Average Error: 30.2 → 0.2
Time: 48.3s
Precision: 64
Internal Precision: 1344
\[\sqrt{x + 1} - \sqrt{x}\]
↓
\[\frac{1}{\sqrt{x + 1} + \sqrt{x}}\]

Error

Bits error versus x

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Results

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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--30.0

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

  5. Applied prod-diff30.2

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

  6. Simplified30.3

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

  7. Simplified0.2

    \[\leadsto \frac{1 + \color{blue}{0}}{\sqrt{x + 1} + \sqrt{x}}\]

  8. Final simplification0.2

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

Runtime

Time bar (total: 48.3s)Debug logProfile

herbie shell --seed 2018219 +o rules:numerics
(FPCore (x)
  :name "2sqrt (example 3.1)"

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

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