Average Error: 19.8 → 0.5
Time: 52.2s
Precision: 64
Internal Precision: 1088
\[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}\]
↓
\[\frac{\sqrt{\frac{1}{\sqrt{x + 1} + \sqrt{x}}}}{\sqrt{x}} \cdot \frac{\sqrt{\frac{1}{\sqrt{x + 1} + \sqrt{x}}}}{\sqrt{x + 1}}\]

Error

Bits error versus x

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original19.8
Target0.7
Herbie0.5
\[\frac{1}{\left(x + 1\right) \cdot \sqrt{x} + x \cdot \sqrt{x + 1}}\]

Derivation

  1. Initial program 19.8

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

  3. Applied frac-sub19.8

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

  4. Applied simplify19.8

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

  6. Applied flip--19.6

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

  7. Applied simplify0.4

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

  9. Applied add-sqr-sqrt0.5

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

  10. Applied times-frac0.5

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

Runtime

Time bar (total: 52.2s)Debug logProfile

herbie shell --seed '#(1072936661 1621281212 3440817831 3219514234 460296804 1258167384)' +o rules:numerics
(FPCore (x)
  :name "2isqrt (example 3.6)"

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

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