Average Error: 14.8 → 0.6
Time: 29.1s
Precision: 64
Internal Precision: 1408
\[r \cdot \frac{\sin b}{\cos \left(a + b\right)}\]
↓
\[r \cdot \frac{\sin b}{\log \left(e^{\cos b \cdot \cos a - \sin b \cdot \sin a}\right)}\]

Error

Bits error versus r

Bits error versus a

Bits error versus b

Derivation

  1. Initial program 14.8

    \[r \cdot \frac{\sin b}{\cos \left(a + b\right)}\]
  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 add-log-exp0.4

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

  6. Applied add-log-exp0.6

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

  7. Applied diff-log0.6

    \[\leadsto r \cdot \frac{\sin b}{\color{blue}{\log \left(\frac{e^{\cos a \cdot \cos b}}{e^{\sin a \cdot \sin b}}\right)}}\]

  8. Applied simplify0.6

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

Runtime

Time bar (total: 29.1s)Debug logProfile

herbie shell --seed '#(1070227846 1561819246 480764335 4016816270 2602869839 2117310382)' 
(FPCore (r a b)
  :name "r*sin(b)/cos(a+b), B"
  (* r (/ (sin b) (cos (+ a b)))))