2nthrt (problem 3.4.6)

Average Error: 32.8 → 23.3

Time: 11.7s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{1}{n} \le -9979.93413369299196:\\ \;\;\;\;\left({\left(\sqrt{x}\right)}^{\left(\frac{1}{n}\right)} + \sqrt{{\left(1 + x\right)}^{\left(\frac{1}{n}\right)}}\right) \cdot \left(\left(1 + \left(0.5 \cdot \left(\frac{\log 1}{n} + \frac{\log x}{n}\right) + \frac{0.5}{n \cdot x}\right)\right) - {\left(\sqrt{x}\right)}^{\left(\frac{1}{n}\right)}\right)\\ \mathbf{elif}\;\frac{1}{n} \le 3.0076420312498511 \cdot 10^{-8}:\\ \;\;\;\;\sqrt{\frac{1}{x}} \cdot \frac{\sqrt{\frac{1}{x}}}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left({\left(1 + x\right)}^{\left(\frac{1}{n}\right)}\right)}^{3} - {\left({x}^{\left(\frac{1}{n}\right)}\right)}^{3}}{{\left(1 + x\right)}^{\left(\frac{1}{n} \cdot 2\right)} + {x}^{\left(\frac{1}{n}\right)} \cdot \left({\left(1 + x\right)}^{\left(\frac{1}{n}\right)} + {x}^{\left(\frac{1}{n}\right)}\right)}\\ \end{array}\]

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)) <= -9979.934133692992)) {
		VAR = ((double) (((double) (((double) pow(((double) sqrt(x)), ((double) (1.0 / n)))) + ((double) sqrt(((double) pow(((double) (1.0 + x)), ((double) (1.0 / n)))))))) * ((double) (((double) (1.0 + ((double) (((double) (0.5 * ((double) (((double) (((double) log(1.0)) / n)) + ((double) (((double) log(x)) / n)))))) + ((double) (0.5 / ((double) (n * x)))))))) - ((double) pow(((double) sqrt(x)), ((double) (1.0 / n))))))));
	} else {
		double VAR_1;
		if ((((double) (1.0 / n)) <= 3.007642031249851e-08)) {
			VAR_1 = ((double) (((double) sqrt(((double) (1.0 / x)))) * ((double) (((double) sqrt(((double) (1.0 / x)))) / n))));
		} else {
			VAR_1 = ((double) (((double) (((double) pow(((double) pow(((double) (1.0 + x)), ((double) (1.0 / n)))), 3.0)) - ((double) pow(((double) pow(x, ((double) (1.0 / n)))), 3.0)))) / ((double) (((double) pow(((double) (1.0 + x)), ((double) (((double) (1.0 / n)) * 2.0)))) + ((double) (((double) pow(x, ((double) (1.0 / n)))) * ((double) (((double) pow(((double) (1.0 + x)), ((double) (1.0 / n)))) + ((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) < -9979.93413369299196

   1. Initial program 0

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

   3. Applied add-sqr-sqrt 0

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

   4. Applied unpow-prod-down 0

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

   5. Applied add-sqr-sqrt 0

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

   6. Applied difference-of-squares 0

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

   7. Simplified 0

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

   8. Taylor expanded around inf 0.4

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

   9. Simplified 0.4

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

   if -9979.93413369299196 < (/ 1.0 n) < 3.0076420312498511e-8

   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 32.0

      \[\leadsto \color{blue}{\frac{1}{x \cdot n}}\]
   4. Using strategy rm

   5. Applied associate-/r* 31.4

      \[\leadsto \color{blue}{\frac{\frac{1}{x}}{n}}\]
   6. Using strategy rm

   7. Applied *-un-lft-identity 31.4

      \[\leadsto \frac{\frac{1}{x}}{\color{blue}{1 \cdot n}}\]

   8. Applied add-sqr-sqrt 31.5

      \[\leadsto \frac{\color{blue}{\sqrt{\frac{1}{x}} \cdot \sqrt{\frac{1}{x}}}}{1 \cdot n}\]

   9. Applied times-frac 31.5

      \[\leadsto \color{blue}{\frac{\sqrt{\frac{1}{x}}}{1} \cdot \frac{\sqrt{\frac{1}{x}}}{n}}\]

   10. Simplified 31.5

       \[\leadsto \color{blue}{\sqrt{\frac{1}{x}}} \cdot \frac{\sqrt{\frac{1}{x}}}{n}\]

   if 3.0076420312498511e-8 < (/ 1.0 n)

   1. Initial program 6.6

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

   3. Applied flip3-- 6.6

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

   4. Simplified 6.6

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

4. Final simplification 23.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \le -9979.93413369299196:\\ \;\;\;\;\left({\left(\sqrt{x}\right)}^{\left(\frac{1}{n}\right)} + \sqrt{{\left(1 + x\right)}^{\left(\frac{1}{n}\right)}}\right) \cdot \left(\left(1 + \left(0.5 \cdot \left(\frac{\log 1}{n} + \frac{\log x}{n}\right) + \frac{0.5}{n \cdot x}\right)\right) - {\left(\sqrt{x}\right)}^{\left(\frac{1}{n}\right)}\right)\\ \mathbf{elif}\;\frac{1}{n} \le 3.0076420312498511 \cdot 10^{-8}:\\ \;\;\;\;\sqrt{\frac{1}{x}} \cdot \frac{\sqrt{\frac{1}{x}}}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left({\left(1 + x\right)}^{\left(\frac{1}{n}\right)}\right)}^{3} - {\left({x}^{\left(\frac{1}{n}\right)}\right)}^{3}}{{\left(1 + x\right)}^{\left(\frac{1}{n} \cdot 2\right)} + {x}^{\left(\frac{1}{n}\right)} \cdot \left({\left(1 + x\right)}^{\left(\frac{1}{n}\right)} + {x}^{\left(\frac{1}{n}\right)}\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020185 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  :precision binary64
  (- (pow (+ x 1.0) (/ 1.0 n)) (pow x (/ 1.0 n))))