Average Error: 29.8 → 0.3
Time: 35.2s
Precision: 64
Internal Precision: 1344
\[\sqrt{x + 1} - \sqrt{x}\]
↓
\[{\left(\sqrt{\sqrt{x + 1} + \sqrt{x}}\right)}^{\left(-1 - 1\right)}\]

Error

Bits error versus x

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original29.8
Target0.2
Herbie0.3
\[\frac{1}{\sqrt{x + 1} + \sqrt{x}}\]

Derivation

  1. Initial program 29.8

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

  3. Applied flip--29.6

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

  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. Applied associate-/r*0.3

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

  9. Applied pow10.3

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

  10. Applied inv-pow0.3

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

  11. Applied pow-div0.3

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

Runtime

Time bar (total: 35.2s)Debug logProfile

herbie shell --seed 2019053 
(FPCore (x)
  :name "2sqrt (example 3.1)"

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

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