\[\sqrt{\frac{-2.839573235346269 \cdot 10^{-37}}{\left(a \cdot b\right) \cdot \left|a\right|}}\]
Test:
(sqrt (/ -2.839573235346269e-37 (* (* a b) (fabs a))))
Bits:
128 bits
Bits error versus a
Bits error versus b
Time: 10.0 s
Input Error: 25.4
Output Error: 8.4
Log: ⚲
Profile: 🕒
\(\begin{cases} \sqrt{-2.839573235346269 \cdot 10^{-37} \cdot \frac{1}{a \cdot b}} \cdot \sqrt{\frac{1}{\left|a\right|}} & \text{when } \left(a \cdot b\right) \cdot \left|a\right| \le -8.532992038844905 \cdot 10^{+260} \\ \sqrt{\frac{\frac{-2.839573235346269 \cdot 10^{-37}}{a}}{b}} \cdot \sqrt{\frac{1}{\left|a\right|}} & \text{otherwise} \end{cases}\)

    if (* (* a b) (fabs a)) < -8.532992038844905e+260

    1. Started with
      \[\sqrt{\frac{-2.839573235346269 \cdot 10^{-37}}{\left(a \cdot b\right) \cdot \left|a\right|}}\]
      41.5
    2. Using strategy rm
      41.5
    3. Applied associate-/r* to get
      \[\sqrt{\color{red}{\frac{-2.839573235346269 \cdot 10^{-37}}{\left(a \cdot b\right) \cdot \left|a\right|}}} \leadsto \sqrt{\color{blue}{\frac{\frac{-2.839573235346269 \cdot 10^{-37}}{a \cdot b}}{\left|a\right|}}}\]
      41.5
    4. Using strategy rm
      41.5
    5. Applied div-inv to get
      \[\sqrt{\color{red}{\frac{\frac{-2.839573235346269 \cdot 10^{-37}}{a \cdot b}}{\left|a\right|}}} \leadsto \sqrt{\color{blue}{\frac{-2.839573235346269 \cdot 10^{-37}}{a \cdot b} \cdot \frac{1}{\left|a\right|}}}\]
      41.5
    6. Applied sqrt-prod to get
      \[\color{red}{\sqrt{\frac{-2.839573235346269 \cdot 10^{-37}}{a \cdot b} \cdot \frac{1}{\left|a\right|}}} \leadsto \color{blue}{\sqrt{\frac{-2.839573235346269 \cdot 10^{-37}}{a \cdot b}} \cdot \sqrt{\frac{1}{\left|a\right|}}}\]
      16.7
    7. Using strategy rm
      16.7
    8. Applied div-inv to get
      \[\sqrt{\color{red}{\frac{-2.839573235346269 \cdot 10^{-37}}{a \cdot b}}} \cdot \sqrt{\frac{1}{\left|a\right|}} \leadsto \sqrt{\color{blue}{-2.839573235346269 \cdot 10^{-37} \cdot \frac{1}{a \cdot b}}} \cdot \sqrt{\frac{1}{\left|a\right|}}\]
      16.7

    if -8.532992038844905e+260 < (* (* a b) (fabs a))

    1. Started with
      \[\sqrt{\frac{-2.839573235346269 \cdot 10^{-37}}{\left(a \cdot b\right) \cdot \left|a\right|}}\]
      18.4
    2. Using strategy rm
      18.4
    3. Applied associate-/r* to get
      \[\sqrt{\color{red}{\frac{-2.839573235346269 \cdot 10^{-37}}{\left(a \cdot b\right) \cdot \left|a\right|}}} \leadsto \sqrt{\color{blue}{\frac{\frac{-2.839573235346269 \cdot 10^{-37}}{a \cdot b}}{\left|a\right|}}}\]
      16.2
    4. Using strategy rm
      16.2
    5. Applied div-inv to get
      \[\sqrt{\color{red}{\frac{\frac{-2.839573235346269 \cdot 10^{-37}}{a \cdot b}}{\left|a\right|}}} \leadsto \sqrt{\color{blue}{\frac{-2.839573235346269 \cdot 10^{-37}}{a \cdot b} \cdot \frac{1}{\left|a\right|}}}\]
      16.3
    6. Applied sqrt-prod to get
      \[\color{red}{\sqrt{\frac{-2.839573235346269 \cdot 10^{-37}}{a \cdot b} \cdot \frac{1}{\left|a\right|}}} \leadsto \color{blue}{\sqrt{\frac{-2.839573235346269 \cdot 10^{-37}}{a \cdot b}} \cdot \sqrt{\frac{1}{\left|a\right|}}}\]
      6.7
    7. Using strategy rm
      6.7
    8. Applied associate-/r* to get
      \[\sqrt{\color{red}{\frac{-2.839573235346269 \cdot 10^{-37}}{a \cdot b}}} \cdot \sqrt{\frac{1}{\left|a\right|}} \leadsto \sqrt{\color{blue}{\frac{\frac{-2.839573235346269 \cdot 10^{-37}}{a}}{b}}} \cdot \sqrt{\frac{1}{\left|a\right|}}\]
      4.7
  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)))))