Average Error: 14.9 → 0.4
Time: 1.1m
Precision: 64
Internal Precision: 1344
\[r \cdot \frac{\sin b}{\cos \left(a + b\right)}\]
↓
\[\left(\frac{r}{\cos b \cdot \cos a - \sin b \cdot \sin a} \cdot \frac{\sin b}{\sin b \cdot \sin a + \cos b \cdot \cos a}\right) \cdot \left(\cos a \cdot \cos b + \sin a \cdot \sin b\right)\]

Error

Bits error versus r

Bits error versus a

Bits error versus b

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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. 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 flip--0.4

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

  9. Applied associate-/r/0.5

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

  10. Applied associate-*r*0.5

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

  11. Applied simplify0.4

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

Runtime

Time bar (total: 1.1m)Debug logProfile

herbie shell --seed 2018208 
(FPCore (r a b)
  :name "r*sin(b)/cos(a+b), B"
  (* r (/ (sin b) (cos (+ a b)))))