2nthrt (problem 3.4.6)

Average Error: 33.0 → 24.5

Time: 14.0s

Precision: 64

• could not determine a ground truth for program body

  1. x = 5.683788269687683e+223
  2. n = 6.200392985854844e-204

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{1}{n} \le -1.6345588493135571 \cdot 10^{-23}:\\ \;\;\;\;\left({\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)} + {\left({x}^{\left(\frac{\sqrt[3]{\frac{1}{n}} \cdot \sqrt[3]{\frac{1}{n}}}{1}\right)}\right)}^{\left(\frac{\sqrt[3]{\frac{1}{n}}}{2}\right)}\right) \cdot \left(\left(\sqrt{{\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)}} + {x}^{\left(\frac{\frac{\frac{1}{n}}{2}}{2}\right)}\right) \cdot \left(\left(\sqrt[3]{\sqrt{{\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)}} - {x}^{\left(\frac{\frac{\frac{1}{n}}{2}}{2}\right)}} \cdot \sqrt[3]{\sqrt{{\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)}} - {x}^{\left(\frac{\frac{\frac{1}{n}}{2}}{2}\right)}}\right) \cdot \sqrt[3]{\sqrt{{\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)}} - {x}^{\left(\frac{\frac{\frac{1}{n}}{2}}{2}\right)}}\right)\right)\\ \mathbf{elif}\;\frac{1}{n} \le 5.53064647223408922 \cdot 10^{-24}:\\ \;\;\;\;\left({\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)} + {\left({x}^{\left(\frac{\sqrt[3]{\frac{1}{n}} \cdot \sqrt[3]{\frac{1}{n}}}{1}\right)}\right)}^{\left(\frac{\sqrt[3]{\frac{1}{n}}}{2}\right)}\right) \cdot \mathsf{fma}\left(-0.25, \frac{1}{{x}^{2} \cdot n} + \frac{\log \left(\frac{1}{x}\right)}{x \cdot {n}^{2}}, \frac{\frac{0.5}{n}}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)}\right)}^{3} - {\left({x}^{\left(\frac{1}{n}\right)}\right)}^{3}}{\mathsf{fma}\left({x}^{\left(\frac{1}{n}\right)}, {\left(x + 1\right)}^{\left(\frac{1}{n}\right)} + {x}^{\left(\frac{1}{n}\right)}, {\left(x + 1\right)}^{\left(2 \cdot \frac{1}{n}\right)}\right)}\\ \end{array}\]

C

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

↓

double code(double x, double n) {
	double temp;
	if (((1.0 / n) <= -1.634558849313557e-23)) {
		temp = ((pow((x + 1.0), ((1.0 / n) / 2.0)) + pow(pow(x, ((cbrt((1.0 / n)) * cbrt((1.0 / n))) / 1.0)), (cbrt((1.0 / n)) / 2.0))) * ((sqrt(pow((x + 1.0), ((1.0 / n) / 2.0))) + pow(x, (((1.0 / n) / 2.0) / 2.0))) * ((cbrt((sqrt(pow((x + 1.0), ((1.0 / n) / 2.0))) - pow(x, (((1.0 / n) / 2.0) / 2.0)))) * cbrt((sqrt(pow((x + 1.0), ((1.0 / n) / 2.0))) - pow(x, (((1.0 / n) / 2.0) / 2.0))))) * cbrt((sqrt(pow((x + 1.0), ((1.0 / n) / 2.0))) - pow(x, (((1.0 / n) / 2.0) / 2.0)))))));
	} else {
		double temp_1;
		if (((1.0 / n) <= 5.530646472234089e-24)) {
			temp_1 = ((pow((x + 1.0), ((1.0 / n) / 2.0)) + pow(pow(x, ((cbrt((1.0 / n)) * cbrt((1.0 / n))) / 1.0)), (cbrt((1.0 / n)) / 2.0))) * fma(-0.25, ((1.0 / (pow(x, 2.0) * n)) + (log((1.0 / x)) / (x * pow(n, 2.0)))), ((0.5 / n) / x)));
		} else {
			temp_1 = ((pow(pow((x + 1.0), (1.0 / n)), 3.0) - pow(pow(x, (1.0 / n)), 3.0)) / fma(pow(x, (1.0 / n)), (pow((x + 1.0), (1.0 / n)) + pow(x, (1.0 / n))), pow((x + 1.0), (2.0 * (1.0 / n)))));
		}
		temp = temp_1;
	}
	return temp;
}

Error

Bits error versus x

Bits error versus n

Derivation

1. Split input into 3 regimes

2. if (/ 1.0 n) < -1.634558849313557e-23

   1. Initial program 5.5

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

   3. Applied sqr-pow 5.6

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

   4. Applied sqr-pow 5.6

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

   5. Applied difference-of-squares 5.6

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

   7. Applied *-un-lft-identity 5.6

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

   8. Applied add-cube-cbrt 5.6

      \[\leadsto \left({\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)} + {x}^{\left(\frac{\color{blue}{\left(\sqrt[3]{\frac{1}{n}} \cdot \sqrt[3]{\frac{1}{n}}\right) \cdot \sqrt[3]{\frac{1}{n}}}}{1 \cdot 2}\right)}\right) \cdot \left({\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)} - {x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right)\]

   9. Applied times-frac 5.6

      \[\leadsto \left({\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)} + {x}^{\color{blue}{\left(\frac{\sqrt[3]{\frac{1}{n}} \cdot \sqrt[3]{\frac{1}{n}}}{1} \cdot \frac{\sqrt[3]{\frac{1}{n}}}{2}\right)}}\right) \cdot \left({\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)} - {x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right)\]

   10. Applied pow-unpow 5.6

       \[\leadsto \left({\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)} + \color{blue}{{\left({x}^{\left(\frac{\sqrt[3]{\frac{1}{n}} \cdot \sqrt[3]{\frac{1}{n}}}{1}\right)}\right)}^{\left(\frac{\sqrt[3]{\frac{1}{n}}}{2}\right)}}\right) \cdot \left({\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)} - {x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right)\]
   11. Using strategy rm

   12. Applied sqr-pow 5.7

       \[\leadsto \left({\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)} + {\left({x}^{\left(\frac{\sqrt[3]{\frac{1}{n}} \cdot \sqrt[3]{\frac{1}{n}}}{1}\right)}\right)}^{\left(\frac{\sqrt[3]{\frac{1}{n}}}{2}\right)}\right) \cdot \left({\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)} - \color{blue}{{x}^{\left(\frac{\frac{\frac{1}{n}}{2}}{2}\right)} \cdot {x}^{\left(\frac{\frac{\frac{1}{n}}{2}}{2}\right)}}\right)\]

   13. Applied add-sqr-sqrt 5.6

       \[\leadsto \left({\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)} + {\left({x}^{\left(\frac{\sqrt[3]{\frac{1}{n}} \cdot \sqrt[3]{\frac{1}{n}}}{1}\right)}\right)}^{\left(\frac{\sqrt[3]{\frac{1}{n}}}{2}\right)}\right) \cdot \left(\color{blue}{\sqrt{{\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)}} \cdot \sqrt{{\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)}}} - {x}^{\left(\frac{\frac{\frac{1}{n}}{2}}{2}\right)} \cdot {x}^{\left(\frac{\frac{\frac{1}{n}}{2}}{2}\right)}\right)\]

   14. Applied difference-of-squares 5.6

       \[\leadsto \left({\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)} + {\left({x}^{\left(\frac{\sqrt[3]{\frac{1}{n}} \cdot \sqrt[3]{\frac{1}{n}}}{1}\right)}\right)}^{\left(\frac{\sqrt[3]{\frac{1}{n}}}{2}\right)}\right) \cdot \color{blue}{\left(\left(\sqrt{{\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)}} + {x}^{\left(\frac{\frac{\frac{1}{n}}{2}}{2}\right)}\right) \cdot \left(\sqrt{{\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)}} - {x}^{\left(\frac{\frac{\frac{1}{n}}{2}}{2}\right)}\right)\right)}\]
   15. Using strategy rm

   16. Applied add-cube-cbrt 5.6

       \[\leadsto \left({\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)} + {\left({x}^{\left(\frac{\sqrt[3]{\frac{1}{n}} \cdot \sqrt[3]{\frac{1}{n}}}{1}\right)}\right)}^{\left(\frac{\sqrt[3]{\frac{1}{n}}}{2}\right)}\right) \cdot \left(\left(\sqrt{{\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)}} + {x}^{\left(\frac{\frac{\frac{1}{n}}{2}}{2}\right)}\right) \cdot \color{blue}{\left(\left(\sqrt[3]{\sqrt{{\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)}} - {x}^{\left(\frac{\frac{\frac{1}{n}}{2}}{2}\right)}} \cdot \sqrt[3]{\sqrt{{\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)}} - {x}^{\left(\frac{\frac{\frac{1}{n}}{2}}{2}\right)}}\right) \cdot \sqrt[3]{\sqrt{{\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)}} - {x}^{\left(\frac{\frac{\frac{1}{n}}{2}}{2}\right)}}\right)}\right)\]

   if -1.634558849313557e-23 < (/ 1.0 n) < 5.530646472234089e-24

   1. Initial program 44.6

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

   3. Applied sqr-pow 44.6

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

   4. Applied sqr-pow 44.6

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

   5. Applied difference-of-squares 44.6

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

   7. Applied *-un-lft-identity 44.6

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

   8. Applied add-cube-cbrt 44.6

      \[\leadsto \left({\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)} + {x}^{\left(\frac{\color{blue}{\left(\sqrt[3]{\frac{1}{n}} \cdot \sqrt[3]{\frac{1}{n}}\right) \cdot \sqrt[3]{\frac{1}{n}}}}{1 \cdot 2}\right)}\right) \cdot \left({\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)} - {x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right)\]

   9. Applied times-frac 44.6

      \[\leadsto \left({\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)} + {x}^{\color{blue}{\left(\frac{\sqrt[3]{\frac{1}{n}} \cdot \sqrt[3]{\frac{1}{n}}}{1} \cdot \frac{\sqrt[3]{\frac{1}{n}}}{2}\right)}}\right) \cdot \left({\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)} - {x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right)\]

   10. Applied pow-unpow 44.6

       \[\leadsto \left({\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)} + \color{blue}{{\left({x}^{\left(\frac{\sqrt[3]{\frac{1}{n}} \cdot \sqrt[3]{\frac{1}{n}}}{1}\right)}\right)}^{\left(\frac{\sqrt[3]{\frac{1}{n}}}{2}\right)}}\right) \cdot \left({\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)} - {x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right)\]

   11. Taylor expanded around inf 32.6

       \[\leadsto \left({\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)} + {\left({x}^{\left(\frac{\sqrt[3]{\frac{1}{n}} \cdot \sqrt[3]{\frac{1}{n}}}{1}\right)}\right)}^{\left(\frac{\sqrt[3]{\frac{1}{n}}}{2}\right)}\right) \cdot \color{blue}{\left(0.5 \cdot \frac{1}{x \cdot n} - \left(0.25 \cdot \frac{1}{{x}^{2} \cdot n} + 0.25 \cdot \frac{\log \left(\frac{1}{x}\right)}{x \cdot {n}^{2}}\right)\right)}\]

   12. Simplified 32.1

       \[\leadsto \left({\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)} + {\left({x}^{\left(\frac{\sqrt[3]{\frac{1}{n}} \cdot \sqrt[3]{\frac{1}{n}}}{1}\right)}\right)}^{\left(\frac{\sqrt[3]{\frac{1}{n}}}{2}\right)}\right) \cdot \color{blue}{\mathsf{fma}\left(-0.25, \frac{1}{{x}^{2} \cdot n} + \frac{\log \left(\frac{1}{x}\right)}{x \cdot {n}^{2}}, \frac{\frac{0.5}{n}}{x}\right)}\]

   if 5.530646472234089e-24 < (/ 1.0 n)

   1. Initial program 13.3

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

   3. Applied flip3-- 13.3

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

   4. Simplified 13.3

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

4. Final simplification 24.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \le -1.6345588493135571 \cdot 10^{-23}:\\ \;\;\;\;\left({\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)} + {\left({x}^{\left(\frac{\sqrt[3]{\frac{1}{n}} \cdot \sqrt[3]{\frac{1}{n}}}{1}\right)}\right)}^{\left(\frac{\sqrt[3]{\frac{1}{n}}}{2}\right)}\right) \cdot \left(\left(\sqrt{{\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)}} + {x}^{\left(\frac{\frac{\frac{1}{n}}{2}}{2}\right)}\right) \cdot \left(\left(\sqrt[3]{\sqrt{{\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)}} - {x}^{\left(\frac{\frac{\frac{1}{n}}{2}}{2}\right)}} \cdot \sqrt[3]{\sqrt{{\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)}} - {x}^{\left(\frac{\frac{\frac{1}{n}}{2}}{2}\right)}}\right) \cdot \sqrt[3]{\sqrt{{\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)}} - {x}^{\left(\frac{\frac{\frac{1}{n}}{2}}{2}\right)}}\right)\right)\\ \mathbf{elif}\;\frac{1}{n} \le 5.53064647223408922 \cdot 10^{-24}:\\ \;\;\;\;\left({\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)} + {\left({x}^{\left(\frac{\sqrt[3]{\frac{1}{n}} \cdot \sqrt[3]{\frac{1}{n}}}{1}\right)}\right)}^{\left(\frac{\sqrt[3]{\frac{1}{n}}}{2}\right)}\right) \cdot \mathsf{fma}\left(-0.25, \frac{1}{{x}^{2} \cdot n} + \frac{\log \left(\frac{1}{x}\right)}{x \cdot {n}^{2}}, \frac{\frac{0.5}{n}}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)}\right)}^{3} - {\left({x}^{\left(\frac{1}{n}\right)}\right)}^{3}}{\mathsf{fma}\left({x}^{\left(\frac{1}{n}\right)}, {\left(x + 1\right)}^{\left(\frac{1}{n}\right)} + {x}^{\left(\frac{1}{n}\right)}, {\left(x + 1\right)}^{\left(2 \cdot \frac{1}{n}\right)}\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020065 +o rules:numerics
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  :precision binary64
  (- (pow (+ x 1) (/ 1 n)) (pow x (/ 1 n))))