2nthrt (problem 3.4.6)

Percentage Accurate: 54.0% → 91.4%

Time: 22.2s

Alternatives: 11

Speedup: 1.0×

• could not determine a ground truth

  1. x = -1.3013929394349376e-278
  2. n = 4.576310716760742e-94

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: 54.0% 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: 91.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.82:\\ \;\;\;\;\frac{x}{n} - \mathsf{expm1}\left(\frac{\log x}{n}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{{x}^{\left({n}^{-1}\right)}}{x}}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 0.82)
   (- (/ x n) (expm1 (/ (log x) n)))
   (/ (/ (pow x (pow n -1.0)) x) n)))

C

double code(double x, double n) {
	double tmp;
	if (x <= 0.82) {
		tmp = (x / n) - expm1((log(x) / n));
	} else {
		tmp = (pow(x, pow(n, -1.0)) / x) / n;
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double tmp;
	if (x <= 0.82) {
		tmp = (x / n) - Math.expm1((Math.log(x) / n));
	} else {
		tmp = (Math.pow(x, Math.pow(n, -1.0)) / x) / n;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 0.82:
		tmp = (x / n) - math.expm1((math.log(x) / n))
	else:
		tmp = (math.pow(x, math.pow(n, -1.0)) / x) / n
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 0.82)
		tmp = Float64(Float64(x / n) - expm1(Float64(log(x) / n)));
	else
		tmp = Float64(Float64((x ^ (n ^ -1.0)) / x) / n);
	end
	return tmp
end

Wolfram

code[x_, n_] := If[LessEqual[x, 0.82], N[(N[(x / n), $MachinePrecision] - N[(Exp[N[(N[Log[x], $MachinePrecision] / n), $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision], N[(N[(N[Power[x, N[Power[n, -1.0], $MachinePrecision]], $MachinePrecision] / x), $MachinePrecision] / n), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.819999999999999951

   1. Initial program 43.1%

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

   3. Taylor expanded in x around 0

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

      1. associate--l+ N/A

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

      2. +-commutative N/A

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

      3. *-rgt-identity N/A

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

      4. associate-*r/ N/A

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

      5. remove-double-neg N/A

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

      6. mul-1-neg N/A

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

      7. distribute-neg-frac N/A

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

      8. mul-1-neg N/A

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

      9. log-rec N/A

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

      10. mul-1-neg N/A

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

      11. associate-+l- N/A

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

      12. lower--.f64 N/A

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

      13. associate-*r/ N/A

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

      14. *-rgt-identity N/A

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

      15. lower-/.f64 N/A

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

   5. Applied rewrites 83.5%

      \[\leadsto \color{blue}{\frac{x}{n} - \mathsf{expm1}\left(\frac{\log x}{n}\right)} \]

   if 0.819999999999999951 < x

   1. Initial program 62.5%

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

   3. Taylor expanded in x around inf

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

      1. associate-/l/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. log-rec N/A

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

      5. mul-1-neg N/A

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

      6. associate-*r/ N/A

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

      7. associate-*r* N/A

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

      8. metadata-eval N/A

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

      9. *-commutative N/A

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

      10. associate-/l* N/A

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

      11. exp-to-pow N/A

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

      12. lower-pow.f64 N/A

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

      13. lower-/.f64 98.6

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

   5. Applied rewrites 98.6%

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

4. Final simplification 89.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.82:\\ \;\;\;\;\frac{x}{n} - \mathsf{expm1}\left(\frac{\log x}{n}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{{x}^{\left({n}^{-1}\right)}}{x}}{n}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 68.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left({n}^{-1}\right)}\\ t_1 := \frac{\frac{t\_0}{x}}{n}\\ t_2 := \frac{x - \log x}{n}\\ \mathbf{if}\;{n}^{-1} \leq -5 \cdot 10^{-165}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;{n}^{-1} \leq -5 \cdot 10^{-213}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;{n}^{-1} \leq -2 \cdot 10^{-259}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;{n}^{-1} \leq 2 \cdot 10^{-12}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;{n}^{-1} \leq 4 \cdot 10^{+176}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{0.5}{n} - 0.5, x, 1\right)}{n}, x, 1\right) - t\_0\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{x}{n \cdot n} \cdot 0.5\right) \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (pow n -1.0)))
        (t_1 (/ (/ t_0 x) n))
        (t_2 (/ (- x (log x)) n)))
   (if (<= (pow n -1.0) -5e-165)
     t_1
     (if (<= (pow n -1.0) -5e-213)
       t_2
       (if (<= (pow n -1.0) -2e-259)
         t_1
         (if (<= (pow n -1.0) 2e-12)
           t_2
           (if (<= (pow n -1.0) 4e+176)
             (- (fma (/ (fma (- (/ 0.5 n) 0.5) x 1.0) n) x 1.0) t_0)
             (* (* (/ x (* n n)) 0.5) x))))))))

C

double code(double x, double n) {
	double t_0 = pow(x, pow(n, -1.0));
	double t_1 = (t_0 / x) / n;
	double t_2 = (x - log(x)) / n;
	double tmp;
	if (pow(n, -1.0) <= -5e-165) {
		tmp = t_1;
	} else if (pow(n, -1.0) <= -5e-213) {
		tmp = t_2;
	} else if (pow(n, -1.0) <= -2e-259) {
		tmp = t_1;
	} else if (pow(n, -1.0) <= 2e-12) {
		tmp = t_2;
	} else if (pow(n, -1.0) <= 4e+176) {
		tmp = fma((fma(((0.5 / n) - 0.5), x, 1.0) / n), x, 1.0) - t_0;
	} else {
		tmp = ((x / (n * n)) * 0.5) * x;
	}
	return tmp;
}

Julia

function code(x, n)
	t_0 = x ^ (n ^ -1.0)
	t_1 = Float64(Float64(t_0 / x) / n)
	t_2 = Float64(Float64(x - log(x)) / n)
	tmp = 0.0
	if ((n ^ -1.0) <= -5e-165)
		tmp = t_1;
	elseif ((n ^ -1.0) <= -5e-213)
		tmp = t_2;
	elseif ((n ^ -1.0) <= -2e-259)
		tmp = t_1;
	elseif ((n ^ -1.0) <= 2e-12)
		tmp = t_2;
	elseif ((n ^ -1.0) <= 4e+176)
		tmp = Float64(fma(Float64(fma(Float64(Float64(0.5 / n) - 0.5), x, 1.0) / n), x, 1.0) - t_0);
	else
		tmp = Float64(Float64(Float64(x / Float64(n * n)) * 0.5) * x);
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[Power[n, -1.0], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(t$95$0 / x), $MachinePrecision] / n), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision]}, If[LessEqual[N[Power[n, -1.0], $MachinePrecision], -5e-165], t$95$1, If[LessEqual[N[Power[n, -1.0], $MachinePrecision], -5e-213], t$95$2, If[LessEqual[N[Power[n, -1.0], $MachinePrecision], -2e-259], t$95$1, If[LessEqual[N[Power[n, -1.0], $MachinePrecision], 2e-12], t$95$2, If[LessEqual[N[Power[n, -1.0], $MachinePrecision], 4e+176], N[(N[(N[(N[(N[(N[(0.5 / n), $MachinePrecision] - 0.5), $MachinePrecision] * x + 1.0), $MachinePrecision] / n), $MachinePrecision] * x + 1.0), $MachinePrecision] - t$95$0), $MachinePrecision], N[(N[(N[(x / N[(n * n), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision] * x), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -4.99999999999999981e-165 or -4.99999999999999977e-213 < (/.f64 #s(literal 1 binary64) n) < -2.0000000000000001e-259

   1. Initial program 74.6%

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

   3. Taylor expanded in x around 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. associate-/l/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. log-rec N/A

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

      5. mul-1-neg N/A

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

      6. associate-*r/ N/A

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

      7. associate-*r* N/A

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

      8. metadata-eval N/A

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

      9. *-commutative N/A

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

      10. associate-/l* N/A

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

      11. exp-to-pow N/A

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

      12. lower-pow.f64 N/A

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

      13. lower-/.f64 86.1

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

   5. Applied rewrites 86.1%

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

   if -4.99999999999999981e-165 < (/.f64 #s(literal 1 binary64) n) < -4.99999999999999977e-213 or -2.0000000000000001e-259 < (/.f64 #s(literal 1 binary64) n) < 1.99999999999999996e-12

   1. Initial program 21.5%

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

   3. Taylor expanded in x around 0

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

      1. associate--l+ N/A

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

      2. +-commutative N/A

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

      3. *-rgt-identity N/A

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

      4. associate-*r/ N/A

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

      5. remove-double-neg N/A

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

      6. mul-1-neg N/A

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

      7. distribute-neg-frac N/A

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

      8. mul-1-neg N/A

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

      9. log-rec N/A

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

      10. mul-1-neg N/A

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

      11. associate-+l- N/A

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

      12. lower--.f64 N/A

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

      13. associate-*r/ N/A

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

      14. *-rgt-identity N/A

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

      15. lower-/.f64 N/A

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

   5. Applied rewrites 62.7%

      \[\leadsto \color{blue}{\frac{x}{n} - \mathsf{expm1}\left(\frac{\log x}{n}\right)} \]

   6. Taylor expanded in n around inf

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

   8. Applied rewrites 62.6%

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

   if 1.99999999999999996e-12 < (/.f64 #s(literal 1 binary64) n) < 4e176

   1. Initial program 72.2%

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

   3. Taylor expanded in x around 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

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower--.f64 N/A

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

      7. associate-*r/ N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. associate-*r/ N/A

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

      13. metadata-eval N/A

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

      14. lower-/.f64 N/A

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

      15. lower-/.f64 57.8

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

   5. Applied rewrites 57.8%

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

   6. Taylor expanded in n around inf

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

   8. Applied rewrites 65.3%

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

   if 4e176 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 8.2%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 57.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x \cdot x, \frac{0.5}{n \cdot n} - \frac{0.5}{n}, \frac{x}{n} - \mathsf{expm1}\left(\frac{\log x}{n}\right)\right)} \]

   6. Taylor expanded in n around 0

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

   8. Applied rewrites 34.2%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 94.9%

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

   12. Taylor expanded in n around 0

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

   14. Applied rewrites 94.9%

       \[\leadsto \left(\frac{x}{n \cdot n} \cdot 0.5\right) \cdot x \]
3. Recombined 4 regimes into one program.

4. Final simplification 76.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{n}^{-1} \leq -5 \cdot 10^{-165}:\\ \;\;\;\;\frac{\frac{{x}^{\left({n}^{-1}\right)}}{x}}{n}\\ \mathbf{elif}\;{n}^{-1} \leq -5 \cdot 10^{-213}:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{elif}\;{n}^{-1} \leq -2 \cdot 10^{-259}:\\ \;\;\;\;\frac{\frac{{x}^{\left({n}^{-1}\right)}}{x}}{n}\\ \mathbf{elif}\;{n}^{-1} \leq 2 \cdot 10^{-12}:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{elif}\;{n}^{-1} \leq 4 \cdot 10^{+176}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{0.5}{n} - 0.5, x, 1\right)}{n}, x, 1\right) - {x}^{\left({n}^{-1}\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{x}{n \cdot n} \cdot 0.5\right) \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 68.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left({n}^{-1}\right)}\\ t_1 := \frac{\frac{t\_0}{x}}{n}\\ t_2 := \frac{x - \log x}{n}\\ \mathbf{if}\;{n}^{-1} \leq -5 \cdot 10^{-165}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;{n}^{-1} \leq -5 \cdot 10^{-213}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;{n}^{-1} \leq -2 \cdot 10^{-259}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;{n}^{-1} \leq 2 \cdot 10^{-12}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\frac{0.5}{n \cdot n} - \frac{0.5}{n}, x, {n}^{-1}\right), x, 1\right) - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (pow n -1.0)))
        (t_1 (/ (/ t_0 x) n))
        (t_2 (/ (- x (log x)) n)))
   (if (<= (pow n -1.0) -5e-165)
     t_1
     (if (<= (pow n -1.0) -5e-213)
       t_2
       (if (<= (pow n -1.0) -2e-259)
         t_1
         (if (<= (pow n -1.0) 2e-12)
           t_2
           (-
            (fma (fma (- (/ 0.5 (* n n)) (/ 0.5 n)) x (pow n -1.0)) x 1.0)
            t_0)))))))

C

double code(double x, double n) {
	double t_0 = pow(x, pow(n, -1.0));
	double t_1 = (t_0 / x) / n;
	double t_2 = (x - log(x)) / n;
	double tmp;
	if (pow(n, -1.0) <= -5e-165) {
		tmp = t_1;
	} else if (pow(n, -1.0) <= -5e-213) {
		tmp = t_2;
	} else if (pow(n, -1.0) <= -2e-259) {
		tmp = t_1;
	} else if (pow(n, -1.0) <= 2e-12) {
		tmp = t_2;
	} else {
		tmp = fma(fma(((0.5 / (n * n)) - (0.5 / n)), x, pow(n, -1.0)), x, 1.0) - t_0;
	}
	return tmp;
}

Julia

function code(x, n)
	t_0 = x ^ (n ^ -1.0)
	t_1 = Float64(Float64(t_0 / x) / n)
	t_2 = Float64(Float64(x - log(x)) / n)
	tmp = 0.0
	if ((n ^ -1.0) <= -5e-165)
		tmp = t_1;
	elseif ((n ^ -1.0) <= -5e-213)
		tmp = t_2;
	elseif ((n ^ -1.0) <= -2e-259)
		tmp = t_1;
	elseif ((n ^ -1.0) <= 2e-12)
		tmp = t_2;
	else
		tmp = Float64(fma(fma(Float64(Float64(0.5 / Float64(n * n)) - Float64(0.5 / n)), x, (n ^ -1.0)), 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]}, Block[{t$95$1 = N[(N[(t$95$0 / x), $MachinePrecision] / n), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision]}, If[LessEqual[N[Power[n, -1.0], $MachinePrecision], -5e-165], t$95$1, If[LessEqual[N[Power[n, -1.0], $MachinePrecision], -5e-213], t$95$2, If[LessEqual[N[Power[n, -1.0], $MachinePrecision], -2e-259], t$95$1, If[LessEqual[N[Power[n, -1.0], $MachinePrecision], 2e-12], t$95$2, N[(N[(N[(N[(N[(0.5 / N[(n * n), $MachinePrecision]), $MachinePrecision] - N[(0.5 / n), $MachinePrecision]), $MachinePrecision] * x + N[Power[n, -1.0], $MachinePrecision]), $MachinePrecision] * x + 1.0), $MachinePrecision] - t$95$0), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -4.99999999999999981e-165 or -4.99999999999999977e-213 < (/.f64 #s(literal 1 binary64) n) < -2.0000000000000001e-259

   1. Initial program 74.6%

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

   3. Taylor expanded in x around 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. associate-/l/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. log-rec N/A

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

      5. mul-1-neg N/A

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

      6. associate-*r/ N/A

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

      7. associate-*r* N/A

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

      8. metadata-eval N/A

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

      9. *-commutative N/A

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

      10. associate-/l* N/A

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

      11. exp-to-pow N/A

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

      12. lower-pow.f64 N/A

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

      13. lower-/.f64 86.1

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

   5. Applied rewrites 86.1%

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

   if -4.99999999999999981e-165 < (/.f64 #s(literal 1 binary64) n) < -4.99999999999999977e-213 or -2.0000000000000001e-259 < (/.f64 #s(literal 1 binary64) n) < 1.99999999999999996e-12

   1. Initial program 21.5%

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

   3. Taylor expanded in x around 0

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

      1. associate--l+ N/A

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

      2. +-commutative N/A

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

      3. *-rgt-identity N/A

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

      4. associate-*r/ N/A

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

      5. remove-double-neg N/A

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

      6. mul-1-neg N/A

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

      7. distribute-neg-frac N/A

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

      8. mul-1-neg N/A

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

      9. log-rec N/A

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

      10. mul-1-neg N/A

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

      11. associate-+l- N/A

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

      12. lower--.f64 N/A

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

      13. associate-*r/ N/A

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

      14. *-rgt-identity N/A

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

      15. lower-/.f64 N/A

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

   5. Applied rewrites 62.7%

      \[\leadsto \color{blue}{\frac{x}{n} - \mathsf{expm1}\left(\frac{\log x}{n}\right)} \]

   6. Taylor expanded in n around inf

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

   8. Applied rewrites 62.6%

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

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

   1. Initial program 45.2%

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

   3. Taylor expanded in x around 0

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

      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

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower--.f64 N/A

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

      7. associate-*r/ N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. associate-*r/ N/A

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

      13. metadata-eval N/A

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

      14. lower-/.f64 N/A

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

      15. lower-/.f64 73.4

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

   5. Applied rewrites 73.4%

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

4. Final simplification 75.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{n}^{-1} \leq -5 \cdot 10^{-165}:\\ \;\;\;\;\frac{\frac{{x}^{\left({n}^{-1}\right)}}{x}}{n}\\ \mathbf{elif}\;{n}^{-1} \leq -5 \cdot 10^{-213}:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{elif}\;{n}^{-1} \leq -2 \cdot 10^{-259}:\\ \;\;\;\;\frac{\frac{{x}^{\left({n}^{-1}\right)}}{x}}{n}\\ \mathbf{elif}\;{n}^{-1} \leq 2 \cdot 10^{-12}:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\frac{0.5}{n \cdot n} - \frac{0.5}{n}, x, {n}^{-1}\right), x, 1\right) - {x}^{\left({n}^{-1}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 68.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left({n}^{-1}\right)}\\ t_1 := \frac{\frac{t\_0}{x}}{n}\\ t_2 := \frac{x - \log x}{n}\\ \mathbf{if}\;{n}^{-1} \leq -5 \cdot 10^{-165}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;{n}^{-1} \leq -5 \cdot 10^{-213}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;{n}^{-1} \leq -2 \cdot 10^{-259}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;{n}^{-1} \leq 5 \cdot 10^{-16}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;{n}^{-1} \leq 4 \cdot 10^{+176}:\\ \;\;\;\;\left(\frac{x}{n} + 1\right) - t\_0\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{x}{n \cdot n} \cdot 0.5\right) \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (pow n -1.0)))
        (t_1 (/ (/ t_0 x) n))
        (t_2 (/ (- x (log x)) n)))
   (if (<= (pow n -1.0) -5e-165)
     t_1
     (if (<= (pow n -1.0) -5e-213)
       t_2
       (if (<= (pow n -1.0) -2e-259)
         t_1
         (if (<= (pow n -1.0) 5e-16)
           t_2
           (if (<= (pow n -1.0) 4e+176)
             (- (+ (/ x n) 1.0) t_0)
             (* (* (/ x (* n n)) 0.5) x))))))))

C

double code(double x, double n) {
	double t_0 = pow(x, pow(n, -1.0));
	double t_1 = (t_0 / x) / n;
	double t_2 = (x - log(x)) / n;
	double tmp;
	if (pow(n, -1.0) <= -5e-165) {
		tmp = t_1;
	} else if (pow(n, -1.0) <= -5e-213) {
		tmp = t_2;
	} else if (pow(n, -1.0) <= -2e-259) {
		tmp = t_1;
	} else if (pow(n, -1.0) <= 5e-16) {
		tmp = t_2;
	} else if (pow(n, -1.0) <= 4e+176) {
		tmp = ((x / n) + 1.0) - t_0;
	} else {
		tmp = ((x / (n * n)) * 0.5) * x;
	}
	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) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = x ** (n ** (-1.0d0))
    t_1 = (t_0 / x) / n
    t_2 = (x - log(x)) / n
    if ((n ** (-1.0d0)) <= (-5d-165)) then
        tmp = t_1
    else if ((n ** (-1.0d0)) <= (-5d-213)) then
        tmp = t_2
    else if ((n ** (-1.0d0)) <= (-2d-259)) then
        tmp = t_1
    else if ((n ** (-1.0d0)) <= 5d-16) then
        tmp = t_2
    else if ((n ** (-1.0d0)) <= 4d+176) then
        tmp = ((x / n) + 1.0d0) - t_0
    else
        tmp = ((x / (n * n)) * 0.5d0) * x
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, Math.pow(n, -1.0));
	double t_1 = (t_0 / x) / n;
	double t_2 = (x - Math.log(x)) / n;
	double tmp;
	if (Math.pow(n, -1.0) <= -5e-165) {
		tmp = t_1;
	} else if (Math.pow(n, -1.0) <= -5e-213) {
		tmp = t_2;
	} else if (Math.pow(n, -1.0) <= -2e-259) {
		tmp = t_1;
	} else if (Math.pow(n, -1.0) <= 5e-16) {
		tmp = t_2;
	} else if (Math.pow(n, -1.0) <= 4e+176) {
		tmp = ((x / n) + 1.0) - t_0;
	} else {
		tmp = ((x / (n * n)) * 0.5) * x;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, math.pow(n, -1.0))
	t_1 = (t_0 / x) / n
	t_2 = (x - math.log(x)) / n
	tmp = 0
	if math.pow(n, -1.0) <= -5e-165:
		tmp = t_1
	elif math.pow(n, -1.0) <= -5e-213:
		tmp = t_2
	elif math.pow(n, -1.0) <= -2e-259:
		tmp = t_1
	elif math.pow(n, -1.0) <= 5e-16:
		tmp = t_2
	elif math.pow(n, -1.0) <= 4e+176:
		tmp = ((x / n) + 1.0) - t_0
	else:
		tmp = ((x / (n * n)) * 0.5) * x
	return tmp

Julia

function code(x, n)
	t_0 = x ^ (n ^ -1.0)
	t_1 = Float64(Float64(t_0 / x) / n)
	t_2 = Float64(Float64(x - log(x)) / n)
	tmp = 0.0
	if ((n ^ -1.0) <= -5e-165)
		tmp = t_1;
	elseif ((n ^ -1.0) <= -5e-213)
		tmp = t_2;
	elseif ((n ^ -1.0) <= -2e-259)
		tmp = t_1;
	elseif ((n ^ -1.0) <= 5e-16)
		tmp = t_2;
	elseif ((n ^ -1.0) <= 4e+176)
		tmp = Float64(Float64(Float64(x / n) + 1.0) - t_0);
	else
		tmp = Float64(Float64(Float64(x / Float64(n * n)) * 0.5) * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = x ^ (n ^ -1.0);
	t_1 = (t_0 / x) / n;
	t_2 = (x - log(x)) / n;
	tmp = 0.0;
	if ((n ^ -1.0) <= -5e-165)
		tmp = t_1;
	elseif ((n ^ -1.0) <= -5e-213)
		tmp = t_2;
	elseif ((n ^ -1.0) <= -2e-259)
		tmp = t_1;
	elseif ((n ^ -1.0) <= 5e-16)
		tmp = t_2;
	elseif ((n ^ -1.0) <= 4e+176)
		tmp = ((x / n) + 1.0) - t_0;
	else
		tmp = ((x / (n * n)) * 0.5) * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[Power[n, -1.0], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(t$95$0 / x), $MachinePrecision] / n), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision]}, If[LessEqual[N[Power[n, -1.0], $MachinePrecision], -5e-165], t$95$1, If[LessEqual[N[Power[n, -1.0], $MachinePrecision], -5e-213], t$95$2, If[LessEqual[N[Power[n, -1.0], $MachinePrecision], -2e-259], t$95$1, If[LessEqual[N[Power[n, -1.0], $MachinePrecision], 5e-16], t$95$2, If[LessEqual[N[Power[n, -1.0], $MachinePrecision], 4e+176], N[(N[(N[(x / n), $MachinePrecision] + 1.0), $MachinePrecision] - t$95$0), $MachinePrecision], N[(N[(N[(x / N[(n * n), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision] * x), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -4.99999999999999981e-165 or -4.99999999999999977e-213 < (/.f64 #s(literal 1 binary64) n) < -2.0000000000000001e-259

   1. Initial program 74.6%

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

   3. Taylor expanded in x around 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. associate-/l/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. log-rec N/A

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

      5. mul-1-neg N/A

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

      6. associate-*r/ N/A

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

      7. associate-*r* N/A

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

      8. metadata-eval N/A

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

      9. *-commutative N/A

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

      10. associate-/l* N/A

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

      11. exp-to-pow N/A

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

      12. lower-pow.f64 N/A

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

      13. lower-/.f64 86.1

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

   5. Applied rewrites 86.1%

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

   if -4.99999999999999981e-165 < (/.f64 #s(literal 1 binary64) n) < -4.99999999999999977e-213 or -2.0000000000000001e-259 < (/.f64 #s(literal 1 binary64) n) < 5.0000000000000004e-16

   1. Initial program 21.7%

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

   3. Taylor expanded in x around 0

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

      1. associate--l+ N/A

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

      2. +-commutative N/A

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

      3. *-rgt-identity N/A

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

      4. associate-*r/ N/A

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

      5. remove-double-neg N/A

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

      6. mul-1-neg N/A

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

      7. distribute-neg-frac N/A

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

      8. mul-1-neg N/A

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

      9. log-rec N/A

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

      10. mul-1-neg N/A

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

      11. associate-+l- N/A

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

      12. lower--.f64 N/A

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

      13. associate-*r/ N/A

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

      14. *-rgt-identity N/A

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

      15. lower-/.f64 N/A

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

   5. Applied rewrites 63.4%

      \[\leadsto \color{blue}{\frac{x}{n} - \mathsf{expm1}\left(\frac{\log x}{n}\right)} \]

   6. Taylor expanded in n around inf

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

   8. Applied rewrites 63.3%

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

   if 5.0000000000000004e-16 < (/.f64 #s(literal 1 binary64) n) < 4e176

   1. Initial program 69.7%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-rgt-identity N/A

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

      3. associate-*r/ N/A

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

      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 62.7

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

   5. Applied rewrites 62.7%

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

   if 4e176 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 8.2%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 57.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x \cdot x, \frac{0.5}{n \cdot n} - \frac{0.5}{n}, \frac{x}{n} - \mathsf{expm1}\left(\frac{\log x}{n}\right)\right)} \]

   6. Taylor expanded in n around 0

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

   8. Applied rewrites 34.2%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 94.9%

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

   12. Taylor expanded in n around 0

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

   14. Applied rewrites 94.9%

       \[\leadsto \left(\frac{x}{n \cdot n} \cdot 0.5\right) \cdot x \]
3. Recombined 4 regimes into one program.

4. Final simplification 76.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{n}^{-1} \leq -5 \cdot 10^{-165}:\\ \;\;\;\;\frac{\frac{{x}^{\left({n}^{-1}\right)}}{x}}{n}\\ \mathbf{elif}\;{n}^{-1} \leq -5 \cdot 10^{-213}:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{elif}\;{n}^{-1} \leq -2 \cdot 10^{-259}:\\ \;\;\;\;\frac{\frac{{x}^{\left({n}^{-1}\right)}}{x}}{n}\\ \mathbf{elif}\;{n}^{-1} \leq 5 \cdot 10^{-16}:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{elif}\;{n}^{-1} \leq 4 \cdot 10^{+176}:\\ \;\;\;\;\left(\frac{x}{n} + 1\right) - {x}^{\left({n}^{-1}\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{x}{n \cdot n} \cdot 0.5\right) \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 80.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left({n}^{-1}\right)}\\ \mathbf{if}\;{n}^{-1} \leq -5 \cdot 10^{-165}:\\ \;\;\;\;\frac{\frac{t\_0}{x}}{n}\\ \mathbf{elif}\;{n}^{-1} \leq 2 \cdot 10^{-12}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right) - \log x}{n}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\frac{0.5}{n \cdot n} - \frac{0.5}{n}, x, {n}^{-1}\right), x, 1\right) - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (pow n -1.0))))
   (if (<= (pow n -1.0) -5e-165)
     (/ (/ t_0 x) n)
     (if (<= (pow n -1.0) 2e-12)
       (/ (- (log1p x) (log x)) n)
       (-
        (fma (fma (- (/ 0.5 (* n n)) (/ 0.5 n)) x (pow n -1.0)) 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) <= -5e-165) {
		tmp = (t_0 / x) / n;
	} else if (pow(n, -1.0) <= 2e-12) {
		tmp = (log1p(x) - log(x)) / n;
	} else {
		tmp = fma(fma(((0.5 / (n * n)) - (0.5 / n)), x, pow(n, -1.0)), 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) <= -5e-165)
		tmp = Float64(Float64(t_0 / x) / n);
	elseif ((n ^ -1.0) <= 2e-12)
		tmp = Float64(Float64(log1p(x) - log(x)) / n);
	else
		tmp = Float64(fma(fma(Float64(Float64(0.5 / Float64(n * n)) - Float64(0.5 / n)), x, (n ^ -1.0)), 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], -5e-165], N[(N[(t$95$0 / x), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[Power[n, -1.0], $MachinePrecision], 2e-12], N[(N[(N[Log[1 + x], $MachinePrecision] - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], N[(N[(N[(N[(N[(0.5 / N[(n * n), $MachinePrecision]), $MachinePrecision] - N[(0.5 / n), $MachinePrecision]), $MachinePrecision] * x + N[Power[n, -1.0], $MachinePrecision]), $MachinePrecision] * x + 1.0), $MachinePrecision] - t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -4.99999999999999981e-165

   1. Initial program 76.7%

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

   3. Taylor expanded in x around inf

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

      1. associate-/l/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. log-rec N/A

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

      5. mul-1-neg N/A

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

      6. associate-*r/ N/A

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

      7. associate-*r* N/A

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

      8. metadata-eval N/A

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

      9. *-commutative N/A

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

      10. associate-/l* N/A

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

      11. exp-to-pow N/A

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

      12. lower-pow.f64 N/A

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

      13. lower-/.f64 87.9

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

   5. Applied rewrites 87.9%

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

   if -4.99999999999999981e-165 < (/.f64 #s(literal 1 binary64) n) < 1.99999999999999996e-12

   1. Initial program 27.8%

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

   3. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

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

   1. Initial program 45.2%

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

   3. Taylor expanded in x around 0

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

      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

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower--.f64 N/A

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

      7. associate-*r/ N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. associate-*r/ N/A

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

      13. metadata-eval N/A

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

      14. lower-/.f64 N/A

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

      15. lower-/.f64 73.4

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

   5. Applied rewrites 73.4%

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

4. Final simplification 82.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{n}^{-1} \leq -5 \cdot 10^{-165}:\\ \;\;\;\;\frac{\frac{{x}^{\left({n}^{-1}\right)}}{x}}{n}\\ \mathbf{elif}\;{n}^{-1} \leq 2 \cdot 10^{-12}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right) - \log x}{n}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\frac{0.5}{n \cdot n} - \frac{0.5}{n}, x, {n}^{-1}\right), x, 1\right) - {x}^{\left({n}^{-1}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 53.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left({n}^{-1}\right)}\\ \mathbf{if}\;{n}^{-1} \leq -0.2:\\ \;\;\;\;1 - t\_0\\ \mathbf{elif}\;{n}^{-1} \leq 5 \cdot 10^{-16}:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{elif}\;{n}^{-1} \leq 4 \cdot 10^{+176}:\\ \;\;\;\;\left(\frac{x}{n} + 1\right) - t\_0\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{x}{n \cdot n} \cdot 0.5\right) \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (pow n -1.0))))
   (if (<= (pow n -1.0) -0.2)
     (- 1.0 t_0)
     (if (<= (pow n -1.0) 5e-16)
       (/ (- x (log x)) n)
       (if (<= (pow n -1.0) 4e+176)
         (- (+ (/ x n) 1.0) t_0)
         (* (* (/ x (* n n)) 0.5) x))))))

C

double code(double x, double n) {
	double t_0 = pow(x, pow(n, -1.0));
	double tmp;
	if (pow(n, -1.0) <= -0.2) {
		tmp = 1.0 - t_0;
	} else if (pow(n, -1.0) <= 5e-16) {
		tmp = (x - log(x)) / n;
	} else if (pow(n, -1.0) <= 4e+176) {
		tmp = ((x / n) + 1.0) - t_0;
	} else {
		tmp = ((x / (n * n)) * 0.5) * x;
	}
	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 ** (n ** (-1.0d0))
    if ((n ** (-1.0d0)) <= (-0.2d0)) then
        tmp = 1.0d0 - t_0
    else if ((n ** (-1.0d0)) <= 5d-16) then
        tmp = (x - log(x)) / n
    else if ((n ** (-1.0d0)) <= 4d+176) then
        tmp = ((x / n) + 1.0d0) - t_0
    else
        tmp = ((x / (n * n)) * 0.5d0) * x
    end if
    code = tmp
end function

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) <= -0.2) {
		tmp = 1.0 - t_0;
	} else if (Math.pow(n, -1.0) <= 5e-16) {
		tmp = (x - Math.log(x)) / n;
	} else if (Math.pow(n, -1.0) <= 4e+176) {
		tmp = ((x / n) + 1.0) - t_0;
	} else {
		tmp = ((x / (n * n)) * 0.5) * x;
	}
	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) <= -0.2:
		tmp = 1.0 - t_0
	elif math.pow(n, -1.0) <= 5e-16:
		tmp = (x - math.log(x)) / n
	elif math.pow(n, -1.0) <= 4e+176:
		tmp = ((x / n) + 1.0) - t_0
	else:
		tmp = ((x / (n * n)) * 0.5) * x
	return tmp

Julia

function code(x, n)
	t_0 = x ^ (n ^ -1.0)
	tmp = 0.0
	if ((n ^ -1.0) <= -0.2)
		tmp = Float64(1.0 - t_0);
	elseif ((n ^ -1.0) <= 5e-16)
		tmp = Float64(Float64(x - log(x)) / n);
	elseif ((n ^ -1.0) <= 4e+176)
		tmp = Float64(Float64(Float64(x / n) + 1.0) - t_0);
	else
		tmp = Float64(Float64(Float64(x / Float64(n * n)) * 0.5) * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = x ^ (n ^ -1.0);
	tmp = 0.0;
	if ((n ^ -1.0) <= -0.2)
		tmp = 1.0 - t_0;
	elseif ((n ^ -1.0) <= 5e-16)
		tmp = (x - log(x)) / n;
	elseif ((n ^ -1.0) <= 4e+176)
		tmp = ((x / n) + 1.0) - t_0;
	else
		tmp = ((x / (n * n)) * 0.5) * x;
	end
	tmp_2 = 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], -0.2], N[(1.0 - t$95$0), $MachinePrecision], If[LessEqual[N[Power[n, -1.0], $MachinePrecision], 5e-16], N[(N[(x - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[Power[n, -1.0], $MachinePrecision], 4e+176], N[(N[(N[(x / n), $MachinePrecision] + 1.0), $MachinePrecision] - t$95$0), $MachinePrecision], N[(N[(N[(x / N[(n * n), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision] * x), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -0.20000000000000001

   1. Initial program 99.9%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   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 63.6%

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

   if -0.20000000000000001 < (/.f64 #s(literal 1 binary64) n) < 5.0000000000000004e-16

   1. Initial program 26.5%

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

   3. Taylor expanded in x around 0

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

      1. associate--l+ N/A

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

      2. +-commutative N/A

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

      3. *-rgt-identity N/A

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

      4. associate-*r/ N/A

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

      5. remove-double-neg N/A

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

      6. mul-1-neg N/A

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

      7. distribute-neg-frac N/A

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

      8. mul-1-neg N/A

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

      9. log-rec N/A

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

      10. mul-1-neg N/A

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

      11. associate-+l- N/A

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

      12. lower--.f64 N/A

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

      13. associate-*r/ N/A

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

      14. *-rgt-identity N/A

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

      15. lower-/.f64 N/A

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

   5. Applied rewrites 53.7%

      \[\leadsto \color{blue}{\frac{x}{n} - \mathsf{expm1}\left(\frac{\log x}{n}\right)} \]

   6. Taylor expanded in n around inf

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

   8. Applied rewrites 53.3%

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

   if 5.0000000000000004e-16 < (/.f64 #s(literal 1 binary64) n) < 4e176

   1. Initial program 69.7%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-rgt-identity N/A

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

      3. associate-*r/ N/A

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

      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 62.7

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

   5. Applied rewrites 62.7%

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

   if 4e176 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 8.2%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 57.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x \cdot x, \frac{0.5}{n \cdot n} - \frac{0.5}{n}, \frac{x}{n} - \mathsf{expm1}\left(\frac{\log x}{n}\right)\right)} \]

   6. Taylor expanded in n around 0

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

   8. Applied rewrites 34.2%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 94.9%

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

   12. Taylor expanded in n around 0

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

   14. Applied rewrites 94.9%

       \[\leadsto \left(\frac{x}{n \cdot n} \cdot 0.5\right) \cdot x \]
3. Recombined 4 regimes into one program.

4. Final simplification 60.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{n}^{-1} \leq -0.2:\\ \;\;\;\;1 - {x}^{\left({n}^{-1}\right)}\\ \mathbf{elif}\;{n}^{-1} \leq 5 \cdot 10^{-16}:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{elif}\;{n}^{-1} \leq 4 \cdot 10^{+176}:\\ \;\;\;\;\left(\frac{x}{n} + 1\right) - {x}^{\left({n}^{-1}\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{x}{n \cdot n} \cdot 0.5\right) \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 53.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - {x}^{\left({n}^{-1}\right)}\\ \mathbf{if}\;{n}^{-1} \leq -0.2:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;{n}^{-1} \leq 2 \cdot 10^{-12}:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{elif}\;{n}^{-1} \leq 2 \cdot 10^{+169}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{x}{n \cdot n} \cdot 0.5\right) \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (- 1.0 (pow x (pow n -1.0)))))
   (if (<= (pow n -1.0) -0.2)
     t_0
     (if (<= (pow n -1.0) 2e-12)
       (/ (- x (log x)) n)
       (if (<= (pow n -1.0) 2e+169) t_0 (* (* (/ x (* n n)) 0.5) x))))))

C

double code(double x, double n) {
	double t_0 = 1.0 - pow(x, pow(n, -1.0));
	double tmp;
	if (pow(n, -1.0) <= -0.2) {
		tmp = t_0;
	} else if (pow(n, -1.0) <= 2e-12) {
		tmp = (x - log(x)) / n;
	} else if (pow(n, -1.0) <= 2e+169) {
		tmp = t_0;
	} else {
		tmp = ((x / (n * n)) * 0.5) * x;
	}
	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 = 1.0d0 - (x ** (n ** (-1.0d0)))
    if ((n ** (-1.0d0)) <= (-0.2d0)) then
        tmp = t_0
    else if ((n ** (-1.0d0)) <= 2d-12) then
        tmp = (x - log(x)) / n
    else if ((n ** (-1.0d0)) <= 2d+169) then
        tmp = t_0
    else
        tmp = ((x / (n * n)) * 0.5d0) * x
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = 1.0 - Math.pow(x, Math.pow(n, -1.0));
	double tmp;
	if (Math.pow(n, -1.0) <= -0.2) {
		tmp = t_0;
	} else if (Math.pow(n, -1.0) <= 2e-12) {
		tmp = (x - Math.log(x)) / n;
	} else if (Math.pow(n, -1.0) <= 2e+169) {
		tmp = t_0;
	} else {
		tmp = ((x / (n * n)) * 0.5) * x;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = 1.0 - math.pow(x, math.pow(n, -1.0))
	tmp = 0
	if math.pow(n, -1.0) <= -0.2:
		tmp = t_0
	elif math.pow(n, -1.0) <= 2e-12:
		tmp = (x - math.log(x)) / n
	elif math.pow(n, -1.0) <= 2e+169:
		tmp = t_0
	else:
		tmp = ((x / (n * n)) * 0.5) * x
	return tmp

Julia

function code(x, n)
	t_0 = Float64(1.0 - (x ^ (n ^ -1.0)))
	tmp = 0.0
	if ((n ^ -1.0) <= -0.2)
		tmp = t_0;
	elseif ((n ^ -1.0) <= 2e-12)
		tmp = Float64(Float64(x - log(x)) / n);
	elseif ((n ^ -1.0) <= 2e+169)
		tmp = t_0;
	else
		tmp = Float64(Float64(Float64(x / Float64(n * n)) * 0.5) * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = 1.0 - (x ^ (n ^ -1.0));
	tmp = 0.0;
	if ((n ^ -1.0) <= -0.2)
		tmp = t_0;
	elseif ((n ^ -1.0) <= 2e-12)
		tmp = (x - log(x)) / n;
	elseif ((n ^ -1.0) <= 2e+169)
		tmp = t_0;
	else
		tmp = ((x / (n * n)) * 0.5) * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(1.0 - N[Power[x, N[Power[n, -1.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Power[n, -1.0], $MachinePrecision], -0.2], t$95$0, If[LessEqual[N[Power[n, -1.0], $MachinePrecision], 2e-12], N[(N[(x - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[Power[n, -1.0], $MachinePrecision], 2e+169], t$95$0, N[(N[(N[(x / N[(n * n), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision] * x), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -0.20000000000000001 or 1.99999999999999996e-12 < (/.f64 #s(literal 1 binary64) n) < 1.99999999999999987e169

   1. Initial program 93.4%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   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 64.2%

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

   if -0.20000000000000001 < (/.f64 #s(literal 1 binary64) n) < 1.99999999999999996e-12

   1. Initial program 26.3%

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

   3. Taylor expanded in x around 0

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

      1. associate--l+ N/A

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

      2. +-commutative N/A

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

      3. *-rgt-identity N/A

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

      4. associate-*r/ N/A

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

      5. remove-double-neg N/A

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

      6. mul-1-neg N/A

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

      7. distribute-neg-frac N/A

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

      8. mul-1-neg N/A

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

      9. log-rec N/A

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

      10. mul-1-neg N/A

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

      11. associate-+l- N/A

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

      12. lower--.f64 N/A

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

      13. associate-*r/ N/A

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

      14. *-rgt-identity N/A

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

      15. lower-/.f64 N/A

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

   5. Applied rewrites 53.4%

      \[\leadsto \color{blue}{\frac{x}{n} - \mathsf{expm1}\left(\frac{\log x}{n}\right)} \]

   6. Taylor expanded in n around inf

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

   8. Applied rewrites 53.0%

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

   if 1.99999999999999987e169 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 12.4%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 57.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x \cdot x, \frac{0.5}{n \cdot n} - \frac{0.5}{n}, \frac{x}{n} - \mathsf{expm1}\left(\frac{\log x}{n}\right)\right)} \]

   6. Taylor expanded in n around 0

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

   8. Applied rewrites 31.3%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 90.8%

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

   12. Taylor expanded in n around 0

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

   14. Applied rewrites 90.8%

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

4. Final simplification 60.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{n}^{-1} \leq -0.2:\\ \;\;\;\;1 - {x}^{\left({n}^{-1}\right)}\\ \mathbf{elif}\;{n}^{-1} \leq 2 \cdot 10^{-12}:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{elif}\;{n}^{-1} \leq 2 \cdot 10^{+169}:\\ \;\;\;\;1 - {x}^{\left({n}^{-1}\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{x}{n \cdot n} \cdot 0.5\right) \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 39.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 3.1 \cdot 10^{+29}:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{x}{n \cdot n} \cdot 0.5\right) \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 3.1e+29) (/ (- x (log x)) n) (* (* (/ x (* n n)) 0.5) x)))

C

double code(double x, double n) {
	double tmp;
	if (x <= 3.1e+29) {
		tmp = (x - log(x)) / n;
	} else {
		tmp = ((x / (n * n)) * 0.5) * 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 <= 3.1d+29) then
        tmp = (x - log(x)) / n
    else
        tmp = ((x / (n * n)) * 0.5d0) * x
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (x <= 3.1e+29) {
		tmp = (x - Math.log(x)) / n;
	} else {
		tmp = ((x / (n * n)) * 0.5) * x;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 3.1e+29:
		tmp = (x - math.log(x)) / n
	else:
		tmp = ((x / (n * n)) * 0.5) * x
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 3.1e+29)
		tmp = Float64(Float64(x - log(x)) / n);
	else
		tmp = Float64(Float64(Float64(x / Float64(n * n)) * 0.5) * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (x <= 3.1e+29)
		tmp = (x - log(x)) / n;
	else
		tmp = ((x / (n * n)) * 0.5) * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 3.1e+29], N[(N[(x - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], N[(N[(N[(x / N[(n * n), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision] * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 3.0999999999999999e29

   1. Initial program 43.0%

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

   3. Taylor expanded in x around 0

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

      1. associate--l+ N/A

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

      2. +-commutative N/A

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

      3. *-rgt-identity N/A

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

      4. associate-*r/ N/A

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

      5. remove-double-neg N/A

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

      6. mul-1-neg N/A

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

      7. distribute-neg-frac N/A

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

      8. mul-1-neg N/A

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

      9. log-rec N/A

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

      10. mul-1-neg N/A

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

      11. associate-+l- N/A

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

      12. lower--.f64 N/A

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

      13. associate-*r/ N/A

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

      14. *-rgt-identity N/A

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

      15. lower-/.f64 N/A

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

   5. Applied rewrites 80.5%

      \[\leadsto \color{blue}{\frac{x}{n} - \mathsf{expm1}\left(\frac{\log x}{n}\right)} \]

   6. Taylor expanded in n around inf

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

   8. Applied rewrites 46.6%

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

   if 3.0999999999999999e29 < x

   1. Initial program 64.6%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 1.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x \cdot x, \frac{0.5}{n \cdot n} - \frac{0.5}{n}, \frac{x}{n} - \mathsf{expm1}\left(\frac{\log x}{n}\right)\right)} \]

   6. Taylor expanded in n around 0

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

   8. Applied rewrites 1.5%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 2.1%

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

   12. Taylor expanded in n around 0

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

   14. Applied rewrites 25.7%

       \[\leadsto \left(\frac{x}{n \cdot n} \cdot 0.5\right) \cdot x \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 21.4% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.82:\\ \;\;\;\;\left(\mathsf{fma}\left(\frac{0.5}{n \cdot n} - \frac{0.5}{n}, -1, \frac{-1}{n \cdot x}\right) \cdot \left(-x\right)\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{x}{n \cdot n} \cdot 0.5\right) \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 0.82)
   (* (* (fma (- (/ 0.5 (* n n)) (/ 0.5 n)) -1.0 (/ -1.0 (* n x))) (- x)) x)
   (* (* (/ x (* n n)) 0.5) x)))

C

double code(double x, double n) {
	double tmp;
	if (x <= 0.82) {
		tmp = (fma(((0.5 / (n * n)) - (0.5 / n)), -1.0, (-1.0 / (n * x))) * -x) * x;
	} else {
		tmp = ((x / (n * n)) * 0.5) * x;
	}
	return tmp;
}

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 0.82)
		tmp = Float64(Float64(fma(Float64(Float64(0.5 / Float64(n * n)) - Float64(0.5 / n)), -1.0, Float64(-1.0 / Float64(n * x))) * Float64(-x)) * x);
	else
		tmp = Float64(Float64(Float64(x / Float64(n * n)) * 0.5) * x);
	end
	return tmp
end

Wolfram

code[x_, n_] := If[LessEqual[x, 0.82], N[(N[(N[(N[(N[(0.5 / N[(n * n), $MachinePrecision]), $MachinePrecision] - N[(0.5 / n), $MachinePrecision]), $MachinePrecision] * -1.0 + N[(-1.0 / N[(n * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * (-x)), $MachinePrecision] * x), $MachinePrecision], N[(N[(N[(x / N[(n * n), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision] * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.819999999999999951

   1. Initial program 43.1%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 73.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x \cdot x, \frac{0.5}{n \cdot n} - \frac{0.5}{n}, \frac{x}{n} - \mathsf{expm1}\left(\frac{\log x}{n}\right)\right)} \]

   6. Taylor expanded in n around 0

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

   8. Applied rewrites 7.9%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 15.7%

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

   12. Taylor expanded in x around -inf

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

   14. Applied rewrites 20.4%

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

   if 0.819999999999999951 < x

   1. Initial program 62.5%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 1.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x \cdot x, \frac{0.5}{n \cdot n} - \frac{0.5}{n}, \frac{x}{n} - \mathsf{expm1}\left(\frac{\log x}{n}\right)\right)} \]

   6. Taylor expanded in n around 0

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

   8. Applied rewrites 1.5%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 2.1%

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

   12. Taylor expanded in n around 0

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

   14. Applied rewrites 23.8%

       \[\leadsto \left(\frac{x}{n \cdot n} \cdot 0.5\right) \cdot x \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 19.1% accurate, 8.6× speedup

Math

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

FPCore

(FPCore (x n) :precision binary64 (* (* (/ x (* n n)) 0.5) x))

C

double code(double x, double n) {
	return ((x / (n * n)) * 0.5) * x;
}

Fortran

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

Java

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

Python

def code(x, n):
	return ((x / (n * n)) * 0.5) * x

Julia

function code(x, n)
	return Float64(Float64(Float64(x / Float64(n * n)) * 0.5) * x)
end

MATLAB

function tmp = code(x, n)
	tmp = ((x / (n * n)) * 0.5) * x;
end

Wolfram

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

Derivation

1. Initial program 50.3%

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

3. Taylor expanded in x around 0

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

5. Applied rewrites 47.1%

   \[\leadsto \color{blue}{\mathsf{fma}\left(x \cdot x, \frac{0.5}{n \cdot n} - \frac{0.5}{n}, \frac{x}{n} - \mathsf{expm1}\left(\frac{\log x}{n}\right)\right)} \]

6. Taylor expanded in n around 0

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

8. Applied rewrites 5.5%

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

9. Taylor expanded in x around 0

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

11. Applied rewrites 10.6%

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

12. Taylor expanded in n around 0

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

14. Applied rewrites 18.3%

    \[\leadsto \left(\frac{x}{n \cdot n} \cdot 0.5\right) \cdot x \]
15. Add Preprocessing

Alternative 11: 4.5% accurate, 19.3× speedup

Math

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

FPCore

(FPCore (x n) :precision binary64 (/ x n))

C

double code(double x, double n) {
	return x / n;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    code = x / n
end function

Java

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

Python

def code(x, n):
	return x / n

Julia

function code(x, n)
	return Float64(x / n)
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 50.3%

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

3. Taylor expanded in x around 0

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

5. Applied rewrites 47.1%

   \[\leadsto \color{blue}{\mathsf{fma}\left(x \cdot x, \frac{0.5}{n \cdot n} - \frac{0.5}{n}, \frac{x}{n} - \mathsf{expm1}\left(\frac{\log x}{n}\right)\right)} \]

6. Taylor expanded in n around 0

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

8. Applied rewrites 5.5%

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

9. Taylor expanded in x around 0

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

11. Applied rewrites 4.6%

    \[\leadsto \frac{x}{n} \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024299 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  :precision binary64
  (- (pow (+ x 1.0) (/ 1.0 n)) (pow x (/ 1.0 n))))