\[\sqrt{\frac{-2.839573235346269 \cdot 10^{-37}}{\left(a \cdot b\right) \cdot \left|a\right|}}\]
Test:
(sqrt (/ -2.839573235346269e-37 (* (* a b) (fabs a))))
Bits:
128 bits
Bits error versus a
Bits error versus b
Time: 7.9 s
Input Error: 25.4
Output Error: 9.7
Log: ⚲
Profile: 🕒
\(\sqrt{\frac{\frac{-2.839573235346269 \cdot 10^{-37}}{b}}{a}} \cdot \sqrt{\frac{1}{\left|a\right|}}\)
  1. Started with
    \[\sqrt{\frac{-2.839573235346269 \cdot 10^{-37}}{\left(a \cdot b\right) \cdot \left|a\right|}}\]
    25.4
  2. Using strategy rm
    25.4
  3. Applied associate-/r* to get
    \[\sqrt{\color{red}{\frac{-2.839573235346269 \cdot 10^{-37}}{\left(a \cdot b\right) \cdot \left|a\right|}}} \leadsto \sqrt{\color{blue}{\frac{\frac{-2.839573235346269 \cdot 10^{-37}}{a \cdot b}}{\left|a\right|}}}\]
    24.0
  4. Using strategy rm
    24.0
  5. Applied div-inv to get
    \[\sqrt{\color{red}{\frac{\frac{-2.839573235346269 \cdot 10^{-37}}{a \cdot b}}{\left|a\right|}}} \leadsto \sqrt{\color{blue}{\frac{-2.839573235346269 \cdot 10^{-37}}{a \cdot b} \cdot \frac{1}{\left|a\right|}}}\]
    24.0
  6. Applied sqrt-prod to get
    \[\color{red}{\sqrt{\frac{-2.839573235346269 \cdot 10^{-37}}{a \cdot b} \cdot \frac{1}{\left|a\right|}}} \leadsto \color{blue}{\sqrt{\frac{-2.839573235346269 \cdot 10^{-37}}{a \cdot b}} \cdot \sqrt{\frac{1}{\left|a\right|}}}\]
    9.8
  7. Using strategy rm
    9.8
  8. Applied associate-/r* to get
    \[\sqrt{\color{red}{\frac{-2.839573235346269 \cdot 10^{-37}}{a \cdot b}}} \cdot \sqrt{\frac{1}{\left|a\right|}} \leadsto \sqrt{\color{blue}{\frac{\frac{-2.839573235346269 \cdot 10^{-37}}{a}}{b}}} \cdot \sqrt{\frac{1}{\left|a\right|}}\]
    9.7
  9. Applied taylor to get
    \[\sqrt{\frac{\frac{-2.839573235346269 \cdot 10^{-37}}{a}}{b}} \cdot \sqrt{\frac{1}{\left|a\right|}} \leadsto \sqrt{\frac{-2.839573235346269 \cdot 10^{-37}}{b \cdot a}} \cdot \sqrt{\frac{1}{\left|a\right|}}\]
    9.8
  10. Taylor expanded around 0 to get
    \[\sqrt{\color{red}{\frac{-2.839573235346269 \cdot 10^{-37}}{b \cdot a}}} \cdot \sqrt{\frac{1}{\left|a\right|}} \leadsto \sqrt{\color{blue}{\frac{-2.839573235346269 \cdot 10^{-37}}{b \cdot a}}} \cdot \sqrt{\frac{1}{\left|a\right|}}\]
    9.8
  11. Applied simplify to get
    \[\sqrt{\frac{-2.839573235346269 \cdot 10^{-37}}{b \cdot a}} \cdot \sqrt{\frac{1}{\left|a\right|}} \leadsto \sqrt{\frac{\frac{-2.839573235346269 \cdot 10^{-37}}{b}}{a}} \cdot \sqrt{\frac{1}{\left|a\right|}}\]
    9.7
  12. Applied final simplification
  13. 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)))))