Average Error: 10.2 → 0.1
Time: 1.5m
Precision: 64
Internal precision: 1152
\[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}\]
⬇
\[\frac{\frac{2}{\left(x + 1\right) \cdot x}}{x - 1}\]

Error

Bits error versus x

Target

Original10.2
Comparison0.3
Herbie0.1
\[ \frac{2}{x \cdot \left(x \cdot x - 1\right)} \]

Derivation

  1. Initial program 10.2

    \[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}\]
  2. Using strategy rm

  3. Applied frac-sub 26.7

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

  4. Applied frac-add 25.9

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

  5. Applied simplify 26.0

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

  6. Taylor expanded around 0 0.3

    \[\leadsto \frac{\color{blue}{2}}{\left(\left(x + 1\right) \cdot x\right) \cdot \left(x - 1\right)}\]
  7. Using strategy rm

  8. Applied associate-/r* 0.1

    \[\leadsto \color{blue}{\frac{\frac{2}{\left(x + 1\right) \cdot x}}{x - 1}}\]
  9. Removed slow pow expressions

Runtime

Time bar (total: 1.5m) Debug log

Please include this information when filing a bug report:

herbie shell --seed '#(2329929097 3210370195 3111198779 2406363002 3511342718 2136436390)'
(FPCore (x)
  :name "3frac (problem 3.3.3)"

  :herbie-target
  (/ 2 (* x (- (* x x) 1)))

  (+ (- (/ 1 (+ x 1)) (/ 2 x)) (/ 1 (- x 1))))