Average Error: 14.9 → 0.1
Time: 33.8s
Precision: 64
Internal Precision: 832
\[\frac{1}{x + 1} - \frac{1}{x - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -249.37112553352097:\\ \;\;\;\;\left(\frac{\frac{-2}{x}}{x} + \frac{-2}{{x}^{6}}\right) - \frac{2}{{x}^{4}}\\ \mathbf{if}\;x \le 225.07246716255815:\\ \;\;\;\;\frac{1}{x + 1} - \frac{1}{x - 1}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\frac{-2}{x}}{x} + \frac{-2}{{x}^{6}}\right) - \frac{2}{{x}^{4}}\\ \end{array}\]

Error

Bits error versus x

Derivation

  1. Split input into 2 regimes

  2. if x < -249.37112553352097 or 225.07246716255815 < x

    1. Initial program 29.8

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

    2. Taylor expanded around inf 0.7

      \[\leadsto \color{blue}{-\left(2 \cdot \frac{1}{{x}^{6}} + \left(2 \cdot \frac{1}{{x}^{4}} + 2 \cdot \frac{1}{{x}^{2}}\right)\right)}\]

    3. Applied simplify0.7

      \[\leadsto \color{blue}{\left(\frac{-2}{x \cdot x} + \frac{-2}{{x}^{6}}\right) - \frac{2}{{x}^{4}}}\]
    4. Using strategy rm

    5. Applied associate-/r*0.1

      \[\leadsto \left(\color{blue}{\frac{\frac{-2}{x}}{x}} + \frac{-2}{{x}^{6}}\right) - \frac{2}{{x}^{4}}\]

    if -249.37112553352097 < x < 225.07246716255815

    1. Initial program 0.0

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

Runtime

Time bar (total: 33.8s)Debug logProfile

herbie shell --seed '#(1071821486 549052472 3784827256 1559736200 3548510075 881134285)' 
(FPCore (x)
  :name "Asymptote A"
  (- (/ 1 (+ x 1)) (/ 1 (- x 1))))