Asymptote C

Average Error: 29.2 → 0.1

Time: 3.1s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -13348.764274700028:\\ \;\;\;\;\frac{-1}{x \cdot x} - \left(\frac{3}{x} + \frac{3}{{x}^{3}}\right)\\ \mathbf{elif}\;x \le 15449.082165169788:\\ \;\;\;\;x \cdot \frac{1}{x + 1} - \frac{x + 1}{x - 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-5}{x \cdot x} - \left(\frac{6}{x} + \frac{16}{{x}^{3}}\right)}{\frac{x + 1}{x - 1} + \frac{x}{x + 1}}\\ \end{array}\]

Error

Bits error versus x

Derivation

1. Split input into 3 regimes

2. if x < -13348.764274700028

   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 0.0

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

   if -13348.764274700028 < x < 15449.082165169788

   1. Initial program 0.1

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

   3. Applied div-inv 0.1

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

   if 15449.082165169788 < x

   1. Initial program 59.3

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

   3. Applied flip-- 59.4

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

   4. Simplified 59.3

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

   5. Taylor expanded around inf 0.3

      \[\leadsto \frac{\color{blue}{-\left(5 \cdot \frac{1}{{x}^{2}} + \left(6 \cdot \frac{1}{x} + 16 \cdot \frac{1}{{x}^{3}}\right)\right)}}{\frac{x}{x + 1} + \frac{x + 1}{x - 1}}\]

   6. Simplified 0.0

      \[\leadsto \frac{\color{blue}{\frac{-5}{x \cdot x} - \left(\frac{6}{x} + \frac{16}{{x}^{3}}\right)}}{\frac{x}{x + 1} + \frac{x + 1}{x - 1}}\]
3. Recombined 3 regimes into one program.

4. Final simplification 0.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -13348.764274700028:\\ \;\;\;\;\frac{-1}{x \cdot x} - \left(\frac{3}{x} + \frac{3}{{x}^{3}}\right)\\ \mathbf{elif}\;x \le 15449.082165169788:\\ \;\;\;\;x \cdot \frac{1}{x + 1} - \frac{x + 1}{x - 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-5}{x \cdot x} - \left(\frac{6}{x} + \frac{16}{{x}^{3}}\right)}{\frac{x + 1}{x - 1} + \frac{x}{x + 1}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020184 
(FPCore (x)
  :name "Asymptote C"
  :precision binary64
  (- (/ x (+ x 1.0)) (/ (+ x 1.0) (- x 1.0))))