\[\sqrt{\frac{-2.839573235346269 \cdot 10^{-37}}{\left(a \cdot b\right) \cdot \left|a\right|}}\]
Test:
(sqrt (/ -2.839573235346269e-37 (* (* a b) (fabs a))))
Bits:
128 bits
Bits error versus a
Bits error versus b
Time: 9.3 s
Input Error: 24.8
Output Error: 8.3
Log: ⚲
Profile: 🕒
\(\begin{cases} \frac{\sqrt{\frac{-2.839573235346269 \cdot 10^{-37}}{a \cdot b}}}{\sqrt{\left|a\right|}} & \text{when } \left(a \cdot b\right) \cdot \left|a\right| \le -3.2336614114229 \cdot 10^{-311} \\ \frac{\sqrt{\frac{\frac{-2.839573235346269 \cdot 10^{-37}}{b}}{a}}}{\sqrt{\left|a\right|}} & \text{otherwise} \end{cases}\)

    if (* (* a b) (fabs a)) < -3.2336614114229e-311

    1. Started with
      \[\sqrt{\frac{-2.839573235346269 \cdot 10^{-37}}{\left(a \cdot b\right) \cdot \left|a\right|}}\]
      15.9
    2. Using strategy rm
      15.9
    3. Applied associate-/r* to get
      \[\sqrt{\color{red}{\frac{-2.839573235346269 \cdot 10^{-37}}{\left(a \cdot b\right) \cdot \left|a\right|}}} \leadsto \sqrt{\color{blue}{\frac{\frac{-2.839573235346269 \cdot 10^{-37}}{a \cdot b}}{\left|a\right|}}}\]
      16.0
    4. Using strategy rm
      16.0
    5. Applied sqrt-div to get
      \[\color{red}{\sqrt{\frac{\frac{-2.839573235346269 \cdot 10^{-37}}{a \cdot b}}{\left|a\right|}}} \leadsto \color{blue}{\frac{\sqrt{\frac{-2.839573235346269 \cdot 10^{-37}}{a \cdot b}}}{\sqrt{\left|a\right|}}}\]
      6.4

    if -3.2336614114229e-311 < (* (* a b) (fabs a))

    1. Started with
      \[\sqrt{\frac{-2.839573235346269 \cdot 10^{-37}}{\left(a \cdot b\right) \cdot \left|a\right|}}\]
      58.6
    2. Using strategy rm
      58.6
    3. Applied associate-/r* to get
      \[\sqrt{\color{red}{\frac{-2.839573235346269 \cdot 10^{-37}}{\left(a \cdot b\right) \cdot \left|a\right|}}} \leadsto \sqrt{\color{blue}{\frac{\frac{-2.839573235346269 \cdot 10^{-37}}{a \cdot b}}{\left|a\right|}}}\]
      52.3
    4. Using strategy rm
      52.3
    5. Applied sqrt-div to get
      \[\color{red}{\sqrt{\frac{\frac{-2.839573235346269 \cdot 10^{-37}}{a \cdot b}}{\left|a\right|}}} \leadsto \color{blue}{\frac{\sqrt{\frac{-2.839573235346269 \cdot 10^{-37}}{a \cdot b}}}{\sqrt{\left|a\right|}}}\]
      21.9
    6. Applied taylor to get
      \[\frac{\sqrt{\frac{-2.839573235346269 \cdot 10^{-37}}{a \cdot b}}}{\sqrt{\left|a\right|}} \leadsto \frac{\sqrt{\frac{-2.839573235346269 \cdot 10^{-37}}{b \cdot a}}}{\sqrt{\left|a\right|}}\]
      21.9
    7. Taylor expanded around 0 to get
      \[\frac{\sqrt{\color{red}{\frac{-2.839573235346269 \cdot 10^{-37}}{b \cdot a}}}}{\sqrt{\left|a\right|}} \leadsto \frac{\sqrt{\color{blue}{\frac{-2.839573235346269 \cdot 10^{-37}}{b \cdot a}}}}{\sqrt{\left|a\right|}}\]
      21.9
    8. Applied simplify to get
      \[\frac{\sqrt{\frac{-2.839573235346269 \cdot 10^{-37}}{b \cdot a}}}{\sqrt{\left|a\right|}} \leadsto \frac{\sqrt{\frac{\frac{-2.839573235346269 \cdot 10^{-37}}{b}}{a}}}{\sqrt{\left|a\right|}}\]
      15.8
    9. Applied final simplification
  1. Removed slow pow expressions

Original test:


(lambda ((a default) (b default))
  #:name "(sqrt (/ -2.839573235346269e-37 (* (* a b) (fabs a))))"
  (sqrt (/ -2.839573235346269e-37 (* (* a b) (fabs a)))))