Asymptote C

Average Error: 29.1 → 0.1

Time: 5.4s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -9873.4280108723797 \lor \neg \left(x \le 11886.148758601244\right):\\ \;\;\;\;\left(\frac{-1}{{x}^{2}} - \frac{3}{x}\right) - \frac{3}{{x}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\log \left(\sqrt{e^{\frac{x}{x + 1} - \frac{x + 1}{x - 1}}}\right) + \log \left(\sqrt{e^{\frac{x}{x + 1} - \frac{x + 1}{x - 1}}}\right)\\ \end{array}\]

Error

Bits error versus x

Derivation

1. Split input into 2 regimes

2. if x < -9873.4280108723797 or 11886.148758601244 < 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 -9873.4280108723797 < x < 11886.148758601244

   1. Initial program 0.1

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

   3. Applied add-log-exp 0.1

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

   4. Applied add-log-exp 0.1

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

   5. Applied diff-log 0.1

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

   6. Simplified 0.1

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

   8. Applied add-sqr-sqrt 0.1

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

   9. Applied log-prod 0.1

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

4. Final simplification 0.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -9873.4280108723797 \lor \neg \left(x \le 11886.148758601244\right):\\ \;\;\;\;\left(\frac{-1}{{x}^{2}} - \frac{3}{x}\right) - \frac{3}{{x}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\log \left(\sqrt{e^{\frac{x}{x + 1} - \frac{x + 1}{x - 1}}}\right) + \log \left(\sqrt{e^{\frac{x}{x + 1} - \frac{x + 1}{x - 1}}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020161 
(FPCore (x)
  :name "Asymptote C"
  :precision binary64
  (- (/ x (+ x 1.0)) (/ (+ x 1.0) (- x 1.0))))