Asymptote C

Average Error: 29.4 → 0.1

Time: 5.4s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -10492.950302238845 \lor \neg \left(x \le 11780.749313135173\right):\\ \;\;\;\;\left(\frac{-1}{{x}^{2}} - \frac{3}{x}\right) - \frac{3}{{x}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{{\left({\left(\frac{x}{x + 1}\right)}^{3}\right)}^{3} - {\left({\left(\frac{x + 1}{x - 1}\right)}^{3}\right)}^{3}}{\left({\left(\frac{x}{x + 1}\right)}^{6} + {\left(\frac{x + 1}{x - 1}\right)}^{6}\right) + {\left(\frac{x}{x + 1}\right)}^{3} \cdot {\left(\frac{x + 1}{x - 1}\right)}^{3}}}{\frac{x + 1}{x - 1} \cdot \left(\frac{x + 1}{x - 1} + \frac{x}{x + 1}\right) + \frac{x}{x + 1} \cdot \frac{x}{x + 1}}\\ \end{array}\]

Error

Bits error versus x

Derivation

1. Split input into 2 regimes

2. if x < -10492.950302238845 or 11780.749313135173 < x

   1. Initial program 59.2

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

   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 -10492.950302238845 < x < 11780.749313135173

   1. Initial program 0.1

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

   3. Applied flip3-- 0.1

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

   4. Simplified 0.1

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

   6. Applied flip3-- 0.1

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

   7. Simplified 0.1

      \[\leadsto \frac{\frac{{\left({\left(\frac{x}{x + 1}\right)}^{3}\right)}^{3} - {\left({\left(\frac{x + 1}{x - 1}\right)}^{3}\right)}^{3}}{\color{blue}{\left({\left(\frac{x}{x + 1}\right)}^{6} + {\left(\frac{x + 1}{x - 1}\right)}^{6}\right) + {\left(\frac{x}{x + 1}\right)}^{3} \cdot {\left(\frac{x + 1}{x - 1}\right)}^{3}}}}{\frac{x + 1}{x - 1} \cdot \left(\frac{x + 1}{x - 1} + \frac{x}{x + 1}\right) + \frac{x}{x + 1} \cdot \frac{x}{x + 1}}\]
3. Recombined 2 regimes into one program.

4. Final simplification 0.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -10492.950302238845 \lor \neg \left(x \le 11780.749313135173\right):\\ \;\;\;\;\left(\frac{-1}{{x}^{2}} - \frac{3}{x}\right) - \frac{3}{{x}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{{\left({\left(\frac{x}{x + 1}\right)}^{3}\right)}^{3} - {\left({\left(\frac{x + 1}{x - 1}\right)}^{3}\right)}^{3}}{\left({\left(\frac{x}{x + 1}\right)}^{6} + {\left(\frac{x + 1}{x - 1}\right)}^{6}\right) + {\left(\frac{x}{x + 1}\right)}^{3} \cdot {\left(\frac{x + 1}{x - 1}\right)}^{3}}}{\frac{x + 1}{x - 1} \cdot \left(\frac{x + 1}{x - 1} + \frac{x}{x + 1}\right) + \frac{x}{x + 1} \cdot \frac{x}{x + 1}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020153 
(FPCore (x)
  :name "Asymptote C"
  :precision binary64
  (- (/ x (+ x 1.0)) (/ (+ x 1.0) (- x 1.0))))