Average Error: 29.5 → 0.0
Time: 4.6m
Precision: 64
Internal Precision: 1344
\[\frac{x}{x + 1} - \frac{x + 1}{x - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -15672.067303827991:\\ \;\;\;\;(\left(\frac{3}{x} + 1\right) \cdot \left(\frac{-1}{x \cdot x}\right) + \left(\frac{-3}{x}\right))_*\\ \mathbf{if}\;x \le 10068.191697185797:\\ \;\;\;\;(\left(\frac{x}{x \cdot x - 1}\right) \cdot \left(x - 1\right) + \left(-\frac{x + 1}{x - 1}\right))_*\\ \mathbf{else}:\\ \;\;\;\;(\left(\frac{3}{x} + 1\right) \cdot \left(\frac{-1}{x \cdot x}\right) + \left(\frac{-3}{x}\right))_*\\ \end{array}\]

Error

Bits error versus x

Derivation

  1. Split input into 2 regimes

  2. if x < -15672.067303827991 or 10068.191697185797 < x

    1. Initial program 59.3

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

    2. Taylor expanded around inf 0.3

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

    3. Applied simplify0.0

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

    if -15672.067303827991 < x < 10068.191697185797

    1. Initial program 0.1

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

    3. Applied flip-+0.1

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

    4. Applied associate-/r/0.1

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

    5. Applied fma-neg0.1

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

  4. Applied simplify0.0

    \[\leadsto \color{blue}{\begin{array}{l} \mathbf{if}\;x \le -15672.067303827991:\\ \;\;\;\;(\left(\frac{3}{x} + 1\right) \cdot \left(\frac{-1}{x \cdot x}\right) + \left(\frac{-3}{x}\right))_*\\ \mathbf{if}\;x \le 10068.191697185797:\\ \;\;\;\;(\left(\frac{x}{x \cdot x - 1}\right) \cdot \left(x - 1\right) + \left(-\frac{x + 1}{x - 1}\right))_*\\ \mathbf{else}:\\ \;\;\;\;(\left(\frac{3}{x} + 1\right) \cdot \left(\frac{-1}{x \cdot x}\right) + \left(\frac{-3}{x}\right))_*\\ \end{array}}\]

Runtime

Time bar (total: 4.6m)Debug logProfile

herbie shell --seed '#(1071373924 2949776965 1885069702 3247780810 90874544 2263903749)' +o rules:numerics
(FPCore (x)
  :name "Asymptote C"
  (- (/ x (+ x 1)) (/ (+ x 1) (- x 1))))