2nthrt (problem 3.4.6)

Average Error: 33.0 → 24.2

Time: 23.9s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -1.7041938189192093 \cdot 10^{-10}:\\ \;\;\;\;2 \cdot \log \left(\sqrt[3]{e^{{\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}}\right) + \left(\log \left(\sqrt[3]{e^{{\left(1 + x\right)}^{\left(\frac{1}{n}\right)}}}\right) + \log \left(\sqrt[3]{e^{-{x}^{\left(\frac{1}{n}\right)}}}\right)\right)\\ \mathbf{elif}\;\frac{1}{n} \leq 6.494796800484382 \cdot 10^{-26}:\\ \;\;\;\;\frac{1}{n \cdot x} + \left(\frac{1}{n \cdot n} \cdot \left(\frac{\log 1}{x} + \frac{\log x}{x}\right) - \frac{0.5}{x \cdot \left(n \cdot x\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left({\left(1 + x\right)}^{\left(\frac{1}{n}\right)}\right)}^{3} - {\left({x}^{\left(\frac{1}{n}\right)}\right)}^{3}}{{\left(1 + x\right)}^{\left(\frac{1}{n} \cdot 2\right)} + {x}^{\left(\frac{1}{n}\right)} \cdot \left({\left(1 + x\right)}^{\left(\frac{1}{n}\right)} + {x}^{\left(\frac{1}{n}\right)}\right)}\\ \end{array}\]

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) <= -1.7041938189192093e-10)) {
		VAR = ((double) (((double) (2.0 * ((double) log(((double) cbrt(((double) exp(((double) (((double) pow(((double) (1.0 + x)), (1.0 / n))) - ((double) pow(x, (1.0 / n))))))))))))) + ((double) (((double) log(((double) cbrt(((double) exp(((double) pow(((double) (1.0 + x)), (1.0 / n))))))))) + ((double) log(((double) cbrt(((double) exp(((double) -(((double) pow(x, (1.0 / n)))))))))))))));
	} else {
		double VAR_1;
		if (((1.0 / n) <= 6.494796800484382e-26)) {
			VAR_1 = ((double) ((1.0 / ((double) (n * x))) + ((double) (((double) ((1.0 / ((double) (n * n))) * ((double) ((((double) log(1.0)) / x) + (((double) log(x)) / x))))) - (0.5 / ((double) (x * ((double) (n * x)))))))));
		} else {
			VAR_1 = (((double) (((double) pow(((double) pow(((double) (1.0 + x)), (1.0 / n))), 3.0)) - ((double) pow(((double) pow(x, (1.0 / n))), 3.0)))) / ((double) (((double) pow(((double) (1.0 + x)), ((double) ((1.0 / n) * 2.0)))) + ((double) (((double) pow(x, (1.0 / n))) * ((double) (((double) pow(((double) (1.0 + x)), (1.0 / n))) + ((double) pow(x, (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) < -1.704193818919209e-10

   1. Initial program 1.9

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

   3. Applied add-log-exp 2.3

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

   4. Applied add-log-exp 2.2

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

   5. Applied diff-log 2.2

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

   6. Simplified 2.2

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

   8. Applied add-cube-cbrt 2.2

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

   9. Applied log-prod 2.2

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

   10. Simplified 2.2

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

   12. Applied sub-neg 2.2

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

   13. Applied exp-sum 2.2

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

   14. Applied cbrt-prod 2.2

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

   15. Applied log-prod 2.2

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

   if -1.704193818919209e-10 < (/ 1.0 n) < 6.4947968004843821e-26

   1. Initial program 44.7

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

   2. Taylor expanded around inf 32.1

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

   3. Simplified 31.9

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

   if 6.4947968004843821e-26 < (/ 1.0 n)

   1. Initial program 12.9

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

   3. Applied flip3-- 13.0

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

   4. Simplified 13.0

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

4. Final simplification 24.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -1.7041938189192093 \cdot 10^{-10}:\\ \;\;\;\;2 \cdot \log \left(\sqrt[3]{e^{{\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}}\right) + \left(\log \left(\sqrt[3]{e^{{\left(1 + x\right)}^{\left(\frac{1}{n}\right)}}}\right) + \log \left(\sqrt[3]{e^{-{x}^{\left(\frac{1}{n}\right)}}}\right)\right)\\ \mathbf{elif}\;\frac{1}{n} \leq 6.494796800484382 \cdot 10^{-26}:\\ \;\;\;\;\frac{1}{n \cdot x} + \left(\frac{1}{n \cdot n} \cdot \left(\frac{\log 1}{x} + \frac{\log x}{x}\right) - \frac{0.5}{x \cdot \left(n \cdot x\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left({\left(1 + x\right)}^{\left(\frac{1}{n}\right)}\right)}^{3} - {\left({x}^{\left(\frac{1}{n}\right)}\right)}^{3}}{{\left(1 + x\right)}^{\left(\frac{1}{n} \cdot 2\right)} + {x}^{\left(\frac{1}{n}\right)} \cdot \left({\left(1 + x\right)}^{\left(\frac{1}{n}\right)} + {x}^{\left(\frac{1}{n}\right)}\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020196 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  :precision binary64
  (- (pow (+ x 1.0) (/ 1.0 n)) (pow x (/ 1.0 n))))