Average Error: 30.0 → 0.3
Time: 18.5s
Precision: 64
Internal Precision: 1408
\[\sqrt{x + 1} - \sqrt{x}\]
↓
\[{\left(\frac{1}{\left(\sqrt{x + 1} + \sqrt{x}\right) \cdot \left(\sqrt{x + 1} + \sqrt{x}\right)}\right)}^{\frac{1}{2}}\]

Error

Bits error versus x

Target

Original30.0
Target0.2
Herbie0.3
\[\frac{1}{\sqrt{x + 1} + \sqrt{x}}\]

Derivation

  1. Initial program 30.0

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

  3. Applied flip--29.8

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

  4. Applied simplify0.2

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

  6. Applied add-sqr-sqrt0.3

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

  8. Applied pow1/20.3

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

  9. Applied pow1/20.3

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

  10. Applied pow-prod-down0.2

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

  11. Applied simplify0.3

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

Runtime

Time bar (total: 18.5s)Debug logProfile

herbie shell --seed '#(1070386091 2509006183 1430610344 1025408621 36622005 1425925650)' +o rules:numerics
(FPCore (x)
  :name "2sqrt (example 3.1)"

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

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