Average Error: 14.9 → 0.4
Time: 34.4s
Precision: 64
Internal Precision: 1344
\[r \cdot \frac{\sin b}{\cos \left(a + b\right)}\]
↓
\[\left(\frac{1}{\cos a \cdot \cos b - \sin a \cdot \sin b} \cdot r\right) \cdot \sin b\]

Error

Bits error versus r

Bits error versus a

Bits error versus b

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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. Using strategy rm

  5. Applied clear-num0.4

    \[\leadsto r \cdot \color{blue}{\frac{1}{\frac{\cos a \cdot \cos b - \sin a \cdot \sin b}{\sin b}}}\]
  6. Using strategy rm

  7. Applied associate-/r/0.4

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

  8. Applied associate-*r*0.4

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

  9. Final simplification0.4

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

Runtime

Time bar (total: 34.4s)Debug logProfile

herbie shell --seed 2018258 +o rules:numerics
(FPCore (r a b)
  :name "r*sin(b)/cos(a+b), B"
  (* r (/ (sin b) (cos (+ a b)))))