\[b - \left({\left(\cot b\right)}^{a} + \sin^{-1} b\right)\]
Test:
(- b (+ (pow (cotan b) a) (asin b)))
Bits:
128 bits
Bits error versus a
Bits error versus b
Time: 44.9 s
Input Error: 2.6
Output Error: 2.5
Log: ⚲
Profile: 🕒
\(\begin{cases} {\left({\left(b - \left({\left(\cot b\right)}^{a} + \sin^{-1} b\right)\right)}^{\frac{1}{3}}\right)}^3 & \text{when } {\left(\cot b\right)}^{a} \le 2.9955618f-05 \\ \frac{{b}^3 - {\left(\sin^{-1} b + {\left(\cot b\right)}^{a}\right)}^3}{{\left(\sin^{-1} b + {\left(\cot b\right)}^{a}\right)}^2 + \left(\left(\sin^{-1} b + \left(b + 1\right)\right) - a \cdot \log b\right) \cdot b} & \text{otherwise} \end{cases}\)

    if (pow (cotan b) a) < 2.9955618f-05

    1. Started with
      \[b - \left({\left(\cot b\right)}^{a} + \sin^{-1} b\right)\]
      8.3
    2. Using strategy rm
      8.3
    3. Applied add-cube-cbrt to get
      \[\color{red}{b - \left({\left(\cot b\right)}^{a} + \sin^{-1} b\right)} \leadsto \color{blue}{{\left(\sqrt[3]{b - \left({\left(\cot b\right)}^{a} + \sin^{-1} b\right)}\right)}^3}\]
      8.3
    4. Using strategy rm
      8.3
    5. Applied pow1/3 to get
      \[{\color{red}{\left(\sqrt[3]{b - \left({\left(\cot b\right)}^{a} + \sin^{-1} b\right)}\right)}}^3 \leadsto {\color{blue}{\left({\left(b - \left({\left(\cot b\right)}^{a} + \sin^{-1} b\right)\right)}^{\frac{1}{3}}\right)}}^3\]
      8.3

    if 2.9955618f-05 < (pow (cotan b) a)

    1. Started with
      \[b - \left({\left(\cot b\right)}^{a} + \sin^{-1} b\right)\]
      0.2
    2. Using strategy rm
      0.2
    3. Applied add-cube-cbrt to get
      \[\color{red}{b - \left({\left(\cot b\right)}^{a} + \sin^{-1} b\right)} \leadsto \color{blue}{{\left(\sqrt[3]{b - \left({\left(\cot b\right)}^{a} + \sin^{-1} b\right)}\right)}^3}\]
      0.4
    4. Using strategy rm
      0.4
    5. Applied flip3-- to get
      \[{\left(\sqrt[3]{\color{red}{b - \left({\left(\cot b\right)}^{a} + \sin^{-1} b\right)}}\right)}^3 \leadsto {\left(\sqrt[3]{\color{blue}{\frac{{b}^{3} - {\left({\left(\cot b\right)}^{a} + \sin^{-1} b\right)}^{3}}{{b}^2 + \left({\left({\left(\cot b\right)}^{a} + \sin^{-1} b\right)}^2 + b \cdot \left({\left(\cot b\right)}^{a} + \sin^{-1} b\right)\right)}}}\right)}^3\]
      0.7
    6. Applied cbrt-div to get
      \[{\color{red}{\left(\sqrt[3]{\frac{{b}^{3} - {\left({\left(\cot b\right)}^{a} + \sin^{-1} b\right)}^{3}}{{b}^2 + \left({\left({\left(\cot b\right)}^{a} + \sin^{-1} b\right)}^2 + b \cdot \left({\left(\cot b\right)}^{a} + \sin^{-1} b\right)\right)}}\right)}}^3 \leadsto {\color{blue}{\left(\frac{\sqrt[3]{{b}^{3} - {\left({\left(\cot b\right)}^{a} + \sin^{-1} b\right)}^{3}}}{\sqrt[3]{{b}^2 + \left({\left({\left(\cot b\right)}^{a} + \sin^{-1} b\right)}^2 + b \cdot \left({\left(\cot b\right)}^{a} + \sin^{-1} b\right)\right)}}\right)}}^3\]
      0.7
    7. Applied simplify to get
      \[{\left(\frac{\color{red}{\sqrt[3]{{b}^{3} - {\left({\left(\cot b\right)}^{a} + \sin^{-1} b\right)}^{3}}}}{\sqrt[3]{{b}^2 + \left({\left({\left(\cot b\right)}^{a} + \sin^{-1} b\right)}^2 + b \cdot \left({\left(\cot b\right)}^{a} + \sin^{-1} b\right)\right)}}\right)}^3 \leadsto {\left(\frac{\color{blue}{\sqrt[3]{{b}^3 - {\left(\sin^{-1} b + {\left(\cot b\right)}^{a}\right)}^3}}}{\sqrt[3]{{b}^2 + \left({\left({\left(\cot b\right)}^{a} + \sin^{-1} b\right)}^2 + b \cdot \left({\left(\cot b\right)}^{a} + \sin^{-1} b\right)\right)}}\right)}^3\]
      0.6
    8. Applied simplify to get
      \[{\left(\frac{\sqrt[3]{{b}^3 - {\left(\sin^{-1} b + {\left(\cot b\right)}^{a}\right)}^3}}{\color{red}{\sqrt[3]{{b}^2 + \left({\left({\left(\cot b\right)}^{a} + \sin^{-1} b\right)}^2 + b \cdot \left({\left(\cot b\right)}^{a} + \sin^{-1} b\right)\right)}}}\right)}^3 \leadsto {\left(\frac{\sqrt[3]{{b}^3 - {\left(\sin^{-1} b + {\left(\cot b\right)}^{a}\right)}^3}}{\color{blue}{\sqrt[3]{{\left(\sin^{-1} b + {\left(\cot b\right)}^{a}\right)}^2 + b \cdot \left({\left(\cot b\right)}^{a} + \left(b + \sin^{-1} b\right)\right)}}}\right)}^3\]
      0.6
    9. Applied taylor to get
      \[{\left(\frac{\sqrt[3]{{b}^3 - {\left(\sin^{-1} b + {\left(\cot b\right)}^{a}\right)}^3}}{\sqrt[3]{{\left(\sin^{-1} b + {\left(\cot b\right)}^{a}\right)}^2 + b \cdot \left({\left(\cot b\right)}^{a} + \left(b + \sin^{-1} b\right)\right)}}\right)}^3 \leadsto {\left(\frac{\sqrt[3]{{b}^3 - {\left(\sin^{-1} b + {\left(\cot b\right)}^{a}\right)}^3}}{\sqrt[3]{{\left(\sin^{-1} b + {\left(\cot b\right)}^{a}\right)}^2 + \left(\left(b + \left({b}^2 + b \cdot \sin^{-1} b\right)\right) - b \cdot \left(a \cdot \log b\right)\right)}}\right)}^3\]
      0.5
    10. Taylor expanded around 0 to get
      \[{\left(\frac{\sqrt[3]{{b}^3 - {\left(\sin^{-1} b + {\left(\cot b\right)}^{a}\right)}^3}}{\sqrt[3]{{\left(\sin^{-1} b + {\left(\cot b\right)}^{a}\right)}^2 + \color{red}{\left(\left(b + \left({b}^2 + b \cdot \sin^{-1} b\right)\right) - b \cdot \left(a \cdot \log b\right)\right)}}}\right)}^3 \leadsto {\left(\frac{\sqrt[3]{{b}^3 - {\left(\sin^{-1} b + {\left(\cot b\right)}^{a}\right)}^3}}{\sqrt[3]{{\left(\sin^{-1} b + {\left(\cot b\right)}^{a}\right)}^2 + \color{blue}{\left(\left(b + \left({b}^2 + b \cdot \sin^{-1} b\right)\right) - b \cdot \left(a \cdot \log b\right)\right)}}}\right)}^3\]
      0.5
    11. Applied simplify to get
      \[{\left(\frac{\sqrt[3]{{b}^3 - {\left(\sin^{-1} b + {\left(\cot b\right)}^{a}\right)}^3}}{\sqrt[3]{{\left(\sin^{-1} b + {\left(\cot b\right)}^{a}\right)}^2 + \left(\left(b + \left({b}^2 + b \cdot \sin^{-1} b\right)\right) - b \cdot \left(a \cdot \log b\right)\right)}}\right)}^3 \leadsto \frac{{b}^3 - {\left(\sin^{-1} b + {\left(\cot b\right)}^{a}\right)}^3}{\left(b \cdot \left(\left(1 + b\right) + \sin^{-1} b\right) - \log b \cdot \left(b \cdot a\right)\right) + {\left(\sin^{-1} b + {\left(\cot b\right)}^{a}\right)}^2}\]
      0.0
    12. Applied final simplification
    13. Applied simplify to get
      \[\color{red}{\frac{{b}^3 - {\left(\sin^{-1} b + {\left(\cot b\right)}^{a}\right)}^3}{\left(b \cdot \left(\left(1 + b\right) + \sin^{-1} b\right) - \log b \cdot \left(b \cdot a\right)\right) + {\left(\sin^{-1} b + {\left(\cot b\right)}^{a}\right)}^2}} \leadsto \color{blue}{\frac{{b}^3 - {\left(\sin^{-1} b + {\left(\cot b\right)}^{a}\right)}^3}{{\left(\sin^{-1} b + {\left(\cot b\right)}^{a}\right)}^2 + \left(\left(\sin^{-1} b + \left(b + 1\right)\right) - a \cdot \log b\right) \cdot b}}\]
      0.0
  1. 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))))