Average Error: 0.6 → 0.6
Time: 17.8s
Precision: 64
Internal Precision: 320
\[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}\]
↓
\[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}\]

Error

Bits error versus x

Derivation

  1. Initial program 0.6

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

  2. Final simplification0.6

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

Reproduce

herbie shell --seed 2019094 +o rules:numerics
(FPCore (x)
  :name "2isqrt (example 3.6)"
  (-.p16 (/.p16 (real->posit16 1) (sqrt.p16 x)) (/.p16 (real->posit16 1) (sqrt.p16 (+.p16 x (real->posit16 1))))))