Average Error: 9.6 → 0.3
Time: 29.0s
Precision: 64
Internal Precision: 1152
\[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}\]
↓
\[\frac{2}{(x \cdot x + x)_* \cdot \left(x - 1\right)}\]

Error

Bits error versus x

Target

Original9.6
Target0.3
Herbie0.3
\[\frac{2}{x \cdot \left(x \cdot x - 1\right)}\]

Derivation

  1. Initial program 9.6

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

  3. Applied frac-sub25.9

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

  4. Applied frac-add25.2

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

  5. Applied simplify25.6

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

  6. Applied simplify25.6

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

  7. Taylor expanded around 0 0.3

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

Runtime

Time bar (total: 29.0s)Debug logProfile

herbie shell --seed '#(1064173506 2580572819 2847706409 4129882574 1125180799 1845288547)' +o rules:numerics
(FPCore (x)
  :name "3frac (problem 3.3.3)"

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

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