Error

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. Taylor expanded around 0 0.2

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

7. Applied add-sqr-sqrt0.2

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

8. Applied sqrt-prod0.3

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

9. Applied fma-def0.2

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

10. Final simplification0.2

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

Runtime

Time bar (total: 14.9s)Debug logProfile

herbie shell --seed 2018355 +o rules:numerics
(FPCore (x)
  :name "2sqrt (example 3.1)"

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

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