\[r \cdot \frac{\sin b}{\cos \left(a + b\right)}\]
Test:
r*sin(b)/cos(a+b), B
Bits:
128 bits
Bits error versus r
Bits error versus a
Bits error versus b
Time: 15.2 s
Input Error: 15.4
Output Error: 0.5
Log: ⚲
Profile: 🕒
\((\left(\cos b \cdot \cos a\right) * \left(\cos b \cdot \cos a\right) + \left(\left(\sin b \cdot \sin a\right) \cdot (\left(\cos b\right) * \left(\cos a\right) + \left(\sin b \cdot \sin a\right))_*\right))_* \cdot \frac{\sin b \cdot r}{{\left(\cos b \cdot \cos a\right)}^3 - {\left(\sin b \cdot \sin a\right)}^3}\)
  1. Started with
    \[r \cdot \frac{\sin b}{\cos \left(a + b\right)}\]
    15.4
  2. Using strategy rm
    15.4
  3. Applied cos-sum to get
    \[r \cdot \frac{\sin b}{\color{red}{\cos \left(a + b\right)}} \leadsto r \cdot \frac{\sin b}{\color{blue}{\cos a \cdot \cos b - \sin a \cdot \sin b}}\]
    0.3
  4. Using strategy rm
    0.3
  5. Applied flip3-- to get
    \[r \cdot \frac{\sin b}{\color{red}{\cos a \cdot \cos b - \sin a \cdot \sin b}} \leadsto r \cdot \frac{\sin b}{\color{blue}{\frac{{\left(\cos a \cdot \cos b\right)}^{3} - {\left(\sin a \cdot \sin b\right)}^{3}}{{\left(\cos a \cdot \cos b\right)}^2 + \left({\left(\sin a \cdot \sin b\right)}^2 + \left(\cos a \cdot \cos b\right) \cdot \left(\sin a \cdot \sin b\right)\right)}}}\]
    0.4
  6. Applied associate-/r/ to get
    \[r \cdot \color{red}{\frac{\sin b}{\frac{{\left(\cos a \cdot \cos b\right)}^{3} - {\left(\sin a \cdot \sin b\right)}^{3}}{{\left(\cos a \cdot \cos b\right)}^2 + \left({\left(\sin a \cdot \sin b\right)}^2 + \left(\cos a \cdot \cos b\right) \cdot \left(\sin a \cdot \sin b\right)\right)}}} \leadsto r \cdot \color{blue}{\left(\frac{\sin b}{{\left(\cos a \cdot \cos b\right)}^{3} - {\left(\sin a \cdot \sin b\right)}^{3}} \cdot \left({\left(\cos a \cdot \cos b\right)}^2 + \left({\left(\sin a \cdot \sin b\right)}^2 + \left(\cos a \cdot \cos b\right) \cdot \left(\sin a \cdot \sin b\right)\right)\right)\right)}\]
    0.5
  7. Applied associate-*r* to get
    \[\color{red}{r \cdot \left(\frac{\sin b}{{\left(\cos a \cdot \cos b\right)}^{3} - {\left(\sin a \cdot \sin b\right)}^{3}} \cdot \left({\left(\cos a \cdot \cos b\right)}^2 + \left({\left(\sin a \cdot \sin b\right)}^2 + \left(\cos a \cdot \cos b\right) \cdot \left(\sin a \cdot \sin b\right)\right)\right)\right)} \leadsto \color{blue}{\left(r \cdot \frac{\sin b}{{\left(\cos a \cdot \cos b\right)}^{3} - {\left(\sin a \cdot \sin b\right)}^{3}}\right) \cdot \left({\left(\cos a \cdot \cos b\right)}^2 + \left({\left(\sin a \cdot \sin b\right)}^2 + \left(\cos a \cdot \cos b\right) \cdot \left(\sin a \cdot \sin b\right)\right)\right)}\]
    0.5
  8. Applied simplify to get
    \[\color{red}{\left(r \cdot \frac{\sin b}{{\left(\cos a \cdot \cos b\right)}^{3} - {\left(\sin a \cdot \sin b\right)}^{3}}\right)} \cdot \left({\left(\cos a \cdot \cos b\right)}^2 + \left({\left(\sin a \cdot \sin b\right)}^2 + \left(\cos a \cdot \cos b\right) \cdot \left(\sin a \cdot \sin b\right)\right)\right) \leadsto \color{blue}{\frac{\sin b \cdot r}{{\left(\cos a \cdot \cos b\right)}^3 - {\left(\sin a \cdot \sin b\right)}^3}} \cdot \left({\left(\cos a \cdot \cos b\right)}^2 + \left({\left(\sin a \cdot \sin b\right)}^2 + \left(\cos a \cdot \cos b\right) \cdot \left(\sin a \cdot \sin b\right)\right)\right)\]
    0.5
  9. Applied taylor to get
    \[\frac{\sin b \cdot r}{{\left(\cos a \cdot \cos b\right)}^3 - {\left(\sin a \cdot \sin b\right)}^3} \cdot \left({\left(\cos a \cdot \cos b\right)}^2 + \left({\left(\sin a \cdot \sin b\right)}^2 + \left(\cos a \cdot \cos b\right) \cdot \left(\sin a \cdot \sin b\right)\right)\right) \leadsto \frac{\sin b \cdot r}{{\left(\cos a \cdot \cos b\right)}^3 - {\left(\sin b \cdot \sin a\right)}^3} \cdot \left({\left(\cos a \cdot \cos b\right)}^2 + \left({\left(\sin a \cdot \sin b\right)}^2 + \left(\cos a \cdot \cos b\right) \cdot \left(\sin a \cdot \sin b\right)\right)\right)\]
    0.5
  10. Taylor expanded around 0 to get
    \[\frac{\sin b \cdot r}{{\left(\cos a \cdot \cos b\right)}^3 - \color{red}{{\left(\sin b \cdot \sin a\right)}^3}} \cdot \left({\left(\cos a \cdot \cos b\right)}^2 + \left({\left(\sin a \cdot \sin b\right)}^2 + \left(\cos a \cdot \cos b\right) \cdot \left(\sin a \cdot \sin b\right)\right)\right) \leadsto \frac{\sin b \cdot r}{{\left(\cos a \cdot \cos b\right)}^3 - \color{blue}{{\left(\sin b \cdot \sin a\right)}^3}} \cdot \left({\left(\cos a \cdot \cos b\right)}^2 + \left({\left(\sin a \cdot \sin b\right)}^2 + \left(\cos a \cdot \cos b\right) \cdot \left(\sin a \cdot \sin b\right)\right)\right)\]
    0.5
  11. Applied simplify to get
    \[\frac{\sin b \cdot r}{{\left(\cos a \cdot \cos b\right)}^3 - {\left(\sin b \cdot \sin a\right)}^3} \cdot \left({\left(\cos a \cdot \cos b\right)}^2 + \left({\left(\sin a \cdot \sin b\right)}^2 + \left(\cos a \cdot \cos b\right) \cdot \left(\sin a \cdot \sin b\right)\right)\right) \leadsto \frac{\sin b \cdot r}{{\left(\cos b \cdot \cos a\right)}^3 - {\left(\sin b \cdot \sin a\right)}^3} \cdot (\left(\cos b \cdot \cos a\right) * \left(\cos b \cdot \cos a\right) + \left(\left(\sin b \cdot \sin a\right) \cdot \left(\cos b \cdot \cos a + \sin b \cdot \sin a\right)\right))_*\]
    0.5
  12. Applied final simplification
  13. Applied simplify to get
    \[\color{red}{\frac{\sin b \cdot r}{{\left(\cos b \cdot \cos a\right)}^3 - {\left(\sin b \cdot \sin a\right)}^3} \cdot (\left(\cos b \cdot \cos a\right) * \left(\cos b \cdot \cos a\right) + \left(\left(\sin b \cdot \sin a\right) \cdot \left(\cos b \cdot \cos a + \sin b \cdot \sin a\right)\right))_*} \leadsto \color{blue}{(\left(\cos b \cdot \cos a\right) * \left(\cos b \cdot \cos a\right) + \left(\left(\sin b \cdot \sin a\right) \cdot (\left(\cos b\right) * \left(\cos a\right) + \left(\sin b \cdot \sin a\right))_*\right))_* \cdot \frac{\sin b \cdot r}{{\left(\cos b \cdot \cos a\right)}^3 - {\left(\sin b \cdot \sin a\right)}^3}}\]
    0.5

Original test:


(lambda ((r default) (a default) (b default))
  #:name "r*sin(b)/cos(a+b), B"
  (* r (/ (sin b) (cos (+ a b)))))