Average Error: 29.9 → 0.2
Time: 17.9s
Precision: 64
Internal Precision: 1344
\[\sqrt{x + 1} - \sqrt{x}\]
↓
\[\frac{1}{\sqrt{1 + x} + \sqrt{x}}\]

Error

Bits error versus x

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Your Program's Arguments

Results

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Target

Original29.9
Target0.2
Herbie0.2
\[\frac{1}{\sqrt{x + 1} + \sqrt{x}}\]

Derivation

  1. Initial program 29.9

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

  3. Applied flip--29.7

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

  5. Applied add-log-exp31.5

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

  6. Applied add-log-exp30.9

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

  7. Applied diff-log30.9

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

  8. Simplified0.2

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

  9. Final simplification0.2

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

Runtime

Time bar (total: 17.9s)Debug logProfile

herbie shell --seed 2018221 
(FPCore (x)
  :name "2sqrt (example 3.1)"

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

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