Average Error: 15.1 → 0.3
Time: 3.0m
Precision: 64
Internal Precision: 128
\[r \cdot \frac{\sin b}{\cos \left(a + b\right)}\]
↓
\[\frac{\sin b}{\frac{\sin a \cdot \sin b + \cos a \cdot \cos b}{\frac{\sin a \cdot \sin b + \cos a \cdot \cos b}{\cos a \cdot \cos b - \sin a \cdot \sin b}}} \cdot r\]

Error

Bits error versus r

Bits error versus a

Bits error versus b

Derivation

  1. Initial program 15.1

    \[r \cdot \frac{\sin b}{\cos \left(a + b\right)}\]
  2. Using strategy rm

  3. Applied cos-sum0.3

    \[\leadsto r \cdot \frac{\sin b}{\color{blue}{\cos a \cdot \cos b - \sin a \cdot \sin b}}\]
  4. Using strategy rm

  5. Applied flip--0.4

    \[\leadsto r \cdot \frac{\sin b}{\color{blue}{\frac{\left(\cos a \cdot \cos b\right) \cdot \left(\cos a \cdot \cos b\right) - \left(\sin a \cdot \sin b\right) \cdot \left(\sin a \cdot \sin b\right)}{\cos a \cdot \cos b + \sin a \cdot \sin b}}}\]

  6. Taylor expanded around inf 0.4

    \[\leadsto r \cdot \frac{\sin b}{\color{blue}{\frac{{\left(\cos a\right)}^{2} \cdot {\left(\cos b\right)}^{2} - {\left(\sin b\right)}^{2} \cdot {\left(\sin a\right)}^{2}}{\sin b \cdot \sin a + \cos a \cdot \cos b}}}\]

  7. Simplified0.3

    \[\leadsto r \cdot \frac{\sin b}{\color{blue}{\frac{\sin b \cdot \sin a + \cos b \cdot \cos a}{\frac{\sin b \cdot \sin a + \cos b \cdot \cos a}{\cos b \cdot \cos a - \sin b \cdot \sin a}}}}\]

  8. Final simplification0.3

    \[\leadsto \frac{\sin b}{\frac{\sin a \cdot \sin b + \cos a \cdot \cos b}{\frac{\sin a \cdot \sin b + \cos a \cdot \cos b}{\cos a \cdot \cos b - \sin a \cdot \sin b}}} \cdot r\]

Reproduce

herbie shell --seed 2019090 
(FPCore (r a b)
  :name "r*sin(b)/cos(a+b), B"
  (* r (/ (sin b) (cos (+ a b)))))