2nthrt (problem 3.4.6)

Average Error: 32.8 → 24.4

Time: 16.1s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{1}{n} \le -2.24942559102901105 \cdot 10^{-18}:\\ \;\;\;\;\left({\left(\sqrt{x + 1}\right)}^{\left(\frac{1}{n}\right)} + {\left(\sqrt{x}\right)}^{\left(\frac{2 \cdot \frac{1}{n}}{2}\right)}\right) \cdot \left(\left(-{\left(\sqrt{x}\right)}^{\left(\frac{1}{n}\right)}\right) + {\left(\sqrt{x + 1}\right)}^{\left(\frac{1}{n}\right)}\right) + \mathsf{fma}\left({\left(\sqrt{x}\right)}^{\left(\frac{1}{n}\right)}, -{\left(\sqrt{x}\right)}^{\left(\frac{1}{n}\right)}, {\left(\sqrt{x}\right)}^{\left(2 \cdot \frac{1}{n}\right)}\right)\\ \mathbf{elif}\;\frac{1}{n} \le 2.41213401008201 \cdot 10^{-10}:\\ \;\;\;\;\mathsf{fma}\left(1, \frac{1}{x \cdot n}, -\mathsf{fma}\left(0.5, \frac{1}{{x}^{2} \cdot n}, 1 \cdot \frac{\log \left(\frac{1}{x}\right)}{x \cdot {n}^{2}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)}\right)}^{3} - {\left({x}^{\left(\frac{1}{n}\right)}\right)}^{3}}{\mathsf{fma}\left({x}^{\left(\frac{1}{n}\right)}, {\left(x + 1\right)}^{\left(\frac{1}{n}\right)} + {x}^{\left(\frac{1}{n}\right)}, {\left(x + 1\right)}^{\left(2 \cdot \frac{1}{n}\right)}\right)}\\ \end{array}\]

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) <= -2.249425591029011e-18)) {
		VAR = (((pow(sqrt((x + 1.0)), (1.0 / n)) + pow(sqrt(x), ((2.0 * (1.0 / n)) / 2.0))) * (-pow(sqrt(x), (1.0 / n)) + pow(sqrt((x + 1.0)), (1.0 / n)))) + fma(pow(sqrt(x), (1.0 / n)), -pow(sqrt(x), (1.0 / n)), pow(sqrt(x), (2.0 * (1.0 / n)))));
	} else {
		double VAR_1;
		if (((1.0 / n) <= 2.412134010082013e-10)) {
			VAR_1 = fma(1.0, (1.0 / (x * n)), -fma(0.5, (1.0 / (pow(x, 2.0) * n)), (1.0 * (log((1.0 / x)) / (x * pow(n, 2.0))))));
		} else {
			VAR_1 = ((pow(pow((x + 1.0), (1.0 / n)), 3.0) - pow(pow(x, (1.0 / n)), 3.0)) / fma(pow(x, (1.0 / n)), (pow((x + 1.0), (1.0 / n)) + pow(x, (1.0 / n))), pow((x + 1.0), (2.0 * (1.0 / n)))));
		}
		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) < -2.249425591029011e-18

   1. Initial program 4.4

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

   3. Applied add-sqr-sqrt 4.4

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

   4. Applied unpow-prod-down 4.5

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

   5. Applied *-un-lft-identity 4.5

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

   6. Applied unpow-prod-down 4.5

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

   7. Applied prod-diff 4.5

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

   8. Simplified 4.4

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

   9. Simplified 4.5

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

   11. Applied sqr-pow 4.5

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

   12. Applied add-sqr-sqrt 4.5

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

   13. Applied unpow-prod-down 4.5

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

   14. Applied difference-of-squares 4.5

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

   15. Simplified 4.5

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

   if -2.249425591029011e-18 < (/ 1.0 n) < 2.412134010082013e-10

   1. Initial program 44.8

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

   2. Taylor expanded around inf 32.7

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

   3. Simplified 32.7

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

   if 2.412134010082013e-10 < (/ 1.0 n)

   1. Initial program 5.8

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

   3. Applied flip3-- 5.9

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

   4. Simplified 5.9

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

4. Final simplification 24.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \le -2.24942559102901105 \cdot 10^{-18}:\\ \;\;\;\;\left({\left(\sqrt{x + 1}\right)}^{\left(\frac{1}{n}\right)} + {\left(\sqrt{x}\right)}^{\left(\frac{2 \cdot \frac{1}{n}}{2}\right)}\right) \cdot \left(\left(-{\left(\sqrt{x}\right)}^{\left(\frac{1}{n}\right)}\right) + {\left(\sqrt{x + 1}\right)}^{\left(\frac{1}{n}\right)}\right) + \mathsf{fma}\left({\left(\sqrt{x}\right)}^{\left(\frac{1}{n}\right)}, -{\left(\sqrt{x}\right)}^{\left(\frac{1}{n}\right)}, {\left(\sqrt{x}\right)}^{\left(2 \cdot \frac{1}{n}\right)}\right)\\ \mathbf{elif}\;\frac{1}{n} \le 2.41213401008201 \cdot 10^{-10}:\\ \;\;\;\;\mathsf{fma}\left(1, \frac{1}{x \cdot n}, -\mathsf{fma}\left(0.5, \frac{1}{{x}^{2} \cdot n}, 1 \cdot \frac{\log \left(\frac{1}{x}\right)}{x \cdot {n}^{2}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)}\right)}^{3} - {\left({x}^{\left(\frac{1}{n}\right)}\right)}^{3}}{\mathsf{fma}\left({x}^{\left(\frac{1}{n}\right)}, {\left(x + 1\right)}^{\left(\frac{1}{n}\right)} + {x}^{\left(\frac{1}{n}\right)}, {\left(x + 1\right)}^{\left(2 \cdot \frac{1}{n}\right)}\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020106 +o rules:numerics
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  :precision binary64
  (- (pow (+ x 1) (/ 1 n)) (pow x (/ 1 n))))