Asymptote C

Average Error: 29.6 → 0.1

Time: 6.4s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -14977.683653710101 \lor \neg \left(x \le 12828.8725117218401\right):\\ \;\;\;\;\left(\frac{-1}{{x}^{2}} - \frac{3}{x}\right) - \frac{3}{{x}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(x \cdot \frac{x}{x + 1}\right) \cdot \left(x - 1\right) - \left(x + 1\right) \cdot \left(\left(x + 1\right) \cdot \frac{x + 1}{x - 1}\right)}{x \cdot x - 1 \cdot 1}}{\frac{x}{x + 1} + \frac{x + 1}{x - 1}}\\ \end{array}\]

Error

Bits error versus x

Derivation

1. Split input into 2 regimes

2. if x < -14977.683653710101 or 12828.8725117218401 < x

   1. Initial program 59.1

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

   2. Taylor expanded around inf 0.3

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

   3. Simplified 0.0

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

   if -14977.683653710101 < x < 12828.8725117218401

   1. Initial program 0.1

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

   3. Applied flip-- 0.1

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

   5. Applied associate-*r/ 0.1

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

   6. Applied associate-*r/ 0.1

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

   7. Applied frac-sub 0.1

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

   8. Simplified 0.1

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

   9. Simplified 0.1

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

4. Final simplification 0.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -14977.683653710101 \lor \neg \left(x \le 12828.8725117218401\right):\\ \;\;\;\;\left(\frac{-1}{{x}^{2}} - \frac{3}{x}\right) - \frac{3}{{x}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(x \cdot \frac{x}{x + 1}\right) \cdot \left(x - 1\right) - \left(x + 1\right) \cdot \left(\left(x + 1\right) \cdot \frac{x + 1}{x - 1}\right)}{x \cdot x - 1 \cdot 1}}{\frac{x}{x + 1} + \frac{x + 1}{x - 1}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020162 
(FPCore (x)
  :name "Asymptote C"
  :precision binary64
  (- (/ x (+ x 1.0)) (/ (+ x 1.0) (- x 1.0))))