2nthrt (problem 3.4.6)

Average Error: 33.0 → 24.1

Time: 19.0s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{1}{n} \le -6.5424017185802929 \cdot 10^{-7}:\\ \;\;\;\;\log \left(e^{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {\left(\sqrt{x}\right)}^{\left(2 \cdot \frac{1}{n}\right)}}\right) + \mathsf{fma}\left({\left(\sqrt{x}\right)}^{\left(\frac{1}{n}\right)}, -{\left(\sqrt{x}\right)}^{\left(\frac{1}{n}\right)}, {\left(\sqrt{x}\right)}^{\left(2 \cdot \frac{1}{n}\right)}\right)\\ \mathbf{elif}\;\frac{1}{n} \le 4.7895543843896953 \cdot 10^{-16}:\\ \;\;\;\;\mathsf{fma}\left(1, \frac{1}{x \cdot n}, -\mathsf{fma}\left(0.5, \frac{1}{{x}^{2} \cdot n}, 1 \cdot \frac{\log \left(\frac{1}{x}\right)}{x \cdot {n}^{2}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)}\right)}^{3} - {\left({\left(\sqrt{x}\right)}^{\left(2 \cdot \frac{1}{n}\right)}\right)}^{3}}{\mathsf{fma}\left({\left(\sqrt{x}\right)}^{\left(2 \cdot \frac{1}{n}\right)}, {\left(x + 1\right)}^{\left(\frac{1}{n}\right)} + {\left(\sqrt{x}\right)}^{\left(2 \cdot \frac{1}{n}\right)}, {\left(x + 1\right)}^{\left(2 \cdot \frac{1}{n}\right)}\right)} + \mathsf{fma}\left({\left(\sqrt{x}\right)}^{\left(\frac{1}{n}\right)}, -{\left(\sqrt{x}\right)}^{\left(\frac{1}{n}\right)}, {\left(\sqrt{x}\right)}^{\left(2 \cdot \frac{1}{n}\right)}\right)\\ \end{array}\]

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)) <= -6.542401718580293e-07)) {
		VAR = ((double) (((double) log(((double) exp(((double) (((double) pow(((double) (x + 1.0)), ((double) (1.0 / n)))) - ((double) pow(((double) sqrt(x)), ((double) (2.0 * ((double) (1.0 / n)))))))))))) + ((double) fma(((double) pow(((double) sqrt(x)), ((double) (1.0 / n)))), ((double) -(((double) pow(((double) sqrt(x)), ((double) (1.0 / n)))))), ((double) pow(((double) sqrt(x)), ((double) (2.0 * ((double) (1.0 / n))))))))));
	} else {
		double VAR_1;
		if ((((double) (1.0 / n)) <= 4.789554384389695e-16)) {
			VAR_1 = ((double) fma(1.0, ((double) (1.0 / ((double) (x * n)))), ((double) -(((double) fma(0.5, ((double) (1.0 / ((double) (((double) pow(x, 2.0)) * n)))), ((double) (1.0 * ((double) (((double) log(((double) (1.0 / x)))) / ((double) (x * ((double) pow(n, 2.0))))))))))))));
		} else {
			VAR_1 = ((double) (((double) (((double) (((double) pow(((double) pow(((double) (x + 1.0)), ((double) (1.0 / n)))), 3.0)) - ((double) pow(((double) pow(((double) sqrt(x)), ((double) (2.0 * ((double) (1.0 / n)))))), 3.0)))) / ((double) fma(((double) pow(((double) sqrt(x)), ((double) (2.0 * ((double) (1.0 / n)))))), ((double) (((double) pow(((double) (x + 1.0)), ((double) (1.0 / n)))) + ((double) pow(((double) sqrt(x)), ((double) (2.0 * ((double) (1.0 / n)))))))), ((double) pow(((double) (x + 1.0)), ((double) (2.0 * ((double) (1.0 / n)))))))))) + ((double) fma(((double) pow(((double) sqrt(x)), ((double) (1.0 / n)))), ((double) -(((double) pow(((double) sqrt(x)), ((double) (1.0 / n)))))), ((double) pow(((double) sqrt(x)), ((double) (2.0 * ((double) (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) < -6.542401718580293e-07

   1. Initial program 1.3

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

   3. Applied add-sqr-sqrt 1.2

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

   4. Applied unpow-prod-down 1.3

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

   5. Applied *-un-lft-identity 1.3

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

   6. Applied unpow-prod-down 1.3

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

   7. Applied prod-diff 1.3

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

   8. Simplified 1.3

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

   9. Simplified 1.3

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

   11. Applied add-log-exp 1.6

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

   12. Applied add-log-exp 1.6

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

   13. Applied diff-log 1.6

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

   14. Simplified 1.6

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

   if -6.542401718580293e-07 < (/ 1.0 n) < 4.789554384389695e-16

   1. Initial program 45.2

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

   2. Taylor expanded around inf 32.4

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

   3. Simplified 32.4

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

   if 4.789554384389695e-16 < (/ 1.0 n)

   1. Initial program 8.9

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

   3. Applied add-sqr-sqrt 8.9

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

   4. Applied unpow-prod-down 9.1

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

   5. Applied *-un-lft-identity 9.1

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

   6. Applied unpow-prod-down 9.1

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

   7. Applied prod-diff 9.1

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

   8. Simplified 9.1

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

   9. Simplified 9.1

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

   11. Applied flip3-- 9.2

       \[\leadsto \color{blue}{\frac{{\left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)}\right)}^{3} - {\left({\left(\sqrt{x}\right)}^{\left(2 \cdot \frac{1}{n}\right)}\right)}^{3}}{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} \cdot {\left(x + 1\right)}^{\left(\frac{1}{n}\right)} + \left({\left(\sqrt{x}\right)}^{\left(2 \cdot \frac{1}{n}\right)} \cdot {\left(\sqrt{x}\right)}^{\left(2 \cdot \frac{1}{n}\right)} + {\left(x + 1\right)}^{\left(\frac{1}{n}\right)} \cdot {\left(\sqrt{x}\right)}^{\left(2 \cdot \frac{1}{n}\right)}\right)}} + \mathsf{fma}\left({\left(\sqrt{x}\right)}^{\left(\frac{1}{n}\right)}, -{\left(\sqrt{x}\right)}^{\left(\frac{1}{n}\right)}, {\left(\sqrt{x}\right)}^{\left(2 \cdot \frac{1}{n}\right)}\right)\]

   12. Simplified 9.2

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

4. Final simplification 24.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \le -6.5424017185802929 \cdot 10^{-7}:\\ \;\;\;\;\log \left(e^{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {\left(\sqrt{x}\right)}^{\left(2 \cdot \frac{1}{n}\right)}}\right) + \mathsf{fma}\left({\left(\sqrt{x}\right)}^{\left(\frac{1}{n}\right)}, -{\left(\sqrt{x}\right)}^{\left(\frac{1}{n}\right)}, {\left(\sqrt{x}\right)}^{\left(2 \cdot \frac{1}{n}\right)}\right)\\ \mathbf{elif}\;\frac{1}{n} \le 4.7895543843896953 \cdot 10^{-16}:\\ \;\;\;\;\mathsf{fma}\left(1, \frac{1}{x \cdot n}, -\mathsf{fma}\left(0.5, \frac{1}{{x}^{2} \cdot n}, 1 \cdot \frac{\log \left(\frac{1}{x}\right)}{x \cdot {n}^{2}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)}\right)}^{3} - {\left({\left(\sqrt{x}\right)}^{\left(2 \cdot \frac{1}{n}\right)}\right)}^{3}}{\mathsf{fma}\left({\left(\sqrt{x}\right)}^{\left(2 \cdot \frac{1}{n}\right)}, {\left(x + 1\right)}^{\left(\frac{1}{n}\right)} + {\left(\sqrt{x}\right)}^{\left(2 \cdot \frac{1}{n}\right)}, {\left(x + 1\right)}^{\left(2 \cdot \frac{1}{n}\right)}\right)} + \mathsf{fma}\left({\left(\sqrt{x}\right)}^{\left(\frac{1}{n}\right)}, -{\left(\sqrt{x}\right)}^{\left(\frac{1}{n}\right)}, {\left(\sqrt{x}\right)}^{\left(2 \cdot \frac{1}{n}\right)}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020123 +o rules:numerics
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  :precision binary64
  (- (pow (+ x 1) (/ 1 n)) (pow x (/ 1 n))))