\[\sqrt{\frac{-2.839573235346269 \cdot 10^{-37}}{\left(a \cdot b\right) \cdot \left|a\right|}}\]
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: 24.5
Output Error: 8.0
Log: ⚲
Profile: 🕒
\(\begin{cases} \frac{\sqrt{\frac{-2.839573235346269 \cdot 10^{-37}}{a \cdot b}}}{\sqrt{\left|a\right|}} & \text{when } a \le -6.638968057369996 \cdot 10^{-13} \\ \sqrt{\frac{\frac{-2.839573235346269 \cdot 10^{-37}}{a}}{b}} \cdot \sqrt{\frac{1}{\left|a\right|}} & \text{when } a \le 3.086244101517683 \cdot 10^{+19} \\ \frac{\sqrt{\frac{-2.839573235346269 \cdot 10^{-37}}{a \cdot b}}}{\sqrt{\left|a\right|}} & \text{otherwise} \end{cases}\)

    if a < -6.638968057369996e-13 or 3.086244101517683e+19 < a

    1. Started with
      \[\sqrt{\frac{-2.839573235346269 \cdot 10^{-37}}{\left(a \cdot b\right) \cdot \left|a\right|}}\]
      24.4
    2. Using strategy rm
      24.4
    3. Applied associate-/r* to get
      \[\sqrt{\color{red}{\frac{-2.839573235346269 \cdot 10^{-37}}{\left(a \cdot b\right) \cdot \left|a\right|}}} \leadsto \sqrt{\color{blue}{\frac{\frac{-2.839573235346269 \cdot 10^{-37}}{a \cdot b}}{\left|a\right|}}}\]
      24.3
    4. Using strategy rm
      24.3
    5. Applied sqrt-div to get
      \[\color{red}{\sqrt{\frac{\frac{-2.839573235346269 \cdot 10^{-37}}{a \cdot b}}{\left|a\right|}}} \leadsto \color{blue}{\frac{\sqrt{\frac{-2.839573235346269 \cdot 10^{-37}}{a \cdot b}}}{\sqrt{\left|a\right|}}}\]
      8.8

    if -6.638968057369996e-13 < a < 3.086244101517683e+19

    1. Started with
      \[\sqrt{\frac{-2.839573235346269 \cdot 10^{-37}}{\left(a \cdot b\right) \cdot \left|a\right|}}\]
      24.6
    2. Using strategy rm
      24.6
    3. Applied associate-/r* to get
      \[\sqrt{\color{red}{\frac{-2.839573235346269 \cdot 10^{-37}}{\left(a \cdot b\right) \cdot \left|a\right|}}} \leadsto \sqrt{\color{blue}{\frac{\frac{-2.839573235346269 \cdot 10^{-37}}{a \cdot b}}{\left|a\right|}}}\]
      22.0
    4. Using strategy rm
      22.0
    5. Applied div-inv to get
      \[\sqrt{\color{red}{\frac{\frac{-2.839573235346269 \cdot 10^{-37}}{a \cdot b}}{\left|a\right|}}} \leadsto \sqrt{\color{blue}{\frac{-2.839573235346269 \cdot 10^{-37}}{a \cdot b} \cdot \frac{1}{\left|a\right|}}}\]
      22.0
    6. Applied sqrt-prod to get
      \[\color{red}{\sqrt{\frac{-2.839573235346269 \cdot 10^{-37}}{a \cdot b} \cdot \frac{1}{\left|a\right|}}} \leadsto \color{blue}{\sqrt{\frac{-2.839573235346269 \cdot 10^{-37}}{a \cdot b}} \cdot \sqrt{\frac{1}{\left|a\right|}}}\]
      9.6
    7. Using strategy rm
      9.6
    8. Applied associate-/r* to get
      \[\sqrt{\color{red}{\frac{-2.839573235346269 \cdot 10^{-37}}{a \cdot b}}} \cdot \sqrt{\frac{1}{\left|a\right|}} \leadsto \sqrt{\color{blue}{\frac{\frac{-2.839573235346269 \cdot 10^{-37}}{a}}{b}}} \cdot \sqrt{\frac{1}{\left|a\right|}}\]
      7.1
  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)))))