\[b - \left({\left(\cot b\right)}^{a} + \sin^{-1} b\right)\]
Test:
(- b (+ (pow (cotan b) a) (asin b)))
Bits:
128 bits
Bits error versus a
Bits error versus b
Time: 28.3 s
Input Error: 4.0
Output Error: 3.9
Log: ⚲
Profile: 🕒
\(\frac{\left(b + \sin^{-1} b\right) \cdot \left(b - \sin^{-1} b\right) - {\left(\cot b\right)}^{a} \cdot \left(2 \cdot \sin^{-1} b + {\left(\cot b\right)}^{a}\right)}{\left(b + \sin^{-1} b\right) + {\left(\cot b\right)}^{a}}\)
  1. Started with
    \[b - \left({\left(\cot b\right)}^{a} + \sin^{-1} b\right)\]
    4.0
  2. Using strategy rm
    4.0
  3. Applied add-cbrt-cube to get
    \[\color{red}{b - \left({\left(\cot b\right)}^{a} + \sin^{-1} b\right)} \leadsto \color{blue}{\sqrt[3]{{\left(b - \left({\left(\cot b\right)}^{a} + \sin^{-1} b\right)\right)}^3}}\]
    4.1
  4. Using strategy rm
    4.1
  5. Applied flip-- to get
    \[\sqrt[3]{{\color{red}{\left(b - \left({\left(\cot b\right)}^{a} + \sin^{-1} b\right)\right)}}^3} \leadsto \sqrt[3]{{\color{blue}{\left(\frac{{b}^2 - {\left({\left(\cot b\right)}^{a} + \sin^{-1} b\right)}^2}{b + \left({\left(\cot b\right)}^{a} + \sin^{-1} b\right)}\right)}}^3}\]
    4.1
  6. Applied cube-div to get
    \[\sqrt[3]{\color{red}{{\left(\frac{{b}^2 - {\left({\left(\cot b\right)}^{a} + \sin^{-1} b\right)}^2}{b + \left({\left(\cot b\right)}^{a} + \sin^{-1} b\right)}\right)}^3}} \leadsto \sqrt[3]{\color{blue}{\frac{{\left({b}^2 - {\left({\left(\cot b\right)}^{a} + \sin^{-1} b\right)}^2\right)}^3}{{\left(b + \left({\left(\cot b\right)}^{a} + \sin^{-1} b\right)\right)}^3}}}\]
    4.1
  7. Applied taylor to get
    \[\sqrt[3]{\frac{{\left({b}^2 - {\left({\left(\cot b\right)}^{a} + \sin^{-1} b\right)}^2\right)}^3}{{\left(b + \left({\left(\cot b\right)}^{a} + \sin^{-1} b\right)\right)}^3}} \leadsto \sqrt[3]{\frac{{\left({b}^2 - \left({\left(\sin^{-1} b\right)}^2 + \left(2 \cdot \left(\sin^{-1} b \cdot {\left(\cot b\right)}^{a}\right) + {\left({\left(\cot b\right)}^{a}\right)}^2\right)\right)\right)}^3}{{\left(\sin^{-1} b + \left(b + {\left(\cot b\right)}^{a}\right)\right)}^3}}\]
    4.1
  8. Taylor expanded around 0 to get
    \[\sqrt[3]{\color{red}{\frac{{\left({b}^2 - \left({\left(\sin^{-1} b\right)}^2 + \left(2 \cdot \left(\sin^{-1} b \cdot {\left(\cot b\right)}^{a}\right) + {\left({\left(\cot b\right)}^{a}\right)}^2\right)\right)\right)}^3}{{\left(\sin^{-1} b + \left(b + {\left(\cot b\right)}^{a}\right)\right)}^3}}} \leadsto \sqrt[3]{\color{blue}{\frac{{\left({b}^2 - \left({\left(\sin^{-1} b\right)}^2 + \left(2 \cdot \left(\sin^{-1} b \cdot {\left(\cot b\right)}^{a}\right) + {\left({\left(\cot b\right)}^{a}\right)}^2\right)\right)\right)}^3}{{\left(\sin^{-1} b + \left(b + {\left(\cot b\right)}^{a}\right)\right)}^3}}}\]
    4.1
  9. Applied simplify to get
    \[\color{red}{\sqrt[3]{\frac{{\left({b}^2 - \left({\left(\sin^{-1} b\right)}^2 + \left(2 \cdot \left(\sin^{-1} b \cdot {\left(\cot b\right)}^{a}\right) + {\left({\left(\cot b\right)}^{a}\right)}^2\right)\right)\right)}^3}{{\left(\sin^{-1} b + \left(b + {\left(\cot b\right)}^{a}\right)\right)}^3}}} \leadsto \color{blue}{\frac{\left(b + \sin^{-1} b\right) \cdot \left(b - \sin^{-1} b\right) - {\left(\cot b\right)}^{a} \cdot \left(2 \cdot \sin^{-1} b + {\left(\cot b\right)}^{a}\right)}{\left(b + \sin^{-1} b\right) + {\left(\cot b\right)}^{a}}}\]
    3.9

Original test:


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