Average Error: 0.5 → 0.3
Time: 8.6s
Precision: 64
Internal Precision: 128
\[\sqrt{x - 1} \cdot \sqrt{x}\]
↓
\[\frac{\frac{-1}{8}}{x} - \left(\frac{1}{2} - x\right)\]

Error

Bits error versus x

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 0.5

    \[\sqrt{x - 1} \cdot \sqrt{x}\]

  2. Taylor expanded around inf 0.3

    \[\leadsto \color{blue}{x - \left(\frac{1}{8} \cdot \frac{1}{x} + \frac{1}{2}\right)}\]

  3. Simplified0.3

    \[\leadsto \color{blue}{\frac{\frac{-1}{8}}{x} - \left(\frac{1}{2} - x\right)}\]

  4. Final simplification0.3

    \[\leadsto \frac{\frac{-1}{8}}{x} - \left(\frac{1}{2} - x\right)\]

Runtime

Time bar (total: 8.6s)Debug logProfile

herbie shell --seed 2018346 +o rules:numerics
(FPCore (x)
  :name "sqrt times"
  (* (sqrt (- x 1)) (sqrt x)))