\[b - \left({\left(\cot b\right)}^{a} + \sin^{-1} b\right)\]
Test:
(- b (+ (pow (cotan b) a) (asin b)))
Bits:
128 bits
Bits error versus a
Bits error versus b
Time: 29.2 s
Input Error: 9.8
Output Error: 9.8
Log: ⚲
Profile: 🕒
\(\frac{{b}^2}{{\left(\cot b\right)}^{a} + \left(b + \sin^{-1} b\right)} - \frac{\left(\sin^{-1} b + {\left(\cot b\right)}^{a}\right) \cdot \left(\sin^{-1} b + {\left(\cot b\right)}^{a}\right)}{{\left(\cot b\right)}^{a} + \left(b + \sin^{-1} b\right)}\)
  1. Started with
    \[b - \left({\left(\cot b\right)}^{a} + \sin^{-1} b\right)\]
    9.8
  2. Using strategy rm
    9.8
  3. Applied flip-- to get
    \[\color{red}{b - \left({\left(\cot b\right)}^{a} + \sin^{-1} b\right)} \leadsto \color{blue}{\frac{{b}^2 - {\left({\left(\cot b\right)}^{a} + \sin^{-1} b\right)}^2}{b + \left({\left(\cot b\right)}^{a} + \sin^{-1} b\right)}}\]
    9.8
  4. Using strategy rm
    9.8
  5. Applied flip-+ to get
    \[\frac{{b}^2 - {\color{red}{\left({\left(\cot b\right)}^{a} + \sin^{-1} b\right)}}^2}{b + \left({\left(\cot b\right)}^{a} + \sin^{-1} b\right)} \leadsto \frac{{b}^2 - {\color{blue}{\left(\frac{{\left({\left(\cot b\right)}^{a}\right)}^2 - {\left(\sin^{-1} b\right)}^2}{{\left(\cot b\right)}^{a} - \sin^{-1} b}\right)}}^2}{b + \left({\left(\cot b\right)}^{a} + \sin^{-1} b\right)}\]
    10.4
  6. Applied square-div to get
    \[\frac{{b}^2 - \color{red}{{\left(\frac{{\left({\left(\cot b\right)}^{a}\right)}^2 - {\left(\sin^{-1} b\right)}^2}{{\left(\cot b\right)}^{a} - \sin^{-1} b}\right)}^2}}{b + \left({\left(\cot b\right)}^{a} + \sin^{-1} b\right)} \leadsto \frac{{b}^2 - \color{blue}{\frac{{\left({\left({\left(\cot b\right)}^{a}\right)}^2 - {\left(\sin^{-1} b\right)}^2\right)}^2}{{\left({\left(\cot b\right)}^{a} - \sin^{-1} b\right)}^2}}}{b + \left({\left(\cot b\right)}^{a} + \sin^{-1} b\right)}\]
    30.3
  7. Applied taylor to get
    \[\frac{{b}^2 - \frac{{\left({\left({\left(\cot b\right)}^{a}\right)}^2 - {\left(\sin^{-1} b\right)}^2\right)}^2}{{\left({\left(\cot b\right)}^{a} - \sin^{-1} b\right)}^2}}{b + \left({\left(\cot b\right)}^{a} + \sin^{-1} b\right)} \leadsto \frac{{b}^2 - \frac{{\left({\left({\left(\cot b\right)}^{a}\right)}^2 - {\left(\sin^{-1} b\right)}^2\right)}^2}{{\left({\left(\cot b\right)}^{a} - \sin^{-1} b\right)}^2}}{b + \left({\left(\cot b\right)}^{a} + \sin^{-1} b\right)}\]
    30.3
  8. Taylor expanded around 0 to get
    \[\frac{{b}^2 - \frac{{\left({\left({\left(\cot b\right)}^{a}\right)}^2 - \color{red}{{\left(\sin^{-1} b\right)}^2}\right)}^2}{{\left({\left(\cot b\right)}^{a} - \sin^{-1} b\right)}^2}}{b + \left({\left(\cot b\right)}^{a} + \sin^{-1} b\right)} \leadsto \frac{{b}^2 - \frac{{\left({\left({\left(\cot b\right)}^{a}\right)}^2 - \color{blue}{{\left(\sin^{-1} b\right)}^2}\right)}^2}{{\left({\left(\cot b\right)}^{a} - \sin^{-1} b\right)}^2}}{b + \left({\left(\cot b\right)}^{a} + \sin^{-1} b\right)}\]
    30.3
  9. Applied simplify to get
    \[\frac{{b}^2 - \frac{{\left({\left({\left(\cot b\right)}^{a}\right)}^2 - {\left(\sin^{-1} b\right)}^2\right)}^2}{{\left({\left(\cot b\right)}^{a} - \sin^{-1} b\right)}^2}}{b + \left({\left(\cot b\right)}^{a} + \sin^{-1} b\right)} \leadsto \frac{{b}^2}{\left(b + \sin^{-1} b\right) + {\left(\cot b\right)}^{a}} - \frac{\left(\left({\left(\cot b\right)}^{a} - \sin^{-1} b\right) \cdot \left({\left(\cot b\right)}^{a} + \sin^{-1} b\right)\right) \cdot \left(\left({\left(\cot b\right)}^{a} - \sin^{-1} b\right) \cdot \left({\left(\cot b\right)}^{a} + \sin^{-1} b\right)\right)}{\left(\left(b + \sin^{-1} b\right) + {\left(\cot b\right)}^{a}\right) \cdot \left(\left({\left(\cot b\right)}^{a} - \sin^{-1} b\right) \cdot \left({\left(\cot b\right)}^{a} - \sin^{-1} b\right)\right)}\]
    30.5
  10. Applied final simplification
  11. Applied simplify to get
    \[\color{red}{\frac{{b}^2}{\left(b + \sin^{-1} b\right) + {\left(\cot b\right)}^{a}} - \frac{\left(\left({\left(\cot b\right)}^{a} - \sin^{-1} b\right) \cdot \left({\left(\cot b\right)}^{a} + \sin^{-1} b\right)\right) \cdot \left(\left({\left(\cot b\right)}^{a} - \sin^{-1} b\right) \cdot \left({\left(\cot b\right)}^{a} + \sin^{-1} b\right)\right)}{\left(\left(b + \sin^{-1} b\right) + {\left(\cot b\right)}^{a}\right) \cdot \left(\left({\left(\cot b\right)}^{a} - \sin^{-1} b\right) \cdot \left({\left(\cot b\right)}^{a} - \sin^{-1} b\right)\right)}} \leadsto \color{blue}{\frac{{b}^2}{{\left(\cot b\right)}^{a} + \left(b + \sin^{-1} b\right)} - \frac{\left(\sin^{-1} b + {\left(\cot b\right)}^{a}\right) \cdot \left(\sin^{-1} b + {\left(\cot b\right)}^{a}\right)}{{\left(\cot b\right)}^{a} + \left(b + \sin^{-1} b\right)}}\]
    9.8
  12. Removed slow pow expressions

Original test:


(lambda ((a default) (b default))
  #:name "(- b (+ (pow (cotan b) a) (asin b)))"
  (- b (+ (pow (cotan b) a) (asin b))))