Average Error: 18.9 → 0.3
Time: 25.8s
Precision: 64
Internal precision: 1152
\[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}\]
⬇
\[\frac{\frac{1}{1 + x}}{\frac{x}{\sqrt{x}} + \frac{x}{\sqrt{1 + x}}}\]

Error

Bits error versus x

Target

Original18.9
Comparison0.6
Herbie0.3
\[ \frac{1}{\left(x + 1\right) \cdot \sqrt{x} + x \cdot \sqrt{x + 1}} \]

Derivation

  1. Initial program 18.9

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

  3. Applied flip-- 19.0

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

  4. Applied simplify 19.0

    \[\leadsto \frac{\color{blue}{\frac{1}{x} - \frac{1}{1 + x}}}{\frac{1}{\sqrt{x}} + \frac{1}{\sqrt{x + 1}}}\]
  5. Using strategy rm

  6. Applied frac-sub 18.4

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

  7. Applied simplify 5.5

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

  8. Applied taylor 5.5

    \[\leadsto \frac{\frac{1}{x + {x}^2}}{\frac{1}{\sqrt{x}} + \frac{1}{\sqrt{x + 1}}}\]

  9. Taylor expanded around 0 5.5

    \[\leadsto \frac{\frac{1}{\color{blue}{x + {x}^2}}}{\frac{1}{\sqrt{x}} + \frac{1}{\sqrt{x + 1}}}\]

  10. Applied simplify 0.3

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

Runtime

Time bar (total: 25.8s) Debug log

Please include this information when filing a bug report:

herbie shell --seed '#(3283856077 3183919125 3399458751 396155847 3688194147 1862413033)'
(FPCore (x)
  :name "2isqrt (example 3.6)"

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

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