\[b - \left({\left(\cot b\right)}^{a} + \sin^{-1} b\right)\]
Test:
(- b (+ (pow (cotan b) a) (asin b)))
Bits:
128 bits
Bits error versus a
Bits error versus b
Time: 1.1 m
Input Error: 9.6
Output Error: 9.9
Log: ⚲
Profile: 🕒
\(\begin{cases} \frac{{b}^3 - {\left(\sin^{-1} b + {\left(\cot b\right)}^{a}\right)}^3}{b \cdot \left({\left(\cot b\right)}^{a} + \left(b + \sin^{-1} b\right)\right) + {\left(\sin^{-1} b + {\left(\cot b\right)}^{a}\right)}^2} & \text{when } b \le -6.666946136074473 \cdot 10^{-165} \\ \frac{{b}^2 - \sqrt[3]{{\left({\left({\left(\cot b\right)}^{a} + \sin^{-1} b\right)}^2\right)}^3}}{b + \left({\left(\cot b\right)}^{a} + \sin^{-1} b\right)} & \text{when } b \le 7.6539915406140805 \cdot 10^{-165} \\ \frac{{b}^2 + {\left(\sin^{-1} b + {\left(\left(\frac{1}{b} - b \cdot \frac{1}{3}\right) - \frac{1}{45} \cdot {b}^3\right)}^{a}\right)}^2}{\frac{{b}^2 + {\left(\sin^{-1} b + {\left(\cot b\right)}^{a}\right)}^2}{\left(b + \sin^{-1} b\right) + {\left(\left(\frac{1}{b} - b \cdot \frac{1}{3}\right) - \frac{1}{45} \cdot {b}^3\right)}^{a}} \cdot \frac{\left(\sin^{-1} b + b\right) + {\left(\cot b\right)}^{a}}{\left(b - \sin^{-1} b\right) - {\left(\left(\frac{1}{b} - b \cdot \frac{1}{3}\right) - \frac{1}{45} \cdot {b}^3\right)}^{a}}} & \text{otherwise} \end{cases}\)

    if b < -6.666946136074473e-165

    1. Started with
      \[b - \left({\left(\cot b\right)}^{a} + \sin^{-1} b\right)\]
      36.9
    2. Using strategy rm
      36.9
    3. Applied flip3-- to get
      \[\color{red}{b - \left({\left(\cot b\right)}^{a} + \sin^{-1} b\right)} \leadsto \color{blue}{\frac{{b}^{3} - {\left({\left(\cot b\right)}^{a} + \sin^{-1} b\right)}^{3}}{{b}^2 + \left({\left({\left(\cot b\right)}^{a} + \sin^{-1} b\right)}^2 + b \cdot \left({\left(\cot b\right)}^{a} + \sin^{-1} b\right)\right)}}\]
      38.0
    4. Applied simplify to get
      \[\frac{\color{red}{{b}^{3} - {\left({\left(\cot b\right)}^{a} + \sin^{-1} b\right)}^{3}}}{{b}^2 + \left({\left({\left(\cot b\right)}^{a} + \sin^{-1} b\right)}^2 + b \cdot \left({\left(\cot b\right)}^{a} + \sin^{-1} b\right)\right)} \leadsto \frac{\color{blue}{{b}^3 - {\left(\sin^{-1} b + {\left(\cot b\right)}^{a}\right)}^3}}{{b}^2 + \left({\left({\left(\cot b\right)}^{a} + \sin^{-1} b\right)}^2 + b \cdot \left({\left(\cot b\right)}^{a} + \sin^{-1} b\right)\right)}\]
      38.0
    5. Applied simplify to get
      \[\frac{{b}^3 - {\left(\sin^{-1} b + {\left(\cot b\right)}^{a}\right)}^3}{\color{red}{{b}^2 + \left({\left({\left(\cot b\right)}^{a} + \sin^{-1} b\right)}^2 + b \cdot \left({\left(\cot b\right)}^{a} + \sin^{-1} b\right)\right)}} \leadsto \frac{{b}^3 - {\left(\sin^{-1} b + {\left(\cot b\right)}^{a}\right)}^3}{\color{blue}{b \cdot \left({\left(\cot b\right)}^{a} + \left(b + \sin^{-1} b\right)\right) + {\left(\sin^{-1} b + {\left(\cot b\right)}^{a}\right)}^2}}\]
      37.9

    if -6.666946136074473e-165 < b < 7.6539915406140805e-165

    1. Started with
      \[b - \left({\left(\cot b\right)}^{a} + \sin^{-1} b\right)\]
      0.1
    2. Using strategy rm
      0.1
    3. Applied flip-- to get
      \[\color{red}{b - \left({\left(\cot b\right)}^{a} + \sin^{-1} b\right)} \leadsto \color{blue}{\frac{{b}^2 - {\left({\left(\cot b\right)}^{a} + \sin^{-1} b\right)}^2}{b + \left({\left(\cot b\right)}^{a} + \sin^{-1} b\right)}}\]
      0.1
    4. Using strategy rm
      0.1
    5. Applied add-cbrt-cube to get
      \[\frac{{b}^2 - \color{red}{{\left({\left(\cot b\right)}^{a} + \sin^{-1} b\right)}^2}}{b + \left({\left(\cot b\right)}^{a} + \sin^{-1} b\right)} \leadsto \frac{{b}^2 - \color{blue}{\sqrt[3]{{\left({\left({\left(\cot b\right)}^{a} + \sin^{-1} b\right)}^2\right)}^3}}}{b + \left({\left(\cot b\right)}^{a} + \sin^{-1} b\right)}\]
      0.2

    if 7.6539915406140805e-165 < b

    1. Started with
      \[b - \left({\left(\cot b\right)}^{a} + \sin^{-1} b\right)\]
      12.0
    2. Using strategy rm
      12.0
    3. Applied flip-- to get
      \[\color{red}{b - \left({\left(\cot b\right)}^{a} + \sin^{-1} b\right)} \leadsto \color{blue}{\frac{{b}^2 - {\left({\left(\cot b\right)}^{a} + \sin^{-1} b\right)}^2}{b + \left({\left(\cot b\right)}^{a} + \sin^{-1} b\right)}}\]
      12.0
    4. Using strategy rm
      12.0
    5. Applied add-cube-cbrt to get
      \[\frac{{b}^2 - \color{red}{{\left({\left(\cot b\right)}^{a} + \sin^{-1} b\right)}^2}}{b + \left({\left(\cot b\right)}^{a} + \sin^{-1} b\right)} \leadsto \frac{{b}^2 - \color{blue}{{\left(\sqrt[3]{{\left({\left(\cot b\right)}^{a} + \sin^{-1} b\right)}^2}\right)}^3}}{b + \left({\left(\cot b\right)}^{a} + \sin^{-1} b\right)}\]
      18.1
    6. Using strategy rm
      18.1
    7. Applied flip-- to get
      \[\frac{\color{red}{{b}^2 - {\left(\sqrt[3]{{\left({\left(\cot b\right)}^{a} + \sin^{-1} b\right)}^2}\right)}^3}}{b + \left({\left(\cot b\right)}^{a} + \sin^{-1} b\right)} \leadsto \frac{\color{blue}{\frac{{\left({b}^2\right)}^2 - {\left({\left(\sqrt[3]{{\left({\left(\cot b\right)}^{a} + \sin^{-1} b\right)}^2}\right)}^3\right)}^2}{{b}^2 + {\left(\sqrt[3]{{\left({\left(\cot b\right)}^{a} + \sin^{-1} b\right)}^2}\right)}^3}}}{b + \left({\left(\cot b\right)}^{a} + \sin^{-1} b\right)}\]
      12.6
    8. Applied associate-/l/ to get
      \[\color{red}{\frac{\frac{{\left({b}^2\right)}^2 - {\left({\left(\sqrt[3]{{\left({\left(\cot b\right)}^{a} + \sin^{-1} b\right)}^2}\right)}^3\right)}^2}{{b}^2 + {\left(\sqrt[3]{{\left({\left(\cot b\right)}^{a} + \sin^{-1} b\right)}^2}\right)}^3}}{b + \left({\left(\cot b\right)}^{a} + \sin^{-1} b\right)}} \leadsto \color{blue}{\frac{{\left({b}^2\right)}^2 - {\left({\left(\sqrt[3]{{\left({\left(\cot b\right)}^{a} + \sin^{-1} b\right)}^2}\right)}^3\right)}^2}{\left(b + \left({\left(\cot b\right)}^{a} + \sin^{-1} b\right)\right) \cdot \left({b}^2 + {\left(\sqrt[3]{{\left({\left(\cot b\right)}^{a} + \sin^{-1} b\right)}^2}\right)}^3\right)}}\]
      19.9
    9. Applied taylor to get
      \[\frac{{\left({b}^2\right)}^2 - {\left({\left(\sqrt[3]{{\left({\left(\cot b\right)}^{a} + \sin^{-1} b\right)}^2}\right)}^3\right)}^2}{\left(b + \left({\left(\cot b\right)}^{a} + \sin^{-1} b\right)\right) \cdot \left({b}^2 + {\left(\sqrt[3]{{\left({\left(\cot b\right)}^{a} + \sin^{-1} b\right)}^2}\right)}^3\right)} \leadsto \frac{{\left({b}^2\right)}^2 - {\left({\left(\sqrt[3]{{\left({\left(\frac{1}{b} - \left(\frac{1}{3} \cdot b + \frac{1}{45} \cdot {b}^{3}\right)\right)}^{a} + \sin^{-1} b\right)}^2}\right)}^3\right)}^2}{\left(b + \left({\left(\cot b\right)}^{a} + \sin^{-1} b\right)\right) \cdot \left({b}^2 + {\left(\sqrt[3]{{\left({\left(\cot b\right)}^{a} + \sin^{-1} b\right)}^2}\right)}^3\right)}\]
      19.9
    10. Taylor expanded around 0 to get
      \[\frac{{\left({b}^2\right)}^2 - {\left({\left(\sqrt[3]{{\left({\color{red}{\left(\frac{1}{b} - \left(\frac{1}{3} \cdot b + \frac{1}{45} \cdot {b}^{3}\right)\right)}}^{a} + \sin^{-1} b\right)}^2}\right)}^3\right)}^2}{\left(b + \left({\left(\cot b\right)}^{a} + \sin^{-1} b\right)\right) \cdot \left({b}^2 + {\left(\sqrt[3]{{\left({\left(\cot b\right)}^{a} + \sin^{-1} b\right)}^2}\right)}^3\right)} \leadsto \frac{{\left({b}^2\right)}^2 - {\left({\left(\sqrt[3]{{\left({\color{blue}{\left(\frac{1}{b} - \left(\frac{1}{3} \cdot b + \frac{1}{45} \cdot {b}^{3}\right)\right)}}^{a} + \sin^{-1} b\right)}^2}\right)}^3\right)}^2}{\left(b + \left({\left(\cot b\right)}^{a} + \sin^{-1} b\right)\right) \cdot \left({b}^2 + {\left(\sqrt[3]{{\left({\left(\cot b\right)}^{a} + \sin^{-1} b\right)}^2}\right)}^3\right)}\]
      19.9
    11. Applied simplify to get
      \[\frac{{\left({b}^2\right)}^2 - {\left({\left(\sqrt[3]{{\left({\left(\frac{1}{b} - \left(\frac{1}{3} \cdot b + \frac{1}{45} \cdot {b}^{3}\right)\right)}^{a} + \sin^{-1} b\right)}^2}\right)}^3\right)}^2}{\left(b + \left({\left(\cot b\right)}^{a} + \sin^{-1} b\right)\right) \cdot \left({b}^2 + {\left(\sqrt[3]{{\left({\left(\cot b\right)}^{a} + \sin^{-1} b\right)}^2}\right)}^3\right)} \leadsto \frac{{b}^2 + {\left(\sin^{-1} b + {\left(\left(\frac{1}{b} - b \cdot \frac{1}{3}\right) - \frac{1}{45} \cdot {b}^3\right)}^{a}\right)}^2}{\frac{{b}^2 + {\left(\sin^{-1} b + {\left(\cot b\right)}^{a}\right)}^2}{\left(b + \sin^{-1} b\right) + {\left(\left(\frac{1}{b} - b \cdot \frac{1}{3}\right) - \frac{1}{45} \cdot {b}^3\right)}^{a}} \cdot \frac{\left(\sin^{-1} b + b\right) + {\left(\cot b\right)}^{a}}{\left(b - \sin^{-1} b\right) - {\left(\left(\frac{1}{b} - b \cdot \frac{1}{3}\right) - \frac{1}{45} \cdot {b}^3\right)}^{a}}}\]
      12.3
    12. Applied final simplification
  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))))