Asymptote C

Average Error: 29.2 → 0.1

Time: 2.6s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -11102.538296250892 \lor \neg \left(x \le 14549.2157139929786\right):\\ \;\;\;\;\frac{-\left(3 + \left(\frac{3}{x \cdot x} + \frac{1}{x}\right)\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x \cdot x - 1 \cdot 1} \cdot \left(x - 1\right) - \frac{x + 1}{x - 1}\\ \end{array}\]

Error

Bits error versus x

Derivation

1. Split input into 2 regimes

2. if x < -11102.538296250892 or 14549.2157139929786 < x

   1. Initial program 59.3

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

   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 0.3

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

   5. Applied associate-*l/ 0.0

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

   6. Simplified 0.0

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

   if -11102.538296250892 < x < 14549.2157139929786

   1. Initial program 0.1

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

   3. Applied flip-+ 0.1

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

   4. Applied associate-/r/ 0.1

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

4. Final simplification 0.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -11102.538296250892 \lor \neg \left(x \le 14549.2157139929786\right):\\ \;\;\;\;\frac{-\left(3 + \left(\frac{3}{x \cdot x} + \frac{1}{x}\right)\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x \cdot x - 1 \cdot 1} \cdot \left(x - 1\right) - \frac{x + 1}{x - 1}\\ \end{array}\]

Reproduce

herbie shell --seed 2020181 
(FPCore (x)
  :name "Asymptote C"
  :precision binary64
  (- (/ x (+ x 1.0)) (/ (+ x 1.0) (- x 1.0))))