Error

Target

Derivation

1. Initial program 20.1

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

3. Applied flip--20.2

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

4. Applied simplify20.2

   \[\leadsto \frac{\color{blue}{\frac{1}{x} - \frac{1}{x + 1}}}{\frac{1}{\sqrt{x}} + \frac{1}{\sqrt{x + 1}}}\]
5. Using strategy rm

6. Applied frac-sub19.6

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

7. Applied simplify5.7

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

8. Applied simplify5.7

   \[\leadsto \frac{\frac{1}{\color{blue}{(x \cdot x + x)_*}}}{\frac{1}{\sqrt{x}} + \frac{1}{\sqrt{x + 1}}}\]
9. Using strategy rm

10. Applied clear-num5.7

    \[\leadsto \color{blue}{\frac{1}{\frac{\frac{1}{\sqrt{x}} + \frac{1}{\sqrt{x + 1}}}{\frac{1}{(x \cdot x + x)_*}}}}\]

11. Applied simplify0.8

    \[\leadsto \frac{1}{\color{blue}{\frac{\frac{x}{\sqrt{x + 1}} + \frac{x}{\sqrt{x}}}{\frac{1}{x + 1}}}}\]
12. Using strategy rm

13. Applied div-inv0.8

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

14. Applied associate-/r*0.4

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

Runtime

Time bar (total: 1.3m)Debug logProfile

herbie shell --seed '#(1070864556 424010669 783715395 1203517814 4070606583 4107618214)' +o rules:numerics
(FPCore (x)
  :name "2isqrt (example 3.6)"

  :herbie-target
  (/ 1 (+ (* (+ x 1) (sqrt x)) (* x (sqrt (+ x 1)))))

  (- (/ 1 (sqrt x)) (/ 1 (sqrt (+ x 1)))))