2nthrt (problem 3.4.6)

Average Error: 33.1 → 24.4

Time: 14.8s

Precision: 64

• could not determine a ground truth for program body

  1. x = 3.8081982856699253e+298
  2. n = 9.300677034083424e-116

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{1}{n} \le -7.27487421476467367 \cdot 10^{-6}:\\ \;\;\;\;\left(\log \left(\sqrt{\sqrt{e^{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}}}\right) + \log \left(\sqrt{\sqrt{e^{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}}}\right)\right) + \log \left(\sqrt{e^{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}}\right)\\ \mathbf{elif}\;\frac{1}{n} \le 2.8892748376481164 \cdot 10^{-10}:\\ \;\;\;\;\mathsf{fma}\left(1, \frac{1}{x \cdot n}, -\mathsf{fma}\left(0.5, \frac{1}{{x}^{2} \cdot n}, 1 \cdot \frac{\log \left(\frac{1}{x}\right)}{x \cdot {n}^{2}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)}\right)}^{3} - {\left({x}^{\left(\frac{1}{n}\right)}\right)}^{3}}{\mathsf{fma}\left({x}^{\left(\frac{1}{n}\right)}, {\left(x + 1\right)}^{\left(\frac{1}{n}\right)} + {x}^{\left(\frac{1}{n}\right)}, {\left(x + 1\right)}^{\left(2 \cdot \frac{1}{n}\right)}\right)}\\ \end{array}\]

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)) <= -7.274874214764674e-06)) {
		VAR = ((double) (((double) (((double) log(((double) sqrt(((double) sqrt(((double) exp(((double) (((double) pow(((double) (x + 1.0)), ((double) (1.0 / n)))) - ((double) pow(x, ((double) (1.0 / n)))))))))))))) + ((double) log(((double) sqrt(((double) sqrt(((double) exp(((double) (((double) pow(((double) (x + 1.0)), ((double) (1.0 / n)))) - ((double) pow(x, ((double) (1.0 / n)))))))))))))))) + ((double) log(((double) sqrt(((double) exp(((double) (((double) pow(((double) (x + 1.0)), ((double) (1.0 / n)))) - ((double) pow(x, ((double) (1.0 / n))))))))))))));
	} else {
		double VAR_1;
		if ((((double) (1.0 / n)) <= 2.8892748376481164e-10)) {
			VAR_1 = ((double) fma(1.0, ((double) (1.0 / ((double) (x * n)))), ((double) -(((double) fma(0.5, ((double) (1.0 / ((double) (((double) pow(x, 2.0)) * n)))), ((double) (1.0 * ((double) (((double) log(((double) (1.0 / x)))) / ((double) (x * ((double) pow(n, 2.0))))))))))))));
		} else {
			VAR_1 = ((double) (((double) (((double) pow(((double) pow(((double) (x + 1.0)), ((double) (1.0 / n)))), 3.0)) - ((double) pow(((double) pow(x, ((double) (1.0 / n)))), 3.0)))) / ((double) fma(((double) pow(x, ((double) (1.0 / n)))), ((double) (((double) pow(((double) (x + 1.0)), ((double) (1.0 / n)))) + ((double) pow(x, ((double) (1.0 / n)))))), ((double) pow(((double) (x + 1.0)), ((double) (2.0 * ((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) < -7.274874214764674e-06

   1. Initial program 1.1

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

   3. Applied add-log-exp 1.5

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

   4. Applied add-log-exp 1.4

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

   5. Applied diff-log 1.4

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

   6. Simplified 1.4

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

   8. Applied add-sqr-sqrt 1.4

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

   9. Applied log-prod 1.4

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

   11. Applied add-sqr-sqrt 1.4

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

   12. Applied sqrt-prod 1.5

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

   13. Applied log-prod 1.4

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

   if -7.274874214764674e-06 < (/ 1.0 n) < 2.8892748376481164e-10

   1. Initial program 45.3

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

   2. Taylor expanded around inf 32.9

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

   3. Simplified 32.9

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

   if 2.8892748376481164e-10 < (/ 1.0 n)

   1. Initial program 7.1

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

   3. Applied flip3-- 7.1

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

   4. Simplified 7.1

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

4. Final simplification 24.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \le -7.27487421476467367 \cdot 10^{-6}:\\ \;\;\;\;\left(\log \left(\sqrt{\sqrt{e^{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}}}\right) + \log \left(\sqrt{\sqrt{e^{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}}}\right)\right) + \log \left(\sqrt{e^{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}}\right)\\ \mathbf{elif}\;\frac{1}{n} \le 2.8892748376481164 \cdot 10^{-10}:\\ \;\;\;\;\mathsf{fma}\left(1, \frac{1}{x \cdot n}, -\mathsf{fma}\left(0.5, \frac{1}{{x}^{2} \cdot n}, 1 \cdot \frac{\log \left(\frac{1}{x}\right)}{x \cdot {n}^{2}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)}\right)}^{3} - {\left({x}^{\left(\frac{1}{n}\right)}\right)}^{3}}{\mathsf{fma}\left({x}^{\left(\frac{1}{n}\right)}, {\left(x + 1\right)}^{\left(\frac{1}{n}\right)} + {x}^{\left(\frac{1}{n}\right)}, {\left(x + 1\right)}^{\left(2 \cdot \frac{1}{n}\right)}\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020114 +o rules:numerics
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  :precision binary64
  (- (pow (+ x 1) (/ 1 n)) (pow x (/ 1 n))))