sqrt B

Average Error: 30.2 → 0.4

Time: 2.2s

Precision: binary64

Math

\[\sqrt{\left(2 \cdot x\right) \cdot x}\]

↓

\[\begin{array}{l} \mathbf{if}\;x \le 2.73314458313503 \cdot 10^{-311}:\\ \;\;\;\;x \cdot \left(-\sqrt{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x \cdot 2} \cdot \sqrt{x}\\ \end{array}\]

Error

Bits error versus x

Derivation

1. Split input into 2 regimes

2. if x < 2.73314458313503e-311

   1. Initial program 30.6

      \[\sqrt{\left(2 \cdot x\right) \cdot x}\]

   2. Taylor expanded around -inf 0.4

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

   3. Simplified 0.4

      \[\leadsto \color{blue}{x \cdot \left(-\sqrt{2}\right)}\]

   if 2.73314458313503e-311 < x

   1. Initial program 29.8

      \[\sqrt{\left(2 \cdot x\right) \cdot x}\]
   2. Using strategy rm

   3. Applied sqrt-prod 0.4

      \[\leadsto \color{blue}{\sqrt{2 \cdot x} \cdot \sqrt{x}}\]
3. Recombined 2 regimes into one program.

4. Final simplification 0.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le 2.73314458313503 \cdot 10^{-311}:\\ \;\;\;\;x \cdot \left(-\sqrt{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x \cdot 2} \cdot \sqrt{x}\\ \end{array}\]

Reproduce

herbie shell --seed 2020184 
(FPCore (x)
  :name "sqrt B"
  :precision binary64
  (sqrt (* (* 2.0 x) x)))