Average Error: 29.8 → 0.2
Time: 14.9s
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

Enter valid numbers for all inputs

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. Using strategy rm
  5. Applied add-log-exp31.4

    \[\leadsto \frac{\sqrt{x + 1} \cdot \sqrt{x + 1} - \color{blue}{\log \left(e^{\sqrt{x} \cdot \sqrt{x}}\right)}}{\sqrt{x + 1} + \sqrt{x}}\]
  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: 14.9s)Debug log

herbie shell --seed '#(2775764126 3555076145 3898259844 1891440260 2599947619 1948460636)' 
(FPCore (x)
  :name "sqrt(x+1) - sqrt(x)"
  (- (sqrt (+ x 1)) (sqrt x)))