Average Error: 14.9 → 0.4
Time: 47.5s
Precision: 64
Internal Precision: 1344
\[r \cdot \frac{\sin b}{\cos \left(a + b\right)}\]
↓
\[\frac{r}{(\left(\frac{\cos a}{\sin b}\right) \cdot \left(\cos b\right) + \left(-\sin a\right))_*}\]

Error

Bits error versus r

Bits error versus a

Bits error versus b

Derivation

  1. Initial program 14.9

    \[r \cdot \frac{\sin b}{\cos \left(a + b\right)}\]
  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. Taylor expanded around inf 0.3

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

  5. Applied simplify0.4

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

Runtime

Time bar (total: 47.5s)Debug logProfile

herbie shell --seed '#(1071948828 1180510430 2986424009 997076509 406109801 420189285)' +o rules:numerics
(FPCore (r a b)
  :name "r*sin(b)/cos(a+b), B"
  (* r (/ (sin b) (cos (+ a b)))))