Average Error: 15.1 → 0.4
Time: 49.5s
Precision: 64
Internal Precision: 1344
\[\frac{r \cdot \sin b}{\cos \left(a + b\right)}\]
↓
\[\frac{r}{(\left(\frac{\cos a}{\sin b}\right) \cdot \left(\cos b\right) + \left(-\sin a\right))_*}\]

Error

Bits error versus r

Bits error versus a

Bits error versus b

Derivation

  1. Initial program 15.1

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

  5. Applied associate-/l*0.4

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

  7. Applied *-un-lft-identity0.4

    \[\leadsto \frac{r}{\frac{\cos a \cdot \cos b - \sin a \cdot \sin b}{\color{blue}{1 \cdot \sin b}}}\]

  8. Applied *-un-lft-identity0.4

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

  9. Applied times-frac0.4

    \[\leadsto \frac{r}{\color{blue}{\frac{1}{1} \cdot \frac{\cos a \cdot \cos b - \sin a \cdot \sin b}{\sin b}}}\]

  10. Applied associate-/r*0.4

    \[\leadsto \color{blue}{\frac{\frac{r}{\frac{1}{1}}}{\frac{\cos a \cdot \cos b - \sin a \cdot \sin b}{\sin b}}}\]

  11. Simplified0.4

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

  12. Final simplification0.4

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

Runtime

Time bar (total: 49.5s)Debug logProfile

herbie shell --seed 2018217 +o rules:numerics
(FPCore (r a b)
  :name "r*sin(b)/cos(a+b), A"
  (/ (* r (sin b)) (cos (+ a b))))