2nthrt (problem 3.4.6)

Average Error: 32.5 → 24.2

Time: 16.1s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;n \le -46.9258573395225582 \lor \neg \left(n \le 54666412.879635885\right):\\ \;\;\;\;\mathsf{fma}\left(1, \frac{1}{x \cdot n}, -\mathsf{fma}\left(0.5, \frac{1}{{x}^{2} \cdot n}, 1 \cdot \frac{\log \left(\frac{1}{x}\right)}{x \cdot {n}^{2}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{e}^{\left(\sqrt[3]{{\left(\log \left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\right)\right)}^{3}}\right)}\\ \end{array}\]

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 (((n <= -46.92585733952256) || !(n <= 54666412.879635885))) {
		VAR = 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 = pow(((double) M_E), cbrt(pow(log((pow((x + 1.0), (1.0 / n)) - pow(x, (1.0 / n)))), 3.0)));
	}
	return VAR;
}

Error

Bits error versus x

Bits error versus n

Derivation

1. Split input into 2 regimes

2. if n < -46.92585733952256 or 54666412.879635885 < n

   1. Initial program 45.0

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

   2. Taylor expanded around inf 33.0

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

   3. Simplified 33.0

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

   if -46.92585733952256 < n < 54666412.879635885

   1. Initial program 2.1

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

   3. Applied add-exp-log 2.6

      \[\leadsto \color{blue}{e^{\log \left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\right)}}\]
   4. Using strategy rm

   5. Applied add-cbrt-cube 2.6

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

   6. Simplified 2.6

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

   8. Applied pow1 2.6

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

   9. Applied log-pow 2.6

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

   10. Applied unpow-prod-down 2.6

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

   11. Applied cbrt-prod 2.6

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

   12. Applied exp-prod 2.6

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

   13. Simplified 2.6

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

4. Final simplification 24.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \le -46.9258573395225582 \lor \neg \left(n \le 54666412.879635885\right):\\ \;\;\;\;\mathsf{fma}\left(1, \frac{1}{x \cdot n}, -\mathsf{fma}\left(0.5, \frac{1}{{x}^{2} \cdot n}, 1 \cdot \frac{\log \left(\frac{1}{x}\right)}{x \cdot {n}^{2}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{e}^{\left(\sqrt[3]{{\left(\log \left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\right)\right)}^{3}}\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020103 +o rules:numerics
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  :precision binary64
  (- (pow (+ x 1) (/ 1 n)) (pow x (/ 1 n))))