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Target

Derivation

1. Initial program 29.8

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

2. Initial simplification29.8

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

4. Applied flip--29.6

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

6. Applied add-log-exp31.5

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

7. Applied add-log-exp31.0

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

8. Applied diff-log31.0

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

9. Simplified0.2

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

10. Final simplification0.2

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

Runtime

Time bar (total: 14.0s)Debug logProfile

herbie shell --seed 2018230 
(FPCore (x)
  :name "2sqrt (example 3.1)"

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

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