Average Error: 14.5 → 0.1
Time: 22.5s
Precision: 64
Internal Precision: 896
\[\frac{1}{x + 1} - \frac{1}{x}\]
↓
\[\frac{\frac{-1}{1 + x}}{x}\]

Error

Bits error versus x

Derivation

  1. Initial program 14.5

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

  3. Applied frac-sub13.8

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

  4. Applied simplify13.8

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

  6. Applied associate-/r*13.8

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

  7. Applied simplify0.1

    \[\leadsto \frac{\color{blue}{\frac{0 - 1}{1 + x}}}{x}\]

  8. Applied simplify0.1

    \[\leadsto \frac{\frac{\color{blue}{-1}}{1 + x}}{x}\]
  9. Removed slow pow expressions.

Runtime

Time bar (total: 22.5s)Debug log

herbie shell --seed '#(1567391828 2030694642 2833800258 828025724 3004380912 3532991858)' +o setup:early-exit +o reduce:binary-search
(FPCore (x)
  :name "2frac (problem 3.3.1)"
  (- (/ 1 (+ x 1)) (/ 1 x)))