Average Error: 15.4 → 0.4
Time: 40.6s
Precision: 64
Internal Precision: 1408
\[\frac{r \cdot \sin b}{\cos \left(a + b\right)}\]
↓
\[\frac{r}{(\left(\frac{\cos a}{\sin b}\right) \cdot \left(\cos b\right) + \left(-\sin a\right))_*}\]

Error

Bits error versus r

Bits error versus a

Bits error versus b

Derivation

  1. Initial program 15.4

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

  3. Applied cos-sum0.3

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

  4. Taylor expanded around inf 0.3

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

  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: 40.6s)Debug logProfile

herbie shell --seed '#(1064300848 3212030778 2049303162 3567222883 2277747821 1384278011)' +o rules:numerics
(FPCore (r a b)
  :name "r*sin(b)/cos(a+b), A"
  (/ (* r (sin b)) (cos (+ a b))))