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Target

Derivation

1. Initial program 19.6

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

3. Applied frac-sub19.6

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

4. Applied simplify19.6

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

6. Applied flip--19.4

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

7. Applied simplify0.4

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

9. Applied *-un-lft-identity0.4

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

10. Applied times-frac0.4

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

12. Applied pow1/20.4

    \[\leadsto \frac{1}{\color{blue}{{x}^{\frac{1}{2}}}} \cdot \frac{\frac{1}{\sqrt{1 + x} + \sqrt{x}}}{\sqrt{x + 1}}\]

13. Applied pow-flip0.2

    \[\leadsto \color{blue}{{x}^{\left(-\frac{1}{2}\right)}} \cdot \frac{\frac{1}{\sqrt{1 + x} + \sqrt{x}}}{\sqrt{x + 1}}\]

Runtime

Time bar (total: 1.2m)Debug logProfile

herbie shell --seed 2020178 
(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)))))