\[b - \left({\left(\cot b\right)}^{a} + \sin^{-1} b\right)\]
Test:
(- b (+ (pow (cotan b) a) (asin b)))
Bits:
128 bits
Bits error versus a
Bits error versus b
Time: 23.3 s
Input Error: 4.0
Output Error: 4.2
Log: ⚲
Profile: 🕒
\(\frac{{b}^2 - (\left({\left(\cot b\right)}^{a}\right) * \left((2 * \left(\sin^{-1} b\right) + \left({\left(\cot b\right)}^{a}\right))_*\right) + \left({\left(\sin^{-1} b\right)}^2\right))_*}{\left(\sin^{-1} b + b\right) + {\left(\cot b\right)}^{a}}\)
  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-cube-cbrt to get
    \[\color{red}{b - \left({\left(\cot b\right)}^{a} + \sin^{-1} b\right)} \leadsto \color{blue}{{\left(\sqrt[3]{b - \left({\left(\cot b\right)}^{a} + \sin^{-1} b\right)}\right)}^3}\]
    4.1
  4. Using strategy rm
    4.1
  5. Applied add-log-exp to get
    \[{\color{red}{\left(\sqrt[3]{b - \left({\left(\cot b\right)}^{a} + \sin^{-1} b\right)}\right)}}^3 \leadsto {\color{blue}{\left(\log \left(e^{\sqrt[3]{b - \left({\left(\cot b\right)}^{a} + \sin^{-1} b\right)}}\right)\right)}}^3\]
    4.4
  6. Using strategy rm
    4.4
  7. Applied flip-- to get
    \[{\left(\log \left(e^{\sqrt[3]{\color{red}{b - \left({\left(\cot b\right)}^{a} + \sin^{-1} b\right)}}}\right)\right)}^3 \leadsto {\left(\log \left(e^{\sqrt[3]{\color{blue}{\frac{{b}^2 - {\left({\left(\cot b\right)}^{a} + \sin^{-1} b\right)}^2}{b + \left({\left(\cot b\right)}^{a} + \sin^{-1} b\right)}}}}\right)\right)}^3\]
    4.7
  8. Applied cbrt-div to get
    \[{\left(\log \left(e^{\color{red}{\sqrt[3]{\frac{{b}^2 - {\left({\left(\cot b\right)}^{a} + \sin^{-1} b\right)}^2}{b + \left({\left(\cot b\right)}^{a} + \sin^{-1} b\right)}}}}\right)\right)}^3 \leadsto {\left(\log \left(e^{\color{blue}{\frac{\sqrt[3]{{b}^2 - {\left({\left(\cot b\right)}^{a} + \sin^{-1} b\right)}^2}}{\sqrt[3]{b + \left({\left(\cot b\right)}^{a} + \sin^{-1} b\right)}}}}\right)\right)}^3\]
    4.7
  9. Applied taylor to get
    \[{\left(\log \left(e^{\frac{\sqrt[3]{{b}^2 - {\left({\left(\cot b\right)}^{a} + \sin^{-1} b\right)}^2}}{\sqrt[3]{b + \left({\left(\cot b\right)}^{a} + \sin^{-1} b\right)}}}\right)\right)}^3 \leadsto {\left(\log \left(e^{\frac{\sqrt[3]{{b}^2 - \left({\left(\sin^{-1} b\right)}^2 + \left(2 \cdot \left(\sin^{-1} b \cdot {\left(\cot b\right)}^{a}\right) + {\left({\left(\cot b\right)}^{a}\right)}^2\right)\right)}}{\sqrt[3]{\sin^{-1} b + \left(b + {\left(\cot b\right)}^{a}\right)}}}\right)\right)}^3\]
    4.7
  10. Taylor expanded around 0 to get
    \[{\left(\log \left(e^{\color{red}{\frac{\sqrt[3]{{b}^2 - \left({\left(\sin^{-1} b\right)}^2 + \left(2 \cdot \left(\sin^{-1} b \cdot {\left(\cot b\right)}^{a}\right) + {\left({\left(\cot b\right)}^{a}\right)}^2\right)\right)}}{\sqrt[3]{\sin^{-1} b + \left(b + {\left(\cot b\right)}^{a}\right)}}}}\right)\right)}^3 \leadsto {\left(\log \left(e^{\color{blue}{\frac{\sqrt[3]{{b}^2 - \left({\left(\sin^{-1} b\right)}^2 + \left(2 \cdot \left(\sin^{-1} b \cdot {\left(\cot b\right)}^{a}\right) + {\left({\left(\cot b\right)}^{a}\right)}^2\right)\right)}}{\sqrt[3]{\sin^{-1} b + \left(b + {\left(\cot b\right)}^{a}\right)}}}}\right)\right)}^3\]
    4.7
  11. Applied simplify to get
    \[{\left(\log \left(e^{\frac{\sqrt[3]{{b}^2 - \left({\left(\sin^{-1} b\right)}^2 + \left(2 \cdot \left(\sin^{-1} b \cdot {\left(\cot b\right)}^{a}\right) + {\left({\left(\cot b\right)}^{a}\right)}^2\right)\right)}}{\sqrt[3]{\sin^{-1} b + \left(b + {\left(\cot b\right)}^{a}\right)}}}\right)\right)}^3 \leadsto \frac{{b}^2 - (\left({\left(\cot b\right)}^{a}\right) * \left((2 * \left(\sin^{-1} b\right) + \left({\left(\cot b\right)}^{a}\right))_*\right) + \left(\sin^{-1} b \cdot \sin^{-1} b\right))_*}{\left(b + \sin^{-1} b\right) + {\left(\cot b\right)}^{a}}\]
    4.2
  12. Applied final simplification
  13. Applied simplify to get
    \[\color{red}{\frac{{b}^2 - (\left({\left(\cot b\right)}^{a}\right) * \left((2 * \left(\sin^{-1} b\right) + \left({\left(\cot b\right)}^{a}\right))_*\right) + \left(\sin^{-1} b \cdot \sin^{-1} b\right))_*}{\left(b + \sin^{-1} b\right) + {\left(\cot b\right)}^{a}}} \leadsto \color{blue}{\frac{{b}^2 - (\left({\left(\cot b\right)}^{a}\right) * \left((2 * \left(\sin^{-1} b\right) + \left({\left(\cot b\right)}^{a}\right))_*\right) + \left({\left(\sin^{-1} b\right)}^2\right))_*}{\left(\sin^{-1} b + b\right) + {\left(\cot b\right)}^{a}}}\]
    4.2
  14. 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))))