2nthrt (problem 3.4.6)

Average Error: 32.2 → 23.3

Time: 12.3s

Precision: 64

• could not determine a ground truth for program body

  1. x = 4.8886932001197025e+227
  2. n = 1.9414494761577414e-197

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{1}{n} \le -8.28957440978626294 \cdot 10^{-4}:\\ \;\;\;\;\left({\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)} + {x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right) \cdot \left({\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)} - {x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right)\\ \mathbf{elif}\;\frac{1}{n} \le 2.7376649343181104 \cdot 10^{-11}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left({x}^{\left(\frac{\frac{1}{n}}{2}\right)}, {x}^{\left(\frac{\frac{1}{n}}{2}\right)} - {\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)}, {\left(x + 1\right)}^{\left(2 \cdot \frac{\frac{1}{n}}{2}\right)}\right), \left(-\left({\left({\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)}\right)}^{3} + {\left({x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right)}^{3}\right)\right) \cdot \mathsf{fma}\left(0.25, \frac{\log \left(\frac{1}{x}\right)}{x \cdot {n}^{2}}, \frac{\frac{0.25}{n}}{{x}^{2}}\right), \left(\frac{\frac{0.5}{n}}{x} \cdot \left({\left({\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)}\right)}^{3} + {\left({x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right)}^{3}\right)\right) \cdot \left({x}^{\left(\frac{\frac{1}{n}}{2}\right)} \cdot \left({x}^{\left(\frac{\frac{1}{n}}{2}\right)} - {\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)}\right) + {\left(x + 1\right)}^{\left(2 \cdot \frac{\frac{1}{n}}{2}\right)}\right)\right)}{\left({\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)} \cdot {\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)} + \left({x}^{\left(\frac{\frac{1}{n}}{2}\right)} \cdot {x}^{\left(\frac{\frac{1}{n}}{2}\right)} - {\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)} \cdot {x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right)\right) \cdot \left({\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)} \cdot {\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)} + \left({x}^{\left(\frac{\frac{1}{n}}{2}\right)} \cdot {x}^{\left(\frac{\frac{1}{n}}{2}\right)} - {\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)} \cdot {x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left({\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)}\right)}^{3} + {\left({x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right)}^{3}}{\mathsf{fma}\left({x}^{\left(\frac{\frac{1}{n}}{2}\right)}, {x}^{\left(\frac{\frac{1}{n}}{2}\right)} - {\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)}, {\left(x + 1\right)}^{\left(2 \cdot \frac{\frac{1}{n}}{2}\right)}\right)} \cdot \left({\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)} - {x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right)\\ \end{array}\]

C

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

↓

double code(double x, double n) {
	double VAR;
	if (((1.0 / n) <= -0.0008289574409786263)) {
		VAR = ((pow((x + 1.0), ((1.0 / n) / 2.0)) + pow(x, ((1.0 / n) / 2.0))) * (pow((x + 1.0), ((1.0 / n) / 2.0)) - pow(x, ((1.0 / n) / 2.0))));
	} else {
		double VAR_1;
		if (((1.0 / n) <= 2.7376649343181104e-11)) {
			VAR_1 = (fma(fma(pow(x, ((1.0 / n) / 2.0)), (pow(x, ((1.0 / n) / 2.0)) - pow((x + 1.0), ((1.0 / n) / 2.0))), pow((x + 1.0), (2.0 * ((1.0 / n) / 2.0)))), (-(pow(pow((x + 1.0), ((1.0 / n) / 2.0)), 3.0) + pow(pow(x, ((1.0 / n) / 2.0)), 3.0)) * fma(0.25, (log((1.0 / x)) / (x * pow(n, 2.0))), ((0.25 / n) / pow(x, 2.0)))), ((((0.5 / n) / x) * (pow(pow((x + 1.0), ((1.0 / n) / 2.0)), 3.0) + pow(pow(x, ((1.0 / n) / 2.0)), 3.0))) * ((pow(x, ((1.0 / n) / 2.0)) * (pow(x, ((1.0 / n) / 2.0)) - pow((x + 1.0), ((1.0 / n) / 2.0)))) + pow((x + 1.0), (2.0 * ((1.0 / n) / 2.0)))))) / (((pow((x + 1.0), ((1.0 / n) / 2.0)) * pow((x + 1.0), ((1.0 / n) / 2.0))) + ((pow(x, ((1.0 / n) / 2.0)) * pow(x, ((1.0 / n) / 2.0))) - (pow((x + 1.0), ((1.0 / n) / 2.0)) * pow(x, ((1.0 / n) / 2.0))))) * ((pow((x + 1.0), ((1.0 / n) / 2.0)) * pow((x + 1.0), ((1.0 / n) / 2.0))) + ((pow(x, ((1.0 / n) / 2.0)) * pow(x, ((1.0 / n) / 2.0))) - (pow((x + 1.0), ((1.0 / n) / 2.0)) * pow(x, ((1.0 / n) / 2.0)))))));
		} else {
			VAR_1 = (((pow(pow((x + 1.0), ((1.0 / n) / 2.0)), 3.0) + pow(pow(x, ((1.0 / n) / 2.0)), 3.0)) / fma(pow(x, ((1.0 / n) / 2.0)), (pow(x, ((1.0 / n) / 2.0)) - pow((x + 1.0), ((1.0 / n) / 2.0))), pow((x + 1.0), (2.0 * ((1.0 / n) / 2.0))))) * (pow((x + 1.0), ((1.0 / n) / 2.0)) - pow(x, ((1.0 / n) / 2.0))));
		}
		VAR = VAR_1;
	}
	return VAR;
}

Error

Bits error versus x

Bits error versus n

Derivation

1. Split input into 3 regimes

2. if (/ 1.0 n) < -0.0008289574409786263

   1. Initial program 0.5

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
   2. Using strategy rm

   3. Applied sqr-pow 0.5

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

   4. Applied sqr-pow 0.5

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

   5. Applied difference-of-squares 0.5

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

   if -0.0008289574409786263 < (/ 1.0 n) < 2.7376649343181104e-11

   1. Initial program 44.6

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
   2. Using strategy rm

   3. Applied sqr-pow 44.7

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

   4. Applied sqr-pow 44.6

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

   5. Applied difference-of-squares 44.6

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

   6. Taylor expanded around inf 32.6

      \[\leadsto \left({\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)} + {x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right) \cdot \color{blue}{\left(0.5 \cdot \frac{1}{x \cdot n} - \left(0.25 \cdot \frac{1}{{x}^{2} \cdot n} + 0.25 \cdot \frac{\log \left(\frac{1}{x}\right)}{x \cdot {n}^{2}}\right)\right)}\]

   7. Simplified 32.1

      \[\leadsto \left({\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)} + {x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right) \cdot \color{blue}{\mathsf{fma}\left(-0.25, \frac{1}{{x}^{2} \cdot n} + \frac{\log \left(\frac{1}{x}\right)}{x \cdot {n}^{2}}, \frac{\frac{0.5}{n}}{x}\right)}\]
   8. Using strategy rm

   9. Applied fma-udef 32.1

      \[\leadsto \left({\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)} + {x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right) \cdot \color{blue}{\left(\left(-0.25\right) \cdot \left(\frac{1}{{x}^{2} \cdot n} + \frac{\log \left(\frac{1}{x}\right)}{x \cdot {n}^{2}}\right) + \frac{\frac{0.5}{n}}{x}\right)}\]

   10. Applied distribute-lft-in 32.1

       \[\leadsto \color{blue}{\left({\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)} + {x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right) \cdot \left(\left(-0.25\right) \cdot \left(\frac{1}{{x}^{2} \cdot n} + \frac{\log \left(\frac{1}{x}\right)}{x \cdot {n}^{2}}\right)\right) + \left({\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)} + {x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right) \cdot \frac{\frac{0.5}{n}}{x}}\]

   11. Simplified 32.0

       \[\leadsto \color{blue}{\left(-\left({\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)} + {x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right)\right) \cdot \mathsf{fma}\left(0.25, \frac{\log \left(\frac{1}{x}\right)}{x \cdot {n}^{2}}, \frac{\frac{0.25}{n}}{{x}^{2}}\right)} + \left({\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)} + {x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right) \cdot \frac{\frac{0.5}{n}}{x}\]

   12. Simplified 32.0

       \[\leadsto \left(-\left({\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)} + {x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right)\right) \cdot \mathsf{fma}\left(0.25, \frac{\log \left(\frac{1}{x}\right)}{x \cdot {n}^{2}}, \frac{\frac{0.25}{n}}{{x}^{2}}\right) + \color{blue}{\frac{\frac{0.5}{n}}{x} \cdot \left({\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)} + {x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right)}\]
   13. Using strategy rm

   14. Applied flip3-+ 32.0

       \[\leadsto \left(-\left({\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)} + {x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right)\right) \cdot \mathsf{fma}\left(0.25, \frac{\log \left(\frac{1}{x}\right)}{x \cdot {n}^{2}}, \frac{\frac{0.25}{n}}{{x}^{2}}\right) + \frac{\frac{0.5}{n}}{x} \cdot \color{blue}{\frac{{\left({\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)}\right)}^{3} + {\left({x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right)}^{3}}{{\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)} \cdot {\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)} + \left({x}^{\left(\frac{\frac{1}{n}}{2}\right)} \cdot {x}^{\left(\frac{\frac{1}{n}}{2}\right)} - {\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)} \cdot {x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right)}}\]

   15. Applied associate-*r/ 32.0

       \[\leadsto \left(-\left({\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)} + {x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right)\right) \cdot \mathsf{fma}\left(0.25, \frac{\log \left(\frac{1}{x}\right)}{x \cdot {n}^{2}}, \frac{\frac{0.25}{n}}{{x}^{2}}\right) + \color{blue}{\frac{\frac{\frac{0.5}{n}}{x} \cdot \left({\left({\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)}\right)}^{3} + {\left({x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right)}^{3}\right)}{{\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)} \cdot {\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)} + \left({x}^{\left(\frac{\frac{1}{n}}{2}\right)} \cdot {x}^{\left(\frac{\frac{1}{n}}{2}\right)} - {\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)} \cdot {x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right)}}\]

   16. Applied flip3-+ 32.0

       \[\leadsto \left(-\color{blue}{\frac{{\left({\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)}\right)}^{3} + {\left({x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right)}^{3}}{{\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)} \cdot {\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)} + \left({x}^{\left(\frac{\frac{1}{n}}{2}\right)} \cdot {x}^{\left(\frac{\frac{1}{n}}{2}\right)} - {\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)} \cdot {x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right)}}\right) \cdot \mathsf{fma}\left(0.25, \frac{\log \left(\frac{1}{x}\right)}{x \cdot {n}^{2}}, \frac{\frac{0.25}{n}}{{x}^{2}}\right) + \frac{\frac{\frac{0.5}{n}}{x} \cdot \left({\left({\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)}\right)}^{3} + {\left({x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right)}^{3}\right)}{{\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)} \cdot {\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)} + \left({x}^{\left(\frac{\frac{1}{n}}{2}\right)} \cdot {x}^{\left(\frac{\frac{1}{n}}{2}\right)} - {\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)} \cdot {x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right)}\]

   17. Applied distribute-neg-frac 32.0

       \[\leadsto \color{blue}{\frac{-\left({\left({\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)}\right)}^{3} + {\left({x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right)}^{3}\right)}{{\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)} \cdot {\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)} + \left({x}^{\left(\frac{\frac{1}{n}}{2}\right)} \cdot {x}^{\left(\frac{\frac{1}{n}}{2}\right)} - {\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)} \cdot {x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right)}} \cdot \mathsf{fma}\left(0.25, \frac{\log \left(\frac{1}{x}\right)}{x \cdot {n}^{2}}, \frac{\frac{0.25}{n}}{{x}^{2}}\right) + \frac{\frac{\frac{0.5}{n}}{x} \cdot \left({\left({\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)}\right)}^{3} + {\left({x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right)}^{3}\right)}{{\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)} \cdot {\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)} + \left({x}^{\left(\frac{\frac{1}{n}}{2}\right)} \cdot {x}^{\left(\frac{\frac{1}{n}}{2}\right)} - {\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)} \cdot {x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right)}\]

   18. Applied associate-*l/ 32.0

       \[\leadsto \color{blue}{\frac{\left(-\left({\left({\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)}\right)}^{3} + {\left({x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right)}^{3}\right)\right) \cdot \mathsf{fma}\left(0.25, \frac{\log \left(\frac{1}{x}\right)}{x \cdot {n}^{2}}, \frac{\frac{0.25}{n}}{{x}^{2}}\right)}{{\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)} \cdot {\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)} + \left({x}^{\left(\frac{\frac{1}{n}}{2}\right)} \cdot {x}^{\left(\frac{\frac{1}{n}}{2}\right)} - {\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)} \cdot {x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right)}} + \frac{\frac{\frac{0.5}{n}}{x} \cdot \left({\left({\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)}\right)}^{3} + {\left({x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right)}^{3}\right)}{{\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)} \cdot {\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)} + \left({x}^{\left(\frac{\frac{1}{n}}{2}\right)} \cdot {x}^{\left(\frac{\frac{1}{n}}{2}\right)} - {\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)} \cdot {x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right)}\]

   19. Applied frac-add 32.0

       \[\leadsto \color{blue}{\frac{\left(\left(-\left({\left({\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)}\right)}^{3} + {\left({x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right)}^{3}\right)\right) \cdot \mathsf{fma}\left(0.25, \frac{\log \left(\frac{1}{x}\right)}{x \cdot {n}^{2}}, \frac{\frac{0.25}{n}}{{x}^{2}}\right)\right) \cdot \left({\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)} \cdot {\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)} + \left({x}^{\left(\frac{\frac{1}{n}}{2}\right)} \cdot {x}^{\left(\frac{\frac{1}{n}}{2}\right)} - {\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)} \cdot {x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right)\right) + \left({\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)} \cdot {\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)} + \left({x}^{\left(\frac{\frac{1}{n}}{2}\right)} \cdot {x}^{\left(\frac{\frac{1}{n}}{2}\right)} - {\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)} \cdot {x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right)\right) \cdot \left(\frac{\frac{0.5}{n}}{x} \cdot \left({\left({\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)}\right)}^{3} + {\left({x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right)}^{3}\right)\right)}{\left({\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)} \cdot {\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)} + \left({x}^{\left(\frac{\frac{1}{n}}{2}\right)} \cdot {x}^{\left(\frac{\frac{1}{n}}{2}\right)} - {\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)} \cdot {x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right)\right) \cdot \left({\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)} \cdot {\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)} + \left({x}^{\left(\frac{\frac{1}{n}}{2}\right)} \cdot {x}^{\left(\frac{\frac{1}{n}}{2}\right)} - {\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)} \cdot {x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right)\right)}}\]

   20. Simplified 32.0

       \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left({x}^{\left(\frac{\frac{1}{n}}{2}\right)}, {x}^{\left(\frac{\frac{1}{n}}{2}\right)} - {\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)}, {\left(x + 1\right)}^{\left(2 \cdot \frac{\frac{1}{n}}{2}\right)}\right), \left(-\left({\left({\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)}\right)}^{3} + {\left({x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right)}^{3}\right)\right) \cdot \mathsf{fma}\left(0.25, \frac{\log \left(\frac{1}{x}\right)}{x \cdot {n}^{2}}, \frac{\frac{0.25}{n}}{{x}^{2}}\right), \left(\frac{\frac{0.5}{n}}{x} \cdot \left({\left({\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)}\right)}^{3} + {\left({x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right)}^{3}\right)\right) \cdot \left({x}^{\left(\frac{\frac{1}{n}}{2}\right)} \cdot \left({x}^{\left(\frac{\frac{1}{n}}{2}\right)} - {\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)}\right) + {\left(x + 1\right)}^{\left(2 \cdot \frac{\frac{1}{n}}{2}\right)}\right)\right)}}{\left({\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)} \cdot {\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)} + \left({x}^{\left(\frac{\frac{1}{n}}{2}\right)} \cdot {x}^{\left(\frac{\frac{1}{n}}{2}\right)} - {\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)} \cdot {x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right)\right) \cdot \left({\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)} \cdot {\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)} + \left({x}^{\left(\frac{\frac{1}{n}}{2}\right)} \cdot {x}^{\left(\frac{\frac{1}{n}}{2}\right)} - {\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)} \cdot {x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right)\right)}\]

   if 2.7376649343181104e-11 < (/ 1.0 n)

   1. Initial program 6.9

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
   2. Using strategy rm

   3. Applied sqr-pow 7.0

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

   4. Applied sqr-pow 6.9

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

   5. Applied difference-of-squares 6.9

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

   7. Applied flip3-+ 7.0

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

   8. Simplified 7.0

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

4. Final simplification 23.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \le -8.28957440978626294 \cdot 10^{-4}:\\ \;\;\;\;\left({\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)} + {x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right) \cdot \left({\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)} - {x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right)\\ \mathbf{elif}\;\frac{1}{n} \le 2.7376649343181104 \cdot 10^{-11}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left({x}^{\left(\frac{\frac{1}{n}}{2}\right)}, {x}^{\left(\frac{\frac{1}{n}}{2}\right)} - {\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)}, {\left(x + 1\right)}^{\left(2 \cdot \frac{\frac{1}{n}}{2}\right)}\right), \left(-\left({\left({\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)}\right)}^{3} + {\left({x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right)}^{3}\right)\right) \cdot \mathsf{fma}\left(0.25, \frac{\log \left(\frac{1}{x}\right)}{x \cdot {n}^{2}}, \frac{\frac{0.25}{n}}{{x}^{2}}\right), \left(\frac{\frac{0.5}{n}}{x} \cdot \left({\left({\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)}\right)}^{3} + {\left({x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right)}^{3}\right)\right) \cdot \left({x}^{\left(\frac{\frac{1}{n}}{2}\right)} \cdot \left({x}^{\left(\frac{\frac{1}{n}}{2}\right)} - {\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)}\right) + {\left(x + 1\right)}^{\left(2 \cdot \frac{\frac{1}{n}}{2}\right)}\right)\right)}{\left({\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)} \cdot {\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)} + \left({x}^{\left(\frac{\frac{1}{n}}{2}\right)} \cdot {x}^{\left(\frac{\frac{1}{n}}{2}\right)} - {\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)} \cdot {x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right)\right) \cdot \left({\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)} \cdot {\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)} + \left({x}^{\left(\frac{\frac{1}{n}}{2}\right)} \cdot {x}^{\left(\frac{\frac{1}{n}}{2}\right)} - {\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)} \cdot {x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left({\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)}\right)}^{3} + {\left({x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right)}^{3}}{\mathsf{fma}\left({x}^{\left(\frac{\frac{1}{n}}{2}\right)}, {x}^{\left(\frac{\frac{1}{n}}{2}\right)} - {\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)}, {\left(x + 1\right)}^{\left(2 \cdot \frac{\frac{1}{n}}{2}\right)}\right)} \cdot \left({\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)} - {x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020078 +o rules:numerics
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  :precision binary64
  (- (pow (+ x 1) (/ 1 n)) (pow x (/ 1 n))))