Input Error: 29.6b
Output Error: 0.2b
Time: 7.8s
Precision: 64b
Ground Truth: 128b
\[\sqrt{x + 1} - \sqrt{x}\]
⬇
\[\frac{1}{\sqrt{x + 1} + \sqrt{x}}\]

Error

Bits error versus x

Derivation

  1. Initial program 29.6b

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

  3. Applied flip-- 29.4b

    \[\leadsto \color{blue}{\frac{{\left(\sqrt{x + 1}\right)}^2 - {\left(\sqrt{x}\right)}^2}{\sqrt{x + 1} + \sqrt{x}}}\]

  4. Applied simplify 0.2b

    \[\leadsto \frac{\color{blue}{1}}{\sqrt{x + 1} + \sqrt{x}}\]
  5. Removed slow pow expressions

Runtime

Total time: 7.8s Debug log

herbie --seed '#(1359497142 1730085586 643824324 1835989010 333004570 216197836)'
(FPCore (x)
  :name "NMSE example 3.1"
  
  :target
  (/ 1 (+ (sqrt (+ x 1)) (sqrt x)))(- (sqrt (+ x 1)) (sqrt x)))