Error

Derivation

1. Initial program 0.8

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

3. Applied p16-flip--0.6

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

5. Applied sqrt-sqrd.p160.5

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

7. Applied sqrt-sqrd.p160.4

   \[\leadsto \frac{\left(\left(\frac{x}{\left(real->posit(1)\right)}\right) - \color{blue}{x}\right)}{\left(\frac{\left(\sqrt{\left(\frac{x}{\left(real->posit(1)\right)}\right)}\right)}{\left(\sqrt{x}\right)}\right)}\]
8. Using strategy rm

9. Applied associate--l+0.3

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

10. Final simplification0.3

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

Reproduce

herbie shell --seed 2019094 +o rules:numerics
(FPCore (x)
  :name "2sqrt (example 3.1)"
  (-.p16 (sqrt.p16 (+.p16 x (real->posit16 1))) (sqrt.p16 x)))