\[\sqrt{\frac{-2.839573235346269 \cdot 10^{-37}}{\left(a \cdot b\right) \cdot \left|a\right|}}\]
Test:
(sqrt (/ -2.839573235346269e-37 (* (* a b) (fabs a))))
Bits:
128 bits
Bits error versus a
Bits error versus b
Time: 7.5 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-cube-cbrt to get
    \[\color{red}{\frac{\sqrt{\frac{-2.839573235346269 \cdot 10^{-37}}{a \cdot b}}}{\sqrt{\left|a\right|}}} \leadsto \color{blue}{{\left(\sqrt[3]{\frac{\sqrt{\frac{-2.839573235346269 \cdot 10^{-37}}{a \cdot b}}}{\sqrt{\left|a\right|}}}\right)}^3}\]
    10.4
  8. Applied taylor to get
    \[{\left(\sqrt[3]{\frac{\sqrt{\frac{-2.839573235346269 \cdot 10^{-37}}{a \cdot b}}}{\sqrt{\left|a\right|}}}\right)}^3 \leadsto {\left(\sqrt[3]{\frac{\sqrt{\frac{-2.839573235346269 \cdot 10^{-37}}{b \cdot a}}}{\sqrt{\left|a\right|}}}\right)}^3\]
    10.4
  9. Taylor expanded around 0 to get
    \[{\left(\sqrt[3]{\frac{\sqrt{\color{red}{\frac{-2.839573235346269 \cdot 10^{-37}}{b \cdot a}}}}{\sqrt{\left|a\right|}}}\right)}^3 \leadsto {\left(\sqrt[3]{\frac{\sqrt{\color{blue}{\frac{-2.839573235346269 \cdot 10^{-37}}{b \cdot a}}}}{\sqrt{\left|a\right|}}}\right)}^3\]
    10.4
  10. Applied simplify to get
    \[{\left(\sqrt[3]{\frac{\sqrt{\frac{-2.839573235346269 \cdot 10^{-37}}{b \cdot a}}}{\sqrt{\left|a\right|}}}\right)}^3 \leadsto \frac{\sqrt{\frac{\frac{-2.839573235346269 \cdot 10^{-37}}{b}}{a}}}{\sqrt{\left|a\right|}}\]
    9.6
  11. Applied final simplification
  12. 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)))))