Asymptote C

Average Error: 28.7 → 0.1

Time: 13.2s

Precision: binary64

• Falling back on racket egraph because egg-herbie package not installed

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -0.999978910091019357 \lor \neg \left(x \le 12520.5956478627577\right):\\ \;\;\;\;\left(-\frac{1}{{x}^{2}}\right) - \left(\frac{3}{{x}^{3}} + \frac{3}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{\sqrt{x + 1}}}{\sqrt{x + 1}} - \frac{x + 1}{x - 1}\\ \end{array}\]

Error

Bits error versus x

Derivation

1. Split input into 2 regimes

2. if x < -0.999978910091019357 or 12520.5956478627577 < x

   1. Initial program 58.7

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

   2. Taylor expanded around inf 0.5

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

   3. Simplified 0.2

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

   if -0.999978910091019357 < x < 12520.5956478627577

   1. Initial program 0.1

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

   3. Applied add-sqr-sqrt 0.1

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

   4. Applied associate-/r* 0.1

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

4. Final simplification 0.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.999978910091019357 \lor \neg \left(x \le 12520.5956478627577\right):\\ \;\;\;\;\left(-\frac{1}{{x}^{2}}\right) - \left(\frac{3}{{x}^{3}} + \frac{3}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{\sqrt{x + 1}}}{\sqrt{x + 1}} - \frac{x + 1}{x - 1}\\ \end{array}\]

Reproduce

herbie shell --seed 2020182 
(FPCore (x)
  :name "Asymptote C"
  :precision binary64
  (- (/ x (+ x 1.0)) (/ (+ x 1.0) (- x 1.0))))