2nthrt (problem 3.4.6)

Average Error: 33.1 → 23.6

Time: 13.2s

Precision: 64

• could not determine a ground truth for program body

  1. x = 3.8081982856699253e+298
  2. n = 9.300677034083424e-116

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{1}{n} \le -7.27487421476467367 \cdot 10^{-6}:\\ \;\;\;\;\left({\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)} + {\left(\sqrt{x}\right)}^{\left(\frac{1}{n}\right)}\right) \cdot \left({\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)} - {\left(\sqrt{x}\right)}^{\left(\frac{1}{n}\right)}\right)\\ \mathbf{elif}\;\frac{1}{n} \le -2.58521681167098109 \cdot 10^{-52}:\\ \;\;\;\;\frac{\frac{1}{n}}{x} - \left(\frac{\frac{0.5}{n}}{{x}^{2}} - \frac{\log x \cdot 1}{x \cdot {n}^{2}}\right)\\ \mathbf{elif}\;\frac{1}{n} \le 2.82170920789161167 \cdot 10^{-21}:\\ \;\;\;\;\left({\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)} + {\left(\sqrt{x}\right)}^{\left(\frac{1}{n}\right)}\right) \cdot \left(\frac{\frac{\frac{0.5}{n}}{\sqrt{x}}}{\sqrt{x}} + \left(\left(1 - 0.5 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}\right) - {\left(\sqrt{x}\right)}^{\left(\frac{1}{n}\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)}\right)}^{3} - {\left({x}^{\left(\frac{1}{n}\right)}\right)}^{3}}{{x}^{\left(\frac{1}{n}\right)} \cdot \left({x}^{\left(\frac{1}{n}\right)} + {\left(x + 1\right)}^{\left(\frac{1}{n}\right)}\right) + {\left(x + 1\right)}^{\left(2 \cdot \frac{1}{n}\right)}}\\ \end{array}\]

C

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

↓

double code(double x, double n) {
	double VAR;
	if ((((double) (1.0 / n)) <= -7.274874214764674e-06)) {
		VAR = ((double) (((double) (((double) pow(((double) (x + 1.0)), ((double) (((double) (1.0 / n)) / 2.0)))) + ((double) pow(((double) sqrt(x)), ((double) (1.0 / n)))))) * ((double) (((double) pow(((double) (x + 1.0)), ((double) (((double) (1.0 / n)) / 2.0)))) - ((double) pow(((double) sqrt(x)), ((double) (1.0 / n))))))));
	} else {
		double VAR_1;
		if ((((double) (1.0 / n)) <= -2.585216811670981e-52)) {
			VAR_1 = ((double) (((double) (((double) (1.0 / n)) / x)) - ((double) (((double) (((double) (0.5 / n)) / ((double) pow(x, 2.0)))) - ((double) (((double) (((double) log(x)) * 1.0)) / ((double) (x * ((double) pow(n, 2.0))))))))));
		} else {
			double VAR_2;
			if ((((double) (1.0 / n)) <= 2.8217092078916117e-21)) {
				VAR_2 = ((double) (((double) (((double) pow(((double) (x + 1.0)), ((double) (((double) (1.0 / n)) / 2.0)))) + ((double) pow(((double) sqrt(x)), ((double) (1.0 / n)))))) * ((double) (((double) (((double) (((double) (0.5 / n)) / ((double) sqrt(x)))) / ((double) sqrt(x)))) + ((double) (((double) (1.0 - ((double) (0.5 * ((double) (((double) log(((double) (1.0 / x)))) / n)))))) - ((double) pow(((double) sqrt(x)), ((double) (1.0 / n))))))))));
			} else {
				VAR_2 = ((double) (((double) (((double) pow(((double) pow(((double) (x + 1.0)), ((double) (1.0 / n)))), 3.0)) - ((double) pow(((double) pow(x, ((double) (1.0 / n)))), 3.0)))) / ((double) (((double) (((double) pow(x, ((double) (1.0 / n)))) * ((double) (((double) pow(x, ((double) (1.0 / n)))) + ((double) pow(((double) (x + 1.0)), ((double) (1.0 / n)))))))) + ((double) pow(((double) (x + 1.0)), ((double) (2.0 * ((double) (1.0 / n))))))))));
			}
			VAR_1 = VAR_2;
		}
		VAR = VAR_1;
	}
	return VAR;
}

Error

Bits error versus x

Bits error versus n

Derivation

1. Split input into 4 regimes

2. if (/ 1.0 n) < -7.274874214764674e-06

   1. Initial program 1.1

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

   3. Applied add-sqr-sqrt 1.1

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

   4. Applied unpow-prod-down 1.1

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

   5. Applied sqr-pow 1.1

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

   6. Applied difference-of-squares 1.1

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

   if -7.274874214764674e-06 < (/ 1.0 n) < -2.585216811670981e-52

   1. Initial program 55.0

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

   2. Taylor expanded around inf 31.6

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

   3. Simplified 31.2

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

   if -2.585216811670981e-52 < (/ 1.0 n) < 2.8217092078916117e-21

   1. Initial program 44.4

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

   3. Applied add-sqr-sqrt 44.4

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

   4. Applied unpow-prod-down 44.4

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

   5. Applied sqr-pow 44.4

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

   6. Applied difference-of-squares 44.4

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

   7. Taylor expanded around inf 44.4

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

   8. Simplified 44.4

      \[\leadsto \left({\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)} + {\left(\sqrt{x}\right)}^{\left(\frac{1}{n}\right)}\right) \cdot \left(\color{blue}{\left(\left(1 - 0.5 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}\right) + \frac{\frac{0.5}{n}}{x}\right)} - {\left(\sqrt{x}\right)}^{\left(\frac{1}{n}\right)}\right)\]
   9. Using strategy rm

   10. Applied +-commutative 44.4

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

   11. Applied associate--l+ 31.6

       \[\leadsto \left({\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)} + {\left(\sqrt{x}\right)}^{\left(\frac{1}{n}\right)}\right) \cdot \color{blue}{\left(\frac{\frac{0.5}{n}}{x} + \left(\left(1 - 0.5 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}\right) - {\left(\sqrt{x}\right)}^{\left(\frac{1}{n}\right)}\right)\right)}\]
   12. Using strategy rm

   13. Applied add-sqr-sqrt 31.6

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

   14. Applied associate-/r* 31.6

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

   if 2.8217092078916117e-21 < (/ 1.0 n)

   1. Initial program 11.9

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

   3. Applied flip3-- 11.9

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

   4. Simplified 11.9

      \[\leadsto \frac{{\left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)}\right)}^{3} - {\left({x}^{\left(\frac{1}{n}\right)}\right)}^{3}}{\color{blue}{{x}^{\left(\frac{1}{n}\right)} \cdot \left({x}^{\left(\frac{1}{n}\right)} + {\left(x + 1\right)}^{\left(\frac{1}{n}\right)}\right) + {\left(x + 1\right)}^{\left(2 \cdot \frac{1}{n}\right)}}}\]
3. Recombined 4 regimes into one program.

4. Final simplification 23.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \le -7.27487421476467367 \cdot 10^{-6}:\\ \;\;\;\;\left({\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)} + {\left(\sqrt{x}\right)}^{\left(\frac{1}{n}\right)}\right) \cdot \left({\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)} - {\left(\sqrt{x}\right)}^{\left(\frac{1}{n}\right)}\right)\\ \mathbf{elif}\;\frac{1}{n} \le -2.58521681167098109 \cdot 10^{-52}:\\ \;\;\;\;\frac{\frac{1}{n}}{x} - \left(\frac{\frac{0.5}{n}}{{x}^{2}} - \frac{\log x \cdot 1}{x \cdot {n}^{2}}\right)\\ \mathbf{elif}\;\frac{1}{n} \le 2.82170920789161167 \cdot 10^{-21}:\\ \;\;\;\;\left({\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)} + {\left(\sqrt{x}\right)}^{\left(\frac{1}{n}\right)}\right) \cdot \left(\frac{\frac{\frac{0.5}{n}}{\sqrt{x}}}{\sqrt{x}} + \left(\left(1 - 0.5 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}\right) - {\left(\sqrt{x}\right)}^{\left(\frac{1}{n}\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)}\right)}^{3} - {\left({x}^{\left(\frac{1}{n}\right)}\right)}^{3}}{{x}^{\left(\frac{1}{n}\right)} \cdot \left({x}^{\left(\frac{1}{n}\right)} + {\left(x + 1\right)}^{\left(\frac{1}{n}\right)}\right) + {\left(x + 1\right)}^{\left(2 \cdot \frac{1}{n}\right)}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020114 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  :precision binary64
  (- (pow (+ x 1) (/ 1 n)) (pow x (/ 1 n))))