2cbrt (problem 3.3.4)

Average Error: 29.0 → 11.6

Time: 8.6s

Precision: 64

Math

\[\sqrt[3]{x + 1} - \sqrt[3]{x}\]

↓

\[\begin{array}{l} \mathbf{if}\;x \le -4.5086911354082924 \cdot 10^{61}:\\ \;\;\;\;\mathsf{fma}\left({\left(\frac{1}{{x}^{2}}\right)}^{\frac{1}{3}}, 0.333333333333333315, 0.061728395061728392 \cdot {\left(\frac{1}{{x}^{8}}\right)}^{\frac{1}{3}} - 0.1111111111111111 \cdot {\left(\frac{1}{{x}^{5}}\right)}^{\frac{1}{3}}\right)\\ \mathbf{elif}\;x \le 3932.66029113376089:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt[3]{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}}, \sqrt[3]{\sqrt[3]{x + 1}} \cdot \sqrt[3]{x + 1}, -\left(\sqrt[3]{\sqrt[3]{x}} \cdot \sqrt[3]{x}\right) \cdot \left(\sqrt[3]{\sqrt[3]{\sqrt[3]{x} \cdot \sqrt[3]{x}} \cdot \sqrt[3]{\sqrt[3]{x} \cdot \sqrt[3]{x}}} \cdot \sqrt[3]{\sqrt[3]{\sqrt[3]{x}} \cdot \sqrt[3]{\sqrt[3]{x}}}\right)\right) + \left(\sqrt[3]{x} \cdot \sqrt[3]{\sqrt[3]{x} \cdot \sqrt[3]{x}}\right) \cdot \left(\left(-\sqrt[3]{\sqrt[3]{x}}\right) + \sqrt[3]{\sqrt[3]{x}}\right)}{\sqrt[3]{x + 1} + \sqrt[3]{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left({\left(\frac{1}{{x}^{7}}\right)}^{\frac{1}{3}}, 0.04938271604938271, 0.66666666666666663 \cdot {\left(\frac{1}{x}\right)}^{\frac{1}{3}} - 0.1111111111111111 \cdot {\left(\frac{1}{{x}^{4}}\right)}^{\frac{1}{3}}\right)}{\sqrt[3]{x + 1} + \sqrt[3]{x}}\\ \end{array}\]

C

double code(double x) {
	return ((double) (((double) cbrt(((double) (x + 1.0)))) - ((double) cbrt(x))));
}

↓

double code(double x) {
	double VAR;
	if ((x <= -4.5086911354082924e+61)) {
		VAR = ((double) fma(((double) pow(((double) (1.0 / ((double) pow(x, 2.0)))), 0.3333333333333333)), 0.3333333333333333, ((double) (((double) (0.06172839506172839 * ((double) pow(((double) (1.0 / ((double) pow(x, 8.0)))), 0.3333333333333333)))) - ((double) (0.1111111111111111 * ((double) pow(((double) (1.0 / ((double) pow(x, 5.0)))), 0.3333333333333333))))))));
	} else {
		double VAR_1;
		if ((x <= 3932.660291133761)) {
			VAR_1 = ((double) (((double) (((double) fma(((double) cbrt(((double) (((double) cbrt(((double) (x + 1.0)))) * ((double) cbrt(((double) (x + 1.0)))))))), ((double) (((double) cbrt(((double) cbrt(((double) (x + 1.0)))))) * ((double) cbrt(((double) (x + 1.0)))))), ((double) -(((double) (((double) (((double) cbrt(((double) cbrt(x)))) * ((double) cbrt(x)))) * ((double) (((double) cbrt(((double) (((double) cbrt(((double) (((double) cbrt(x)) * ((double) cbrt(x)))))) * ((double) cbrt(((double) (((double) cbrt(x)) * ((double) cbrt(x)))))))))) * ((double) cbrt(((double) (((double) cbrt(((double) cbrt(x)))) * ((double) cbrt(((double) cbrt(x)))))))))))))))) + ((double) (((double) (((double) cbrt(x)) * ((double) cbrt(((double) (((double) cbrt(x)) * ((double) cbrt(x)))))))) * ((double) (((double) -(((double) cbrt(((double) cbrt(x)))))) + ((double) cbrt(((double) cbrt(x)))))))))) / ((double) (((double) cbrt(((double) (x + 1.0)))) + ((double) cbrt(x))))));
		} else {
			VAR_1 = ((double) (((double) fma(((double) pow(((double) (1.0 / ((double) pow(x, 7.0)))), 0.3333333333333333)), 0.04938271604938271, ((double) (((double) (0.6666666666666666 * ((double) pow(((double) (1.0 / x)), 0.3333333333333333)))) - ((double) (0.1111111111111111 * ((double) pow(((double) (1.0 / ((double) pow(x, 4.0)))), 0.3333333333333333)))))))) / ((double) (((double) cbrt(((double) (x + 1.0)))) + ((double) cbrt(x))))));
		}
		VAR = VAR_1;
	}
	return VAR;
}

Error

Bits error versus x

Derivation

1. Split input into 3 regimes

2. if x < -4.5086911354082924e+61

   1. Initial program 61.2

      \[\sqrt[3]{x + 1} - \sqrt[3]{x}\]

   2. Taylor expanded around inf 40.0

      \[\leadsto \color{blue}{\left(0.333333333333333315 \cdot {\left(\frac{1}{{x}^{2}}\right)}^{\frac{1}{3}} + 0.061728395061728392 \cdot {\left(\frac{1}{{x}^{8}}\right)}^{\frac{1}{3}}\right) - 0.1111111111111111 \cdot {\left(\frac{1}{{x}^{5}}\right)}^{\frac{1}{3}}}\]

   3. Simplified 40.0

      \[\leadsto \color{blue}{\mathsf{fma}\left({\left(\frac{1}{{x}^{2}}\right)}^{\frac{1}{3}}, 0.333333333333333315, 0.061728395061728392 \cdot {\left(\frac{1}{{x}^{8}}\right)}^{\frac{1}{3}} - 0.1111111111111111 \cdot {\left(\frac{1}{{x}^{5}}\right)}^{\frac{1}{3}}\right)}\]

   if -4.5086911354082924e+61 < x < 3932.660291133761

   1. Initial program 4.9

      \[\sqrt[3]{x + 1} - \sqrt[3]{x}\]
   2. Using strategy rm

   3. Applied flip-- 4.9

      \[\leadsto \color{blue}{\frac{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1} - \sqrt[3]{x} \cdot \sqrt[3]{x}}{\sqrt[3]{x + 1} + \sqrt[3]{x}}}\]
   4. Using strategy rm

   5. Applied add-cube-cbrt 4.9

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

   6. Applied cbrt-prod 4.8

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

   7. Applied associate-*r* 4.8

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

   8. Applied add-cube-cbrt 4.8

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

   9. Applied cbrt-prod 4.9

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

   10. Applied associate-*r* 4.9

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

   11. Applied prod-diff 4.9

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

   12. Simplified 4.9

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

   13. Simplified 4.8

       \[\leadsto \frac{\mathsf{fma}\left(\sqrt[3]{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}}, \sqrt[3]{\sqrt[3]{x + 1}} \cdot \sqrt[3]{x + 1}, -\left(\sqrt[3]{\sqrt[3]{x}} \cdot \sqrt[3]{x}\right) \cdot \sqrt[3]{\sqrt[3]{x} \cdot \sqrt[3]{x}}\right) + \color{blue}{\left(\sqrt[3]{x} \cdot \sqrt[3]{\sqrt[3]{x} \cdot \sqrt[3]{x}}\right) \cdot \left(\left(-\sqrt[3]{\sqrt[3]{x}}\right) + \sqrt[3]{\sqrt[3]{x}}\right)}}{\sqrt[3]{x + 1} + \sqrt[3]{x}}\]
   14. Using strategy rm

   15. Applied add-cube-cbrt 4.8

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

   16. Applied cbrt-prod 4.8

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

   17. Applied add-cube-cbrt 4.8

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

   18. Applied cbrt-prod 4.8

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

   19. Applied swap-sqr 4.8

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

   20. Applied cbrt-prod 4.8

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

   if 3932.660291133761 < x

   1. Initial program 60.2

      \[\sqrt[3]{x + 1} - \sqrt[3]{x}\]
   2. Using strategy rm

   3. Applied flip-- 60.2

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

   4. Taylor expanded around inf 5.1

      \[\leadsto \frac{\color{blue}{\left(0.04938271604938271 \cdot {\left(\frac{1}{{x}^{7}}\right)}^{\frac{1}{3}} + 0.66666666666666663 \cdot {\left(\frac{1}{x}\right)}^{\frac{1}{3}}\right) - 0.1111111111111111 \cdot {\left(\frac{1}{{x}^{4}}\right)}^{\frac{1}{3}}}}{\sqrt[3]{x + 1} + \sqrt[3]{x}}\]

   5. Simplified 5.1

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\left(\frac{1}{{x}^{7}}\right)}^{\frac{1}{3}}, 0.04938271604938271, 0.66666666666666663 \cdot {\left(\frac{1}{x}\right)}^{\frac{1}{3}} - 0.1111111111111111 \cdot {\left(\frac{1}{{x}^{4}}\right)}^{\frac{1}{3}}\right)}}{\sqrt[3]{x + 1} + \sqrt[3]{x}}\]
3. Recombined 3 regimes into one program.

4. Final simplification 11.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -4.5086911354082924 \cdot 10^{61}:\\ \;\;\;\;\mathsf{fma}\left({\left(\frac{1}{{x}^{2}}\right)}^{\frac{1}{3}}, 0.333333333333333315, 0.061728395061728392 \cdot {\left(\frac{1}{{x}^{8}}\right)}^{\frac{1}{3}} - 0.1111111111111111 \cdot {\left(\frac{1}{{x}^{5}}\right)}^{\frac{1}{3}}\right)\\ \mathbf{elif}\;x \le 3932.66029113376089:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt[3]{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}}, \sqrt[3]{\sqrt[3]{x + 1}} \cdot \sqrt[3]{x + 1}, -\left(\sqrt[3]{\sqrt[3]{x}} \cdot \sqrt[3]{x}\right) \cdot \left(\sqrt[3]{\sqrt[3]{\sqrt[3]{x} \cdot \sqrt[3]{x}} \cdot \sqrt[3]{\sqrt[3]{x} \cdot \sqrt[3]{x}}} \cdot \sqrt[3]{\sqrt[3]{\sqrt[3]{x}} \cdot \sqrt[3]{\sqrt[3]{x}}}\right)\right) + \left(\sqrt[3]{x} \cdot \sqrt[3]{\sqrt[3]{x} \cdot \sqrt[3]{x}}\right) \cdot \left(\left(-\sqrt[3]{\sqrt[3]{x}}\right) + \sqrt[3]{\sqrt[3]{x}}\right)}{\sqrt[3]{x + 1} + \sqrt[3]{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left({\left(\frac{1}{{x}^{7}}\right)}^{\frac{1}{3}}, 0.04938271604938271, 0.66666666666666663 \cdot {\left(\frac{1}{x}\right)}^{\frac{1}{3}} - 0.1111111111111111 \cdot {\left(\frac{1}{{x}^{4}}\right)}^{\frac{1}{3}}\right)}{\sqrt[3]{x + 1} + \sqrt[3]{x}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020113 +o rules:numerics
(FPCore (x)
  :name "2cbrt (problem 3.3.4)"
  :precision binary64
  (- (cbrt (+ x 1)) (cbrt x)))