\[\sqrt{\frac{-2.839573235346269 \cdot 10^{-37}}{\left(a \cdot b\right) \cdot \left|a\right|}}\]
Test:
(sqrt (/ -2.839573235346269e-37 (* (* a b) (fabs a))))
Bits:
128 bits
Bits error versus a
Bits error versus b
Time: 8.8 s
Input Error: 24.6
Output Error: 9.2
Log: ⚲
Profile: 🕒
\(\begin{cases} \frac{\sqrt{\frac{\frac{-2.839573235346269 \cdot 10^{-37}}{a}}{b}}}{\sqrt{\left|a\right|}} & \text{when } b \le -1.2391807006856761 \cdot 10^{-204} \\ \sqrt{\frac{-2.839573235346269 \cdot 10^{-37}}{a \cdot b}} \cdot \sqrt{\frac{1}{\left|a\right|}} & \text{otherwise} \end{cases}\)

    if b < -1.2391807006856761e-204

    1. Started with
      \[\sqrt{\frac{-2.839573235346269 \cdot 10^{-37}}{\left(a \cdot b\right) \cdot \left|a\right|}}\]
      25.0
    2. Using strategy rm
      25.0
    3. Applied associate-/r* to get
      \[\sqrt{\color{red}{\frac{-2.839573235346269 \cdot 10^{-37}}{\left(a \cdot b\right) \cdot \left|a\right|}}} \leadsto \sqrt{\color{blue}{\frac{\frac{-2.839573235346269 \cdot 10^{-37}}{a \cdot b}}{\left|a\right|}}}\]
      23.7
    4. Using strategy rm
      23.7
    5. Applied sqrt-div to get
      \[\color{red}{\sqrt{\frac{\frac{-2.839573235346269 \cdot 10^{-37}}{a \cdot b}}{\left|a\right|}}} \leadsto \color{blue}{\frac{\sqrt{\frac{-2.839573235346269 \cdot 10^{-37}}{a \cdot b}}}{\sqrt{\left|a\right|}}}\]
      8.1
    6. Using strategy rm
      8.1
    7. Applied associate-/r* to get
      \[\frac{\sqrt{\color{red}{\frac{-2.839573235346269 \cdot 10^{-37}}{a \cdot b}}}}{\sqrt{\left|a\right|}} \leadsto \frac{\sqrt{\color{blue}{\frac{\frac{-2.839573235346269 \cdot 10^{-37}}{a}}{b}}}}{\sqrt{\left|a\right|}}\]
      8.4

    if -1.2391807006856761e-204 < b

    1. Started with
      \[\sqrt{\frac{-2.839573235346269 \cdot 10^{-37}}{\left(a \cdot b\right) \cdot \left|a\right|}}\]
      24.3
    2. Using strategy rm
      24.3
    3. Applied associate-/r* to get
      \[\sqrt{\color{red}{\frac{-2.839573235346269 \cdot 10^{-37}}{\left(a \cdot b\right) \cdot \left|a\right|}}} \leadsto \sqrt{\color{blue}{\frac{\frac{-2.839573235346269 \cdot 10^{-37}}{a \cdot b}}{\left|a\right|}}}\]
      22.9
    4. Using strategy rm
      22.9
    5. Applied div-inv to get
      \[\sqrt{\color{red}{\frac{\frac{-2.839573235346269 \cdot 10^{-37}}{a \cdot b}}{\left|a\right|}}} \leadsto \sqrt{\color{blue}{\frac{-2.839573235346269 \cdot 10^{-37}}{a \cdot b} \cdot \frac{1}{\left|a\right|}}}\]
      22.9
    6. Applied sqrt-prod to get
      \[\color{red}{\sqrt{\frac{-2.839573235346269 \cdot 10^{-37}}{a \cdot b} \cdot \frac{1}{\left|a\right|}}} \leadsto \color{blue}{\sqrt{\frac{-2.839573235346269 \cdot 10^{-37}}{a \cdot b}} \cdot \sqrt{\frac{1}{\left|a\right|}}}\]
      9.8
  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)))))