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Target

Derivation

1. Initial program 29.5

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

2. Initial simplification29.5

   \[\leadsto \sqrt{1 + x} - \sqrt{x}\]
3. Using strategy rm

4. Applied flip--29.3

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

6. Applied *-un-lft-identity29.3

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

7. Applied *-un-lft-identity29.3

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

8. Applied distribute-lft-out29.3

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

9. Applied *-un-lft-identity29.3

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

10. Applied times-frac29.3

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

11. Simplified29.3

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

12. Simplified0.2

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

13. Final simplification0.2

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

Runtime

Time bar (total: 36.8s)Debug logProfile

herbie shell --seed 2018225 
(FPCore (x)
  :name "2sqrt (example 3.1)"

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

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