2nthrt (problem 3.4.6)

Average Error: 32.7 → 23.1

Time: 10.8s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{1}{n} \le -1.0314329489541403 \cdot 10^{-6}:\\ \;\;\;\;\sqrt[3]{{\left({\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\right)}^{3}}\\ \mathbf{elif}\;\frac{1}{n} \le 5.9317209469061504 \cdot 10^{-23}:\\ \;\;\;\;\frac{\frac{1}{n}}{x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{{\left(\frac{{\left(1 + x\right)}^{\left(\frac{1}{n} \cdot 2\right)} - {x}^{\left(\frac{1}{n} \cdot 2\right)}}{{\left(1 + x\right)}^{\left(\frac{1}{n}\right)} + {x}^{\left(\frac{1}{n}\right)}}\right)}^{3}}\\ \end{array}\]

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)) <= -1.0314329489541403e-06)) {
		VAR = ((double) cbrt(((double) pow(((double) (((double) pow(((double) (1.0 + x)), ((double) (1.0 / n)))) - ((double) pow(x, ((double) (1.0 / n)))))), 3.0))));
	} else {
		double VAR_1;
		if ((((double) (1.0 / n)) <= 5.93172094690615e-23)) {
			VAR_1 = ((double) (((double) (1.0 / n)) / x));
		} else {
			VAR_1 = ((double) cbrt(((double) pow(((double) (((double) (((double) pow(((double) (1.0 + x)), ((double) (((double) (1.0 / n)) * 2.0)))) - ((double) pow(x, ((double) (((double) (1.0 / n)) * 2.0)))))) / ((double) (((double) pow(((double) (1.0 + x)), ((double) (1.0 / n)))) + ((double) pow(x, ((double) (1.0 / n)))))))), 3.0))));
		}
		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) < -1.0314329489541403e-6

   1. Initial program 1.2

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

   3. Applied add-cbrt-cube 1.3

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

   4. Simplified 1.3

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

   if -1.0314329489541403e-6 < (/ 1.0 n) < 5.9317209469061504e-23

   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 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

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

   5. Applied *-un-lft-identity 31.7

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

   6. Applied times-frac 31.2

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

   8. Applied associate-*l/ 31.2

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

   9. Simplified 31.2

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

   if 5.9317209469061504e-23 < (/ 1.0 n)

   1. Initial program 11.4

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

   3. Applied add-cbrt-cube 11.4

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

   4. Simplified 11.4

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

   6. Applied flip-- 11.5

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

   7. Simplified 11.4

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

4. Final simplification 23.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \le -1.0314329489541403 \cdot 10^{-6}:\\ \;\;\;\;\sqrt[3]{{\left({\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\right)}^{3}}\\ \mathbf{elif}\;\frac{1}{n} \le 5.9317209469061504 \cdot 10^{-23}:\\ \;\;\;\;\frac{\frac{1}{n}}{x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{{\left(\frac{{\left(1 + x\right)}^{\left(\frac{1}{n} \cdot 2\right)} - {x}^{\left(\frac{1}{n} \cdot 2\right)}}{{\left(1 + x\right)}^{\left(\frac{1}{n}\right)} + {x}^{\left(\frac{1}{n}\right)}}\right)}^{3}}\\ \end{array}\]

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))))