\[\sqrt{\frac{-2.839573235346269 \cdot 10^{-37}}{\left(a \cdot b\right) \cdot \left|a\right|}}\]
Test:
(sqrt (/ -2.839573235346269e-37 (* (* a b) (fabs a))))
Bits:
128 bits
Bits error versus a
Bits error versus b
Time: 6.5 s
Input Error: 25.6
Output Error: 9.6
Log: ⚲
Profile: 🕒
\(\frac{\sqrt{\frac{\frac{-2.839573235346269 \cdot 10^{-37}}{b}}{a}}}{\sqrt{\left|a\right|}}\)
  1. Started with
    \[\sqrt{\frac{-2.839573235346269 \cdot 10^{-37}}{\left(a \cdot b\right) \cdot \left|a\right|}}\]
    25.6
  2. Using strategy rm
    25.6
  3. Applied associate-/r* to get
    \[\sqrt{\color{red}{\frac{-2.839573235346269 \cdot 10^{-37}}{\left(a \cdot b\right) \cdot \left|a\right|}}} \leadsto \sqrt{\color{blue}{\frac{\frac{-2.839573235346269 \cdot 10^{-37}}{a \cdot b}}{\left|a\right|}}}\]
    24.3
  4. Using strategy rm
    24.3
  5. Applied sqrt-div to get
    \[\color{red}{\sqrt{\frac{\frac{-2.839573235346269 \cdot 10^{-37}}{a \cdot b}}{\left|a\right|}}} \leadsto \color{blue}{\frac{\sqrt{\frac{-2.839573235346269 \cdot 10^{-37}}{a \cdot b}}}{\sqrt{\left|a\right|}}}\]
    9.7
  6. Using strategy rm
    9.7
  7. Applied add-exp-log to get
    \[\frac{\sqrt{\frac{-2.839573235346269 \cdot 10^{-37}}{a \cdot b}}}{\color{red}{\sqrt{\left|a\right|}}} \leadsto \frac{\sqrt{\frac{-2.839573235346269 \cdot 10^{-37}}{a \cdot b}}}{\color{blue}{e^{\log \left(\sqrt{\left|a\right|}\right)}}}\]
    12.8
  8. Applied add-exp-log to get
    \[\frac{\color{red}{\sqrt{\frac{-2.839573235346269 \cdot 10^{-37}}{a \cdot b}}}}{e^{\log \left(\sqrt{\left|a\right|}\right)}} \leadsto \frac{\color{blue}{e^{\log \left(\sqrt{\frac{-2.839573235346269 \cdot 10^{-37}}{a \cdot b}}\right)}}}{e^{\log \left(\sqrt{\left|a\right|}\right)}}\]
    13.4
  9. Applied div-exp to get
    \[\color{red}{\frac{e^{\log \left(\sqrt{\frac{-2.839573235346269 \cdot 10^{-37}}{a \cdot b}}\right)}}{e^{\log \left(\sqrt{\left|a\right|}\right)}}} \leadsto \color{blue}{e^{\log \left(\sqrt{\frac{-2.839573235346269 \cdot 10^{-37}}{a \cdot b}}\right) - \log \left(\sqrt{\left|a\right|}\right)}}\]
    13.8
  10. Applied taylor to get
    \[e^{\log \left(\sqrt{\frac{-2.839573235346269 \cdot 10^{-37}}{a \cdot b}}\right) - \log \left(\sqrt{\left|a\right|}\right)} \leadsto e^{\log \left(\sqrt{\frac{-2.839573235346269 \cdot 10^{-37}}{b \cdot a}}\right) - \log \left(\sqrt{\left|a\right|}\right)}\]
    13.8
  11. Taylor expanded around 0 to get
    \[e^{\log \left(\sqrt{\color{red}{\frac{-2.839573235346269 \cdot 10^{-37}}{b \cdot a}}}\right) - \log \left(\sqrt{\left|a\right|}\right)} \leadsto e^{\log \left(\sqrt{\color{blue}{\frac{-2.839573235346269 \cdot 10^{-37}}{b \cdot a}}}\right) - \log \left(\sqrt{\left|a\right|}\right)}\]
    13.8
  12. Applied simplify to get
    \[e^{\log \left(\sqrt{\frac{-2.839573235346269 \cdot 10^{-37}}{b \cdot a}}\right) - \log \left(\sqrt{\left|a\right|}\right)} \leadsto \frac{\sqrt{\frac{\frac{-2.839573235346269 \cdot 10^{-37}}{b}}{a}}}{\sqrt{\left|a\right|}}\]
    9.6
  13. Applied final simplification

Original test:


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