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In Handbook of Floating-Point Arithmetic (Birkhäuser, 2010, Chapter 6) Muller et al. presented the following absolute forward error bound for the floating-point recursive dot product: $$ \left|RecursiveDot (a, b) - \sum_{i=1}^n a_ib_i\right| \le \gamma_n \sum_{i=1}^n |a_ib_i|, $$ where $RecursiveDot (a, b)$ is the result obtained in floating-point arithmetic, $\sum_{i=1}^n a_ib_i$ is the exact result, and $\gamma_n \approx n\mathbf{u}$. Here $\mathbf{u}$ is the unit roundoff: $\mathbf{u} = 1/2\cdot 2^{1-p}$ in round-to-nearest mode, and $\mathbf{u} = 2^{1-p}$ in the directed rounding modes (round up, round down, etc.), $p$ is the precision.

Suppose that there are two values:

  • $\underline{RecursiveDot(a,b)}$ is the floating-point recursive dot product computed with rounding down;
  • $\overline{RecursiveDot(a,b)}$ is the floating-point recursive dot product computed with rounding up.

I cannot understand, what expression I should use to estimate the maximum difference between these two values: $$ \Biggl| \overline{RecursiveDot (a, b)} - \underline{RecursiveDot (a, b)}\Biggr| \le \gamma_n \sum_{i=1}^n |a_ib_i| $$ or $$ \Biggl| \overline{RecursiveDot (a, b)} - \underline{RecursiveDot (a, b)}\Biggr| \le \mathbf{2}\gamma_n \sum_{i=1}^n |a_ib_i|. $$

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