Average Error: 30.3 → 0.3
Time: 1.8s
Precision: binary64
\[\sqrt{x \cdot x + x \cdot x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le 3.59275856969827 \cdot 10^{-310}:\\ \;\;\;\;\sqrt{\sqrt[3]{\sqrt{2}}} \cdot \left(x \cdot \left({\left(\sqrt[3]{\sqrt{2}}\right)}^{2} \cdot \left(-\sqrt{\sqrt[3]{\sqrt{2}}}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot \sqrt{x + x}\\ \end{array}\]

Error

Bits error versus x

Derivation

  1. Split input into 2 regimes

  2. if x < 3.59275856969827e-310

    1. Initial program 30.0

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

    2. Simplified30.0

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

    3. Taylor expanded around -inf 0.4

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

    4. Simplified0.4

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

    6. Applied add-cube-cbrt0.4

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

    7. Applied distribute-lft-neg-in0.4

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

    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. Simplified0.4

      \[\leadsto \color{blue}{\left(x \cdot \left(\sqrt[3]{\sqrt{2}} \cdot \left(-\sqrt[3]{\sqrt{2}}\right)\right)\right)} \cdot \sqrt[3]{\sqrt{2}}\]
    10. Using strategy rm

    11. Applied add-sqr-sqrt0.4

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

    12. Applied associate-*r*0.7

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

    13. Simplified0.3

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

    if 3.59275856969827e-310 < x

    1. Initial program 30.6

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

    2. Simplified30.6

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

    4. Applied sqrt-prod0.3

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

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le 3.59275856969827 \cdot 10^{-310}:\\ \;\;\;\;\sqrt{\sqrt[3]{\sqrt{2}}} \cdot \left(x \cdot \left({\left(\sqrt[3]{\sqrt{2}}\right)}^{2} \cdot \left(-\sqrt{\sqrt[3]{\sqrt{2}}}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot \sqrt{x + x}\\ \end{array}\]

Reproduce

herbie shell --seed 2020181 
(FPCore (x)
  :name "sqrt A"
  :precision binary64
  (sqrt (+ (* x x) (* x x))))