\[\sqrt{\frac{-2.839573235346269 \cdot 10^{-37}}{\left(a \cdot b\right) \cdot \left|a\right|}}\]
Test:
(sqrt (/ -2.839573235346269e-37 (* (* a b) (fabs a))))
Bits:
128 bits
Bits error versus a
Bits error versus b
Time: 11.7 s
Input Error: 24.9
Output Error: 8.0
Log: ⚲
Profile: 🕒
\(\begin{cases} \sqrt{\frac{-2.839573235346269 \cdot 10^{-37}}{a \cdot b}} \cdot \sqrt{\frac{1}{\left|a\right|}} & \text{when } \left(a \cdot b\right) \cdot \left|a\right| \le -1.3280784008142608 \cdot 10^{+187} \\ \sqrt{\frac{\frac{-2.839573235346269 \cdot 10^{-37}}{a}}{b}} \cdot \sqrt{\frac{1}{\left|a\right|}} & \text{otherwise} \end{cases}\)

    if (* (* a b) (fabs a)) < -1.3280784008142608e+187

    1. Started with
      \[\sqrt{\frac{-2.839573235346269 \cdot 10^{-37}}{\left(a \cdot b\right) \cdot \left|a\right|}}\]
      34.9
    2. Using strategy rm
      34.9
    3. Applied associate-/r* to get
      \[\sqrt{\color{red}{\frac{-2.839573235346269 \cdot 10^{-37}}{\left(a \cdot b\right) \cdot \left|a\right|}}} \leadsto \sqrt{\color{blue}{\frac{\frac{-2.839573235346269 \cdot 10^{-37}}{a \cdot b}}{\left|a\right|}}}\]
      34.9
    4. Using strategy rm
      34.9
    5. Applied div-inv to get
      \[\sqrt{\color{red}{\frac{\frac{-2.839573235346269 \cdot 10^{-37}}{a \cdot b}}{\left|a\right|}}} \leadsto \sqrt{\color{blue}{\frac{-2.839573235346269 \cdot 10^{-37}}{a \cdot b} \cdot \frac{1}{\left|a\right|}}}\]
      34.9
    6. Applied sqrt-prod to get
      \[\color{red}{\sqrt{\frac{-2.839573235346269 \cdot 10^{-37}}{a \cdot b} \cdot \frac{1}{\left|a\right|}}} \leadsto \color{blue}{\sqrt{\frac{-2.839573235346269 \cdot 10^{-37}}{a \cdot b}} \cdot \sqrt{\frac{1}{\left|a\right|}}}\]
      13.3

    if -1.3280784008142608e+187 < (* (* a b) (fabs a))

    1. Started with
      \[\sqrt{\frac{-2.839573235346269 \cdot 10^{-37}}{\left(a \cdot b\right) \cdot \left|a\right|}}\]
      19.3
    2. Using strategy rm
      19.3
    3. Applied associate-/r* to get
      \[\sqrt{\color{red}{\frac{-2.839573235346269 \cdot 10^{-37}}{\left(a \cdot b\right) \cdot \left|a\right|}}} \leadsto \sqrt{\color{blue}{\frac{\frac{-2.839573235346269 \cdot 10^{-37}}{a \cdot b}}{\left|a\right|}}}\]
      17.1
    4. Using strategy rm
      17.1
    5. Applied div-inv to get
      \[\sqrt{\color{red}{\frac{\frac{-2.839573235346269 \cdot 10^{-37}}{a \cdot b}}{\left|a\right|}}} \leadsto \sqrt{\color{blue}{\frac{-2.839573235346269 \cdot 10^{-37}}{a \cdot b} \cdot \frac{1}{\left|a\right|}}}\]
      17.1
    6. Applied sqrt-prod to get
      \[\color{red}{\sqrt{\frac{-2.839573235346269 \cdot 10^{-37}}{a \cdot b} \cdot \frac{1}{\left|a\right|}}} \leadsto \color{blue}{\sqrt{\frac{-2.839573235346269 \cdot 10^{-37}}{a \cdot b}} \cdot \sqrt{\frac{1}{\left|a\right|}}}\]
      7.1
    7. Applied taylor to get
      \[\sqrt{\frac{-2.839573235346269 \cdot 10^{-37}}{a \cdot b}} \cdot \sqrt{\frac{1}{\left|a\right|}} \leadsto \sqrt{\frac{-2.839573235346269 \cdot 10^{-37}}{a \cdot b}} \cdot \sqrt{\frac{1}{\left|a\right|}}\]
      7.1
    8. Taylor expanded around 0 to get
      \[\sqrt{\frac{-2.839573235346269 \cdot 10^{-37}}{a \cdot b}} \cdot \color{red}{\sqrt{\frac{1}{\left|a\right|}}} \leadsto \sqrt{\frac{-2.839573235346269 \cdot 10^{-37}}{a \cdot b}} \cdot \color{blue}{\sqrt{\frac{1}{\left|a\right|}}}\]
      7.1
    9. Applied simplify to get
      \[\sqrt{\frac{-2.839573235346269 \cdot 10^{-37}}{a \cdot b}} \cdot \sqrt{\frac{1}{\left|a\right|}} \leadsto \sqrt{\frac{\frac{-2.839573235346269 \cdot 10^{-37}}{a}}{b}} \cdot \sqrt{\frac{1}{\left|a\right|}}\]
      5.0
    10. 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)))))