2nthrt (problem 3.4.6)

Average Error: 33.0 → 23.6

Time: 11.5s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;n \le -92.670369249432468:\\ \;\;\;\;\frac{1}{n \cdot x} + \left(0.27777777777777779 \cdot \frac{{\left(\log 1\right)}^{2}}{n \cdot n} + \left(\log 1 \cdot \frac{0.33333333333333337}{n} + 0.33333333333333326 \cdot \left(\log x \cdot \frac{\log 1}{n \cdot n}\right)\right)\right)\\ \mathbf{elif}\;n \le 238139942086233056:\\ \;\;\;\;{\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - {\left(\sqrt[3]{x} \cdot e^{\log \left(\sqrt[3]{x}\right)}\right)}^{\left(\frac{1}{n}\right)} \cdot {\left(\sqrt[3]{x}\right)}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(0.27777777777777779 \cdot \frac{{\left(\log 1\right)}^{2}}{n \cdot n} + \left(\log 1 \cdot \frac{0.33333333333333337}{n} + 0.33333333333333326 \cdot \left(\log x \cdot \frac{\log 1}{n \cdot n}\right)\right)\right) + \frac{1}{n} \cdot \frac{1}{x}\\ \end{array}\]

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 ((n <= -92.67036924943247)) {
		VAR = ((double) (((double) (1.0 / ((double) (n * x)))) + ((double) (((double) (0.2777777777777778 * ((double) (((double) pow(((double) log(1.0)), 2.0)) / ((double) (n * n)))))) + ((double) (((double) (((double) log(1.0)) * ((double) (0.33333333333333337 / n)))) + ((double) (0.33333333333333326 * ((double) (((double) log(x)) * ((double) (((double) log(1.0)) / ((double) (n * n))))))))))))));
	} else {
		double VAR_1;
		if ((n <= 2.3813994208623306e+17)) {
			VAR_1 = ((double) (((double) pow(((double) (1.0 + x)), ((double) (1.0 / n)))) - ((double) (((double) pow(((double) (((double) cbrt(x)) * ((double) exp(((double) log(((double) cbrt(x)))))))), ((double) (1.0 / n)))) * ((double) pow(((double) cbrt(x)), ((double) (1.0 / n))))))));
		} else {
			VAR_1 = ((double) (((double) (((double) (0.2777777777777778 * ((double) (((double) pow(((double) log(1.0)), 2.0)) / ((double) (n * n)))))) + ((double) (((double) (((double) log(1.0)) * ((double) (0.33333333333333337 / n)))) + ((double) (0.33333333333333326 * ((double) (((double) log(x)) * ((double) (((double) log(1.0)) / ((double) (n * n)))))))))))) + ((double) (((double) (1.0 / n)) * ((double) (1.0 / x))))));
		}
		VAR = VAR_1;
	}
	return VAR;
}

Error

Bits error versus x

Bits error versus n

Derivation

1. Split input into 3 regimes

2. if n < -92.670369249432468

   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 add-cube-cbrt 44.6

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

   4. Applied unpow-prod-down 44.6

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

   5. Taylor expanded around inf 32.0

      \[\leadsto \color{blue}{\left(1 \cdot \frac{1}{x \cdot n} + \left(0.27777777777777779 \cdot \frac{{\left(\log 1\right)}^{2}}{{n}^{2}} + 0.33333333333333337 \cdot \frac{\log 1}{n}\right)\right) - 0.33333333333333326 \cdot \frac{\log 1 \cdot \log \left(\frac{1}{x}\right)}{{n}^{2}}}\]

   6. Simplified 32.0

      \[\leadsto \color{blue}{\frac{1}{x \cdot n} + \left(0.27777777777777779 \cdot \frac{{\left(\log 1\right)}^{2}}{n \cdot n} + \left(\log 1 \cdot \frac{0.33333333333333337}{n} + 0.33333333333333326 \cdot \left(\log x \cdot \frac{\log 1}{n \cdot n}\right)\right)\right)}\]

   if -92.670369249432468 < n < 238139942086233056

   1. Initial program 4.2

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

   3. Applied add-cube-cbrt 4.2

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

   4. Applied unpow-prod-down 4.3

      \[\leadsto {\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - \color{blue}{{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}^{\left(\frac{1}{n}\right)} \cdot {\left(\sqrt[3]{x}\right)}^{\left(\frac{1}{n}\right)}}\]
   5. Using strategy rm

   6. Applied add-exp-log 4.3

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

   if 238139942086233056 < n

   1. Initial program 45.3

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

   3. Applied add-cube-cbrt 45.3

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

   4. Applied unpow-prod-down 45.3

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

   5. Taylor expanded around inf 31.8

      \[\leadsto \color{blue}{\left(1 \cdot \frac{1}{x \cdot n} + \left(0.27777777777777779 \cdot \frac{{\left(\log 1\right)}^{2}}{{n}^{2}} + 0.33333333333333337 \cdot \frac{\log 1}{n}\right)\right) - 0.33333333333333326 \cdot \frac{\log 1 \cdot \log \left(\frac{1}{x}\right)}{{n}^{2}}}\]

   6. Simplified 31.8

      \[\leadsto \color{blue}{\frac{1}{x \cdot n} + \left(0.27777777777777779 \cdot \frac{{\left(\log 1\right)}^{2}}{n \cdot n} + \left(\log 1 \cdot \frac{0.33333333333333337}{n} + 0.33333333333333326 \cdot \left(\log x \cdot \frac{\log 1}{n \cdot n}\right)\right)\right)}\]
   7. Using strategy rm

   8. Applied *-un-lft-identity 31.8

      \[\leadsto \frac{\color{blue}{1 \cdot 1}}{x \cdot n} + \left(0.27777777777777779 \cdot \frac{{\left(\log 1\right)}^{2}}{n \cdot n} + \left(\log 1 \cdot \frac{0.33333333333333337}{n} + 0.33333333333333326 \cdot \left(\log x \cdot \frac{\log 1}{n \cdot n}\right)\right)\right)\]

   9. Applied times-frac 31.1

      \[\leadsto \color{blue}{\frac{1}{x} \cdot \frac{1}{n}} + \left(0.27777777777777779 \cdot \frac{{\left(\log 1\right)}^{2}}{n \cdot n} + \left(\log 1 \cdot \frac{0.33333333333333337}{n} + 0.33333333333333326 \cdot \left(\log x \cdot \frac{\log 1}{n \cdot n}\right)\right)\right)\]
3. Recombined 3 regimes into one program.

4. Final simplification 23.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \le -92.670369249432468:\\ \;\;\;\;\frac{1}{n \cdot x} + \left(0.27777777777777779 \cdot \frac{{\left(\log 1\right)}^{2}}{n \cdot n} + \left(\log 1 \cdot \frac{0.33333333333333337}{n} + 0.33333333333333326 \cdot \left(\log x \cdot \frac{\log 1}{n \cdot n}\right)\right)\right)\\ \mathbf{elif}\;n \le 238139942086233056:\\ \;\;\;\;{\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - {\left(\sqrt[3]{x} \cdot e^{\log \left(\sqrt[3]{x}\right)}\right)}^{\left(\frac{1}{n}\right)} \cdot {\left(\sqrt[3]{x}\right)}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(0.27777777777777779 \cdot \frac{{\left(\log 1\right)}^{2}}{n \cdot n} + \left(\log 1 \cdot \frac{0.33333333333333337}{n} + 0.33333333333333326 \cdot \left(\log x \cdot \frac{\log 1}{n \cdot n}\right)\right)\right) + \frac{1}{n} \cdot \frac{1}{x}\\ \end{array}\]

Reproduce

herbie shell --seed 2020184 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  :precision binary64
  (- (pow (+ x 1.0) (/ 1.0 n)) (pow x (/ 1.0 n))))