Average Error: 14.9 → 0.3
Time: 44.6s
Precision: 64
Internal Precision: 1344
\[r \cdot \frac{\sin b}{\cos \left(a + b\right)}\]
↓
\[\frac{\sin b}{\cos b \cdot \cos a - (e^{\log_* (1 + \sin a \cdot \sin b)} - 1)^*} \cdot r\]

Error

Bits error versus r

Bits error versus a

Bits error versus b

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Results

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Derivation

  1. Initial program 14.9

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

  2. Initial simplification14.9

    \[\leadsto \frac{r \cdot \sin b}{\cos \left(b + a\right)}\]
  3. Using strategy rm

  4. Applied cos-sum0.3

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

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

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

  7. Applied times-frac0.3

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

  8. Simplified0.3

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

  10. Applied expm1-log1p-u0.3

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

  11. Final simplification0.3

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

Runtime

Time bar (total: 44.6s)Debug logProfile

herbie shell --seed 2018250 +o rules:numerics
(FPCore (r a b)
  :name "r*sin(b)/cos(a+b), B"
  (* r (/ (sin b) (cos (+ a b)))))