2nthrt (problem 3.4.6)

Average Error: 32.7 → 23.4

Time: 15.5s

Precision: binary64

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;n \le -4984.24792040262128:\\ \;\;\;\;\left(\sqrt[3]{\frac{1}{n}} \cdot \sqrt[3]{\frac{1}{n}}\right) \cdot \frac{\sqrt[3]{\frac{1}{n}}}{x}\\ \mathbf{elif}\;n \le 837.61817414980806:\\ \;\;\;\;{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{1}{\sqrt[3]{n} \cdot \sqrt[3]{n}}}{\sqrt{\sqrt{x}}}}{\sqrt{\sqrt{x}}} \cdot \frac{\frac{\frac{1}{\sqrt[3]{n}}}{\sqrt{\sqrt{x}}}}{\sqrt{\sqrt{x}}}\\ \end{array}\]

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 ((n <= -4984.247920402621)) {
		VAR = ((double) (((double) (((double) cbrt((1.0 / n))) * ((double) cbrt((1.0 / n))))) * (((double) cbrt((1.0 / n))) / x)));
	} else {
		double VAR_1;
		if ((n <= 837.618174149808)) {
			VAR_1 = ((double) (((double) pow(((double) (x + 1.0)), (1.0 / n))) - ((double) pow(x, (1.0 / n)))));
		} else {
			VAR_1 = ((double) ((((1.0 / ((double) (((double) cbrt(n)) * ((double) cbrt(n))))) / ((double) sqrt(((double) sqrt(x))))) / ((double) sqrt(((double) sqrt(x))))) * (((1.0 / ((double) cbrt(n))) / ((double) sqrt(((double) sqrt(x))))) / ((double) sqrt(((double) sqrt(x)))))));
		}
		VAR = VAR_1;
	}
	return VAR;
}

Error

Bits error versus x

Bits error versus n

Derivation

1. Split input into 3 regimes

2. if n < -4984.24792040262128

   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 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 31.6

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

   5. Applied associate-/r* 30.9

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

   7. Applied *-un-lft-identity 30.9

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

   8. Applied add-cube-cbrt 31.2

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

   9. Applied times-frac 31.2

      \[\leadsto \color{blue}{\frac{\sqrt[3]{\frac{1}{n}} \cdot \sqrt[3]{\frac{1}{n}}}{1} \cdot \frac{\sqrt[3]{\frac{1}{n}}}{x}}\]

   10. Simplified 31.2

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

   if -4984.24792040262128 < n < 837.61817414980806

   1. Initial program 2.3

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

   if 837.61817414980806 < n

   1. Initial program 44.8

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

   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 32.5

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

   5. Applied associate-/r* 31.9

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

   7. Applied add-sqr-sqrt 32.0

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

   8. Applied associate-/r* 32.0

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

   10. Applied add-sqr-sqrt 32.0

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

   11. Applied sqrt-prod 32.0

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

   12. Applied add-sqr-sqrt 32.0

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

   13. Applied sqrt-prod 32.1

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

   14. Applied add-cube-cbrt 32.2

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

   15. Applied *-un-lft-identity 32.2

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

   16. Applied times-frac 32.2

       \[\leadsto \frac{\frac{\color{blue}{\frac{1}{\sqrt[3]{n} \cdot \sqrt[3]{n}} \cdot \frac{1}{\sqrt[3]{n}}}}{\sqrt{\sqrt{x}} \cdot \sqrt{\sqrt{x}}}}{\sqrt{\sqrt{x}} \cdot \sqrt{\sqrt{x}}}\]

   17. Applied times-frac 32.2

       \[\leadsto \frac{\color{blue}{\frac{\frac{1}{\sqrt[3]{n} \cdot \sqrt[3]{n}}}{\sqrt{\sqrt{x}}} \cdot \frac{\frac{1}{\sqrt[3]{n}}}{\sqrt{\sqrt{x}}}}}{\sqrt{\sqrt{x}} \cdot \sqrt{\sqrt{x}}}\]

   18. Applied times-frac 32.2

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

4. Final simplification 23.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \le -4984.24792040262128:\\ \;\;\;\;\left(\sqrt[3]{\frac{1}{n}} \cdot \sqrt[3]{\frac{1}{n}}\right) \cdot \frac{\sqrt[3]{\frac{1}{n}}}{x}\\ \mathbf{elif}\;n \le 837.61817414980806:\\ \;\;\;\;{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{1}{\sqrt[3]{n} \cdot \sqrt[3]{n}}}{\sqrt{\sqrt{x}}}}{\sqrt{\sqrt{x}}} \cdot \frac{\frac{\frac{1}{\sqrt[3]{n}}}{\sqrt{\sqrt{x}}}}{\sqrt{\sqrt{x}}}\\ \end{array}\]

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))))