Average Error: 29.5 → 0.0
Time: 1.2m
Precision: 64
\[\frac{x}{x + 1} - \frac{x + 1}{x - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -12964.802277507168:\\ \;\;\;\;\frac{-1}{x \cdot x} + \left(\frac{-3}{\left(x \cdot x\right) \cdot x} + \frac{-3}{x}\right)\\ \mathbf{elif}\;x \le 13345.433606935452:\\ \;\;\;\;\frac{\left(x - 1\right) \cdot x - \left(1 + x\right) \cdot \left(1 + x\right)}{\left(x - 1\right) \cdot \left(1 + x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{x \cdot x} + \left(\frac{-3}{\left(x \cdot x\right) \cdot x} + \frac{-3}{x}\right)\\ \end{array}\]

Error

Bits error versus x

Derivation

  1. Split input into 2 regimes

  2. if x < -12964.802277507168 or 13345.433606935452 < x

    1. Initial program 59.4

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

    2. Taylor expanded around -inf 0.3

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

    3. Simplified0.3

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

    4. Taylor expanded around -inf 0.3

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

    5. Simplified0.0

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

    if -12964.802277507168 < x < 13345.433606935452

    1. Initial program 0.1

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

    3. Applied frac-sub0.1

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

  4. Final simplification0.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -12964.802277507168:\\ \;\;\;\;\frac{-1}{x \cdot x} + \left(\frac{-3}{\left(x \cdot x\right) \cdot x} + \frac{-3}{x}\right)\\ \mathbf{elif}\;x \le 13345.433606935452:\\ \;\;\;\;\frac{\left(x - 1\right) \cdot x - \left(1 + x\right) \cdot \left(1 + x\right)}{\left(x - 1\right) \cdot \left(1 + x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{x \cdot x} + \left(\frac{-3}{\left(x \cdot x\right) \cdot x} + \frac{-3}{x}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019100 
(FPCore (x)
  :name "Asymptote C"
  (- (/ x (+ x 1)) (/ (+ x 1) (- x 1))))