Average Error: 15.2 → 0.3
Time: 29.7s
Precision: 64
Internal Precision: 1344
\[r \cdot \frac{\sin b}{\cos \left(a + b\right)}\]
↓
\[\frac{\sin b}{(\left(\cos a\right) \cdot \left(\cos b\right) + \left(-\log_* (1 + (e^{\sin a \cdot \sin b} - 1)^*)\right))_*} \cdot r\]

Error

Bits error versus r

Bits error versus a

Bits error versus b

Derivation

  1. Initial program 15.2

    \[r \cdot \frac{\sin b}{\cos \left(a + b\right)}\]
  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 fma-neg0.3

    \[\leadsto r \cdot \frac{\sin b}{\color{blue}{(\left(\cos a\right) \cdot \left(\cos b\right) + \left(-\sin a \cdot \sin b\right))_*}}\]
  6. Using strategy rm

  7. Applied log1p-expm1-u0.3

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

  8. Final simplification0.3

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

Runtime

Time bar (total: 29.7s)Debug logProfile

herbie shell --seed 2018255 +o rules:numerics
(FPCore (r a b)
  :name "r*sin(b)/cos(a+b), B"
  (* r (/ (sin b) (cos (+ a b)))))