2nthrt (problem 3.4.6)

Average Error: 32.7 → 23.5

Time: 22.9s

Precision: binary64

• Falling back on racket egraph because egg-herbie package not installed

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{1}{n} \le -18.09077906127504 \lor \neg \left(\frac{1}{n} \le 0.001178309117642572\right):\\ \;\;\;\;{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x}}{\sqrt[3]{n} \cdot \sqrt[3]{n}} \cdot \frac{1}{\sqrt[3]{n}}\\ \end{array}\]

C

double code(double x, double n) {
	return ((double) (((double) pow(((double) (x + 1.0)), (1.0 / n))) - ((double) pow(x, (1.0 / n)))));
}

↓

double code(double x, double n) {
	double VAR;
	if ((((1.0 / n) <= -18.090779061275043) || !((1.0 / n) <= 0.0011783091176425717))) {
		VAR = ((double) (((double) pow(((double) (x + 1.0)), (1.0 / n))) - ((double) pow(x, (1.0 / n)))));
	} else {
		VAR = ((double) (((1.0 / x) / ((double) (((double) cbrt(n)) * ((double) cbrt(n))))) * (1.0 / ((double) cbrt(n)))));
	}
	return VAR;
}

Error

Bits error versus x

Bits error versus n

Derivation

1. Split input into 2 regimes

2. if (/ 1.0 n) < -18.09077906127504 or 0.001178309117642572 < (/ 1.0 n)

   1. Initial program 1.8

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

   if -18.09077906127504 < (/ 1.0 n) < 0.001178309117642572

   1. Initial program 44.4

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

   2. Taylor expanded around -inf 64.0

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

   3. Simplified 32.2

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

   5. Applied *-un-lft-identity 32.2

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

   6. Applied times-frac 31.6

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

   8. Applied add-cube-cbrt 31.8

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

   9. Applied *-un-lft-identity 31.8

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

   10. Applied times-frac 31.8

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

   11. Applied associate-*r* 31.8

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

   12. Simplified 31.8

       \[\leadsto \color{blue}{\frac{\frac{1}{x}}{\sqrt[3]{n} \cdot \sqrt[3]{n}}} \cdot \frac{1}{\sqrt[3]{n}}\]
3. Recombined 2 regimes into one program.

4. Final simplification 23.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \le -18.09077906127504 \lor \neg \left(\frac{1}{n} \le 0.001178309117642572\right):\\ \;\;\;\;{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x}}{\sqrt[3]{n} \cdot \sqrt[3]{n}} \cdot \frac{1}{\sqrt[3]{n}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020182 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  :precision binary64
  (- (pow (+ x 1.0) (/ 1.0 n)) (pow x (/ 1.0 n))))