Average Error: 30.1 → 0.2
Time: 9.9s
Precision: 64
Ground Truth: 128
\[\sqrt{x + 1} - \sqrt{x}\]
⬇
\[\frac{1}{\sqrt{x + 1} + \sqrt{x}}\]

Error

Bits error versus x

Target

Original30.1
Comparison0.2
Herbie0.2
\[ \frac{1}{\sqrt{x + 1} + \sqrt{x}} \]

Derivation

  1. Initial program 30.1

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

  3. Applied flip-- 29.8

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

  4. Applied simplify 0.2

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

Runtime

Total time: 9.9s Debug log

Please report a bug with the following info:

herbie --seed '#(792327344 2201558356 1278027338 1418945471 2048560777 3261111377)'
(FPCore (x)
  :name "NMSE example 3.1"

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

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