Error

Target

Derivation

1. Initial program 19.5

   \[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}\]
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 simplify19.6

   \[\leadsto \frac{\color{blue}{\frac{1}{x} - \frac{1}{x + 1}}}{\frac{1}{\sqrt{x}} + \frac{1}{\sqrt{x + 1}}}\]
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 simplify5.3

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

8. Applied simplify5.3

   \[\leadsto \frac{\frac{1 - 0}{\color{blue}{(x \cdot x + x)_*}}}{\frac{1}{\sqrt{x}} + \frac{1}{\sqrt{x + 1}}}\]
9. Using strategy rm

10. Applied *-un-lft-identity5.3

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

11. Applied add-sqr-sqrt5.3

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

12. Applied add-cube-cbrt5.3

    \[\leadsto \frac{\frac{\color{blue}{\left(\sqrt[3]{1 - 0} \cdot \sqrt[3]{1 - 0}\right) \cdot \sqrt[3]{1 - 0}}}{\sqrt{(x \cdot x + x)_*} \cdot \sqrt{(x \cdot x + x)_*}}}{1 \cdot \left(\frac{1}{\sqrt{x}} + \frac{1}{\sqrt{x + 1}}\right)}\]

13. Applied times-frac5.3

    \[\leadsto \frac{\color{blue}{\frac{\sqrt[3]{1 - 0} \cdot \sqrt[3]{1 - 0}}{\sqrt{(x \cdot x + x)_*}} \cdot \frac{\sqrt[3]{1 - 0}}{\sqrt{(x \cdot x + x)_*}}}}{1 \cdot \left(\frac{1}{\sqrt{x}} + \frac{1}{\sqrt{x + 1}}\right)}\]

14. Applied times-frac5.2

    \[\leadsto \color{blue}{\frac{\frac{\sqrt[3]{1 - 0} \cdot \sqrt[3]{1 - 0}}{\sqrt{(x \cdot x + x)_*}}}{1} \cdot \frac{\frac{\sqrt[3]{1 - 0}}{\sqrt{(x \cdot x + x)_*}}}{\frac{1}{\sqrt{x}} + \frac{1}{\sqrt{x + 1}}}}\]

15. Applied simplify5.2

    \[\leadsto \color{blue}{\left(\frac{\sqrt[3]{1 - 0}}{\sqrt{(x \cdot x + x)_*}} \cdot \sqrt[3]{1 - 0}\right)} \cdot \frac{\frac{\sqrt[3]{1 - 0}}{\sqrt{(x \cdot x + x)_*}}}{\frac{1}{\sqrt{x}} + \frac{1}{\sqrt{x + 1}}}\]

Runtime

Time bar (total: 57.0s)Debug logProfile

herbie shell --seed '#(1070131407 1246090267 3027482374 2150728003 2026520792 2347815650)' +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)))))