Average Error: 15.1 → 0.3
Time: 30.2s
Precision: 64
Internal Precision: 128
\[r \cdot \frac{\sin b}{\cos \left(a + b\right)}\]
↓
\[\frac{r \cdot \sin b}{\cos a \cdot \cos b - \log_* (1 + (e^{\sin b \cdot \sin a} - 1)^*)}\]

Error

Bits error versus r

Bits error versus a

Bits error versus b

Derivation

  1. Initial program 15.1

    \[r \cdot \frac{\sin b}{\cos \left(a + b\right)}\]
  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 div-inv0.4

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

  6. Applied associate-*r*0.4

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

  8. Applied associate-*r/0.3

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

  10. Applied log1p-expm1-u0.3

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

  11. Final simplification0.3

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

Reproduce

herbie shell --seed 2019089 +o rules:numerics
(FPCore (r a b)
  :name "r*sin(b)/cos(a+b), B"
  (* r (/ (sin b) (cos (+ a b)))))