Average Error: 30.2 → 0.2
Time: 12.0s
Precision: 64
Internal Precision: 1408
\[\sqrt{x + 1} - \sqrt{x}\]
↓
\[{\left(\left(\sqrt{x + 1} + \sqrt{x}\right) \cdot \left(\sqrt{x + 1} + \sqrt{x}\right)\right)}^{\left(-\frac{1}{2}\right)}\]

Error

Bits error versus x

Target

Original30.2
Target0.2
Herbie0.2
\[\frac{1}{\sqrt{x + 1} + \sqrt{x}}\]

Derivation

  1. Initial program 30.2

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

  3. Applied flip--29.9

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

  4. Applied simplify0.2

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

  6. Applied add-sqr-sqrt0.4

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

  8. Applied pow1/20.4

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

  9. Applied pow1/20.4

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

  10. Applied pow-prod-down0.2

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

  11. Applied pow-flip0.2

    \[\leadsto \color{blue}{{\left(\left(\sqrt{x + 1} + \sqrt{x}\right) \cdot \left(\sqrt{x + 1} + \sqrt{x}\right)\right)}^{\left(-\frac{1}{2}\right)}}\]
  12. Removed slow pow expressions.

Runtime

Time bar (total: 12.0s)Debug logProfile

herbie shell --seed '#(1063282112 2455465480 4141627379 3773598652 1647277307 776739644)' 
(FPCore (x)
  :name "2sqrt (example 3.1)"

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

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