2nthrt (problem 3.4.6)

Average Error: 29.0 → 22.2

Time: 15.5s

Precision: 64

• could not determine a ground truth for program body

  1. x = 6.317567976138939e+21
  2. n = 4.904694764599502e-148

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{1}{n} \le -9.91421720864949135 \cdot 10^{-7} \lor \neg \left(\frac{1}{n} \le 1.63989863279117905 \cdot 10^{-10}\right):\\ \;\;\;\;\left(\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}} \cdot \sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right) \cdot \sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(1, \frac{1}{x \cdot n}, -\mathsf{fma}\left(0.5, \frac{1}{{x}^{2} \cdot n}, 1 \cdot \frac{\log \left(\frac{1}{x}\right)}{x \cdot {n}^{2}}\right)\right)\\ \end{array}\]

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) <= -9.914217208649491e-07) || !((1.0 / n) <= 1.639898632791179e-10))) {
		temp = ((cbrt((pow((x + 1.0), (1.0 / n)) - pow(x, (1.0 / n)))) * cbrt((pow((x + 1.0), (1.0 / n)) - pow(x, (1.0 / n))))) * cbrt((pow((x + 1.0), (1.0 / n)) - pow(x, (1.0 / n)))));
	} else {
		temp = fma(1.0, (1.0 / (x * n)), -fma(0.5, (1.0 / (pow(x, 2.0) * n)), (1.0 * (log((1.0 / x)) / (x * pow(n, 2.0))))));
	}
	return temp;
}

Error

Bits error versus x

Bits error versus n

Derivation

1. Split input into 2 regimes

2. if (/ 1.0 n) < -9.914217208649491e-07 or 1.639898632791179e-10 < (/ 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-cube-cbrt 8.4

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

   if -9.914217208649491e-07 < (/ 1.0 n) < 1.639898632791179e-10

   1. Initial program 44.4

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

   2. Taylor expanded around inf 32.5

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

   3. Simplified 32.5

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

4. Final simplification 22.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \le -9.91421720864949135 \cdot 10^{-7} \lor \neg \left(\frac{1}{n} \le 1.63989863279117905 \cdot 10^{-10}\right):\\ \;\;\;\;\left(\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}} \cdot \sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right) \cdot \sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(1, \frac{1}{x \cdot n}, -\mathsf{fma}\left(0.5, \frac{1}{{x}^{2} \cdot n}, 1 \cdot \frac{\log \left(\frac{1}{x}\right)}{x \cdot {n}^{2}}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020049 +o rules:numerics
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  :precision binary64
  (- (pow (+ x 1) (/ 1 n)) (pow x (/ 1 n))))