Error

Target

Derivation

1. Initial program 19.4

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

3. Applied flip--19.5

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

4. Applied simplify19.5

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

6. Applied frac-sub18.9

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

7. Applied simplify5.6

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

8. Applied simplify5.6

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

10. Applied add-sqr-sqrt5.6

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

11. Applied associate-/r*5.6

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

Runtime

Time bar (total: 53.1s)Debug logProfile

herbie shell --seed '#(1070386091 2509006183 1430610344 1025408621 36622005 1425925650)' +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)))))