Error

Derivation

1. Initial program 0.8

   \[\left(\sqrt{\left(\frac{x}{\left(1\right)}\right)}\right) - \left(\sqrt{x}\right)\]
2. Using strategy rm

3. Applied p16-flip--0.6

   \[\leadsto \color{blue}{\frac{\left(\left(\left(\sqrt{\left(\frac{x}{\left(1\right)}\right)}\right) \cdot \left(\sqrt{\left(\frac{x}{\left(1\right)}\right)}\right)\right) - \left(\left(\sqrt{x}\right) \cdot \left(\sqrt{x}\right)\right)\right)}{\left(\frac{\left(\sqrt{\left(\frac{x}{\left(1\right)}\right)}\right)}{\left(\sqrt{x}\right)}\right)}}\]
4. Using strategy rm

5. Applied p16-flip--1.0

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

6. Applied associate-/l/1.0

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

7. Simplified0.5

   \[\leadsto \frac{\color{blue}{\left(\left(\frac{\left(0.0\right)}{\left(1\right)}\right) \cdot \left(\frac{\left(\frac{x}{x}\right)}{\left(1\right)}\right)\right)}}{\left(\left(\frac{\left(\sqrt{\left(\frac{x}{\left(1\right)}\right)}\right)}{\left(\sqrt{x}\right)}\right) \cdot \left(\frac{\left(\left(\sqrt{\left(\frac{x}{\left(1\right)}\right)}\right) \cdot \left(\sqrt{\left(\frac{x}{\left(1\right)}\right)}\right)\right)}{\left(\left(\sqrt{x}\right) \cdot \left(\sqrt{x}\right)\right)}\right)\right)}\]

8. Simplified0.2

   \[\leadsto \color{blue}{\left(\frac{\left(1\right)}{\left(\frac{\left(\sqrt{\left(\frac{\left(1\right)}{x}\right)}\right)}{\left(\sqrt{x}\right)}\right)}\right) \cdot \left(1.0\right)}\]

9. Final simplification0.2

   \[\leadsto \frac{1}{\sqrt{1 + x} + \sqrt{x}} \cdot 1.0\]

Reproduce

herbie shell --seed 2019100 +o rules:numerics
(FPCore (x)
  :name "2sqrt (example 3.1)"
  (-.p16 (sqrt.p16 (+.p16 x (real->posit16 1))) (sqrt.p16 x)))