Average Error: 15.2 → 0.5
Time: 25.4s
Precision: 64
Internal Precision: 128
\[r \cdot \frac{\sin b}{\cos \left(a + b\right)}\]
↓
\[\frac{\sin b}{\cos a \cdot \cos b - \sqrt[3]{\sin b} \cdot \left(\left(\sqrt[3]{\sin b} \cdot \sqrt[3]{\sin b}\right) \cdot \sin a\right)} \cdot r\]

Error

Bits error versus r

Bits error versus a

Bits error versus b

Derivation

  1. Initial program 15.2

    \[r \cdot \frac{\sin b}{\cos \left(a + b\right)}\]
  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 add-cube-cbrt0.5

    \[\leadsto r \cdot \frac{\sin b}{\cos a \cdot \cos b - \sin a \cdot \color{blue}{\left(\left(\sqrt[3]{\sin b} \cdot \sqrt[3]{\sin b}\right) \cdot \sqrt[3]{\sin b}\right)}}\]

  6. Applied associate-*r*0.5

    \[\leadsto r \cdot \frac{\sin b}{\cos a \cdot \cos b - \color{blue}{\left(\sin a \cdot \left(\sqrt[3]{\sin b} \cdot \sqrt[3]{\sin b}\right)\right) \cdot \sqrt[3]{\sin b}}}\]

  7. Final simplification0.5

    \[\leadsto \frac{\sin b}{\cos a \cdot \cos b - \sqrt[3]{\sin b} \cdot \left(\left(\sqrt[3]{\sin b} \cdot \sqrt[3]{\sin b}\right) \cdot \sin a\right)} \cdot r\]

Reproduce

herbie shell --seed 2019093 
(FPCore (r a b)
  :name "r*sin(b)/cos(a+b), B"
  (* r (/ (sin b) (cos (+ a b)))))