2nthrt (problem 3.4.6)

Average Error: 32.6 → 23.6

Time: 29.9min

Precision: binary64

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{1}{n} \le -0.8758774385902903:\\ \;\;\;\;2 \cdot \left(\sqrt{\log \left(\sqrt[3]{e^{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}}\right)} \cdot \sqrt{\log \left(\sqrt[3]{e^{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}}\right)}\right) + \log \left(\sqrt[3]{e^{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}}\right)\\ \mathbf{elif}\;\frac{1}{n} \le 3.70607248943366594 \cdot 10^{-15}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(x + 1\right)}^{\left(2 \cdot \frac{1}{n}\right)} - {x}^{\left(2 \cdot \frac{1}{n}\right)}}{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} + {x}^{\left(\frac{1}{n}\right)}}\\ \end{array}\]

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) <= -0.8758774385902903)) {
		VAR = ((double) (((double) (2.0 * ((double) (((double) sqrt(((double) log(((double) cbrt(((double) exp(((double) (((double) pow(((double) (x + 1.0)), (1.0 / n))) - ((double) pow(x, (1.0 / n))))))))))))) * ((double) sqrt(((double) log(((double) cbrt(((double) exp(((double) (((double) pow(((double) (x + 1.0)), (1.0 / n))) - ((double) pow(x, (1.0 / n))))))))))))))))) + ((double) log(((double) cbrt(((double) exp(((double) (((double) pow(((double) (x + 1.0)), (1.0 / n))) - ((double) pow(x, (1.0 / n)))))))))))));
	} else {
		double VAR_1;
		if (((1.0 / n) <= 3.706072489433666e-15)) {
			VAR_1 = ((1.0 / x) / n);
		} else {
			VAR_1 = (((double) (((double) pow(((double) (x + 1.0)), ((double) (2.0 * (1.0 / n))))) - ((double) pow(x, ((double) (2.0 * (1.0 / n))))))) / ((double) (((double) pow(((double) (x + 1.0)), (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) < -0.8758774385902903

   1. Initial program 0.0

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

   3. Applied add-log-exp 0.3

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

   4. Applied add-log-exp 0.2

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

   5. Applied diff-log 0.2

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

   6. Simplified 0.2

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

   8. Applied add-cube-cbrt 0.2

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

   9. Applied log-prod 0.2

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

   10. Simplified 0.2

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

   12. Applied add-sqr-sqrt 0.2

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

   if -0.8758774385902903 < (/ 1.0 n) < 3.70607248943366594e-15

   1. Initial program 45.0

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

   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 32.1

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

   if 3.70607248943366594e-15 < (/ 1.0 n)

   1. Initial program 8.7

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

   3. Applied flip-- 8.7

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

   4. Simplified 8.6

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

4. Final simplification 23.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \le -0.8758774385902903:\\ \;\;\;\;2 \cdot \left(\sqrt{\log \left(\sqrt[3]{e^{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}}\right)} \cdot \sqrt{\log \left(\sqrt[3]{e^{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}}\right)}\right) + \log \left(\sqrt[3]{e^{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}}\right)\\ \mathbf{elif}\;\frac{1}{n} \le 3.70607248943366594 \cdot 10^{-15}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(x + 1\right)}^{\left(2 \cdot \frac{1}{n}\right)} - {x}^{\left(2 \cdot \frac{1}{n}\right)}}{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} + {x}^{\left(\frac{1}{n}\right)}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020181 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  :precision binary64
  (- (pow (+ x 1.0) (/ 1.0 n)) (pow x (/ 1.0 n))))