2nthrt (problem 3.4.6)

Average Error: 33.2 → 24.1

Time: 13.8s

Precision: 64

• could not determine a ground truth for program body

  1. x = 1.6710656672296886e+109
  2. n = 3.229095618059491e-301

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;n \le -1660.74369281715803:\\ \;\;\;\;\left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} + \sqrt{{x}^{\left(\frac{1}{n}\right)}}\right) \cdot \left(\left(-0.25 \cdot \left(\frac{1}{{x}^{2} \cdot n} + \frac{\log \left(\frac{1}{x}\right)}{e^{\log \left(x \cdot {n}^{2}\right)}}\right)\right) + \frac{\frac{0.5}{n}}{x}\right)\\ \mathbf{elif}\;n \le 7633188339970.2734:\\ \;\;\;\;\left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} + \sqrt{{x}^{\left(\frac{1}{n}\right)}}\right) \cdot \left(\left(\sqrt[3]{\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} - \sqrt{{x}^{\left(\frac{1}{n}\right)}}} \cdot \sqrt[3]{\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} - \sqrt{{x}^{\left(\frac{1}{n}\right)}}}\right) \cdot \sqrt[3]{\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} - \sqrt{{x}^{\left(\frac{1}{n}\right)}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} + \sqrt{{x}^{\left(\frac{1}{n}\right)}}\right) \cdot \left(\left(-0.25 \cdot \left(\frac{1}{\log \left(e^{{x}^{2} \cdot n}\right)} + \frac{\log \left(\frac{1}{x}\right)}{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \left(\sqrt[3]{x} \cdot {n}^{2}\right)}\right)\right) + \frac{\frac{0.5}{n}}{x}\right)\\ \end{array}\]

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 ((n <= -1660.743692817158)) {
		temp = ((sqrt(pow((x + 1.0), (1.0 / n))) + sqrt(pow(x, (1.0 / n)))) * (-(0.25 * ((1.0 / (pow(x, 2.0) * n)) + (log((1.0 / x)) / exp(log((x * pow(n, 2.0))))))) + ((0.5 / n) / x)));
	} else {
		double temp_1;
		if ((n <= 7633188339970.273)) {
			temp_1 = ((sqrt(pow((x + 1.0), (1.0 / n))) + sqrt(pow(x, (1.0 / n)))) * ((cbrt((sqrt(pow((x + 1.0), (1.0 / n))) - sqrt(pow(x, (1.0 / n))))) * cbrt((sqrt(pow((x + 1.0), (1.0 / n))) - sqrt(pow(x, (1.0 / n)))))) * cbrt((sqrt(pow((x + 1.0), (1.0 / n))) - sqrt(pow(x, (1.0 / n)))))));
		} else {
			temp_1 = ((sqrt(pow((x + 1.0), (1.0 / n))) + sqrt(pow(x, (1.0 / n)))) * (-(0.25 * ((1.0 / log(exp((pow(x, 2.0) * n)))) + (log((1.0 / x)) / ((cbrt(x) * cbrt(x)) * (cbrt(x) * pow(n, 2.0)))))) + ((0.5 / n) / x)));
		}
		temp = temp_1;
	}
	return temp;
}

Error

Bits error versus x

Bits error versus n

Derivation

1. Split input into 3 regimes

2. if n < -1660.743692817158

   1. Initial program 46.0

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

   3. Applied add-sqr-sqrt 46.1

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

   4. Applied add-sqr-sqrt 46.1

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

   5. Applied difference-of-squares 46.1

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

   6. Taylor expanded around inf 32.9

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

   7. Simplified 32.4

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

   9. Applied add-exp-log 64.0

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

   10. Applied pow-exp 64.0

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

   11. Applied add-exp-log 64.0

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

   12. Applied prod-exp 64.0

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

   13. Simplified 32.4

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

   if -1660.743692817158 < n < 7633188339970.273

   1. Initial program 3.2

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

   3. Applied add-sqr-sqrt 3.3

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

   4. Applied add-sqr-sqrt 3.3

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

   5. Applied difference-of-squares 3.3

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

   7. Applied add-cube-cbrt 3.3

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

   if 7633188339970.273 < n

   1. Initial program 45.3

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

   3. Applied add-sqr-sqrt 45.3

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

   4. Applied add-sqr-sqrt 45.3

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

   5. Applied difference-of-squares 45.3

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

   6. Taylor expanded around inf 33.6

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

   7. Simplified 33.1

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

   9. Applied add-cube-cbrt 33.1

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

   10. Applied associate-*l* 33.1

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

   12. Applied add-log-exp 33.1

       \[\leadsto \left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} + \sqrt{{x}^{\left(\frac{1}{n}\right)}}\right) \cdot \left(\left(-0.25 \cdot \left(\frac{1}{\color{blue}{\log \left(e^{{x}^{2} \cdot n}\right)}} + \frac{\log \left(\frac{1}{x}\right)}{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \left(\sqrt[3]{x} \cdot {n}^{2}\right)}\right)\right) + \frac{\frac{0.5}{n}}{x}\right)\]
3. Recombined 3 regimes into one program.

4. Final simplification 24.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \le -1660.74369281715803:\\ \;\;\;\;\left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} + \sqrt{{x}^{\left(\frac{1}{n}\right)}}\right) \cdot \left(\left(-0.25 \cdot \left(\frac{1}{{x}^{2} \cdot n} + \frac{\log \left(\frac{1}{x}\right)}{e^{\log \left(x \cdot {n}^{2}\right)}}\right)\right) + \frac{\frac{0.5}{n}}{x}\right)\\ \mathbf{elif}\;n \le 7633188339970.2734:\\ \;\;\;\;\left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} + \sqrt{{x}^{\left(\frac{1}{n}\right)}}\right) \cdot \left(\left(\sqrt[3]{\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} - \sqrt{{x}^{\left(\frac{1}{n}\right)}}} \cdot \sqrt[3]{\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} - \sqrt{{x}^{\left(\frac{1}{n}\right)}}}\right) \cdot \sqrt[3]{\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} - \sqrt{{x}^{\left(\frac{1}{n}\right)}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} + \sqrt{{x}^{\left(\frac{1}{n}\right)}}\right) \cdot \left(\left(-0.25 \cdot \left(\frac{1}{\log \left(e^{{x}^{2} \cdot n}\right)} + \frac{\log \left(\frac{1}{x}\right)}{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \left(\sqrt[3]{x} \cdot {n}^{2}\right)}\right)\right) + \frac{\frac{0.5}{n}}{x}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020066 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  :precision binary64
  (- (pow (+ x 1) (/ 1 n)) (pow x (/ 1 n))))