2nthrt (problem 3.4.6)

Average Error: 33.0 → 23.9

Time: 13.9s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{1}{n} \le -2.55425582413963677 \cdot 10^{-9}:\\ \;\;\;\;\log \left(e^{{\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right)\\ \mathbf{elif}\;\frac{1}{n} \le 3.2984038541839436 \cdot 10^{-18}:\\ \;\;\;\;\frac{1}{n \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left({\left({\left(\sqrt{1 + x}\right)}^{\left(\frac{1}{n}\right)}\right)}^{3} - {\left(\sqrt{{x}^{\left(\frac{1}{n}\right)}}\right)}^{3}\right) \cdot \left({\left(\sqrt{1 + x}\right)}^{\left(\frac{1}{n} \cdot 2\right)} - {x}^{\left(\frac{1}{n}\right)}\right)}{\left({\left(\sqrt{1 + x}\right)}^{\left(\frac{1}{n}\right)} - \sqrt{{x}^{\left(\frac{1}{n}\right)}}\right) \cdot \left({x}^{\left(\frac{1}{n}\right)} + {\left(\sqrt{1 + x}\right)}^{\left(\frac{1}{n}\right)} \cdot \left({\left(\sqrt{1 + x}\right)}^{\left(\frac{1}{n}\right)} + \sqrt{{x}^{\left(\frac{1}{n}\right)}}\right)\right)}\\ \end{array}\]

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)) <= -2.5542558241396368e-09)) {
		VAR = ((double) log(((double) exp(((double) (((double) pow(((double) (1.0 + x)), ((double) (1.0 / n)))) - ((double) pow(x, ((double) (1.0 / n))))))))));
	} else {
		double VAR_1;
		if ((((double) (1.0 / n)) <= 3.2984038541839436e-18)) {
			VAR_1 = ((double) (1.0 / ((double) (n * x))));
		} else {
			VAR_1 = ((double) (((double) (((double) (((double) pow(((double) pow(((double) sqrt(((double) (1.0 + x)))), ((double) (1.0 / n)))), 3.0)) - ((double) pow(((double) sqrt(((double) pow(x, ((double) (1.0 / n)))))), 3.0)))) * ((double) (((double) pow(((double) sqrt(((double) (1.0 + x)))), ((double) (((double) (1.0 / n)) * 2.0)))) - ((double) pow(x, ((double) (1.0 / n)))))))) / ((double) (((double) (((double) pow(((double) sqrt(((double) (1.0 + x)))), ((double) (1.0 / n)))) - ((double) sqrt(((double) pow(x, ((double) (1.0 / n)))))))) * ((double) (((double) pow(x, ((double) (1.0 / n)))) + ((double) (((double) pow(((double) sqrt(((double) (1.0 + x)))), ((double) (1.0 / n)))) * ((double) (((double) pow(((double) sqrt(((double) (1.0 + x)))), ((double) (1.0 / n)))) + ((double) sqrt(((double) pow(x, ((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) < -2.55425582413963677e-9

   1. Initial program 2.4

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

   3. Applied add-log-exp 2.9

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

   4. Applied add-log-exp 2.7

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

   5. Applied diff-log 2.7

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

   6. Simplified 2.7

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

   if -2.55425582413963677e-9 < (/ 1.0 n) < 3.2984038541839436e-18

   1. Initial program 44.9

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

   2. Taylor expanded around -inf 64.0

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

   3. Simplified 31.7

      \[\leadsto \color{blue}{\left(\frac{1}{x \cdot n} + 0\right) - 0}\]

   if 3.2984038541839436e-18 < (/ 1.0 n)

   1. Initial program 10.8

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

   3. Applied add-sqr-sqrt 10.9

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

   4. Applied add-sqr-sqrt 10.9

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

   5. Applied unpow-prod-down 10.9

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

   6. Applied difference-of-squares 10.9

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

   8. Applied flip3-- 10.9

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

   9. Applied flip-+ 11.2

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

   10. Applied frac-times 11.2

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

   11. Simplified 11.1

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

   12. Simplified 11.1

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

4. Final simplification 23.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \le -2.55425582413963677 \cdot 10^{-9}:\\ \;\;\;\;\log \left(e^{{\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right)\\ \mathbf{elif}\;\frac{1}{n} \le 3.2984038541839436 \cdot 10^{-18}:\\ \;\;\;\;\frac{1}{n \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left({\left({\left(\sqrt{1 + x}\right)}^{\left(\frac{1}{n}\right)}\right)}^{3} - {\left(\sqrt{{x}^{\left(\frac{1}{n}\right)}}\right)}^{3}\right) \cdot \left({\left(\sqrt{1 + x}\right)}^{\left(\frac{1}{n} \cdot 2\right)} - {x}^{\left(\frac{1}{n}\right)}\right)}{\left({\left(\sqrt{1 + x}\right)}^{\left(\frac{1}{n}\right)} - \sqrt{{x}^{\left(\frac{1}{n}\right)}}\right) \cdot \left({x}^{\left(\frac{1}{n}\right)} + {\left(\sqrt{1 + x}\right)}^{\left(\frac{1}{n}\right)} \cdot \left({\left(\sqrt{1 + x}\right)}^{\left(\frac{1}{n}\right)} + \sqrt{{x}^{\left(\frac{1}{n}\right)}}\right)\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020179 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  :precision binary64
  (- (pow (+ x 1.0) (/ 1.0 n)) (pow x (/ 1.0 n))))