2nthrt (problem 3.4.6)

Average Error: 31.9 → 22.4

Time: 16.0s

Precision: binary64

• could not determine a ground truth for program body

  1. x = 5.495291702870734e+94
  2. n = 3.179899380440183e-117

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{1}{n} \le -1.7479825902971762 \cdot 10^{-6}:\\ \;\;\;\;{\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - \left(\sqrt{{\left({x}^{\left(\frac{1}{n}\right)}\right)}^{\frac{2}{3}} \cdot \sqrt[3]{{x}^{\left(\frac{1}{n}\right)}}} \cdot \left|\sqrt[3]{{\left({x}^{\left(\frac{1}{n}\right)}\right)}^{\frac{2}{3}} \cdot \sqrt[3]{{x}^{\left(\frac{1}{n}\right)}}}\right|\right) \cdot \sqrt{\sqrt[3]{{x}^{\left(\frac{1}{n}\right)}}}\\ \mathbf{elif}\;\frac{1}{n} \le 1.0160411307279881 \cdot 10^{-15}:\\ \;\;\;\;\frac{1}{n \cdot x}\\ \mathbf{else}:\\ \;\;\;\;{\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - \sqrt{{x}^{\left(\frac{1}{n}\right)}} \cdot \sqrt{{x}^{\left(\frac{1}{n}\right)}}\\ \end{array}\]

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)) <= -1.7479825902971762e-06)) {
		VAR = ((double) (((double) pow(((double) (1.0 + x)), ((double) (1.0 / n)))) - ((double) (((double) (((double) sqrt(((double) (((double) pow(((double) pow(x, ((double) (1.0 / n)))), 0.6666666666666666)) * ((double) cbrt(((double) pow(x, ((double) (1.0 / n)))))))))) * ((double) fabs(((double) cbrt(((double) (((double) pow(((double) pow(x, ((double) (1.0 / n)))), 0.6666666666666666)) * ((double) cbrt(((double) pow(x, ((double) (1.0 / n)))))))))))))) * ((double) sqrt(((double) cbrt(((double) pow(x, ((double) (1.0 / n))))))))))));
	} else {
		double VAR_1;
		if ((((double) (1.0 / n)) <= 1.016041130727988e-15)) {
			VAR_1 = ((double) (1.0 / ((double) (n * x))));
		} else {
			VAR_1 = ((double) (((double) pow(((double) (1.0 + x)), ((double) (1.0 / n)))) - ((double) (((double) sqrt(((double) pow(x, ((double) (1.0 / n)))))) * ((double) sqrt(((double) pow(x, ((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) < -1.7479825902971762e-6

   1. Initial program 0.8

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

   3. Applied add-sqr-sqrt 0.9

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

   5. Applied add-cube-cbrt 1.0

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

   6. Applied sqrt-prod 1.0

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

   7. Applied associate-*r* 1.0

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

   8. Simplified 1.0

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

   10. Applied add-cube-cbrt 1.0

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

   11. Simplified 1.0

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

   13. Applied add-cube-cbrt 1.0

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

   14. Simplified 1.0

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

   if -1.7479825902971762e-6 < (/ 1.0 n) < 1.0160411307279881e-15

   1. Initial program 44.3

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

   3. Applied add-sqr-sqrt 44.3

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

   4. Taylor expanded around -inf 64.0

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

   5. Simplified 30.6

      \[\leadsto \color{blue}{\left(\frac{1}{x \cdot n} + 0\right) - 0}\]

   if 1.0160411307279881e-15 < (/ 1.0 n)

   1. Initial program 8.4

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

   3. Applied add-sqr-sqrt 8.6

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

4. Final simplification 22.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \le -1.7479825902971762 \cdot 10^{-6}:\\ \;\;\;\;{\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - \left(\sqrt{{\left({x}^{\left(\frac{1}{n}\right)}\right)}^{\frac{2}{3}} \cdot \sqrt[3]{{x}^{\left(\frac{1}{n}\right)}}} \cdot \left|\sqrt[3]{{\left({x}^{\left(\frac{1}{n}\right)}\right)}^{\frac{2}{3}} \cdot \sqrt[3]{{x}^{\left(\frac{1}{n}\right)}}}\right|\right) \cdot \sqrt{\sqrt[3]{{x}^{\left(\frac{1}{n}\right)}}}\\ \mathbf{elif}\;\frac{1}{n} \le 1.0160411307279881 \cdot 10^{-15}:\\ \;\;\;\;\frac{1}{n \cdot x}\\ \mathbf{else}:\\ \;\;\;\;{\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - \sqrt{{x}^{\left(\frac{1}{n}\right)}} \cdot \sqrt{{x}^{\left(\frac{1}{n}\right)}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020184 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  :precision binary64
  (- (pow (+ x 1.0) (/ 1.0 n)) (pow x (/ 1.0 n))))