2nthrt (problem 3.4.6)

Average Error: 32.6 → 23.3

Time: 12.6s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;n \leq -250.7434974379807:\\ \;\;\;\;\frac{1}{x} \cdot \frac{1}{n} - \left(0.2777777777777778 \cdot \frac{{\left(\log 1\right)}^{2}}{n \cdot n} + 0.33333333333333337 \cdot \left(\frac{\log 1}{n} + \frac{\log 1}{n} \cdot \frac{\log x}{n}\right)\right)\\ \mathbf{elif}\;n \leq 18206649765.93102:\\ \;\;\;\;\sqrt[3]{{\left(\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}\right)}^{\left(\frac{1}{n}\right)} \cdot {\left(\sqrt[3]{x + 1}\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}} \cdot \left(\sqrt[3]{{\left(\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}\right)}^{\left(\frac{1}{n}\right)} \cdot {\left(\sqrt[3]{x + 1}\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}} \cdot \sqrt[3]{{\left(\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}\right)}^{\left(\frac{1}{n}\right)} \cdot {\left(\sqrt[3]{x + 1}\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x}}{n} - \left(0.2777777777777778 \cdot \frac{{\left(\log 1\right)}^{2}}{n \cdot n} + 0.33333333333333337 \cdot \left(\frac{\log 1}{n} + \frac{\log 1}{n} \cdot \frac{\log x}{n}\right)\right)\\ \end{array}\]

C

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

↓

double code(double x, double n) {
	double VAR;
	if ((n <= -250.7434974379807)) {
		VAR = ((double) (((double) ((1.0 / x) * (1.0 / n))) - ((double) (((double) (0.2777777777777778 * (((double) pow(((double) log(1.0)), 2.0)) / ((double) (n * n))))) + ((double) (0.33333333333333337 * ((double) ((((double) log(1.0)) / n) + ((double) ((((double) log(1.0)) / n) * (((double) log(x)) / n)))))))))));
	} else {
		double VAR_1;
		if ((n <= 18206649765.93102)) {
			VAR_1 = ((double) (((double) cbrt(((double) (((double) (((double) pow(((double) (((double) cbrt(((double) (x + 1.0)))) * ((double) cbrt(((double) (x + 1.0)))))), (1.0 / n))) * ((double) pow(((double) cbrt(((double) (x + 1.0)))), (1.0 / n))))) - ((double) pow(x, (1.0 / n))))))) * ((double) (((double) cbrt(((double) (((double) (((double) pow(((double) (((double) cbrt(((double) (x + 1.0)))) * ((double) cbrt(((double) (x + 1.0)))))), (1.0 / n))) * ((double) pow(((double) cbrt(((double) (x + 1.0)))), (1.0 / n))))) - ((double) pow(x, (1.0 / n))))))) * ((double) cbrt(((double) (((double) (((double) pow(((double) (((double) cbrt(((double) (x + 1.0)))) * ((double) cbrt(((double) (x + 1.0)))))), (1.0 / n))) * ((double) pow(((double) cbrt(((double) (x + 1.0)))), (1.0 / n))))) - ((double) pow(x, (1.0 / n)))))))))));
		} else {
			VAR_1 = ((double) (((1.0 / x) / n) - ((double) (((double) (0.2777777777777778 * (((double) pow(((double) log(1.0)), 2.0)) / ((double) (n * n))))) + ((double) (0.33333333333333337 * ((double) ((((double) log(1.0)) / n) + ((double) ((((double) log(1.0)) / n) * (((double) log(x)) / n)))))))))));
		}
		VAR = VAR_1;
	}
	return VAR;
}

Error

Bits error versus x

Bits error versus n

Derivation

1. Split input into 3 regimes

2. if n < -250.74349743798069

   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

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

   4. Applied unpow-prod-down 44.6

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

   5. Taylor expanded around inf 32.2

      \[\leadsto \color{blue}{\left(1 \cdot \frac{1}{x \cdot n} + 0.33333333333333337 \cdot \frac{\log 1 \cdot \log \left(\frac{1}{x}\right)}{{n}^{2}}\right) - \left(0.2777777777777778 \cdot \frac{{\left(\log 1\right)}^{2}}{{n}^{2}} + 0.33333333333333337 \cdot \frac{\log 1}{n}\right)}\]

   6. Simplified 32.2

      \[\leadsto \color{blue}{\frac{1}{x \cdot n} + \left(0.33333333333333337 \cdot \left(\frac{\log 1}{n} \cdot \frac{-\log x}{n} - \frac{\log 1}{n}\right) - 0.2777777777777778 \cdot \frac{{\left(\log 1\right)}^{2}}{n \cdot n}\right)}\]
   7. Using strategy rm

   8. Applied *-un-lft-identity 32.2

      \[\leadsto \frac{\color{blue}{1 \cdot 1}}{x \cdot n} + \left(0.33333333333333337 \cdot \left(\frac{\log 1}{n} \cdot \frac{-\log x}{n} - \frac{\log 1}{n}\right) - 0.2777777777777778 \cdot \frac{{\left(\log 1\right)}^{2}}{n \cdot n}\right)\]

   9. Applied times-frac 31.5

      \[\leadsto \color{blue}{\frac{1}{x} \cdot \frac{1}{n}} + \left(0.33333333333333337 \cdot \left(\frac{\log 1}{n} \cdot \frac{-\log x}{n} - \frac{\log 1}{n}\right) - 0.2777777777777778 \cdot \frac{{\left(\log 1\right)}^{2}}{n \cdot n}\right)\]

   if -250.74349743798069 < n < 18206649765.9310188

   1. Initial program 2.9

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

   3. Applied add-cube-cbrt 3.0

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

   4. Applied unpow-prod-down 3.0

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

   6. Applied add-cube-cbrt 3.0

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

   if 18206649765.9310188 < n

   1. Initial program 45.1

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

   3. Applied add-cube-cbrt 45.1

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

   4. Applied unpow-prod-down 45.1

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

   5. Taylor expanded around inf 32.6

      \[\leadsto \color{blue}{\left(1 \cdot \frac{1}{x \cdot n} + 0.33333333333333337 \cdot \frac{\log 1 \cdot \log \left(\frac{1}{x}\right)}{{n}^{2}}\right) - \left(0.2777777777777778 \cdot \frac{{\left(\log 1\right)}^{2}}{{n}^{2}} + 0.33333333333333337 \cdot \frac{\log 1}{n}\right)}\]

   6. Simplified 32.6

      \[\leadsto \color{blue}{\frac{1}{x \cdot n} + \left(0.33333333333333337 \cdot \left(\frac{\log 1}{n} \cdot \frac{-\log x}{n} - \frac{\log 1}{n}\right) - 0.2777777777777778 \cdot \frac{{\left(\log 1\right)}^{2}}{n \cdot n}\right)}\]
   7. Using strategy rm

   8. Applied associate-/r* 31.9

      \[\leadsto \color{blue}{\frac{\frac{1}{x}}{n}} + \left(0.33333333333333337 \cdot \left(\frac{\log 1}{n} \cdot \frac{-\log x}{n} - \frac{\log 1}{n}\right) - 0.2777777777777778 \cdot \frac{{\left(\log 1\right)}^{2}}{n \cdot n}\right)\]
3. Recombined 3 regimes into one program.

4. Final simplification 23.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -250.7434974379807:\\ \;\;\;\;\frac{1}{x} \cdot \frac{1}{n} - \left(0.2777777777777778 \cdot \frac{{\left(\log 1\right)}^{2}}{n \cdot n} + 0.33333333333333337 \cdot \left(\frac{\log 1}{n} + \frac{\log 1}{n} \cdot \frac{\log x}{n}\right)\right)\\ \mathbf{elif}\;n \leq 18206649765.93102:\\ \;\;\;\;\sqrt[3]{{\left(\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}\right)}^{\left(\frac{1}{n}\right)} \cdot {\left(\sqrt[3]{x + 1}\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}} \cdot \left(\sqrt[3]{{\left(\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}\right)}^{\left(\frac{1}{n}\right)} \cdot {\left(\sqrt[3]{x + 1}\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}} \cdot \sqrt[3]{{\left(\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}\right)}^{\left(\frac{1}{n}\right)} \cdot {\left(\sqrt[3]{x + 1}\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x}}{n} - \left(0.2777777777777778 \cdot \frac{{\left(\log 1\right)}^{2}}{n \cdot n} + 0.33333333333333337 \cdot \left(\frac{\log 1}{n} + \frac{\log 1}{n} \cdot \frac{\log x}{n}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020199 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  :precision binary64
  (- (pow (+ x 1.0) (/ 1.0 n)) (pow x (/ 1.0 n))))