Average Error: 15.1 → 0.4
Time: 24.9s
Precision: 64
Internal Precision: 128
\[\frac{r \cdot \sin b}{\cos \left(a + b\right)}\]
↓
\[\frac{r}{\frac{(\left(\cos a\right) \cdot \left(\cos b\right) + \left(-(e^{\log_* (1 + \sin a \cdot \sin b)} - 1)^*\right))_*}{\sin b}}\]

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 fma-neg0.3

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

  7. Applied associate-/l*0.4

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

  9. Applied expm1-log1p-u0.4

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

  10. Final simplification0.4

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

Reproduce

herbie shell --seed 2019053 +o rules:numerics
(FPCore (r a b)
  :name "r*sin(b)/cos(a+b), A"
  (/ (* r (sin b)) (cos (+ a b))))