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Derivation

1. Initial program 14.5

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

2. Initial simplification13.9

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

4. Applied *-un-lft-identity13.9

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

5. Applied times-frac13.9

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

6. Simplified0.1

   \[\leadsto \frac{1}{x + 1} \cdot \color{blue}{\frac{0 - 1}{x}}\]
7. Using strategy rm

8. Applied pow10.1

   \[\leadsto \frac{1}{x + 1} \cdot \color{blue}{{\left(\frac{0 - 1}{x}\right)}^{1}}\]

9. Applied pow10.1

   \[\leadsto \color{blue}{{\left(\frac{1}{x + 1}\right)}^{1}} \cdot {\left(\frac{0 - 1}{x}\right)}^{1}\]

10. Applied pow-prod-down0.1

    \[\leadsto \color{blue}{{\left(\frac{1}{x + 1} \cdot \frac{0 - 1}{x}\right)}^{1}}\]

11. Simplified0.1

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

12. Final simplification0.1

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

Runtime

Time bar (total: 1.6m)Debug logProfile

herbie shell --seed 2018214 +o rules:numerics
(FPCore (x)
  :name "2frac (problem 3.3.1)"
  (- (/ 1 (+ x 1)) (/ 1 x)))