sqrt A

Average Error: 30.7 → 0.4

Time: 6.2s

Precision: binary64

• Falling back on racket egraph because egg-herbie package not installed

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \le 1.978600608064798 \cdot 10^{-310}:\\ \;\;\;\;-x \cdot \sqrt{2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot \sqrt{x + x}\\ \end{array}\]

Error

Bits error versus x

Derivation

1. Split input into 2 regimes

2. if x < 1.978600608064798e-310

   1. Initial program 30.3

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

   2. Simplified 30.3

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

   3. Taylor expanded around -inf 0.4

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

   4. Simplified 0.4

      \[\leadsto \color{blue}{-x \cdot \sqrt{2}}\]

   if 1.978600608064798e-310 < x

   1. Initial program 31.1

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

   2. Simplified 31.1

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

   4. Applied sqrt-prod 0.4

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

4. Final simplification 0.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le 1.978600608064798 \cdot 10^{-310}:\\ \;\;\;\;-x \cdot \sqrt{2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot \sqrt{x + x}\\ \end{array}\]

Reproduce

herbie shell --seed 2020182 
(FPCore (x)
  :name "sqrt A"
  :precision binary64
  (sqrt (+ (* x x) (* x x))))