2nthrt (problem 3.4.6)

Percentage Accurate: 53.8% → 85.1%

Time: 24.6s

Alternatives: 19

Speedup: 4.6×

• Could not converge on a ground truth

  1. x = -8.985405270083088e+224
  2. n = -2.888281818567458e-170

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 53.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 85.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{-78}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n} + -1\right)}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-85}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 4000000:\\ \;\;\;\;\frac{t\_0}{n \cdot x}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -2e-78)
     (/ (pow x (+ (/ 1.0 n) -1.0)) n)
     (if (<= (/ 1.0 n) 2e-85)
       (/ (log (/ (+ 1.0 x) x)) n)
       (if (<= (/ 1.0 n) 4000000.0)
         (/ t_0 (* n x))
         (- (exp (/ (log1p x) n)) t_0))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -2e-78) {
		tmp = pow(x, ((1.0 / n) + -1.0)) / n;
	} else if ((1.0 / n) <= 2e-85) {
		tmp = log(((1.0 + x) / x)) / n;
	} else if ((1.0 / n) <= 4000000.0) {
		tmp = t_0 / (n * x);
	} else {
		tmp = exp((log1p(x) / n)) - t_0;
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -2e-78) {
		tmp = Math.pow(x, ((1.0 / n) + -1.0)) / n;
	} else if ((1.0 / n) <= 2e-85) {
		tmp = Math.log(((1.0 + x) / x)) / n;
	} else if ((1.0 / n) <= 4000000.0) {
		tmp = t_0 / (n * x);
	} else {
		tmp = Math.exp((Math.log1p(x) / n)) - t_0;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	tmp = 0
	if (1.0 / n) <= -2e-78:
		tmp = math.pow(x, ((1.0 / n) + -1.0)) / n
	elif (1.0 / n) <= 2e-85:
		tmp = math.log(((1.0 + x) / x)) / n
	elif (1.0 / n) <= 4000000.0:
		tmp = t_0 / (n * x)
	else:
		tmp = math.exp((math.log1p(x) / n)) - t_0
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -2e-78)
		tmp = Float64((x ^ Float64(Float64(1.0 / n) + -1.0)) / n);
	elseif (Float64(1.0 / n) <= 2e-85)
		tmp = Float64(log(Float64(Float64(1.0 + x) / x)) / n);
	elseif (Float64(1.0 / n) <= 4000000.0)
		tmp = Float64(t_0 / Float64(n * x));
	else
		tmp = Float64(exp(Float64(log1p(x) / n)) - t_0);
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -2e-78], N[(N[Power[x, N[(N[(1.0 / n), $MachinePrecision] + -1.0), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-85], N[(N[Log[N[(N[(1.0 + x), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 4000000.0], N[(t$95$0 / N[(n * x), $MachinePrecision]), $MachinePrecision], N[(N[Exp[N[(N[Log[1 + x], $MachinePrecision] / n), $MachinePrecision]], $MachinePrecision] - t$95$0), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -2e-78

   1. Initial program 83.5%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   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. lower-/.f64 N/A

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

      2. log-rec N/A

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

      3. mul-1-neg N/A

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

      4. associate-*r/ N/A

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

      5. associate-*r* N/A

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

      6. metadata-eval N/A

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

      9. exp-to-pow N/A

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

      10. lower-pow.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 93.0

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

   5. Applied rewrites 93.0%

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

   7. Applied rewrites 93.1%

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

   if -2e-78 < (/.f64 #s(literal 1 binary64) n) < 2e-85

   1. Initial program 35.9%

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

   3. Taylor expanded in n around inf

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
   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 82.1

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

   5. Applied rewrites 82.1%

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

   7. Applied rewrites 82.4%

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

   if 2e-85 < (/.f64 #s(literal 1 binary64) n) < 4e6

   1. Initial program 21.6%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   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. lower-/.f64 N/A

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

      2. log-rec N/A

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

      3. mul-1-neg N/A

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

      4. associate-*r/ N/A

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

      5. associate-*r* N/A

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

      6. metadata-eval N/A

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

      9. exp-to-pow N/A

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

      10. lower-pow.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 81.0

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

   5. Applied rewrites 81.0%

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

   if 4e6 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 63.7%

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

      1. lift-pow.f64 N/A

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

      2. pow-to-exp N/A

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

      3. lower-exp.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-log1p.f64 99.8

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

   4. Applied rewrites 99.8%

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

4. Final simplification 88.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{-78}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n} + -1\right)}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-85}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 4000000:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{n \cdot x}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 81.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{-78}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n} + -1\right)}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-85}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 4000000:\\ \;\;\;\;\frac{t\_0}{n \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{0.5}{n \cdot n} + \frac{-0.5}{n}, \frac{1}{n}\right), 1\right) - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -2e-78)
     (/ (pow x (+ (/ 1.0 n) -1.0)) n)
     (if (<= (/ 1.0 n) 2e-85)
       (/ (log (/ (+ 1.0 x) x)) n)
       (if (<= (/ 1.0 n) 4000000.0)
         (/ t_0 (* n x))
         (-
          (fma x (fma x (+ (/ 0.5 (* n n)) (/ -0.5 n)) (/ 1.0 n)) 1.0)
          t_0))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -2e-78) {
		tmp = pow(x, ((1.0 / n) + -1.0)) / n;
	} else if ((1.0 / n) <= 2e-85) {
		tmp = log(((1.0 + x) / x)) / n;
	} else if ((1.0 / n) <= 4000000.0) {
		tmp = t_0 / (n * x);
	} else {
		tmp = fma(x, fma(x, ((0.5 / (n * n)) + (-0.5 / n)), (1.0 / n)), 1.0) - t_0;
	}
	return tmp;
}

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -2e-78)
		tmp = Float64((x ^ Float64(Float64(1.0 / n) + -1.0)) / n);
	elseif (Float64(1.0 / n) <= 2e-85)
		tmp = Float64(log(Float64(Float64(1.0 + x) / x)) / n);
	elseif (Float64(1.0 / n) <= 4000000.0)
		tmp = Float64(t_0 / Float64(n * x));
	else
		tmp = Float64(fma(x, fma(x, Float64(Float64(0.5 / Float64(n * n)) + Float64(-0.5 / n)), Float64(1.0 / n)), 1.0) - t_0);
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -2e-78], N[(N[Power[x, N[(N[(1.0 / n), $MachinePrecision] + -1.0), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-85], N[(N[Log[N[(N[(1.0 + x), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 4000000.0], N[(t$95$0 / N[(n * x), $MachinePrecision]), $MachinePrecision], N[(N[(x * N[(x * N[(N[(0.5 / N[(n * n), $MachinePrecision]), $MachinePrecision] + N[(-0.5 / n), $MachinePrecision]), $MachinePrecision] + N[(1.0 / n), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] - t$95$0), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -2e-78

   1. Initial program 83.8%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. log-rec N/A

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

      3. mul-1-neg N/A

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

      4. associate-*r/ N/A

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

      5. associate-*r* N/A

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

      6. metadata-eval N/A

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

      9. exp-to-pow N/A

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

      10. lower-pow.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 90.9

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

   5. Applied rewrites 90.9%

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

   7. Applied rewrites 90.9%

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

   if -2e-78 < (/.f64 #s(literal 1 binary64) n) < 2e-85

   1. Initial program 34.5%

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

   3. Taylor expanded in n around inf

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

      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 82.2

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

   5. Applied rewrites 82.2%

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

   7. Applied rewrites 82.5%

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

   if 2e-85 < (/.f64 #s(literal 1 binary64) n) < 4e6

   1. Initial program 20.9%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. log-rec N/A

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

      3. mul-1-neg N/A

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

      4. associate-*r/ N/A

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

      5. associate-*r* N/A

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

      6. metadata-eval N/A

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

      9. exp-to-pow N/A

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

      10. lower-pow.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 49.8

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

   5. Applied rewrites 49.8%

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

   if 4e6 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 52.5%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-fma.f64 N/A

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

      4. sub-neg N/A

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

      5. lower-+.f64 N/A

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

      6. associate-*r/ N/A

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

      7. metadata-eval N/A

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

      8. lower-/.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. associate-*r/ N/A

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

      12. metadata-eval N/A

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

      13. distribute-neg-frac N/A

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

      14. metadata-eval N/A

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

      15. lower-/.f64 N/A

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

      16. lower-/.f64 73.4

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

   5. Applied rewrites 73.4%

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

4. Final simplification 81.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{-78}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n} + -1\right)}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-85}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 4000000:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{n \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{0.5}{n \cdot n} + \frac{-0.5}{n}, \frac{1}{n}\right), 1\right) - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024228 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  :precision binary64
  (- (pow (+ x 1.0) (/ 1.0 n)) (pow x (/ 1.0 n))))