0.5 / x), $MachinePrecision]), $MachinePrecision] / N[(n * x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 51.9%

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

3. Taylor expanded in x around inf

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

5. Applied rewrites 41.0%

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

6. Taylor expanded in n around inf

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

8. Applied rewrites 46.0%

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

Reproduce

herbie shell --seed 2024305 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  :precision binary64
  (- (pow (+ x 1.0) (/ 1.0 n)) (pow x (/ 1.0 n))))