sqrt A

Average Error: 30.6 → 0.3

Time: 1.8s

Precision: binary64

Math

\[\sqrt{x \cdot x + x \cdot x}\]

↓

\[\begin{array}{l} \mathbf{if}\;x \le 1.817538864223962 \cdot 10^{-310}:\\ \;\;\;\;\sqrt[3]{\sqrt{2}} \cdot \left(x \cdot \left(\sqrt[3]{\sqrt{2}} \cdot \left(-\sqrt[3]{\sqrt{2}}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot \sqrt{x + x}\\ \end{array}\]

Error

Bits error versus x

Derivation

1. Split input into 2 regimes

2. if x < 1.817538864223962e-310

   1. Initial program 30.6

      \[\sqrt{x \cdot x + x \cdot x}\]

   2. Simplified 30.6

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

   3. Taylor expanded around -inf 0.4

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

   4. Simplified 0.4

      \[\leadsto \color{blue}{x \cdot \left(-\sqrt{2}\right)}\]
   5. Using strategy rm

   6. Applied add-cube-cbrt 0.4

      \[\leadsto x \cdot \left(-\color{blue}{\left(\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}\right) \cdot \sqrt[3]{\sqrt{2}}}\right)\]

   7. Applied distribute-lft-neg-in 0.4

      \[\leadsto x \cdot \color{blue}{\left(\left(-\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}\right) \cdot \sqrt[3]{\sqrt{2}}\right)}\]

   8. Applied associate-*r* 0.4

      \[\leadsto \color{blue}{\left(x \cdot \left(-\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}\right)\right) \cdot \sqrt[3]{\sqrt{2}}}\]

   9. Simplified 0.4

      \[\leadsto \color{blue}{\left(x \cdot \left(\sqrt[3]{\sqrt{2}} \cdot \left(-\sqrt[3]{\sqrt{2}}\right)\right)\right)} \cdot \sqrt[3]{\sqrt{2}}\]

   if 1.817538864223962e-310 < x

   1. Initial program 30.6

      \[\sqrt{x \cdot x + x \cdot x}\]

   2. Simplified 30.6

      \[\leadsto \color{blue}{\sqrt{x \cdot \left(x + x\right)}}\]
   3. Using strategy rm

   4. Applied sqrt-prod 0.3

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

4. Final simplification 0.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le 1.817538864223962 \cdot 10^{-310}:\\ \;\;\;\;\sqrt[3]{\sqrt{2}} \cdot \left(x \cdot \left(\sqrt[3]{\sqrt{2}} \cdot \left(-\sqrt[3]{\sqrt{2}}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot \sqrt{x + x}\\ \end{array}\]

Reproduce

herbie shell --seed 2020179 
(FPCore (x)
  :name "sqrt A"
  :precision binary64
  (sqrt (+ (* x x) (* x x))))