2nthrt (problem 3.4.6)

Average Error: 32.2 → 23.4

Time: 10.9s

Precision: 64

• could not determine a ground truth for program body

  1. x = 3.864806516844435e+224
  2. n = 2.278825339876756e-210

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;n \le -35172.9164463997804:\\ \;\;\;\;\frac{1 \cdot \left(\frac{1}{n} - \frac{\log \left(\frac{1}{x}\right)}{{n}^{2}}\right)}{x} + \frac{-0.5}{\log \left(e^{{x}^{2} \cdot n}\right)}\\ \mathbf{elif}\;n \le 2309.1261634880311:\\ \;\;\;\;\left({\left(\sqrt{x + 1}\right)}^{\left(\frac{1}{n}\right)} + \sqrt{{x}^{\left(\frac{1}{n}\right)}}\right) \cdot \left({\left(\sqrt{x + 1}\right)}^{\left(\frac{1}{n}\right)} - \sqrt{{x}^{\left(\frac{1}{n}\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\frac{1}{x \cdot n} + \frac{\log x}{x \cdot {n}^{2}}\right) + \frac{-0.5}{{x}^{2} \cdot n}\\ \end{array}\]

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 ((n <= -35172.91644639978)) {
		VAR = ((double) (((double) (((double) (1.0 * ((double) (((double) (1.0 / n)) - ((double) (((double) log(((double) (1.0 / x)))) / ((double) pow(n, 2.0)))))))) / x)) + ((double) (((double) -(0.5)) / ((double) log(((double) exp(((double) (((double) pow(x, 2.0)) * n))))))))));
	} else {
		double VAR_1;
		if ((n <= 2309.126163488031)) {
			VAR_1 = ((double) (((double) (((double) pow(((double) sqrt(((double) (x + 1.0)))), ((double) (1.0 / n)))) + ((double) sqrt(((double) pow(x, ((double) (1.0 / n)))))))) * ((double) (((double) pow(((double) sqrt(((double) (x + 1.0)))), ((double) (1.0 / n)))) - ((double) sqrt(((double) pow(x, ((double) (1.0 / n))))))))));
		} else {
			VAR_1 = ((double) (((double) (1.0 * ((double) (((double) (1.0 / ((double) (x * n)))) + ((double) (((double) log(x)) / ((double) (x * ((double) pow(n, 2.0)))))))))) + ((double) (((double) -(0.5)) / ((double) (((double) pow(x, 2.0)) * n))))));
		}
		VAR = VAR_1;
	}
	return VAR;
}

Error

Bits error versus x

Bits error versus n

Derivation

1. Split input into 3 regimes

2. if n < -35172.91644639978

   1. Initial program 44.3

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

   2. Taylor expanded around inf 31.9

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

   3. Simplified 31.4

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

   5. Applied associate-*l/ 31.3

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

   7. Applied add-log-exp 31.4

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

   if -35172.91644639978 < n < 2309.126163488031

   1. Initial program 2.0

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

   3. Applied add-sqr-sqrt 2.0

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

   4. Applied add-sqr-sqrt 2.0

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

   5. Applied unpow-prod-down 2.0

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

   6. Applied difference-of-squares 2.0

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

   if 2309.126163488031 < n

   1. Initial program 44.5

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

   2. Taylor expanded around inf 32.6

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

   3. Simplified 31.9

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

   5. Applied associate-*l/ 31.9

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

   6. Taylor expanded around 0 32.6

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

   7. Simplified 32.6

      \[\leadsto \color{blue}{1 \cdot \left(\frac{1}{x \cdot n} + \frac{\log x}{x \cdot {n}^{2}}\right)} + \frac{-0.5}{{x}^{2} \cdot n}\]
3. Recombined 3 regimes into one program.

4. Final simplification 23.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \le -35172.9164463997804:\\ \;\;\;\;\frac{1 \cdot \left(\frac{1}{n} - \frac{\log \left(\frac{1}{x}\right)}{{n}^{2}}\right)}{x} + \frac{-0.5}{\log \left(e^{{x}^{2} \cdot n}\right)}\\ \mathbf{elif}\;n \le 2309.1261634880311:\\ \;\;\;\;\left({\left(\sqrt{x + 1}\right)}^{\left(\frac{1}{n}\right)} + \sqrt{{x}^{\left(\frac{1}{n}\right)}}\right) \cdot \left({\left(\sqrt{x + 1}\right)}^{\left(\frac{1}{n}\right)} - \sqrt{{x}^{\left(\frac{1}{n}\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\frac{1}{x \cdot n} + \frac{\log x}{x \cdot {n}^{2}}\right) + \frac{-0.5}{{x}^{2} \cdot n}\\ \end{array}\]

Reproduce

herbie shell --seed 2020121 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  :precision binary64
  (- (pow (+ x 1) (/ 1 n)) (pow x (/ 1 n))))