Average Error: 14.2 → 0.1
Time: 3.1s
Precision: binary64
\[\frac{1}{x + 1} - \frac{1}{x - 1} \]
↓
\[\frac{\frac{-2}{x + 1}}{x + -1} \]
\frac{1}{x + 1} - \frac{1}{x - 1}
↓
\frac{\frac{-2}{x + 1}}{x + -1}

(FPCore (x) :precision binary64 (- (/ 1.0 (+ x 1.0)) (/ 1.0 (- x 1.0))))
↓
(FPCore (x) :precision binary64 (/ (/ -2.0 (+ x 1.0)) (+ x -1.0)))
double code(double x) {
	return (1.0 / (x + 1.0)) - (1.0 / (x - 1.0));
}
↓
double code(double x) {
	return (-2.0 / (x + 1.0)) / (x + -1.0);
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 14.2

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

  3. Applied flip-+_binary6428.5

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

  4. Applied associate-/r/_binary6428.5

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

  5. Simplified28.5

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

  7. Applied associate-*l/_binary6428.5

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

  8. Applied frac-sub_binary6429.6

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

  9. Simplified13.6

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

  10. Simplified13.6

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

  12. Applied associate-/r*_binary6413.6

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

  13. Simplified0.1

    \[\leadsto \frac{\color{blue}{\frac{-2}{1 + x} \cdot 1}}{x + -1} \]
  14. Using strategy rm

  15. Applied *-un-lft-identity_binary640.1

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

  16. Applied associate-/r*_binary640.1

    \[\leadsto \color{blue}{\frac{\frac{\frac{-2}{1 + x} \cdot 1}{1}}{x + -1}} \]

  17. Simplified0.1

    \[\leadsto \frac{\color{blue}{\frac{-2}{x + 1}}}{x + -1} \]

  18. Final simplification0.1

    \[\leadsto \frac{\frac{-2}{x + 1}}{x + -1} \]

Reproduce

herbie shell --seed 2021190 
(FPCore (x)
  :name "Asymptote A"
  :precision binary64
  (- (/ 1.0 (+ x 1.0)) (/ 1.0 (- x 1.0))))