Asymptote A

Average Error: 14.7 → 0.1

Time: 3.2s

Precision: 64

Math

\[\frac{1}{x + 1} - \frac{1}{x - 1}\]

↓

\[\left(\frac{\sqrt{1}}{x + 1} \cdot \frac{\sqrt{1}}{x - 1}\right) \cdot \left(-2\right)\]

C

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

↓

double code(double x) {
	return (((sqrt(1.0) / (x + 1.0)) * (sqrt(1.0) / (x - 1.0))) * -2.0);
}

Error

Bits error versus x

Derivation

1. Initial program 14.7

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

3. Applied flip-- 29.3

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

4. Applied associate-/r/ 29.3

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

5. Applied flip-+ 14.8

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

6. Applied associate-/r/ 14.7

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

7. Applied distribute-lft-out-- 14.1

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

9. Applied add-cbrt-cube 14.1

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

10. Applied add-cbrt-cube 14.1

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

11. Applied cbrt-undiv 14.1

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

12. Simplified 14.1

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

13. Taylor expanded around 0 10.3

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

15. Applied difference-of-squares 10.3

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

16. Applied add-sqr-sqrt 10.3

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

17. Applied times-frac 10.3

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

18. Applied unpow-prod-down 10.3

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

19. Applied cbrt-prod 5.4

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

20. Simplified 5.3

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

21. Simplified 0.1

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

22. Final simplification 0.1

    \[\leadsto \left(\frac{\sqrt{1}}{x + 1} \cdot \frac{\sqrt{1}}{x - 1}\right) \cdot \left(-2\right)\]

Reproduce

herbie shell --seed 2020065 
(FPCore (x)
  :name "Asymptote A"
  :precision binary64
  (- (/ 1 (+ x 1)) (/ 1 (- x 1))))