Error

Target

Derivation

1. Initial program 30.3

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

3. Applied flip--30.1

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

4. Taylor expanded around inf 0.2

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

6. Applied add-sqr-sqrt0.3

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

7. Applied fma-def0.2

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

8. Final simplification0.2

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

Reproduce

herbie shell --seed 2019090 +o rules:numerics
(FPCore (x)
  :name "2sqrt (example 3.1)"

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

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