Average Error: 14.9 → 0.4
Time: 30.8s
Precision: 64
Internal Precision: 1344
\[r \cdot \frac{\sin b}{\cos \left(a + b\right)}\]
↓
\[\sin b \cdot \frac{r}{(e^{\log_* (1 + \left(\cos a \cdot \cos b - \sin b \cdot \sin a\right))} - 1)^*}\]

Error

Bits error versus r

Bits error versus a

Bits error versus b

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

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

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

  6. Applied *-un-lft-identity0.3

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

  7. Applied times-frac0.3

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

  8. Simplified0.3

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

  10. Applied expm1-log1p-u0.4

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

  11. Final simplification0.4

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

Runtime

Time bar (total: 30.8s)Debug logProfile

herbie shell --seed 2018273 +o rules:numerics
(FPCore (r a b)
  :name "r*sin(b)/cos(a+b), B"
  (* r (/ (sin b) (cos (+ a b)))))