Asymptote C

Average Error: 29.4 → 0.2

Time: 3.4s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{x}{x + 1} - \frac{x + 1}{x - 1} \le 8.4395697880213 \cdot 10^{-6}:\\ \;\;\;\;-\left(\frac{3}{x} + \frac{1 + \frac{3}{x}}{x \cdot x}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{1}{x + 1} - \frac{x + 1}{x - 1}\\ \end{array}\]

Error

Bits error versus x

Derivation

1. Split input into 2 regimes

2. if (- (/ x (+ x 1.0)) (/ (+ x 1.0) (- x 1.0))) < 8.4395697880213e-6

   1. Initial program 58.9

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

   3. Applied add-log-exp 58.9

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

   4. Applied add-log-exp 58.9

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

   5. Applied diff-log 58.9

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

   6. Simplified 58.9

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

   7. Taylor expanded around inf 0.6

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

   8. Simplified 0.3

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

   if 8.4395697880213e-6 < (- (/ x (+ x 1.0)) (/ (+ x 1.0) (- x 1.0)))

   1. Initial program 0.1

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

   3. Applied div-inv 0.1

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

4. Final simplification 0.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{x + 1} - \frac{x + 1}{x - 1} \le 8.4395697880213 \cdot 10^{-6}:\\ \;\;\;\;-\left(\frac{3}{x} + \frac{1 + \frac{3}{x}}{x \cdot x}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{1}{x + 1} - \frac{x + 1}{x - 1}\\ \end{array}\]

Reproduce

herbie shell --seed 2020179 
(FPCore (x)
  :name "Asymptote C"
  :precision binary64
  (- (/ x (+ x 1.0)) (/ (+ x 1.0) (- x 1.0))))