Average Error: 15.1 → 0.4
Time: 39.6s
Precision: 64
Internal Precision: 128
\[r \cdot \frac{\sin b}{\cos \left(a + b\right)}\]
↓
\[\frac{1}{\frac{(\left(\cos a\right) \cdot \left(\cos b\right) + \left(\left(-\sin b\right) \cdot \sin a\right))_*}{\sin b}} \cdot r\]

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 fma-neg0.3

    \[\leadsto r \cdot \frac{\sin b}{\color{blue}{(\left(\cos a\right) \cdot \left(\cos b\right) + \left(-\sin a \cdot \sin b\right))_*}}\]
  6. Using strategy rm

  7. Applied *-un-lft-identity0.3

    \[\leadsto r \cdot \frac{\color{blue}{1 \cdot \sin b}}{(\left(\cos a\right) \cdot \left(\cos b\right) + \left(-\sin a \cdot \sin b\right))_*}\]

  8. Applied associate-/l*0.4

    \[\leadsto r \cdot \color{blue}{\frac{1}{\frac{(\left(\cos a\right) \cdot \left(\cos b\right) + \left(-\sin a \cdot \sin b\right))_*}{\sin b}}}\]

  9. Final simplification0.4

    \[\leadsto \frac{1}{\frac{(\left(\cos a\right) \cdot \left(\cos b\right) + \left(\left(-\sin b\right) \cdot \sin a\right))_*}{\sin b}} \cdot r\]

Reproduce

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